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This is to certify that the

dissertation entitled

ELECTRON SPIN ECHO ENVELOPE MODULATION STUDIES OF SOME
TRANSITION METAL MODEL COMPLEXES

presented by

HONG IN LEE

has been accepted towards fulfillment
of the requirements for

PH . D CHEMISTRY

degree in

 

 

JOHN L. MCCRACKEN

pg; (77%:“4;

Major professor

Date 7/; /€g/

MSU is an Affirmative Action/Equal Opportunity Institution 0-12771

 

 

 

 

 

llllllllllllllllllllllllllllllHill“llHlllllllllllllllllll

31293 010530784

 

LIBRARY
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University

 

 

 

PLACE IN RETURN BOX to remove this checkout from your record.
TO AVOID F INES return on or before date due

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MSU Is An Affirmative Action/Equal Opportunity Institution
WMJ

 

 

 

 

 

ABSTRACT

ELECTRON SPIN ECHO ENVELOPE MODULATION STUDIES 0]
SOME TRANSITION METAL MODEL COMPLEXES

By

Hong In Lee

Two different lines of pulsed-EPR experiments linked by 1
common theme of further developing electron spin echo envelc
modulation (ESEEM) methods for the determination of structure abi
paramagnetic centers in randomly ordered samples are described. The h
set of measurements involve the extension of a recently developed fo
pulse ESEEM method to measure the proton hyperfine couplings
strongly bound water ligands to paramagnetic transition ions. The secc
project involves the use of standard ESEEM methods to characterize
solution structures of some Cu(II)-pteridine complexes.

Four-pulse ESEEM studies aimed at characterizing the hyperf
interactions between protons of axially bound water molecules and
nickel ion of Ni(III)(CN)4(H20)2' were carried out. Because the ligz
hyperfine coupling of the strongly bound water protons to Ni(III)
characterized by a large anisotropic interaction, an ESEEM feature at
sum combination frequency, va+v5, that shows pronounced shifts fri

twice the proton Larmor frequency is observed. Theoretical simulation

the magnetic field dependence of the va+v5 lineshapes and frequency shift
from the twice the Larmor frequency gave an effective Ni-I—I dipole-dipol
distance of 2.33i0.03 A and a 6n, the orientation for the principal axi
system of the 1H hyperfine coupling tensor with respect to the g3 axis c
the Ni(lII) g—tensor, of 18i30. The r-suppression behavior of the va+v
lineshape at a fixed magnetic field position was used to place more exac
constraints on the isotropic hyperfine coupling constant than possible with
simple two-pulse approach. An isotropic hyperfine coupling constant c
25:05 MHz was found for the bound axial water protons.

Four-pulse ESEEM studies aimed at characterizing hyperfin
interactions between protons of equatorially and axially bound wate
molecules and the copper ion of Cu(II)(H20)62+ were carried ow
Theoretical simulation of the magnetic field dependence of the proton sur
combination lineshape revealed an effective Cu—H distance of 2.49 A, an
showed that ON, the angle between the principal axis of the ligan
hyperfine tensor and the g3 of the Cu(ll) electron g-tensor, can b
discribed by a distribution of values ranging from 71° to 900. An isotropi
hyperfine coupling constant of S 4 MHz for the equatorially bound watc
protons was estimated. Attempts were made to distinguish the S111
combination peaks of the axially bound water protons by using the 1
suppression behavior of the four-pulse ESEEM spectra. It was found th:
the intensities and lineshapes of the sum combination peaks of the axi:
protons are relatively well developed and distinguished from the sui
combination peaks of the equatorial protons when r is around values give
by n/vI (n=integer). Simulation of the frequency shift of the axial proto

sum combination peaks vs. magnetic field strength yielded an effective Cl

H distance of 3.05 A, 3N of 110 and an isotropic hyperfine coupling of S
MHz.

ESEEM measurements were also used to characterize the ligation I
pteridine ligands to Cu(Il) in a variety of complexes prepared in aqueor
and non—aqueous solvents. These studies were aimed at understanding it
structural relationship between the Cu(II) and pterin cofactors i
Phenylalanine Hydroxylase (PAH) from Chromobacterium Violaceum. Fr
the model compound, Cu(II)(ethp)2(H20)2 (ethp = 2-ethylthio-4
hydroxypterin), X-ray crystallographic studies showed bidentai
coordination of ethp through 0—4 and N-S. ESEEM spectra obtained fr
this compound show intense, sharp peaks at 0.6, 2.4, 3.0 MHz and a bro;
peak at 5.4 MHz. The spectrum is indicative of two identically coupled 14
nuclei and is assigned to N-3 of the coordinated ethp ligands. Simil:
spectra were obtained for a Cu(II)-folic acid (FA) complex at pH=9.5 i
aqueous media. ESEEM studies of compounds where mixed equatoria
axial ligation of the pterin moiety show little or no 14N contributions ‘3
their ESEEM spectra. Examples of a complex of this type incluc
Cu(II)(bpy)(PC) where N—5 is equatorially bound while O-4 is axial]
coordinated (PC = 6-carboxy pterin). Because the 14N ESEEM respons
from equatorially coordinated pterin is intense and the peaks occur in
spectral region where there would be little interference from the protei:
the experimental results are most consistent with a mixed equatorial-axi;

ligation of this cofactor at the Cu(II) site of PAH.

 

I To My Parents and Wife

 

 

ACKNOWLEDGMENT

Somebody said, "Life is a voyage." I feel that I am finishing :
portion of the voyage. For the last five years Prof. John McCracken ha:
been my private captain. Without his brilliant guidance and support I migh
not have finished the voyage. I would like to express my deep thanks t(
Prof. John McCracken, my captain. Also I would like to thank m}
committee members, Prof. G. Babcock, Prof. C. K. Chang, and Prof. M
Kanatzidis. I would like to thank Prof. J. Jackson, Prof. R. Cukier, am
Prof. J. Yesinowski for their help. I would like to thank my colleagues, Dr
Kurt Wamcke, Michelle Mac, Kerry Reidy, Gyorgy Filep, and Vladimi
Bouchev for their help and discussions.

I would like to give my heartful thanks to my parents and sisters fo
their endless love. Special thanks should be given to my wife and daughte

for their love and ability to create happiness.

vi

 

 

 

 

LIST OF CONTENTS

LIST OF TABLES ........................................................................... x
LIST OF FIGURES .......................................................................... xi
LIST OF ABBREVIATIONS ............................................................. xviii
I. INTRODUCTION ......................................................................... 1
References .............................................................................. 6

II. BASIC PRINCIPLES OF ELECTRON SPIN ECHO ENVELOPE

MODULATION ............................................................................... 7
1. Basic Principles of Electron Spin Echo (ESE) ......................... 7
1-1. Two-Pulse Spin Echo ............................................... 7
1-2. Three-Pulse Spin Echo (Stimulated Spin Echo) ........... 10
1-3. Four-Pulse Spin Echo ............................................... 14
2. Basic Principles of Electron Spin Echo Envelope
Modulation (ESEEM) .............................................................. 16
2-1. Two-Pulse ESEEM .................................................. 17
2-2. Three-Pulse ESEEM ................................................ 21
2-3. Four-Pulse ESEEM .................................................. 23
References .............................................................................. 26
III. FORMALISM OF ESEEM .......................................................... 28
1. Density Matrix .................................................................... 28
2. The Density Matrix Formalism of ESEEM ............................. 31
2-1. Two-Pulse ESEEM .................................................. 31
2-2. Three-Pulse ESEEM ................................................ 35
2-3. Four-Pulse ESEEM .................................................. 36
3. Formulas of ESEEM ............................................................ 38
3—1. Formulas for S=1/2 and I=1/2 ................................... 39
3-2. Formulas for S=1/2 and 121 ...................................... 42
References .............................................................................. 46

vii

IV. INSTRUMENTATION ............................................................... 47
References .............................................................................. 5]

V. FOUR-PULSE ELECTRON SPIN ECHO ENVELOPE
MODULATION STUDIES OF AXIAL WATER LIGATION TO

BIS—AQUO TETRACYANONICKELATE(III) ................................... 52
1. Abstract .............................................................................. 52
2. Introduction ........................................................................ 53
3. Experimental ....................................................................... 56
4. Theory ............................................................................... 57
5. Results and Discussion .......................................................... 71
References .............................................................................. 97

VI. FOUR-PULSE ELECTRON SPIN ECHO ENVELOPE

 

 

MODULATION STUDIES OF Cu(II)(HzO)62+ IO
1. Abstract .............................................................................. 10
2. Introduction ........................................................................ 10
3. Experimental ....................................................................... 10
4. Theoretical Aspects .............................................................. 10
4-1. Angle Selection 10

4-2. Lineshape Properties of Four—Pulse ESEEM Sum
Combination Band .......................................................... 10
5. Results and Discussion .......................................................... ll
References .............................................................................. 13

VII. ELECTRON SPIN ECHO ENVELOPE MODULATION

STUDIES OF COPPER(II)-PT ERIN MODEL COMPLEXES ............... 14
1. Abstract .............................................................................. 14
2. Introduction ........................................................................ l4
3. Experimental ....................................................................... l4
4. Results and Discussion .......................................................... 14
References .............................................................................. 16
APPENDIX ..................................................................................... 16
A1. Hamitonian Matrices for Hyperfine (HFI) and Nuclear
Zeeman Interactions (NZI) ....................................................... 16
A2. M Matrices for the Hamiltonian containing only Hyperfine
and Nuclear Zeeman Interactions .............................................. 16

A3. Hamiltonian Matrices for Nuclear Quadrupole Interaction

(NQI) along Hyperfine Interaction Principle Axis System (HFI
A) ...................................................................................... 17
A3- 1. Q elements in Tables A3-1, 2, 3, and 4 .................... 17

viii

 

 

A4. Computer Programs for Two—Pulse and Three-Pulse

ESEEM Time Domain Simulations for 1:], 3/2, 5/2, and 7/2 ....... 174
A4-l. Main Program for Two—Pulse ESEEM .................... 174
A4-2. Main Program for Three-Pulse ESEEM .................. 175
A4-3. Subprograms ........................................................ 176
A5. Pulse Logic Circuits .......................................................... 196

A6. Computer Interfacing Programs for Performing 4-pulse
ESEEM and HYSCORE (hyperfine sublevel correlation

spectroscopy) Experiments ....................................................... 211
A6-1. Main Program for 4-Pulse ESEEM ......................... 211
A6-2. Main Program for HYSCORE ................................ 215
A6—3. Subprograms ........................................................ 220

A7. Computer Programs for Searching Angle Sets from an

Anisotropic EPR Spectrum to Interprete ESEEM Spectra ............ 234
A7-1. Main Program ...................................................... 234
A7-2. Subprograms ........................................................ 236

A8. Computer Programs for Calculating Orientation Selective
Four-Pulse ESEEM Frequency Domain Spectrum for S=1/2

and I=1/2 Spin System .............................................................. 241
A8-1. Main Program ...................................................... 241
A8-2. Subprograms ........................................................ 242

ix

 

Table A1-1.
Table Al-2.
Table Al-3.
Table A1-4.
Table A2—1.
Table A2-2.
Table A2-3.
Table A2-4.
Table A3-1.
Table A3-2.
Table A3-3.

LIST OF TABLES

Table 11-1. Four step phase cycle for four-pulse ESEEM ...................... 26
Matrix elements of HFI and NZI of 1:1 ............................ 164
Matrix elements of HFI and NZI of I=3/2 ......................... 165
Matrix elements of HFI and NZI of I=5/2 ......................... 165
Matrix elements of HFI and NZI of I=7/2 ......................... 166
M matrix elements for [:1 .............................................. 167
M matrix elements for I=3/2 ........................................... 167
M matrix elements for I=5/2 ........................................... 168
M matrix elements for I=7/2 ........................................... 169
Matrix elements of NQI of 1:1 ........................................ 170
Matrix elements of NQI of I=3/2 ..................................... 170
Matrix elements of NQI of I=5/2 ..................................... 171
Matrix elements of NQI of I=7/2 ..................................... 172

Table A3-4.

 

LIST OF FIGURES

Figure 11-1. Resolution of a linearly polarized microwave field into
two circularly polarized components, Bl(r) and Bl(l). X, Y and x, y
denote the laboratory and the rotating frames, respectively ...................

Figure II-2. Two—pulse spin echo sequence and formation of spin
echo in the rotating frame. ................................................................

Figure II-3. Three-pulse or stimulated spin echo sequence and
formation of spin echo. .....................................................................

Figure 11-4. Four-pulse spin echo sequence and formation of spin
echo. ...............................................................................................

Figure 11-5. Two-pulse ESEEM experiment scheme .............................
Figure 11-6. Nuclear modulation effect of two—pulse electron spin echo.
(a) Energy level diagram for I=1/2 nucleus coupled to S=1/2

electron spin. (b)—(d) Behavior of magnetization of the allowed and
the semiforbidden spin packets during two-pulse sequence ....................

Figure 11-7. Three—pulse ESEEM experiment scheme ...........................
Figure 11-8. Four-pulse ESEEM experiment scheme. ...........................

Figure III-1. Relative orientation of the external magnetic field with
repect to the principle hyperfine axes. ................................................

Figure IV-l. Schematic diagram of pulsed-EPR spectrometer built in
Michigan State University

 

Figure V-1. Simulated field profiles of the frequency shifts,
(va+v[5)-2v1, for the turning points of the proton sum combination

xi

 

 

bands across the EPR absorption spectrum as a function of On.
Parameters common to all simulations were g I, 2.198; g”, 2.007
microwave frequency, 8.789 GHz; effective dipole—dipole distant
2.4 A; and isotropic hyperfine coupling, 0 MHz. For (a) 6n=00;
0n=300; (c) 0n=450; (d) 0n=600; and (e) 0n=900. The solid and
dashed curves represent singularities calculated for d>=0 and 71',
respectively, while "++++" and "****" patterns are for numerica
calculated features. The frequency shifts were calculated at 46
magnetic field positions across the EPR absorption spectrum. Th
arrows in (b) indicate the field positions where the lineshapes an
computed in Fig. V—2 .............................................................

Figure V-2. Lineshapes of proton sum combination bands at fieli
positions across the EPR absorption spectrum indicated by the

arrows in Fig. V-1 (b). The magnetic field strengths represented
(a) 2857.0 G (g I), (b) 2870.0 G, (c) 2910.0 G, (d) 2975.0 G, (e
3120.0 G, and (f) 3128.8 G (gH). An intrinsic linewidth (FWHM'
0.03 MHz was used in the calculation to reveal all of the discrete
lineshape features. .................................................................

Figure V-3. Four—pulse ESEEM (a) time domain data and (b)

corresponding magnitude FT spectrum. The experimental condit
were magnetic field strength, 2981 G; microwave frequency, 8.7
GHz; microwave pulse powers of n/2 and 7t pulses, 31.5 W and t
W; pulse width, 16 ns (FWHM); sample temperature, 4.2 K; puls
sequence repetition rate, 10 Hz; events averaged/pt, 12; and 1, 1S
ns .........................................................................................

Figure V-4. A plot of the four-pulse ESEEM experimental protc
sum frequency shifts (circles) from twice the proton Larmor
frequency, (va+v5)-2v1, vs. magnetic field strength for

N i(III)(CN)4(H20)2'. Experimental conditions were same as in l
V-3 except magnetic field strengths and r values which were 313
186 ns; 3041 G, 193 ns; 3011 G, 195 ns; 2981 G, 197 ns; 2951G
199 ns; 2921 G, 201 ns; 2891 G, 203 ns; 2869 G, 205 ns; and 28
G, 206 ns. Error bars represent the intrinsic uncertainty in
measuring frequencies from four-pulse ESEEM spectra. ...........

Figure V-5. A comparison of the field profile of the axial water
proton sum frequencies (circles) with simulated field profiles of
turning points of the proton sum combination peak lineshape for
different effective dipole—dipole distances. The simulation param

 

 

 

 

were gr, 2.198; g”, 2.007; microwave frequency, 8.789 GHz; a, 0
MHz, an, 180; reg, (a) 2.28 A, (b) 2.33 A, (c) 2.38 A. The frequenc,
shifts were calculated at 46 magnetic field positions across the EPR
absorption spectrum .....................................................................

Figure V—6. A comparison of the field profile of the axial water
proton sum combination frequencies (circles) with the simulated
field profile of the turning points of the proton sum combination
peak lineshape for an of (a) 13°, (b) 18° and (c) 23°. Other
simulation parameters were identical to those of Fig. V—5 except I'eff
2.33 A; and a, 0 MHz ...................................................................

Figure V-7. Proton sum combination peak lineshapes obtained from
four—pulse magnitude FT -ESEEM spectra of Ni(III)(CN)4(HzO)2‘.
Experimental conditions were same as in Fig. V—3 except magnetic
field strength, 2944G; microwave frequency, 8.691 GHz; r, (a) 239
ns and (b) 260 ns. ........................................................................

Figure V-8. A comparison of the measured proton sum frequency
shifts (circles) from twice the proton Larmor frequency in Fig. V-7
with the simulated field profile of the turning points of proton sum
combination band. The parameters for simulation were microwave
frequency, 8.691 GHz; l‘eff, 2.31 A, On, 18°; a, (a) 0 MHz, and (b) 4
MHz

ooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooo

Figure V—9. Simulations of the proton sum combination peak
lineshapes the in frequency domain without the 1' dependent
coefficient of Eq. V-8. The simulation parameters were magnetic
field strength, 2944 G, microwave frequency, 8.691 GHz, I'eff, 2.31
A, 0“, 18° and a, 0 MHz. Intrinsic gaussian linewidths (FWHM) wer
(a) 0.03 MHz, and (b) 0.4 MHz. ....................................................

Figure V-lO. Simulated proton sum combination peak lineshapes
including the 1" dependent coefficient of Eq. V-8 and different
isotropic hyperfine coupling constants. The simulation parameters
were identical to those of Fig. V-9 except that the isotropic hyperfin
coupling constants shown in figure were varied from 0 to 4.0 MHz,
and intrinsic gaussian linewidths (FWHM) of 0.4 MHz were utilized.
For the simulated lineshapes of Fig. V-10 (a), 1:239 ns, while for
Fig. V-lO (b), =260 ns

 

xiii

 

 

 

 

 

 

 

Figure V-ll. Computed field profiles of the frequency shifts of the
turning points of the proton sum combination bands from twice the
proton Larmor frequencies in rhombic g-tensor system. The
parameters for the simulations were g1=2.19; g2=2.15; g3=2.02;
microwave frequency, 9.0 GHz; reff, 2.33 A; a, 2.5 MHz; (a) 0n,

180; (in, 00; (b) 9n, 180; (bu. 450; (C) 6n, 180; cbn, 900; (d) 6n, 300;

4)", 0°; (e) (in, 30°; (1)“, 45°; and (f) (in, 30°; (1)“, 90°. ...........................

Figure VI-l. Electron spin echo detected-EPR (solid) and simulated
EPR (dashed) absorption spectra of Cu(II)(H20)62+. The stimulated
echo (1r/‘2-r-7r/2-T-7r/2) microwave pulse sequence was used to
generate electron spin echo signal with r=250 ns and T=2000 ns.
Other experimental conditions were microwave frequency, 8.820
GHz; microwave pulse power, 52.5 W; microwave pulse width
(FNHM), 16 ns; averaging number, 10; pulse repetition rate, 12 Hz,
and temperature, 4.2 K. The simulated spectrum was obtained with
EPR parameters of g_1_=2.083, g||=2.411, A _|_=10 MHz, and A||=418

NIHZ ................................................................................................

Figure VI-2. Four-pulse magnitude FT-ESEEM spectra of
Cu(II)(l-120)62+ at proton sum combination frequency region.
Experimental conditions were magnetic field strength, 2452G;
microwave frequency, 8.795 GHz; microwave pulse powers of it-
pulse, 141 W and n/2-pulse, 71 W; averaging number, 30; pulse
repetition rate, 30. r values are (a) 239 ns and (b) 287 ns. ....................

Figure VI-3. Proton sum combination frequency shifts (circles)
[Av=va+v5-2v1] of four-pulse ESEEM spectra obtained at magnetic
field strengths, 2431 G, 2452 G, 2473 G, 2494 G, 2515 G, and 2536

G. Other experimental conditions were same as in Fig. VI—2 except r
value. At each field strength, r values are were changes between

2.5/v1 and 3/v1 with 10 ns or 8 ns step ................................................

Figure VI-4. A comparison of the measured proton sum combination
frequency shifts (circles) with simulated field profile of the turning
points of proton sum combination peak lineshape. The EPR
parameters for angle selection were g_L, 2.083; g", 2.411; A _|_, 10
MHz; A", 418 MHz; copper M[,3/2; and microwave frequency, 8.795
GHz. The ESEEM simulation (ligand hyperfine) parameters were
reg, 2.49 A; a, 0 MHz; em, (a) 900, (b) 800, and (c) 710. The sum
frequency shifts were calculated with 46 0, the angle between the

xiv

 

 

external magnetic field and g3 axis, values (0° ~ 90°) which cover

the EPR absorption spectrum. In the simulated field profile, solid

and dashed curves correspond to the turning points occurring at

¢=0, and it, respectively. The turning points of plus signs were
numerically determined .....................................................................

Figure Vl-5. Experimental proton sum frequency shifts (circles) of
lower-frequency peak of the two sum frequency peaks obtained with

r values of around 3/v1 (See text). Experimental conditions were

same as in Fig. Vl—2 except r values of 282 ns, 290 ns at 2431 G; 279
ns, 287 ns at 2451 G; 277 us at 2473 G; 282 us at 2494 G; 280 ns at
2515 G; and 271 ns at 2536 G ............................................................

Figure Vl-6. A comparison of the measured proton sum frequency
shifts (circles) of Fig. VI-5 with the simulated filed profile of the
turning points of the proton sum combination peak lineshape. The
ligand hyperfine parameters for the simulation were reff=3.05 A,

a=0 MHz, and 0N=l 1°. The other parameters were same as in Fig.
VI-4 ................................................................................................

Figure VI-7. Simulated Lineshapes and relative intensities of proton
’ sum combination band for data of Fig. VI-5 (solid curves) and the
equatorially bound water protons (the other curves) with T=3/VI.
Parameters for the angle selection were same as in Fig. VI—4. Ligand
hyperfine parameters for the simulation of the solid curves were
same as in Fig. VI-6. Ligand hyperfine parameters for the other
curves were reff=2.49 A, a=1 MHz and 0N values are in the figures.
r values and magnetic field strengths were (a) 290 ns, 2431 G; (b)
287 ns, 2452 G; (c) 285 ns, 2473 G; (d) 282 ns, 2494 G; (e) 280 ns,
2515 G; and (f) 278 ns, 2536 G. The arrows indicate the frequency
of 2V}. Intrinsic gaussian linewidth (FWHM) of 0.2 MHz were used
in the calculation. .............................................................................

Figure VI-8. Simulated Lineshapes and relative intensities of proton
sum combination band for the axially bound water protons (solid
curves) and the equatorially bound water protons (the other curves)
with 1:2.5/v1. Parameters for the simulations were same as in Fig.

Vl—7 except 1 values and magnetic field strengths of (a) 242 ns and

2431 G; (b) 233 ns and 2515 G. The arrows indicate the frequency

of Zn. Intrinsic gaussian linewidth (FWHM) of 0.2 MHz were used

in the calculation. .............................................................................

XV

 

 

 

Figure VI-9. Electron spin echo detected-EPR spectrum of axial-

HzO ligand-containing Cu(II)TPP. The stimulated echo sequence was
used with r=220 ns and T=2000 ns. Other experimental conditions
were microwave frequency, 8.978 GHz; microwave pulse power,

29.5 W; microwave pulse width (FWHM), 16 ns; averaging number,
30; pulse repetition rate, 10 Hz; and temperature, 4.2 K. The arrow
indicates the field position where four-pulse ESEEM experiments

were performed. .............................................................................

Figure VI—10. Proton sum combination peak region of four-pulse
magnitude FT-ESEEM spectrum of axial-H20 ligand-containing
Cu(II)TPP. Experimental conditions were microwave frequency,

8.798 GHz; magnetic field strength, 3131 G; r, 225 ns; microwave
pulse powers of it-pulse, 178 W and n/2-pulse, 89 W; averaging
number, 30; and pulse repetition rate, 10 Hz. .....................................

 

Figure VII-1. Pterin derivatives. ...............

Figure VII-2. (a) Three—pulse ESEEM data and (b) cosine fourier
transformation spectrum of Cu(II)(ethp)2(H20)2. Experimental
conditions are magnetic field strength, 3265 G; microwave

frequency, 9.406 GHz, microwave power, 63 W; scanning number,

30; pulse repetition rate, 80 Hz; 1, 140 ns; and temperature, 4.2 K. ......

Figure VII-3. Electron spin enegy level diagram for 14N near eaxct
cancellation regime ..........................................................................

Figure VII—4. Crystal structure of Cu(II)(ethp)2(H20)2 .......................

Figure VII-5. (a) Time domain 14N-ESEEM simulation and (b)
Fourier transformation. Hamiltonian parameters for the simulations
are AXX=2.12; Ayy=2.12 MHZ; Azz=2.60 MHZ, €2QQ=3.63 MHZ;
n=0.30; Euler angles, (1:810, 6:900, y=0°; magnetic field strength,
3265 G; and r=140 ns. Two 14N nitrogen contribution to ESEEM
was asmlmed

 

ooooooooooo

FlgUIe VII-6. (a) Three-pulse ESEEM data and (b) cosine fourier
transformation spectrum of Cu(II)(FA)2(H20)2. Experimental
cOIIditions are magnetic field strength, 3100 G; microwave
fre‘luellcy, 8.926 GHz, microwave power, 63 W; seaming number,

; PUISe repetition rate, 30 Hz; 1, 152 ns; and temperature, 4.2 K. ......

xvi

 

Figure VII-7. Crystal structure of Cu(II)(bpy)(PC)(HzO) ..................... 1.

Figure VII-8. Three-pulse ESEEM data of Cu(II)(bpy)(PC)(HzO).
Experimental conditions are magnetic field strength, 3050 G;

microwave frequency, 8.774 GHz, microwave power, 36 W;

scanning number, 100; pulse repetition rate, 30 Hz 1, 155 ns; and
temperature, 4.2 K. .......................................................................... 1;

Figure VII-9. (a) Three-pulse ESEEM data and (b) cosine fourier
transformation spectrum of Cu(II)(bpy)(PC)(im). Experimental
conditions are magnetic field strength, 3050 G; microwave
frequency, 8.897 GHz, microwave power, 45 W; scanning number,

30; pulse repetition rate, 30 Hz; r, 155 ns; and temperature, 4.2 K. ....... It
Figure A5-l. PIN 1 Circuit ............................................................... 1‘
Figure A5-2. PIN 2 Circuit ............................................................... 1‘
Figure A5-3. Phase Control Circuit .................................................... 21
Figure A5-4. Receiver Control Circuit ............................................... 21

Figure A5-5. Timing diagram of from To delay input to 4 input (8-
NAND gate) in PIN 1 ....................................................................... 21

Figure A5—6. Timing diagram of from delay inputs to TWT input ......... 21

Figure A5-7. Timing diagram of from To delay input to phase
circuit input. .................................................................................... 21

Figure A5—8. Timing diagram of from delay inputs to phase circuit

inputs. ............................................................................................. 21
Figure A5-9. Timing diagram of from To delay input to 10 input ......... 21
Figure A5-10. Timing diagram of from To, A, B delay input to PIN

1 input. ............................................................................................ 21
Figure A5-11. Timing diagram of phase control circuit ........................ 21
Figure A5-l2. Timing diagram of receiver switch control circuit .......... 2

xvii

 

 

LIST OF ABBREVIATIONS

bpy 2,2'-bipyridine

CW Continuous Wave

DEER Double Electron Electron Resonance
dien Diethylenetriamine

ENDOR Electron Nuclear Double Resonance
EPR Electron Paramagnetic Resonance
ESE Electron Spin Echo

ethp 2~ethylthio-4-hydroxypterin
ESE-ENDOR Electron Spin Echo Detected-ENDOR
ESE—EPR Electron Spin Echo Detected—EPR
ESEEM Electron Spin Echo Envelope Modulation
FA Folic acid

FID Free Induction Decay

FT Fourier Transformation

I“"WHM Full Width at Half Maximum

HFI Hyperfine Interaction

HY SCORE Hyperfine Sublevel Correlation Spectroscopy
MW Microwave

NQI Nuclear Quadrupole Interaction

NZI Nuclear Zeeman Interaction

PAH Phenylalanine Hydroxylase

xviii

 

 

PAS

TPP
TWT

 

Principal Axis System
6—carboxy pterin

Tetraphenylporphyrin
Travelling Wave Tube

 

 

 

I. INTRODUCTION

Although nuclear spin echoes resulting from the application 1
resonant radio frequency pulses were first reported by Hahn in 1950,1 ii
application of pulse methods to electron spin resonance has been slow 1
develop. Reasons for this slow development included the lack of fast digit
electronics, required to generate electron spin echoes using microwai
(MW) pulses, and restrictions imposed by microwave pulse bandwidti
Most of the early experiments utilized ESE to measure electron sp
relaxation times.2 Nuclear modulations of the electron spin echo amplitu<
were unexpectedly observed by two independent groups during these ear
experiments on electron spin relaxation.3-4 Further experimental ar
theoretical studies of these nuclear modulation effects, commonly referre
to as electron spin echo envelope modulation (ESEEM), showed that t1
modulation depths and frequencies are related to the electron-nucle:
C0uplings.5'8 Many technical problems in MW pulse generation and da
collection have been solved within the past 15 years. As a result, electrt
paramagnetic resonance (EPR) studies utilizing MW pulse techniques a
being rapidly developed to open a new renaissance in the study .
paramagnetic species.9'11 Several different pulse sequences and da
CC’HCCtion schemes have been developed for pulsed-EPR spectroscop
EaCh SCheme has its own characteristics and applications depending on tl

types of radical centers and the desired information.12

1

 

2

Theoretical studies have revealed that electron spin echo envelo
modulation effects are observed when the nuclear hyperfine states 2
quantized along effective fields that differ between electron spin manifoh
In other words, when the hyperfine interactions between unpaired electr
and magnetic nuclei contain anisotropic couplings, the high-field nucle
hyperfine eigenstates are mixed so that all EPR transitions (AMs=-|_
AM1=O) between or and B electron spin manifolds can simultaneously occ
in response to a resonant MW pulse. The resulting interferences betwe
the EPR transitions give rise to nuclear modulation effects. The effecti
frequency width of the fourier transform components of the MW puls
used in ESEEM experiments depend on the pulse width and are genera
ca. 80 MHz . Therefore, only weak hyperfine interactions ( < 40 MHz ) 2
observed in ESEEM making it complementary to conventional CW—El
where small hyperfine couplings are not resolved due to inhomogeneo
line broadening. ESEEM can be applied to not only crystalline samples I
also randomly oriented powder or glassy paramagnetic compoum
Because ESEEM spectra are most sensitive to anisotropic hyperfi
interactions between magnetic nuclei and the unpaired electron, th
provide structural information about the paramagnetic center.

Two-pulse and three—pulse or stimulated echo ESEEM are m
routine pulsed—EPR methods and have been very well studied. Two-pui
ESEEM measurements are carried out with a n/2-r-rr pulse sequence whr
T is the time interval between the first and second MW pulses. Electr
spin echoes are produced at a time, 1, after the second pulse. In a two—pui
ESEEM experiment, echo intensities are recorded as a function of the til

interval, T. The stimulated echo is generated at time T after the third pul

Of a three ir/2 pulse sequence (7r/2—r-ir/2-T-Tr/2). The echo amplitudes z

3

collected as a function of T, the time interval between the second and third
pulses. The spectral resolution of stimulated ESEEM is generally better
than that of two-pulse ESEEM because the decay time of two-pulse echo is
restricted by the relatively short phase memory time (TM) while the
stimulated echo decay is determined by the electron spin-lattice relaxation
time, T1, or nuclear spin-spin relaxation time, T2“. In ESEEM
experiments, the choice of pulse scheme depends on the purpose of the
experiment. Even though the stimulated ESEEM experiment has better
resolution, the spectra suffer from r- suppression effects which cause
distortion of the ESEEM lineshapes. However, the suppression effect can
be used to suppress unwanted echo modulations and identify correlations
between peaks. In two pulse ESEEM, the lineshapes are not distorted, but
the spectral resolution is limited as mentioned above. The two-pulse scheme
is applicable when the hyperfine interaction is strongly anisotropic. Two—
pulse ESEEM spectra have sum (va+v5) and difference (Va-VB)
combination bands as well as fundamental frequencies (Va, v5), while
stimulated ESEEM experiments show only the fundamental frequencies.
For the case of strong anisotropic hyperfine coupling, the fundamental
bands are spread over a wide frequency range for randomly oriented
samples, making them difficult to detect. The peak positions of the sum
combination bands are directly related to the anisotropic portion of the
hyperfine interactions, and have a narrow linewidth making them easy to
dam. The two—pulse sequence can be most useful for such cases.13,14
A Four-pulse (rt/2-t-rr/2—T/2-rr-T/2-rr/2) ESEEM method has also
been suggested recently.15 In this experiment, a n-pulse is inserted in the
middle of the second and third pulses of the three-pulse ESEEM sequence

and the time domain ESEEM data are collected as a function of T, the time

 

 

4
interval between the second and fourth pulses. Here, sum (va+v[3) and
difference (Va-v5) combination bands are also found, but with improved
spectral resolution because the spin coherences decay by a T1 process as in
a stimulated echo experiment. Few results from four-pulse experiments
have been reported.16 For detection of the combination bands, four-pulse
experiments have the advantage of better spectral resolution when
compared to two-pulse ESEEM results. Also, the r—suppression behavior of
the four-pulse echo modulation experiment can give more information. A
problem in using the four-pulse scheme is echo amplitude diffusion. As the
number of pulses increases, so does the number of unwanted two— and
three—pulse echoes. These additional echoes lead to loss of four-pulse echo
intensity. As a result, relatively high concentrations or increased signal
averaging is required.16 Another problem of four-pulse ESEEM
measurements is that the stimulated echo made of the first, second and
fourth pulses occurs at the same position and phase of four-pulse echo.
Hence, four-pulse ESEEM data always contains some stimulated echo
modulation data.15

Two-dimensional (2D) ESEEM experiments can be carried out using
three-pulse or four—pulse sequences.17s18 2D ESEEM experiments provide
information about the correlation between nuclear hyperfine lines
belonging to the same paramagnetic center and the relative signs of
hyperfine couplings. In the three-pulse 2D ESEEM experiment, time
domain data are collected in r and T time axes for the it/2-r-7r/2-T-rr/2
pulse sequence. Four-pulse 2D ESEEM or hyperfine sublevel correlation
(HYSCORE) spectroscopy is performed with a 7r/2-r-ir/2-t1-n-t2-1r/2 pulse
sequence. The data are collected as a function of t1 and t2. In the three-

pulse 2D experiments, the echo decay rates on the r and T axes are

5

determined by the phase memory time (TM) and spin-lattice relaxation tii
(T1), respectively. Therefore, linewidths along the 1' axis are severe
broadened. In the HYSCORE experiment, spectral resolution along t1 a
t2 axes are the same because the echo amplitude decays are determined
T1. However, the HYSCORE experiments can suffer from the
suppression effects while three-pulse 2D experiments are free of ti
distortion. There are few publications concerning the application of 2
ESEEM methods to paramagnetic samples.19 The main reason for this
the data collection time to produce a 2D spectrum. A new detection scher
using coherent "Raman Beat" detection is expected to provide solution
this problem.20

This thesis will present the applications of ESEEM to some transiti
metal ion complexes. In chapter II, the basic principles of electron 3;
echo formation and the ESEEM experiment will be explained. T
quantum mechanical formalism of ESEEM will be derived in chapter I
In chapter IV, the pulsed—EPR spectrometer which was used in t1
research will be described with an emphasis on modifications made
perform the experiments described in chapters V and VI. In chapter
angle selection four-pulse ESEEM studies of bis-aq
tetracyanonickelateflll) [Ni(III)(CN)4(HzO)2'] will be discussed
characterize the hyperfine interactions between the axially bound wa
protons and the nickel ion. In chapter VI, the hyperfine interactions
equatorially and axially bound water protons in Cu(II)(H20)62+ will
distinguished using four-pulse ESEEM experiments. In chapter VII, thr
pulse ESEEM studies of copper-pterin complexes as a model complexes
the Cu(II) site of Phenylalanine Hydroxylase (PAH) fri

6

Chromobacterium Violaceum21 will be described to understanc

structural relationship between Cu(II) and the pterin cofactor.

References

1. E. L. Hahn, Phys. Rev., 80, 580 (1950).

2. K. D. Bowers, and W. B. Mims, Phys. Rev., 115, 285 (1959).

3. W. B. Mims, K. Nassau, and J. D. McGee, Phys. Rev., 123, 205
(1961).

4. J. A. Cowen, and D. E. Kaplan, Phys. Rev., 115, 285 (1959).

5. LG. Rowan, E. L. Hahn, and W. B, Mims, Phys. Rev., A137, 6
(1965).

6. D. Grischkowsky, and S. R. Hartmann, Phys. Rev., BZ, 60 (196

7. W. B. Mims, Phys. Rev., B5, 2409 (1972).

8. W. B. Mims, Phys. Rev., B6, 3543 (1972).

9. L. Kevan, and M. Bowman, Ed., in Modern Pulsed and Continm
Wave Electron Spin Resonance, John Wiley & Sons, New York
(1990).

10. A. J. Hoff, Ed., in Advanced EPR, Elsevier, Amsterdam (1989).

11. C. P. Keijers, E. J. Reijerse, and J. Schmidt, Ed., in Pulsed EPR
North-Holland, Amsterdam (1989).

12. A. Schweiger, Angew. Chem. Int. Ed. Engl, 30, 265 (1991).

13. A. V. Astashkin, S. A. Dikanov, and Yu. D. Tsvetkov, Chem. Pl
Lett., 136, 204 (1987).

14. E. J. Reijerse, and S. A. Dikanov, J. Chem. Phys, 95, 836 (199

15. C. Gemperle, G. Abeli, A, Schweiger, and R. R. Ernst, J. Magn.
Reson., 88, 241 (1990).

16. A. M. Tryshkin, S. A. Dikanov, and D. Goldfarb, J. Magn. Resc
A105, 271 (1993).

17. R. P. J. Merks, and R. de Beer, J. Phys. Chem, 83, 3319 (1979

18. P. Hofer, A. Grupp, H. Nebenfiihr, and M. Mehring, Chem. Ph}
Lett., 132, 279 (1986).

19. P. Hofer, Bruker Report, 118, 1 (1993).

20. M. K. Bowman, Israel J. Chem, 32, 339 (1992).

21. S. O. Pember, J. J. Villafranca, and S. J. Benkovic, Biochemistr

25, 6611 (1986).

II. BASIC PRINCIPLES OF ELECTRON SPIN ECHO
ENVELOPE MODULATION

1. Basic Principles of Electron Spin Echo 038E)
1-1. Two-Pulse Spin Echo

Two-pulse spin echoes were first observed for nuclear spins usir
ril2-r-7r/2 pulse sequence by Hahn.1 Later, a more idealized two-p1
sequence of ir/2-r-ir was described by Carr and Purcell.2 The formation
spin echoes can be explained using a magnetization vector model.

In a static external magnetic field, the magnetic moments of unpa
electron spins are aligned either antiparallel or parallel to the external 1
according to their spin states, a or B, respectively. The populations of
states follow the Boltzmann distribution law. For electrons, the 6 spin 5
is more stable resulting in a total magnetization, M, parallel to the exte
field direction. When a linearly polarized MW is applied to the :
system, a torque is applied to the magnetization and it is turned into
plane perpendicular to the external magnetic field direction. This line
polarized MW magnetic field can be represented in the rotating frame
letting one of two circular polarized components of the MW be along
x-axis of rotating frame with frequency of <01 [B1(r) in Fig. II-l]. Ther

field mm is time-independent. The other component Bl(l) has a freque

7

8
of -2w1 in the rotating frame but has little effect on a magnetic reson
experiment. From now on, X,Y and Z indicate axes of the labora
frame and x, y, and 2 indicate the axes of the rotating frame.

In this rotating frame, when a short and intense MW pulse, w‘
center frequency, (00, and bandwidth cover the precession frequency r:
of the individual spin magnetic moments that contribute to
magnetization, is applied, the magnetization vector, M, is deflected i
the external field direction due to the torque between the time indepen
MW field of the rotating frame and the magnetization vector. The angi
deflection ,or flip angle, is given by 0fl1p=w1tp=yeB1tp. Here, tp is the p
width, ye is the gyromagnetic ratio for an electron, and B1 is the magi
field amplitude of the applied MW pulse. The relationship between
applied MW power and the 31 field depends on the cavity or sample pr

 

 

Figure II-l. Resolution of a linearly polarized microwave field into
c1rcular1y polarized components, mm and B1(l). X, Y and x, y denote
laboratory and the rotating frames, respectively.

If one choose a 90° pulse, the magnetization vector ends up along
Y-axis of the rotating frame immediately after the pulse is turned off I

11-2 (a)]. Because of inhomogeneity in external and internal magr

9

fields, the individual electron spin magnetic moments precess about the z-
axis with the frequencies of w-wo. If one installs the detection system along
the y-axis of the rotating coordinate, the amplitude of the total transverse
i magnetization decays rapidly (Free Induction Decay: FID) [Fig. II-2 (b)].
After a time T a 180° pulse is applied about the x-axis so that the each
magnetization vector is turned into the -y direction [Fig. II—2 (0)]. The
precession behavior of each vector component or spin packet remains the

same as before the

n/2 7T

       

l

 

 

 

(a) (b) (C) (d)

900

 

 

 

 

 

 

Figure lI-2. Two—pulse spin echo sequence and formation of spin echo in
the rotating frame.

10

refocussing pulse. Hence, all magnetization vectors are aligned along the
direction at time 1' after the second pulse to build up a "spin echo" [Fig,
2 (d)].

The amplitude of the two-pulse spin echo is a function of T, the ti:
interval between the first and second MW pulses. During r, i
magnetization is in the xy plane so that relaxation processes are due to sp
spin relaxation. The decay rate of the two-pulse electron spin ec
amplitude can be represented by a phase memory time, TM. TM is , in p2
a function of the spin-spin relaxation time, T2, which corresponds to 1
time during which individual spin packets maintain their phases in I
diffused pattern of Figure II—2 (b).

1-2. Three—Pulse Spin Echo (Stimulated Spin Echo)

Three-pulse or stimulated spin echoes are produced by a 7r/2—r-1r
T-rr/2 pulse sequence. The spin echo is formed along the -y direction of 1
rotating frame as depicted in Fig. II-3. The formation of the spin echo c
be easily explained by following the behavior of a component of the to
magnetization vector M.

The character of the magnetization after the first pulse is the same
in the two-pulse spin echo sequence. At time I after the first pulse
COmponent 6M of the total transverse magnetization M makes an angle
(OJ-cook with respect to the y-axis [Fig. II—3 (a)]. Here, too is the M
Pillse frequency and w is the precession frequency of the component 5
the laboratory field axis (Z-axis). The second 90° MW pulse, applied alo
the x-axis at time T after the first pulse, brings 5M into the xz pla
making an angle between 6M and the -z-axis of (co-won" [Fig. II-3 (b)].

 

 

    

 

   

ll

Tr/2

  

 

 

 

   

 

 

 

 

 

 

 

 

(a) (b) (C) (d) (e
(a) 2 (b) 1
6M
‘7 " W"
X I
-v ‘ i
5MZ - -
5M
(c) Z (d)
5My
x y 4
stv
X
(e) (0
6M,- 5Mi
wk
‘3' 3 'y
We Mr (M,
X X

Figure II-3. Three-pulse or stimulated spin echo sequence and formation

sPin echo.

 

1 2
At this moment, 5M can be resolved into two components, a transverse
component, be, and a longitudinal component, 5M1. The transverse
magnetization 5Mx makes a spin echo of 7T/2-T-TI/2, so called "eight ball", at
time t after the second pulse. The "eight ball spin echo" was described by
Hahn in his original discovery of the spin echo.1 The longitudinal
magnetization 5M2 begins decaying on the z-axis by spin-lattice relaxation
process [Fig II-3 (e)]. The third 7r/2 pulse applied at time T after the
second pulse torques 6Mz leaving it along the —y—axis to generate a
transverse magnetization component 6M), [Fig. II-3 (d)]. Assuming that
there is no decay and the magnitude of 5M is unity, the projection of EM),
along the -y direction is cos(w-wo)r at this moment. After the third pulse
is turned off, the magnetization component 5M), starts precessing about the
z-axis with the frequency of w-wo. At an arbitrary time t after the third
pulse, the magnetization component 6M6 is located at the angle of (w-wo)t
from the -y-axis [Fig. II-3 (e)] and a projection along the -y-axis of
cos{(w-wo)rlcos{(w—wo)t}. Until now, we have considered only one
component of the total magnetization. The total magnetization My along the
-y direction at this time t can be estimated as the integration of cos{(w—
wo)r}cos{(w-wo)t} over the range of w. Assuming the width of w is 2Aw
and the distribution function of to is constant over the considered range

(f(w)=1),

My: 030+Aw

coo-Aw cos{(w—wo)r}cos{(w-wo)t}dw. [II-1]

The timing for the maximum transverse magnetization My is obtained

satisfying the condition, dMy/6t=0.

 

0M! mmmwfim
6t — (‘r+t)2(r-t)2
+-(r+t)Zsin{Aw(r-t)}-Aw(T-t)(r+t)2cos{Aw(r-t)} 12
(ooze-02 ‘ [1']

Therefore, when t=r for t>0, the dMy/6t=0 condition is satisfied and 1
total magnetization along the -y-axis is maximized to form the three-pu
or stimulated spin echo. At the time of spin echo formation, the componi
5Me of the total magnetization can be divided into two components, x an<
components, BMW“ and 5Me,y. Their sizes along x and y axes are given
5Me,x=cos{(w—wo)r}sin{(w—wo)r} and 5Me,y=-cosz{(w-wo)1
respectively. Hence, the relationship between SM,” and 6Me,y is obtain

as
[bMey]2 + [5Me,y + 1/2]2=1/2. [II-3]

Therefore, when the stimulated spin echo is formed all the individual sr
moment vectors of the total magnetization are placed on the cir<
expressed by Eq. II-3 [ Fig. II-3 (0], which results in echo formation alo
the -y direction.

The amplitude of the stimulated echo is a function of r, the tii
interval between the first and second pulses, and T, the time inten
between the second and third pulses. During 1', the magnetization dephas

in the xy plane so that the decay rate should follow the phase memory til

 

l 4
TM as in the two—pulse spin echo sequence. The second ir/2 pulse serves 1
store the precession frequency offsets (co—Loo) of each spin packet as
longitudinal magnetization which forms the stimulated echo. Because t1
amplitudes of these longitudinal magnetization components are governed I:
a relatively slow spin—lattice, T1, relaxation time, the overall the decay ra

of the amplitude of the stimulated echo is reduced.
1-3. Four-Pulse Spin Echo

The four-pulse spin echo sequence and the formation of the fou
pulse echo are similar to that described above for the three-pulse spin ech<
Here, a it pulse is inserted between the second and third n/2-pulses wil
overall pulse sequence being ir/2-r-7r/2—T/2—rr-T/2-7r/2. The four-pulse sp:
echo is formed at time T after the fourth pulse.

Fig. II-4 shows the pulse sequence and a vector diagram for t1
formation of the four-pulse spin echo. Following the description given ft
Fig. II-3, we focus on a spin packet with an electron spin magnet
moment, located along the -z-axis with z-projection of cos(w-woj
immediately following the second rt/2 pulse [Fig. II-4 (a)]. By applying t1
it pulse at T2 after the second pulse, the longitudinal component BMZ (
the magnetization turns into the z—axis [Fig. II-4 (b)]. This component go<
I0 y-axis by the fourth pulse of 7r/2 which is applied at time T/2 after thii
Pulse [Fig. II-4 (c)]. Immediately after the fourth pulse, the transvers
COmponent 5My precesses about the z-axis with a frequency of 03-030. 1
analogy to the formation of the stimulated spin echo, the tot:
magnetization along the y-axis is maximized at time I after the fourth pul:

forming the four-pulse spin echo [Fig. II-4 (d)—(e)]. Here, the spin echo

 

.—
LII

n72 n72

  

lllllllllllllll :j

 

    

V Illllllllllllll

(d)

(C)

.3

'~<

(c) (d)
5
9
X

 

Figure II-4. Four-pulse spin echo sequence and formation of spin 60110-

 

I

16

produced along the y-axis in contrast to the two-pulse and the three-p1
spin echoes where echoes are formed along the -y-axis. The decay rate
the amplitude of the four-pulse spin echo is determined by both TM and
as in the stimulated echo because the magnetization is stored on the

plane for the time T and along the z-axis during T.
2. Basic Principles of Electron Spin Echo Envelope Modulation (ESEEIV

ESEEM experiments are carried out by measuring the integra
intensity of electron spin echoes as a function of the time intervals betw
pulses. The modulation frequencies and depths reflect the hyperf
couplings between electron and nuclear spins and ,in some cases, nucl
quadrupole interactions.3'8 The modulation effect occurs when allowed ;
semiforbidden EPR transitions are simultaneously excited by a MW pi
with enough bandwidth to cover the transition energies. Interfere
between the EPR transitions results in the nuclear frequencies wh
appear as modulations of the electron spin echo decay envelope. So,
ESEEM can be applied only to samples where this "branching" in
energy level scheme occurs.

ESEEM spectra generated by Fourier transformation are analog
to electron nuclear double resonance (ENDOR) spectra because both sh
the nuclear hyperfine frequencies. The intensity of ESEEM frequenc
depend on the product of EPR transition probabilities and the number
nuclei coupled to the electron spin. In contrast, the amplitude of
ENDOR peak depends on a balance of electron and nuclear relaxat
Processes in addition to these factors. In practice, these techniques

complimentary to each other.14

2-1. Two-Pulse ESEEM

In the two-pulse ESEEM experiment, the echo intensity is
as a function of r, the time interval between the pulses [Fig.
modulation of the electron spin echo can be explained semiclassic
spin system consisting of a single unpaired electron spin (S=1/2) i

an I=1/2 nucleus, the hyperfine energy levels are split as in Fig. I
I i<— T ->fl<— T —>/\
re . au— . —» A

l ‘f\
s‘ I ‘
_ I \

h \ ‘3 ‘

,\ s
\ ‘ ‘

I I

I “I\
I\
.1

I
I I

‘~.l

I
I I I,

Figure II-5. Two-pulse ESEEM experiment scheme

Assuming that the energy levels within each electron spin ma

admixtures of the high-field nuclear states that are a COHSCt

 

 

 

(b) (C)

Flgllre H—6. Nuclear modulation effect of two-pulse electron spin echo.

(a) Energy level diagram for I=1/2 nucleus coupled to S=1/2 electron spin.
(bHd) Behavior of magnetization of the allowed and the semiforbidden
spin packets during two-pulse sequence.

 

l9

anisotropic hyperfine couplings, all four EPR transitions, 1-3, 2-4, 2-3,
and 1-4, are allowed. If the Fourier frequency range of the MW pulse is
wide enough, all four transitions are excited simultaneously. Letting the
MW frequency correspond to the energy difference between levels 1 and 3
(wo=w13) in Fig. II-6, the modulation effect can be explained using the
magnetization vectors corresponding to the 1-3 and 2-3 transitions. If the
first ir/2 pulse is directed along the x-axis, the vectors for the 1-3 and 2-3
transitions point along the y—axis of the rotation frame which has the
frequency of (so with respect to the laboratory frame. Letting the
transition probabilities of 1-3 and 2-3 be 1113]2 and |123|2, the magnitudes of
the vectors, 0013 and 0923, correspond to their transition probabilities.
After the pulse is turned off, the vector (n23 precesses about 2 axis with
frequency of w23-w13 which is the hyperfine frequency of the or electron
spin state, tea. At time 1‘ after the first pulse, the vector 0323 makes the
angle war with y-axis [Fig. II-6 (b)]. At this time, a it pulse is applied
along the x-axis. This torques the magnetization vectors, (1013 and 0323, to
the -y direction. The 7r pulse also generates new vectors, c013 and 0323',
along (023 and (013 directions due to the "branching of the transitions"
[Fig II-6 (c)]. The length of the vector any is the product of the transition
probabilities of 2-3 and 1—3 transitions, i.e., |I23|2|I13|2. By analogy, the
magnitudes of the vectors, 0323' , c013, and (1)23, are |I13l2|123|2, |I13|4 and
“2314, respectively. During the second evolution time, 0013 and (1313'
remain stationary while w23 and 0323' precess with an angular velocity of
wa- At the moment of the spin echo, c013 and w23 are refocused on the -y—
axis and (.023' and (013' make an angle of war with respect to this axis [Fig.

“‘6 (d)]. Since we detect the magnitude of the total magnetization along the

 

20

-y-direction, the amplitude of the echo signal which is produced by 1

transitions, 0313 and c023, will be given by
E(w13,w23,r)= I113|4 + |123l4 + 2|113I2|123l2008war. III-4]

In Eq. II-4, we can see two components of the echo amplitude. One is a I

component which is not modulated and the other is an ac component whi

is modulated at a frequency of tea. The above mechanism of the [111011

modulation effect can be applied to all other "branching" combinations

the energy level diagram of Fig. II-6. Because |I13|2 = |Izal2 = Ilal2 a
|I14|2 = |123|2 = IIflz, the overall normalized amplitude of the two-pulse s1
echo is expressed by4

E(r)=1-k/4{2-ZCos(war)-2cos(w5r)+cos(w-r)+cos(w+r)]. [II-5]

Here, k = 4IIaI2IIfI2, w— = (ca-cop, and co.» = wanna, As seen in Eq. I]
the two-pulse echo amplitude is modulated by sum, 00+, and difference,
combination frequencies as well as the fundamental frequencies, on; 2
035. The modulation amplitude is determined by the depth parameter, k.

Including the decay rate of the echo amplitude, V(r), the total e<
amplitude is given by

Etotrr) = V(T)E(r). [II-6]

As described in the previous section, the decay rate of the two-pulse echu

determined by the phase memory time, TM. During the evolution time

the electron spin coherences which give rise to the modulation also de

2 l
with the phase memory time. Hence, the decay rates of the dc ampli
and the ac amplitude of the two-pulse echo are the same. In a solid, T
only a few microseconds so that the linewidths of two—pulse ESEEM [)1
are severely broadened.

In ESEEM experiments, the echo signal can not be recorded
about one hundred nanoseconds after the final pulse due to cavity-ring
This time is called the dead time, tD. During the dead time, the microv
pulse power in the cavity is so high that the echo signal can not be deter
The dead time determines the minimum value of I that can be used 11
ESEEM experiment. The affect of the dead time on ESEEM experim
can be very serious. For example, if the dead time is 200 nanoseconds
the echo signal persists for only 2 microseconds (a typical TM), frequc
domain spectra derived from Fourier transformation can be seve
distorted. To reduce the dead time or the effect of the dead time, SCV
instrumental,15'17 mathematical,18'20 pulse sequence,21,22

detection”,24 methods have been proposed.
2-2. Three-Pulse ESEEM

In three—Pulse ESEEM experiments, the stimulated echo sign:
recorded as a function of T, the time interval between the second and
third rt/2 pulses, with I being fixed for a given measurement [Fig. I
Here, the first pulse generates the electron spin coherence as in the 1
pulse scheme. The second pulse transfers the electron spin coherence to
nuclear spin coherence. During the time T, nuclear spin coherences
developed. The third pulse takes these nuclear spin coherences back to

electron spin coherence which is detected. Hence, the three-pulse ESE

 

 

2 2
has only the fundamental frequencies, (0a an (.05, without the combina'

freQuencies.
For the S=l/2, I=1/2 system [Fig. II-6 (a)], the echo amplitudi
expressed by4

'E(T,T)=1-k/4[{l-cos(war)}{1—cos(wfi(T+T))}
+{1-cos(co[3'r)}{1-cos(wa(r+T))}]. [II-7]

As expected, only the fundamental frequencies are shown in Eq. II-7. '
modulation amplitudes of the nuclear frequencies depend on not only

modulation depth parameter, k, but also the time interval, 1, between the

nephew
awn—Ham
hit—T ——»H«—»A

 

Figure II-7. Three—pulse ESEEM experiment scheme.

2 3

first and second pulses. Because of this r-suppression effect, the linesh:
of three-pulse ESEEM peaks can be distorted. When r=2nrr/wa,
amplitude of the cop modulation component completely disappears or
versa. Even though the r-suppression effect can prevent observation of
entire lineshape for a modulation component, it can be used to rerr
unwanted peaks from complicated spectra or to identify correlati
between peaks.

During the time T in three pulse ESEEM experiments, the nuc
spin coherence decays with the transverse nuclear relaxation time, '
Hence, the decay of the echo amplitude depends on Tzn or Tle in the i
of T1e<T2n. Because Tzn and T16 are usually much longer than TM,
resolution of three-pulse ESEEM spectra is much better than that of t
pulse ESEEM spectra. In the case of randomly oriented samp
destructive interference between different frequencies can cause a fa
decay of the modulation. For these samples, the decay of d.c amplituc
much longer than that of ac amplitude.

The three~pulse spin echo sequence generates several two-p
echoes. These two-pulse echoes overlap with the three-pulse echo posi
when T=T or 2r to create pronounced distortions in the ESEEM pati
To remove the influence of these unwanted two-pulse echoes, the phas
the MW pulses is cycled to produce alteranate negative and positive ‘
pulse echoes while the three—pulse echoes always remain posit
Summation of the echoes from an even numbers of phase cycles exti
the pure three-pulse echo amplitude free from distortions caused by

pulse echoes.25

2-3. Four—Pulse ESEEM

The four-pulse ESEEM experiment is performed by varying th
time interval, T, between the second and fourth pulses with a fixed I [Fig
II—8]. The third pulse is always centered between the second and fourt}
pulses. The formula for an S=l/2, I=1/2 spin system is given by26

 

 

k waT waT wfsT mgr
E(T,T)= 1'Z[C0+CQCOS(T+T) + CBCOS( 2 +7)
to T cc r w_T [04-

where

Co = 3-COS((.0aT)-COS(wfiT)-SZCOS(OJ+T)-CZCOS((.O—T),

 

Ca = 2[CZCOS(wfiT-w;T)+S2COS(wfiT+%)-COS(%)L
C5 = 2[czcos(waT-926—T)+szcos(war+%E)—COS(9¥)L
C+ = ~4c2sin(%)sin(%),
C- = —4szsin(%)sin(%),

1
lw12-;(wa+wa)2|

2=__._._ = 2
8 (mm M ,
Iwrz-ima—wafll
c2 = ————— = [1.112, and
waCO6
k = 4s2c2. [II-9]

AS seen in Eq. II-8, the four-pulse echo is modulated with frequencies 0
war/2, dog/2, (wa+w5)/2, and (marrow/2. An advantage of the four-pulsr

25

experiment is that the combination bands are detected with high frequency
resolution. Even though the combination bands are detected in the two-
pulse experiment, the echo amplitude decay broadens the linewidth. Ir
four-pulse measurements, the decay of the echo amplitude follows Tzn o
Tle as in the three-pulse experiment. Therefore, the four-pulse experimen
provides a more accurate method for measuring the frequency ant
lineshape of the sum combination peak. This case will be described ir
chapter V. Four-pulse ESEEM experiments are also subject to T-

suppression effects.

 

Figure II—8. Four-pulse ESEEM experiment scheme.

The four-pulse ESEEM experiment suffers from interference due tr
the unwanted two- and three- pulse echoes. All of These unwanted echoes

Can be removed by a phase cycling except for the three—pulse echo derivec

26

from the first, second and third pulses.26 This echo can not be removed by
phase cycling because the phases of this echo and the four-pulse echo are
always the same. The phase cycle suggested by ref. 26 results in the
alteration of the signs of the four-pulse echo. This scheme was somewhat
inconvenient for our data collection system and a new four-step phase cycle
of +(0,0,0,0), +(T[,T[,0,0), +(0,0,tr,0), +(tr,7r,7r,0) was developed. Here, the

sign of the four-pulse echo is unaltered [Table II—l].

Table II-l. ESEEM

    

References

l. E. L. Hahn, Phys. Rev., 80, 580 (1950).

2. H. Y. Carr, and E. M. Purcell, Phys. Rev., 94, 630 (1954).

3. LG. Rowan, E. L. Hahn, and W. B, Mims, Phys. Rev., A137, 61

(1965).

4. W. B. Mims, Phys. Rev., B5, 2409 (1972).

5. W. B. Mims, and J. Peisach, J. Chem. Phys, 69, 4921 (1978).

6. A. A. Shubin, and S. A. Dikanov, J. Magn. Reson., 52, l (1983).

7. M. Romanelli, D. Goldfarb, and L. Kevan, Magn. Reson. Rev., 13,
179 (1988).

8. W. B. Mims, and J. Peisach, in Biological Magnetic Resonance, L.

J. Berliner, and J. Reuben, Ed., Plenum, New York, 213 (1981).
9. L. Kevan, in Hme Domain Electron Spin Resonace, L. Kevan, and
R. N. Schwartz, Ed., Wiley-Interscience, New York, 279 (1979).

27

W. B. Mims, and J. Peisach, in Advanced EPR, A. J. Hoff, Ed.,
Elsevier, Amsterdam, 1 (1989).

S. A. Dikanov, and A. V. Astashkin, in Advanced EPR, A. J. Hoff,
Ed., Elsevier, Amsterdam, 59 (1989).

A. Schweiger, in Advanced EPR, A. J. Hoff, Ed., Elsevier,
Amsterdam, 243 (1989).

A. Schweiger, in Modern Pulsed and Continuous—Wave Electron
Spin Resonance, L. Kevan, and M. Bowman, Ed., John Wiley &
Sons, New York, 43 (1990).

E. J. Reijerse, N. A. J. M. Van Aerle, and C. P. Keijers, J. Magn.
Reson., 67, 114 (1986).

J. L. Davis, and W. B. Mims, Rev. Sci. Instrum, 52, 131 (1981).
P. A. Narayana, R. J. Massoth, and L. Kevan, Rev. Sci. Instrum,
53, 624 (1981).

W. Barendswaard, J. A. J. M. Disselhorst, and J. Schmidt, J. Magn.
Reson., 58,477 (1984).

D. Van Ormondt, and K. Nederveen, Chem. Phys. Lett., 82, 443
(1981).

W. B. Mims, J. Magn Reson., 59,291 (1984).

R. de Beer, and D. Ormondt, in Advanced EPR, A. J. Hoff, Ed.,
Elsevier, Amsterdam, 135 (1989).

H. Cho, S. Pfenninger, C. Gemperle, A. Schweiger, and R. R. Ernst,
Chem. Phys. Lett., 160, 391 (1989).

C. Gemperle, A. Schweiger, and R. R. Ernst, Chem. Phys. Lett.,
178, 565 (1991).

A. Schweiger, and R. R. Ernst, J. Magn. Reson., 77,512 (1988).

J. -M. Fauth, A. Schweiger, and R. R. Ernst, J. Magn. Reson.,
81,262 (1989).

J. -M. Fauth, A. Schweiger, L. Braunschweiler, J. Forrer, and R. R.
Ernst, J. Magn. Reson., 66, 74 (1986).

C. Gemperle, G. Abeli, A. Schweiger, and R. R. Ernst, J. Magn.
Reson., 88, 241 (1990).

III. FORMALISM OF ESEEM

If a system evolves with time, the density matrix forrr
especially useful to describe the time dependence of the physical p
of the system},2 In ESEEM experiments, the echo modulation res
the time evolution of the coupled spin system in response to
microwave pulses and free precession. This chapter will explain th

matrix or density operator and derive the formulas of ESEEM.
1. Density Matrix

Consider a system which consists of a time dependent or

cn(t) and a complete set of orthonormal functions, {|¢n>}.

l‘p(t)> = 20n(t)l¢n> [1
n

where I‘P(t)>, is normalized so that

):ICn(t)I2 = 1. [1
1’]

The expectation value of an observable A is given by
<A>(t) = <‘I’(t)|A|‘I’(t)> [1

28

 

2 9
XCn*(t)Cm(t)<¢nlAl¢m>
nm

ch*(t)cm(I)<fllAlm>.
nm

Now, define density operator, p, as
00) = I‘I’(t)><‘1’(t)l. [111-4

The density operator is represented in the {¢n>} basis set by a dc

matrix whose elements, pnm(t), are expressed as
phm(t) = <nlp(t)lrn> = Cm*(t)cn(t). [III-f

Therefore, from Eq. III-2 and III-5,

zlcn(t)l2 = zpnnu) = Tr{p(t)} = 1, [III-t
I1 [1

the sum of the diagonal elements of the density matrix is equal tc
Substitution of Eq. III-5 into Eq. III-3 shows that the expectation vali
be expressed by

<A>(t) men(t)<nlAlm> [1115
nm

z<m|p(t)ln><nIAIm>-
m

Since the set, {|¢n>}, is orthonormal ( |n><n| = I, I is an identity ma‘

 

3 0
<A>(t) = Z<mlp(t)A|m> [111-8]
m

= Tr{p(t)A}.

Since the evolution of I‘I’(t)> is described by the time-depem

Schrodinger equation,

ihadil‘l’(t)> = H(t)|‘P(t)>, [III-9]

where H(t) is the Hamiltonian of the system, the time evolution of

density operator p(t) is derived from Eq. III-4 as

find) = <§IW<0>><W<0I + I‘P(t)>(d%<‘1’(t)l) [III-10‘
=%H(t)l‘1’(t)><‘11(t)l + filwmxvmlum
1 l
= gH(t)p(t) - gp(t)H(t)

= figment».

In summary, the expectation value of a physical observable .

obtained by using the density operator,
<A>(t) = Tr{p(t)A}, [III-11

and the time evolution of the density operator p(t) can be derived by

equation of motion,

(1 ' ..
d—tpc) = -f‘—,[H(t>.p(t)1. [111-1.

31

p(O) is the thermal equilibrium density matrix or operator prior

application of H(t)
2. The Density Matrix Forrnalism of ESEEM
2-1. Two—Pulse ESEEM

In two-pulse ESEEM experiments, the transverse electron :
magnetization is observed along the —y—direction of a rotating fra
Hence, the normalized expectation value of the echo magnetization whic

generated at time 2r after the first pulse is expressed from Eq. III-8 by

_ Tr{ p(2r)Sy}

where p(2r) is the density matrix at time 21 and Sy is the electron :
angular momentum operator along the y-axis in the rotating frame.
Hamiltonian for ESE experiments consists of time independent and I

dependent parts,
Htot = Ho + Ht, [III-14

Where Ho and Ht describe interactions between an electron spin sy:
including coupling nuclear spins and a static external magnetic field,
interactions between the spin system and a MW pulse which is apt

perpendicular to the external field, respectively. For simplicity, ass

32

that the Hamiltonian is already expressed in the rotating frame abou
external field direction with the carrier frequency of the MW p1
During the free precession period, Htot = Ho. Assuming that
interactions between the electron spin and the MW field are much 12
than Ho, Htot = Ht when the spin system is subject to MW pulse excitati

From Eq. III-12, the density matrix p(t) is obtained by
p(t) = exp[-th/h]p(0)exp[th/f1]. [III-15

If we let the exponential operators for the precession time (pulse off)

the nutation time (pulse on) be as

Rt = exp[-iHot/f1], and [III-1t
Rip = exp[-thtip/fi], [III-1'.

then, the density matrix, p(2r), at the time of two-pulse echo is given b
p(ZT) = Rp(0)R+ [III-11
where
R = RTRszTR1p [III-1‘
Assuming that there is no mixing between a and 6 electron

states, and letting the eigenvalues and eigenvectors of the time indepei

Hamiltonian, Ho, be given by

3 3
HoWa,i = wa,i1Por,i . [III-13]
Howak = wfi,k\l)6,k .
19a = Mala,m1>, and

up = Malfi.mr>,
where
Ma 0 M+a 0
Ho= [0 M5] 0 Wei [III~14]

then Ma and M5 represent the eigenvectors, tpa and 1495, which span the
complete sets of "high-field" spin functions, lamp and H3,m1>,
respectively. The dimension of Ma and M5 are 2I+l where I is a nuclear
spin number. The eigenvectors, tpa and ms, are the linear combinations of
mods and llamas, respectively while H0 is diagonalized by the unitary
transformation of Eq. III-4. Now Rt, Rip, p(O) and Sy can be obtained in

terms of the diagonalized H0 by the unitary transformations,

M 0 0 P+ 0
Rt zi: OaMfl]eXpI‘iI_IOt/I11[M+OaM+fi] E[6 Qt+:i’ [III-15]

or 0 0

P=Mi Malexpi'flttlp/hliM 0 awfi] [III-l6]

Ma 0 +0 0

=[0 MB:lexp[ —inn/2l;M 0 M+Bi

=Moi 0 ['Icostt/Z iIsinn/Z] Mtg 0
0 [3 iIsinTI/ZIcostI/Z 0 M+B

II
:3

“M1
—iM+I

34
M o , 1w 0
I OaMBiexP["‘H“2P/mi 0aM+Bi

M 0 M+ O
[ OaMfi]exp[-isx7r][ 0a ]

. M+fi
=l-i§4+“3“l

5y

[Ma 0] M+a 0 ]

0 M5 0 MW}

Ma 0 O -iI/2 M+a 0
0 M5 iI/2 0 0 M+5

_[ 0 —iM/2]

‘ iM+/2 0 ’and

pm) =[igagfiip(otMgaM°+6]
[ I

_ a O]
— ObI
where
exp[—iwa1t] O 0
[3+ = O I 0 ,
0 O exp[—iwa(21+1)t]

exp[-iwf31t] O 0
Q+ = 0 ' 0 ,

O 0 exp[-iw5(21+1)t]

[III-17]

[III-18]

[III-19]

[III-20]

[III-21]

M=MaM5+, and I is an identity matrix. a and b represent the populati

Of a and 5 spin states in the external magnetic field.

Now the density matrix at time Zr is

DIZT) = Rp(0)R+

[III-22

 

3 5
where

Therefore, the normalized two-pulse echo amplitude is given by3

TI'{p(2T)Sy}
13(1) = Tr{p(0)Sy}

= 211+]Rerrr{QT+M+PT+MQTM+PTM}]. [111-24]

 

 

If more than one nuclear spin is coupled to the electron spin,

total echo amplitude is given by product rule.3

13m = 35m). [111-25]
1

2-2. Three-Pulse ESEEM

The three-pulse spin echo is produced at time 2r+T after the 1
pulse. Here, I and T are the time intervals between the first and secc
and the second and third MW pulses, respectively. Since all three pulses

1r/2 pulses, the density matrix at time, 21+T is expressed by
p(2T+T) = R 3p(0)R 3+ [III-26

where

3 6
Here, R/p, R2p, and R3p are identical to ij of Eq. Ill-16. The
amplitude can be described by3
Tr{p(2T+T) j}_

Tlr'{p(0}sy}
—Re[Tr{QT+M+PT+PT+MQTM+PTPTM

 

E(T,T) =

‘ 2(21+1)
+Qr+QT+M+PT+MQTQTM+PTM}]. [III-21

For the cases where several nuclei are involved, the product rule is ap

only within the same electron spin states.4

1 n n
E(T,T) = ’Z‘IHEa,i(T,T) + HEp,j(T,T)l- [III—2‘
1 J

2-3. Four-Pulse ESEEM

The four—pulse echo amplitude can be obtained using the :
formalism. Let the time intervals between first and second, second
third, and third and forth pulses be 1', t1, and t2, respectively. The

amplitude is expressed as

l
E(r,t1,t2) = 4—~(21+1)Re[Tr{ [III-3'
MQ+TQ+t2M+P+t1P+TMQTM+PthQt2M+PT
' MQ+TQ+t2M+P+th+TMQTQt1M+Pt2PT
+ MQ+TQ+QM+P+tlMQ+TM+PTPtIMQt2M+PT

3 7
+ MQ+TQ+t2M+P+t1MQ+TM+PTMQt1M+Pt2PT
+ MQ+TM+P+t2MQ+t1Q+TM+PTPtlMQt2M+PT
+ MQ+TM+P+t2MQ+tlQ+TM+PTMQt1M+Pt2Pt
- MQ+TM+P+t2MQ+t1M+P+TMQTM+Pt1MQt2M+PT
+ MQ+TM+P+t2MQ+th+P+TMQTQt1M+Pt2PT}]

or

E(T,t1,t2) = 2Re[ XMiIMlj+Mijmk+MlmMni+ [III-31]
ijklmn

X exp{-i(wijr+wikt2+winr+wimtv}

+ XMi1M1jJ"MjmMmk+MlleIniJr
i jklmn

X eXp{-i(wikT+wjkt2+w1nr+wmnt1fl}-
For four-pulse ESEEM experiments, the echo modulation function
obtained by inserting t1=t2=T/2. If an electron spin is coupled to sevr

nuclear spins and t1=t2, the echo modulation is obtained using the forrr

derived for two-pulse ESEEM.6

n
E(T,T/2) = HEi(T,T/2). [III-32]
1

1ft] and t2 are independently varied as they are in HYSCORE experime
the echo amplitude is given the by three—pulse formulation of the proc

rule6 because this experiment is basically a three-pulse experiment7

1 n n .
E(T9t19t2) = §[HEaal(1—ytl9t2) + HEB,j(Tat19t2)] [III-33.
1 J

3 8
3. Formulas of ESEEM

General formulas for two— and three—pulse ESEEM were deri

from Eq.'s III-24 and III—28 by Mims.8 They are

E(T) =§Il+—1[Xo+2);_£X(a)ijcosw°‘ijT [III-34:
1 1<j
+22 2X(l3)kncosw5kn’r
kk<n

+22 2 Z ZXW’WUMcoswaij+w5kn)r+cos(wa1j-wfikn)r}j
i i<j kk<n

and

E(T,T) = $30 [III-35]

+2: .2X(G)ij {coswaijr+coswaij(r+T)}
1 1<j
+2 zX(l5)1m{coswfiknr+cosw5kn(r+T)}
kk<n
+22 2 2 {Xmafihjkn{coswaij(r+T)cosw5knr
i i<jkk<n

+coswl3kn(r+T)coswaiJ-r}]
where 00(00ij = w(a)i - w(a)j, 03(3)“ = 03(B)k - com)“, and
X0 = zleikI4, [III-36
i k
X(0‘)ij = EIMikllejklz,
X<B>kn = ZlMiklleinlza
1

X(a’fi)ijkn = Re[M*ikMinM*jnMij-

39

3-1. Formulas for S=l/2 and [=1/2

The spin Hamiltonian for electron spin, S=l/2, is weakly coupled ‘
nuclear spin, I=l/2, consists of electronic Zeeman, electron-nuci

hyperfine (HFI) and nuclear Zeeman interactions and is given by
Ho = BS'g'Bo + S'A’I " ganBo'I. [[11'37

[3, in Eq. III-37, is the Bohr magneton, S is the electron spin ang1

momentum operator, g is the electron g-tensor, Bo is the external magn

field vector, A is the hyperfine interaction tensor, I is the nuclear s

angular momentum operator, gn is nuclear g-value, and [in is the nuc
magneton. Assuming that the electron g-tensor is isotropic, the elect

spin is quantized along the field direction. In the rotating frame witt

angular velocity of too, the Hamiltonian can be rewritten in the form of
Ho: ‘h[(ws-wo)Sz + A8212 + BSZIX - 1011;] [111-38

where cos is the electron Larmor frequency, A and B are the secular
off-diagonal components of the electron-nuclear hyperfine field, and u
the nuclear Larmor frequency. The hyperfine eigenvalues and eigenvec

of 01 and [3 electron spin states of the operator,

Ka,5fh .-.-. -wIIz j: (1/2)[Alz + 81x], [111-39

along the field can be obtained as

4 0
501.13 = Twas = f1[(iA/2-w1)2+|32/411/2
Ma = exp[-i<ply]
MB 2 €Xp[-i€Iy]

where

_ i d ”ELL
tancp — A/2—w1 an tanE — A/2+w1'

[III-4C
[III-42
[III-43

[III-44

Because the quantidization axes for a and [3 states are obtained by rote

of the [Z axis by 4p and E, about Iy, respectively, the eigenvectors ca1

expressed as the rotation operators (Eq.'s 111—42 and III—43). Then

eigenvalues and eigenvectors for the Hamiltonian, Eq. III—38, are

E(ms,m1) = f1(w-wo)ms + ’fi[(1111A/-ooi)2+32/4]1’2
w(1/2,m1) = Mall/2,m1>
1p(—l/2,m[) = MBI-1/2,ml>

where
cos? ring?
Ma = exp[-i(ply] = <2 SB and
sin‘2 c o s 2
cosg —sin%
M5 = exp[—i§ly] = E .
sini c osi

[III—45
[III-4f
[III—4'1

[III-4f

[III—4‘.

41

cosgag —sin%§
Hence, M = MaM+5 = exp[-i(<p-E)Iy] = [III-50:
sing; coscp—é

Now, we have all elements for Eq.'s III-34, III-35, and III—36. For

S=1/2, I=1/2 spin system, the following results are obtained
X0 = 2-sin2(<p-§), [III-51:

X(°‘)12 = X(5)12 = (1/2)sin7-(cp-E), and
Xm’lelZ = -(1/4)sin2(<p-€)

where

sin2(<P- E) — (57(9sz - [III-52?

Inserting Eq.'s 111-51 and III-52 into Eq's, III-34 and III-35, the two-p'
and three-pulse echo amplitude functions for an S=l/2, I=1/2 spin sys

are given by
E(1')=1-ld4{2-2cos(war)-ZCos(w5r)+cos(w-r)+cos(w+r)} [III—53
and

E(T,T)=1-k/4[{1-COS(waT)}{ l-cos(wfi(T+T))}
+{1-cos(wfir)}{1-cos(wa(r+T))}], [III—54

42
respectively. Here, w+=wa+wg, mam-cog. For the four-pulse ESEEM,

the echo amplitude is expressed by,9

k waT war
E(T,T) = 1-1[C0+Cacosfi2‘—+ w; _—T) + CBCOS(L 2T T+T)

+T+w
+ C+cos(@2 wig) + C cos(‘2£+9*‘)] [111-55]

where

Co = 3-cos(war)-cos(w5T)-52cos(w+r)—c2cos(w-r),

 

co r w r w r
C“ = 2[C2C°S(wfiT'T:)+szcos(wpr+*:—)-cos(w7a)],

 

C5 = 2[CZCOS(waT-w2 Mfiszcosmafi my cos(— 26 T)],
0.)

Q, = -4c2sin(—20L)sin(7),
u) r w r

C- = -4szsin(‘g—)sin(‘zfi_),

1
|w12-;(wa+wp)zl

 

$2 = ———— ,
waCOfi
lwIZ-iwa-wpfll
02 = —, and
(Dawfi
k = 432c2. [III-56]

3—2. Fortnulas for S=l/2 and [21

For a spin system of 121 nuclei, the spin Hamiltonian includes the

nuclear quadrupole interaction (NQI). Hence, the total Hamiltonian is given
by

 

 

 

43

Ho = BS-g-Bo + S-A'l — ganBo-l + I-Q-I [III-57

where Q is nuclear quadrupole interaction tensor. Approximate analyt
expressions of the echo modulation function where electron-nuc‘:
hyperfine and nuclear quadrupole interactions can be treated us
pertubation theory have been derived.4s10'13 Completely gent
expressions for this case must be obtained numerically.
In the case of an isotropic electron g-tensor, the eigenvalues

eigenvectors of the Hamiltonian without the electron Zeeman interact
can be used for calculation of the echo modulation. Hence, we need c

consider the nuclear spin Hamiltonian,
HN = S~A~I - ganBoJ + I'Q-I. [III—58

If only HFI and nuclear Zeeman interactions are involved, the Hamiltor

is

HHFI = S'A'I ' ganBo'I. [III-59
For spin systems where the electron g-tensor is isotropic the electron s
Operator can be replaced by a ficticious spin operator, where S = 8';

1y, 12), and

“HFI/h = (S'zAxx‘wIanx + (S'szy'wIflny [III-60]
+ (S'ZAZZ'wIflzIz-

 

 

44

lx=sinficos¢, ly=sinesin¢, and lz=cosfi are direction cosines that desc1
the orientation of Bo with respect to the HFI principal axis system (I
PAS) [Fig III-1]. Axx, Ayy, and All are the principle values of
hyperfine tensor. The matrix elements, <ms,m1|HHF1|ms,m1>, of
Hamiltonian, Eq. III-60, for I=l, 3/2, 5/2, and 7/2 spin systems are liste<
Appendix, A1. The matrices of Al-l, 2, 3, and 4 are readily diagonali
and the

-
~“
‘_

    

 

 

_1_--—--

Figure III-l. Relative orientation of the external magnetic field with re;
to the principle hyperfine axes.

 

diagonalized matrix elements are mn/X2+Y2+Zz. Here, X=(hmsA
gnfinBo)sinficos¢, Y=(hmsAyy-gnBnBo)sinf)sin¢, and Z=(hmsA
ganBo)cosfi. The eigenvectors of the Hamiltonian can be taken as

rotational operator matrices whose elements are given by14

 

[III-61]

 

45

where i=1, m and n = m1's, t=-m1 -> m, and v is the rotation angle. Fort

 

M matrices of our calculation, y=<p~€ where tancpzza/‘JXZwYZa a
tan€=ZB/\/X25+Y25 . The M matrices obtained from Eq. III-61 are list

 

in Appendix, A2. The echo modulation can be obtained by inserting t
eigenvalues of mr\/X2+Y2+Z2 and the M matrices of Table A2-1, 2, 3, a
4 into the Eq.'s, III-34 and III-35 for two-pulse and three-pulse ESEE
and the Eq. III-31 for four-pulse ESEEM.

 

If NQI are involved in the spin Hamiltonian, the above eigenvah

and the M matrices are no longer valid. The NQI Hamiltonian is given by

HNQI = For , [III-62]

‘he2
= fling-Wm(I'XZ-I'yzn

where equ and n are the quadrupole coupling constant and asymme
parameter, respectively. The prime notation means that the spin operate
refer to the NQI principal axis system (NQI PAS). If the HFI PAS is 1
standard coordinate system for our calculation, the NQI PAS must
rotated into the HFI PAS by an Euler rotation. The NQI Hamiltoni
matrices for 1:1, 3/2, 5/2, and 7/2 in the HFI PAS are listed in Append
A3. The eigenvalues and eigenvectors of the total Hamiltonian, HHF]
HNQI, can be numerically obtained from the Tables in Appendix, A1 a
A3. From the results, the echo modulation can be calculated.

When the molecules are randomly distributed with respect to t
external magnetic field, the modulation functions must be orientational
averaged over all possible orientations of the spin system with respect

the lab axis system.

 

46

<E>e = 2111; Jgrr J' 6E(6)sin8d8d¢. [III-63

Appendix, A4 shows the computer programs for two-pulse

three-pulse echo modulation functions in the time domain for the case 0

isotropic electron g-tensor. The programs are written in the C comp

language (Symantec, v.4.0). The diagonalization subroutines are ts

from the NuTools numerical package (Metaphor, v.1.02).

References

\000

10.
11.

12.

13.
14.

C. Cohen-Tannoudji, B. Diu, and F. Laloé, Quantum Mechanics,
John Wiley & Sons, New York, 295 (1977).

C. P. Slichter, Principles of Magnetic Resonance, 3rd Ed.,
Springer-Verlag, Berlin, 157 (1989).

W. B. Mims, Phys. Rev., BS, 2409 (1972).

S. A. Dikanov, A. A. Shubin, and V. N. Parmon, J. Magn. Reson
42, 474 (1981).

E. J. Reijerse, and S. A. Dikanov, Pure & Appl. Chem, 64, 789
(1992).

A. M. Tyryshkin, S. A. Dikanov, and D. Goldfarb, J. Magn. Resc
A105, 271 (1993). .

P. deer, A. Grupp, H. Nebenfiihr, and M. Mehring, Chem. Phys
Lett., 132, 279 (1986).

W. B. Mims, Phys. Rev., B6, 3543 (1972).

C. Gemperle, G. Abeli, A. Schweiger, and R. R. Ernst, J. Magn.
Reson., 88, 241 (1990).

A. A. Shubin, and S. A. Dikanov, J. Magn. Reson., 52, 1 (1983).
M. Heming, M. Narayana, and L. Kevan, J. Chem. Phys, 83, 14’
(1985). ‘

M. Romanelli, M. Narayana, and L. Kevan, J. Chem. Phys, 83,
4395 (1985).

K. Matar, and D. Goldfarb, J. Chem. Phys, 96, 6464 (1992).

D. M. Brink, and G. R. Satchler, Angular Momemtum, Clarendo
Press, Oxford University, London, 22 (1968).

IV. INSTRUMENTATION

The pulsed~EPR spectrometer used for the research in this thesis was
built by Professor I. McCracken at Michigan State University in 1990. The
details of its design have been described elsewhere.1 As part of this thesis
work, several modifications have been made to the pulse programming
portion of the spectrometer to provide a more open structure for the
incorporation of new pulsed—EPR methods. Currently, the instrument can
perform 2-, 3-, and 4—pulse ESEEM?‘4 HYSCORE (hyperfine sublevel
correlation spectroscopy),5 Mims pulsed-ENDOR,6 Davis pulsed-
ENDOR,7 ESE-EPR (electron spin echo detected EPR),8 "2+1" ESE,9 and
DEER (double electron electron resonance) ESE.10 A schematic diagram
of the spectrometer is shown in Fig. lV-l.

The spectrometer was constructed with wideband frequency
components and covers a frequency range from 6 to 18 GHz. A microwave
Synthesizer (Gigatronics, Model 610) serves as the microwave source. The
synthesizer output is divided by a directional coupler (Omni Spectra,
Model PN2025-6018-10) so that 90 % of the MW power goes to a
reference arm and 10 % is directed to two microwave pulse-forming
arms. The reference arm has an adjustable phase shifter (ARRA, Model
9828A) and makes detection of the magnetization in the rotating frame
possible by phase sensitive detection. A power divider (Omni Spectra,

Model PN2089-6209-OO) splits the remaining source power into two

47

48

independent pulse channels through an isolator (Innowave, Model 1119IR).

Each pulse channel consists of a low-power, high-speed PIN diode switch

Reference Arm (1) £1.22 x/><>

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

    
 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

4043 (17181? Bo
mos— /- ——-> E]
*— 1 an:
Il/ - J 23dB
f I __
: A : Filter
I I ,
m I l -20dB “10108
I | Monitor
64861-12 : 1AM,
Synthesner | l
: : 23dB
11 2 2 11 \ *1—
¢mod pm
Control Control Limitefi

 

 

Pulse Logic

 

 

 

Module 3‘9““

- n ”V 30...;
111111 mm... U I +

' i ' Boxcar Switch

TOABCDE >

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Dela Spare II Field Gated
Geneyrators Work- «mm M30 11X Controller Integrator

- Station

t IEEE-488 Bus t j
Figure IV-1. Schematic diagram of pulsed-EPR spectrometer built in
Michigan State University

 

 

 

 

 

 

 

 

 

 

 

 

 

 

(General Microwave, Model FM864-BH), a high-speed 00/1800 wideband
digital phase modulator (General Microwave, Model F1938) and an
isolator. The pulse widths and phases of each channel are controlled by a

home-built pulse logic circuit whose circuit drawings and timing diagrams

 

49

are in Appendix, A5. One of the pulse channels has an attenuator (ARRi
Model P9804—20) and the other has an adjustable phase shifter. By th
adaptation of the two MW pulse channels, the phases and the amplitude 1
the MW pulses are controlled independently to do phase-cycling ar
"pulse-swapping".ll The MW pulses of each channel are combined at
power divider and fed into a medium power GaAs FET amplifit
(Avantek, Model SWL—89-O437) which amplifies the low power M‘
pulses to the required minimum input power of the travelling wave tut
amplifier. The MW pulse power is fed into a 1kW pulsed travelling wa1
tube (TWT) amplifier (Applied Systems Engineering, Model 117) via 2
attenuator. The high power MW pulses from the TWT are passed throug
a rotary vane attenuator (Hewlett Packard, Model X382A) and fed into
reflection type resonator through a circulator (MACOM, Model Ml
8K269). The cavity used for the experiments described in this thes
employs a folded half-wave resonator12 and an X-band Gordon coupler.
Signals from the resonator are fed to a low-noise GaAs FET amplifit
(Avantek, Model AWT-l8635). This amplifier is protected from the higi
power excitation pulses by a medium-power, high-speed PIN diode limit
(Innowave, Model VPL-6018) and a fast PIN diode switch (Gener
Microwave, Model F9114) which is controlled by the pulse—logic circuit.
% of the amplified signal is fed into an oscilloscope (Hewlett Packar
Model 54520A) for spectrometer tuning via a directional coupler. TI
signal is fed into a double balanced mixer (RHG, Model DMZ-18 A]
through a bandpass filter (K & L Microwave, Model 3HlO—2000/1800
0/0) which serves to remove radio frequency noise introduced by the Pl
swithches. The mixing scheme is a conventional homodyne arrangemei

At the mixer, the signals from the reference arm and the receiver a

50

combined to generate signals at the sum and difference of the two inpt
frequencies. The amplitude of the signal is proportional to the two inp1
amplitudes and their relative phases. In our instrument, the amplitude 1
the difference frequency is chosen to be amplified and adjusted by sever
video amplifiers (Comlinear, Model E220) and a variable attenuator (IFV
Model 50DR-003). The amplified signal is fed into a gated integrat<
(Stanford Research, Model 250) which is triggered by a signal from
delay generator (Stanford Research, Model DG535).

The spectrometer is controlled by a Macintosh lIx microcomputt
(Apple Computer) via an IEEE-488 interface (National Instruments) to tl
delay generator, the gated integrator and a computer interface modu
(Stanford Research, Model SR245). The delay generator is used to genera
the accurate timing signals which are transferred to the pulse logic circu
to control microwave pulse generation and spacing, trigger the gate
integrator, and control the overall repetition rate of the experiment. Tl
computer interface module serves to digitize the output of the gate
integrator, to count the number of pulse sequences repeated at a given MI
pulse spacing, and to clear the gated integrator after its analog output h:
been read. Data collection and analysis software are written using the
Computer language (Symantec). In Appendix, A6, computer programs f1
performing four-pulse ESEEM and HYSCORE are listed. The Macintos
microcomputer is connected via ethemet to a SPARCstation 2 workstatic
(Sun Microsystems) where more sophisticated analysis programs a1
Written in the MATLAB computer language (The Mathworks, Inc).

ESE-ENDOR experiments can be performed using an RF generatt
(Hewlett Packard, Model 8656B). The low power RF signal is pulsed by
RF GaAs switch (Mini Circuits, Model YSW—50DR) which can I:

51

controlled by another home-built pulse logic circuit. The RF pulse pow:
amplified by a RF amplifier (ENI, Model 3200L) and fed into
resonator via a variable attenuator (J FW, Model 50DR-003).

The magnetic field strength is provided by an electro-mag

(Walker Scientific, Model HF-12H) and controlled over a range from 5
to 15000 G by a Hall-effect field controller (Bruker, Model B-H15). ‘

field controller is interfaced to the spectrometer computer via the [E

488 interface to perform ESE-EPR experiments.

References

l.

2.
3.
4

owe .U'

>9

13.

J. McCracken, D. -H. Shin, and J. L. Dye, Appl. Magn. Reson., 3
305 (1992).

W. B. Mims, Phys. Rev. , B5, 2409 (1972).

W. B. Mims, Phys. Rev. , B6, 3543 (1972).

C. Gemperle, G. Abeli, A. Schweiger, and R. R. Ernst, J. Magn.

Reson., 88, 241 (1990).

P. Hofer, A. Grupp, H. Nebenfiihr, and M. Mehring, Chem. Phys
Lett., 132, 279 (1986).

W. B. Mims, Proc. Roy. Soc., A283, 452 (1965).

E. R. Davis, Phys. Lett., A47, 1 (1974).

R. T. Weber, J. A. J. M. Disselhorst, L. J. Prevo, J. Schmidt, and
Th. Wenckebach, J. Magn. Reson., 81, 129 (1989).

V. V. Kurshev, A. M. Raitsimring, and Yu. D. Tsvetkov, J. Magi
Reson., 81, 441 (1989).

A. D. Milov, K. M. Salikhov, and M. D. Shchirov, Sov. Phys. S01
State, 23, 565 (1981)

J. -M. Fauth, A. Schweiger, L. Braunschweiler, J. Forrer, and R.
Ernst, J. Magn. Reson., 66, 74 (1986).

C. P. Lin, M. K. Bowman, Norris, J. R., J. Magn. Reson., 65, 36
(1985).

R. D. Britt, M. K. Klein, J. Magn. Reson., 74, 535 (1987).

V. FOUR-PULSE ELECTRON SPIN ECHO ENVELOPE
MODULATION STUDIES OF AXIAL WATER LIGATION TO
BIS-AQUO TETRACYANONICKELATE(III)

1. Abstract

Four-pulse electron spin echo envelope modulation (ESEEM) studies
aimed at enhancing our previous characterization of the hyperfine
interactions between protons of axially bound water molecules and nickel
ion in Ni(III)(CN)4(H20)2' were carried out. Because the ligand hyperfine
Coupling of the strongly bound water protons to Ni(III) is characterized by
a large anisotropic interaction, an ESEEM feature at the sum combination
band, va+v5, that shows pronounced shifts from twice the proton Larmor
frequency is observed. In contrast to our previous study where a two-pulse
(n/Z-r-rr) microwave pulse sequence was used, four-pulse data show deep
modulation with reduced damping for the sum combination feature that
results in a 10-fold increase in spectral resolution. The corresponding
ESEEM spectra provide lineshape and frequency constraints that allow for
a more accurate and complete characterization of the bound water proton
hyperfine coupling tensor. Theoretical simulation of the magnetic field
profile dependence of the va+v5 lineshapes and frequency shifts from the
twice the Larmor frequency gave an effective Ni-H dipole-dipole distance

of 2.33i0.03 A and a 0n, the orientation for the principal axis system of

52

 

53

the 1H hyperfine coupling tensor with respect to the g3 axis of the Ni(
g-tensor, of 18130. The T-suppression behavior of the va+v5 lineshape :
fixed magnetic field position was used to place more exact constraints
the isotropic hyperfine coupling constant than possible with a simple t1
pulse approach. An isotropic hyperfine coupling constant of 2.52105 M

was found for the bound axial water protons.
2. Introduction

The electron spin echo envelope modulation (ESEEM) technique
pulsed EPR spectroscopy is a useful tool for characterizing weak hyperf
interactions (HFI) that are not resolved in conventional continuous w:
(CW) EPR due to inhomogeneous line broadening}3 When the hyperf
interaction is largely anisotropic, the fundamental hyperfine frequenc
(Va and v5) are broadened and marked by lineshape singularities t
provide for the accurate determination of hyperfine couplings in
orientationally disordered system. Unfortunately in ESEEM spectrosco
the anisotropy in the modulation amplitudes obscures these singularities :

Often prevents or hampers the determination of hyperfine ten
Components.4 In two-pulse ESEEM spectra, fundamental, sum (vow-v5) a

difference (Va-v5) combination frequencies are observed for coup

nuclei.5a6 Of these spectral features, the sum combination peak has recei1
much attention in recent studies because of the sensitivity of its frequer
Position to the anisotropic portion of the hyperfine interaction.7‘9 T
advantages of using the proton sum combination peak in ESEEM spec
for characterizing large anisotropic hyperfine couplings are that

spectral lines are narrow and occur at frequencies where there is

54

relatively clean spectral window,10 and for spin systems where g- o
hyperfine anisotropy dominate the EPR absorption spectrum, the variatior
in the sum combination frequency shift from twice the nuclear Larmo
frequency as a function of magnetic field strength allows accuratt
determination of the orientation of the ligand hyperfine tensor principa
axes.‘11'13 However, the spectral resolution of a two-pulse ESEEIV
experiment is limited by relatively short electron spin phase memory time:
(TM). To overcome these problems, a four-pulse ESEEM experiment (7I/2
r-rr/2—T/2-7r—T/2—1r/2) has been suggested.14'16 In these experiments, a it
pulse is applied in the middle of the free precession period between tht
second and third pulses of a stimulated echo ESEEM sequence. The timt
domain ESEEM data are collected as a function of T, the time interva
between the second and fourth pulses of the sequence. The sun
combination band can also be detected using this sequence, but with 1
spectral resolution that is dependent on T1.

The lineshape properties of the sum combination band have beer
derived for systems with an isotropic electron g-tensor and an axia
electron nuclear hyperfine tensor.9 When the hyperfine coupling is small
SO that |T+2a| << 2v1 (where the dipolar part of the hyperfine tensor i
represented by principal values (-T, -T, 2T), "a" is the isotropic coupling
aIlcl V} is the nuclear Larmor frequency), the position of the lineshapt
SiIlgularity is expressed by va+v5 = 2v1+(9/16)T2/v1.9,17 For systems witfi
aXial g-tensors, analytical expressions for the sum combination peal
Positions at magnetic field positions corresponding to g“ and g J__ extrema c
an EPR spectrum have also been developed”,18

Recently, two-pulse ESEEM studies have been used to characteriz

the proton hyperfine couplings of the axially bound water ligand 0

5 5

Ni(III)(CN)4(H20)2'.13 Because the proton hyperfine interaction for the
strongly-bound water ligands was mostly anisotropic, the sum combinatit
band provided a useful tool for the study. Field profiles constructed by t]
plotting the frequency shifts of the sum combination peaks from twice tl
Larmor frequency as a function of magnetic field strength across the EF
absorption spectrum were used as a foundation for a more comple
analysis of the anisotropic portion of hyperfine tensor and the orientatit
of the hyperfine tensor principal axes with respect to the g-tensor. Fro
the analysis of the field profile, an effective Ni-H dipole-dipole distance
2.4 i 0.1 A, a scalar or isotropic hyperfine coupling of S 4 MHz, and
relative orientation of principal hyperfine axis with respect to g3 axis (0
of 12 i 30 were determined.13 The chief shortcomings of this field profi
analysis are that the frequency shifts are only marginally sensitive to tl
isotropic portion of the hyperfine coupling,9s11112 and the reduction of t]
sum combination peak to a single frequency shift value precludes recove
of the information contained in the lineshape.

Recently, four-pulse ESEEM studies of 15N, 1H, and 2H have bet
reported. In these experiments, more accurate hyperfine and/or nucle
quadrupole interaction (NQI) coupling parameters could be determint
uSing an analysis based on sum combination peaks and their improv:
SPectral resolution.18,19 In this paper, our previous characterization of t]
1H~ligand hyperfine coupling for the axially bound water ligands 1
Ni(IIl)(CN)4(H20)2' is refined using the four-pulse ESEEM technique ar
an enhanced sum combination peak analysis. Our analysis extends the fie
Profile strategy introduced in the previous study to include the behavior 1
the sum combination peak lineshape singularities for a randomly orient:

sample. This method is developed for a spin system described by axial ;

56

and ligand hyperfine tensors, but can be easily extended to conside
rhombic interactions. The selection of r, the time interval between the fir:
and second pulses, can be used to tailor the amplitudes of the ESEEII
components in an analogous way to that frequently used in convention;
three-pulse ESEEM studies. For the sum combination band, the suppressio
effect is a function of the fundamental modulation frequencies which are
in turn, sensitive to the isotropic hyperfine coupling.1"~16a18 The isotropi
hyperfine coupling for the axially bound water protons was determine
using the r suppression behavior of the sum combination peak lineshape
The results reveal a more refined hyperfine coupling tensor for thes

protons.
3. Experimental

Bis—aquo tetracyanonickelate(III), Ni(III)(CN)4(HzO)2‘, wa
prepared according to the literature procedureszos21 Samples for CW-EPI
and ESEEM experiments were mixed with an equal volume of ethylen
glycol prior to freezing in liquid nitrogen.

CW-EPR spectra were obtained on a Varian E-4 X-band EP]
SPectrometer at 77K. ESEEM experiments were executed on a home~bui
PUIsed-EPR spectrometer which has been described in detail elsewhere.2
An EIP model 25B frequency counter and a Micro-Now model 515B NMl
gaussmeter were used to calibrate the microwave frequency and magneti
f161d strength of the pulsed-EPR spectrometer. The ESEEM data wer
COIIected at X—band using a reflection cavity that employed a folde
Stripline resonant element and a Gordon coupling arrangement suitable fc

Studies in a cryogenic immersion dewar.23,24

 

5 7

A four—pulse (rt/2-T—7r/2-T/2-rr—T/2—rr/2) microwave pulse sequence
was used to collect the ESEEM data. A second microwave pulse channel
was employed for the n-pulse so that its width and amplitude could be
controlled independently. The length (full-width at half maximum) of all
four microwave pulses was 16 ns with the peak power of the n-pulse being
the twice that of the rr/2—pulses.18 A four step phase cycle, +(0,0,0,0),
+(1r,1r,0,0), +(0,0,1r,0), +(7r,7r,rr,0), was used to eliminate unwanted echo
modulations. Four-pulse ESEEM data were collected as a function of T, the
time interval between second and fourth pulses. Fourier transformation of
the time domain ESEEM data were done without dead-time reconstruction.
The fourier transformation and computer simulations were performed on
a SPARCstation 2 workstation (Sun Microsystems) using software written

in MATLAB v 4.2 (The Mathworks, Inc.).
4. Theory
The intensity of the sum combination peak derived from two-pulse

ESEEM experiments on an orientationally disordered, S=1/2, I=l/2 spin

System described by isotropic g- and axial hyperfine tensors is given by9

 

1 . 60
IVQ+V5(6) = §k(0)srn0|a(va+vfi)l [V-l]
where
VIB
- 2
k _ (W6) .

Voc<6>= [(v211a1m- vziatnflcoszfi + vziarfiflm,

58

a+2T
V11a<151= -VI i ‘2 ,

a-T
VJ_0t(B) = "VI 112’,

B = 3Tsin0c050,
T = gegnBeBn/hr3, and
vI = ganBo/h. [V-2]

The terms in the above equations are: v1, the Larmor frequency of th
coupled nucleus; vow), the fundamental hyperfine frequencies in or and
electron spin manifolds; a, the isotropic hyperfine coupling; 0, the angl
between the external magnetic field and the principal hyperfine axis; ge, th
electron g-value; gn, the nuclear g-value; Be, the Bohr magneton; fin, th
nuclear magneton; r, the electron-nuclear distance; h, Planck's constan
and Bo, the external magnetic field strength. The lineshape features at
governed by turning points of the sum combination frequency determine

. . . . 6(V01+V[3) . .
by calculating the conditions for whrch T=O' For tumrng porn‘

that occur at 0 = 0 and n/2, the modulation depth parameter, k, is equal t
zero, so that no intensity singularities are observed. A third turning poir
condition, for which an intensity singularity does exist, was derived 1:

Reijerse and Dikanov,9 and occurs when

1 %T(T+2a)
2 __ —__ -
COS 6 — 2 + 16V12 _ (T+2a)2 , [V 3]
for 0 S cos20 S 1 and IT + 2a| < 4v1. These authors also derived t1

expression for the sum frequency of this turning point, va+vg = 2v1

5 9
(9/16)T2/v1, allowing for direct determination of the dipolar hyperf
coupling for isotropic electron spin systems.
For systems with an anisotropic g-tensor, the fundamental ESEE

or ENDOR frequencies of an S=l/2 and I=1/2 spin system are giv
by25,26,27

V01(B) = [(msA1-vrl1)2+(msAz-11112)2+(ms.A3-vrls)2]1/2 [V-4]

where ms=1/2 or -1/2 for the or and 6 spin manifolds, respectively. "1

other terms in Eq. V-4 are

11 = sin0cosd),

12 = sin0sind),

13 = cos0,

A1 = [g111(g1D(3n12-1)+a) + 3g2212Dn1n2 + 3g37-13Dn1n31/ge,
A2 = I3g1211Dn1n2 + g212(g2D(3n22-1)+a) + 3g3213Dn2n31/ge,
A3 = [3g1211Dn1n3 + 3g2212Dn2n3 + g313(g3D(3n32-1)+a)]/ge.
n1 = sin0ncos¢n,

n2 = sin0nsin¢n,

n3 = cos0n, and

D = -gnf5ef3n/hr3- [V-5I

11,12,and 13 are direction cosines that define the orientation of the exter
magnetic field with respect to the g-tensor axes. n1, n2, and n3 are 1
direction cosines of the principal hyperfine tensor axis with respect to ‘
g-tensor. g1, g2, and g3 are the principal values of the electron g-tensor.

Is the effective electron g—value for a discrete measurement with

 

6 0
ge = [(grlr)2 + (gzlz)2 + (g3l3)2]1/2.

In orientation selective ESEEM experiments with axial g-
hyperfine tensors, the angle between the external magnetic field and th
axis, 0, is fixed by the external magnetic field strength and the micror
pulse frequency. The hyperfine frequencies are also independent of d)"
azimuthal angle used to describe the orientation of the hyperfine print
axis system with respect to the g-tensor. Because 0n, the angle betweet
principal hyperfine axis and the g3 axis, is constant for a given 1i;
geometry, the lineshape functions of the hyperfine frequencies will
function of d), the azimuthal angle that describes the external magnetic
direction with respect to the g-tensor axes.

For an S=1/2, I=1/2 spin system, the echo modulation function
results from a four-pulse ESEEM experiment was derived by Gemp

et. al.,14,16,18 and is given by

 

 

k w T w r w T (.1) r
Emod(T,T)=1‘Z[Co+CaCOS( ; + 5‘ )+ Cpcost 25 + 25 )
w+T w+r WaxT no-1"

 

 

+C+cos( 2 + 2—)+C cos( 2 +2 )] [V— 6]

where

Co = 3-cos(war)-cos(w5r)-szcos(w+r)-c2cos(w—r),

 

w r w r 1'
Ca: 2[CZCOS(wfiT- 2a )+szcos(wfir+—g—)-cos(*—a*)],

C5 = 2[czcos(war-92E)+szcos(war+9’@“)- cos(-9:61)],

C+- = -4c23in(—2—)sin(w—2—),

 

6 1
. w r . w r
C- = -4s251n(7a‘)s1n(“26*),
1
lwrz-gtwawefll
S = _-—<oa—wfi_ _
1
2 1w12-;(wa-wa>21
C = ‘W' and
k = 4s2c2- [V-71

0301(6) are the angular frequencies of vow) which are given in Eq. V-4 a
(.0: = um i tog. c2 and s2 are the EPR transition probabilities. Hence, t
lineshape function of the four-pulse sum combination band is deterrnin
by the depth parameter, k(d>), the r-dependent coefficient, C+ in Eq. V-
and the line integral weighting factor, [(60/(9cl>)2+sin20]1/2dcl).28 F
systems described by axial g and ligand hyperfine tensors, 0 is constant 1
a given measurement and the lineshape function of the sum combinati

band detected in four-pulse ESEEM experiments can be expressed as

IVa+Vfi(¢) = k(¢)c25in(%)sin(%)sinel5(—i9—i

+v6)i‘ [V's]

The frequency distribution range of the sum combination band
determined by the turning points of the sum frequencies where t

. . 0(Va+V6) . . .
cond1tron, T10, 1s satisfied. For the case where both g- and lrga

hyperfine tensors are axial, (1)“ can be set to be zero and the derivative,

av01(6) _ 3msD12n1

ms
_ 2 _ — -
04> gevtxw) [g1 n11112V1 geg1(g1D+Za)} [V 9]

— W-’ —_

62

m
+ n313{(g12+g32)vr-gistg3(g1g3[>+(g1+g3)a)}]

is obtained. When 12 = sin0sind) = 0 and sin0 i 0, there are the tur
points for the sum frequency when d) = 0 and 7r. The other turning pt
can not be derived analytically, but can be obtained numerically for a g

. . 0(Va+vn)
0 by calculation of d) values for whrch T = 0. At these tur

points, the intensities of the sum frequencies show singularities exce
va+vg=2v1 where branching of the EPR transitions ceases (s7- and 1
zero in Eq. V-7 and V-8).

These turning points can also be estimated using Eq. V-2 ir
strong field limit. For axial g- and ligand hyperfine tensors,

fundamental frequencies are given by
Vatfi) = [(vzuaw) - VziarmkoszK + Vzmtwim [V-IO‘

where K defines the angle between the external magnetic field directior

the principal axis of the hyperfine tensor with
cosrc = sin0sin0ncosd) + cosBcoan [V—l 1:

and ¢n=0. The turning points of the sum combination peak obtained

0 0
Eq. V-10 occur when d>=0, 71', cosd) = — W (for sin0sin0n it 0 a1

%T(T+2a)
S cosd) S 1), and when COSZK = 2 + TWT-W (for COS(B+E

COSK S cos(B—Gn) and |T+2a| < 4v1 ). The latter condition is identical tc

63

of Eq. V-3, but the limits have changed because of the relative orientatio

between the magnetic field and the hyperfine principal axis system. Th
cosficosfin
sin05in0n

 

turning point for cosd> = — is also obtained when g1=g3 in Ec

V-9. At this turning point, an intensity singularity is not observed becaus
the sum frequency is equal to twice the Larmor frequency. Tyryshkin e
a112,18 have derived analytical expressions for the sum frequencies of th
singularity points at the g“ and g1 extremes of the EPR absorptio
spectrum utilizing Eq. V-10. These expressions are useful fc
determination of the dipolar hyperfine coupling and the relative orientatio
of the hyperfine principal axis with respect to the g-tensor, but are les
sensitive to the isotropic hyperfine couplings.

Fig. V-1 shows plots of the simulated sum frequency shi:
[Av=(va+v[3)-2v1] profiles of the turning or lineshape singularity points a
a function of magnetic field strength across the EPR absorption line as
function of 0,1 for a spin system described by axial g— and ligand hyperfin
tensors. The sum frequencies of the turning points were calculated usin
Eq. V-4 and V-5, principal g values of 2.198, 2.198, and 2.007, and
microwave frequency of 8.789 GHz. The solid line represents th
singularity point for which ¢=0 and the dashed line shows the field profil
of the ¢=7r singularity. The curves plotted using "+++" or "***" pattern
represent singularities that were determined numerically. The dipole—dipol
distance and the isotropic hyperfine coupling were fixed at 2.4 A and
MHz, respectively.

The field profile patterns of Fig. V—l show a strong dependence 0
the angle between the principal axis of the hyperfine coupling tensor an

the g3 axis, 0n. For 0n=0 (Fig. V-1 (a)), the hyperfine tensor principal axi

 

64

Figure V-l. Simulated field profiles of the frequency shifts, (va+v5)-2v1,
for the turning points of the proton sum combination bands across the EPR
absorption spectrum as a function of 0,1. Parameters common to all
simulations were g1, 2.198; g”, 2.007; microwave frequency, 8.789 GHz;
effective dipole-dipole distance, 2.4 A; and isotropic hyperfine coupling, 0
MHz. For (a) 0n=00; (b) 0n=300; (c) 0n=450; (d) 0n=600; and (e) 0n=900.
The solid and dashed curves represent singularities calculated for d>=0 and
II, respectively, while "++++" and "****" patterns are for numerically
calculated features. The frequency shifts were calculated at 46 magnetic
field positions across the EPR absorption spectrum. The arrows in (b)

indicate the field positions where the lineshapes are computed in Fig. V—2.

Frequency Shift (MHz)

 

 

    

\
\ /
IIIIIIIIIIID - ’

i 1 1 L

 

 

 

 

'7 v r v v

\O
\‘*‘4
\ “+“* C

\

*§§‘
oxg.
*4-
t¢+oxgg..
e.,‘
o...
0.2,

0“
I

  

 

 

2850 2900 2950 3000 3%0 3100

Field Strength (0)

3150

 

 

Frequency Shift (MHz)

66

 

0.8 '-

0.6 ~

0.2 -

 

 

 

1.81"
1.6-

1.4"

0.81-
0.61

0.4 1-

 

l

 

l

 

 

2850

2900

2950 3000

Field Strength (G)

3050

3100

 

 

 

6 7

is along g3, so that the angle, K, is equal to 0 which varies from 900 to OC
across EPR absorption envelope. Hence. the d) dependence of the sum
combination band (Eq. V-4 and V-5) is removed and the lineshape maxima
occur at a single frequency for each magnetic field value. At g” and gj
extremes in Fig. V-l (a), the angles, K, are 00 or 90°, and the sum
frequency shifts are zero. The resulting field profile is identical to that
obtained using two—pulse ESEEM methods.13

At the high field extreme for each orientation shown in Fig. V—l, the
magnetic field direction is aligned with g3 axis. For this orientation, 1c is
determined by 0“ and the d) dependence of sum combination band is
removed. Hence, the sum frequencies have only one singularity, or turning
point, and show shifts from 2v1 that vary from 0 MHz when 0n=00 or 90C
to a maximum when 0n=450. At the g1 extreme, 0:900 and the angle, K.
will range from 0—0n to 0+0“ with the sum frequency lineshape showing
features determined by On. The sum combination peak lineshape will cover
frequencies from 2v1 to singularity points described by d>=0, it or

d v +v
numerical solution of Lgérj’)“: .

As 0,, is increased in Fig. V-1 (b)-(e), K ranges from 0-0n to 0+0r
and the sum frequency peak can show more than one singularity or turning
point at a single magnetic field position. The field profiles marked witt

asterisks were numerically determined, and correspond to turning points
cos0cos0n

sin0sin0 , as derivec
n

that can be estimated from the relationship cosd) = -

from Eq. V—10. For this feature the sum frequency shift is zero. These
turning points become intensity singularity points with non-zero frequency
shifts from 2v1 when isotropic ligand hyperfine couplings are large anc

Eq. V-10 is no longer valid. With an electron—nuclear distance of 2.4 A,

68

Figure V-2. Lineshapes of proton sum combination bands at field positions

across the EPR absorption spectrum indicated by the arrows in Fig. V-l
(b). The magnetic field strengths represented are (a) 2857.0 G (g .L), (b)

2870.0 G, (c) 2910.0 G, ((1) 2975.0 G, (e) 3120.0 G, and (0 3128.8 G (gu).
An intrinsic linewidth (FWHM) of 0.03 MHz was used in the calculation to

reveal all of the discrete lineshape features.

69

 

 

 

 

 

 

- 1
_ b .
" 4
’ 1
- .
. 4
.
r n A 1
v 1 v ,
h .
. 4
. q
. .
a
.
.
-

 

 

 

-0.5

Frequency Shift (MHz)

 

70

 

 

 

 

 

 

 

 

 

 

 

 

 

——.
- 1
P <
-0.5 0 0.5 1 1.5 2.5

Frequency Shift (MHz)

 

71

these turning points show intensity singularities when a Z 18 MHz. The

turning or singularity points marked with plus signs in Fig. V-l can be
2T(T+2a)

estimated using the expression, cos21< = 2 + 16Wfi+22172 from Eq. V-

10. One interesting feature is shown in Fig.V—l (d) (0n=600) where a

foldover of this turning point is observed at low magnetic field.

The predicted sum combination peak lineshapes that correspond to
the various field values indicated by arrows on Fig. V-1 (b) (0n=30) are
shown in Fig. V—2. The frequency shift, (va+v5)-2v1, is displayed on the
x-axis. At 2857.0 G (g I), there are two turning points (Fig. V-l (b)). One
is an intensity singularity corresponding to d)=0 and 7t, and the other is
numerically determined and occurs at zero frequency shift. Hence, the sum
frequencies are spread from twice the Larmor frequency where the
intensity is zero to the singularity point (Fig. V-2 (a)). At 2870G, the
predicted sum frequency peak lineshape singularities occur at frequency
shifts of 0.6 MHz and 1.7 MHz (Fig. V-2 (b)). The lineshapes shown in
Fig. V-2 (c)-(f) mirror the changing pattern of singularity points across the
EPR spectrum (Fig. V—l (b)) in an analogous fashion. At the g” extreme
(3128.8 G), all singularity points occur at the same frequency position and

only a single peak is predicted (Fig. V—l (b) and Fig. V-2 (f)).
5. Results and Discussion

The CW-EPR spectrum of Ni(lII)(CN)4(HzO)2' showed g” and g _L
features at 2.198 and 2.007, respectively, in agreement with previous
findings.13,21 Four-pulse ESEEM time domain data and the corresponding
magnitude FT-ESEEM spectrum obtained at g=2.107 with r=l97 ns are

 

 

72

Figure V-3. Four-pulse ESEEM (a) time domain data and (b)
corresponding magnitude FT spectrum. The experimental conditions were
magnetic field strength, 2981 G; microwave frequency, 8.789 GHz;
microwave pulse powers of n/2 and it pulses, 31.5 W and 63 W; pulse
width, 16 ns (FWHM); sample temperature, 4.2 K; pulse sequence

repetition rate, 10 Hz; events averaged/pt, 12; and r, 197 ns.

73

 

 

 

 

0 5 1 0 1 5 20 25 30

r + T (usec)

 

 

 

 

 

 

30

Frequency (MHz)

 

74

shown in Fig. V-3. Low frequency peaks are observed at 2.66, 2.8, and 5
MHz and are assigned to 14N of the equatorially bound cyanide group
The broad peaks that extend from 10 to 16 MHz are the fundament
frequency bands of the axially bound water protons. The sharp, inten:
peak at 25.38 MHz, twice the proton Larmor frequency On) at 2981G,

the second harmonic frequency of the matrix water protons. The su:

combination peak for the axially bound water protons is observed at 27.1

MHz, shifted 1.76 MHz from 2111. Two well resolved, sharp peaks centere
about 2v1 at 25.68 and 25.06 MHz can be assigned to combinatic
frequencies that arise from the product of ESEEM contributions at 0.2
MHz due to 14N and the matrix proton sum combination line1329'31 Th
0.26 MHz 14N ESEEM component is not shown in Fig. V-3 (b) but can t
seen in expanded spectra as a dominant feature.

Four—pulse ESEEM spectra collected at several magnetic fiel
positions across the EPR absorption band showed that the sum frequenc
peak positions of the axial water protons are shifted 0.8-1.9 MHz abov
twice the Larmor frequency. The field profile constructed by plotting th
measured frequency shifts of the sum frequencies vs. magnetic fiel
strength is shown in Fig. V-4. The data were collected using a microwav
frequency of 8.789 GHz and various 1 values. The analysis of th
experimental field profile was achieved using the singularity concept 2
explained in the above theory section. The frequency shifts in Fig. V-4 wi
be on one of the singularity points at the experimental field strength. Fig
V-5 shows a comparison of the experimental frequency shifts and th
simulated singularity curves across the EPR absorption spectrum fc
different dipole-dipole distances and fixed values of 0n=18° and a=0 MH:

As expected the frequency shifts are strongly dependent on the dipole-

75

Figure V-4. A plot of the four-pulse ESEEM experimental proton sum
frequency shifts (circles) from twice the proton Larmor frequency,
(va+v5)-2v1, vs. magnetic field strength for Ni(III)(CN)4(HzO)2'.
Experimental conditions were same as in Fig. V-3 except magnetic field
strengths and r values which were 3131 G, 186 ns; 3041 G, 193 ns; 3011
G, 195 ns; 2981 G, 197 ns; 2951G, 199 ns; 2921 G, 201 ns; 2891 G, 203
ns; 2869 G, 205 ns; and 2844 G, 206 ns. Error bars represent the intrinsic

uncertainty in measuring frequencies from four-pulse ESEEM spectra.

 

 

 

Frequency Shift (MHz)

76

 

0.8

 

1

r

 

 

2850

2900

2950 3000

Field Strength (G)

3050

 

77

Figure V-5. A comparison of the field profile of the axial water proton
sum frequencies (circles) with simulated field profiles of the turning points

of the proton sum combination peak lineshape for different effective

dipole-dipole distances. The simulation parameters were g1, 2.198; g“,
2.007; microwave frequency, 8.789 GHz; a, 0 MHz, 0“, 180; feff, (a) 2.28
A, (b) 2.33 A, (c) 2.38 A. The frequency shifts were calculated at 46

magnetic field positions across the EPR absorption spectrum.

 

Frequency Shift (MHZ)

 

 

 

 

 

 

 

 

 

 

2900 2950 3000

Field Strength (G)

3050

 

79

dipole distance. Within the experimental resolution limit, the effective Ni-l—
dipole-dipole distance was found to be 2.33i0.03 A. In Fig. V-6, the
experimental field profile is compared to the calculated profiles for 0,
values of 130, 180 and 230 holding reff=2.33 A and a=0 MHz. The
calculated field profiles show that the singularity curve for the ¢=0 feature
(solid line) rises more steeply in the low field region as 0n increases withir
a reasonable range of values. A 0,, range of 18:30 is satisfactory fo:
characterization of the axially bound water protons. The profile patterns
and the frequency shifts were relatively insensitive to isotropic hyperfine
couplings as expected.

For orientation selective ESEEM experiments on spin systems sucl
as Ni(III)(CN)4(HzO)2-, where axial g- and ligand hyperfine tensors apply
multiple lineshape singularities can be observed for the sum combinatior
band. The nature of these singularities depend on the relative orientation:
of the tensors and are best observed in studies of the r dependence of four-
pulse ESEEM spectra. Fig. V-7 shows the sum combination peak regions 01
two four-pulse FT-ESEEM spectra obtained at g=2.109 with (a) 1:239 ns
and (b) @260 ns. In the spectra, the sharp second harmonic peaks of the
matrix protons are observed at 25.06 MHz, twice the proton Larmor
frequency at 2944 G. The sum combination peaks are shown at (a) 26.66
MHz (1:239 ns) and (b) 26.94 MHz (r=260 ns) shifted 1.60 MHz and 1.88
MHz from twice the proton Larmor frequency, respectively. These
lineshapes are broadened toward higher frequency because of the overlap
of the sum combination band with a combination peak near 27.70 MHz
resulting from the product of the matrix proton second harmonic ESEEM
frequency with an 14N fundamental peak at 2.6 MHz. The sum

combination peak positions obtained using two different r values at

80

Figure V-6. A comparison of the field profile of the axial water proton
sum combination frequencies (circles) with the simulated field profile of
the turning points of the proton sum combination peak lineshape for On of
(a) 130, (b) 180 and (c) 230. Other simulation parameters were identical to
those of Fig. V-5 except I'eff, 2.33 A; and a, 0 MHz.

Frequency Shift (MHz)

 

0.8

0.6

O.‘

0.2

O

)-

 

 

0.8

0.6

0.4

0.2

 

 

0.81-

0.6 *-

0.‘ I-

0.2 *-

 

IIIIIIII v ’

l

2900

A

2850

A A I

2950 3000
Field Strength (G)

 

 

82

Figure V-7. Proton sum combination peak lineshapes obtained from four-
pulse magnitude FT-ESEEM spectra of Ni(III)(CN)4(HzO)2'. Experimental
conditions were same as in Fig. V-3 except magnetic field strength, 29440;
microwave frequency, 8.691 GHz; 1', (a) 239 ns and (b) 260 ns.

83

 

 

 

- d
.- -t
h .
t- <
I d
1 I i 1 1 L I
T T I I y T f
D d
b
I- 1

 

 

 

 

 

 

Frequency (MHz)

 

84

g=2.109 are plotted in Fig. V-8 along with the simulated singularitj
curves for refF2.31 A, 6n=180, (a) a=0 MHz, and (b) a=4 MHz. Fig. V-t
shows that both measured frequency shifts of sum combination peaks are
accounted for by predicted lineshape singularities.

A major shortcoming of this field profile strategy as implementee
using two-pulse ESEEM measurements was the lack of sensitivity of the
method to the isotropic hyperfine coupling.13 A comparison of the profile:
shown in Fig. V-8 (a) and (b) show this same problem because they are
calculated from Eq. V-4 and V-5 without the inclusion of ESEEM intensitj
weightings or the predicted singularity amplitudes. Because the r
suppression behavior of the sum combination peak amplitude depends or
the isotropic hyperfine coupling, it is possible to use four-pulse ESEEIV
studies to refine its determination. In Fig. V-9, two sum combination peal
lineshapes calculated with different intrinsic linewidths of (a) 0.03 MH:
and (b) 0.4 MHz along with common hyperfine coupling parameters 0
refi=2.31 A, 0n=180, and a=0 MHz are compared. The spectra in Fig. V-‘.

. . . waT . (OBT
were calculated usrng Eq. V-8 wrth srn(—2—)sm(—2—)=l. As a result, the}

show the entire lineshape without distortion from r-suppression effects. It
Fig. V-9 (a), all three singularity points at 26.06, 26.68, and 26.98 MH:
are observed. To facilitate comparison between experiment and theory, :
proper gauge of the intrinsic linewidth must be utilized. Fig. V-8 (a) show:
that at the g” extreme, the sum frequency lineshape features collapse to’ :
single frequency. Using the ESEEM spectrum collected at that magnetie
field strength, we estimated the intrinsic linewidth to be 0.4 MHz. A:

shown in Fig. V-9 (b), when this intrinsic linewidth is used to calculate the

85

Figure V-8. A comparison of the measured proton sum frequency shifts
(circles) from twice the proton Larmor frequency in Fig. V-7 with the
simulated field profile of the turning points of proton sum combination
band. The parameters for simulation were microwave frequency, 8.691
GHz; reff, 2.31 A, an, 180; a, (a) 0 MHz, and (b) 4 MHz.

 

 

Frequency Shift (MHz)

 

1.5"

05%

 

 

05-

 

1 l

 

 

2800 2850 2900 2950 3000 3050

Field Strength (G)

3100

 

 

87

Figure V-9. Simulations of the proton sum combination peak lineshapes the
in frequency domain without the 1‘ dependent coefficient of Eq. V-8. The
simulation parameters were magnetic field strength, 2944 G, microwave
frequency, 8.691 GHz, reff, 2.31 A, 0,1, 180 and a, 0 MHz. Intrinsic
gaussian linewidths (FWHM) were (a) 0.03 MHz, and (b) 0.4 MHz.

88

 

 

 

 

 

 

 

 

 

 

25

,
25.5 25 25.5 27 27.5 28
Frequency (MHz)

89

sum combination peak lineshape, the two higher frequency singularit
points are not resolved.

Predicted sum combination peak lineshapes at g=2.109 using 1' value

of 239 and 260 ns are shown for different values of the scalar hyperfin
coupling in Fig. V-10. For a=0 MHz with t=239 ns (Fig. V-10 (a), bottor
trace), two peaks are predicted at 26.14 MHz and 26.90 MHz. When 1' 1
increased to 260 ns leaving a=0 MHz (Fig. V-10 (b), bottom trace), only
single peak at 26.76 MHz is found. As the isotropic hyperfine coupling 1
increased while holding 0n and I'eff constant, pronounced changes in th
number and frequency positions of the lineshape extrema are predicted fc
the two r values considered. In Fig. V—10 (a), with 1:239 ns, the peak du
to the (13:0 singularity point appears when a Z 2 MHz, and for 1:260 111
the feature that arises from the singularity point corresponding to cos21< =
1 2T(T+2a)
2 + W of Eq. V—10 is seen when a 2 1.5 MHz. Th
lineshape feature corresponding to the eb=7r singularity point is resolve
when a 2 3 MHz. A comparison of these simulations with the data of Fig
V-7 allow an isotropic hyperfine coupling of 2.5i0.5 MHz for the axiall
bound water protons to be determined.

The results reported in this work are more refined than those of th
previous two-pulse ESEEM studies where reff=2.4i0.1A, a S 4 MHz an

n=12i50 were obtained from a field profile analysis.13 The improvemer
arises from the higher resolution of the four-pulse ESEEM measurement
extension of the field profile analysis to include the details of the sur
combination peak lineshape, and analysis of the r-dependence of th
lineshape. Since the hyperfine coupling mechanism of the axial wate

protons to Ni(III)(CN)4(H20)2' is similar to that of the equatorially

90

Figure V-10. Simulated proton sum combination peak lineshapes including
the 1' dependent coefficient of Eq. V-8 and different isotropic hyperfine
coupling constants. The simulation parameters were identical to those of
Fig. V-9 except that the isotropic hyperfine coupling constants shown in
figure were varied from 0 to 4.0 MHz, and intrinsic gaussian linewidths
(FW HM) of 0.4 MHz were utilized. For the simulated lineshapes of Fig. V—
10 (a), r=239 ns, while for Fig. V-10 (b), 1:260 ns.

91

am

ms“

32

um

Sc .85...»er

ad“ on

 

 

 

 

 

om

Aux—‘6 35:35

hm meow on

ma

 

 

 

 

 

 

9 2

bound water protons of Cu(II)(H2O)62+, their ligand hyperfine couplings
are expected to be comparable. Both scalar and dipolar portions of the
proton hyperfine couplings found for Ni(III)(CN)4(H2O)2‘ in this study
are slightly larger than those reported previously for the equatorial protons
of Cu(II)(H2O)62+ where isotropic ligand hyperfine couplings of S 1.2
MHz and dipolar couplings of approximately 5 MHz were measured in
single crystal ENDOR experiments.32 The difference in isotropic hyperfine
coupling is consistent with the larger crystal field splittings expected for
CN' vs. H2O ligation.21

EPR methods have played an important role in characterizing the
structure and biological function of the nickel cofactor of Ni-Fe
hydrogenases which catalyze the reversible oxidation of dihydrogen.33'35
Two oxidized and inactive forms (Ni-A and Ni—B) of the enzyme give
distinct EPR signals.36’37 The dominant species, Ni-A, has principle g-
values of 2.31, 2.26, and 2.02, and the second species, Ni-B, has principle
g-values of 2.33, 2.16, and 2.02. When the enzyme is activated under a H2
atmosphere, only a third Ni EPR signal, Ni-C, characterized by a rhombic
g-tensor with principle values of 2.19, 2.15 and 2.02 is observed.36
Previous Q-band ENDOR studies of the Ni-C form of D. Gigas Ni-Fe
hydrogenase at g1=2.19 revealed that two kinds of exchangeable protons
are bound to the Ni-C site with the hyperfine couplings of 4.4 MHz and
16.8 MHz assigned to bound H2O or 0H ligands and a proton directly
coordinated to Ni, respectively.38 Q-band ENDOR simulation with the
results obtained from the present studies showed that the 4.4 MHz
hyperfine coupling is in good agreement with the HFI couplings of the
axial water proton of Ni(III)(CN)(H2O)2'.13 Recent Q-band studies of the

Ni—C sites of the hydrogenase from T. Roseopersicina resolved a 20 MHz

 

93

hyperfine coupling from solvent exchangeable protons that originated from
dihydrogen.39 This extent of proton hyperfine coupling could not be
realized using the hyperfine coupling parameters of Ni(IlI)(CN)4(HzO)2.13

The Ni-C EPR signal from a Ni-Fe hydrogenase isolated from T.
Roseopersicina showed a clean, rhombic EPR signal while that from D.
Gigas was overlapped with spectral contributions from a reduced 4Fe-4S
center at high field.38,39 In the case of T. Roseopersicina, four-pulse
ESEEM studies of the field profile of the proton sum combination bands
will be helpful to identify more detailed structural information than
available from the previous ENDOR and XAS studies of the enzyme.39
Fig. V—11 shows numerically calculated field profiles of the turning points
of the proton sum frequencies with g1=2.19, g2=2.15, and g3=2.02 as
observed for the Ni-C form of the enzyme and with feff=2.33 A, a=2.5
MHz, (a) 0n=180 and (b) 0n=300. The frequency shifts are (va+v5)-2v1 as
in the above field profiles. To determine the frequencies of the turning
points, the magnetic field strength and microwave frequency were first
used to calculate continuous angle sets (0, (b) that were selected25'28 and
then the sum frequencies were calculated as a function of these angle sets
using the ligand hyperfine couplings based on Eq. V-4 and numerically
determined d(va+v5)=0 points. As shown in Fig. V-11, the overall field
profile patterns are mainly determined by 0". In the calculated field
profiles, there are ¢n-sensitive regions around g2, so it should be possible
to determine 0n and (bn separately. The effective dipole-dipole distances did
not affect the profile shape, but frequency shifts were strongly dependent
on reff as in the axial g-tensor case. The frequency shifts and field profile

pattern were weakly sensitive to the isotropic HFI coupling, as expected. If

 

 

94

Figure V-l 1. Computed field profiles of the frequency shifts of the turning
points of the proton sum combination bands from twice the proton Larmor
frequencies in rhombic g-tensor system. The parameters for the
simulations were g1=2.19; g2=2.15; g3=2.02; microwave frequency, 9.0
GHz; reff, 2.33 A; a, 2.5 MHz; (a) en, 180; d)“, 00; (b) on, 180; 4),, 450; (c)
Gm, 180; d)“, 900; (d) 011, 300; (bu, 00; (e) 0“, 30°; (1)“, 450; and (0 0n, 300;
cbn, 90°.

 

 

Frequency Shift (MHz)

 

 

 

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O O O

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0 o O
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v ‘ I r

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0 O
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Field Strength (G)

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Frequency Shift (Mllz)

 

 

 

 

 

 

 

 

 

 

 

 

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0000
1 a - 0° °'8<,,>°eac>¢g,,,,o e .
0 oo 9 :°°°°o
o °8°°°o
I 6 i' O 0 0° 0 7
0 ° 0°
0 o
1 4- 0 0° 0 .
o 0 o
y 2 - 08° 0 .
o
1 . 0 O O .
o
o o o
0.3- ° 0 o o 1
o o o
0.5» ° 0 o o i
o o o
0 a» o o o O 4
o 0 o o
0 2+ 0 0 ° °0 -
o o o O o
o o o
0}- 908800308080‘) 000 i
A A; L A 4
2 r v . Y . 1
0 f l
1 8 1' Geo 00800000000 .
o °°°o°o i
0 08°00 .
1 5' 9 Q 0 ‘1
o o o o
o
1 4v- 8 o 00 0° 0 .
o o o
1 2- 8 g o .
0 o
9
1 - 0 o o <
o o
0 O
o a - ° 0 .
9 o
o O
0.6 t- O O -
9 o
o.» o 0 o .
o
9 o o
0.2 1' o O ‘
o o o O
00 O
01- Boe8o8°oooeoBoO° 00° i
2900 2950 SW 3050 3100 3150 32%
Field Strength (G)

97

more than one singularity point is found at a single magnetic field position,

the r—suppression behavior of the sum combination band lineshapes will

provide a means for measuring the isotropic HFI couplings. Therefore,

four-pulse ESEEM field profile studies of the proton sum combination

band of the Ni-C species of T. Roseopersicina should allow for a more

complete characterization of the strong proton couplings reported from

EN DOR studies.

References

1. L. Kevan, in Time Domain Electron Spin Resonance, L. Kevan, and
R. N. Schwartz, Ed., Wiley-Interscience, New York, Chapter 8
(1979).

2. A. J. Hoff, Ed., in Advanced EPR. Applications in Biology and
Biochemistry, Elsevier, New York, Chapters 1, 2, 3 and 6. (1989).

3. L. Kevan, in Modern Pulsed and Continuous -Wave Electron Spin
Resonance, L. Kevan, and M. K. Bowman, Ed., John Wiley & Sons,
New York, Chapter 5 (1990).

4. A. de Groot, R. Evelo, and A. J. Hoff, J. Magn. Reson., 66, 331
(1986).

5. W. B. Mims, Phys. Rev., B5, 2409 (1972).

6. W. B. Mims, Phys. Rev., B6, 3543 (1972).

7. A. V. Astashkin, S. A. Dikanov, and Yu. D. Tsvetkov, Chem. Phys.
Lett., 136, 204 (1987).

8. S. A. Dikanov, A. V. Astashkin, and Yu. D. Tsvetkov, Chem. Phys.
Lett., 144, 251 (1988).

9. E. J. Reijerse, and S. A. Dikanov, J. Chem. Phys, 95, 836 (1991).

10. W. B. Mims, J. Peisach, and J. L. Davis, J. Chem. Phys, 66, 5536
(1977).

1 1. G. J. Gerfen, P. M. Hanna, N. D. Chasteen, and D. J. Singel, J. Am.
Chem. Soc, 113, 9513 (1991).

12. A. M. Tyryshkin, S. A. Dikanov, R. G. Evelo, and A. J. Hoff, J.
Chem. Phys, 97, 42 (1992).

13. J. McCracken and S. Friedenberg, J. Phys. Chem, 98, 467 (1994).

15.
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20.
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28.
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30.
31.
32.
33.

34.
35.

 

98

C. Gemperle, G. Abeli, A. Schweiger, and R. R. Ernst, J. Magn.
Reson., 88, 241 (1990).

A. Schweiger, Angew. Chem. Int. Ed. Engl., 30, 265 (1991).

A. Schweiger, in Modern Pulsed and Continuous —Wave Electron
Spin Resonance, L. Kevan, and M. K. Bowman, Ed., John Wiley &
Sons, New York, Chapter 2 (1990).

E. J. Reijerse, and S. A. Dikanov, Pure & Appl. Chem, 64, 789
(1992).

A. M. Tyryshkin, S. A. Dikanov, and D. Goldfarb, J. Magn. Reson.,
A105, 271 (1993).

S. A. Dikanov, C. Burgard, and J. Hutterman, Chem. Phys. Lett.,
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T. L. Pappenhagen, and D. W. Margerum, J. Am. Chem. Soc, 107,
4576 (1985).

Y. L. Wang, M. W. Beach, T. L. Pappenhagen, and D. W.
Margerum, Inorg. Chem, 27, 27, 4464 (1988).

J. McCracken, D. H. Shin, and J. L. Dye, Appl. Magn. Reson., 3,
205 (1992).

C. P. Lin, M. K. Bowman, and J. L. Norris, J. Magn. Reson., 65,
369 (1985).

R. D. Britt and M. P. Klein, J. Magn. Reson., 74, 535 (1987).

C. A. Hutchison and D. B. McKay, J. Chem. Phys, 66, 3311 (1977).
G. C. Hurst, T. A. Henderson, and R. W. Kreilick, J. Am. Chem.
Soc, 107, 7294 (1985).

T. A. Henderson, G. C. Hurst, and R. W. Kreilick, J. Am. Chem.
Soc., 107, 7299(1985).

B. M. Hoffman, J. Martinsen, and R. A. Venters, J. Magn. Reson.,
59, 110 (1984).

L. G. Rowan, E. L. Hahn, and W. B. Mims, Phys. Rev., A137, 61
(1965).

D. J. Kosman, J. Peisach, and W. B. Mims, Biochemistry,19, 1306
(1980).

C. P. Lin, M. K. Bowman, and J. R. Norris, J. Chem. Phys, 85, 56
(1986)

N. M. Atherton and A. J. Horsewill, J. Molec. Phys, 37, 1349
(1979).

J. C. Salerno, in The Bioinorganic Chemistry of Nickel, J. R.
Lancaster Jr., Ed., VCH Publication, New York, Chapters 3, 8
(1988).

M. W. W. Adams, Biochim. Biophys. Acta, 1020, 115 (1990).

A. E. Przybyla, J. Robbins, N. Menon, and H. D. J. Peck, FEMS
Microbiol. Rev., 88, 109 (1992).

 

 

 

  

36.

37.
38.
39.

2 2.. waggegjiéééifigém .

99

M. Teixeira, I. Moura, A. V. Xevier, B. H. Hyunh, D. V. Der
Vartanian, H. D. Peck, J. LeGall, and J. J. G. Moura, J. Biol Chem,
260, 8942 (1985).

V. M. Femadez, C. E. Hatchikian, D. S. Patil, and R. Cammack,
Biochim. Biophys. Acta, 883, 145 (1986).

C. Fan, M. Teixeira, J. J. G. Moura, B. H. Hyunh, J. LeGall, H. D.
Peck, and B. M. Hoffman, J. Am. Chem. Soc, 113, 20 (1991).

J. P. Whitehead, R. J. Gurbiel, C. Bagyinka, B. M. Hoffman, and

‘ M. J. Maroney, J. Am. Chem. Soc., 115, 5629 (1993).

r;.g.. '

VI. FOUR-PULSE ELECTRON SPIN ECHO ENVELOPE
MODULATION STUDIES OF Cu(II)(HzOhs2+

1. Abstract

Four-pulse electron spin echo envelope modulation (ESEEM) studies
aimed at characterizing hyperfine interactions between protons of
equatorially and axially bound water molecules and the copper ion in
Cu(II)(H2O)¢52+ were carried out. Four-pulse ESEEM experiments showed
that these hyperfine couplings are characterized by large anisotropic
interactions resulting in the shifts of the proton sum combination peaks,
va+v5, from twice the Larmor frequency, 212]. For the equatorially bound
water protons, a distribution of sum combination peak frequency shifts that
spanned the range from twice the Larmor frequency to 1.4-1.7 MHz above
twice the Larmor frequency at each magnetic field strength was found. At
a given magnetic field strength the sum combination frequency shift from
these strongly bound water protons varied with the 1' value used in the
measurement so that a discrete field profile pattern was not found. These
features result from the rotational distribution of the equatorially bound
water molecules about Cu-O bonds at 4.2 K in frozen solution. Theoretical
simulation of the field profile of the turning points of the proton sum
combination lineshape revealed an effective Cu-H distance of 2.49 A, an

angle between the principal axis of the ligand hyperfine tensor and the g3

100

lOl

axis (0N), that was distributed between 710 and 90°, and an isotrop
hyperfine coupling constant of S 4 MHz for the equatorially bound wate
protons. Attempts were made to distinguish the sum combination peaks 1
the axially bound water protons by using the r—suppression behavior of ti
four-pulse ESEEM. It was found that the intensities and lineshapes of ti
sum combination peaks for these protons were relatively well develope
and distinguished from the sum combination peaks of the equatori:
protons when r was set to values near n/v1 (n=integer). Simulation of th
frequency shifts of the axial proton sum combination peaks vs. magneti
field yielded an effective Cu-H distance of 3.05 A, 0N=110 and an isotropi
hyperfine coupling of S 4 MHz.

2. Introduction

Copper ions are found to play important roles in sever:
metalloenzymes.1 Native copper proteins may contain Cu(I) and/or Cu(l
forms. Cu(II) forms are paramagnetic and give rise to electro
paramagnetic resonance (EPR) signals. In electronic spectra of coppe
complexes, weak copper d-d transitions are often masked by stronge
ligand electronic transitions. For this reason, EPR has played an importar
role in characterizing the structure and biological function of copper i
copper proteins. In these proteins, copper ligands may consist of group
supplied by the protein backbone, amino acid side chains, enzymati
cofactors, or solvent molecules.

Although EPR studies of copper sites give useful information, mor
detailed ligand structural information is not easily obtained since the ligan

hyperfine interactions are generally much weaker than the hyperfin

102

interaction between the copper nucleus and the unpaired electron. Electrc
spin echo envelope modulation (ESEEM)}4 is especially sensitive 1
Cu(II)-ligand interactions because the weak hyperfine interactions that a:
hidden by the broad Cu-EPR lineshape can be detected. Many ESEEI
studies of copper containing proteins have demonstrated the capabilities 1
this method for the identification of Cu(II) ligands and their coordinatic
geometries}9 In ESEEM studies of copper proteins, the binding 1
histidine side chains as equatorial copper ligands has been we
characterized because at X-band microwave frequencies, the hyperfir
interaction of the remote nitrogen of the equatorially bound imidazole rir
is most distinctive. This characteristic 1"(N-ESEEM pattern results from
coupling regime known as "exact cancellation" where characteristic 14
nuclear quadrupole resonance lines are well resolved“),11 14N-ESEEI
studies of copper model complexes that support the assignment of
characteristic Cu-His interaction have been reported.12,13

Several studies have also shown that the copper sites of mar
proteins are accessible to solvent water molecules thought to have critic
functions in catalytic processes. ESEEM studies of the accessibility t
solvent H20 molecules to copper sites and the direct measurement of wat
ligand hyperfine couplings have been most often done with D20 exchang
experiments.7,9 The hyperfine interactions between Cu(II) and the protor
or deuterons of coordinated H2O or D20 are characterized as large
anisotropic by CW-ENDOR studies on single crystals. Even at X-bar
frequencies, the nuclear Larmor frequencies of protons and deuterons a

larger than the isotropic hyperfine couplings. As a result, these stror

anisotropic hyperfine interactions prevent the fundamental frequencies (V

ya) from splitting and cause only broadened lineshapes centered about [I

103

Larmor frequencies. This lack of resolution gives rise to difficulties
characterizing the ligand hyperfine interactions of bound wa
molecules.14 Previous attempts at ESEEM characterization of bound we
molecules involved the use of data ratioing methods”,16 to remove
large background provided by matrix protons or deuterons. Often,
contribution from the matrix deuterons could not be separated fr
ESEEM due to bound D20 so that the accurate measurement of
deuteron hyperfine interaction and water geometry were difficult. For t
reason, the sum combination frequency (va+v[3), where pronoune
differences have been observed between axial and equatorially bound D
in two-pulse ESEEM, was studied],9

The properties of ESEEM sum combination peaks have been ve
characterized in the recent literature.17-22 The advantage in using the s
combination band is that the sum combination peak positions are sensit
to the anisotropic portion of the ligand hyperfine interaction and that
spectral line is narrow enough to be easily resolved. Sum combinat
peaks are characterized by frequency shifts from twice the nuclear Lari
frequency. For a spin system described by an isotropic electron g-ten
and an axial ligand hyperfine tensor, the sum combination peak freque'
shift is given by Av=va+v5-2v1=(9/16)T2/v1 where va and v5 are
fundamental hyperfine frequencies of 01 and 0 electron spin manifolds
is the Larmor frequency of the coupled nucleus; and T is the anisotrc
portion of the perpendicular component of the ligand hyperfine tenso
For this simple spin system, the frequency shifts are dependent on
relative orientation between external magnetic field and principal lig
hyperfine axis.17'19 The sum combination frequency shifts found for

nuclei are approximately 6.5 times less than those observed for identic

104

coupled protons due to the difference in their gyromagnetic ratios. Hen
anisotropic hyperfine couplings are better resolved using the proton sr
combination peak. Also, the proton sum combination band is found 11
region where there is a relatively clean spectral window.23

We have recently introduced sum frequency field profiles obtain
by plotting the frequency shifts of the sum combination peaks from tw.
the Larmor frequencies as a function of magnetic field strength across I
EPR absorption spectrum to serve as a foundation for a more systema
analysis of the ligand hyperfine tensor in anisotropic electron 31
systems.” This method was applied to the Ni(lII)(CN)4(H20)2- system
characterize the ligand proton hyperfine couplings of the axially bou
water molecules by using two-pulse ESEEM experiments.” More recent
we have extended this field profile analysis to include the behavior of 1
sum combination peak lineshape singularities.” This enhanced field prof
analysis strategy was applied to the same Ni(III)(CN)4(H2O)2' sample
using four-pulse ESEEM24J5 where the sum combination band lineshape
better resolved.” The r-suppression behavior2425 of the sum combinati
lineshape of four-pulse ESEEM was used to place more exact constrai:
on the isotropic hyperfine coupling constant than possible using two—pu
methods.”

In the present chapter, the enhanced sum frequency field prol
analysis of the proton sum combination peak is applied to Cu(II)(H20)(
in frozen solution to distinguish the ligand hyperfine interactions of axia
and equatorially bound water protons using four-pulse ESEE
experiments. Our goals in undertaking this work were to address I
usefulness of four-pulse ESEEM experiments for studies of me

complicated spin systems, and to test the analysis procedures developed

105

the Ni(III)(CN)4(H2O)2' system on a compound where single crystal EPl
and ENDOR results are available. For Cu(II) in a tetragonally distorte
octahedral ligand field, the unpaired electron is mostly distributed in dx22
orbital. Hence, the ligand hyperfine interaction in the equatorial plane i
expected to be stronger and give rise to larger sum frequency shifts fror
2v1 than interactions placed along the axial direction. However, a portio:
of the lineshape of the equatorial proton sum combination band wil
overlap the lineshape of the axial protons making the analysis mor
difficult.” Attempts at distinguishing the the sum combination peaks an
the ligand hyperfine interactions of these two different pOpulations 0
protons were made using the r-suppression behavior of the four-puls
ESEEM lineshapes. Four-pulse ESEEM spectra of axial water ligand
bound to Cu(II)-tetraphenylporphyrin [Cu(II)TPP] were also obtained t
compare with those of Cu(II)(H2O)62+.

3. Experimental

The Cu(II)(H2O)62+ sample was prepared by dissolvin
Cu(II)(NO3)2 [Aldrich] salt in distilled water. The sample for electron spi
echo experiments was mixed with an equal volume of ethylene glycol prie
to freezing in liquid nitrogen. The resulting concentration c
Cu(lI)(H20)62+ was 6 mM. Cu(II)-tetraphenylporphyrin [Cu(II)TPI
complexes with axially bound water ligands were prepared by dissolvin
Cu(II)TPP [Aldrich] into dehydrated CH2Cl2. The resulting solution we

mixed with excess amount of water and vigorously shaken. The CH2C?

layer where Cu(II)TPP is dissolved was separated and frozen in liqui

106

nitrogen. The final concentration of the Cu(II)TPP was 0.45 mM to avoi
aggregation or dimerization.26

Electron Spin Echo detected-EPR (ESE-EPR) and ESEEII
experiments were executed on a home-built pulsed-EPR spectromete
which has been explained in detail elsewhere27 and in chapter IV. 1
stimulated echo microwave pulse sequence (tr/2-r-rr/2-T-n/2) was used t
collect ESE-EPR data by varying external field strength at fixed r and '
values. A four-pulse sequence (7r/2-r-rr/2-T/2-1r-T/2-rr/2) was used t
produce the ESEEM data. The length (full-width at half maximum) of a.
four microwave pulses was 16 ns. The peak power of the n—pulse was twic
that of 7r/2-pulse.”a25 A four step phase cycle, +(0,0,0,0), +(0,0,7r,7r:
+(0,0,7r,0), +(7r,7r,7r,0), was used to eliminate unwanted echo modulation.2
The ESEEM data were collected as a function of T, the time interva
between the second and fourth pulses. Fourier transformation of the tim
domain ESEEM data were done without dead-time reconstruction. Th
fourier transformation and computer simulations were performed 0
SPARCstation 2 workstation (Sun Microsystems) using software written i
MATLAB v.4.2 (The Mathworks, Inc.).

4. Theoretical Aspects
4-1. Angle Selection

For CW-EPR spectra dominated by large anisotropic g- and/or met:
hyperfine tensors such as in Cu(II)(H20)62+, each portion of the EP

absorption spectrum of randomly oriented samples represents EP

transitions from complexes which have specific orientations with respect t

107

the external magnetic field direction. Specific orientations that contrit
to the EPR absorption are determined by the microwave frequen
magnetic field strength, and the nature of the anisotropic magn:
interactions. To interpret ESEEM spectra of randomly oriented samr
obtained at a fixed external field strength and microwave frequency,
orientation or angle selection analysis is needed.

The angle selection analysis portion of this study made use of
approach of Hurst, Henderson, and Kreilick for the interpretation
ENDOR spectra of a bis(2,4-pentanedionato)copper(II) powder.28,29
Cu(II)(H20)62+, the metal quadrupole and ligand hyperfine interacti
were not included because these interactions are small enough to be tree
by the ESEEM lineshape function or background decay. Also, differer
resulting from the natural abundance of 63Cu (69%) and 65Cu (31%)
too small to be distinguished in ESEEM studies. The Hamiltonian for
Cu(II) EPR transition is then expressed as

H = BeS-g-H + s.A.1 [VI-1]

where Be is the electron Bohr magneton, S is electronic spin ang
momentum Operator, g is the electron g-tensor, H is the external f
vector, A is the metal hyperfine tensor, and I is the nuclear spin ang
momentum operator. From Eq. Vl-l, the resonance field position 01

EPR transition is given by

_ hV-MIA(6,¢)
I' — fieg(ea¢)

 

[VI-2]

 

l 0 8
where h is Planck's constant, v is the microwave frequency, M1 is the

nuclear spin quantum number,

3
edit!» = [, 2101102110. and [VI-3

1:

3 3
1. 2(2 Ajigjlj)2]1/2
1: =

“6"") = —‘ g<6.¢>

Here, l1=sin0cosd>, l2=sin0sin<b, l3=cos0, g1, g2, g3 are the princir
tensor values, 0 is the angle between the external field and g3 axis, 4)
azimuthal angle of the field with respect to g-tensor axes, and the A
the metal hyperfine tensor matrix elements. The orientations (0,4):
satisfy the resonace condition at a fixed field strength and micro
frequency can be obtained analytically in the case where the both g
metal hyperfine tensors are axial and coincident.28 For other general 1
the angle set can be obtained numerically. Appendix, A7 lists corn
programs written for this purpose. The program calculates Eq. VI-2
around the roots of Eq. VI-2 so that the calculation time can be ree
when compared to parabolic search algorithms commonly used by

researchers in the field.
4-2. Lineshape Properties of Four-Pulse ESEEM Sum Combination B:

For systems with an anisotropic g-tensor, the fundamental ES

or ENDOR frequencies of an S=l/2, I=1/2 spin system are given by”-

 

109
3 m, 3
vaep) = [i 32233821ngAjD-ln/IIZ l“2 [VI-4]
where
Aji = fifgfiggianinj-hij) + ahij, [VI-51

ms is the electron spin quantum number, gN is the nuclear g value, [N i;
Bohr magneton, r is the electron—nuclear dipole-dipole distance
n1=sin0Ncos¢N, n2=sin0Nsin¢N, n3=cos0N, 0N is the angle between the
ligand hyperfine principal axis and g3 axis, 0N is the azimuthal angle of the
ligand hyperfine principal axis with respect to g-tensor axes and "a" is the
isotropic ligand hyperfine coupling constant. (Note: Aji of Eq. VI-5 i:
different from Aji of Eq. VI-3 which is the metal hyperfine tenso
component.)

For the system where both g- and metal hyperfine tensors are axia
and coincident, the angle 0 is determined by the microwave frequency ane
external field strength (Eq. VI-2). If the ligand hyperfine interaction 1:
axial, the hyperfine frequencies are independent of dJN. Because 0N i;
constant for a given ligand geometry, the lineshape of an orientatior
selective ESEEM peak is a function of d) for the above spin system and the
lineshape function of the sum combination band as obtained by four-pulse

ESEEM is given by22

. . . a
IVa+vp(¢) = k(cb)02s1n(%)sm(%5—T)Slnelg(_va¢_

+We [VI-61

110

 

 

where
2 _ 10012-211'(w01+w0)21
s — mu, 0 . [v1-7]
c2 _ IwIZ-gfiwa-wpfll
wawp
k = 45202, and

£001 and 005 are the angular frequencies of Va and v5, respectively. The
frequency distribution range of the sum combination band is determined b;

the turning points of the sum frequencies where the condition

d(v +v ) . . . . .
3 <1) 6 =0, IS satrsfied.19a” Provrded ¢N=0, the tumrng p01nts occur a

 

¢=0, it and values determined numerically by solution of the abov
relationship.” At these turning points, the intensities of the sun

frequencies show singularities except at va+v5=2v1 where branching of the

EPR transition ceases (k and s2 are zero.).

The frequency shift [Av=va+vB-2v1] field profile”,” of the turnin;
points of the sum combination band is sensitive to the effective electron
nuclear distance and 0N, but less sensitive to isotropic ligand hyperfin
coupling. The ir—suppression effect on the lineshape of the sum combinatio:
band of four-pulse ESEEM shows the sensitivity to the isotropic hyperfin
coupling.” The r-suppression behavior of the sum combination lineshap
of four-pulse ESEEM requires a more complete theoretical treatment
Appendix, A8, shows the computer program to calculate orientatio:

selective four-pulse ESEEM spectra for S=1/2, I=1/2 spin system.

5. Results and Discussion

The electron spin echo-detected EPR (ESE-EPR) spectrum (so
curve) of Cu(II)(H20)62+ is shown along with the simulated E

absorption spectrum (dashed curve) in Fig. VI-l. The simulation revea

that electron g- and copper hyperfine tensor components of g_1_=2.01
g”=2.411, A_1_=10 MHz and AH=418 MHz are well fit to the experimen-
result. The low-field region, where the only EPR transition arises from
copper M1=3l2 spin manifold, is most suitable for Cu(II) ESEE
orientation selection experiments because only a single value of 0, the an,
between the external field and g3 axis, is selected. At higher field strengt
more than one copper hyperfine spin manifold is involved in the E
absorption so that more than one value of 0 is selected. ESEEM spec
collected in the higher-field region may become too complicated to
analyzed.

Fig. VI-2 (a) and (b) show the proton sum combination peak regie
of four-pulse FT-ESEEM spectra obtained at g=2.563 with (a) 1:239
and (b) r=287 ns. In the spectra, sharp peaks shown at 20.88 MHz are
second harmonic frequency of matrix protons at 2452G. For r=239
[Fig. VI-2 (a)], a peak was resolved at 21.20 MHz, shifted 0.32 MHz fr
2v1 and one broad feature was found at higher frequency with maxim
intensity position of 22.27 MHz. These peaks are assigned to the s
combination peaks of axially and/or equatorially bound water protons. 'l
four-pulse ESEEM spectrum obtained with 1:287 ns [Fig VI-2 (b)] sho

Figure VI—l. Electron spin echo detected-EPR (solid) and simula
(dashed) absorption spectra of Cu(II)(H2O)62+. The stimulated ech
n/2-T-rr/2) microwave pulse sequence was used to generate elect
echo signal with r=250 ns and T=2000 ns. Other experimental ce
were microwave frequency, 8.820 GHz; microwave pulse power,
microwave pulse width (FNHM), 16 ns; averaging number, 1
repetition rate, 12 Hz, and temperature, 4.2 K. The simulated spect
obtained with EPR parameters of g i=2.083, gll=2.411, A _L=10 IV
A“=418 MHz.

Echo Amplitude

 

 

 

 

2500 2600 2700 2800 2900 3000 3100 3200

Field Strength (0)

 

 

 

 

114

Figure VI—2. Four—pulse magnitude FT—ESEEM spectra of Cu(II)(H2O)62+
at proton sum combination frequency region. Experimental conditions
were magnetic field strength, 2452G; microwave frequency, 8.795 GHz;
microwave pulse powers of Tr-pulse, 141 W and rt/2-pulse, 71 W;
averaging number, 30; pulse repetition rate, 30. r values are (a) 239 ns and
(b) 287 ns.

 

 

 

 

1 1

 

 

21 22

Frequency (MHz)

24

 

 

L.

116

different features from that with r=239 ns. In Fig. VI-2 (b), the SCCOH!
harmonic frequency peak of the matrix proton is partially suppressor
because the I value is set to 3/v1.18~31 The sum combination peaks of th.
coordinated water protons are resolved at 21.34 and 22.00 MHz shifter
0.46 and 1.12 MHz from 2v1, respectively. The frequency shift
[Av=va+v5-2v1] of the proton sum combination peak positions obtainec
for several 1 values at six different field positions, where only EPI
transitions of copper M1=3/2 spin manifold occur, are plotted as a functi01
of magnetic field strength in Fig. VI-3. As shown in Fig. VI-3, the protor
sum frequency shifts are almost continuously distributed at each magnetic
field strength. Previously, we have utilized this field profile approach t<
study the proton ligand hyperfine interactions of axially bound wate
protons in Ni(III)(CN)4(H20)2'.21,22 In those experiments, the sun
combination peaks were placed on one of the turning points of the sun
combination lineshape. The same field profile approach should be
applicable to the results of this study.

Fig. VI-4 shows comparisons of the experimental proton sun
frequency shifts (circles) with the simulated field profiles of the turning
POints of the sum combination bands across the EPR spectrum for a1
effective Cu-H distance (reff) of 2.49 A, an isotropic hyperfine coupling
Constant ("a") 'of 0 MHz, and 6N, the angle between the principal axis 0
the proton ligand hyperfine interaction and g3 axis, of (a) 900, (b) 800, am
(0) 710. As shown in Fig. VI-4, there are always experimental proton sun
Combination peaks that correspond to the simulated turning point curves.

The features found in the studies of Cu(II)(H20)62+ can be explaine<

by the rotation of coordinated water molecules about Cu-O bond axis. A

room temperature, the coordinated water molecules freely rotate abou

 

 

117

Figure VI-3. Proton sum combination frequency shifts (circles)
[Av=va+v[3-2v1] of four-pulse ESEEM spectra obtained at magnetic field

strengths, 2431 G, 2452 G, 2473 G, 2494 G, 2515 G, and 2536 G. Other

experimental conditions were same as in Fig. VI-2 except 1 value. At each

field strength, r values are were changes between 2.5/v1 and 3/v1 with 10

us or 8 ns step.

118

O 0080
0 000850

08 00

 

 

 

1.8~
1.61-
0&-

fizzv .Em 3532".

2500 2600 2700 2800 2900 3000
Field Strength (G)

2400

119

Figure VI-4. A comparison of the measured proton sum combinati
frequency shifts (circles) with simulated field profile of the turning poi]

of proton sum combination peak lineshape. The EPR parameters for an;
selection were gt, 2.083; g”, 2.411; A_|_, 10 MHz; All, 418 MHz; cop;

M1,3/2; and microwave frequency, 8.795 GHz. The ESEEM simulati
(ligand hyperfine) parameters were reff, 2.49 A; a, 0 MHz; 6N, (a) 900, 1
800, and (c) 710. The sum frequency shifts were calculated with 46 6, 1
angle between the external magnetic field and g3 axis, values (00 ~ 9(
Which cover the EPR absorption spectrum. In the simulated field profi
Solid and dashed curves correspond to the turning points occurring at d;
and 1t, respectively. The turning points of plus signs were numerica

determined.

 

03f
Ofir
OAr

02*

o} +++++¢++o

 

 

18*

LG?

lfih

1.21-

00+

05?

Frequency Shift (M111)

0.4)-

Q
N
1

 

 

05k

1

04

0.21-

 
  
 

\

\ I
0)- ‘*++++++*+++++++++++++

\

++wH*

 

% 1 1

2“» 23” 23» 2ND 2mm 2%» 3“”

Field Strength (G)

 

121

corresponding Cu-O bonds. But at the ESEEM experimental temperat‘
4.2K, the rotation of the water molecules is restricted and the geomet
of the water molecules take on thermodynamically stable orientations.
equatorially bound water molecules, the rotation of the ligand brings at
a change of 6N and cm, the polar and azimuthal angles that describe
orientation of the principal axis of the proton ligand hyperfine tensor v
respect to g-tensor axes. In case of a randomly oriented Cu(II)(H20)
sample where both the g- and metal hyperfine tensors are axial
coincident and the ligand hyperfine interaction is also axial, the variation
ch has no effect on ESEEM spectra, but the variation of 6N gives ris«
changes in the lineshapes and the field profiles of the sum combinat
peak.21,22 Because rotation of the axially bound water molecules affc
only 475:, a specific pattern or field profile of the sum combination pea
expected as in the Ni(III)(CN)4(H20)2- case.21,22 Hence, the four-pl
ESEEM data of Cu(II)(H20)62+ and the simulated field profiles of
turning points of the proton sum combination band show that 6N values
the equatorially bound water protons are distributed from 900 to ’.
Considering the structure of Cu(II)(H20)62+,32 the above feature indic
that the dihedral angles between the g3 axis and H-0 bonds of
equatorially bound waters are distributed from 00 to 3600 at 4.
However, these four-pulse ESEEM experiments are not sufficient to 1
Conclusive information about this distribution of the dihedral angles.
Considering the intrinsic uncertainty in the determination of the
frequency peakpositions by four-pulse ESEEM experiments, an re1
2.49:1:0.04 A can be assigned to the equatorial protons. Because the 1
profile is not sensitive to the isotropic hyperfine coupling, thc

suppression behavior of the sum combination peak lineshape has I

122

previously used to determine the isotrOpic ligand proton hyperfine
coupling in Ni(III)(CN)4(H20)2‘.22 For equatorially bound water protons
of Cu(II)(H20)62+, determination of the isotropic ligand hyperfine
coupling constant is not possible due to the rotational distribution of the
equatorial ligands. However, based on simulations with refF2.49 A and
6N=710 for most of the shifted peaks, an isotropic hyperfine coupling
constant of less than 4 MHz can be estimated.

Another problem of our field profile approach for analyzing the
ligand hyperfine interactions of protons in Cu(II)(HzO)62+ is that the sum
combination peaks arising from axially bound water protons, of which the
ligand hyperfine interactions are expected to be weaker than that of the
equatorial protons, are not resolved because of the rotational distributior
of the equatorial water molecules about the Cu-O bonds. Attempts tc
extract the sum combination peaks of the axial water protons were made
using the r—suppression behavior of four-pulse ESEEM spectra. The protor
sum combination frequency regions of four-pulse ESEEM spectra collectec
with r values of n/v1 (n=integer value), like in Fig. VI-2 (b), generally
showed three peaks. One of these peaks is the second harmonic frequency
of the matrix protons and the other two are the sum combination peaks 0]
the coordinated water protons. The field profile obtained by plotting the
frequency shift of the lower-frequency component of the sum combinatior
peaks against magnetic field strength showed a trend where the frequency
shift was found to increase with increasing field strength (Fig. VI-S). The
simulated field profile of the turning points of the proton sum combinatior
frequency for the ligand hyperfine coupling parameters of ref1=3.05 A
6N=11°, and a=O MHz are best fit to the experimental frequency shifts 0

Fig. VI-S (Fig. VI-6). The simulation, considering the errors in measuring

123

Figure VI-5. Experimental proton sum frequency shifts (circles) of lower-
frequency peak of the two sum frequency peaks obtained with r values of
around 3/v1 (See text). Experimental conditions were same as in Fig. VI-2
except r values of 282 ns, 290 ns at 2431 G; 279 ns, 287 ns at 2451 G; 277
ns at 2473 G; 282 us at 2494 G; 280 ns at 2515 G; and 271 ns at 2536 G.

 

 

Frequency Shift (MHz)

0.6

0.5

0.4

0.3

0.2

0.1

124

 

 

J;

 

 

2400 2500 2600 2700 2000 2900 3000
Field Strength (G)

3100

Figure VI-6. A comparison of the measured proton sum frequency shifts
(circles) of Fig. VI-5 with the simulated filed profile of the turning points
of the proton sum combination peak lineshape. The ligand hyperfine
parameters for the simulation were refF3.05 A, a=0 MHz, and 6N=110.

The other parameters were same as in Fig. VI—4.

 

 

Frequency Shift (MHz)

0.1

126

   

.,.. / . .n..
-; aax‘kmgfzéfliksf a 3 ~

31.38:.

   

 

 

 

 

 

2500

26m

27‘00 2800
Field Strength (G)

29%

3000

3100

 

 

127

the sum combination frequencies, revealed that the components of the
ligand hyperfine tensor showing the trend of Fig. VI-5 are expected to be
reff=3.05i0.1 A, 6N=11i60, and a s 4 MHz.

Fig. VI-7 shows a comparison of the relative intensities of the sum

combination bands simulated with the ligand hyperfine parameters obtained
for the equatorial protons and Fig. VI-5 at six field positions where the
four-pulse ESEEM experiments were performed. In the calculation of Fig.
VI-7, the lineshapes and intensities were calculated from Eq. VI- 4 and Eq.
VI-6, and r values are fixed to 3/v1. Solid curves are calculated with the
parameters of Fig. VI—6 and the other curves are calculated with
parameters for the equatorially bound water protons. Fig. VI-7 shows that
for low-frequency shift regions, the intensities of the lineshapes
represented by solid curve dominate the spectra. Even though there are
contributions from the simulated sum combination bands for the
equatorially bound water protons (non-solid curves) in the low—frequency
shift regions, the intensities are much less than those represented by solid
curve. Considering the intensities and peak positions of the experimental
SUm combination bands obtained with r values near 3/v1 (Fig. VI—2 (b) and
Other spectra at different field positions), the low-frequency shift peaks
Could not result from ligand hyperfine interactions of the equatorially
bound water protons. The only possible assignment for the low frequency
lirleshapes used to construct the field profile of figures VI-5 and VI—6 is to
the sum combination peaks arising from the axially bound water protons.

Fig. VI-8 shows the lineshapes and relative intensities of the sum
Combination bands predicted with r=2.5/v1 and the ligand hyperfine

parameters of axial (solid curves) and equatorial (other curves) protons at

(a) 2431G and (b) 2515G. Large intensity contributions for the equatorial

 

 

Figure VI-7. Simulated Lineshapes and relative intensities of proton sum
combination band for data of Fig. Vl—5 (solid curves) and the equatorially
bound water protons (the other curves) with r=3/v1. Parameters for the
angle selection were same as in Fig. VI-4. Ligand hyperfine parameters for
the simulation of the solid curves were same as in Fig. VI-6. Ligand
hyperfine parameters for the other curves were reff=2.49 A, a=1 MHz and
6N values are in the figures. r values and magnetic field strengths were (a)
290 ns, 2431 G; (b) 287 ns, 2452 G; (c) 285 ns, 2473 G; (d) 282 ns, 2494
G; (e) 280 ns, 2515 G; and (f) 278 ns, 2536 G. The arrows indicate the
frequency of 2v1. Intrinsic gaussian linewidth (FWHM) of 0.2 MHz were

 

used in the calculation.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

20 20.5 21 21,5 22 22.5 23 23.5 24

Frequency (MHz)

 

 

 

 

 

 

 

 

 

 

 

20 20.5 21 21.5 22 22.5 23 23.5 2‘
Frequency (MHz)

 

131

Figure VI-8. Simulated Lineshapes and relative intensities of proton sum
combination band for the axially bound water protons (solid curves) and
the equatorially bound water protons (the other curves) with r=2.5/v1.
Parameters for the simulations were same as in Fig. VI-7 except 1' values
and magnetic field strengths of (a) 242 ns and 2431 G; (b) 233 ns and 2515
G. The arrows indicate the frequency of Zn. Intrinsic gaussian linewidth

(FWHM) of 0.2 MHz were used in the calculation.

 

"‘ LL. IA; .~:-.- - .,

 

.m—

 

 

132

 

 

 

 

 

 

 

 

24

23.5

20.5 21 21.5 22 22.5
Frequency (MI-11)

20

 

proton sum combination frequencies are expected to overlap the axial
proton sum frequency. Therefore, extraction of the sum combination peaks
of the axial water protons may be difficult in the four-pulse ESEEM
experiments. Simulations with different r values between 2.5/v1 and 3/v1
showed that the relative contribution of the equatorial proton sum
combination band in the axial proton sum combination band region
decreases as r values increase.

The hyperfine interactions for equatorially bound water protons
obtained in this study (refF2.49 A and a S 4 MHz) are slightly larger than
those previously reported from single crystal ENDOR study of
Cu(II)(H20)62+ where the isotropic hyperfine coupling of S 1.2 MHz and
the dipolar couplings of approximately 5.6 MHz were measured.33 The
discrepancy arises from the difficulty of the sum frequency field profile
approach due to the rotational distribution of the equatorial water
molecules at 4.2 K. But the hyperfine couplings of the axially bound water
protons from this study (refF3.05 A, a S 4 MHz) are in good agreement
with those from the ENDOR study where the dipolar couplings of
approximately 3.6 MHz and the isotropic couplings of S 0.5 MHz were
reported.33 The inaccuracy of the isotropic coupling constant of the axial
protons from this study comes from the weak sensitivity of the sum
frequency field profile to the isotropic hyperfine interaction. When four-
pulse ESEEM experiments are performed in the higher field regions of the
Cu(II) EPR spectrum where more than one 0 value is selected, more
COnstraints on the isotropic hyperfine coupling are expected by using the r-
SUppression effect.

To compare the ligand hyperfine couplings of the axial water

protons obtained from the four-pulse ESEEM experiments of

A ‘1“. h. 549$”wa '

Figure VI—9. Electron spin echo detected—EPR spectrum of axial-H20
1i gand—containing Cu(II)TPP. The stimulated echo sequence was used with
T=220 ns and T=2000 ns. Other experimental conditions were microwave
frequency, 8.978 GHz; microwave pulse power, 29.5 W; microwave pulse
wi dth (FWHM), 16 ns; averaging number, 30; pulse repetition rate, 10 Hz;

and temperature, 4.2 K. The arrow indicates the field position where four-

pulse ESEEM experiments were performed.

 

Echo Amplitude

2494

135

1
2994
Field Strength (G)

 

A.“ umm‘r', -‘ ’

 

 

 

136

Figure VI-10. Proton sum combination peak region of four—pulse
magnitude FT-ESEEM spectrum of axial—H20 ligand-containing
Cu(II)TPP. Experimental conditions were microwave frequency, 8.798
GHz; magnetic field strength, 3131 G; r, 225 ns; microwave pulse powers
of fl-pulse, 178 W and 7r/2-pulse, 89 W; averaging number, 30; and pulse

repetition rate, 10 Hz.

 

137

 

 

 

 

 

25 25.5 26 26.5 27 27.5 20 20.5 29
Frequency (MHz)

29.5

30

 

 

 

138

results from a more discrete system, four-pulse

Cu(II)(HzO)62+ with
ESEEM experiments of axial H20 ligation to Cu(II)TPP were carried out.
Fig. VI-9 shows the ESE-EPR spectrum of Cu(II)TPP(HzO)2. The
spectrum shows strong copper hyperfine couplings along g“ and hyperfine
splittings of equatorially coordinated nitrogen nuclei at gi.34a35 The
proton sum combination frequency region of the four—pulse ESEEM
spectrum of the Cu(II)TPP(HzO)2 obtained at g J_ is shown in Fig. VI-10.
In the spectrum, the second harmonic frequency peak of the matrix protons
is found at 26.66 MHz and the sum combination peak of the axial water
protons appears at 27.11 MHz, shifted 0.45 MHz from 2v1. The ESEEM
spectra obtained at g1 will give rise to a lineshape that approximates that
expected for an isotropic electron g-tensor spin system because of the
distribution of 0 values that are selected due to the hyperfine interactions of
the copper and the coordinated nitrogen and because spin packets at several
different magnetic field positions around the external field value will be
effectively excited due to the microwave pulse bandwidth. Using the well
known equation of the sum frequency shift [Av=va+v5-2v1] drived for
iso tropic g— and axial ligand hyperfine tensor spin systems,19
Av=(9/l6)T2/vr, T=3.3 MHz and rear-2.9 A for the axially bound water
protons. This value is in good agreement with the dipolar hyperfine
couplings of the axial water protons obtained from the four-pulse ESEEM
presented above and previous ENDOR33 studies of Cu(II)(HzO)62+.

References

1‘ R. Lontie, Ed., in Copper Proteins and Copper Enzymes, Vol. I. II.
III., CRC Press, Boca Roton (1984).

 

 

 

15-
16-
17-
18.

19.
20.

21.
23.
24.

l 3 9
L. Kevan, in Time Domain Electron Spin Resonance, L. Kevan and
R. N. Schwartz, Ed., Wiley-Interscience, New York, Chpter 8
(1979).
A. J. Hoff, Ed., in Advanced EPR. Applications in Biology and
Biochemistry, Elsevier, New York, Chapters 1, 2, 3, and 6 (1989).
L. Kevan, in Modern Pulsed and Continuous —Wave Electron Spin
Resonance, L. Kevan and M. K. Bowman, Ed., John Wiley & Sons,
New York, Chapter 5 (1990).
W. B. Mims, and J. Peisach, in Biological Magnetic Resonace, L. J.
Berliner, J. Reuben, Ed., Plenum, New York, Vol. 3, 213 (1981).
J. L. Zweier, J. Peisach, and W. B. Mims, J. Biol. Chem, 257,

10314 (1982)
J. McCracken, J. Peisach, and D. M. Dooley, J. Am. Chem. Soc.,

109, 4046 (1987).

J. McCracken, S. Pember, S. J. Benkovic, J. J. Villafranca, R. J.
Miller, and J. Peisach, J. Am. Chem. Soc., 110, 1069 (1988).

F. Jiang, J. Peisach, L. -J. Ming, L. Que, Jr., and V. J. Chen,
Biochemistry, 30, 11437 (1991).

W. B. Mims, and J. Peisach, J. Chem. Phys, 19, 4921 (1978).

K. L. Flagan, and D. J. Singel, J. Chem. Phys, 87, 5606 (1987).
J. B. Cornelius, J. McCracken, R. B. Clarkson, R. B. Belford, and J.
Peisach, J. Phys. Chem, 94, 6977 (1990).

D. Goldfarb, J. —M. Fauth, Y. Tor, and A. Shanzer, J. Am. Chem.
Soc., 113, 1941 (1991).

A. De Groot, R. Evelo, and A. J. Hoff, J. Magn. Reson., 66, 331

(1986).

W. B. Mims, J. L. Davis, and J. Peisach, Biophys. J., 45, 755
(1984).

J. Peisach, W. B. Mims, and J. L. Davis, J. Biol. Chem, 259, 2704
(1984).

A. V. Astashkin, S. A. Dikanov, and Yu. D. Tsvetkov, Chem. Phys.

Lett., 136,204 (1987).
S. A. Dikanov, A. V. Astashkin, and Yu. D. Tsvetkov, Chem. Phys.

Lett., 144, 251 (1988).

E. J. Reijerse, and S. A. Dikanov, J. Chem. Phys, 95, 836 (1991).
A. M. Tyryshkin, S. A. Dikanov, R. G. Evelo, and A. J. Hoff, J.
Chem. Phys, 97, 42 (1992).

J. McCracken and S. Friedenberg, J. Phys. Chem, 98, 467 (1994).
H. -1. Lee, and J. McCracken, submitted to J. Phys. Chem.

W. B. Mims, J. Peisach, and J. L. Davis, J. Chem. Phys, 66, 5536
( 1 977).

C. Gemperle, G. Abeli, A. Schweiger, and R. R. Ernst, J. Magn.
Reson., 88, 241 (1990).

 

25.
26.
27.
28.
29.
30.
3 1.

32.
33.

34.
35-

1 40
A. M. Tyryshkin, S. A. Dikanov, and D. Goldfarb, J. Magn. Reson.,
A105, 271 (1993).
W. J. White, in The Porphins, D. Dolphin, Ed., Academic Press,
New York, Vol. V, Chapter 7 (1978).
J. McCracken, D. -H. Shin, and J. L. Dye, Appl. Magn. Reson., 3,
205 (1992).
G. C. Hurst, T. A. Henderson, and R. W. Kreilick, J. Am. Chem.
Soc., 107, 7294 (1985).
T. A. Henderson, G. C. Hurst, and R. W. Kreilick, J. Am. Chem.
Soc., 107, 7299(1985).
C. A. Hutchison, and D. B. McKay, J. Chem. Phys, 66, 3311
(1977).
C. Germperle, G. Abeli, A. Schweiger, and R. R. Ernst, J. Magn.
Reson., 88, 241 (1990).
G. M. Brown, and R. Chidambaram, Acta. Cryst, B25, 676 (1969).
N. M. Atherton, and A. J. Horsewill, J. Molec. Phys, 37, 1349
(1979).
W. C. Lin, in The Porphyrins, D. Dolphin, Ed., Academic Press,
New York, Vol. IV, Chapter 7 (1978).
J. Subramanian, in Porphyrins and Metalloporphyrins, K. M. Smith,
Ed., Elsevier, Amsterdam, Chapter 13 (1975).

 

 

VII. ELECTRON SPIN ECHO ENVELOPE MODULATION
STUDIES OF COPPER(II)-PTERIN MODEL COMPLEXES

1 . Abstract

The Electron Spin Echo Envelope Modulation (ESEEM) technique
of pulsed EPR spectroscopy has been used to characterize the ligation of
pteridine ligands to Cu(II) in a variety of complexes prepared in aqueous
and non-aqueous solvents. These studies were aimed at understanding the
structural relationship between the Cu(II) and pterin cofactors in
Phenylalanine Hydroxylase (PAH) from Chromobacterium Violaceum
where previous continuous wave EPR results have provided evidence for
equatorial binding of 6,7-dimethyltetrahydropterin to the metal through N-
5 - ESEEM studies of the protein show deep 1"'N modulations originating
from two equatorially bound histidyl imidazole groups strongly bound to
Cu(II), but provided no evidence for coordinated pterin. For the model
compound, Cu(II)(ethp)2(HzO)2 (ethp = 2-ethylthio-4-hydroxypterin), X-
ray crystallographic studies showed bidentate coordination of ethp through
0—4 and N-5. ESEEM spectra obtained for this compound show intense,
Sharp peaks at 0.6, 2.4, 3.0 MHz and a broad peak at 5.4 MHz. The
Spectrum is indicative of two identically coupled 14N nuclei and is assigned
to N —3 of the coordinated ethp ligands. Similar spectra were obtained for a
Cu(II)—folic acid (FA) complex at pH=9.5 in aqueous media. ESEEM

141

 

 

 

 

 

142

studies of compounds where mixed equatorial-axial ligation of the pterin
moiety has been found by X-ray analysis, show little or no 14N
contributions in their ESEEM spectra. This complex includes
Cu(II)(bpy)(PC) where N—5 is equatorially bound while O—4 is axially
coordinated (PC = 6—carboxy pterin). Because the 14N ESEEM response
from equatorially coordinated pterin is intense and the peaks occur in a
spectral region where there would be little interference from the protein,
the experimental results are most consistent with a mixed equatorial-axial

l i gation of this cofactor at the Cu(II) site of PAH.

2 - Introduction

Phenylalanine hydroxylase (PAH)1,2 catalyzes the hydroxylation of
phenylalnine to tyrosine in the presence of molecular oxygen and a reduced
pterin cofactor. One atom of the oxygen molecule is in corporated into
tyrosine and the other forms water. In the reaction, four electrons are
transfered to molecular oxygen, two electrons are from breaking of the C-
H bond of phenylalanine and the other from a reduced pterin cofactor.
The overall reaction is depicted in Scheme 1. Mammalian PAH contains a
single iron per monomer1 while PAH isolated from Chromobacterium
Violaceum has copper.2 The iron is required for the enzymatic activity in
all mammalian pterin-dependent monooygenases such as PAH, tyrosine
hydroxylase, and tryptophan hydroxylase.3-5 In both mammalian and
bacterial enzymes, the metal ions must be reduced to lower oxidation states
(F e( I I), Cu(l)) to show activity.2,6,7 It has been thought that in both the

enzymes the transition metal ions play a critical role in the overall catalytic

reaetion. However, recent studies have shown that copper is not required

¥

 

143

for bacterial PAH enzymatic activity.8 In addition, evidence that Cu(II)
inhibits or regulates catalysis was presented. The overall mechanism

including the redox chemistry and ligation structure of the copper is not

clear for bacterial PAH.

Scheme I.

02 o
0 g Phe Ty r 3 2
HN 4 4a 5 U ”N
——>
I PAH
2 \ 7 \ \
HZN If N HZN N N
H

0

H20 N

#4 i /
J\ /

HZN N N

H
The study of the coordination of Cu(II) is important for
understanding the mechanism of inhibition and the redox chemistry of
bacterial PAH. S—band continuous wave (CW) EPR studies of 5—14N and 5-
1 5 N—labeled 6,7-dimethyltetrahydropterin bound to PAH have revealed that
the pterin cofator is coordinated to Cu(II) via N-5 of the pterin.9 Electron
Spin echo envelope modulation (ESEEM)11-13 studies of the enzyme have

Shown two equatorially bound histidyl imidazole groups strongly bound to

Cu(l I), but provided no evidence for coordinated pterin.10 EXAFS studies

¥

 

 

144

of the enzyme has also suggested a Cu(II) coordination of two histidines
and two additional O/N-donor groups.1421

In this investigation, we undertake ESEEM studies of a variety of
Cu(II)—pterin model complexes prepared in aqueous and non-aqueous
solvents to provide a better understanding of the ligation of pteridine

ligands to Cu(II).

3 . Experimental

Cu(II)(ethp)2(HzO)2 [ethp = 2-ethylthio-4—hydroxypterin (Fig. VII-1
(c))] was prepared as described by Perkinson et al.14b Other complexes
were synthesized in water. Cu(II)(FA)2(HzO)2 [FA=folic acid (Fig. VII-1
(d))] was made by adding 10-fold excess of FA into 2.9 mM Cu(II)(NO3)2.
The final pH of the solution was adjusted to 9.5 with NaOH. Cu(II)(bpy)
[bpy=2,2'-bipyridine] complexes were made by adding a 1:1 stoichiometric
ra ti o of bpy to an aqueous solution of Cu(II)(NO3)2.
Cu(II)(bpy)(PC)(HzO)2 [PC=6-carboxy pterin (Fig. VII-1 (b))] was made
by adding an 1.1:1 molar ratio of PC to the Cu(II)(bpy) solution and
adjusting the pH to 7.5. Cu(II)(bpy)(PC)(im) [im=imidazole] was prepared
by adding a stoichiometric amount of imidazole to an aqueous solution of
Cu(II)(bpy)(PC)(HzO). The pH of the resulting solution was adjusted to 8.
C u(II)(ethp)2(H20)2 powder samples were dissolved in a DMF
[dimethylformamideVi‘oluene (50/50) mixture prior to freezing in liquid
nitrogen for ESEEM experiments. The presence of an equal volume of
to! uene promoted formation of a glass upon freezing. The other aqueous
Compounds were mixed with an equal volume of ethylene glycol prior to
f“I'eezing in liquid nitrogen. The CW-EPR spectrum of Cu(II)(ethp)2(HzO)2

 

 

 

145

showed a clean axially symmetric electronic g-tensor dominated by a single
speciesl‘ib The CW—EPR spectrum of Cu(II)(FA)2(HzO)2 showed also a
clean spetrum dominated by a single species (gj_=2.114, gH=2.373, and
AH=111G) . About 70% and 50% of populations of Cu(II)(byp)(PC)(HzO)
and Cu(II)(byp)(PC)(im) were judged by Odani et al. in their aqueous

solutions, respectively. 14b,25

5
4a N R
HN3 4 2
k \
R1 N N
1 8

(a) pterin (R1=NH2, R2=H)

(b) PC = 6-carboxy pterin (R1=NH2, R2=COOH)

(c) ethp = 2-ethylthio-4—hydroxypterin (Rl=SCH2CH3, R2=H)

((1) FA = folic acid (R1=NH2, R2=CH2NH-C6H4—CONH-
CH(COOH)CH2CH2-COOH)

Figure VII-1. Pterin derivatives.

ESEEM experiments were executed on a home—built pulsed-EPR
spectrometer which has been described in detail elsewhere15 and in chapter
IV. A three—pulse (7r/2-r-7r/2-T/7r/2) microwave pulse sequence was used to
collect the ESEEM data. The length (full-width at half maximum) of all
three microwave pulses was 16 ns. A two step phase cycle, +(0,0,0),
+ (71.37130), was used to eliminate unwanted echo modulations. Three-pulse
ESEEM data were collected as a function of T, the time interval between
Second and third pulses. Fourier transformation of the time domain

ES EEM data was done using a modified dead-time reconstruction method

 

 

146

developed by Mims.16 Simulation programs were written in C (Symantec)
using a density matrix formalism”,18 and executed on a Macintosh 1ch

microcomputer (Apple Computer Co.).
4. Results and Discussion

Fig. VII-2 shows the time—domain ESEEM and FT—ESEEM spectrum
of Cu(II)(ethp)2(HzO)2 in DMF/Toluene as obtained in the g _L region. The
spectrum shows four major frequency components, three narrow lines at
0.6, 2.4, and 3.0 MHz and a broad peak at 5.4 MHz as well as weaker lines
at 1.2, 1.8, 3.6 and 6.0 MHz. These ESEEM features are similar to those
reported for Cu(II)—2-methylimidazole where deep 14N modulations were
assigned to the remote 14N of the equatorially coordinated 2-
methylimidazole liganle. 14N ESEEM spectra of this type that show three
strong sharp lines, where the two lower frequencies add to give the third,
and one broad line positioned near 2v1+Aiso, where v1 is the nuclear
Larmor frequency and Aiso is the isotropic hyperfine coupling constant,
are characteristic of a coupling regime known as "exact cancellation".19a20
Exact cancellation occurs when the isotropic hyperfine coupling is
approximately equal to twice the nuclear Zeeman interaction and the
electron-nuclear dipolar interaction is weak.20 As a result, for one of the
electron spin manifolds the hyperfine and nuclear Zeeman interactions
cancel each other so that the energy level splittings are determined by the
14N nuclear quadrupole interaction (NQI) as in Fig. VII-3. Because the
NQI is independent of the magnetic field, this spin manifold gives rise to
three sharp lines at low frequency. For the other electron spin manifold,

the hyperfine and nuclear Zeeman interactions are additive and give rise to

¥

 

147

Figure VII-2. (a) Three-pulse ESEEM data and (b) cosine fourier
transformation spectrum of Cu(II)(ethp)2(HzO)2. Experimental conditions
are magnetic field strength, 3265 G; microwave frequency, 9.406 GHz,

microwave power, 63 W; scanning number, 30; pulse repetition rate, 80

Hz; r, 140 ns; and temperature, 4.2 K.

 

148
vrrv—rv'vrr

 

T I‘V’Y’V Vgr 7

r7! V

Y I T77 Y Y

Y T I V V Y

Y Y’Y’ngit

Y T I Y

 

 

 

 

 

L l 1_L,L L_171 A L,‘

l l l

 

b
slslslll l L,L,L,L,L LngJ 1

 

1001'

MDDHHJQZ< 010m

0

10

tau+T(usec)

 

 

 

 

 

 

 

 

 

llllllllll

 

 

320

A.:.ms>uHmzmezH

10

FREQUENCYiMHZ)

 

 

 

 

149
M5 M1 ,’ ‘E
I
1 ,'
,/————’ broad
1/2 ,’ 0 line
fi\__ _ x
\
x —1
~——~.\ i
\ I

' ‘ I
-1/2 ’I \‘ I E
— l
x ,——I. NQI

\ ‘1 ,’ ‘\ l 1 lines

 

 

 

 

am + I |+ layperrmel + “ml

Figure VII—3. Electron spin enegy level diagram for 14N near eaxct
Cancellation regime.

broad peaks for a randomly oriented sample. (Fig. VII—3). Another
feature in spectrum of Fig. VII-2 is the appearance of four narrow and
weak lines at 1.2, 1.8, 3.6 and 6.0 MHz. These peaks are combination bands
0f the three strong peaks at 0.6, 2.4, and 3.0 MHz that result from the
Product rule when more than one nucleus, with electron-nuclear couplings
that result in strong ESEEM, is coupled to the same electron spin“),21

The crystal structure of Cu(II)(ethp)2(H20)2 has been determined
and is shown in Fig. VII-4. The ethp ligand is bidentate and coordinated to

 

1;.- +59" ’-.ib-_ -§_

150

Cu(II) through equatorial N-5 and O-4.14 For ESEEM spectra of
Cu(II)(ethp)2(HzO)2, the hyperfine coupling of the directly coordinated

HOH
CH3CH25)/N\ Om“. ....... 'C ........ “(j N
N / /N/ u\o \N/KSCH2CH3
N\ HOH

Figure VII-4. Crystal structure of Cu(II)(ethp)2(HzO)2.

N-5 nucleus is expected to be too large to give rise to envelope modulation
as found for Cu(II)-imidazole. There are three possible nitrogens of the
equatorially-bound ethp ligand that could give rise to the modulation of
Fig. VII-2. The ESEEM pattern of Fig. VII—2 is dominated by hyperfine
coupling from N-3. The reasons for this assignment will be explained later
with a comparison of the ESEEM features of Cu(II)(ethp)2(HzO)2 with
those of Cu(II)(FA)2(H20)2. Quantitative analysis of the ESEEM data from
Cu(II)(ethp)2(HzO)2 was carried out with computer simulations based on
the density matrix formalism”,18 The spin Hamiltonian used to describe

electron-nuclear coupling for 1‘iN-ESEEM is given by

H = - gnfinBo-I + S'_A_-I + I~Q~I [VII-l]

Where S is the electron spin angular momentum operator, Bo is the
external magnetic field vector, A is the hyperfine interaction tensor, I is
the nuclear spin angular momentum operator, gn is nuclear g-value, and fin

is the nuclear magneton, and Q is nuclear quadrupole interaction tensor.

151

When 14N ESEEM shows near-cancellation character, the isotrop
hyperfine coupling constant can be estimated as Aiso _: 2v1 and the N(
parameters can be estimated from the frequencies of the three sha

nuclear quadrupole resonance 1inesl9 using Eq. VII-2.

‘ vi = (3/4)e2qQ (1 :t n/3) [VII-2]
Vo = (1/2)CZQQ0

q, in Eq. VII-2, is the principle value of electric field gradient tensor”,

is the nuclear quadrupole moment, and n is the asymmetry paramete
Simulations for the ESEEM of Cu(II)(ethp)2(HzO)2 were started usir

estimated hyperfine and NQI parameters of Aiso=2.0 MHz, eZqQ=3.6 MH
and n=0.30 and the simulation parameters were refined until a best fit
the experimental result was obtained. Fig. VII-5 shows the time doma
and FF spectra simulated with the parameters which are given in t1
caption of Fig. VII-5. In the simulation, it was assumed that tv
magnetically equivalent nitrogens are coupled to Cu(II). The simulatir
results indicated that the crystal structure of Cu(II)(ethp)2(H20)2 was st
valid in DMF/Toluene media.

To study the ligation of pterin derivatives of Cu(II) in aqueo
solution, ESEEM experiments on Cu(II)(FA)x(HzO)[6-x or 6-2x]~ Sampl
were made as described in the experimental section. A typical time-doma
ESEEM pattern and F1“ spectrum collected in the g_1_ region of the Cu(l
EPR spectrum are shown in Fig. VII-6. The spectrum shows very simil
features to those obtained from Cu(II)(ethp)2(H20)2. Three strong sha
Peaks were found at 0.9, 1.9, and 2.8 MHz along with smaller bands at 3
, 4.7, 5.2, and 5.6 MHz. ESEEM data obtained at a second magnetic fie

152

Figure VII-5. (a) Time domain 14N-ESEEM simulation and (b) Fourier
transformation. Hamiltonian parameters for the simulations are Axx=2.12;
Ayy=2.12 MHz; Azz=2.60 MHz; e2qQ=3.63 MHz; n=0.30; Euler angles,
a=8 lo, 6:900, y=00; magnetic field strength, 3265 G; and r=140 ns. Two
14N nitrogen contribution to ESEEM was assumed.

EON AMITUOE

INTENSITY (l.u.1

246

153

 

p

 

r a .

rY—V—TYVVVIVY'VIVIVVIIUIIIVVVIVIYIIIIIVYVTYY’Y'IIV'VI'

 

 

 

4...!.‘xxliitiLLLAilxilxliix-l....l....l.xi.l..L

 

0

1 2 3 4 5 5 7 3 9 10

mm tune)

 

 

 

 

 

 

 

154

Figure VII-6. (a) Three-pulse ESEEM data and (b) cosine fourier
transformation spectrum of Cu(II)(FA)2(HzO)2. Experimental conditions
are magnetic field strength, 3100 G; microwave frequency, 8.926 GHz,
microwave power, 63 W; scanning number, 30; pulse repetition rate, 30

Hz; 1', 152 ns; and temperature, 4.2 K.

155

 

r I V 771 T’Y’r’r’r’r T
1

I V

I Y

r r T’Y Y

r TiTTT T'I V V I

V T V I

 

‘—

.—-‘H‘AA‘AAL.

 

 

 

 

Y Y’T’T I

J lnj l I

lsJ l

l 1,J,L 1 l

A L L l I

All

LzAUL.L,L,L 1 l

 

 

100

mDDHHndzxa 010m.

10

tau+Tlusecl

 

V'U'U"

IIITYT‘I

‘TTUYTVII'

 

 

 

 

llll

AIJILLLLAAIILI

111

ALLLLLALLLLJLIIILLIL

 

 

5

:.me>HHmZMHZH

10

FREQUENCYiMHZ)

156
strength showed that the 5.2 MHz peak was shifted by approximately 2Av1,
twice the change in the nitrogen Larmor frequency. The frequencies of the
other ESEEM lines remained the same. These observations are
characteristic of the "exact cancellation" regime of 14N hyperfine coupling.
Simulations using parameters of Aiso=2.45 MHz, Adip=0.12 MHz,
eZqO=3.05 MHz, and r1=0.57 adequately described the three strong sharp
lines at 0.9, 1.9, and 2.8 MHz and the broad peak at 5.2 MHz. When two
equivalent nitrogens were assumed to be coupled to Cu(II), the
experimental modulation depth and the other combination bands were
properly predicted. These hyperfine parameters are close to those obtained
from our analysis of Cu(II)(ethp)2(HzO)2 where an Aiso of 2.28 MHz and
Adip of 0.16 MHz were found. Hence, it can be concluded that two FA
ligands are coordinated to Cu(II) through equatorial N-5 and O-4 as in the
Cu(II)(ethp)2(HzO)2 case. The differences in the NQI parameters are due
to the different groups substituted at the C-2 position.23 Because the
electronic environments of N-l and N-8 are similar for ethp and FA, if the
modulations arose from hyperfine couplings to these nuclei, the ESEEM
results would show similar NQI parameters for Cu(II)(ethp)2(HzO)2 and
Cu(II)(FA)2(HzO)2. Because our experimental results show completely
different spectra for two samples, the modulations must be dominated by
hyperfine coupling to N-3.

Time-domain ESEEM data of Cu(II)(bpy)(PC)(HzO), where X-ray
crystalograpic analysis has shown that the N-S nitrogen, O-4 oxygen, and
carboxylate oxygen are bound to Cu(II) as shown in Fig. VII-7,14,24 is
shown in Fig. VII-8. For this compound only shallow modualtions are
observed. The large change in the 1‘iN-ESEEM observed for this

compound shows that N-8 can not be the source of the deep modulations

157

observed for the folic acid and ethp complexes and strenghens our above
assignment to N-3. The equatorially bound water molecule of

Cu(II)(bpy)(PC)(HzO) can be replaced by imidazole under slightly basic

 

 

Figure VII-7. Crystal structure of Cu(II)(bpy)(PC)(HzO).

conditions.25 Time and frequency domain ESEEM results obtained for
Cu(II)(bpy)(PC)(im) are shown in Fig. VII-9. These ESEEM features are
identical to those of Cu(II)(dien)(im) [dien = diethylenetriamine] where
imidazole is equatorially bound to Cu(II).19,25 The modulation arises from
hyperfine couplings between the remote nitrogen of equatorially bound
imidazole and Cu(II).19 Fig. VII-9 shows that the magnetic couplings of the
remote nitrogen of equatorially bound imidazole are not perturbed by the
coordination of PC. ESEEM patterns obtained from
Cu(II)(dien)(FAXHzO) and Cu(lI)(bpy)(pterin)(H20)2 were the same as
those found for Cu(II)(bpy)(PC)(HzO). 14N-ESEEM spectra of
Cu(II)(bpy)(pterin)(im)(H20) were also identical to those of
Cu(II)(bpy)(PC)(im) and Cu(II)(dien)(im). Therefore, FA and pterin
ligands seem to be coordinated to Cu(II) equatorially through N-S and

158

Figure VII-8. Three—pulse ESEEM data of Cu(II)(bpy)(PC)(HzO).
Experimental conditions are magnetic field strength, 3050 G; microwave
frequency, 8.774 GHz; microwave power, 36 W; scanning number, 100;

pulse repetition rate, 30 Hz; r, 155 ns; and temperature, 4.2 K.

 

 

159

 

 

 

vrvww

 

. l innmlnnull

 

 

100

mQDHHiEz/x OIum

10

tau+T(usecl

Figure VII-9. (a) Three-pulse ESEEM data and (b) cosine fourier
transformation spectrum of Cu(II)(bpy)(PC)(im). Experimental conditions
are magnetic field strength, 3050 G; microwave frequency, 8.897 GHz,
microwave power, 45 W; scanning number, 30; pulse repetition rate, 30

Hz; 1', 155 ns; and temperature, 4.2 K.

161

rttf'rr

'Yr‘r—‘TVY

YTY—rTVrr'I'YV

 

 

L

lLLLLl

[411

[L1

annlnnnmln

lllJeLlL

llJLLlllll

 

 

100 P

mQDHHJQZ< OIom

1C

tau+T(usec)

 

 

407_

A.3.me>hHmzmHZH

Si 1.

FREQUENCYiMHZ)

162

axially through O-4 in a similar fashion to PC in crystallin
CU(II)(bpy)(PC)(H20).

Previous ESEEM studies of PAH have revealed that the Cu(II) i
coordinated to two histidine ligands in an equatorial fashion, but hav
provided no evidence for the coordination of the pterin cofactor.10 CW
EPR studies of identical samples showed equatorial coordination of N-5 c
6,7-dimethyltetrahydropterin cofactor. Hence, the present ESEEM studie
of our Cu(II)-pterin model complexes are consistent with the equatoria
coordination of N-S and axial coordination O-4 of the pterin cofactor t
Cu(II) at the active site of the enzyme. This study shows that the ligation c
oxidized pterin derivatives to Cu(II) prefers axial O-4 and equatorial N-
coordination when two nitrogen donors are present. This bidentat
coordination of the pterin cofactor may prevent the formation of a 43
hydroxypterin intermediate and bear some reponsibility for the role c
Cu(II) as an inhibitor of PAH.

References

l. D. Gottschall, R. F. Dietrich, S. J. Benkovic, and R. Shiman, J. Bio
Chem, 257, 845 (1982).

2. S. O. Pember, J. J. Villafranca, and S. J. Benkovic, Biochemistry,

25, 6611 (1986).

D. B. Fischer, R. Kirkwood, and S. Kaufman, J. Biol. Chem, 247,

5161 (1972).

T. A. Dix, and S. J. Benkovic, Biochemistry, 24, 5839 (1985).

D. M. Kuhn, W. Lovenberg, in Folates and Pterins, R. L. Blakely, :

J. Benkovic, Ed., Vol.2, Wiley-Interscience, New York, 363 (1985]

D. E. Wallick, L. M. Bloom, B. J .- Gaffney, and S. J. Benkovic,

Biochemistry, 23, 1295 (1984).

J. J. A. Marota, and R. Shiman, Biochemistry, 23, 1304 (1984).

b)

so?»

14a.

14b.

15.
16.
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18.

20.
21.

22.
23.
24.
25.

163

R. T. Carr, and S. J. Benkovic, Biochemistry, 32, 14132 (1993).
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J. McCracken, S.Pember, S.J. Benkovic, J.J. Villafranca, R.J. Miller,
and J. Peisach, J. Am. Chem. Soc. 110, 1069 (1988).

L. Kevan, in Time Domain Electron Spin Resonance, L. Kevan, and
R. N. Schwartz, Ed., Wiley-Interscience, New York, Chapter 8
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A. J. Hoff, Ed., in Advanced EPR. Applications in Biology and
Biochemistry, Elsevier, New York, Chapters 1, 2, 3 and 6. (1989).
L. Kevan, in Modern Pulsed and Continuous -Wave Electron Spin
Resonance, L. Kevan, and M. K. Bowman, Ed., John Wiley & Sons,
New York, Chapter 5 (1990).

N. J. Blackburn, R. W. Strange, R.T. Carr, and S. J. Benkovic,
Biochemistry, 31, 5298 (1992).

J. Perkinson, S. Brodie, K. Yoon, K. Mosny, P. J. Caroll, T. V.
Morgan, and S. J. N. Burgmayer, Inorg. Chem, 30, 719 (1991).
J. McCracken, D. H. Shin, and J. L. Dye, Appl. Magn. Reson., 3,
205 (1992).

W. B. Mims, J. Magn Reson., 59,291 (1984).

W. B. Mims, Phys. Rev., BS, 2409 (1972).

W. B. Mims, Phys. Rev., B6, 3543 (1972).

W. B. Mims, and J. Peisach, J. Chem. Phys, 69, 4921 (1978).

K. L. Flagan, and D. J. Singel, J. Chem. Phys, 87, 5606 (1987).
S. A. Dikanov, A. A. Shubin, and V. N. Parmon, J. Magn. Reson.,
42, 474 (1981).

E. A. C. Lucken, Ed., in Nuclear Quadrupole Coupling Constants,
Academic Press, New York, 156 (1966).

F. J iang, J. McCracken, and J. Peisach, J. Am. Chem. Soc., 112,
9035 (1990).

T. Kohzuma, H. Masuda, and O. Yamauchi, J. Am. Chem. Soc.,
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Soc., 114, 6294 (1992).

 

APPENDIX

A1. Hamitonian Matrices for Hyperfine (HFI) and Nuclear Zeemai

Interactions (NZI)

Table Al-l. Matrix elements of HFI and NZI of I=1*

 

 

 

 

 

 

 

 

 

* x=(hmsAxx'gnfinBo)SiHBCOSd)
Y=(hmsAyy'ganBo)SiflfiSin)
Z=--"(hmsAzz'gnfinBo)COS0

164

165

Table A1-2. Matrix elements of HFI and NZI of I=3/2

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

m1 3/2 1/2 -1/2 -3/2
3 3 .
3/2 ‘2 Z %(X-1Y) 0 0
3 . 1 .
1/2 %(X+1Y) 2 Z X-lY — 0
. 1
-1/2 0 X+1Y :2“ Z ‘12—3-(X4y)
3 . 2
.-3/2 0 O lé:(X-l-1Y) -2 Z
Table A1-3. Matrix elements of HFI and NZI of I=5/2
m1 5/2 3/2 1/2 -1/2 -3/2 -5/2
5 5 .
5/2 2 gang) 0 0 0 0
3/2 \é—§(X+iY) 3‘2 \12(X-iY) 0 0 0
1/2 0 J2(X+iY) $2 %(X-iY) 0 0
-1/2 0 o %(X+iY) $2 \f2(X-iY) 0
-3/2 0 0 0 J2(X+iY) g2 #09”:
1 5
-5/2 0 0 O 0 #004“ 52

 

 

 

 

 

 

 

 

 

1 6 6
Table Al-4. Matrix elements of HFI and NZI of I=7/2

7/2 5/2 3/2 1/2 -1/2 -3/2

7
2Z ‘IYJ o 0 o o

5,
+iY) 2Z \/§(x-1Y) 0

2 .
0 «Emma 2Z '00
1
o 2Z

0 2(X+iY)

2
o '22

o \/§<x+m

0

-5/2
0
0
0
0
0
Vim-m

5

-§Z

(X+1Y)

—7/2

 

 

167

A2. M Matrices for the Hamiltonian containing only Hyperfine and Nucle

Zeeman Interactions

Table A2-1. M matrix elements for I=l*

 

 

 

 

 

 

 

 

 

 

m1 0 -1
1 1 +1 J- ' 1 +1
2(cosy ) V2511” 2(-cosy )
0 Lsin cos -Lsin
‘15 Y Y \5 Y
1 1 . 1
-1 §(-cosy+ 1) \/—2$mY §(cosy+ 1)
* See Eq. III-61 and Text.
Table A2-2. M matrix elements for I=3/2
ml 3/2 1/2 -1/2 -3/2
3/2 cos3(%) -\/3cos2(§-) \f3cos(%) -sin3(\2—()
x Sin(%) x sinzé)
1/2 (500.2%) cos(§)13cos3(§) sin(§)[3sm2(§) 150086)
x sing) -21 —21 x sinzé)
-1/2 «icos(%) sin(%)[3sin2(%) cos(%)[3cos3(%) -\/3cosz(%)
x sinzé) —2] —2] x sing)
on sin3(;) «Ecosrp «Boca; c0902")
x sinzé) x sing)

 

 

 

 

 

168

Table A2-3. M matrix elements for I=5/2

 

 

 

 

 

 

 

 

 

 

 

 

ml 5/2 3/2 1/2 -1/2 -3/2 -5/2
5/2 cossé) -\[5-cos4(%) Mcos3é) ~flcoszé) VgCOSCZX) -sin5(%)
x sing) x 9,1112%) x sin3(§) x sin4(%)
3/2 V's—coy“; cos3(%)[5 ficos2(':‘)x V200“? x sin3(%)[5 N’s—cos(;
. Y Y . . Y . Y . Y
X Sing) cosz(g)-4] smé‘flS sm2(;2‘)[5 sm2('2')-4] X sm4(”2')
sin2(‘:‘)-2] c032%)-2] .
1/2 \ll—O-cos3(21) -‘/2-cos2('§') cos(g'fllo -sin(§)[10 ficosé") x meoszé
x sin2(g‘) x sax-2‘55 cos4(§)-12 swig-pm sin2(‘:‘)[5 x sin3(%)
sin2(2ly2] cosz(%)+3] sin2%)+3] cos2(%)-2]
-1/2 ficoszé) ficosé) x sinéfllo cos(?)[lo ficoszéht ”005391)
X sin3(2X) sin2(%)[5 sin4(%)-12 cos4(g‘)-12 sing)” X sinzé")
coszé)-2] sin2(‘2Y‘)+3] coszé)+3] sin2%)-2]
-3/2 ficosé) -sin3(21)[5 N/Zcosé) x l—N/Zcos2(%)x cos3(%)[5 -'\/—5—cos4(%)
, Y
X sin4(%) sin2(éy‘)-4] sin2("2Y')[5 sin(g')[5 cosz(§)-4] X sm(E)
cosz(%)-2] sin2%)-2]
-5/2 sinsé) ficosé) mooszé) mcosag) \l-S-cos4é) 00556)
X 81041;) X sin3(%) X sinzé) X sin(2l)

 

 

 

 

 

 

 

 

 

 

 

 

 

 

l 69
Table A2-4. M matrix elements for I=7/2 .
m1 7/2 5/2 3/2 1/2 -1/2 -3/2 ~5/2 ~7/2
7/2
cos7(;) -J7cos6(% ficossg) - 35cos4(%) ficos3é) -¢_lcosz(%) «700M; sin-7(9)::
x sing) x sinzé) x sin3(%) x sin4(:,}) x sinsé') x sinéé)
5/2
«Flcos6(%) cos5(%)[ ficos4é) «Ecos3(%) \Ecoszg') ficosg) sin5(%)[ V7008<§
x sing) 6cosz(% x sing) x sinzé‘ x sin3(% x sin4(% 63in2(% x sing;
-5] ~51
1731:1202! -)-21 [7cosz(%)-4] [7sin2(%)-4] [7c032(%)-2]
«(Z—leossé) -ficos4(%) cos3(§)[ J—Scoszé) «TS-cos(? -sin3(%)[ ficosé) -flcos21
x sinzé‘) x sing) 2lcos4(%) x sing“) x sinzé) 213in4(%) x sin4(%) x sinsg
[7sin2 (21.)2] ~30c032(2) [3cos4(%)-2] [331n4(%)-2] -30sin2(% [7cosz(%)-2]
+101 +10] 1
ficos4€ 45cos3(,§) mcoszé) cos(%)[ sin(2 Y)[ 1,3008% 45c032(%) 43500331
x 1.119%) x 31.12%) x sing) 55c036(%) 5531n6(‘2-) x sinzé) x sin3(-:‘) x sin”;
[7c032(%)-4] [3cos4('2Y*)-2] -60¢os4(§")+ -6031n4(%)+ [33in4(%)-2] [7sin2(%)-4]
30¢os4(%—)4]3031n4(:‘-)4]
-1/2 I
ficos3é) -1/5cosz(g‘) mcosé) ~sin(2 r)[ cos(: —)[ -V—50032(%) 45cos3(::*) ~ 350034-
x sinfig) x sin3(%) x sinzé) 55sin6(§) 55C086(§) x sing) I Sinzé) x ““39:
[79181; -)41 [331mg —)2] -60sin4(;: + -60¢os4(: -)+ [3cos4(%)-2] [7cosz(%)-4]
303in4(-:')-4] 30cos4(y “Zn-4]
'-3/2
ficoszé) ficosé) sin3(%)[ 4T5cos(%) 47500325) 0083(%)[ 6003‘”? 4500351
x sinsé) x sin4(%) 213in4("2Y“) x sinzé) x sing") 21cos4(%) x sing) x sin2(%
[7c032(%)-2] -303in2(:7 [3sin4(‘:‘)-2] [3cos4(%)-2] -30cosz(g‘ [7sin2(%)-2]
+10] ‘ +10]
-S/2
\f7cos6é) -sin5(y §)[ 45cc“; —V5cosz(':- 1,5cos3(% chosd'é) cossg —)[ -‘J7c0361
xsin(*:-) 61111125) x sin4(g-) x sin3(g) xsinzé) x sing) 6cosz(§) x sin(%
-5] -5]
[7cosz(%)-2] [7sin2(g‘)-4] [7cosz(;:‘)-4] [7sin2(g')-2]
-7/2
sin7('2- ficoséé ficoszé 430083é) ficos‘té) 500356) ficoséé) 0087(%
x sin('2" x sinsé x sin4(,f:‘) x sin-3%) x sin-(3:: ) x sing

 

 

 

 

 

 

 

 

170

A3. Hamiltonian Matrices for Nuclear Quadrupole Interaction (NQI) alon

Hyperfine Interaction Principle Axis System (HFI PAS)

Table A3-1. Matrix elements of NQI of I=l*

 

 

 

 

 

 

 

 

 

 

 

m1 1 0 -1
1 %(Q11+Q22) ng3-1Q23) §(Q11-Q22)
_ +Q33 4iQ12 g
0 1é-i-(Q13’riQ23) Q11+Q22 \éz-Qw‘rins)
- 1 %(Q1 I’M «75(‘Q13‘iQ23) %(Q11+Q22)
+iQ12 +Q33
* Qij values are listed in Section A3-1.
Table A3-2. Matrix elements of NQI of I=3/2
m 3/2 1/2 -1/2 -3/2
§(Q +(b2) . \5 .
3/2 4 11 V5(Q13-1Q23) ~2—(Qll-Q22) 0
9
+ZQ33 _ -i\/§Q121 L .
. Z + 73
1/2 \I'B'(Q13+1Q23) 4(Q11 Q22) 0 7(Qii-Q22)
+1{233
r 4 __ -i\/3Q12
7
-1/2 @0102.) 0 1<Qll+Qzfl V3(-Q13+iQ23)
1
+i\/3Q12 +ZQ33 ,
J 3
‘3’2 0 QQll-QZZ) \f3-(-Q13-iQ23) Z<Q191+Q22)
+i\/§Qig +ZQ33 t.

 

 

 

171

Table A3-3. Matrix elements of NQI of I=5/2

 

 

 

 

 

 

 

 

m1 5/2 3/2 1/2 -1/2 —3/2 -5/2
5
5/2 3(Q11+Q23) 242((213 izl—an-sz) o o o
25
+4033 -ng3) _ile2
Eu) +0”) fl
3/2 2V§(Qis+ 4 11 -- 2V§<Qis 2“?“ 0 0
. .2Q .
1023) 4 33 -an3) .Q,,)
-i3\/2Q12
0 fl 3
1/2 ngl' 2V2(Q13 4(Qll 0 #0211 0
1
Q22) 'iQ23) +Q22)+5Q33 -sz)
+i‘/—IT)Q12 -i3\/2Q12
17
-1/2 0 #(Q11 0 4—(Qll 2\/§(-Q13 @(Qn-sz)
1
4222) +Q22)+4Q33 +iQ23) 4%le
Hat/5012
3 2 12
-3/2 0 o —2—(Qn Ni(-013 4<Q11+Q33> 2‘/5(-Q13
.2Q _
.Q22) -in3) 4 33 +1Q23)
+i31/2Q12
5
-5/2 0 0 0 @(Qll'Qfl) 2V5(-Q13 3011142217
25
m/EQ12 -in3) +7033

 

 

 

 

 

 

 

 

 

 

172

Table A3-4. Matrix elements of NQI of I=7/2

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

m] 7/2 5/2 3/2 1/2 -1/2 -3/2 —5/2 -7/2
7/2 i‘QlWQZ?) 347(Q13- «751(011- o o o o 0
+393 1023) Q22}
ifiQlZ
5/2 347(le 14—9(Q11"sz) 445(Q13 321—5-(Q11-Q22) 0 0 0 0
+in3) +2351?” 4023) 4343012
3/2 gm” 445(Q13 %Z(Q11+sz) 43ml; VII—5(Qn- o o o
Qn) “‘in3) +393 4023) Q22)
Adz—1012 4241—5012
In 0 1:1—2((211922) Nil—5(Q13 T‘QWQn’ o «ll—5(Qu- o o
+134§Q12 +in3) +1I403: 922)
421/3012
-1/2 0 o 43(011- o %Q11*Q22) «IE-Qt; pzfiwlt‘w 0
C222) +1/4033 +1023) 431/3012
+i2‘1IEQ12
-3/2 0 o o YT;(Q11' «ITS-(4213 ¥(Q11+Qn) 443-«Q13 ngl‘sz}
Q22) 4023) +393 +iQ23) 4451-012
“243012
-5/2 0 o o o EPQUQP) 443(013 179(Q”+Q22) 347(—Q13
+i3J5-Q12 -iQ23) +71%” +in3)
-7/2 0 o o o o gmll-Qw Ni(-013 %(Q“+Q’-’)
+iJZ—1Qi2 ~iQ23) +712Q’3

 

 

 

l 73
A3-1. Q elements in Tables A3-l, 2, 3, and 4

Qij = QlAliAlj + Q2A2iA2j + Q2A3iA3j

where

_ heng
Q1 ‘ 41(21-1)(”'1)

h 2
Q2=gfé§_%(n+l)

_ heng
Q3 ‘ 21(21—1)

A11 = cosycosficosa - sinysincx
A12 = cosycosfisina + sinycosor
A13 = -cosysin6

A21 = -sinycos[3cosor - cosysina
A22 = -sinycosfisina + cosycosa
A23 = sinysinfi

A31 = sinflcosa

A32 = sinBsina

A33 = cosfi

A4. Computer Programs for Two-Pulse and Three-Pulse ESEEM Time

174

Domain Simulations for I=l, 3/2, 5/2, and 7/2

v.4.0). The diagonalization routine for Hermitian matrices was taken from
a numerical package called "NuToolszNumerical Methods in C" (Metaphor,
v.1.02).

These Programs are written in the C computer language (Symantec

A4-1. Main Program for Two-Pulse ESEEM

/**

at:

* 2pulsesim.c = 2-pulse ESEEM simulation (1:1, 3/2, 5/2, and 7/2 ,gq’sotropic) in time domain
*

**/

#include <console.h>
#include <NuToolsNumericaLh>
#include <nu_test.h>

mainO

{

int i,parai[6],err[l];
float paraf[10];
vector sinthe,costhe,sinphi,cosphi;
float qr[M+l][M+1],qi[M+1][M+l]; /* M=21+1, I=nuclear spin number */
float emod[1024],oremod[1024];

/** parametersetting
at

* Floating Parametersetting

*******************

paraiIO]=nuclear g-value
paraf[1]=field strength (G)
parafl21=Axx (MHz)

pamtl31=Ayy (MHz)

parafl4l=Azz (MHZ)

Parafl5l=°6qQ (MHZ)

parafI6]=eta (asymmetry parameter)
parai[7]=alpha (radian)
paraf[8]=beta (radian)
paraiI9I=gamma (radian)

Integer parameter setting

parai[0]=tau (ns)

parai(l]=starting T (ns) set to zero

parai[2]=time increment (ns)

parai[3]=number of data points desired

parai[4]=number of theta increments (# of theta between 0 and tr)

175

* parai[5]=number ofphi increments (11' ofphi between 0 and 71')

**/
paraset(paraf.parai);
/* sine and cosine table */
sinthc = costhe : sinphi = cosphi = InitV();
sincostab1e(parai,&sinthe.&costhe,&sinphi,&cosphi);
/* quadrupole energy */
quadmpoleZhlf(paraf.qr.qi); /* I=l */
quadrupole3hlf(paraf,qr,qi); /* I=3/2 */
quadmpole5h1f(paraf,qr,qi); /* [=5/2 */
quadrupole7hlf(paraf,qr,qi); /* I=7/2 */
/* hyperfine energy + quadrupole energy */
emodulation(paraf,parai,&sinthe.&costhe,&sinphi,&cosphi.qr,qi.emod);
/* save simulation data */

J “‘pamipamfi‘

A4—2. Main Program for Three-Pulse ESEEM

/**

*

* 3pulsesim.c = 3—pulse ESEEM simulation (I=1,3/2,5/2,7/2.g=isotropic)
*

*au/
#include <console.h>
#include <NuToolsNumerica1.h>
#include <nu_test.h>
main()
int i,parai[6];
float paraiIIO];
vector sinthe,costhe,sinphi,cosphi;
float qrfM+l][M+l],qi[M+l][M+l]; /* M=2I+I, I=nuclear spin number */
float emod[2][1024],emodsum[1024];
/* parametersetting */
pamsetSE(pamf,pa1ai);
/* sine and cosine table */
sinthe = costhe = sinphi = cosphi = InitV();

sincostable(parai,&sinthe,&costhe,&sinphi,&cosphi);

/* quadrupole energy */

176

 

quadrupole2hlf(paraf.qr.qi); /* 1:1 */

quadrupole3hlf(paraf.qr.qi); /* I=3/2 */
quadrupole5h1f(paraf.qr.qi); /* 1=5l2 */
quadrupole7hlf(paraf.qr.qi); /* 127/2 */

/* hyperfine energy + quadrupole energy */

cmodulationSE(paraf.parai.&sinthe.&costhe,&sinphi &cosphi.qr,q1 emod)
/* Save the simulation data */

‘ “'7’ ‘pamipamfi'

A4-3. Subprograms

 

/**
)1:

* *echoamp. c- — echo amplitude at a certain theta and phi and

integration of the amplitude overall orientation of phi for 2-pulse
*
**/

echoamp(nu,parai.da.db,x0,xa,xb.xab.echamp sphi delphi)
int nu ,;pa1ai[]
float x0[] ,xa[], xb[] xab[] echampU, sphi delphi;
vector *da.*

int ii,jj,kk;
int i,j k, nd;
int ind,n1n

float wag][9],wb[9][9],tau.temp echo

float temp1.temp2,temp3,temp4;
float fnu;

 

fnu=nu;
for(_fi=2;jj<=nu;jj++)

temp 1=da—>baseflj- 1 ];
temp3=db—>base[jj— 1 ];

fortii=1;ii<=ij—1;ii++)

temp2=da->base[ii- 1];
temp4=db->base[ii-1];

wa[ii][jj]=temp2—temp I;
wb[ii][jj]=temp4—temp3;

}

for(kk=0;kk<=parai[3]-1;kk++)
1

tau=(pa.rai[0]+parai[2]*kk)*0.00l; /* unit correction (*0001) */
ech =0.0;
inda=0;

l 7 7
ind=0;
f0rti=2;j<=nu;j++)
f0r11=l‘.iej-l ;i++)
for(n=2;n<=nu;n++)
foflk=l;k<=n-l;k++)

temp=cos((wa[illjl+Wblkllnl)*Laut+COS((WalilUl-Wblkllnl)*taU):
echo+=temp*xab[ind];
ind++;
} 1
echo+=xa[inda]*cos(tau*wa[i1U])+xb[indal*c05(taU*Wbii1111);
inda++;
}

echo=(x0[0] +2*echo)/fnu;

/* integration of Modulation amplitude overall orientation of phi */

echamp[kk]+=echo*sphi*de1phi;

 

return;
/*>i=
*
* echoampSE.c = echo amplitude at a certain theta and phi and
* integration of the amplitude overall orientation of phi
*
* for 3-pulse ESEEM

#include <NuToolsNumerical.h>
#include <math.h>

echoampS E(nu.parai.da.db,x0.xa,xb.xab.echamp,sphi.deiphi)

int nu.parai[];
float x0[],x [I.xb[],xab[],echamp[2][1024],sphi,delphi;
r*da,* ;

a
vecto db
int ii,jj.kk;
int i.j,k,n;
int ind,inda;
float wa[9][9],wb[9][9],tau,tauplusT,echoO,echo1;
float cosAtau[9][9].cothau[9][9],cosAtauT[9][9],cothauT[9][9];
float fnu;

tau=parai[0]*0.001; /* unit correction (*0.001) */
fnumu;
for(jj=2;jj<=nu;jj++)
{
for(ii=1;ii<=jj-1;ii++)

wa[ii]flj]=da->base[ii-l]~da—>base[ij— 1 ];
wb[ii][_'1j]=db->base[ii-1]—db->base[jj-1];

cosAtau[ii][jj]=cos(wa[ii]L'1j]*tau);
”‘ ("21.le ""‘Liii‘wur.

\ l I

 

 

178

l
}

for(kk=0;kk<=pami[3]—1;kk+ +)
{

tauplusT=(pa1ai[0]+parai[1]+parai[2]*kk)*0.001;
echoO=0. ;
echol=0.0;
inda=0:

for(jj=2;jj <=nu;jj++)
for(ii=1:iiejj-1;ii++)
cosAtauT[ii][jj]=cos(wa[ii]fij]*tauplusT);
cothauTIii][fi]=cos(wb[ii][jj]*tauplusT);
1
for(j=2;j<=nu;j++)
for(i=l;i<=j—1;i++)
for(n=2;n enu;n++)
{for(k=l;k<=n—l;k++)

echoO+=2.0*cothau[k][n]*cosAtauT[i][j]*xab[ind];

echo 1 +=2.0*cosAtau[i][j]*cothauT[k] [n]*xab[ind];
ind++;

}

echoO+=cothau[i][j]*xb[inda]+cosAtauT[i][j]*xa[inda];
echo 1 +=cosAtau[i][j]*xa[inda]+cothauT[i][j]*xb[inda];
inda++;
}
ech00=(x0[0]*0.5+echoO)/fnu;
echo l=(x0[0]*0.5+echo 1 )lfnu;

/* integration of Modulation amplitude overall orientation of phi */

echamp[0][kk]+=ech00* sphi*delphi;
echamp[ l ][kk]+=echol*sphi*delphi;
}

retum;
/>i<*
*
* emodulationc = caculation of Emodulation for 2—pulse
:1:

)lnk/

#include <math.h>
#include <NuToolsNumerical.h>

’ ‘ r fpm-ni qinthe cnqthe sinphi vnqphi qr qi emod)

int paraifl;

float parafl].qr{M+1][M+l],qi[M+1][M+1].emod[]; /* M=21+1. I=nuclear spin number */
vector *sinthe.*costhe.*sinphi,*cosphi;

 

179

int i.j,k,n,kk,ii,m,mm,p;
float gn,h0,Axx,Ayy,Azz;
float r2,vn;
float xaa,yaa,zaa.xbb,ybb,zbb;
float x,y,z;
cmatrix ha,hb,ma,mb,mc;
vector da,db;
float x0[1],xa[28],xb[28],xab[784];
float sthe,sphi,de1the,de1phi,eamp[1024];
float st,ct,sp,cp;

gn=para110]; /* nuclear g—value */
h0=paraiIl]; /* field strength (G) */
Axx=parafl2]; /* Axx (MHz) */
AYY=Pamtl3k /* Ayy (MHZ) */
Azz=pa1aiI4]; /* Azz (MHz) */
12:1.414213562; /* square root 2 */

vn=gn*5.0507866*h0*0.001/6.6260755;
/***** Hyperfine + Quadrupole energy in MHz unit *****/

xaa=(0.5*Axx-vn)*6.283185306;
yaa=(0.5*Ayy-vn)*6.283185306;
zaa=(0.5*Azz-vn)*6.283185306;

xbb=(-0.5*Axx-vn)*6.283185306;
ybb4-0.5*Ayy-vn)*6.283185306;
zbb=(-0.5*Azz-vn)*6.283 1 85 306;

delthe=(3.14159265358979/parai[4])/3.0; /* for Simpson's integration */
delphi=(3.14159265358979/parai[5])/3.0; /* for Simpson's integration */

for(kk=0;kk<=parai[3]—l;kk++) emod[kk]=0.0;

h3. = hb = ma = mb = me = IthGMO;

da = db = InitV();
for(i=0;i<=parai[4];i++) /* theta range */
if(i=0 ll i=parai[4]) sthe=1.0; /* for Simpson's integration */

else if((kk=fmod(i,2))=1) sthe=2.0;
else sthe=4.0;

st=sinthc->baselil;
ct=costhe—>ba861i];

for(kk=0;kk<=parai[3]-l;kk++) eamp[kk]=0.0;

for(j=0;j<=parai[5];j++) /* phi range */

if(i=0 || jgaraiISD sphi=1.0; /* for Simpson's integration */

else if((kk=fmod(j,2))=1) sphi=2.0;
else sphi=4.0;

/* Ms=ll2 */

sp=sinphi->base[j];
cp=cosphi->base[j];

=xaa*st*cp;
y=yaa*st*sp;
z=zaa*ct;

180

HF2h11'(&ha,qr.qi,x,y,z); /* I=l */
HF3hIf(&ha.qr.qi.x,y.z); /* I=3/2 */
HFShlf(&ha.qr.qi,x,y,z); /* I=5/2 */
HF7hlf(&ha.qr,qi,x,y.z); /* I=7/2 */
EigCHM(&da.&ha,NU_ALL_VECTORS,&ma); /* eigenvaule and eigen vector for alpha state */
TransposeCGM(&ma);
ConjCGM(&ma); /* Ma+ */
/* Ms=-1/2 */
x=xbb*st*cp;
y=ybb*st*sp;
z=zbb*ct;
HF2h1f(&ha,qr,qi,x,y,z); /* I=1 */
HF3hlf(&ha,qr,qi,x,y,z); /* I=3/2 */
HF5h1f(&ha,qr.qi,x,y,z); /* I=5/2 */
HF7h1f(&ha,qr,qi,x,y,z); /* I=7/2 */
EigCHM(&db.&hb.NU_ALL_VECTORS,&mb); /* eigenvaule and eigen vector for beta state */
MulCGM(&mc,&ma.&mb); /* M matrix */

xmatrix(M,x0,xa,xb.xab,&mc); /* M=21+1, I:nuclear spin number */

/* echo amplitude at a certain phi and theta */

 

echoamp(M,parai,&da,&db,x0,xa,xb,xab.eamp,sphi,delphi); /* M=21+l, I=nuclear spin number */

/* integration Modulation amplitude overall orientation of theta */
for(p=0:p<=pamil3l-1;p++)
emod[p]=emod[p]+eamp[p]*st*sthe*delthe;
}
/* neutralization of the integration */
for(ii=0;ii<=parai[3]-1;ii++)

emod[ii]=emod[ii]/(2.0*3. 14159265358979);
return;

/**

*

* emodulationSE.c = caculation of Emodulation for 3—pulse
*

**/

#include <math.h>
#include <NuToolsNumerical.h>

emodulationSE(paraf,parai,sinthe,costhe,sinphi,cosphi.qr,qi,emod)

int paraiU;

181

float parafI],qr{M+ll[M+l].qi[M+l HM+1|,emod[2][ 1024]; /* M:2[+l. l=nuclear spin number */
vector *sinthe,*costhe.*sinphi,*cosphi;

int i.j,k,n,kk.ii,m,mm.p;

float gn.h0.Axx,Ayy,Azz;

float (Am;

float xaa.yaa.zaa.xbb,ybb.zbb;

float x,y,z;

cmatrix ha,hb,ma.mb.mc;
vector da,db;

float x0[l].xa[28].xb[28],xab[784];

float sthe,sphi.delthe.delphi.eamp[2][1024];
float st.ct,sp.cp;

gn=paraf10]; /* nuclear g—value */
h0=parafIl]; /* field strength (G) */
Axx=parafl2]; /* Axx (MHz) */
Ayy=parafl3]; /* Ayy (MHz) */
Azzzparafl4]; /* Azz (MHz) */
r9.=l.4l4213562; /* square root 2 */

vn=gn*5.0507866*h0*0.001/6.6260755;

/***** Hyperfine + Quadrupole energy in MHz unit *****/
xaa=(0.5*Axx-vn)*6.283185306;
yaa=(0.5*Ayy-vn)*64283185306;
zaa=(0.5*Azz-vn)*6.283185306;
xbb=(-0.5*Axx-vn)*6.283185306;
ybb=(-0.5*Ayy-vn)*6.283185306;
zbb=(-O.5*Azz-vn)*6.283185306;

delthe=(3.l4159265358979/parai[4])/3.0; /* for Simpson's integration */
delphi=(3.14159265358979/parai[5])/3.0; /* for Simpson's integration */

for(kk=0;kk<=parai[3]-I ;kk++)
emod[0][kk]=0.0;
emod[l l[kk]=0.0;

ha = hb = ma = mb 2 me : lnitCGM();

da = db = lnitV();
for(i=0;i<=paiai[4];i++) /* theta range */
if(i=0 || i=parai[4]) sthe=l.0; /* for Simpson‘s integration */

else if((kk=fmod(i.2))=l) sthe=2.0;
else sthe=4.0;

st=sinthe->base[i];
ct=costhe->base[i];

for(kk=0;kk<=parai[31-l ;kk++)
eamp[0][kk]=0.0;
eamp[1][kk]=0.0;
}
for(j=0;jeparai[5];j++) /* phi range */

if(j=0 llj=parai[5]) sphi=l.0; /* for Simpson's integration */

 

 

 

1 8 2
else if((kk=fmod(j,2))=1) sphi=2.0;
else sphi=4.0;

/* Ms=l/2 */

sp=sinphi—>base[j];
cp=cosphi—>base[j];

x=xaa*st*cp:
y=ya.a*st*sp;
z:zaa*ct;
HF2hlf(&ha,qr.qi,x.y.z); /* I=1 */
HF3hlf(&ha.qr,qi,x.y,z); /* I=3/2 */
HF5h1f(&ha,qr,qi,x,y.z); /* I=5/2 */
HF7hlf(&ha.qr,qi,x.y,z); /* [=7/2 */
EigCHM(&da,&ha.N U_ALL_VECI'ORS.&ma);
TransposeCGM(&ma);
ConjCGM(&ma);
/* Ms=-l/2 */
x=xbb*st*cp;
y=ybb*st*sp;
z=zbb*ct;
l-lF2hlf(&ha,qr,qi,x,y,z); /* [=1 */
HF3h1f(&ha,qr.qi,x,y,z); /* l=3/‘2 */
HFShlf(&ha,qr,qi,x,y,z); /* l=5l2 */
l‘fl“7hlf(&ha.qr,qi,x,y,z); /* l=7l2 */
EigCHM(&db,&hb,NU_ALL_VECTORS.&mb);

MulCGM(&mc,&ma,&mb);

 

xmatrix(M,x0,xa,xb,xab,&mc); /* M=2l+l, l=nuclear spin number */
/* echo amplitude at a certain phi and theta */

echoampSE(M,parai,&da,&db,x0,xa,xb,xab,eamp.sphi,delphi); /* M=21+l. l=nuclear spin number */

/* integration Modulation amplitude overall orientation of theta */
for(p=0:p<=pamil3]-l;p++)
{ .Irlnr r. unr 1... a. I «J In.
emodhiigjmmgttjt;j*2:*::1;2*3:izi;;;
}
/* normalization of the integration */
for(ii=0;ii<=parai[3}-l;ii++)
emod[O]{ii]/=(2.0*3.14159265358979);
emod[l][ii]/=(2.0*3.14159265358979);

retum;

/****
*

* euler.c = Euler Angle Rotation => Change Quadrupole Principle Axis to Hyperfine PAS
*

****/

#include <console.h>
#include <math.h>

euler( para f.q)

float paranJll‘tll‘lll

int i.j:

float eta,alpha.beta.gamma;
float csa,csb,csg,sna,snb,sng;
float matr[4][4];

eta=parafI6];
alpha=paraf[7];
betazpar'atIS];
gamma=parafl9];

csa=cos(a1pha);
csb=cos(beta);
csg=cos(gamma);

sna=sin(a1pha);
snb=sin(beta);
sng=sin( gamma);

matr[1][1]=csa*csb*csg—sna*sng;
matr[l ][2 l=sna*csb*csg+csa*sng;
matr[1][3]=-snb*csg;

matr[2][l ]=—csa*csb*sng—sna*csg;
matrf2][2]=vsna*csb*sng+csa*csg;
matr[2J[3]=snb*sng;
matr[3][1]q:sa*snb;
matr[3][2]=sna*snb;
matr[3][3]=csb;

/* q[][] = Q matrix in Hyperfine Principle Axis System */
fortj=l;j<=3;j++)
for(i=l;i<=3;i++)
{

q[i][j]=(eta—1.0)*matr[l][i]*matr{1][j]-(eta+l.0)*matr{2][i]*matr[2][_j];
q[iltil+=2.0*matrt3lti]*matrt3]til;

return;

/**

*

* HF2hlf.c = Hyperfine & Quadrupole Energy (I=l)
*

**/

184

#include <NuToolsNumerical.h>
#include <math.h>

HF2h1f(a,qr,qi.x.y.Z)

cmatrix *a;
float qd4][4],qi[4ll4l.xyy,l;

int i,j.D1M:
float 1’2;

DIM=3;
r2:sqrt(2);

AllocateCGM(a.DIM,DIM);
if(nu_error) return;

else

for(i=0;i<DlM;i++)

for(i=0;j<DIM;j++)
{

a->mat[ilU]-re=qdi+1 1U+1];
a—>mat[i][_i].im=qi[i+ l ][j+ l];

}

a—>mat[0][0].re+=z;
a->mat[0][l].re+=x/r’2;
a->mat[l][0].re+=x/r’l;
a->mat[1][2].re+=x/r’2;
a->mat[2][1].re+=x/1’.Z;
a->mat[2][2].re-n;

a->mat[0][l].im—=y/r’.2;
a—>mat[1][0].im+=y/r’2;
a—>mat[1][2].im—=y/r’2;
a—>mat[2][l].im +=y/rQ;

 

return;

/**
*

* l-[F3hlf.c = Hyperfine & Quadrupole Energy (123/2)
*

**/

#include <NuToolsNumerical.h>
#include <math.h>

HF3hlf(a,qr,qi.x,y,Z)

cm trrx *a;
float qr151151‘qi151151,x.y,z;

int i,j,DlM;
float r3;

DIM=4;
r3=sqrt(3);

AllocateCGM(a,DlM, DIM);

if(nu_error) return;
else
for(i=0;i<DlM;i++)
for0=0;j<DlM;j++)
{

a->matli1li14re=qui+llli+1];
a->mat[i][j].im=qi[i+1][j+ l];

a->mat[0][0]tre+=l 5*2;
a->mat[0][ l ].re+=r3*0,5 *x;
a—>mat[1][0].re=a—>mat[0][l]rre;
a—>mat[ l ] [ l ].re+=0.5*z:
a->mat[1][2].re+=x;
a->mat[2][1].re=a—>mat[1][2].re;
a—>mat[2][2].re—=0.5 *z;
a—>mat[2][3].re+#).5*r3 *x;
a—>mat[3][2].re==a—>mat[2][3].re;
a—>mat[3][3].re-:l .5 *z;

a->mat[0][ l ] .im-=0,5 *r3*y;
a->mat[1][0].im=—a->mat[0][l].im;
a->mat[1][2].im—=y;
a->mat[2][1].im=—a->mat[1][2].im;
a->mat[2][3].im—=0.5*r3*y;
a—>mat[3][2].im=~a->mat[2][3].im;

return;

pm

*

* HFShlf.c = Hyperfine & Quadrupole Energy (I=5/2)
*

**/

#include <NuToolsNumerica1.h>
#include <math.h>

HFShlf(a,qr,qi.x.y.Z)
cmatrix *a;
float qr[7][7],qi[7][7],x,y.Z;
int iJDIM;
float r’Z,r5;
DIM=6;

{2% qrt(2);
15=sqrt(5 );

AllocateCGM(a.DIM,DlM);
if(nu_error) return;

else{
for(i=0;i<DlM;i++)
forQ=0;j<DlM;j++)

a->mat[i1ti]-re=qrti+llti+1];
a—>mat[i]Li].im=qi{i+ 1 ][j+ l ];

 

186

a->mat[0] [O].re+=2.5*z;
a->mat[0][1].re+:1'5*0.5*x;
a->mat[1][0].re=a->mat[0][1].re;
a—>mat[1][1].re+=l .5*z;

a->mat[ l. ][2] .re+=rZ*x;
a->mat[2][1].re=a->mat[1][2].re;
a->mat[2][2].re+=0.5 *z;
a->mat[2][3].re+=1.5*x;
a->mat[3 ][2].re=a->mat[2][3].re;
a->.mat[3][3].re-=O.5*z;
a->mat[3][4].re+=r’2*x;
a->mat[4][3].rm—>mat[3][4].re;
a->mat[4][4].re-=1.5*z;
a->mat[4][5].re+=0.5*r5*x;
a->mat[5][4].re=a->mat[4][5].re;
a->mat[5][5]Je-=2.5 *z;

a->mat[0][l].im-=0.5*r5*y;
a->mat[1][0].im=—a->mat[0][1].im;
a—>mat[ 1 ][2].im—=r2*y;
a->mat[2][1].im=—a->mat[l][2].im;
a->mat[2][3].im-=15*y;
a->mat[3][2].im=—a->mat[2][3].im;
a->mat[3][4].im-=12*y;
a->mat[4][3].im=—a—>mat[3][4].im;
a—>mat[4][5] .im-=O.5 *r5*y;
a—>mat[5][4].im=—a->mat[4][S].irn;

return ;

}

/*=I=
a:

* HF7hlf.c = Hyperfine & Quadrupole Energy (127/2)
*

**/

#include <NuToolsNumerical.h>
#include <math.h>

HF7hlf(a,qr,qi,x,y,z)

cmatrix *a;
float qr[9][9],qi[9][9],x,y,z;

{
int i,j,DIM;
float r3,r7,r15;

DlM=8;

r3=l .732050808;
r7=2.64575131 1;
r15=3.872983346;

AllocateCGM(a,DlM,DlM);
if(nu_error) return;

else{

{for(i=0;i<DIM;i++)

for(i=0;j <DIM:j++)

a—>mattilli1-n’;qr[i+llti+ll;
a->mat[i][_j].im=qi[i+l ][j+1];

a‘>mat[0][0].re+=3.5*z;
a->mat[0][l]rre+=r7*0.5*x;
a->mat[1][0].re=a—>mat[0][l].re;
a—>mat[1][l].re+=2.5*z;
a—>mat[l][2].re+=r3*x;
a—>mat[2][1].re=a->mat[l][2].re;
a->mat[2][2].re+=l .5 *z;
a->mat[2][3].re+=0.5*r15*x;
a->mat[3][2].re=a->mat[2][3].re;
a—>mat[3][3].re+=0.5*z;

a— >mat[3][4] .re+=2 0*x;
a—>mat[4][3]. re=a- ->mat[3][4]. re;
a— >mat[4][4] .re—#) .*5

a- >mat[4][5]. re+=0. 5*r15*x;
a—>mat[5][4].ro:a->mat[4][5].re;
a->mat[5][5].re-=l.5*z;
a->mat[5][6].re+=r3*x;
a—>mat[6][5].re=a—>mat[5][6].re;
a->mat[6][6].re-=2.5*z;
a->mat[6][7].re+=0.5*r7*x;
a->mat[7][6].re:a->mat[6][7].re;
a->mat[7][7].re-=3.5*z;

a->mat[0][1].im-=0.5*r7*y;
a—>mat[ l ][0].im=—a->mat[0][1].im;
a->mat[1][2].im-=1'3*y:

a—>mat[2][l].im=—a—>mat[1][2].im;

a->mat[2][3].im-=05*r15*y;
a->mat[3][2].im=-a—>mat[2][3 ].im;
a->mat[3][4] .im-=2.0*y;
a—>mat[4][3].im=—a—>mat[3][4].im;
a->mat[4][5]tim—=0.5*r15*y;
a—>mat[5][4].im=-a—>mat(4][5].im;
a->mat[5][6].im-=r3*y;
a—>mat[6][5].im:a->mat[5][6].im;
a->mat[6][7] .im-=0.5*r7*y;
a->mat[7][6].im=—a—>mat[6][7].im;

return;

/**

187

* quadrupoleZhlf.c 2 Nuclear Quadrupole Energy in Hyperfine Axis (I=l, MHz unit)
*

* qulll = Re (quadrupole energy)

* qi[][] = 1m(quadrupole energy) in Hyperfine PAS (MHz unit)
*

**/

#include <math.h>

quadmpole2h1f( para f,qr,qi)

float Parafilyqu4ll4lflil4ll4];

int i,j;

 

l 8 8
float equ,q[4][4l;

euler(paraf.q);

c(’vth=Peu'¢lf151;

/* q matrix in MHz unit */
for( i=1 ;i <=3 ;i+ +)

for(j=l;j<=3;j++) q[i][j]=0.25*equ*q[i][j]*6.’283185306;

/* qr[][] = Re (quadrupole energy) qi[][] = Imtquadrupole energy) in Hyperfine PAS */

qrtlIlll=05*(qllltl|+q[2][21)+q[3l[31;
qr[1][2]=0.707106781*q[l n3];
qr111131=0.5*(q[1111HIZIIZD;
qrt21t11=qrt11t21;
qr[21121=q[1][1]+q[21[2l;
qr[2][3]=-0.707106781*q[1][3];
qr[3I[11=qr[1][3];

qr[3][2]=qrt2][3];

qrt31t31=qrt11t11;

qillltll=0.0;
qi[l][2]=-0.707106781*q[2][3];
Qi[1][31=-qlll[21;
qi[21[11=<1it11[2];

qi[2][2]=0.0;
qi[2][3]=0.707106781*q[2][3];
qit3ltll=qit 1113];
qi[3][2]=-qi12113];

qi[3][3]=0.0;

return;

/**

a:

* quadrupole3hlf.c = Nuclear Quadrupole Energy in Hyperfine Axis (I=3/2, MHz unit)
*

* qr[][] = Re (quadrupo1e energy)
* qi[][] = lm(quadrupole energy) in Hyperfine PAS (MHz unit)
*

**/
#include <math.h>
quadrupole3hlf(paraf.qr,qi)
float parafl],qr[5][5],qil5][5];
int i. 3':

float equ,q[4][4],r3;
euler(paraf,q);

r3=sqrt(3.0);
equwamfISh

/* q matrix in MHz unit */

189

for(i=l;i<=3;i++)

for(j=1;j<=3;j++) q[ile]=cqu*q[i]Lj]*6.283185306/12.0;

/* qr[][] = Re (quadrupole energy) qi[][] = lm(quadr‘upole energy) in Hyperfine PAS */

qulllll=075*(q{11[11+q121[21)+2- 25*ql31131;
qr111121*r3*q111131:

qul 1131=(r3/2 0)*<Ctlllll 1-q{21[21)

qr[1114 1=00 -

qr121111=q 11l2 '
qr[21121=175*<Ql11111+Ql21121)+0-5*'Ql31131
qr[2][3]=0.0.

qI121141=0-5*r3*(q{11ll1-q121121);
qu31111=qulll31;

qti31121=0.0;

qu3ll31=qr121121;

qu3ll41='r3*Ql11[31;

q1141111=0-0;

qu41121=qd21141;

qu4ll31=qu3ll41;

qr[4][4]=qr[l][1];

qi111111=0.0;
q1111121=-r3*q[21[31;
qilll[3]= r3"‘<1[11121;
qill 1141=0-
qi121[ll=-qi[11[21;
qi 2 2 =03;
qi121131=0
qi[21l41=-r3*q[11[21;
qil31lll=<tillll31
qi[3 1121=0.0;
qil3ll31=0.0;
qil31141=r3*q[21l31;
qi[4][1]=0.0;
qil4][2]=-qi[2][4];
qil41131=fli13ll41;
qil4ll41=0.0'.

return;

[an

t

* quadmpoleShlf.c = Nuclear Quadrupole Energy in Hyperfine Axis (1:5/2. MHz unit)
*

* qr[][] = Re (quadrupole energy)
* qillll = lm(qualdrur>ole energy) in Hyperfine PAS (MHz unit)
a:

**/
#include <math.h>
quadrupole5h1f(paraf,qr.qi)

float paraf[].qr[7][7],qi[7][7];

int i.j;
float equ,q[4][4],12.r5.r10;

euler(paraf,q);

190

r2=sqrt(2);
r5=sqrt(5);
r10:sqrt( 10);
equ=parafISk

/* q matrix in MHZ unit */
for(i=l;i<=3;i++)

for(j=l;j<=3;j++) q[i][j]=equ*q[i]Lj]*6.283185306/40.0;

for(i=l;i<=6;i++)
for(j=l;j<=6;j++)

quiltj1=0.0;
qi[i][j]=0.0;

}
/* qr[][] = Re (quadrupole energy) qi[][] = lm(quadrupole energy) in Hyperfine PAS */

qulllll=l-25*(q[11[11+ql21121)+6-25*ql31[31;
qr[11121=2-0*r5*q{11[31;
qul113l=0-5*r10*(q11111141121121);
qr[2][1]=qr[1][2]'

qu21121= 3 25*(Ql11111+ql21121)+2 25*Ql31131
qr[21131=2 0*1” *q111131

qtl21141= 15*12*(Ql11111fl121l21)
qu31111=qrt11131;
qr131121=qu2 113 1'

qr[3ll31=4 25*(C1111l1 1+Ql21121)+0 25*q[31[31;
qu31151= 15*I2*(q111111-ql21[21);

qu41151= -20*1'2*q[11l31;
q-l'l41[61=0 5*r10*(q[l 111141121121)
qr151131=qu31151;
qtl51141=qu41151;
qu51151=qr121121;
qr151161=-2-0*1'5*q{11[31:
qr161141=qu4ll6lz
qr[61151=qr151[61:
qri61161=qr111111;

qi[1][2]=-2.0*r5*q[2][3];
qi[11131=r10*q{11[2];
qil21[11=-qi[11[21;
qil2ll31=~2-0*I2*<1[21[31;
qil21141=-3-0*12*q{11[21;
qi[31[11=-qi[11l3];
qi[3ll21=-qil21[31:
qil3ll5l =-3 -0*12*q{11[21;
qi[41121—-qil2 114 1'
qi[4l[5]=2 0*12*q[21[31;
qi141161=r10*q[11[21;
qi[51[3]=—qi[3][5];
qi151l41=41il4115h
qi{51[61=2-0*r5*q[21131;
qi161l41=-qi[4ll61;
qi[61151=qi[51[61;

 

1 9 l
Mum;

}

/**
*

* quadrupole7hlf.c = Nuclear Quadrupole Energy in Hyperfine Axis (I=7/2, MHz unit)
*

* qr[][] = Re (quadrupole energy)
* qi[][] = lm(quadrupole energy) in Hyperfine PAS (MHZ unit)
*

**/
”include <math.h>
quadrupole7hlf(paraf,qr.qi)

float parafl]lqr[91[91.qi191191;

inti
float equ, q[4][4], r3, r5 r7 .,r15 r21;

eulertpamflq);

r3=sqrt(3);
l’5=8<1fl(5);
r7=sqrt(7);
r15=sqrt(15);
rQl:sqrt(21);

aqu=pamf15k
/* q matrix in MHz unit */
for(i=l;i<=3;i++)

for(i=l;j<=3;j++) q[i]Li]=equ*q[i][j]*6.283l85306/84.0;

for(i=1;i<=8;i++)
for(j=l ;j<=8"j++)
{

quiltil=0.0:
qi[ile]=0.0;

}
/* qr[][] = Re (quadrupole energy) qi[][] = lm(quadrupole energy) in Hyperfine PAS */

qf111111= 1-75*(Q[11111+Q121121)+12- 25*q[31[31;
qr111121=3 -70*T*1CI1 1131;

qr111131=0 5*r21*(Q111111-Q[21121);
qr121111=qr111121;

qr121121=4 75*(‘1111111+Ctl211211+6 25*9131131;
qu21131=4-30*r*q{11131;
qr121141=1-:5*r5*(9111111-q{21121)
qr[31111=qr111131;

q1131121=q 2311 1‘
qr[3][3]=6.75*(q[1][l]+q[2][2])+2. 25*q[3][3];
qu31141=715*q [11131;
qr131151=T15*(Q111111fl121121);
qr141121=qu21[41;
qr141131=qr131141;

192

qr[41141=7.75*(Q111111+q121121)+0-25*Q131131;
qr141161=Tl$*(<I111111-q121121):
qr151131=q1131151;
qr151151=qr141141;

qr151161=-r15*Q11 1131;

qr151171=1 -5*1'5*(CI11111 1-q121121);
c11’161141=qr141161;

qr161151=qr151161;

q1161161=qr131131:

91161171=-4-0*r3 *Q111131;
qr161181=0.5*f21*(q[11111-q{21[21);
qr171151=qr151171;

qr171161=qr161171;
qr171171=qr12112l;
qr171181=~3-0*1‘7*CI111131;
qr131161=qu6118k

qr181171=qr171181;

qr[81181=qr111[11;

qi[l][2]=—3.0*r7*q[2][3];
qi111131=r21*q111[21;
qi[211 11=-qi111121;
qi[21131=40*r3*q[21[31;
qi[21141=-3.0*r5*q11 1121;
qi[31111=<1i111[31;
qil31121=<1i121131;
qi131141=-r15*<1121[31;
qi[31151=-2-0*r15*q[11121;
qi[41121=-qi[21[41;
qi141131=qil31141;
qi[41161=-2.0*r15*q111121;
qi[51131=<1il31[51;
qi151161=rl$*q[2113]:
qi151171=3 ~0*r5*q[1 1121;
qi161141=-qil41[6];
qi[6115]=qi[5][6l;
qil61171=4.0*r3*q{21131;
qi[61181=-r21*q111[2];
qil71151=qi151171;
qi171161=qil61171;
qil71181=3 -0*r7*q[21[3];
qi[81161=*1i161[81;
qil81171=<1i171181;

return;

}

/***

*

* sincostablec = generating SlN and COS table
*

*

***/

#include <NuToolsNurnerical.h>
#include <math.h>

sincostable(parai,sint,cost,sinp,cosp)
int paraifl;
vector *sint,*cost,*sinp,*cosp;

{
int i,SIZE;

193

float theta.phi,pi;
pi=3.14159265358979; /* constant 7r */
/* caculation of theta and phi increments */

theta=parai[4];
phi=parai[5];

theta=pi/theta;

phizpi/phi;
SIZE=parai[4]+l;
AllocateV(sint,SIZE);
if(nu_error) return;
AllocateV(cost,SlZE);
if(nu_error) return;

/* making tables (theta) */
for(i=0;i<=parai[4];i++)
1

sint—>base[i]=sin(i*theta);
cost->base[i]=cos(i*theta);

}

/* making tables (phi) */
SIZE=parai[51+1;
AllocateV(sinp,SIZE);
if(nu_error) return;
AllocateV(cosp,SIZE);
if(nu_error) return;

for(i=0;i<=parai[5];i++)

sinp->base[i]ain(i*phi);
cosp—>base[i]=cos(i*phi);

return;
/**
*
* xmatrix.c = xmatrix(nu,x0,xa,xab,mr,mi)
*
* calculate the X coefficients
:k
*x/

xmatrix(nu,x0,xa.xb,xab,m)

1

int nu;
float x0[],xa[1,xb[],xab[];
cmatrix *m;

int i,j,k,n;
int inda,'mdb,indab;
float x,temp1,temp2,temp3.temp4,temp5,temp6,temp7.temp8,temp;

194

/* Xo */
x0[0]=0.0;
for(i=1;i<=nu;i++)
for(k=l;kenu;k++)
templ=m->mat[i-l][k-1].re;
temp2=1n->mat[i-l][k—l].im;
tempztemp l *templ +temp2*tem p2;
x0[0]+=temp*temp;
1
/* Xij */
inda=0;
for(i=2;jenu;j++)
for(i=1 ;i<=j-1;i++)
x=0.0;
for(k=1;k<=nu;k++)
{
templ=m->mat[i-1][k-l].re;
temp2=m—>mat[i- 1][k-1 1.1m;
temp3=m->mat[j-1][k-l].re;

temp4— ->mat[j- l ][k-l ] .im;
x +=(temp1*templ +temp2*temp2)*(temp3 *temp3 +temp4*temp4);

xa[inda]=x;
inda++;
1
1

indb=0;
for(n=2;n <=nu;n ++)
for(k=1;k<=n—l ;k++)
x=0.0;

for(i=l :iL-nu;i+ +)
{ ,
templ=m~>mat[i-1][k—1].re;
temp2-—-m- >mat[i- 1 ][k— l J .irn;
temp3=m->mat[i- 1][n-1].re;
temp4— ->mat[i—1][n—1].im;
x+=(templ *templ +temp2*temp2)* (temp3 *temp3 +temp4 *temp4);

xb[indb]=x;
indb++;
1
1

/* Xijkn */

 

195

indab=0;

for(i=2;jenmj++)
1
for(i=1;i<=j— 1 ;i++)
{

for(n=2;n <=nu;n++)
for(k=l;ken-1;k++)

templzm->mat[i-l][k-1].re;
temp2=1n—>mat[i—l ][k— 1 ].im;
temp3=m~>mat[i—l][n-1].re;
temp4— ->mat[i-l][n—l].im;
temp5=m->mat[j-1][k-l].re;
temp6=m~>matLj-1][k—1 ].im;
temp7=m->mat[j—1][n-1].re;
temp8=m—>mat[j-1][n—1].im;

temp=(temp l *temp3 +temp2*temp4)*(temp5 *temp7+temp6*ternp8);
temp-=(temp1 *temp4-temp2*temp3 )* (temp7*temp6-temp8 *tempS );

xab[indab]=temp;
indab++;

l 96
A5. Pulse Logic Circuits

The following drawings are of the pulse logic circuits which are
installed in the pulsed-EPR spectrometer at Michigan State University. The
ideas for the logic circuits were given by Professor J. McCracken. The
original circuits were designed by Mr. Marty Rabb in the electronics shop
of the Chemistry department and modified by Prof. McCracken and
myself.

In the figures, the [OS are numbered in bold type. The IC's used in
these circuits are 7400 (1, 34, 205, Quad 2-input positive NAND gate),
7404 (2, 209, Hex inverter), 74123 ( 3, 4*, 6, 11, 15, 17, 19, 20, 36,
203, Dual retriggerable monostable mutivibrators with clear), 7430 (4, 8-
input positive-NAND gate), 74121 (16, 18, Monostable multivibrator),
7405 (7, 21, Hex inverter with open-collector output), 7402 (8, Quad 2-
input positive NOR gate), 74128 (9, 31, 500 Line driver; NOR gate),
74140 (9*, 207, Dual 4-input positive—NAND 500 Line Driver), 7432
(10, 208, Quad 2-input positive-OR gate), 74373 (21, 22, 23, 24, 25, 26,
27, 28 in phase module, Octal D-type latches), 7454 (29, 30, 4-wide
AND-OR-Invert Ggate), 7493 (32, 4-Bit binary counter), 74138 (33, 3 to
8-Line decorder), 7474 (35, 206, Dual-D-type positive-edge-triggered
flip-flops with preset and clear), 7408 (205, Quad 2—input positive-AND
gate), PPG-33F—5 (12, 13, 14, Programmable pulse generator), and ECG
8520 (201, 202, Modulo-n divider).

The logic circuits shown on the next four pages are Figure AS-l,
PIN 1 Circuit; Figure A5-2, PIN 2 Circuit; Figure A5-3, Phase Control

Circuit; and Figure A5-4, Receiver Control Circuit.

Figure AS-l. PIN 1 Circuit

197

 

 

<1 . d) CONT.

T0 in l 3 l 2 2 13 9 12 REC. CONT.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

199

Figure A5-2. PIN 2 Circuit

>
S>47k 30p 301’
I: ‘ 20k 50k
- l
C'“ 1’ 2 3113934 12 PINlMOD.4-4
7 5
47 _ lk Lg 10.11

4;
\1
K‘
3:.
AAA J
vv ()3
AAA I I

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

REC.
CONT.

 

 

 

 

201

Figure A5-3. Phase Control Circuit

 

 

33-11

  
  
 

1k
(8)

 

 

6

 

 

 

 

   

12341213 5
30

 

#mééé
NNNNN 79

'1 JJ 0 90.9k 1' 0.004711 —

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

203

         

 

 

 

 

PIN 1
lt’IOD. 1 3 4 6
-3 .g
7 5 14
203-10 7 T ' ,
H2M1301149$5 4%; 41
14 _
PIN 1 2 143 1‘1 1‘1 1'1
MOD. 20 8 1 14 3 15 3 13 9 12 1°

 

10-6——i’
7 201

 

 

 

 

 

 

 

    

l_11_ll_1

114315313912

 

 

 

 

 

 

 

 

 

 

 

 

 

_ Pulse
Coun-
PlN 2

Figure A5-4. Receiver Control Circuit

 

204

The timing diagrames of the above circuits are depicted as

followings.

 

To Input _.l
13 _1
2-2 _j

3-13 _l‘I’l
3-12 t—ZSOM—j

 

 

 

 

 

Figure A5-5. Timing diagram of from To delay input to 4 input (8-NAND
gate) in PIN 1

T0 Input —l:o

4-1

A Input
4-2

B Input
4-3

C Input
4-4

D Input
4-5

E Input
4-6

4-8
2—4

9-1

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

/100+a+t°
—*l_l$..
:
_gJ.,
_l
Fall
:
l
_.___: 1,,
___|
1_.._1_;
:
_l—l.,
:
I
I
a 1,
l
a . Jr
1
I
e 1:
l
_, J
: ‘e 1_l

500
TWT

Figure A5-6. Timing diagram of from delay inputs to TWT input

 

2 0 6
To Input _.l

1-3

 

 

 

BIN SW (Assuming that at least one switch is high.)
8- 1

 

 

 

 

 

4*..5 —3mm—~

 

 

 

 

6-13 +——380m ———~ :

(:1 0f To (29-2)
Figure A5-7. Timing diagram of from To delay input to phase circuit
input.

To Input _lto

6-13
PIN l

A Input
6-5
PIN- 1
B Input
19-13
PIN 1
C Input
6-13
PIN 2
D Input
6-5
PIN 2

E Input

6-19
PIN 2

207

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

/320+a+10
_.u—Pio'“ (P‘To (2.9-2)
| .
I
I
_.:_ t.
I
:<-t,:l I 0*!“ (294)
E >
2 1,
I
I
:«— t. _.j_‘| as (29-13) g
l
l t
l
g.— ., _.l—‘1 ac (30-13) g
. ,
I
I ..
I
g 1, fl d>D (30—4)
5
I
te
l
:1 ,e ,j I d>-E (30-2) _

 

 

‘7

Figure A5-8. Timing diagram of from delay inputs to phase circuit inputs.

208
T0 Input _1

1-3

 

 

BIN SW (Assuming that at least one switch is high.)

 

 

 

 

 

 

 

 

 

84 __

4km «am—u

5-6 JP?“—
122 ~W”

 

 

 

 

1— Pulse width determined by the
BIN SW and 12 (PPG).

Figure A5-9. Timing diagram of from To delay input to 10 input

 

 

T0 Input _

 

 

 

122 hwe

 

 

A Input

 

 

 

we HWJ

 

 

 

B Input

 

 

 

14"2 fi—tbm—w

 

 

 

 

 

 

10-6

 

 

 

 

 

 

 

 

 

 

 

 

9-13 '7—75051
PIN 1
Figure A5-10. Timing diagram of from To, A, B delay input to PIN l

 

 

 

 

 

 

 

 

 

input

209
Assuming 4-step phse cylcle (To, A, B - 000, 011, 110, 101)

34-2

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

35-3 '

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

35-5 l 1
, RESET"
34-3 __1_7_1—___|E _1—
32-out LLL 1 LLH 1 LHL 1 111111 I LLL
33-4 1 1
I
36-12 154:1 I 1
I
36-4 '
<1»TwA.B
29-1 1 1
29-3 | 1
29-12 I '1 | |
”'6 U11 1111 1111 HIV
31-1 1111 1111 11.11 1111 .
50!)
(1)1

Figure AS-l 1. Timing diagram of phase control circuit

210
203-1 ‘ F—l

203-4 —: a.

55m '—

 

 

 

 

 

 

 

 

 

203-13

 

 

Assuming that 3-pulses are turned on at each PIN MOD.

208-1 W
208-5 W
_____5, 1—1
208-3 1—1 W l—l__l—l_l_l_
208-6 I l m 1—1

201-10

 

 

208-2,4

 

 

 

 

 

 

 

 

 

 

202-10

 

 

 

 

 

 

 

205-6

 

 

 

 

 

 

 

203-12

 

 

 

 

 

 

204-10

204-8 —__l_1
206-5 1 l l

207-6 I 500
REC. SW.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Figure A5-12. Timing diagram of receiver switch control circuit

211

A6. Computer Interfacing Programs for Performing 4-pulse ESEEM and
HYSCORE (hyperfine sublevel correlation spectroscopy) Experiments

The programs are written in the C computer language. The main
programs were written by myself. The subroutines were originally written
by Professor J. McCracken for 2- and 3-pulse ESEEM experiments and
modified for 4-pulse ESEEM and HYSCORE experiments when

appropriate.

A6-1. Main Program for 4-Pulse ESEEM

\
*-

4—pulse.c - this routine will be used to govern the collection
of 4-pulse experiment data.

definition
1imits(xmin.xmax,ymin,ymax,ixtic,iytic) — the box limit for the live
gpibO - gpib controller
dgl — dg535 pulse generator (#1)
dg2 - ng35 pulse generator (#2)
sri - 31245 computer interface

mloc - mouse location (pointer)

scan control parameters

cval[O] - tau (nsec)
cva1[ 1] — starting T/Z (nsec)
cval[2] - T/‘2 increment (nsec)

cval[3] - repetitions/time point
cval[4] - sequence repetition rate (Hz)
cavl[5] - plot scale factor

cval[6] - A timing shim (nsec)
cval[7] - B timing shim (nsec)
cval[8] - C timing shim (nsec)
cval[9] - D timing shim (nsec)

cval[10]- # ofdata
delay generator set up parameters

tpar[0] - trigger rate

tparl 1] - A delay time (tau)
tpar[2] - B delay time (tau + T0)
tpar[3] - C delay time (tau + T)
tpar[4] - D delay time (2*tau +T)

***********************************

/

#include <stdio.h>

#include
#include
#include

212

<console.h>
<string.h>
<stdlib.h>

#include "decl.h”

#include

"matrixh"

int gpib0,dg1,dg2,sri;
char spr.pbuf1121;

typedef struct{int v,h;} POINT;
POINT *mloc;

main() -

int i,limits[6],cval[1 l ].tpar[5],specparm[4];
int *data,iopt,sizeofdata;

float fval[4],tl,dtl;

char chopt[3];

sizeofdata = -l ;

limits[0]=100;
1imits[l ]=575;
limits[2]=40;
limits[3]=300;
limits[4]=10;
limits[5]=10;

fval[0]=9.0;
fval[1]=3000.0;
fval[2]=50.0;
fval[3]=4.2;

/* set up console screen */

pa geset(stdout);

/* read cval[] from time4p.dat */

tparin4p(cval);

/* open and clear gpib controller. Then open and clear 31245 and dg535 */

if((gpib0=ibfind(”gpib0"))<0)

printf("gpib0 not foundlnlr");
exit(-l);
if(ibsic(gpib0) & ERR) error();
if((sri=ibfind("er45"))<0)
1

printf("sr’245 device not foundlnlr");

exit(-l);

1

if((dg1=ibfind("dg535"))<o)

{
printf(" first dg535 device not foundlnlr");
exit(-l);

1

if(ibclr(sri) & ERR) error();

2 1 3
if(ibclr(dg1) & ERR) errort);

/* get scanning parameters */
change: getparrn4p(cva1);
/* Initilize data array. */

if( sizeofdata > -1 ) free(data);
sizeofdata = cval[lO] + 20;

data =( int * ) calloc( sizeofdata , sizeof( int ) );

/* set up initial timing parameters */

tpar[0]=cva1[4]; /* trigger rate */

tpar[1]=cva1[0]+cval[6]; /* A delay time */
tpar[2]=cval[0]+cval[1]+cval{7]; /* B delay time */
tpar{3]=cval[0]+2*cval[l]+cval[8]; /* C delay time */
tpa.r[4]=2*cval[0]+2*cval[l]+cva1[9]; /* D delay time *I

/* set up the scanning parameters for 31245 and dg535 devices */
srsethy(cval[3]);
repeat: dgseth y l (tpar);
/* set up the screen for data aquisition */
eraseplot();
1plabel4p(cval);
box(limits);
/* set timing parameters for display */
specparm[0]=cval[0];
spec parm[l ]=cval[0] +cva1[ l ];
specparm[2]=cval[0]+2*cval[1];
specparm[3]=2*cval[0]+2*cval[l ];
dgupdate4p(0,0,specpann[0]);
dgupdate4p( l,0.specparm[1]);
dgupdate4p(2,0,specpann[2]);
dgupdate4p(3,0,specparm[3]);

/* data collection - the first point as collected from the integrator

* is usually bad. the first twice to make up for this. 10 data

* points are for baseline.

*/
trstarthyl(); /* trigger start */
data[0]=getpoint(sri,0); /* data aquisition */
trstophyl(); /* trigger stop */

for(i=0;i<cval[10];i++)

cgotoxy(49,34,stdout); /* put channel # on screen*/
printf("%d",i);

if(kbhit()) goto compl; /* check for keystroke */

214

trstarthle; /* trigger start */
data[i]=getpoint(sri,0); /* data aquisition and type signal value */
trstophyl(); /* trigger stop */

dplot4p(cval[5],i,cva1[10]+20,data[i]); /* plot data on screen */

specparm[l]=dgupdate4p(l,cva1[2],specparm[1l); /* update delay generator */
specparm[2]=dgupdate4p(2,2*cval[2],specparm[2]); /* for 3rd, 4th pulses and */

speeparm[3]=dgupdate4p(3,2*cval[2],specparrn[3]); /* boxcar gate */
1
/* get baseline points */
specparm[3]=dgupdate4p(3,100,specparm[3]);
for(i=cval[10];i<cval[10]+20;i+ +)

cgotoxy(49,34,stdout); /* put channel # on screen*/
printf("%d",i);

trstarthy 1();
data[i]=getpoint(sri,0);
trstophyl();

dplot4p(cval[5],i,cval[l 0] +20,data[i]);

/* select exit option */
compl: iopt=0;

cgotoxy(68,34,stdout);

printf("l. save data set");
cgotoxy(68,35,stdout);

printf("2. repeat scan - same setting");
cgotoxy(68,36,stdout);

printf("3. repeat scan - change setting");
cgotoxy(68,37,stdout);

printf("4. manual scan");
cgotoxy(68,38,stdout);

printf("S. exit program");

again:

cgotoxy(68,32,stdout);
printf("ENT ER OPTION : ");
gets(chopt);
iopt=atoi(chopt);
if(iopt<1 liopt>5)
cgotoxy(63,32,stdout);
ccleol(stdout);
goto again;

1
if(iopt=l )

savedata4p(cval,fval,data); /* save comment,fval,cval,date,data */
goto compl;

if(iopt=2) goto repeat;

215

if(iopt=3) goto change;
if(ioptfl)

trstarthy l 0;

ibloc(dg1);

cgotoxy(68,39,stdout);

printf("CLICK MOUSE WHEN FINISHEDlr");
SysBeep( 1 );

f0r(;;)

if(ButtonO) break;

1

dgl=ibfind("dg535");
trstophyl();
cgotoxy(63,32,stdout);
ccleol(stdout);

goto again;

1

free(data);
ibonl(sri,0);
ibonl(dg1,0);
tparout4p(cval);

crror()

printf("GPIB function call errorlnlr");
printf("ibsta=0x%x, iberr = 0x%x\n\r",ibsta,iberr);
printf("ibcnt=0x%x Inlr",ibcnt);

exit(-l);

A6-2. Main Program for HYSCORE

\
*

hyscorelc - this routine will be used to govern the collection
of HYSCORE (2-D ESEEM) data.

there are two different data collection scheme.

lst — direct detection
(4—pulse sequence <— This program
5-pulse sequence)

2nd - remote detection

definition
limits(xmin,xmax,ymin,ymax,ixtic,iytic) — the box limit for the live

gpibO - gpib controller

dgl - dg535 pulse generator (#1)
dg2 - dg535 pulse generator (#2)
sri - 31’245 computer interface
mloc - mouse location (pointer)

***********************

scan control parameters

"u‘r

216

cval[O] - tau (nsec)

cval[1]- starting tl (nsec)

cval[2] - starting t2 (nsec)

cval[3] - t1 increment (nsec)

cval[4] - t2 increment (nsec)

cval[5] - long time [tL] (nsec) [remote]
cval[6] - tauR (nsec) [remote]

cval[7] - # of data points oftl

cval[8] - # of data points oft2

cval[9] - repetitions/time point
cval[lO] - sequence repetition rate (Hz)
cavl[l 1] - plot scale factor

cval[12] - A timing shim (nsec)
cval[l3] - B timing shim (nsec)
cval[14] - C timing shim (nsec)
cval[lS] — D timing shim (nsec)
cval[16] - E timing shim (nsec) [remote]
cval[l7] - F timing shim (nsec) [remote]
cval[l8] - G timing shim (nsec) [remote]

delay generator set up parameters

tpar[O] - trigger rate

tpar[l] - A delay time (tau)

tpar[2] - B delay time (tau + t1)
tpar[3] - C delay time (tau + t1 + t2)

tpar[4] - D delay time (2*tau +tl + t2) [direct - boxcar gate]
[remote - 5th pulse]
tpar[S] - E delay time (2*tau + t1 + t2 + tL) [remote]

tpar[6] - F delay time (2*tau + t1 + t2 + tL + tauR) [rempte]
tpar[7] - G delay time (2*tau + t1 + t2 + tL + 2*tauR) [remote boxcar gate]

*****************************96*

*-
\

#include <stdio.h>
#include <console.h>
#include <string.h>
#include <stdlib.h>
#include "decl.h"
#include "matrixh"

int gpib0,dg l ,dg2,sri;
char spr,pbuff[2];

typedef struct{int v,h;} POINT;
POINT *mloc;

main()

int i,j,limits[6],cval[19],tpar[8],specparm[4];
int **data,iopt;

float fval[4],tl,dtl;

char chopt[3];

limits[0]=100;
limits[1]=575;
limits[2]=40;
1imits[3]=300;
limits[4]=10;
limits[5]=10;

fval[0]=9.0;
fval[1]=3000.0;
fval[2]=50.0;
fval[3]=4.2;

/* set up console screen */
pageset(stdout);
/* read cval[] from timehy.dat */
tparinhy(cval);
I“ open and clear gpib controller. Then open and clear srC’.45 and dg535 */
if((gpib0=ibfmd("gpib0"))<0)
{ .
printf("gpibO not found\n\r");
exit(—l);
if(ibsic(gpib0) & ERR) error();
if((sri=ibfind("sr245"))<0)
{
printf("sr245 device not found\n\r");
exit(-1);
1
if((dg1=ibfind("dg535"))<0)
{
printf(" first dg535 device not found\n\r”);
exit(-l );
1

if(ibclr(sri) & ERR) error();
if(ibclr(dgl) & ERR) error();

/* get scanning parameters */

change: getparmhyl(cval);

/* allocate data array */
data=imatrix(cva1[7],cva1[8]+10);

/* set up initial timing parameters */

tpar[0]=cval[10]; /* trigger rate */
tpar[l]=cval[0]+cval[12]; /* A delay time */
tpar[2]=cval[0]+cval[1]+cval[l3]; /* B delay time */

tpar[3]=cval[0]+cval[l]+cval[2]+cval[l4]; /* C delay time */
tpar[4]=2*cval[0]+cval[1]+cval[2]+cval[15]; /* D delay time */

/* set up the scanning parameters for sr245 and dg535 devices */
srsethy(cval[91):
repeat: dgsethy 1 (tpar);
/* set up the screen for data aquisition */
eraseplotO:
lplabelhy1(cva1);

box(limits);

um‘fl.m.l&l r... : . : .-' ..

218

/* set timing parameters for display */

specparm[0]=cval[0];
specpann[1]=cval[0]+cval[1];
specparm[2]=cval[0] +cva1[ l]+cval[2];
specparm[3]=2*cval[0]+cva1[1]+cval[2];

dgupdate hyl (0,0,specpa1m[0]);
dgupdatehy l ( l ,0,specparm[1]);
dgupdatehy 1 (2,0.specparrn[2] );
dgupdatehyl (3,0,specparm[3]);

/* data collection - the first point as collected from the integrator

* is usually bad. We will collect cval[7]*(cval[8]+10)

* points, the first twice to make up for this. 10 data

* points are for baseline.

*/
trstarthyl(); /* trigger start */
data[0][0]=getpoint(sri,0); /* data aquisition */
trstophyl(); /* trigger stop */
tl=cval[1];
dt1=cval[3];

t1=t1/1000.0;
dt1=dt1/1000.0;

for(i=0;i<cva1[7];i++) /* t1 timing */
cgotoxy(49,34,stdout); /* put channel # on screen*/

printf("%d",i);
cgotoxy(48,35,stdout); /* put t1 (nsec) on screen */
printf("%.3l",t1);
t1 +=dt1;
for(j=0;j<cva1[8];j++) /* t2 timing */
1

cgotoxy(56,34,stdout); /* put channel # on screen*/
printf("%d",i);

if(kbhit()) goto compl; /* check for keystroke */

trstarthyl(); /* trigger start */
data[i][i]=getpoint(sri.0); /* data aquisition and type signal value */
trstophyl(); /* trigger stop */
dplothy(cval[11],j,cval[8]+10,data[i][j]); /* plot data on screen */
SPCCPm121=ngPdatChy1(2.0Vall41.8pecpann[2]); /* update delay generator */

specparm[3]=dgupdatehyl(3,cval[4],specpann[3]); /* for 3rd pulse and gate */
1

/* get baseline points */
specparm[3]=dgupdatehy1 (3, 100,specparm[3]);

for(i=cval[8];j <cval[8]+ 103+ +)
l

219

cgotoxy(56,34.stdout); /* put channel # on screen*/
printf("%d",j);

trstarthyl();
data[i][j]=getpoint(sri,0);
trstophyl();

dplothy(cval[11],j,cval[8]+10.data[i][j]);

eraseplofl 1; /* set up the screen for data aquisition */
lplabelhyl(cva1);
box(1imits);

/* update for new t1 value */

cgotoxy(28,33,stdout);

printf( " % d" ,specparm[0]);
specparm[l]=dgupdatehy1(1,cva1[3],specparm[1]);
specparm[2]=dgback1(2,cval[8]*cva[[4]—cval[3],specparm[2]);
specparrn [3]=dgbackl (3,cva1[8]*cva1[4] + l OO—cval[3],specpann[3]);

/* display data[(cval[7]-1)][j] set (the last t1 set) */

for(j=0;j<cval[8]+10;j++) dplothy(cval[1l].j,cval[8]+lO,data[(cval[7]-l)][j]);
/* select exit option */
compl: iopt=0;

cgotoxy(68,34,stdout);

printf("l. save data set");
cgotoxy(68,35,stdout);

printf("2. repeat scan - same setting");
cgotoxy(68,36,stdout);

printf("3. repeat scan — change setting");
cgotoxy(68,37,stdout);

printf("4. manual scan");
cgotoxy(68,38,stdout);

printf("S. exit program");

again:

cgotoxy(68,32,stdout);
printf("ENTER OPTION : ");
gets(chopt);
iopt=atoi(chopt);
if(iopt<l I iopt>5)

cgotoxy(63,32,stdout);
ccleol(stdout);
goto again;

if(iopt=l)

savedatahyl(cval,fval,data); /* save comment,fval.cval,date,data */
free_imatrix(data,cval[71): /* dellocate data array */
goto compl;

if(iopt=2) goto repeat;
if(iopt=3) goto change;
if(ioptfl)

error()

{

220

trstarthyl();

ibloc(dg1 );

cgotoxy(68,39,stdout);

printf("CLICK MOUSE WHEN FINISHEDII");
SysBeep( 1 );

for(;;)

{

if(Button()) break;

1
dgl=ib find("dg535");
trstophy 1();

. cgotoxy(63,32,stdout);

ccleol(stdout);
goto again;

ibon1( sri.0);
ibonl(dgl,0);
tparouthy(cval);

printf("GPIB function call errorlnlr");
printf("ibsta=0x%x, iberr = 0x%x1nlr",ibsta,iberr);
printf("ibcnt=0x%x \nIr",ibcnt);

exit(-l);

A6-3. Subprograms

/*************
*
* box.c constructs a box with tick marks on the screen. The
* characteristics of this box are governed by values placed in
* an array of limits whose address is passed to box when called
* from the main prog. The structure is an integer array with elements
* defined as xmin, xmax, ymin, ymax, xtic, ytic
*
**************/

box(plm)

int *plm;

/*

*/

int i,xdel,ydel,xval,yval;
int xmin,xmax,ymin,ymax,ixtic,iytic;

xmin=*plm;
xmax=*++plm;
ymin=*++plm;
ymax=*++plm;
ixtic=*++plm;
iytic=*++plm;

draw the box

Iine(xmin,ymin,xmin,ymax);
cont(xmax,ymax);

/*

/*

*/

\
*

*****

\
*-

*****

221

cont(xmax.ymin);
cont(xmin,ymin);

xdel=(xmax-xmin)/ixtic;
ydel=(ymax-ymin)/iytic;

draw the tic marks */

for(i=1;i<ixtic;i++){
xval=xmin+i*xdel;
1ine(xva1,ymin,xval,ymin+xde1/10);
line(xval,ymax,xval,ymax-xdel/10);

for(i=l ;i<iytic;i++){
yva1=ymin+i*ydel;
line(xmin,yval,xmin+ydel/5,yval);
Iine(xmax,yval,xmax-ydel/S,yval);

return to calling routine

return;

cont.c — this function draws a line on the screen using quickdraw
routines

it is meant to take the place of the routine cont(x1,yl)
that existed in the unix library of think c3.0.

#include <console.h>
cont(x l ,y 1)

int x1.yl;

GrafPtr oldport;

GetPort(&oldport);
SetPort(stdout->window);

LineTo(x l ,y l );
SetPort(oldport);

return;

delaysethyc - this function takes in the delay value and sets it using the
the cursor mode of the dg535. the routine assumes that prior

the function call the user has set the display to the approriate

channel and the the sursor is in position 16, the nanosecond

position. upon return, the cursor is resored to the nanosecond.

#include "decl.h"
extem int dgl;

222

delaysethy(ival)
int ival;

{

int i,k;

long cval,temp;

/* ival can range from 1 nanosecond to lmsec, the leftmost digit is

* tested first, and the dg535 cursor adusted if digit incrementation
* is required.
*/

ibwrt(dg1,"sc 10\r",6L);
if(ibsta & ERR) error();

cval= 1000000;
t‘or(k=0;k<7;k++){
temp=ivallcval;
if(temp){
for(i=0;i<temp;i++){
ibwrt(dgl,"ic lIr",5L);
if(ibsta & ERR) error();
1;
ival=ival-temp*cval;
cval=cvaI/10;
ibwrt(dgl,"MC 11r",5L);
if(ibsta & ERR) error();

/* make sure cursor is left at position 16 */

ibwrt(dgl,"sc 16\r",6L);
if(ibsta & ERR) errorO;

return;

1
/*
* dgback1.c - this routine makes delays of dg535 move backward for new
* time set of t2 in HYSCORE (4—pulse), displays new time
* position on mac, and put new time values into specparm
* array. Assuming dg535 cursor is on nanosecond unit.
*
* j = delay number
* nd = moving backward by nd nanosecond
*

sp = specparmU]

*
\

#include "decl.h"
#include <console.h>

extem int dgl;
dgbackltj,nd,sp)
int j,nd,sp;
int msec,nlOOsec,nlOsec,nsec,value,i;

float fnd,fmsec,fnlOOsec,fn10sec;
char s[12];

223

8101='d';
8111='l';
$121=' ';
8131='1’;
8141='.';
s151='0’;
$161='.';
5171=48+j;
s[8]='\r';
s[9]=‘\0';

/***************

* select device and update it by nd nsec
* note: the timing shims will be taken care of in the initialization

* of the generators upon exiting from the setup table
***********/

ibwrt(dgl,s,9L);
if(ibsta & ERR) error();

/* move backward on 1.1 sec unit */

fnd=nd;
fnd=fnd/1000.0;
msec=fnd;

ibwrt(dgl,"MC 0;MC 0;MC Olr",15L);
if(ibsta & ERR) error();

for(i=0;i<msec;i++)

ibwr1(dg1,”IC Olr",5L);
if(ibsta & ERR) error();
1

/* move backward on lOOn sec unit */

fmsec=msec;
fnd=( fnd— fmsec)* 10;
n lOOseczfnd;

ibwrt(dgl,"MC lIr",5L);
if(ibsta & ERR) error();

for(i=0;i<nlOOsec;i++)

{

ibwrt(dgl,"IC Olr",5L);
if(ibsta & ERR) error();
} .

/* move backward on lOn sec unit */

fn lOOsec=n lOOsec;
fnd=(fnd-fn100sec)*10;
n103ec=fnd;

ibwrt(dgl,"MC llr",5L);
if(ibsta & ERR) error();

for(i=0;i<nlOsec;i++)

{
ibwrt(dgl,"lC 011'".5L);
if(ibsta & ERR) error();

224

/* move backward on n sec unit */

fnlOsec=n105ec;
fnd=(fnd—fnlOsec)* 10;
nsec=fnd',

ibwrt(dgl."MC l\r",5L);
if(ibsta & ERR) error();

for(i=0;i<nsec;i++)
ibwrt(dg1,"IC 0\r",51_.);

if(ibsta & ERR) error();
1

/* update specparm and display on mac */

********************:

*
\

value=sp-nd;
cgotoxy(28,33 +j,stdout);
printf("%d",va1ue);

retum(value);

dgsethyl.c - this rouitne is responsible for setting up the dg535
delay and gate generator. It first finds and returns the
integer descriptor for the device, then the device is
cleared, the trigger rate is set, the outputs are changed
to 50 ohms, the starting times are computed and set. the unit is
placed in cursor mode for timing updates during a scan, and
finally, the display is set to the appropriate channel so
that the operator can monitor the progress of a scan from the
front panel.

the setup parameters are in an input array, that is passed
to the routine in the call statement. the definitions of the
5 values in this array are as follows

tpar[O] - int trigger rate (Hz)
tpar[l] - Adelaytirne

tpar[2] — B delay time

tpar[3] - C delay time
tpar[4] - D delay time

#include "decl.h"
extem int dgl;
extem char spr,

dgsethy l (tpar)
int *tpar;

int trate.ast.bst.cst,dst;
char sl[10].s[15];

int 1;

long rent;
trate=tpar[0];

ast=tpar[ l ];
bst=tpar[2];

 

225

cst=tpar[3 ];
dst=tpar[4];

/* the device has been opened prior to calling this routine,
* clear it by sending some caraiage returns
*/

for(i=0;i<5;i++){
ibwrt(dgl."\r".lL);

if(ibsta & ERR) error();

};

. if(ibrsp(dgl,&spr) &ERR) error();

/* do a device clear and then set the internal trigger rate
* and the to,a,b,c, and d outputs to 50 ohms
*/

ibwrt(dg1,"cl\r",3L);

if(ibsta & ERR) error();

s[0]='t';
s[1]='r';
s[21=' ';
s[3]=‘0';
s[4]=',';

rcnt=stci_d(s 1,trate,6);
for(i=0;i<rcnt;i++){
s[5+i]=s l[i];

’

s[rcnt+5]=’lr';
s[rcnt+6]='\0';

rcnt=rcnt+6;

ibwrt(dg1,s,rcnt);
if(ibsta & ERR) error();

ibwrt(dg1."dl O.l,0\r",9L);
if(ibsta & ERR) error();

ibwrt(dg1,"tz 1,0;tz 2,0;tz 3,0;tz 5.1;tz 6,1\r",35L);
if(flmta &ERR) error();

/* initialize delays to zero */

ibwrt(dg1,”dt 2,1 ,0\r".9L);
if(ibsta & ERR) error();

ibwrt(dgl,"dt 3,1,0lr",9L);
if(ibsta & ERR) error();

ibwrt(dgl,"dt 5,1,0\r”,9L);
if(ibsta & ERR) error();

ibwrt(dgl,"dt 6,1,0lr",9L);
if(ibsta & ERR) error();

/* set cursor mode and place c ursor on nsec digit */

ibwrt(dg1 ,"dl l,0,0\r",9L);
if(ibsta & ERR) error();

/*

* * * * * * *':

*
\

226

ibwrt(dgl,"cs 0;sc l6lr",1 1L);
if(ibsta & ERR) error();

now set initial delay values */
if(ast) delaysethy(ast);

ibwrt(dgl,"dl l.O,l\r".9L);
if(ibsta & ERR) error();

if(bst) delaysethy(bst);

ibwrt(dg1,"d1 1,0,2lr",9L);
if(ibsta & ERR) error();

if(cst) delaysethy(cst);

ibwrt(dg1,"dl 1,0,3\r",9L);
if(ibsta & ERR) error();

if(dst) delaysethy(dst);

return;

}

dgupdate4p.c — this function is responsible for updating the
delay values of the 4 delay channels in the SRS dg535
digital delay generator. [I also updates the display with the new
value and returns this value to the specparrn array. the routine uses
the dg535's cursor mode and assumes that the cursor is at
the nanosecond digit prior to the function call.

#include "decl.h"
#include <console.h>

extem int dgl;
dgupdate4p(j,nd,sp)
int j,nd,sp;

j: delay channel selected
nd: # of 1 nsec increments to be made
sp: specpann value

{

int value,i;
char s[12];

s[0]='d';
s[1]='l';
8[2]=' ‘;
s[3]=‘1°;
Sl4]='.';
s[5]='0';
Sl6l='.';
8l7l=48+j;
s[8]=‘\r';
s[9]='\0';

/***************

227

* select device and update it by nd nsec

* note: the timing shims will be taken care of in the initialization

* of the generators upon exiting from the setup table
***********/

ibwrt(dgl.s.9L);
if(ibsta & ERR) error();

for(i=0;i<nd;i++){

ibwrt(dgl,“lC llr".5L);
if(ibsta & ERR) error();
};

/* update specparm and the display on mac */
value=sp+nd;
cgotoxy(28,34+j,stdout);
printf("%d",value);

return(value);

/*
* dplot4p.c - plot 4-pulse ESEEM data point on screen
a:

* fmax = maximum value of data set

* Emin = minimum value of data set

* fi = current data position on x-axis

* fnum = total data point If

* fdata = current data value

*/

dplot4p(max.i,num,data)
int max,i,num,data;

{ .
int x,y;
float fmax,fmin,fi.fnum,fdata,fx,fy;
float bxmax,bxmin,bymax,bymin,bdelx,bdely;

bxmax=575;
bxmin=100;
bymax=300;
bymin=40;
bdelx=bxmax-bxmin;
bdely=bymax~bymin;

fmax=max;
fmin=-fmax/S;
fi=i;

fnum=num;
fdata=data;
fdata=fdatavfmin;

/* fraction of x,y axis */

fdata=1-fdata/( fmax-fmin);
fi=fi/(fnum-1.0);

/* plot */

fx=fi*bdelx+bxmin;

228

fy=fdata*bdely +bymin;

x=fxz

Y=fy;

point(x,y);

return;

/*

/*

/*

/*

/*

/*

getpoint(sri.iarr) - gets data point from sr245 boxcar interface

after the device has been triggered. The result is
transmitted as a two byte string which is then
assembled into integer form, displayed on the

and returned for storage in the approprite anay.

#include <stdio.h>
#include "decl.h"
#include <console.h>

extem char spr,pbuft{2];
extem int sr'i,dg535;

getpoint(sri,iarr)

int iarr;

int value,sval,vall,val2;
int rqs.cmpl,end;
rqs=0x800;
end=0x2000;
cmpl=0x100;

value=0;

enable the running average on the erSO by swinging sb2 high */

ibwrt(sri,"sb2=1\r",6L);
if(ibsta & ERR) error();

scan port 1 and set rqs alter l trigger */

ibwrt(sri,"ssl :1lr",6L);
if(ibsta & ERR) error();

wait until rqs set on sr245 */
ibwait(sri,rqs);
read status byte and check data validity */

if(ibrsp(sri,&spr) & ERR) error();
if(spr & 2) printf(”data value out of rangelnlr");

read in value */

ibrd( sri,pbuff,2L);
if(ibsta & ERR) error();

clear the integrator running average */

/*

/*

/*

\
it

*****

 

229

ibwrt(sri,"sb2=0\r",6L);
if(ibsta & ERR) error();

reconstruct data value */

sval=1;

vall=pbuffl1];

va12=pbufflO];

if(va12 & l6) sva1=—1;

value=sval*((va12 & 15)*256+((vall>>4) & 15)*16+(vall & 15));

clear the status byte - check for scan end condition */

for( ; ; ){
ibrsp(sri,&spr);
if(spr & 15) printf("sr245 error alter read");
if(spr & 16) break;

9

update the screen */
cgotoxy(10,34+iarr,stdout);

printf("%d ",value);
retum(value);

kbhit() - this function returns a non-zero result if the keyboard has been
depressed

It will be usd to prematurely stop data collection

kbhit()
EventRecord the_Event;
EventAvail(40,&the_Event);

retum(the_Event.what);

}

#include <stdlib.h>
#include <stdio.h>
#include "matrix.h"

void matrixerrodchar *text)

{

}

printf(text);
exit( 1);

int **imatrix(int row,int col)

{

int i, **m;

m = ( int **) malloc( (size_t) row * sizeof(int*));
if(lm) matrixerror(" Allocation Faliure For imatrixln");

for(i =0; i <row; i++ ) {
m[i]=(int*) malloc( (size_t) col*sizeof(int));
if(!m[i]) matrixerror(" Allocation Faliure For imatrixln");

230

return m;

}

void free_imatrix(int **m, int row)
int i;

for(i = O; i < row; i++ ) free( m[i] );

free( m );
}
/=r:
* lplabel4p.c - labelling the grid for the dac live plot
* in 4—pulse ESEEM experiment
*/

#include <console.h>

lplabel4p(cval)

int *cval;
{

int i;

float t,t2,t4,t8.n;

Char ampl[4]={'E','C','H‘,'O'};
char amp2[4]={' ','A’.'M','P'};
char amp3[4]={’L','I',T','U'};

char amp4[4]={'D','E'.' 3' 'i:

cgotoxy(45,2,stdout);
printf("KEYSTROKE ABORTS SCAN");
/* y-axis */
for(i=0;i<=3;i++)

cgotoxy(5,9+i,stdout);
printf("% .ls",&amp l [i]);

for(i=0;i<=3;i++)

cgotoxy(5,13 +i,stdout);
printf("%.1s".&amp2[i]);

for(i=0;i<=3;i++)

{
cgotoxy(5,l7+i,stdout);
printf("% .1s",&amp3[i]);

for(i=0;i<=3;i++)

{
cgotoxy(5,21+i.stdout);
printf("% . ls ",&amp4[ i1);

cgotoxy(12,27,stdout);
printf("%d”,-cval[5]/5);

f:

 

231

cgotoxy(12.4,stdout);
printf("% d",cval[5] );

/* x-axis */

t2=cval[0];
t4=cval[1];
t8=cval[2]:
n=cva1[10];

t=(t2+2.0*t4)/1000.0;
cgotoxy(15,29.stdout);
printf(" % .21" ,t);

t=(t2+2.0*t4+(n+19.0)*2.0*t8)/1000.0;
cgotoxy(94,29,stdout);
printf("%.2t",t);

t=(2.0*t2+4.0*t4+(n+19.0)*2.0*t8)/2000.0;
cgotoxy(54,29,stdout);
printf("%.2t",t);

cgotoxy(49,31,stdout);
printf("tau + T (usec)");

/* information */

 

for(i=0;i<=3;i++)

{
cgotoxy(5,34+i,stdout);
printf("S%d = ",i);
}
for(i=0;i<=3;i++)
{

cgotoxy(23,34+i,stdout);
printf("T%d = ".i);
}

cgotoxy(41 .34,stdout);
printf("ch# = ");

cgotoxy(1,1,stdout);

return;

#include <stdlib.h>
#include <stdio.h>
#include "matrix.h"

void matrixermr(char *text)

{
printf(text);
exit(l);
}
int **imatrix(int row,int col)
{
int i, **m;

m = ( int **) malloc( (size_t) row * sizeof(int*));

232

if(lm) matrixerron" Allocation Faliure For imatrixln");

l‘or(i =0; i < row; i++ ){
m[i]=(int*) malloc( (size_t) col*sizeof(int));
if(!m[i]) matrixerror(" Allocation Faliure For imatrixln");

return m;
}
void free_imatrix(int **m, int row)
{
int i;
for(i = 0; i < row; i++ ) free( m[i] );
free< m );
l
/*
* tparin4p.c - read cval[] from "time4p.dat"
*/

#include <stdio.h>
tparin4p(cval)
int cval[];
int i;
FILE *fp,*fopen();
i{f((fFfopen("time4p.dat","r")) == '\0’)

printf("file <time4p.dat> not found ! \n");
for(i=0;i<=10;i++) cval[i]=0;

else

for(i=0;i<=10;i++) fscanf(fp,"%dln",cval++);
l

fclose( fp);
SysBeep( 1 );

return;

* tparout4p.c - save timing parameters cval[] for 4-pu1se
* in time4p.dat file
*/

#include <stdio.h>
#include <console.h>

tparout4p(cval)

int cval[];

\
****

*l

\
***

233

int i;
FILE *fp.*fopen();
if((fp=fopen("time4p.dat","r+")) = 'lO')
{
printf("file <time4p.dat> not found ! ln");
for(i=0;i<=10;i++) cval[i]=0;
1
else
for(i=0;i<=10;i++) fprintf(fp,"%d\n",cval[i]);
}
fclose(fp);
cgotoxy(68,39,stdout);
ccleol(stdout);
printf("file <time4p.dat> updated");
cgotoxy(1,l,stdout);
SysBeep(l);
return;
trstarthylc - this routine starts the dg535 internal rate generator. it is
presumed that the trigger rate has already been appropriately
set by the user.
#include "decl.h"
extem int dgl;
trstarthle
ibwrt(dgl,"tm 0\r”,5L):
if(ibsta & ERR) error();
retum;
}
trstophyl .c - this function stops the internal rate generator on the dg535
so that the unit is not triggered during an update

#include "decl.h"
extem int dgl;

trstophy 1 ()

ibwrt(dgl,"tm 21r".5L);
“if(ibsta & ERR) error();

return;

}

234

A7. Computer Programs for Searching Angle Sets from an Anisotropic

EPR Spectrum to Interprete ESEEM Spectra

The computer program listed below was used to search the angle sets
of samples excited by a microwave pulse from an anisotropic EPR
spectrum of randomly oriented samples. The selected angle sets are used to
interprete orientation selective ESEEM or ENDOR spectra. The program
is written in MATLAB v.4.2 (Mathworks, Inc.).

A7—1. Main Program

%
% anselm - this generate selected theta and phi angle sets from g-anisotropic

% and A (hyperfine enrgy)-anisotopic EPR pattern at some microwave
% frequency and magnetic field strength

%

% This finds out angle set (theta,phi) which satisfy the following
% equation

% hv - ml*A(theta,phi)

% = —-—-—-——-—--——-

% betae*g(theta,phi)

%

%

% Outputs of this subroutine are

%

% —l -1 1

% angset(theta,phLtheta.phi,....)

% l | l l

% columns

%

% angsetl - angle sets which are symmetry of angset

% about the center of sphere

% nurnmi - vector : each culumn has the information of
% the selected angle set number

% ofmI=—l,-l+l,...

% prob — probabilities of each angle set for line integral
% same row size , half culumn size as angset

%

global epr;

% rotate Hyperfine PAS to g—tensor axis by Euler rotation
AA=eulerA(epr(4:6),epr(7:9));
% loop for searching the theta and phi set

phi=-719:7 l9;phi=pi*phi/180:
theta=02180;theta=pi*theta/180;

235

theta(l82)=theta(1);
icul=l;

nmi=0;

nummi=[0 0 0 0 0 0 0 O];
dphi=pi/180;
dphi2=pil360;

for mi=-epr(12):epr( 12).

nmi=nmi+ l;

gor=lz

theind=1;

phiind=720;
temp1=rescalc(AA.mi,theta(theind),phi(phiind));

numset=0;

while (gop>0) & (theind< 182)
theind=theind+ 1;
temp2=rescalc(AA,mi,theta(th eind),phi(phiind) );
gop=temp 1 *temp2;
temp1=temp2;
end

if theind<1 82
numset=numset+ 1;

% the first angle set

an gset( numsetjculzic u1+ 1 )=[(theta(theind) +theta(theind—1 ))/‘2 phi(phiind)];
angsetl(numset,icul:icu1+ 1)=[pi-angset(numset,icul) pi+angset(numset,icul+1)];

% initiate the searching loop

orst=[theind,0];
phfind=phfind+1;
nalphi= 1;
clor=[theind nalphi];
iclopt=1;

temp4=templ ;
temp 1 =rescalc(AA,mi,theta(theind),phi(phiind));
temp2=rescalc(AA,mi,theta(thein d- 1),phi(phiind));
temp3=rescalc(AA,mi,theta(theind—l),phi(phiind- 1));

gop=temp1*temp2;

if temp1*temp2 <0 clopt=1;
elseif temp2*temp3<0 clopt=2;
else clopt=4;

end

probcalc;
selangmvcell;

% searching loop
while orst(1)<clor(1)|orst(1)>clor(1)lorst(2)<clor(2)lorst(2)>clor(2)

if iclopt=1
temp4=temp1 ;
temp3=temp2;
templ=rescalc(AA,mi,theta(theind),phi(phiind));
temp2=rescalc(AA,mi,theta(theind-l),phi(phiind));

236

if temp 1 *temp2<0 clopt=1 ;
elseif temp2*temp3<0 clopt=2;
else clopt=4;

end

elseif iclopt=2
temp 1 dempZ;
temp4=temp3;
temp2=rescalc(AA.mi,theta(theind-1).phi(phiind));
temp3=rescalc(AA,mi,theta(theind-1 ),phi(phiind- 1 ));

if templ *temp2<0 clopt=1;
elseif temp2*temp3 <0 clopt=2;
else clopt=3;
end

elseif iclopt=3
temp l=temp4;
temp2=temp3;
temp3=rescalc(AA,mi,theta(theind—1 ),phi(phiind-l ));
temp4=rescalc(AA,mi,theta(theind),phi(phiind—1));

if temp2*temp3 <0 clopt=2;
elseif temp3*temp4<0 clopt=3;
else clopt=4;
end
else % iclopt=4 case
temp2=templ;
temp3=temp4;
templ=rescalc(AA,mi,theta(theind),phi(phiind));
temp4=rescalc(AA,mi,theta(theind),phi(phiind-1));
if templ *temp2 <0 clopt=1;
elseif temp3*temp4<0 clopt=3;
else clopt=4;
end
end

probcalc;
selangmvcell;

end

clopt=1; % probability calcuation for last angleset
probcalc;

end

ic ul=icul+2;
nummi(nmi)=numset;

end

return;

A7-2. Subprograms

%

2 3 7
% eprinputm - EPR parameters which determine the EPR line shape
%

global epr;
epr=[2.0 2.0 2.5 20 20 400 0 0 0 3000 9.000 1.5];

fprintf(' \n');

fprintf(‘ ----------------------------- \n n';)
fprintf(' Type the number to be changedln );
fpnntf(‘ ------------------------------ \n ';)
fprintf( 1. gx — -%10. Sfin' ,;epr(1))
fprintf('Z. gy — -.%10 Slln' epr(2));
fprintf('3. gz = %10.51\n',epr(3));
fprintf(‘4. Ax(MHz) = %10.5fln'.epr(4));
fprintf('S. Ay(Ml-lz) = %10.5fin'.epr(5));
fprintf(‘6. Az(MHz) = %10.5 fln',epr(6));

fprintf('7. alpha(degree) = %10.2f\n'.epr(7));
fprintf(‘S. beta (degree) = %10.2fin'.epr(8));
fprintf('9. gamma(degree) = %10.2fin',epr(9));
fprintf(‘IO. field(G) = %10.2fln‘,epr(10));
fprintf('ll. frequency(GHz) = %10.5fin',epr(1l));
fprintf('l2. l(nuclear spin): %10.lfln',epr(12));
fprintf('13. listln');

fprintf('l4. calculation\n');

fprintf(' ------------------------------- \n');
fprintf('ln');

n=0;

while n<l4 l n>14
n=input('number to be changed : ');
if n=l epr(l)=input('1. gx : ');
elseif n=2 epr(2):input('2. gy = ‘);
elseif n=3 epr(3)=input('3. gz = ');
elseif n=4 epr(4)=input('4. Ax (MHz) = ');
elseif n=5 epr(5)=input('5. Ay (MHz) = ');
c1seifn=6 epr(6)=input('6. Az (MHz) = ');
elseif n=7 epr(7)=input('7. alpha (degree) = ');
elseif n=8 cpr(8)=input('8. beta (degree) = ');
elseif n=9 epr(9)=input('9. gamma (degree) = ');
elseif n=10 epr(10)=input('10. field (G) = ');
elseif n=ll epr(11)q'nput('11. frequency (6112) = ‘);
elseif n=12 cpr(12)=input.('12. 1 (nuclear spin) = ');
elseif n=l3

fprintf(' \n );

fprintf(' —-——-————————-\n );
fprintf(' Type the number to be changedln' );
fpnntf(' --—-——-—-—-——-———-\n‘ );
fprintf('l. gx - %—10. Sfin' ,epr(1));
fprintf('2. gy =%10.5fin'epr(2));
fprintf('3. gz = %10.5f\n'.epr(3));
fprintf('4. Ax(MHz) = %10.5fln‘,epr(4));
fprintf('S. Ay(MHz) = %10.5fln',epr(5));
fprintf('6. Az(MHz) = %10.5 fln‘.epr(6));

fprintf('7. alpha(degree) = %10.2fln‘,epr(7));
fprintf(‘S. beta (degree) = %10.2fin',epr(8));
fprintf('9. gamma(degree) = %10.2fln'.epr(9));
fprintf(‘lo. field(G) = %10.2fln',epr(10));
fprintf('ll. frequency(GHz) = %10.5fin',epr(11));
fprintf('12. l(nuc1ear spin): %10.5f\n',epr(12));
fprintf('13. listln');

fprintf('14. calculation\n');
fprintf('———--——-——-—————\n');

2 3 8
fprintf('\n');
end

end

epr(4)=epr(4)*1e6;
epr(5)=epr(5)*1e6;
epr(6)=epr(6)*le6;
epr(7)=epr(7)*pi/180;
epr(8)=epr(8)*pi/180:
epr(9)=epr(9)*pi/180;

epr(l 1)=epr(l 1)*1e9;

return;

%

% eulerAm - this rotate compin(l:3) in some axis system to compout(l:3,l:3)
% in other axis by Euler rotation

%

% input compin(l:3)
% eulang(l:3) - alpha.beta,gamma
% output compout(l:3,1:3)

r I A I

function W, . _ Prrlang)

sint=sin(eulang);
cost=cos(eulang);

Rzl=[cost(l) sint(l) 0;—sint(1) cost(1) 0:0 0 1];
Ry=[cost(2) 0 -sint(2):0 1 0;sint(2) 0 cost(2)];
R12=[cost(3) sint(3) 0;—sint(3) cost(3) 0;0 0 1];

R=Rz 1 *Ry*RzZ;
compout=zeros(3);

for i=lz3,
compout(i,i)=compin(i);
end

compout=R'*compout*R;

return;

%

% pmbcalc. m - this calculates the line integral probability value
% in angle selection ESEEM or ENDOR

%

% prob=sqrt((dtheta/dphi)"2+sin(theta)"2)*dphi

%

if iclopt=1
if clopt=4 I clopt=2
prob(numset.(icu1+1)/2)=sqrt(1+(sin(angset(numset.icul)))"2)*dphi2;
else
prob(numset,(icul+l)f7) _‘ ’ ’ ’ iclll\))"2)*dphi;
end
elseif iclopt=2
if clopt=1 I clopt=3
prob(numset,(icul+1)/2)=sqrt(1+(sin(angset(numset.icul)))"2)*dphi2;
e se
prob(numset,(icul+1)/2)=0;
d

 

en

239

elseif iclopt=3
if clopt=2 I clopt=4
prob(numset,(icul+1)/2)=sqrt(1+(sin(angset(numset,icul)))"2)*dphi2;
else
prob(numset,(icu1+1)/2)=sqrt((sin(angset(numset,icul)))A2)*dphi;
end
else
ifclopt=1 l clopt=3
prob(numset,(icul+1)/2)=sqrt(1+(sin(angset(numset,icul)))"2)*dph12;
else
prob(numset,(icul+1)/2)=0;
end
end

return ;

% rescalc.m - calcuate the value of the following equation

 

% hv - mI*A(theta,phi)
% f = Hr— ---——-—-—-— «_____
% betae*g(theta,phi)
%

% at some theta and phi value

function x=rescalc(AAin,miin,thetain,phiin)
global epr;

% calculate g(thetain.phiin)
% direction cosines
h=[sin(thetain)*cos(phiin) sin(thetain)*sin(phiin) cos(thetain)];

gh=epr(1:3).*h;
geff=sqrt(gh*gh');

% calcuate A(thetain,phiin)

Ah=gh*AAin;
Aeff=sqrt(Ah*Ah')/geff;

% calcualte gcalc
x:epr(10)—7.1448e~7*(epr(1l)-miin*Aeff)/geff;

return;

% selangmvcellm - this stores the selected angle set and
% makes the cell, used in root finding of some function,
% move up,down,left or right.

% Actually, this makes the origin of cell move.

% x -————- x

% l 2 |

% l3 1 I <——- cell

% | l 1,2,3,4 - side numbers
% l 4 |

% x —--—-- x <-———-origin

%
numset=numset+1;

if clopt=1
angset(numset,icul:icul+ l )=[(theta(theind)+theta(theind—l ))/2 phi(phiind)];
phiind=phiind+1;
nalphi=nalphi+ 1;

elseif clopt=2
angset(numset.icul:icul+1)=[theta(theind-1) (phi(phiind)+phi(phiind-1))/2];
theindztheind— 1;

elseif clopt=3
angset(numset,icul:icul+1)=[(theta(theind)+theta(theind-1))/2 phi(phiind—l)];
phiind=phiind-1:
nalphi=nalphi-1;

else
angset(numseLicul:icul+1)=[theta(theind) (phi(phiind)+phi(phiind-1))l2];
theind=theind+1:

end

angsetl(numset,icul:icul+l)=[pi-angset(numset,icul) pi+angset(numset,icul+l)l;
iclopt=clopt;

if nalphi=360 nalphi=0;
end

clor=[theind nalphi];

return;

 

241

A8. Computer Programs for Calculating Orientation Selective Four-Pulse

ESEEM Frequency Domain Spectrum for S=l/2 and 1:1/2 Spin System

The computer program used to calculate orientation selective four-
pulse ESEEM frequency domain spectra is listed below. The program is
designed for a spin system described by anisotropic g- and/or metal
hyperfine tensors. The ligand hyperfine interaction is assumed to be axial.
First part of the program determines the selected angle sets (0,4)) from
EPR parameters (Appendix, A7) and the second part of the program
generates the four-pulse ESEEM spectrum. The program is written in
MATLAB v.4.2 (Mathworks, Inc.).

A8-1. Main Program

%

% ansel4ph.m - this program generate angle selection 4—pulse ESEEM frequency
% domain spectrum for S=l/2 and 1:1/2

%

% epr parameter input

global epr;
eprinput;

% angle selection - g anisotropic and A anisotropic
ansel;

eseem=[5.58566 2.0 2.5 o 0];
width=0.l;

eseopt=1 ;
while eseopt=1

% eseem parameter input
eseeminput;

% gaussian function for convolution
gaus;

% eseem frequency calculation

fpfrsp:

2 4 2
fp frdspl;

end

A8—2. Subprograms

‘2: Acalcm - calculate the superHyperfine matrix in g-tensor axis

0

% direction cosine of superhyperfine axis in g-axis
r=[sin(eseem(4))*cos(eseem(5)) sin(eseem(4))*sin(eseem(5)) cos(eseem(4))];
T=7.0692e-18*eseem(l)/(eseem(3))"3;

% superhyperfine matrix in g-axis

gmat=lepr(l) 0 0:0 epr(Z) 0:0 0 epr(3)l;

amat=eye(3)*eseem(2);

rmat=3*r'*r-eye(3);

A=—T*gmat*rmat+amat;

return;

%

% eseeminputm - gets eseem parameters for 1:1/2 case
%

% eseem(l) - nuclear g-value

% eseem(2) - Aiso

% eseem(3) - dipole-dipole distance
% eseem(4) - theta-n

% eseem(S) - phi-n

 

 

%

fprintf('ln');

fprintf(' \n');

fprintf(' type the number you wantln');

fprintf(' \n');
fprintf(‘l. gn = %10.5fln’,eseem(1));
fprintf('12. Aiso (MHz) = %8.3fin',eseem(2));

fprintf('3. dipole distance (A) = %8.3f\n',eseem(3));
fprintf('4. theta—n (dgree) = %7.2fln',eseem(4));
fprintf('S. phi-n (dgree) = %7.21\n',eseem(5));
fprintf('6. gaussian band width = %7.2fln',width);
fprintf(’7. listln’);

fprintf('8. calculation\n');

fprintf(' \n’);
fprintf('ln');

 

eseemopt=0;

while eseemopt<8 | eseemopt>8
eseemopt=input('number to be changed : ');
if eseemopt=l eseem(1)=input('1. gn = ');
elseif eseemopt=2 eseem(2)q‘nput('2. Aiso (MHz) = ');
elseif eseemopt=3 eseem(3)=input('3. dipole distance (A) = ’);
elseif eseemopt==4 eseem(4)=input('4. theta-n (dgree) = ');

243

elseif eseemopt=5 eseem(5)q'nput('5. phi-n (dgree) = ');
elseif eseemopt=6 width=input('6. gaussian band width = ‘);
elseif eseemopt=7

 

 

fprintf('ln');

fprinth \n');

fprintf(' type the number you wantln');

fprintf(' \n');
fprintf(‘l. gn = %10.5fin’.eseem(1));
fprintf('2. Aiso (MHz) = %8.3fln',eseem(2));

fprintf('3. dipole distance (A) = %8.3 fln',eseem(3));
fprintf('4. theta-n (dgree) = %7.2fin',eseem(4));
fprintf('S. phi-n (dgree) = %7.2fin‘,eseem(5));
fprintf('6. gaussian band width = %7.2fin‘,width);
fprintf('7. listln');
fprintf(‘8. calculation\n’);
fprintf(’ \n');
fPl‘intf('\n');
end
end

 

eseem(2)=eseem(2)* 1 e6;
eseem(3)=eseem(3)*1e-8;
eseem(4)=eseem(4)*pi/180;
eseem(5)=eseem(5)*pi/180;

return;

%
% fpfrdsplm — display 4-pulse (S=1f2,l=1/2) simulation results

%

imin=1 ;
imax=n(2);
eseopt=0;

while eseOpt=0

plot(2*fr(imin:imax),l(imin:imax));
xlabel('Frequency (MHz)');

 

 

ylabel('Amplitude’);

% grid;

fprintf(‘ln');

fprintf(' \n');
fprintf(' Type the number you wantln');
fprintf(’ \n’);

fprintf('l. valn');

fprintf(‘2. vbln');

fprintf('3. v(a+b)\n');

fprintf('4. v(a-b)\n');

fprintf('S. va+vb\n‘);

fprintf('6. v(a+b)+v(a-b)ln');

fprintf('7. allln');

fprintf('8. change max,min frequenciesln');
fprintf('9. tau effect (on)\n');

fprintf('lo. tau effect (offiln');

fprintf('ll. change ESEEM parametersln');
fprintf('12. println');

fprintf('13. new plot windowln');
fprintf('14. Larmor £requency\n’);
fprintf('IS. tau suppression plotln');

2 4 4
fprintf('16. quit\n');

fprintf(' \n');

fprintf('ln');

 

iopt=input('Type the number you want : ’);

if iopt=1,
Itot=la;
l=ltot((nn(2)-1)/2+1:(nn(2)-l)/2+n(2));
imin=1;imax=n(2);
elseif iopt=2, ‘
Itot=lb;
l=ltot((nn(2)—1)/2+l:(nn(2)-l)/‘2+n(2));
imin=1;imax=n(2);
elseif iopt=3,
Itot=lapb;
I=Itot((nn(2)— l )1? +1 :(nn(2)-1)/2+n(2));
imin=1;imax=n(2);
elseif iopt=4,
Itot=lamb;
l=ltot((nn(2)-1)/2+1 :(nn(2)-l)/2+n(2));
imin=1 ;imax=n(2);
elseif iopt=5,
ltot=la+lb;
l=ltot((nn(2)-1)/2+l:(nn(2)-l)/2+n(2));
imin=1;imax=n(2);
elseif iopt=6,
ltot=lapb+lamb;
l=ltot((nn(2)—1)/‘2+1:(nn(2)-1)/2+n(2));
imin=1;imax=n(2);
elseif iopt=7,
Itot=la+lb+lamb+lapb;
l=ltot((nn(2)-l)/2+1:(nn(2)-1)/2+n(2));
imin=1;irnax=n(2);
elseif iopt—-=8,
imin=input('Minimum Frequency (MHz) :');
imax=input('Maximum Frequency (MHz) :');
imin=round((imin/2+10)*6000/60)+1;
imax=round((imax/2+ 1 0)*6000/60)+ 1;
elseif iopt=9,
tauq'nputCtau (nsec) :');
tau=tau*1e-9;

ca=c2.*cos(2*pi*(frb-fra/2)*tau)+52.*cos(2*pi*( frb+fra/2)*tau);
ca=ca-cos(pi*fi'a*tau);
ca=2*ca;
cb=c2.*cos(2*pi*(fra-frb/2)*tau)+s2.*cos(2*pi*(fra+frb/2)*tau);
cb=cb—cos(pi*frb*tau);
cb=2*cb;
cp=-4*c2.*sin(pi*tau*fra).*sin(pi*tau*frb);
cm=-4*s2.*sin(pi*tau*fra).*sin(pi*tau*frb);

la=zeros(si2e(fr));
Ib=zeros(size(fr));
Iapb=zeros(size(fr));
Iamb=zeros(size( fr));

n=size( fra);

for i=1 :n(2),
Ia(nfra(i))=Ia(nfra(i))—k(i)/2*probfr(i)*ca(i);
Ib(nfrb(i))=lb(nfrb(i))-k(i)/2*probfr(i)*cb(i);
lamb(nframb(i))=lamb(nframb(i))-k(i)/2*probfr(i)*cm(i);
Iapb(nfrapb(i))=Iapb(nfrapb(i))-k(i)/2*probfr(i)*cp(i);

245

end

lambsaa);
lb=abs(lb);
lamb=abs( lamb);
Iapb=abs(lapb);

% convolution

la=conv(gau,la);
Ib=conv(gau,lb);
lapb=conv(gau,lapb);
Iamb=conv(gau,lamb);

Itot=la+Ib+lapb+lamb;

% rearrange the intensity vector due to convolution

nnqize(gau);
n=size(fr);

l=ltot((nn(2)-l)/2+1:(nn(2)-1)/2+n(2));
imin=1;imax=n(2):

elseif iopt=10;

Ia=zeros(size(fr));
Ib=zeros(size(fr));
lapb=zeros(size(fr));
Iamb=zeros(size( fr));

n=size(fra);

ca=ones(size(k));
cb=ones(size(k));
cm=ones(size(k));
cp=ones(size(k));

for i=1 :n(2),

la(nfra(i))=Ia(nfi'a(i))-k(i)/2*probfr(i)*ca(i);
Ib(nfr’o(i))=Ib(nfib(i))-k(i)/2*probfr(i)*cb(i);

lamb(n framb(i))=lamb(n framb(i))-k(i)/2*prob fr(i)*cm(i);
Iapb(nfrapb(i))=lapb(nfrapb(i))—k(i)/2*pmbfr(i)*cp(i);

end

Ia=abs(Ia);
Ib=abs(Ib);
Iamb=abs(lamb);
lapb=abs(lapb);

% convolution
la=conv(gau,la);
Ib=conv(gau,lb);
lapb=conv(gau,lapb);
Iamb=conv(gau,lamb);
Itot=la+Ib+lapb+Iamb;

% rearrange the intensity vector due to convolution

nn=size(gau);
n=size(fr);

246

l=ltot((nn(2)-1)/2+1:(nn(2)-l)/2+n(2));
imin=1;imax=n(2);

elseif iopt=ll
eseem(2)mseem(2)* 1e-6;
eseem(3)=eseem(3)* 1e8;
eseem(4)=eseem(4)* 180/pi;
eseem(5)=eseem(5)*180/pi;
eseopt=1;
elseif iopt=l2 print;
elseif iopt=l3 figure;
elseif iopt=14,
fprintf('ln');
fprintf('vn/Z = %7.2f MHz : vn = %7.2fMHz',vn*1e-6/2,vn*1e—6);
fprintf('ln');
elseif iopt=15,
tau=input('tau (nsec) : ');
tau=tau*1e-9;

ca=c2.*cos(2*pi*(frb-fra/2)*tau)+52.*cos(2*pi*(frb+fra/2)*tau);
ca=ca—cos(pi*fra*tau);
ca=2*ca;
cb=c2.*cos(2*pi*(fra—frb/2)*tau)+52.*cos(2*pi*(fra+frb/2)*tau);
cb=cb-cos(pi* frb*tau);
cb=2*cb;
cp=-4*c2.*sin(pi*tau*fra).*sin(pi*tau*frb);
cm=—4*s2.*sin(pi*tau*fra).*sin(pi*tau*frb);

la=zeros(size(fr));
Ib=zeros(size(fr));
Iapb—-:zeros(size( fr));
Iamb=zeros(size( fr));

n=size(fra);

for i=1:n(2),
Ia(nfra(i))=la(nfra(i))+ca(i);
Ib(nfr‘b(i))=Ib(nfrb(i))+cb(i);
lamb(nframb(i))=lamb(nframb(i))+cm(i);
Iapb(nfi~apb(i))=lapb(n&apbfi»+cp(i);
end

Ia=abs(Ia);
lb=abs(lb);
lamb=abs(lamb);
Iapb=abs(lapb);

l=la+Ib+Iapb+lamb;
nn=[1 l];
n=size(fr);

imin=1;imax=n(2);

elseif iopt=16 eseopt=2;

return;

% fpfrsp.m - generates 4—pulse frequency domain spectrum

2 4 7

global epr;
Acalc;
% calculation of the superhyperfine frequency on the selected angle set
dummy:0;
frcnt=0;
{rs-10001150; % frequency domain vector
la=zeros(size(fr)); % intensity vectors
Ib=zeros(size(fr));
lapb=zeros(size(fr));
lamb:zeros(size(fr));
for mi=—epr(12):epr(l2),

dummy=dummy+l;

for angi=l :nummi(dummy),

% direction cosine vector of g-tensor to magnetic field

theang:angset(angi,dummy*2- 1 );
phiang=angset(angi,dummy*2);

h1=[sin(theang)*cos(phiang) sin(theang)*sin(phiang) cos(theang)];
gh1=epr(1:3).*hl;

ghleff=sqrt(ghl*gh1');
gh1A=gh1*A;

vn=762.2591*eseem( 1)*epr(10);

% for ms=1/2 spin manifold
ghA1=0.5*ghlA/gh1eff—vn*h1;
frcnt=frcnt+1;
fra(frcnt):sqrt(ghAl*ghA1'); % alpha hyperfine frequency

% for ms=-1/2 spin manifold

ghAl=—0.5*ghlA/gh1eff-vn*hl;
frb(frcnt)=sqrt(ghAl*ghAl'); % beta hyperfine frequency

end
end

% counter partner of angle set ‘ gives same frequencies
% combination band freqency vector

frapb=fra+frb;
framb=abs(fra-frb);

% probability vector for corresponding frequency vector
dummy=0;
for mi=—epr(12):epr(12),

if dummy=0,
ininum=l ;
finnum=nummi( l );
else

5 P - ., 51>;

 

248

end
dummy=dummy+ 1 ;

if finnum ~ 0,
probfr(ininum:finnum)=prob(1:nummi(dummy),dummy);
end

end
% calcuation of k and C vectors
probfr=probfrlsum(probfr); % normalization of probability

$2=abs(vn"2—0.25*(fra+frb)."2)./(fra.*frb);
c2=1-s2;
k=4*82.*c2;

ca=ones(size(k));
cb=ones(size(k));
cm=ones(size(k));
cp=ones(size(k));

% calculation of Intensity of each frequency

nfra=ro und((fra/2+ 10e6)*6000/60e6)+1 ;
nfrb=round((frb/2+10e6)*6000/60e6)+1;
nframb=round(( framb/2+ 10e6)*6000/60e6)+ 1;
nfrapb=round((frapb/2+10e6)*6000/60e6)+1 ;

n=size( fra);

for i=1:n(2),
la(nfra(i))=la(nfra(i))-k(i)/2*prob fi'(i)*ca(i); % divide 2 instead of
lb(nfib(i))=lb(nfi'b(i))-k(i)/2*probfr(i)*cb(i); % 4 because half of
lamb(nframb(i))=lamb(nframb(i))-k(i)/2*probfr(i)*cm(i); % anglse set is
lapb(nfrapb(i))=lapb(nfrapb(i))-k(i)/2*probfr(i)*cp(i); % calculated

end

la=abs(la);

Ib=abs(lb);
lamb=abs(lamb);
Iapb=abs(lapb);

% convolution
la=conv(gau,la);
Ib=conv(gau,lb);
lapb=conv(gau,lapb);.
Iamb=conv(gau,lamb);
ltot=la+lb+lapb+lamb;

% rearrange the intensity vector due to convolution

nn=size(gau);
n=size(fr);

l=ltot((nn(2)—l)/2+1:(nn(2)-1)/2+n(2));

return;

%
‘70 gausm - generate gaussian function for convolution
%

%widthd).01; % for delbb=0.03 MHz in two-pulse scale
%width=0.l23; % for delbb=0.4 MHz in two-pulse scale
%width=0.06; % for delbb=0.2 MHz in two-pulse scale

frgau=-0.25:0.01:0.25;
gau=exp(-frgau."2/width"2);

gau:gau/sum(gau);

% plot( frgau.gau);
% grid

return;

 

 

 

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