THEORY OF THE MAGNETIC RESONANCE SPECTRA OF WATER—OF—HYDRATION AND METHYL PROTONS By Deborah May Roudebush A THESIS Submitted to Michigan State University in partiai fulfiilment of the requirements for the degree of MASTER OF SCIENCE Department of Physics 1978 SIP-'- ABSTRACT THEORY OF THE MAGNETIC RESONANCE SPECTRA OF NATER-OF-HYDRATION AND METHYL PROTONS By Deborah May Roudebush The system of two quantum mechanical dipoles in different local magnetic fields is solved completely for the case of protons. Eigen- values, transition energies, intensities, and spectra are given. The secular equation for three quantum mechanic dipoles in an equilateral triangular configuration in different local magnetic fields is found for protons. The symmetrical canting case, a special case for the three proton configuration, is treated with the eigenvalues found numerically at a number of geometric configurations. ACKNOWLEDGMENTS I wish to acknowledge Dr. Robert Spence for suggesting the problem and assisting with the experimentation. My deepest appre- ciation goes to Dr. Paul Parker for seeing me through to the end. I also wish to thank David Larch for his support and patience and Delores Sullivan for her skillful typing. ii Chapter I II III IV TABLE OF CONTENTS Page INTRODUCTION 1 THE INTERACTION OF THO QUANTUM MECHANICAL DIPOLES LOCATED IN DIFFERENT LOCAL MAGNETIC FIELDS THE INTERACTION OF THREE QUANTUM MECHANICAL DIPOLES IN AN EQUILATERAL TRIANGULAR CONFIGURATION IN DIFFERENT LOCAL MAGNETIC FIELDS 26 4l THE SYMMETRICAL CANTING CASE iii LIST OF TABLES Table Page 1 Definitions of direction cosines, y, in the three- dipole triangular configuration 29 2 Definitions of scalar products, a, between coordinate systems in the three-dipole triangular configuration 29 3 The eigenvalues of the Upper 3 x 3 submatrix as a function of e in units of C=(ge)2R-3 46 iv Figure 1 .1 2.1 .10 .11 .12 LIST OF FIGURES The 4.297 MHz resonance line. Two protons separated by distance R and located in magnetic field Ha and Hb respectively. The fields are not necessarily coplanar. Energy level diagram for the two dipole configuration in the general case. Spectrum for the two dipole configuration in the general case. Geometric configuration for two dipole case. Energy level diagram for the two dipole configuration in limiting case A. Spectrum for the two dipole configuration in limiting case A. Energy level diagram for the two dipole configuration in limiting case B. Spectrum for the two dipole configuration in limiting case B. Energy level diagram and spectrum for the two dipole configuration in equal field case. Plot of v(ea,eb) where ¢ab=0 for the two proton general case. Plot of v(6a,6b) where ¢ab=20° for the two proton general case. Plot of v(ea,6b) where ¢ab=45° for the two proton general case. Page 10 10 12 12 15 15 17 17 19 21 23 25 The equilateral triangular geometric configuration. The Hamiltonian matrix for the three proton general case. Geometric configuration with N perpendicular to and directed out of the page. All vectors are coplanar. Geometric configuration with 3] perpendicular to and directed out of the page. All vectors are coplanar. Geometric configuration with B b perpendicular to and directed into the page. A l vectors are coplanar. The symmetrical canting case as seen in the plane containing Pab and H at the proton a site. The eigenvalues of the symmetrical canting case as a function of e. The geometric configuration for 6=¢=90°. In this case a3=b3=C3 and directed out of the page. The geometric configuration for e=l50°, ¢=l80°. In this case a1= b1= c1 and directed out of the page. vi Page 31 36 37 37 42 47 49 49 CHAPTER I INTRODUCTION In their study of the antiferromagnetic crystal CoCl3((CH3)3NH)-2H20, Spence and Botterman1 observed nuclear magnetic resonances (NMR) for the various protons in the crystal. The l7.64 MHz, the l6.80 MHz, and the 8.454 MHz proton resonances appear to have no splitting. The 5.955 MHz, the 3.359 MHz, and the 1.95 MHz proton resonance lines show dipole-like splitting. The line of particular interest, the 4.297 MHz proton resonance line, shows no typical dipole splitting. Instead it is made up of seven evenly spaced lines of varying intensity as shown in Figure l.l. These resonances were all observed at approximately 2.4°K. The 4.297 MHz lines are field independent over a range from l25 G to 450 G. In an effort to explain these lines, the general case of the dipole-dipole interaction is investigated since this corresponds to the physical situation of the protons of water of hydration units. The present study resulted in a complete solution for this geometry which can yield a maximum of only four lines. The Hamiltonian, secular equation, eigenvalues, eigenfunctions, and intensities are treated in detail. Also included are several limiting cases, including the "case of equal local magnetic fields" treated by Pake.2 Figure l.l. The 4.297 MHz resonance line. FIGURE |.| 4 Since protons exist in the equilateral triangular configuration in methyl groups, this geometry can also offer results of physical significance. This thesis presents an appropriate secular equation, which due to its complexity could not be solved analytically for the general case. For an arbitrarily chosen magnetic field direction, the secular equation was found to reduce to the "equal-field-case" as treated by Andrew and Bersohn.3 A physically significant special case is obtained when the magnetic fields at the three proton sites of the equilateral triangle are taken such that the threefold symmetry of the geometrical configuration is retained. One such case of "symmetrical canting" requires that the three magnetic fields at the proton sites have equal magnitudes, and that their respective directions intersect on the threefold symmetry axis of the proton triangle. Such a case could conceivably apply at the methyl sites due to internal magnetic fields arising at cobalt sites located on or near the above threefold symmetry axis. In this thesis, the secular equation for synmetrical canting is developed. Since only one angle is required to specify a particular canting configuration, it was found possible to solve the secular equation numerically for the eigenvalues for a series of angles of canting. The special case in which the magnetic fields are perpen- dicular to the plane of the methyl group triangle is an equal-field- case and is shown to properly reduce to the results of Andrew and Bersohn. The three-proton case results in eight energy levels, and hence, can give rise to a maximum number of 28 possible transitions. If the dipole-dipole interactions are weak compared to the field-dipole 5 interactions, a maximum of l5 possible transitions can result in the general case. Thus, the observed splitting of the 4.297 MHz tran; sition could possibly be accounted for in the proposed manner, but a much more detailed analysis would be required to be certain. Such an analysis will most likely have to include a study of the respective transition probabilities as well. CHAPTER II THE INTERACTION OF TWO QUANTUM MECHANICAL DIPOLES LOCATED IN DIFFERENT LOCAL MAGNETIC FIELDS The Hamiltonian for two interacting dipoles in local magnetic fields Ba and fib is given by the well-known expression4 H = 98(Ha1a3+HbIb3) + (gmzR'3 [Ta-Tb-sfia-EHTb-ED (2.1) Here 3 is the unit vector between the two dipoles as shown in Figure 2.l. It is convenient to choose two Cartesian coordinate systems such that the local axis of quantization is along the local field at each dipole. These axes will be indicated as the respective 3-axis. The 2-axis will be in the plane determined by the 3-axis and B. The right hand rule then determines the l-axis. The unit vectors of each system will be designated by the letter of the nucleus at the site; i.e. 31 at proton a, and Bi at proton b. 0n defining the scalar products, (2.2) 7 where Yai is the direction cosine of E in the 3i direction, Ybi is the direction cosine of 31in the Si direction and aij angle between 31 and DJ, the Hamiltonian may be written as H = 98(Ha1a3+HbIb3) ‘ Cg; “‘"zn'litfl'o‘ij)I where With the further definition = -1. D.. 4C(3y 13 aiybj'aij) The Hamiltonian becomes H ‘ 98(Ha1a3+HbIb3) + 4% DianiIbJ' I ai bj If I=-12-, which is the case for protons, then in the basis indicated below, the block diagonal matrix of the Hamiltonian is found to be the following: Basis: (++) (+-) (-+) _1_ §93(Ha+Hb)-O O 0 l H-H +D - - 0 79m 6 b) DII*°22"‘(012 ”21) . l 0 D11“[’22"‘ (Dl 2'021) ' 298(Ha'Hb)+D o O 0 IL_. O -]§gB(Ha+Hb)-D is the cosine of the (2.3) (2.5) (2.6) (2.7) 8 where D:"DB3' One can immediately write down two eigenvalues, viz., E1 =%QB(Ha +H b-) D 4 ="298(Ha+Hb)'D In order to obtain the remaining two eigenvalues, let A E (”11*022)+1(912'021) which yields the secular equation 2 (5-0) = %(GBAH)2+|A|2 where AH=Ha-Hb, with roots —I 2' 52,, = 0: (%(OBAH)2+|AIZ) The general solution, therefore, is El = gBHave"D E2 = S+D E3 = -S+D E4 = "gBHaveD where ml—a = ((JZ-geAHTZHAIZ) (2.8) (2.9) (2.10) (2.ll) (2.12) (2.13) and _ l Hawe - 2(Ha+Hb) (2.14) In the diagonalized basis, the corresponding normalized basis functions are found to be I w] = (252+98AHS) 2E(-%—gBAH-S)(+-)-A*(-+)] 1 (2.15) I, = (ZSZ-OBAHS) 2[(-%OBAH+S)(+-)-A*(-+)l The transition probabilities are proportional to ||2 where n and n' are the initial and final states between which magnetic dipole transitions are considered. In the two-dipole interaction, this calculation yields :ZD+S : h] + KZSEQBAHWQA] 25 +gBAHS (2.16) 120-5 : )[I + (ZS'QBAHSWEAJ ZSZ-QBAHS where the left column is the change in energy of transition from the unperturbed value and the right hand column is the corresponding normalized transition probability. The energy levels are shown in Figure 2.2 with the transition energies as follows: 10 Ha II D" 1 'd ‘5 b2 9. 3 9.. V' ’IJ Proton o a Proton b U2 Figure 2.l. Two protons separated by distance R and located in magnetic field Ha and Hb respectively. The fields are not necessarily coplanar. gBHa—ve -0- 5+0 ' 3 AW 1.12—4— Figure 2.2. Energy level diagram for the two dipole configuration in the general case. . 11 hv] = gBHave-ZD-S hv = gBH +ZD+S 2 ave (2.17) hv3 = gBHave-ZD+S hv4 = gBHave+2D-S These energies yield the spectrum shown in Figure 2.3. Referring to Figure 2.l and Figure 2.4, the following relations may be determined: YaI = 0 Ybl ‘ 0 Ya2 = sinea Yb2 = sineb Ya3 = cosea Yb3 = coseb ++_ a1] al-b1 : coscbab (2 18) “22 = 32-32 = sineasineb+coseacosebcos¢ab ++ . . _ O33 a3-b3 - C059a6059b+51"9a51n9b005¢ab = coseab + .4 012 - a1032 - smoabcoseb ++ . a21 - aZ-b1 - -s1n¢abcosea Inserting these results into Equation 2.4 and Equation 2.7, the geometrical relations for D and IAIZ become _ l 2 2 -3 D - 39 B R (3coseacoseb-coseab) 2 _ l 2 2 -3 2 2 [Al - (fig 8 R ) { (3coseacoseb-coseab) (2.l9) 2 + 4(1-coseab) -3(cosea-coseb) } 12 L 40:2; 4 4D— 23 11V 3 T 4 2 QBHove Figure 2.3. Spectrum for the two dipole configuration in the general case. plane of Figure 2.4. Geometric configuration for two dipole case. l3 These results agree with the results given by G.E.G. Hardeman5 in his dissertation and credited to J. D. Poll. Limiting,Cases A. Consider the case where |A|2<<(%gBAH)2, i.e. Ha>>Hb or Ha<>H and the bottom sign refers to H <>(%gBAH)2, i.e. Hasz. Now 5: yielding eigenvalues of where IA! +%IM“(l,gsAm2 El = gBHave"D E2 = D+|A|+e E3 = D-IAI-e E4 ="geHave'D 1 - = -2-IA| ‘(lggeamz (2.24) (2.25) (2.26) and E1 and E2 are exact. The energy level diagram is given as Figure 2.7. 15 9B HW? «D ‘ l/Zg BAH+D+8 I92 gBAH +D-3 35,—— -g B Hove ~01 Figure 2.5. Enérgy level diagram for the two dipole configuration In limiting case A. I 4MBAH£§ a) - gBAH-ZS l a T QB Hove Figure 2.6. Spectrum for the two dipole configuration in limiting case A. 21 ‘;111/ 45. 16 The transition energies are as follows hv1 = gBHave-ZD-lAl-E hvz = gBHave+ZD+|A|+e (2.27) hv3 = gBHave-ZD+|A|+E hv4 = gBHave+2D-|AI-E with the spectrum shown in Figure 2.8. If AH becomes very small, then 6 goes to zero. C. For the special case that the two magnetic fields are equal in direction and magnitude, -> + -> + + Ha - Hb, |Ha| - |Hb| - H, aij - a1 Bj - aij (2.28) Equation 2.5 reduces readily to the Hamiltonian which applies in this case, viz., the one given by Fake.2 It follows that oab=0, Oa=eb=e, and therefore D and |A|2 reduce to D = %C (3cosze-l) Ia|2 = (”11“”22)2 = (%C)2 (3:05.294)? (2.29) IAI = D The eigenvalues reduce to E] = gBH'D E2 = ZD (2.30) E3 = 0 E4 = "gBH-D 17 ’gBH‘We -D PAL—L— Figure 2.7. Energy level diagram for the two dipole configuration in limiting case B. P 4D+21AI+2€ 4D -2|A|-2€l l 3 T . 4 2 Th” QBHove Figure 2.8. Spectrum for the two dipole configuration in limiting case B. 18 as in Pake, with transition energies hv1 = gBH-3D hv2 = gBH+BD (2.31) hv3 = gBH-D hv4 = gBH+D The transitions hv3 and hv4 are the forbidden singlet-triplet tran- sition as shown in Figure 2.9. The intensities reduce to %, %, and 0 as required by the Pake solution. For the general case, the frequencies were plotted using the Hewlett-Packard 91003 three dimensional perspective plot routine. The local magnetic field values and ¢ab were held constant throughout a given plot, while Ga and 0b were varied. Each plot has Oz in black and 03 in red and is centered about l h"(gsH +[(%98AH)2-IATZ)17) (2.32) ave The field dependence of the frequencies is so small that these per- turbations are not observable on the plots. The ¢ab dependence and therefore the eab dependence is small, but can be seen by comparing closely Figure 2.10, Figure 2.11, and Figure 2.12. 19 93H“)? —D_T__T_' :3 ‘ 1 O wL_JW 2 '.. F 60 H >111! u T 2 OBH Figure 2.9. Energy level diagram and spectrum for the two dipole configuration in equal fie d case. 20 Figure 2.10. Plot of 0(Oa,eb) where Oab=0 for the two proton general case. 21 22 Figure 2.11. Plot of v(ea,eb) where ¢ab=20° for the two proton general case. 23 :.N MKDQE 24 Figure 2.12. Plot of v(ea,6b) where ¢ab=45° for the tWO proton general case. 25 N_..N MEDOE «wwwly - . .1.- .1.1... ..,.,_L......-.... \§-kk““‘l I - ‘ D». ‘1 A '3 A ~ "'1 . .e R3311- iii..- ii!) .3.“ ‘M ‘\\$§\h\\\\§ om CHAPTER III THE INTERACTION OF THREE QUANTUM MECHANICAL DIPOLES IN AN EQUILATERAL TRIANGULAR CONFIGURATION IN DIFFERENT LOCAL MAGNETIC FIELDS The Hamiltonian for the interaction of three quantum mechanical dipoles in an equilateral triangular configuration in different local magnetic fields Ha, Rb, and HE follows readily from (2.1) as + C[Ié°Tb'3(Ta;Bab)(TPOEab)] (3 1) + C[TH-I63(THEM)(Tc'Ebcn + CIYc-Ta-3(Tc°3ca)(1 a n aca where C is defined as in Chapter II, and with R the distance between any two dipoles. The vector Bij i to dipole j, as in Figure 3.1. Again, it is convenient to choose is the directional vector from dipole coordinate systems such that the local axis of quantization is along the field at each dipole. These axes will again be indicated as the 3-axis in each system. The unit vectors of each system will be designated by the letter of the nucleus at that site, i.e., 31 at 1 proton a, etc. 26 27 .Proion C p’ca fibc E503 \ R / Proton o Proton b Figure 3.1. The equilateral triangular geometric configuration. 28 On defining the scalar products as in Table l and Table 2, the Hamiltonian, Equation 3.1, may be written H = 98(Ha1a3+HbIb3+HcIc3) ab ab ab ' C EX (3Yainj'aijnaiij J (3.2) bc be be "Ci g (31bich'aij)Ibich ca ca ca ' C E; (3YciYaj'aijnciIaJ’ Let ab _ _l_ ab ab_ ab Dlj - 4C (BYainj 0‘13 (3.3) DC ca ij and Dij' With these definitions the Hamiltonian, Equation 3.2, may be written with similar definitions under cyclic permutations for D H = gB(H I +HbIb3+HcIc3) a a3 (3.4) +422(Oab1 I +DbcI .I +DcaI I ) ij ij ai bj ij b1 cj ij ci aj Figure 3.2 shows the matrix representation of Equation 3.4 for the case of protons, I=%, in the basis indicated and with only the secular terms as described by Andrew4 retained. Two eigenvalues may be written down immediately, viz., _ l ab bc ca E1 ‘ 298(Ha+Hb+Hc)+D33+D33+033 (3.5) _ l ab bc ca E8 —"298(Ha+Hb+Hc)+033+033+D33 29 Table 1. Definitions of direction cosines, y, in the three-dipole triangular configuration Scalar Product Direction cosines of: System Ta°3ab- §Y:?Ia1 isab (31:52 ’33) Th "no= g 1:3le isab (El 5233) Right: E Ygflbi isbe (Bl 52:33) THEM: E *2?ch iSbc (61 32:53) Tc'-)ca= EYEIIci Eca (El 3233) Tc°5ca= 215312) sea (31 32:53) Table 2. Definitions of scalar products, a, between coordinate systems in the three-dipole triangular configuration Scalar Product _ ab aab_+ Ta°Tb'§§ O‘ijlailhi “ii a‘i -J.E bc bc_ .+ ca ca4+ .+ 30 Figure 3.2 The Hamiltonian matrix for the three photon general case. 31 NM mm 39... .mM-JMMQMH. NIEEWWN o o o o o o o J ..me..mo-awqe _. £3-.. .9- O TITJIT NIH—QNOWNNO_ + N3 JFK; O O O o NQMDWW wQ+nnoumon _N.E.NO. SQ? O mmO+~% T..- eI+omeNmWN 20+ org 0 O O O :N on" I. mnNOu umOIMMQIMMO... o 8 . .5 N. 5.1.2.33. o o o o Ammo- $988-33... Ne. _- Ema-Mo? O O O O E%~mwm2 + :Q No + LEV O o 38...... 188...? “we- go N.._o_- o O O O Nowh— + Lon: WIEIIoIHWNWNN on wordy QM -NoQT .mQNoNLQ_+fiwQ-an-wwD+ O O O C 0+ ""0 .WMQ + ordw HfInI+ilew O n n + o o o o o o o 38 3:8 81......imlfl T - -_ T-.. T + -_ T - Ma: 7. -_ T-.. T; 7 .AI 32 Upon defining Dab _ Dab Dbc _ Dbc Dca _ _Dca 33“‘ . 33"’ 33 and ”11m£+‘(012 M21) Aab 01$ Db2+‘(912° D21) ' Abe D$?+Dca+i(0% g?) = AC3 the upper 3 x 3 submatrix becomes ED Abc Aca* Abc* DZ Aab Aca Aab* D3 where D =lgG(Ha +Hb -H c) + (- Dab+Db°+Oca ) 1 N —l DZ= Egema -H b+H ) + (Dab+DbC Dca) D = 2gB(-Ha+Hb+Hc) + (Dab_Dbc+Dca) 3 (3.6) (3.7) (3.8) (3.9) 33 The lower 3 x 3 submatrix becomes F'_ .— 01' Abc* Aca Abc Dz' Aab* (3.10) ca* ab . i. A 0.3.3 where 01'" "12'{~’8(Ha+Hb'H c) + (-D Dab+Dbc+Dca) 02' = -%gB(Ha -H b+H c) + (oab+DbC-D°a) (3.11) 031 = __a%gB( H +H b+H C) + (Dab_ DbC+DC3) The secular equation for the eigenvalues 21 of Equation 3.8 and Equation 3.l0 are obtained in the standard manner, viz., by subtracting A from the diagonal elements and by equating to zero the determinant of the resulting matrix. Using Equation 3.9 and Equation 3.ll, this pro- cedure leads to the very complicated secular equations below, with the upper signs applying to Equation 3.8, and the lower signs to Equation 3.10: -x 3+A2[+198(H +H b+H )+(oab+nbc+oca)1 +A{%(ge)2[Hz+H§ +HE -2(Ha Hb+ HbHC+HcHa)]} (3.12) +gB[Ha (Dab_ Hbc+Dca)+Hb(Dab+bc _Dca)+HC (_ Dab+ Dbc+Dca)] 34 Equation 3.12 continued +E(Dab)2+(Dbc)2+(Dca)2_2(DabDbc+Dbcha+DcaDab)J +t<03% g) +<033- 033) 2+<033+ 033) 2+<033- 033)2+<033+033)2 +(013- 033) 2]} 3 3 3 2 2 2 (g B)3 [Ha +H b+H c+2H acHbH -H a(Hb+Hc)-Hb(Ha+Hc)-Hc(Ha+Hb)] 00l-J ; +3(DB)2[H§(Dab-aobc+Dca)+H§ (Dabmbc- 3caDca‘)+H§(-3Da'b+0bC+D ) +2H H (_ Dab+Dbc+Dca)+2Hb H c(Dab_ Dbcwca)+2HC H a+D(Dab bc _Dc a)] a b :1? (gB){Ha[(Dab)2-3(Dbc)2+(Dca)2+2(DabDwabcha-DcaDab)] +Hb[(Dab)2+(Dbc)2-3(Dca)2+2(-DabDbc+Dbcha+DcaDab)] +HC[-3(Dab 2+(Dbc)2+(Dca)2+2 bc Dbc Dab Dab bc+ Dbc ca ca ab ab -D( )(0112)(0-0 )--(011+022)(012 D21)(013- 033)} 12 021 12 21 :[<031+0 033)z+(013-0 031)21[290 _ + .+ = o + .+ =_ o pab°mab'0 pbc mab c0530 pca mab c0530 -> + _ o + .-> = -> .+ = ° pab-mbc—-cos30 pbc mbc 0 pca mbc c0530 + + O + 0+ =- O + 0+ = pab-mca—COSBO pbc mca c0530 pca mca 0 (3.l4) -> .+ _-> .+ _+ .-+ _ 60° mab mbc'mbc mca'mca mab'"COS -> + -> + o ’ o —+ 0+ " 60° pab pbc' bc pca'pca pab--cos 36 fig» V Proion o Proion b Figure 3.3. Geometric configuration with W perpendicular to and directed out of the page. All vectors are coplanar. 37 Figure 3.4. Geometric configuration with 31 perpendicular to and directed out of the page. All vectors are coplanar. Figure 3.5. Geometric configuration with Bab perpendicular to and d1rected 1nto the page. All vectors are coplanar. 38 and . +- + N-fia-s1n¢a na-mab=cos¢a +=. +0.): fi-nb s1n¢b nb mbc cosob (3.l5) +-> . ++_ N-nc-s1noc nC-mca-cosoc From these relationships, the values of the various yab and cab may be developed and one finds: ab=cose ab=-—c0560°cose -cos30°sine coso Ya3 a Yb3 b b b ab=sine ab=--c0560°si 6 +cos30°cose coso Ya2 a Yb2 ” b b b ab_ ab_ 0 ° Val—0 ybl—cos30 s1n¢b ab_ . . . . o . . a33-s1n6a51n6bs1n¢as1n¢b-c0560 s1neasmebcos¢acos¢b -cos60 coseacoseb+c0530 s1n6acosebcos¢a -c0530 smebcoseacosob ab_ . . o aZZ-coseacosebs1n¢a51n¢b-c0560 coseacosebcosoacosob (3.l6) _ o . . + o . _ o , c0560 s1nea51neb c0530 smeacosebcosob c0530 coseas1nebcos¢a ab_ 0 . . a]]--c0560 s1n¢as1n¢b+cos¢acos¢b ab__ 0 . . _ o . _ . “12‘ cos30 s1nebs1n¢a cos60 cosebs1n¢acos¢b cosebcosoasmob ab_ 0 . . o . . a21-cos30 s1n6a51nob-c0560 coseacos¢a51n¢b-coseas1noacosob Cyclic permutation gives the corresponding expressions for ch’ abc, 39 Consider now the special case of equal fields for which 33533=E . When;3 and 33 are written out in terms of their components in the W, mab, and Bab directions, the following relations apply: ++ ++ (a) b3'pbc - a3'pbc (b) 33-h = 33ofi (3.17) ++_+.+ (C) b3""bc ‘ a3 mbc or more explicitly: (a) coseb =-c0560 cosea+cos30 smeacosoa (b) sinebsinob = sineasinoa (3.l8) - =_ o- _ o (c) s1nebcos¢b c0560 s1neacos¢a c0530 cosea These equations specify the relationship between (ea,¢a) and (6b,¢b) when fiagfib’ and examination of these equations shows that indeed a particular choice of (6a,¢a) uniquely determines (eb,¢b), and vice versa. Cyclic permutation of indices yields the sets of equat1ons which relate (eb,¢b) to (6c,¢c), and (6c,¢c) to (ea,¢a), respectively. ab;+ Substitution of Equations 3.l8 into the expressions for a33—a3-b3 yields unity, as required for equal fields. Furthermore, with vgg = 7:3 = cosea, cyclic (3.19) 40 ab 2 the diagonal elements 0 etc., are found to reduce to %43(cos ea-l), -cyclic, which agrees with the corresponding results of Andrew and Bersohn. Despite much effort, no analytical method was found that would ab bc’ and Aca reduce show that the off-diagonal matrix elements A , A to those of Andrew and Bersohn when Equations 3.l8 hold. Instead, a particular numerical choice for ea and ¢a was made (viz., 6a=43° ¢a=27°), and for that choice, and with Equations 3.l8 applying, it was found that the resulting cubic secular equations agreed with those obtained from Andrew and Bersohn's paper. It seems obvious that an exact analytic solution for Equation 3.12 cannot easily be found. Perturbation theory is not particularly helpful either, since for a degenerate or nearly degenerate zero-order sub- block, perturbation theory calls for diagonalizing that block, i.e., the exact solution of Equation 3.l2. There is, however, a physically interesting special case for which computing numerical solutions becomes practicable. This case will be treated in the next chapter. CHAPTER IV THE SYMMETRICAL CANTING CASE The symmetrical canting case is defined as the case for which ¢a=¢b=¢c=¢ (4.1) In this case, ab_ ab Ya3"'Yb3 (4.2) as seen in Figure 4.1. As a consequence, Equations 3.16 give cote = 2c0530°cos¢ (4.3) This equation determines the angle ¢ for a given 6, and vice versa. The following three special cases of Equation 4.3 may be considered: a) 9: 30°, ¢=0° b) a: 90°, ¢=90° (4 4) c) e=150°, ¢=180° 41 42 I1 9 9 59b Pro’ton o Proion b Figure 4.1. The symmetrical cantin , . 9 case as seen in the lane containing Pab and H at the proton a site. p 43 where a) is the case in which H is in the plane of the proton triangle and directed radially inward; b) is the case in which H is perpendicular to the plane of the triangle; and c) is the case in which H is in the plane of the triangle and directed radially outward. The relevant scalar products become, upon inserting Equation 4.1 and Equation 4.2 into Equation 3.16, y:?= 0 vg?=cos30°sin¢ ab_ ab 0 o y a2 -sin6 yb2=-c0560 sin6+cos30 cosecoso vgg=cose vgg= - cose (4.5) o33=l-2cosze ogg=coszesin2 q».- vlcoszecosz ¢- lsinze a?$= -%sin2¢+coszo ( ag?-aa?) =1/3 3sinesin¢ With Equation 4.3, the following relationship is seen to hold: ab 2 ab 2+ ab 2_ (TM) +(Yb 2) +-(Yb3) (4.6) 44 ab ab The relationships for D and A follow from Equation 4.5, viz., 11:1)a'b=obc=pca - )c (1+cosze) A=Aab=Abc=Aca 2 2 ecoszo) (4.7) C(l-3cos 6+3$in l 8 + (i/8) C ( f3- sinesino) The following eigenvalues may be written down immediately: -3. (4.8) E = --3-gBH+BD 8 2 The remaining six eigenvalues are -1 ._ Ei- ngH+Ai (1-l,2,3) (4.9) 1 . 0:- + . = EJ ‘EgBH x3 (J 1.2.3) where the Ai are the three eigenvalues of the upper 3 x 3 Hamiltonian submatrix with associated secular equation D-A A A* A* 0-1 A = 0 (4.10) A A* D-x In more explicit form, A3-BDA2+3(DZ-|A|2)A-D3+3D|A|2-2Ai+6ArA§=O (4.11) 45 where Ar is the real part of A and Ai is the imaginary part. The associated eigenvalues were determined numerically as a function of 9. They are listed in Table 3 and plotted in Figure 4.2. The lower 3 x 3 submatrix is found to have eigenvalues given by Equation 4.9, with the Aj identical with the xi of Equation 4.10. In the special case when e=¢=90°, it can be seen from Figure 4.3 that 121:0 131:12/3— O‘11: 1? 0‘12: ’1?”— y§2=1 133"]? 0‘22“% 0‘21=']2"/3— (4'12) Y§§=o Y§§=0 0‘33:] The general expressions, Equation 3.16, and the symmetrical canting expressions, Equation 4.5, are found to reduce properly to the above values. Furthermore one has that here (4.13) A= _ 21fC e-‘i (211/3) which checks Andrew and Bersohn's results except for the constant phase factor. It has been verified that the same secular equation is obtained with or without the phase factor, which is connected with the rotation 51, 32, 33 relative to 3], 32, 33 by 2n/3 for this special case. Table 3. The eigenvalues of the upper 3 x 3 submatrix as a function of e in units of C=(gB)2R‘3. 8 o A] A2 A3 (degrees) (degrees) C C C 30 0 .37500 .37500 .56250 35 34.45795 .25334 .49737 .50254 40 46.52332 .20008 .54992 .44031 45 54.73561 .15853 .59147 .37500 50 61.02327 .12371 .62629 .30987 60 70.52878 .06878 .68120 .18752 70 77.86954 .03039 .71954 .08780 80 84.15703 .00745 .74239 .02278 90 90.00000 .00000 .75000 .00000 100 95.84297 .00745 .74239 .02278 110 102.13046 .03039 .71954 .08780 120 109.47122 .06878 .68120 .18752 130 118.97673 .12371 .62629 .30987 135 125.26439 .15853 .59147 .37500 140 133.47668 .20008 .54992 .44031 145 145.54205 .25334 .49737 .50254 150 180.00000 .37500 .37500 .56250 47 Xi units o.o--0 C o .- A 8 6 A £5 __|___ C) C) A A 1'- A 72" O " A A '13“ o o ‘— A o A -.4-- —- -5__. é 6 .. 13 C) -.6--' " .._ A X -.7-- x -- X x x X -.ea- ~- . , 1 9 in De rees , , 25 5% 7'5 1010 1 5 150 Figure 4.2. The eigenvalues of the symmetrical canting case as a function of e. 48 Figure 4.3. The geometric configuration for 6=¢=90°. In this case a3=b3=C3 and directed out of the page. Figure 4.4. The geometric+cogfiguration for 6=150°, ¢=180°. In this case a1=b1=c1 and directed out of the page. I1 49 303 CZ 6, ' .3 >62 .3 63. FIGURE 4 6' 1'. ° 5’2 6 C." _ 9 3‘ H 62. . 52 0'. 9 63 0 FIGURE 44 , _ 50 In the special case where e=150° and ¢=180°, as in Figure 4.4, ab_ ab_ ab_ 1_ ab= 1a1'O Yb1“) O‘33" ’ 2 0‘12 0 ab_l ab_]_ ab= ab= Y(12‘? 1132’ 2 0‘11 1 0‘21 0 (4°14) ab_ ab_ ab: _ l_ Y513' ‘ “72 1(113‘ “Br/2 0‘22 2 again determined by inspection. The symmetrical canting expressions, Equation 4.5, once more reduce properly to the above values. REFERENCES boom 51 REFERENCES R. D. Spence and A. C. Botterman, Phys. Rev. B 9, 2993 (1974). G. E. Pake, J. Chem. Phys. 16, 327 (1948). E. R. Andrew and R. Bersohn, J. Chem. Phys. 18, 159 (1950). E. R. Andrew, Nuclear Magnetic Resonance, (Cambridge at the University Press, London, 1955). G.E.G. Hardeman, Resonantie en Relaxatie Van Protonspins in een Antiferromagnetisch Kristal, (Riksuniveisiteit Te Leiden, 1954). 11111111111111 1111111111 11111111 0 5 6 9 8 6 1 3 0 3 9 2 1 3