“ (IX ABSTRACT TEMPERATURE DEPENDENCE OF THE NQR FREQUENCIES OF PERIODATES AND PERRHENATES AND CALCULATION OF ELECTRIC FIELD GRADIENTS FROM X-RAY DIFFRACTION DATA BY Richard Allen Johnson The temperature dependence of the nuclear quadrupole resonance (NQR) frequencies in NaIO4 and MReO4 (M = Na, K, Ag) have been experimentally studied from liquid helium to room temperature. Theoretical curves were constructed based on the Kushida, Benedek, and Bloembergen theory and compared with experiment in order to determine the importance of lattice changes with temperature on the field gradient, which is often neglected in such treatments. Frequencies, symmetries, and temperature variations of the external lattice vibrational modes were obtained from a detailed laser-Raman study of all the Scheelite-type per- rhenates. Mode assignments were made from depolarization ratio measurements on NaReO4 single crystals and spectra of polycrystalline ND4ReO4 and ND4IO4. Richard Allen Johnson Ammonium perrhenate exhibits an anomalous NQR tempera- ture dependence with a loss of signal upon cooling at 2450K. A model for the motion of the ammonium group in the crystal has been constructed which is consistent with the NQR, Raman, and infrared results. The ammonium group in NH4ReO4 is apparently undergoing torsional motion at 77°K and freely rotating at room temperature. The possibility of calculating electric field gradients using electron densities obtained from accurate X-ray dif- fraction data was also investigated. Electron density maps were calculated from observed structure factors for several compounds, with actual field gradient calculations carried out only for fi-ICl. The reliability of the results in terms of the assumptions made in the calculations is discus- sed in detail. TEMPERATURE DEPENDENCE OF THE NQR FREQUENCIES OF PERIODATES AND PERRHENATES AND CALCULATION OF ELECTRIC FIELD GRADIENTS FROM X-RAY DIFFRACTION DATA BY Richard Allen Johnson A THESIS Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Chemistry Program in Chemical Physics 1971 ACKNOWLEDGMENTS The author would like to express his indebtedness to Professor M. T. Rogers for his guidance, encouragement and patience during the course of this study, which ranged rather far afield from magnetic resonance at times. Special thanks are also due to Professor G. E. Leroi for his help in certain aspects of the laser Raman study. The author would also like to thank Professors E. Carlson and A. Tulinsky, D. A. Hatzenbuhler, Dinesh, K. V. S. Rama R00 and the X-ray crystallography group for helpful discus- sions in various aspects of this work. The support of my wife Kathy, both mentally and financi- ally, in all aspects of this work is deeply appreciated. Finally, the financial support of the Gulf Oil Corpora- tion, the Atomic Energy Commission, and the Chemistry Depart- ment during various parts of this study are gratefully acknowledged. ii II. TABLE OF CONTENTS INTRODUCTION . . . . . . . . . . . . . . . . THEORY . . . . . . . . . . . . . -V- . . . . A. C. General Background of NQR Spectroscopy . 1. Quadrupolar Hamitonian . . . . . . 2. Energy Level Expressions:RX'I = 5/2 . Calculation of Electric Field Gradients . 1. -Non—overlapping Charge Model . . . 2. Townes-Dailey Method . . . . . . . . Temperature Dependence of NQR Frequencies EXPERIMENTAL O C O O C O O I O O O O O O O A. Synthesis and Crystal Growing . . . . . . 1. Synthesis . . . . . . . . . . . . . . 2. Crystal Growth . . . . . . . . . . . Infrared and Raman Spectroscopy . . . . . 1. Far Infrared . . . . . . . . . . . . 2. Laser Raman . . . . . . . . . . . . . -NQR Temperature Studies . . . . . . . . . 1. Constant Temperature Baths . . . . . 2. Dewar Methods . . . . . . . . . . . . 3. Temperature Measurement . . . . . . . 4. NQR Oscillator Modifications and Operation . . . . . . . . . . . . . iii Page 10 11 12 18 21 32 32 32 '33 37 37 39 4O 40 42 48 53 TABLE OF CONTENTS (Cont.) Page III. .RESULTS AND DISCUSSION . . . . . . . . . 57 A. .Far IR and Raman Spectra . . . . . . 57 1. Factor Group Analysis . . . . . . 57 2. Frequencies and Symmetry Assignments 60 B. Temperature Dependence . . . . . . . 96 1. Normal Behavior: NaIO4 and MReO4 (M = Na, K, Ag) . . . . . . . . . 96 2. Anomalous Behavior: NH4ReO4 . . 121 C. Electric Field Gradient Calculations. 133 1. Method of Calculation . . . . . . 133 2. Results and Problems Encountered. 137 LIST OF REFERENCES . . . . . . . . . . . 144 iv 10. 11. 12. 13. LIST OF TABLES Page Site and factor group splittings of Scheelite- type perrhenates . . . . . . . . . . . . . . 59 Activity of polarizability derivative tensor components under C4h point group symmetry. 60 Infrared and Raman active ReO4- fundamentals at 2940K O O O O O O O O O O O O O O O p O 0 72 Lattice modes in Scheelite-type perrhenates at 2940K O O O O O O O O O O O O O O O O O O 73 ~Raman-active lattice modes in Scheelite-type perrhenates near 77°K . . . . . . . . . . . 73 Experimental versus theoretical depolarization ratios in NaReO4 o o o o o o o o o o o o o o 77 NH4+ and ND4+ Raman-active fundamentals in the periodates and perrhenates . . . . . . . . . 91 Isotopic ratios for NH4+ fundamentals compared to valence force model predictions . . . . . 93 Raman-active lattice modes in NH4+ and.ND4+ perrhenates and periodates near liquid nitrogen temperatures . . . . . . . . . . . . . . . . 93 Lattice modes in NH4+ and ND4+ perrhenates and periodates at 298°K . . . . . . . . . . . . 94 Temperature dependences of periodate and perrhenate lattice modes.. . . . . . . . . . 100 Experimental frequencies and temperatures for the (t 3/2 <-> i 1/2) 187Re pure quadrupole reso- nance transition in NaReO4 . . . . . . . . . 102 Experimental frequencies and temperatures for the (t 3/2 <-> i 1/2) 187Re pure quadrupole resonance transition in KReO4 . . . . . . . 103 V LIST OF TABLES (Cont.) TABLE 14. 15. 16. 17. 18. 19. Page uExperimental frequencies and temperatures for the (i 3/2 <—® t 1/2) 187Re pure quadrupole resonance transition in AgReO4 . . . . . . . . 104 .Experimental frequencies and temperatures for the (T 5/2 - T 3/2) 1271 pure quadrupole resonance transition in NaIO4 . . . . . . . . 104 Final parameters obtained in applying the adjusted Bayer theory . . . . . . . . . . . . 120 Experimental frequencies observed at various temperatures for the (t 5/2 <—¢ T 3/2) transition of the 137Re pure quadrupole resonance Spectrum 124 Positive temperature coefficients of NOR frequ- encies observed in NH4ReO4 and several other compounds . . . . . . . . . . . . . . . . . . 132 Calculated electric field gradient in B—Icl . 139 vi LIST OF FIGURES Figure 1. Variable temperature water bath for single crystal growth . . . . . . . . . . . . . . . 2. Liquid helium Dewar with a 4" Pyrex pipe fitting for NOR and Q-Band ESR studies . . . 3. Cartesian manostat for pressure regulation . 4. Cross-section of Swenson-type variable- temperature bomb used with the liquid helium Dewar O O O O O O O I O I O O O O O O O O O 5. Correction voltage between measured and tabu- lated emf's for the copper-constantan thermo- couple used in this work . . . . . . . . . . 6. Rama spectrum of polycrystalline NaReO4 using 5145 Ar(II) laser excitation . . . . . . . 7. .Raman spectrum of the lattic§ region in poly- crystalline KReO4 using 5145 Ar(II) laser excitation . . . . . . . . . . . 8. Raman spectrum of the lattice region in poly- crystalline RbReo4 using 51458 Ar(II) laser excitation . . . . . . . . . . . . . . . . . 9. Raman spectrum of the lattice region in poly- crystalline AgReO4 using 51453 Ar(II) laser excitation . . . . . . . . . . . . . . . . . 10. Infrared spectrum of polycrystalline NaReO4 in Nujol mull at 298°K in the 400-500 cm '1 range 11. Raman spectra of single crystal.NaReO4 in the four unique orientations . . . . . . . . . . 12. Raman spectra of the lattice r§gion in poly- crystalline NH4ReO4 using 5145 Ar(II) laser excitation . . . . . . . . . . . . vii Page 35 43 45 47 50 62 64 66 68 7O 74 79 LIST OF FIGURES (Cont.) Figure 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. Raman spectra of the lattice r§gion in )poly- crystalline ND4ReO4 using 5145 Ar(II) laser excitation . . . . . . . . . . . .Raman spectra of the lattice iegion) in poly- crystalline ND4IO4 using 4880 Ar(II) laser excitation . . . . . . . . . . . . Raman specéra of ammonium fundamentals in NH4ReO4 using 5145 Ar(II) laser excitation . . . . .Raman spectra of deut roammonium fundamentals Page 81 83 85 in ND4ReO4 using 5145 Ar(II) laser excitation 87 Raman spectra of deu eroammonigm fundamentals in ND4IO4 using 4880 and 5145 Ar(II) laser excitation . . . . . . . . . . . . NQR spectrum of the (i 3/2 <-¢ t 1/2) 187Re pure quadrupole resonance transition in poly- crystalline KReO‘ at 300K 0 o o o o o o o 0 Experimental and calculated temperature de- pendence of the.137I quadrupole resonance in Nan4 0 O o I o o o o o o o o o o o o o o 0 Experimental and calculated temperature de- pendence of the 187Re quadrupole resonance in ~NaReO4 o o o o o o o o o o o o o o o o o o 0 Experimental and calculated temperature de- pendence of the 187Re quadrupole resonance in mac 4 o o o o o o o o o o o o o o o o o o 0 Experimental and calculated temperature de- ,pendence of the 137Re quadrupole resonance in AgReO 4 o o o o o o o o o o o o o o o o o o 0 »Experimental tem erature dependence of the (T 5/2 <—> t 3/2 187Re transition in NH‘ReO4 viii 89 106 109 111 113 115 125 INTRODUCTION Although the first quadrupole resonance experiment was performed over twenty years ago, the field has yet to under- go the rapid growth and wide application experienced by the analogous magnetic resonance spectroscopic techniques of NMR and ESR. Some of the reasons for the slow growth of NOR spectroscopy are more readily apparent by comparing the resonance equations relating the energy level splitting to the applied frequency v for the various types of magnetic resonance spectroscopy, hv gBH (NMR or ESR) 2 hv = g_ Qq (NQR) where B, the Bohr magneton, and Q, the nuclear quadrupole moment, are nuclear properties.. In the NMR-ESR case reso- nance may be achieved by varying the frequency v or the ex- ternal magnetic field H, while in the case of NOR only the frequency may be varied since the electric field gradient q is fixed by the electronic environment of the quadrupolar nucleus. The difficulties associated with building high- sensitivity radio-frequency oscillators capable of scanning large frequency ranges and the inability of current theories relating experimental frequencies to molecular structure are 1 2 the principal reasons for the slow growth of NQR. The approaches presently used to calculate electric field gradients have some serious deficiencies as a result of the approximations necessary in their application to solid state calculations. One of the projects in this work involved investigating an alternate, semi-empirical pro- cedure for field gradient calculation based on using elec- tron densities in crystals obtained from X-ray structure determinations. The ability to predict approximate values for quadrupole coupling constants for nuclei not previously studied by NQR would be of great advantage in locating new resonances experimentally. ~In addition to the information concerning molecular structure obtainable from quadrupole coupling constants, studies of the quadrupole frequency shifts with temperature can provide detailed information about intermolecular inter- actions and motional changes in the crystal. The second pro- ject investigated in this work was the temperature dependence of the NQR frequencies in several perrhenates and periodates. According to conventional theories, in order to obtain agree- ment between theory and experiment an independent experiment,. the pressure dependence of the NQR frequency, must be per- formed. Since the pressure experiment is difficult to per- form, several approximations are made in order to account for the lack of pressure data. One of the assumptions in- volved, the neglect of the effect of lattice contraction or expansion on the field gradient, was tested in this work for the series of perrhenates and periodates studied. 3 Experimentally, this involved a detailed infrared and Raman study of the lattice region and the experimental tempera- ture versus NQR frequency from liquid helium to room tem- perature. .During the course of the work a liquid helium dewar system was constructed for both NQR and Q-Band EPR studies. One of the perrhenates, NH4ReO4, showed an anomalous temperature dependence with loss of resonance upon cooling near 2459K. Use of the X-ray structure determination re- sults, the vibrational spectra, and the quadrupole reso- nance results allow a model for the ammonium ion motion in the lattice to be constructed. The mechanism proposed to account for the anomalous behavior in ammonium perrhenate is different from previous hypotheses explaining anomalous NQR temperature dependences, which do not appear to be ap- plicable to ammonium perrhenate. I . THEORY A. General Background of NOR Spectroscopy 1. .guadrupolar Hamiltonian The quadrupole term H in the total Hamiltonian can be Q looked upon as describing the effect of an interaction be- tween a non-spherical nucleus and its environment. The quantum mechanical treatment of the quadrupolar Hamiltonian ‘was first given by Casimir (1). More accessible treatments of the derivation are given by Ramsey (2), Kopfermann (3) and Cohen and Reif (4). An excellent discussion of the application of the Wigner-Eckart theorem to the matrix elements of H0 is given by Slichter (5). The total energy of a system of a-nuclei and i elec- trons can be written classically in the Hamiltonian formu- lism as _ 1 2 1 2 HTOT _ i EMQ Pa + E 2m Pi + vee + Vnn + Vne ’ ‘where Pa and Pi denote the momenta of the nth nucleon and ith electron, respectively, V the electron-elec- ee tron potential, and th the nucleon interaction potential. The potential energy term Vhe for nuclei and electrons is 4 5 given by the electrostatic interaction between point charges according to Coulomb's law N (D Vne = - QM 2 0‘ <1) i 10 H with rid the separation between the a nucleon and the ith electron. As one would expect, however, the nucleus has a finite volume and the effect of this charge distri- bution within the nuclear volume V must be included in the expression for Vne' If a continuous charge distribu- tion p(£f) is substituted for the point charge 20' the potential energy expression becomes vne = Hne = f p(f')V(f - f') d3f' (2) where the electron is at a point I , (fl >> lf'l, and the primed coordinates denote points inside the nuclear volume V'. Writing V(§ — E') explicitly and regrouping terms gives .lf-f ° (3) I H =e|f m'dfilfl ne , l V I Since p(f') is the nuclear charge distribution, contained in a radius a << III, the expression enclosed in the dotted lines may be considered as an electrostatic potential due to an unknown charge distribution. Therefore, carrying out a multipole expansion (6) of this potential we have H =ef [i+§é§;+lm- r'3] + ...}p(ft) d3'r-n. (5) Rewriting this expression in terms of potential energies and their derivatives leads to the expression for Hne given by Cohen and Reif (4): Hne = f d3xp(x){vo + E (%¥')- x. + %- ; (ggyb )_ x.xk+---] v' j=1 j x=o 3 j,k=1~ .j XI x=o 3 (6) and the origin is now the nuclear center of mass, since its motion is unaffected by nuclear reorientation. If the terms independent of i, the electron position vector, are taken from under the integral, it gives (from either Equation (5) 0r (6)) _ 1 . H - zevo + .5; P.V. + 23; jkvjk + , (7) with f d3ip(i) = Ze, the nuclear charge f d3ip(i)xj = Pj, nuclear electric dipole moment - _ (8) f d3Xp(x)ij ? Qfik’ nuclear electric quadrupole moment. The first term is simply the charge of a point nucleus and therefore has already been included in the Hamiltonian. 7 In the quantum mechanical transformation, the charge distri- bution p(i) is replaced by the operator §(i), which de- pends on the position vectors of the A nucleons in the nucleus. However, the energy separation between the ground and excited nuclear states is very large compared to the quadrupole splittings and to a very good approximation the matrix elements of H can be considered diagonal in all quantum numbers characterizing the nuclear ground state, except the nuclear spin projection m. If all unspecified quantum numbers are denoted by n , the matrix elements will have the form - c , (13) where C is a proportionality constant independent of m or m' and I2 = I: + I: + I3. The constant C is related to the nuclear electric quadrupole moment, Q , defined con- ventionally as eQ = <11|033III> - (14) Then, (see Equation (13)) eQ = C(III3I: - I2III> = c{3I2 - I(I + 1)} and C = I(ZIQ- 1) ° (15) By incorporation of the above definitions into HQ . the quadrupolar Hamiltonian can be written as “ e H =+Z Q 61 21-1)J k 0 I 3 -2 [511j1k - IkIj) - éjk I } vjk . (16) The field gradient tensor, g , is also symmetric by its definition, resulting in six independent components rather than nine. By choosing the coordinate system to coincide with the principal axis system of 9', only three components of 3' need to be specified in addition to their direction cosines. In the principal axis system the Hamiltonian becomes iiQ = ZITSI—ZIT'[V22(3I; ' i2) + (Vxx - VYY)(I; - 1%)}.(17) Finally, since only spherically symmetric S state elec- trons have non-zero probability of being at the nuclear center of mass, Laplace's equation gives us the additional relationship V + V + V = 0. The two independent XX YY 22 components needed to characterize V’ are conventionally ~ defined as sz , the field gradient, (D an Ill and n = V . the asymmetry parameter. 10 Since the range of values of I is small, the resulting secular equations have been solved to give energy level ex- pressions and transtion frequencies for most nuclei which it is feasible to study. References to such tables of eigenvalues and energy level expressions are given in Das and Hahn (8). Because of their applicability to nuclei studied here, the expressions for I = g- are developed in the following section. 2. Energy Levels and Frequencies for I =‘% The nuclei studied in this work are all of nuclear spin I = g7 127:, 185Re, 187Re and the energy level and transi- tion frequencies for this case are developed below. Refer- ring to the expression for HQ given above (see Equation 17)) A 2 H = 6 Q {(31 Q. 41 I - + 13 )} - (19) Of the possible matrix elements of Hb only two non-zero types remain, j =E%;%g—:T)' [3m2 - I(I + 1)] and (20) = 4:221_1 311(I+1)-m(mi1)}1/2{I(1+1)—(m:1)(mi2)}1/2. The secular equation which results for spin g-cannot be solved exactly, but Bersohn (9) has derived a general ex- pression convergent for n41 0.25. The energy levels to second order in n, sufficient if n 5.0.1, are 11 Ei5/2 ‘9'}39‘ (10+3n2 + ...) 2 Eta/2 =31? (-2 + 3n2 + ...) (21) .e2 32 n+1/-—.%9<---§-n2+~-> A complete tabulation applicable to any value of n is given by Livingston and Zeldes (10), who give tables of eigenvalues for I = g- and n values from 0 to 1 at intervals of 0.001. Considering only allowed transitions two lines should be observable with frequencies (5 3 _ 3equ 11 wQ(:2—:§-)- 1m. ”WIT” (22) 3 1 -362 5 2 wo'70*%g<1+§zn’- For the compounds considered in this study it has been found (11) that within experimental error q.= 0, resulting in the (i %-<-¢ i go line being twice the frequency of the (i 523- <-—>i -;—) line. B. Calculation of Electric Field Gradients As shown in the preceding section, the observed quad- rupole frequencies permit the determination of two quanti- ties, equ and q. Since values of Q can in most cases be obtained from molecular or atomic beam measurements or gas phase coupling constants, the calculation of quadrupole frequencies reduces to calculating q and n, the electric field gradient (EFG) tensor components. Since pure quadru- pole resonance is observable only in solids, the discussion 12 here will be limited to solid state calculations of EFG's. The non-overlapping charge or ionic method will be dis- cussed first, followed by a brief discussion of the Townes- Dailey theory as background for references to the theory in discussing the anomalous temperature effects in Section C. 1. _Non-overlapping Charge Model The general problem is to calculate the electrostatic potential at the center of mass of the quadrupolar nucleus due to an external charge distribution p(f). The potential is given by d3? 55 V(O) = f _ . (23) r with the integration extending over the range of p(§). Correspondingly, the field gradient tensor components are given by _ - 3K2 "" r2 3" VKICI(O) - f p(r) T d r and (24) = - .K_V 3- VICV(O) f p(r) r5 d r where x, v denote Cartesian components of f and again the integration is over the range of p(f). In cases where accurate wave functions are available, such as isolated simple molecules, this involves the evaluation of one—, two- and three-center integrals. In most cases this re- sults in fairly close agreement between theory and experi- ment. For solids, however, existing wave functions are 13 not of sufficient quality for such calculations and approxi- mations must be made. ,Presently the most widely used ap- proximation is the non-overlapping charge model, in which the charge distribution in the solid is assumed to be centered on the nuclei such that the charge clouds do not overlap. This clearly implies complete ionic bonding and is not expected to give completely realistic results for a crystal with any appreciable amount of covalent bonding. The second, and more restrictive, assumption usually made is that these non-overlapping charge distributions are spherical. As electrostatic theory states that such a distribution behaves as if the charge were concentrated at the center of the sphere, this assumption is referred to as the point-charge model. vIn this model the field gradient components can be calculated by evaluating two types of sums, 3K? - r? v =2e.-—3———-1 3'3 r3 and. (25) K.V. Vkv - z 6j 5 J rj I where ej _is the ionic charge of the jth atom, rj and vj are components of the position vector of the jth atom, and the sum extends over the atoms of an infinite crystal. Contrary to the feeling that the %3 dependence should lead to rapid convergence of these conditionally convergent sums, the number of terms needed to achieve reliable results is 14 quite large and is strongly dependent upon the method of summation employed. .Direct summation requires summing in spheres of increasing radius, first carried out by Bersohn (12), but convergence is slow and spheres as large as 1008 diameter are often needed to get satisfactory convergence. Several rapidly converging methods involving cumbersome neutral units such as the crystallographic unit cell have been developed (13,14), but one of the most elegant and rapidly convergent methods is that given by DeWette and Schacher (15,16) based on transforming the slowly converging series into one which is rapidly convergent. The method is derived from a procedure (17) for calcu- lating internal fields in dipole lattices, an important problem in dielectric constant theory. The sums in Equa- tion (25) are broken up into summations over sublattices of identically charged ions and then summed over the total number of unit cells. -For a crystal containing Ej dif- ferent ions the tensor components are 3 2‘ j) \kK(O) = Z Z s.{’K‘ )“ - -l— } A j 3 r5 . r3 . )\13 x13 and (26) 3x .V . _ AL) ALl. v.(o)-z:>:e.( ) , A,j ‘where 2A is the sum over the unit cells. The number of terms in EA is large, but should converge rather rapidly otherwise the calculated field gradient would depend upon 15 the external crystal shape. .Instead of summing Equation (26), the order of the A and j summation is interchanged and the conditionally convergent sums evaluated planewise rather than spherically. The first sum is over A1, A2 with A3 = 0, then over A1, A2 with A3 = 11, etc. The advantage of such an ordering is that instead of summing the slowly convergent A1, A2 series directly, its Fourier transform is evaluated in the reciprocal lattice. The resulting sum is absolutely convergent with rapid convergence in most cases, leading to consistent results after combining the sums from each of the sublattices. Although this method removes uncertainties in the results due to convergence errors, agreement of the calculated field gradients with those obtained from experiment is still extremely poor, in— dicating that several important effects have not been con- sidered. One of the asSumptions up to now is that the electron shell around the quadrupolar nucleus itself contributes nothing to the field gradient, since it is in an 18 state. In reality, however, this electron cloud is distorted due to polarization by the external field gradient or alterna- tively by the aspherical quadrupolar nucleus, giving rise to a significant, and in some cases exteremely large, contri- bution to the lattice field gradient (18). Regardless of the point of View, the calculated field gradient is altered from 6&at' the calculated external field gradient. The as new tensor V is given by 16 3‘: (1 - y )‘3 co lat (27) where ya) is the Sternheimer shielding factor. This cor- N rection to V1a is often very large, since Va) is usually t negative and ranges up to 100 or more. The Sternheimer factors are not measureable experimentally and the uncer— tainties in calculating values for larger ions makes com- parison between theory and experiment difficult. In many cases, however, unusually large 700's are needed to give agreement with experiment, indicating that the point charge assumption is invalid and some distortion from sphericity in the other ionic charge distributions must be included. Since the site symmetry of nuclei in lattices in which quadrupole resonance can be observed is necessarily less than cubic, it might be expected that easily polarizable ions such as 02- would undergo some distortion. To take this distortion into account the charge distribution is replaced by a multipfle expansion, with the resulting poten- tial (28) e a 1 1 52 1 V = 2 _' + Z P 5"(‘9 +'-n Z q 5—-5-7(-) + '-° A R). an. 0‘ [Xa I 11 ”(1.53. 0‘5 xa XS r a R a where 2A denotes summation over all lattice points, pa is the a component of the dipole moment, qafi is the QB component of the quadrupole moment, the cubic term is as- sumed negligible and the expansion is truncated. The in- duced moments can be written as pa = avg . an = BVQB (29) ‘where 0:5 are the dipolar and quadrupolar polarizabilities, respectively, for each ion, although the smaller cation polarizabilities are usually neglected. Several problems arise in applying_this correction to the calculated field gradient, which makes the significant improvement in agree- ment with experiment questionable. Since alternate indepen- dent experiments don't exist to give a and 6 for these ions they are often treated as adjustable parameters. Values obtained from this procedure usually don't agree with polarizabilities calculated from ab initio wave func— tions, but the calculated values are strongly dependent upon the wave functions used. Even more discouraging, however, the dipolar and quadrupolar terms are of comparable, or even larger magnitude, to the point-charge term, contrary to the assumption made in truncating the series in Equation (28). As a result of such uncertainties in yoo and in the induced polarizabilities, agreement between theory and ex- periment does not necessarily imply validity of the ionic model. The need for such modification is a result of at- tempting to modify the non-overlapping charge model to accommodate some degree of covalent bonding, certainly present even in “ionic" crystals. When the degree of co- valency is high enough to consider the quadrupolar nucleus as forming part of a molecular unit an alternate procedure 18 is used. In this procedure the EFG contribution from the lattice is calculated as outlined above and added to the co- valent contribution calculated from the Townes-Dailey theory discussed in the next section. Unfortunately, the empirical nature of the Townes-Dailey theory usually results in agree- ment with experiment, obscuring the deficiencies in this type of approximation. 2. Townes-Dailey Method This discussion is only a brief outline of the com- plete theory which has been presented elsewhere (8,19,20), with an outline of the main assumptions and resulting equa- tions relating the coupling constant, equ, to the bonding parameters for the molecule containing the quadrupolar nucleus. If the Sternheimer polarization factor is neglected, the value of q in isolated atoms arises only from the electrons outside the closed shells. The magnitude of this atomic field gradient is known from atomic fine or hyperfine structure studies or can be calculated from atomic wave functions. (The central assumption which Townes and Dailey make is that the quadrupolar nucleus retains its outer electrons in hybridized atomic orbitals that take part in forming molecular orbitals, and that the resulting coupling constant is proportional to qat’ the field gradient for the isolated atom. Therefore, Townes and Dailey express qmol as 19 q = f mol qat (30) where the quantity f depends on the electronic structure calculated according to the prescription given by the Townes- Dailey theory. For simplicity consider the case of one valence elec- tron in a molecular orbital mi constructed from a basis set of atomic orbitals Xr’ ri Xr ’ (31) ¢i = 2 C r the contribution to the field gradient from this electron . can be written 3cosae. —1 _ * * 1A (eq)e1 — aflf crier r3 H: CsiXs) dr (32) iA for the field gradient at nucleus A. Rearranging, * * 3c0529i -1 (eq)e1 - e? i [Cricsi] f Xr( r3 ) Xs dT iA (33) q S ,= e Z Z R r s r rs where the Cri's are the expansion coefficients from Equa- tion (31), 91A and ’riA components of the position vector in polar coordinates, and Rrs denotes the term in brackets and qrs the integral involving the atomic basis functions x. For hydrogen-like atomic orbitals contribu- tions to the field gradient from orbitals with g > 1 are 20 relatively unimportant, since 4eZ3 (eq) = 3 n3ao(2£-1)(2z+1)(2£+3) n,£,m=0 and (34) _ -2z-1 (eQ)n,2,m=iz ‘ 2+1 (eQ)n.z,m=0 ' This inverse third power dependence of qat upon 2 is the basis for the assumption that contributions from orbi- tals with E > 1 are negligible, with the result that only valence electrons in p orbitals need to be considered, since the spherically symmetric s electrons don't con- tribute to (eq)at. Following Scrocco (21), consider as an example the field gradient at nucleus A in a diatomic molecule AB. The expression for (eq)mol following the notation given above is ) atom A (eq)mol = (eq)el + (eq)nuc ‘ (26: E Rrsqrs (35) ) + (2e: 2 RI q + (2e; 2 R q ) r 3 rs r s 8 rs overlap AB rs atom B _ 2eZB 3 RAB Two primary assumptions are now made (19) to simplify this expression; 1) Due to the 1/riA term in qrs (see Equation (33)) the second and third terms are rather small and are assumed to be cancelled.by the last term, leaving only the first 21 term contributing to (eq)mol. 2) Assuming contributions from inner shell polariza- tion is negligible compared to the field gradient from the valence p electrons we arrive at the simple result el (36) (eq)mol = (eq)atom A = (nx(eq)px + n-y(eq)py + nz(eq)pz)atom A‘ Assuming the field gradient is equal to the corresponding free atom quantities reduces the relationship to the form given in Equation (30), ezoqmol 8 eQ{[nz - %(nx + ny)](eq)p ] . (37) 2 atom A In treating specific examples the occupation numbers nx, ny of the px and py orbitals are often related to w molecular orbital occupation numbers and n2 to the frac— tion of Sp hybridization in the 0 bonds. For transition metal nuclei a correSponding expression involving nd valence orbitals is used, since the (n + 1)p orbitals are too high in energy to contribute to the bonding. C. Temperature Dependence of NOR Frequencies Since the electric field gradient and hence the fre— quency of absorption are dependent upon the positions of other ions in the crystal lattice, the observed temperature dependence of NOR frequencies is reasonable. In practice two general types of behavior are observed: (a) the NQR frequency increases with decreasing temperature, termed the 22 normal temperature dependence, and (b) the frequency de- creases with decreasing temperature, termed the anomalous temperature dependence. Other types of behavior include changes in the number of absorption lines and the disap- pearance of a line due to phase transitions or the creation of slightly inequivalent sites as the crystal contracts. This discussion will deal primarily with normal temperature variation, but a short discussion of possible origins of the anomalous temperature effect will be given, since NH4ReO4 exhibits this behavior. In the case of normal temperature dependence, the NQR frequencies increase as the temperature is lowered. Since the quadrupole moment, Q, is a nuclear ground state property it should experience no change over normal temperature ranges, therefore changes in the field gradient tensor must be responsible for the shifts of frequency with temperature. A mechanism was first advanced by Bayer, who attributed (22) the observed behavior to low-frequency vibrations of the atoms or molecules in the crystal. In many cases, however, the Bayer theory gives poor agreement with experiment, since it neglects lattice expansion. The theory was extended (23) by Kushida, Benedek and Bloembergen in a phenomenalogical theory including the Bayer theory as one of several terms. An outline of this theory is given below along with a dis- cussion of the model of the crystal vibrations used in calcu- lating the Bayer term. The following discussion is derived mainly from references 8, 22 and 23 with a number of missing steps supplied. 23 As discussed in Section A and above, the quadrupole interaction is completely characterized by two parameters, equ and n. Before examining the microscopic model relating crystal vibrations to these parameters, consider the quadrupole frequency v to be a function of volume, temperature and the low frequency vibrations, with the fre— th quency of the E vibrational mode denoted by v3. The total differential of v, with v = v(v£,V,T) is 8v 5v 5v dv = z ( ) dv + (——) dv + ( ) dT (38) 2 5V2 v,T 2 5" vE,T B‘T v£,V where the sum is over all possible modes of vibration, both internal modes and external lattice modes. Partial differ- entiation with respect to T at constant P,N gives (3%)P,N = i, (%5E)V.T (;¥)P,N + (%%)V£,T (%%)P,N + (3%)V,V£ Similarly, (39) av 5v 5v 5v 8V (FF-)T,N= 5LZZ(5-‘7)€)V,'I.‘ (#)T,N + (EV)V£,T(¥)T,N . Assuming that the lattice vibrations depend only on volume, i.e., that the crystal vibrates isotropically then v = z VZ(V) and (27:14) ' ifs —(5—)P P,N ,N and (40) (3%) = §V&(§§)TN 24 Substitution of these relations into Equation (39), gives the necessary final relationships: dv av _ 5v 2 5v av av () -{2<>—+r) }() +< as”, ‘ 253,3“ dv a7 WT EM as’v‘vg (41) av _ 5v d") av av ' < ) - {2( ) -——- + < ) 3< ) . 5 m1 2 53 M, dv 5? V2” 5‘5 T'N Note that the bracketed terms are identical in the above equations, although they are multiplied by different partial differentials. These differentials are related to two experimentally observable quantities , Coefficient of thermal expansion av ( ) T P,N —.l v 5" and (42) _ 1 av . . . K = - Vlsij N , Isothermal compreSSIbIlIty. In principle, therefore, it should be possible to remove the effects of lattice expansion by performing an additional experiment, the measurement of pressure dependence of the NQR frequency and either measuring, or obtaining from the literature, values of a and KT over the range of tem- peratures covered. Then the bracketed term could be calcu- lated from the pressure-dependence data leaving only the desired Bayer term, (Bv/BT)V v ; to be calculated from an ' £ appropriate microscopic model and compared to experiment. Unfortunately, compressibility data below room temperature 25 are rare, and since 'KT is a function of temperature an equation of state is needed from which KT can be calcu- lated at any desired temperature. Before discussing the method used by Kushida, et al. a short discussion of the harmonic potential model for a crystal is given. The method of treating vibrations in solids (24) is the harmonic approximation in which the displacements of atoms from their equilibrium positions are assumed small and as a result the potential V(Rj) can be written as V(R3 + fij) where the Rg's are equilibrium coordinates and the fij's displacement coordinates. Expanding V(Rj) in a Taylor series and truncating the series after the term second order in uj gives the well-known result for the vibrational Hamilitonian, 3N P? 3N 3N -Hvib=2 5-1-+-1- 2 z A.-u.u. . (43) i=1 m i=1 j=1 13 1 3 The displacement coordinates which diagonalize this Hamili- tonian are called normal coordinates, the resulting Hamili- tonian being identical to that for the harmonic oscillator with the eigenvalues termed normal modes. In a three- dimensional crystal the number of normal modes is very large, on the order of 3nN for a crystal with an n-atom basis (n-atoms per lattice point). These quantized vibra— tions are known as phonons and are conventionally labelled by k_ vectors, which are the position vectors in reciprocal lattice space. Additional details concerning phonons, the harmonic approximation and the Brillouin zone representation 26 can be found elsewhere (24,25). .If the phonon states are labelled by kfvectors, the molecular partition function becomes -mlmk '69). "mwnk’rl/fl e 2 zk = 2 e = 2 e = “—:§fiai'- (44) k nk 1-e The total partition function, Z = szk , becomes 2 = F 1 (45) k Bhuk 2 sinh( 2 ) and, remembering that A 3 -kT ln Z, one obtains 3N flak A "Ucohesive(v) - kTKEI 1n(2 SInh (515)), (46) where the U (V) term arises from a redefinition of cohesive the zero of potential energy. Since 5A _ 82A P - '(W)T N , then “—T- - V(;;)T N (47) 74.1 and the compressibility may be obtained from A at any temperature by differentiating twice with respect to volume. By the above procedure the only remaining unknown is the Bayer term, (BV/BT)V v . Bayer assumed that the field ' 2 gradient would be averaged in the crystal by librational, or torsional, motions which modify the field gradient tensor by mixing field gradient components. Following Das and Hahn (8), the general case of a molecule undergoing 27 torsional motion about all three principal tensor axes, in a system without axial symmetry, will be considered. Small rotations 9X, 9y, 6 about the molecule-fixed axes x, y, z z can be related to the space-fixed system x', y', 2' by a unitary transformation. To find the transformation consider first a rotation 6X about the x axis, then successive rotations about the y and z axes of our coordinate sys- tem. Combining these will give the desired transformation matrix. The rotation 9X gives X'=X y' ycos 9x - zsin 9x (48) , . z zcos 6X + ySIn 6x . Similarly for 6y and 92 II = I I ' III = II _ II ' x x cos 9y + 2 Sin 9y x x cos 62 y SIn 92 u = III 3 II II ° y' y' y y cos 62 + x SIn 92 z" = Z'cos e - x's'n z"' = z" 49 y I 9y ( ) where the axis system to the left in each set corresponds to the rotated system. Substitution and combination of terms gives the relationship between the space—fixed x, y, z axes and the moving x', y', z' axes x' = (coseycosez)x + (coseysinez)y - (sin9y)z (50) y' = (sinexsineycosez—cosexsinez)x + (cosexcosez)y + (sinexcosey)z z: = (Slnexs1n92+COSGXSIn9yCOSGZ)X + (COSSx81n9ySIn92 -sin9xcosez)y + (cosexcosey)z . 28 Use of this transformation and expansion of the Sines and cosines in a MacLaurin series, since the rotations 9X, 6 I Y 62 are assumed small, gives Vx'x' = (l-eY-GZ'MIXX + £9ng + 912!sz VY'Y' = 9; Vxx + (1—9;-9;)VYY + 9;sz VZ,Z, = a; vxx + eftvW + (I-efc-effwzz (51) VX'Y' = (exey'ez)vxx + esz - exeyvzz VY'Z' = ' eyez Vxx + (eYez-eX)VYY + 9szz VX,Z. = (axeZ + eY)vxx - eerVYY - eY vZZ . Only displacements up to second order have been retained in the above products. Thus, the instantaneous field gradient tensor is both time dependent and non-diagonal as a result of these librational motions. If the time-averaged effect on q and n is considered, and it is remembered that = <9Y> = = o . (52) we have «T = q[1 - g-< + )- f.,}< - > (53) + -;-<3 -n>1 and 29 n' = g, [n -'§-< - > - §n< + > %(3 - n)1. For all compounds considered here n has been found to be zero experimentally and, therefore, the effect of n on q' and q' will not be considered further. The (9:) terms can be related to measured vibrational frequencies by use of the harmonic approximation and the quantum mec- I hanical analysis for a harmonic oscillation, leading to mu¢ = (E) = ( + %)‘hw . (54) Recalling that phonons obey Base-Einstein statistics, and by use of rotational counterparts of m and x, the ex- pression becomes 1 > exp (hwz/kT) -1 Maya?) = hwz ($4 (55) for the 3th vibrational mode. Substitution of Equation (53) into the expression for q', the final expression relating the time dependent quadrupole frequency v to v0, the quadrupole frequency of the rigid lattice, becomes 3h 1 1 v(T) = v 1- z: - [—+ )). 56) o( normal 21V»z 2 expfix‘wz/krr) - 1 ( modes ‘ The summation term is written in a general way including other motions besides librations, although the derivation of Equation (55) assumed that only librations along the principal tensor axis modified the EFG. A discussion of 30 this point in this work and its treatment in the literature ‘will be given in Chapter III. As mentioned above, in a few cases an anomalous tem- perature effect is seen in which the quadrupole frequency increases with increasing temperature. In almost all such cases the compounds contain transition metal ions in oxida- tion states in which the t29 orbitals which take part in dw-pv metal-ligand bonding.are partially filled. The anoma- lous behavior has been attributed (26,27) to the effects of fundamental vibrations of the transition metal complex on the degree of w-bonding. In Section B the f factor in the Townes—Dailey theory for interpreting quadrupole coupling constants was shown to be f = N2 - [(NX + NY)/ 2], (57) where the Ni are ligand pi orbital occupation numbers. In the cases involving chlorine ligands, Haas and Marram (26) assumed that the px, p orbitals were involved in Y w-bonding with the central metal ion, giving f=Nz-N7T. (58), Since the vibrational fundamentals are primarily stretching and bending motions the effects of each will be considered. The effect of‘the Stretching motions was shown by Haas and Marram to be zero if a harmonic potential for the stretching vibrations was 31 chosen, since second—order terms in the expansion were then cancelled by deviations from harmonicity. In contrast, they postulated that the bending vibrations perturbed the bonding wave functions such that NW increased linearly with (62>, the time average of the square of the bending angle. Since NV > NC for the halogens, the result was an increase in If] with increasing (92>; (62) is proportional to tem- perature hence they obtained an increase in the coupling constant with increase in temperature which overcame the negative temperature dependence of the Bayer term. Recently this explanation has been disputed (28) by Brown, who con- tends that it is the stretching, rather than the bending, motions which give rise to the observed effects. An excel- lent treatment of the two competing effects has been dis- cussed in a series of papers (29) by Armstrong and coworkers. The above mechanism which give rise to an anomalous NQR temperature dependence are not applicable to NH4ReO4, where the anomalous behavior is believed to result from the freezing out of the rotation of the ammonium ion. The specific mechanism will be discussed in detail in Chapter III. PHI .1: .‘ II. EXPERIMENTAL A. Synthesis and Crystal Growing 1. Synthesis Most of the compounds used in this study had been pre— pared and purified previously (11) and were used without further purification. However, the ammonium and deutero— ammonium salts were prepared for this work and purification continued until impurity bands observed in the Raman spectra disappeared. The method of preparation for each compound is summarized below. M4104 and ND4 104 These salts were prepared by neutralization of ammonium hydroxide or deuteroammonium hydroxide with periodic acid. The ammonium salt was prepared by dissolving periodic acid (H5106, G. F. Smith Chemical Co.) in distilled water and adding ammonium hydroxide solution («'6N) dropwise to pHRd. The deuteroammonium salt was made in a similar manner, ex- cept the H5103 was dissolved in a large amount of D20 to reduce the probability of isotopic mixing and the acid solu- tion neutralized with a 26% solution of ND4OD in D20 32 33 (Diaprep, Inc.). In both cases, if the neutralization was carried out by adding acid to a basic solution strong im- purity peaks, apparently due to 103-, were found in the Raman spectra. No evidence of these impurity peaks was obtained in the samples used. NH4ReO4 and ND4ReO4 Rhenium heptoxide, Re207, was dissolved in water to form perrhenic acid which was then neutralized by the corresponding base. For the deuteroammonium salt the Re207 (Pressure Chemical Co.) was dissolved in D20 forming DReO4 which was then neutralized (pH ~05) by the ND4OD solution. Because of its hygroscopic nature, the rhenium heptoxide was handled in an inert atmosphere until just prior to addition of the H20 or D20. -RbIO4 This compound was prepared by neutralizing a solution of H5103 with a solution of rubidium hydroxide (Alfa In- organics, Inc.). 2. Crystal Growth Large, good quality crystals of both NaIO4 and NaReO4 were needed for Raman polarization studies. A suitable crystal of NaReO4 was obtained by evaporation of an aqueous solution of the salt at room temperature, but several methods were tried before good crystals of sodium periodate 34 were obtained. Neither crystal could be grown from the melt since decomposition occurs before melting. The first attempt to grow the NaIO4 crystals was by slow evaporation of a saturated aqueous solution in a parti- ally covered beaker at room temperature. The seed crystals were either laid on the bottom of the beaker or suspended by thin string. In both cases the resulting crystals were encrusted with small crystallites which formed at the evapor— ating surface and dropped onto the seeds. The latter prob- lem was taken care of by covering the seed crystals with an inverted beaker perforated to allow free circulation of solution. .The resulting crystals were relatively free of attached crystallites but the larger ones became cloudy. In order to prevent the formation of spurious seeds an alternate method (31) employing a completely closed system was used but crystallites still formed on the seed crystals. The procedure finally adopted was based on a method (32) used in growing crystals for X-ray studies. If crys- tallization is carried out by this method in a slightly acidic (H2804) solution between 40°and 45°C the resulting crystals do not have any cloudiness, even when grown rapidly. Also, the number of spurious crystallites formed is smaller and they do not seem to adhere to the seed crystal as much. The crystals were grown in sealed vials immersed in a large water bath (Figure 1) at a temperature of about 500 which was cooled at a rate of 1°/day for 10-14 days. Heat was supplied by a 125 watt blade heater connected to a 35 _ Thyratron 1 Temperature Control .___. Ci it Differential fl Thermoregulator l I Gear f" 125 w tt Arrange- Blade ment L, 1 Heater 1 ) g R / )3 fray \\ 2 é a > \\\\\\J ? ) t \ \\\\\\\\\\\W g 11R \ \\ U "\ u \\\ //////// //// / Figure 1. Variable temperature water bath for single crystal growth. \ 36 thermoregulator circuit containing a Precision Scientific Differential-Range Thermoregulator. The temperature was lowered by slowly moving the contact pointer of the thermo- regulator by means of a gear system connected to a one RPM motor. Some problems were encountered in achieving satura- tion of the initial solutions in the sealed Vials so the seed crystals would not be dissolved. The best procedure was to prepare a saturatedlsolution by filling a vial with excess solid, leaving the vial in the 50° bath for several days, then decanting (rather than filtering) the saturated solution into a separate vial containing the seed crystal; this vial was then quickly capped and returned to the warm bath before crystallization could occur. The seed crystals were usually placed on the bottom of the vials although attempts were made to glue the seeds to the vial walls. However, none of the adhesives tried would hold the seeds for the full time required. The external faces exhibited by crystals with the Scheelite structure are given by Tutton (33). For both NaIO4 and NaReO4 the face lying on the beaker bottom is the (111) first-order bipyramid with (110) faces as the other primary set. A small segment of the third-order (311) set is also present. Measurement of the angles between the crystal faces using an optical goniometer was unsuccessful due to the large size of the faces. The crystals were also examined under polarized light in a polarizing microscope to check the quality of the crystal and to locate the 37 unique (S4) axis. All the crystals used showed no evidence of twinning or enclosed crystallites when examined under polarized light. However, the S4 axis lies along the elon- gated physical axis of the crystal and it was not possible to mount the crystal so the characteristic extinction (34) could be observed. Attempts to cleave or grind crystals to provide faces perpendicular to the S4 axis were unsuc- cessful. Rough measurements made with a makeshift contact goniometer, plus the Raman depolarization results, substani- ate the assignment of (110) and (111) faces as the major external growth faces. B. Infrared and Raman Spectroscopy 1. Far Infrared All of the far-IR spectra reported here were run on a Fourier-transform instrument, a Digilab Inc. Model FTS-16 Far-Infrared Spectrometer with Mylar film beamsplitters. The interferograms were digitized and transformed to con- ventional frequency plots on a Data General Nova computer interfaced with the FTS-16. The computer has a 16-bit word length, 12K of core and a 15-bit A/D converter. The spectrometer records in either a single beam or pseudo- double-beam mode or can store a reference Spectrum to be combined internally to give a double—beam result. A Single scan takes about 1 second and in normal operation a large number of scans, ranging from 500 to 2000, are II ‘V 38 accumulated to improve the single-to-noise ratio before transforming. The samples were run as polycrystalline powders in Nujol mulls between polystyrene plates. To remove water vapor the spectrometer can be either purged with nitrogen or evacuated. In most cases the spectra were run with the spectrometer evacuated which greatly reduced the amount of time required to eliminate water vapor before scanning. In order to characterize the instrument spectra of several periodates were run for comparison with previous results (35). The agreement was excellent, frequencies usually agreeing within 3 cm-l, despite the broad lines. Two methods were used in obtaining the double-beam spectra. The normal method employed in the FTS-16 is to take 20 scans of the sample cell, then 20 scans of the reference cell, etc. Although not a true double-beam technique the pro- cedure minimizes errors due to long term fluctuations such as light source warmup or water vapor purging. For some of the compounds the above technique could not be used as a result of an instrument malfunction and an alternate pro- cedure was used. A single—beam reference spectrum was taken, stored, and compared to a sample spectrum to give a double- beam result. New reference spectra were run at frequent intervals to check for instrumental drifts but no noticeable changes occurred. 39 2. Laser-Raman The general layout and design of the laser Raman spectrometer used in this work has been described else- ‘where (36). The light source was a Coherent Radiation Model 52A Ar(II) ion laser. The scattered light was col- lected at 90° to the incident beam, passed through a Spex Double Monochromator and detected with an IIT FW-130 Photomultiplier tube cooled to -30°. The signal was dc amplified and recorded. Two lines of the Ar(II) ion laser were used, the 48803 and 51458 lines, each with about 1.5 watt output power. In most cases it was necessary to at- tenuate the incident beam to avoid sample decomposition. This was a definite problem with the ammonium perrhenates, which exploded violently after a short time in the unat- tenuated beam. Polycrystalline samples were run using ordinary capil- lary melting point tubes as sample cells. The single crystals were mounted in a piece of modeling clay attached to a glass rod. The crystals were of sufficient size so that no portion of the laser beam was incident on the clay or sample supports. For low temperature work samples were mounted inside an unsilvered Dewar cell (37) through which cold nitrogen gas was passed. Temperature variation was accomplished either by changing the nitrogen gas flow or the liquid nitrogen level in the Dewar used to precool the gas flow. Temperature in the Dewar cell was measured by use of a copper-constantan thermocouple placed near the 40 sample. No special precautions were taken to calibrate the thermocouple and all temperatures are accurate to i5°K except at 77°K, where an appreciable amount of liquid nitrogen collected as a result of Joule—Thompson cooling of the gas entering the cell. Single crystal depolarization spectra were obtained by orienting the single crystals in the plane-polarized laser beam and placing a polarizer in front of the monochromator to admit only the desired component of the scattered light. Rough alignment of the crystal was done by eye then final adjustements made by monitoring the photomultiplier tube current at the v1 fundamental frequency while moving the crystal. Since the crystal had only (110) and (111) ex- ternal faces the incident beam was never perpendicular to a crystal face and both internal and external scattering occurred. The internal reflection destroyed the original polarization and resulted in scattered light contributions that led to deviations from theoretically predicted depolari- zation ratios. This problem could have been prevented by using crystals large enough to grind into cubic shape along the crystallographic axes. C. NQR Temperature Studies 1. Constant Temperature Baths In obtaining the temperature variation of the NQR fre— quencies a variety of techniques were employed for achieving and maintaining low temperatures. The different techniques 41 will be discussed in chronological order. Constant tempera- ture baths were used first because of availability and ease of preparation; finally, a liquid helium Dewar system, which required a considerable amount of work in design and imple- mentation, was employed. .In the constant temperature baths the liquid and solid (phases of a pure substance were placed in equilibrium. The available baths ranged from 273°K (ice-water) to 113°K (isopentane) selected from an extremely useful compendium (38) of such mixtures. A large number of possible liquids could not be used since the oscillator would not oscillate with the sample coil immersed in liquids of high dielectric constant. .For suitable liquids a slush was prepared by adding small amounts of liquid nitrogen to a Dewar containing the liquid until a thick viscous mixture of liquid and solid phases formed. The sample and coil were allowed to equilibrate in the bath for about one-half hour and the spectrum recorded. During this time the temperature was held constant by adding small amounts of liquid nitrogen to the mixture. The tem- perature was monitored with a copper-constantan thermocouple and remained constant during the time required to record the spectra. Although this procedure was accurate it was used only with-NaIO4 since it was extremely time consuming to make measurements at a series of temperatures. Both of the other methods used Dewar containers and liquid nitrogen or liquid helium as coolants. 42 2. Dewar Methods The first Dewar container used was the variable tempera- ture Dewar system for the ENDOR (Varian Associates Model E-700) spectrometer in conjunction with a Varian ESR Variable Temperature Controller, Model V-4540. The samples were (placed in glass tubes and inserted into the unsilvered Dewar \flith the oscillator sample coil wrapped around the outside of the Dewar. This arrangement severely reduced the signal- to-noise ratio due to the low rf power density at the sample but was suitable for NaIO4 and KReO4, where large amounts of sample were available. .Cooling was achieved by blowing cold nitrogen gas past the sample and out the top of the Dewar. The gas temperature could be set on the Varian Temperature Control- ler and sample temperatures were measured by a theromocouple inside the sample with leads extending out the top of the Dewar to a potentiometer. The lowest temperature reached was -180°, with marginal temperature stability at the lower portion of the range. These problems clearly indi- cated the need for a Dewar especially designed for variable temperature NQR measurements. The liquid helium dewar, shown in Figure 2, was de- signed to be used for both NQR measurements and Q-Band ESR measurements and was constructed by H. S. Martin Co., Evanston, Illinois. This double Dewar was constructed as a single unit with a Pyrex pipe fitting which greatly simplified mounting and dismounting of the Dewar from its “—4!“ Brass plate on u g 5 Dewar stand ’1 f \ Liquid nitrogen fill port 42 mm O.D. 24 mm I.D. Figure 2. Liquid helium Dewar with a 4" Pyrex pipe fitting for NOR and Q-Band ESR studies. 44 stand. .In designing the Dewar stand several objectives were taken into account. A good vacuum line was needed for pump- ing the vacuum space between the inner helium section and outer nitrogen section and for pumping down the transfer tube vacuum Space. .To reach temperatures below 4.2°K pro- vision was made for pumping on the liquid, monitoring the pressure, and holding the pressure constant at some reduced vaIue. Flexible metal vacuum hose was used for the con- nection between vacuum line and Dewar. The entire stand can be moved about on wheels or raised off the ground for rigidity by lowering four heavy bolts welded to the frame. (The vacuum pump is mounted separately to minimize vibration, the only connection to the system being rubber vacuum tubing. Pressure control is obtained with a Cartesian manostat (39) Shown in detail in Figure 3. During operation stopcocks A, B,.C are opened until the desired pressure is reached then stopcock C is closed. The pressure of the trapped gas pushes the float up until the rubber stopper pushes against the orifice, isolating the vacuum pump from the system. A build-up of pressure in the system has the opposite effect, pushing the float away from the orifice allowing the pump to again begin evacu- ating the system. Theoretically a 1 mm orifice gives a 5% maximum fluctuation of the pressure at any given pressure. (For the NQR studies the manostat was operated slightly below atmospheric pressure and worked well. Butyl Phthalate was used in the differential manometer for greater sensitivity. 45 .cofiumasmmu ousmmmum uom umumOSME Smflmmuumo .m musmflm {C umumeocmfi amaucmummmao mumamzusm amusm / L (A N X A f 2 : (I ] I . Ewummm OB Edsom> OB 46 -Since a range of temperatures was needed for the NQR measurements direct sample immersion in the cryogenic liquid was not employed, but rather a variable-temperature bomb similar to that of Swenson (40). The sample is placed in a coil inside a 1 1/2" OD copper cylinder with about 20 feet of 3/16" OD copper tubing coiled in three layers around the outside and silver-soldered in place. The bomb is attached to either of two Dewar tops, each with a different length of stainless steel connecting tubing, as shown in Figure 4, and placed inside the helium Dewar. A pressure differential at the top of the apparatus pulls liquid into a 1/8" stainless steel tube which extends into the liquid reservoir, up through the 30 coils of copper tubing acting as a heat exchanger and finally past the sample. For the longer Dewar top the apparatus was too efficient and heat had to be supplied. Therefore, a one- watt 1000 resistor was mounted in the base of the bomb to supply a variable heat influx to balance the cooling. Normally about one-half to one watt of power gave ac- ceptable heat influx with the combination of variable heat input and adjustable gas flow giving a wide range of heating or cooling rates. For the shortest Dewar top used the heat exchanger coils were above the outer nitrogen section and the heat influx was great enough so that no external heating was necessary. The maximum warming rate was about 0.5°K/min with no appreciable broadening of the NQR reso- nance from temperature inhomogeneities. 47 ENG to oscillator To vacuum < E. _ 29% — thermocouple reference bath \\ ,a—-—I “ - I A I Ill l‘J 1000, . <—— Cooling coils 1/2 watt ' resistor . I Oscillator coils To coolant reservoir Figure 4. Cross-section of Swenson-type variable-temperature bomb used with the liquid helium Dewar. 48 3. .Temperature Measurement The problems involved in measuring temperatures ac- curately down to 4.2°K have been thoroughly discussed else- ‘where (39,41-43) and will not be repeated here. It was anticipated that the NQR frequency versus temperature curves 'would flatten out as liquid helium temperatures were ap- proached such that errors in frequency measurement would preclude the need for extremely accurate temperature measurement at these low temperatures. Therefore, thermo- couples were chosen for temperature measurement because of their reproducibility, convenience,and quick response. A copper-constantan thermocouple, which is capable of high accuracy at low temperatures (44), was chosen and con- structed from 40 A.W.G. nylon covered wire (45). Care was taken to avoid flexing or bending the wire which can create strains that produce spurious emf's when placed in a thermal gradient. The lengths of constantan wire employed were passed through a U-tube immersed in liquid nitrogen and the voltage generated across the ends measured to check for chemical inhomogeneities; with the lengths used voltages generated were less than 2 uV. .Before using thermocouples to measure temperature a careful calibration should be carried out, since individual thermocouple readings invariably deviate from those in Standard tables. .Fortunately, the error is usually a linear deviation with temperature and couples made from the same 49 spool of wire can use the same calibration chart. The couples used here were calibrated by measuring emf's at four fixed points and constructing such an error curve by comparing true and tabulated temperatures. The four fixed points used were (1) an icedwater mixture, (2) the boiling point of liquid oxygen, (3) the boiling point of liquid helium, and (4) the sublimation point of C02. Liquid oxygen was produced by passing oxygen gas into a cold trap immersed in liquid nitrogen; oxygen gas was also bubbled through constantly during measurements to avoid superheating. Crushed C02 was allowed to stand over- night to equilibrate before use. ,The boiling and sublima- tion points were corrected to actual atmospheric pressure. All measurements, including the later measurements, were made using the same Leeds and Northrup Model 2745 Potenti- ometer to eliminate errors in absolute emf measurements. -Measured emf's were compared with values in a comprehensive collection of various thermocouple tables by Powell and co- workers (46) covering the range O°K to 3009K. The resulting calibration curve for the copper-constantan couple, Figure 5, is linear within experimental error down to around 709K with appreciable deviations appearing below that temperature. .A serious problem with using thermocouples, especially copper-constantan, at low temperatures is the low thermo- electric power. The copper-constantan couple goes from 41 uV/°K at 300°K to 1.2 uV/°K at 4°K. Obviously, measure- ment errors and spurious emf's can lead to large temperature 50 .Aucwom mcaaflon somOHDHS oflsqfla um GOAHOSSn mosmHmmmmv xHOB menu CH poms mOHQSOUOEHmnu cmucmumGOUImeaou mzu How m.me Umumasnmu can Umuommwa cmm3umn ummmuao> cowuomuuoo .m Tasman 51 .m musmflm A>EV Umusmmma ooo.¢ m ooo.o COHDUTHHOO m + _ owuommmfim I wanna 0 HOT 391.100 (AT1) '3 a C In 52 errors near 49K and several precautions must be taken to minimize these. A reference junction at either liquid nitrogen or liquid helium temperatures is a necessity in order to keep measurement errors a minimum. ,For example, ‘with copper-constantan a reference junction at 779K and cold junction at 49K gives an emf of -0.724 mV contrasted with an emf of -7.327 mv for a 3000K reference junction and the same cold junction. ,For the same absolute error the second measurement must be made with a factor of ten better accuracy. .In addition to measurement errors, leading constantan wires out to room temperature from 4.2°K can generate appreciable induced thermal emf's. Therefore, the reference junction used here consisted of a reservoir containing liquid.nitrogen extending into the helium Dewar (Figure 4). Thermocouple wires ran from the sample to the reference bath through the cooling gas stream such that the warmest temperature they were exposed to was 779K. vCare was taken that the liquid nitrogen used contained no dis- solved oxygen and the boiling point was calculated from a table in the book by_White (41). To take the precautions discussed above is extremely important in obtaining accurate temperature measurements near liquid helium temperatures. ‘During these experiments the lowest measured atmospheric pressure was 732 torr, re- sulting in a 6 uV correction in the reference emf. .Neg- lecting this correction would give a 5°K error at 49K. The need for a calibration curve is even more important, 53 as is evident in the magnitude of the 20 uV correction at 109K. The following example clearly illustrates the mag- nitude of the errors from thermal gradients which were present in experiments attempting to get a liquid helium calibration point for the thermocouple. .With helium in the ADewar a thermocouple was lowered into the liquid with the 779K reference bath for the thermocouple outside the Dewar. The measured emf was -3 uv, indicating an error of only 6 uV from the tabulated valve (+ 3 uv). -However, with the thermocouple mounted completely inside the Dewar using the internal nitrogen reference bath the measured voltage was —77 av, corresponding to an 80 uV error. Apparently this large discrepancy was due to the spurious voltage induced by bringing the wire out to room temperature. 4. .EQB Oscillator Modifications and Operation All the resonances measured in this work were observed using a Dean-type superregenerative oscillator constructed by Ryan (47), who gives a detailed discussion of its circuit and operation. The superregenerative oscillator is dis- tinctive in that in the normal mode of operation the oscil- lation is periodically damped or quenched. As a result of this quenching action the oscillator output contains a number of frequencies-~the center frequency, v0, and a series of sidebands on either side at v0 i v0, v0 i 2v , o--, where the quench frequency v ranges from 20 to 40 Q kHz. .The NQR spectrum will show an absorption for each 54 of these discrete frequencies, resulting in a complex pat- tern. .In theory, the center frequency should be the most intense but in practice the oscillator is difficult to ad- just for this condition and at least one of the sidebands is usually of equal intensity. This leads to problems in frequency measurement as will be discussed below. Despite these disadvantages the superregenerative oscillator is widely used in NQR because of its greater sensitivity than the marginal oscillator commonly used in NMR circuits. The normal physical arrangement of the oscillator described by Ryan was employed with the oscillator coil im- mersed in a slush bath or in the ESR ENDOR Dewar to provide controlled temperatures. .The normal frequency range covered by the oscillator is from 10 to 60 MHz; this is achieved by changing the sample coils. —For use with the liquid helium Dewar, as shown in Figure 4, the sample coil was located in the variable-temperature bomb between 25 and 50 cm from the oscillator. The connection was made by a coaxial line con- structed from 1/2" stainless steel tubing and #24 B- & S. gauge coated copper wire. .Correct spacing of the central copper wire was maintained using small styrofoam separators. The extra length added to the sample coil due to this co- axial lead-in added a large impedance to that of the normal configuration, thus giving a much lower frequency and range of operation. For example, the longest bomb assembly resulted in a maximum obtainable frequency of only 33 MHz with the minimum one-turn sample coil. 55 In order to look at the higher frequency NaReO4 and AgReO4 resonances a lead-in tube of very short length was used giving.a maximum frequency of 48 MHz. .However, this placed the heat exchanger portion of the bomb assembly above the outer nitrogen shield of the helium Dewar re- sulting in poor temperature stability. For both configura- tions the effective tuning range of the sample coils was reduced by a factor of three to about 3 MHz. ,Operation of the oscillator at low temperature was the same as at room temperature except that mounting the oscillator flat rather than on one edge substantially reduced sensitivity to vibra- tion and markedly more stable operation was achieved. -Modu- lation was usually 39 Hz, although frequencies as high as 200 Hz were tried with no appreciable gain in signal-to- noise ratio. -Frequency measurements were made by beating the output of a Hewlett Packard Model 608A Signal Generator against the oscillator frequency and monitoring the oscillator out- put on an oscilloscope. .The signal generator output was then measured by use of an electronic counter. .The tuning capacitor position was monitored by a large dial which turned as the capacitor turned but rotated three times as fast. "During each scan the chart paper was marked at fre- quent intervals with the correSponding dial position so that the center absorption line could be Specified in rela- tion to the dial position. .Immediately after each scan the capacitor was swept manually and the oscillator frequency 56 at various dial positions measured to calibrate the chart paper. From the calibration so obtained the absorption fre- quency could be calculated directly assuming a linear varia- tion of frequency with capacitor position; this is a good assumption over the small frequency ranges between calibra- tion points. .The main source of error in measuring NQR absorption frequencies from superregenerative patterns is the possi- bility of erroneously choosing a sideband rather than the center line as the main absorption. .The oscillator used here gave a characteristic absorption pattern, with the higher frequency sidebands stronger and more resolved than those on the low frequency side. For each compound studied the room temperature lines were well characterized before studying the temperature variation of the frequencies. .The center line was located by changing the quench frequency and observing which line moved least. For all succeeding measurements the pattern was similar enough so that the same line could always be distinguished. The absolute error could therefore be i30 kHz if a sideband, rather than the center line, was chosen. However, assuming correct choice of the center line there would be a relative error of only i 1 kHz for a series of measurements on a particular com- pound. In view of the care taken in the determination of the temperature dependence of the frequencies the relative error of :1 kHz is believed to be appropriate and will be used as the quoted measurement error in this work. III . ~ RESULTS A. Far-IR and Raman Spectra 1. Factor Group Analysis As discussed in Chapter I, in order to apply the Kusida g£_al. theory to predicting the observed NQR tempera- ture dependence, information about the low frequency libra- tional modes is needed. The frequencies of these modes ob— tained from infrared and Raman spectroscopy are those at the center of the Brillioun zone and may be used in Equation (54) to calculate the NQR temperature dependence if the modes are assumed to show negligible dispersion for other .k_values in the zone. This assumption may be invalid in certain cases but unless independent information on the phonon density of states and dispersion curves is available no other approach is possible. For the periodates some pre- vious work has been done (48) in addition to a more detailed study carried out as a joint project by Doug Hatzenbuhler and myself (37,49). Only that portion of the study not previously reported by Hatzenbuhler (37) is included here. In the perrhenate case the fundamentals and some lattice frequencies have been reported (50-52), with most of the earlier results summarized by Ulbricht and Kriegemann (52). 57 58 Further study was needed to assign symmetries and observe the remaining lattice modes. To aid in the symmetry assign- ments, depolarization ratios were measured for a single crystal of NaReO4. For the ammonium salts the Nth analogs were prepared in order to separate the ammonium librations from the other lattice modes. The symmetry and number of k_= 0 modes to be expected in the crystalline IR and Raman spectra can be predicted if the space group and site symmetry are known. All the per- rhenates and periodates studied here have the Scheelite (cawo4) structure, space group C: with both the anions h . and cations occupying S4 sites in the crystal. From this data a factor group analysis (53-56) was carried out. Only the results are given here, since the treatment is identical to the periodate case given elsewhere (37). A very clear discussion of the method, including many illustrative ex- amples, has recently been published (57). The isolated ReO4- group is tetrahedral with 3(5)—6 = 9 fundamental vibrations. These may be classified under the Td group in four irreducible representations, A1 + E + 2T2 (58). These four fundamentals are split in the crystal, first by the S4 site symmetry and then by the coupling of the two inequivalent sites in the primitive unit cell, termed the factor group coupling. In a similar manner the external vibrations of the ReO4_ and the cation are also split. The resulting symmetries and number of vibrational modes are shown in Table 1. Since the ammonium 59 Table 1. Site and factor group Splittings of Scheelite- type perrhenates. Isolated . .Factor Group Spectral Molecule Site Symmetry Symmetry Activity v A A A R 1 1 .l~‘\‘~“~“§“~ g Bu inactive V2 E L B ‘ B R A IR u VS’V4 T2 E \\ Eg R Eu IR ExternalHModes < Ag R x y z anion Bu inactive E R E ::r""”—'—’—fi" g a (Tx'Ty’Tz)anion J/// . Eu IR T2 (Tx'Ty’Tz)cation \\\ B R B//g k“—“““~~-——~__.A IRa u aIncluding three accoustic modes not observable at k_= O° 60 group is also located on an S4 site the predicted split- tings for the ammonium fundamentals are analogous to the ReO4- group. I In order to make symmetry assignments in the observed polycrystalline spectra single crystal spectra of NaReO4 were run at various orientations. Since the laser light is plane polarized, a single crystal may be oriented so that only certain polarizability derivative tensor components contribute to the transition integral (59). The derived polarizability components, grouped according to symmetry for the C4h point group, are given in Table 2. Table 2. -Activity of polarizability derivative tensor com- ponents under .C4h point group symmetry. c1 0 0 c3 c4 0 Ag : 0 c1 o 89 : c4 —c3 0 o 0 c2 0 o o 2. Frequencies and Symmetry Assignments The samples were run as polycrystalline powders in addition to the NaReO4 single crystal work. The spectra 61 were obtained at room temperature and near liquid N2 tempera- ture in order to resolve the lattice modes and get data con— cerning the frequency shifts with temperature. The observed Raman spectra for the Na, K, Rb, and Ag perrhenate salts are given in Figures 6-9, while the ammonium and deuteroammonium salts will be discussed separately below. Only the lattice region is shown in most cases, since the frequencies of the fundamentals (see Table 3) agreed fairly well with the previous work of Ulbricht and Kriegsmann (52). The far-IR spectra of the powdered samples were run as Nujol mulls at room temperature and are of poorer quality overall. A representative infared spectrum, of polycrystalline NaReO4, is shown in Figure 10 for illustration. The powder spectra for the series of salts were very similar in appearance so that assignments based on the NaReO4 single crystal results, which are discussed below, were straightforward. The results are summarized in Table 3 for the fundamentals and Table 4 for the lattice modes at room temperature. Table 5 gives the Raman results obtained near 77°K, which allows several more lattice modes to be resolved. Apparently the highest frequency Bg mode was observable only in KReO4; it was unobservable in all the Scheelite-type periodates also. ,The single crystal results for NaReO4 are shown in Figure 11, with both the lattice region and the fundamentals included. As can be seen from Table 2, there are only four unique orientations among the possible combinations of 62 Figure 6. Rama spectrum of polycrystalline NaReO4 using 5145 Ar(II) laser excitation. F eeeeeee 64 Figure 7. Raman spectrum of the la tice region in polycrys- talline KReO4 using 5145 Ar(II) laser excitation. A K R004 Powder 298 °K | l 40 I 1&0 Frequency (cm-1) 66 Figure 8. Raman spectrum of the lattice re ion in polycrys- talline RbReO4 using 51453 Ar(II laser excitation. 67 RbReCk Powder 298°K 90°K I I 40 120 200 Frequency (cm'q) Figure 8. 68 Figure 9. ARaman spectrum of the lat ice re ion in polycrys- talline AgReO4 using 5145 Ar(II) laser excitation. 69 AgReO4 Powder 77 °K l 40 120 260 Figure 9 . Frequency (cm-l) 7O .mmcmu Hugo comuooe as xammu um Haze Honsz as voomwz osflaamummuomaom mo Esuuowmm DOHMHMGH .oH onsmflm 71 .oH magmas AHIEOV mosmsvmum .uou oov aoueqqrmsuexm 72 muasmmu cmEmm .ommuo>mu somecmwmmm uomuuoosHo ..Um>ummno uoz .Hm .mmm scum muasmmu consumed .Nm .mmm Eoum .mocwsvoum soHusHomm p n m w.mbm mAamummno m.msm mm mamcamvv> mm ... new «an a new mom «em Hem mam .awm Hum new mm w mimmmv > ... Hon wfim «a» can mam can «an sfim vfim ... saw an ... emu mom «mm mum cum mom can won mom ... mun ac ... on vfim n mam mam mam «ma vmm mam vmm sum mm xmfimvm> ... new new a new «mm mam ham mam mam omm mam mm m ... an” Ham ham man ”an can wen Hon van ... mmm :« ... on» onmm a ommm mum comm mam comm sum ummm mam M< mxmmmcus ... mum ocmm n onfim mam ofimm- Ham comm can owns mum m ... mam mom a use «em mmm new mom vmm mam was we axosacs> XHOS. xHOS xHOB xu03 xnoz xuozv xHOS xHO3 MH03 xHO3 MHOS. xno3. mu mafia . ,msoh> mans msoh> «use msow> mass msow> mans msoa> mane msow> mass u m lmHm ¢ IGHQ v IOHQ ImHm IUHQ UIGHQ. + oz + m2 +m< +nm +9 +mz .AerOv xovmm um mamucofimussm woom m>fluom amend was omnmumsH .m magma 73 Table 4. Lattice modes in Scheelite-type perrhenates at 2949K (cm ) Symmetry Na+ K+ Rb+ Ag+ Bg “ 77 55 45 59 139 84 65 48 76 A9 145 113 98 132 E9 181 —-— 100 -— 8g --- --- --- -- Au 98 67 57 En 140 125 98 85 Eh 163 158 104 102 Table 5. ,Raman active lattice modes in Scheelite-type perrhenates near 779K (cm'l). + + + + symmetry 1330K 77°K 1229K 779K 89 79 55 46 63 89 88 7o 56 85 Eg 132 110 77 '95 Ag N153 122 102 148 -Eg 185 147 104 185 39 a 155 a a aNot observed. 74 Figure 11. Raman spectra of single crystal NaReO4 in the four unique orientations. 75 INTENSITY RELATIVE Z(XX)Y M M ‘JJJL X(YX)Z xanv l 50 150 300 400 850 950 FREQUENCY km“) Figure 11. 76 incident and reflected polarizations and the results obtained for these orientations are shown in Figure 11. The spectra are labelled following conventional notation (60) by a series of letters which describe propagation and polariZa tion directions in terms of the crystallographic axes. The symbols inside the parentheses are, left to right, the polarization of the incident and scattered light, while the ones to the left and right of the parentheses are the propa- gation directions of the incident and scattered light re- spectively. The results of the polarization study are sum- marized in Table 6. .The agreement between theoretical and experimental depolarization ratios are consistent in all but a few cases, although the ratios are not as large as theoretically predicted. As discussed previously, the crys— tal's external habit is such that the incident light was never perpendicular to a crystal face and internal reflec- tion always occurred. In cases where experimental values are reported for theoretical ratios of 0/0, the line was extremely weak in both orientations. The single crystal study also resolved the small site splitting of v4, the asymmetric bending vibration. This band is incorrectly as- signed by Ulbricht and Kriegsmann (52) to give v2 > v4, in agreement with the order found for the infrared active modes (see Table 3). The depolarization study, however, shows unambiguously that the order of the Raman active modes is 77 Table 6. Experimental gs theoretical depolarization ratios in NaReO4. Ii = zéxng H= X(YY)Z II = xézzgy =zxzy _|_=x(xy)z i=xxzy Ratio ( I) /_L) Ratio ( I] /_]_) Ratio ( || /_]_) Symmetry Exp. Theo. ‘Exp. Theo. Exp. Theo. Fundamentals Ag (v1) 6.2 00 ~ 4.2 00 4.4 0:) A§(v2) 2.5 00 .4 g: 14.5 00 B9 7.0 00 1.7 - Era - 0/0 Bg (v3) 5.7 00 2.4 a - O/O E9 .14 0 .42 0/0 .54 0 8g (v4) .70 oo .12 a .21 0/0 89 .33 0 -- 0/0 .33 0 Lattice Modes Big' (77 cm-l) .71 oo .22 a'" .41 0/0 89 (84) ~. .10 0 ~ .37 0/0 .44 0 E9 (131) «0 0 -- 0/0 ~0 0 A9 (145) 11.3 to 7.9 CD, 5.0 a) E9 (181) ~ .02 0 -- 0/0 .38 0 _The ammonium salt will be discussed separately for several reasons. .As mentioned earlier, the vibrational spectra should contain additional lines from the ammonium librational modes and absorptions of the ammonium group fundamentals. In addition, the NQR results show that the 187Re temperature dependence in NH4RBO4 is anomalous in 78 contrast to the other Scheelite structure perrhenates studied. This anomalous behavior probably arises from the presence of the ammonium group rather than a simple cation, particularly the possibility of a phase transition from libration to free rotation near ZSQL the temperature at which the quadrupole resonance signal disappears. There- fore a more detailed study of the ammonium salt was indi- cated, which included obtaining spectra for the ND4+ salts of ammonium perrhenate and periodate. The resulting spectra for both the lattice region and the fundamentals are shown in Figures 12-17; the analogous NH4IO4 spectra are given elsewhere (37). In several of the Spectra Ar(II) fluorescence lines are also present and are marked by asterisks. Assignments for the ammonium fundamentals were made by comparison to the solution fre- quencies for ammonium ion (58). The experimentally ob- served frequencies for the ammonium ion fundamentals are given in Table 7 along with the NH4IO4 results. In addi- tion to the fundamentals additional bands were observed in several of the spectra. In NH4ReO4 (Figure 15) there 1 which can be assigned as is a weak absorption at 3064 cm- a ammonium combination band "v2 + v4"( k_= 0 sum = 3095). In ND4IO4 (Figure 17) there is a fairly strong line present at 2306 cm"1 which may be the "2v2" overtone (k_- 0 sum = 2376) in Fermi resonance with v1 or less likely the "v; + v4" combination (k_= 0 sum = 2276) in Fermi reso— nance with v3, as in the NH4ReO4 case. These overtones 79 Figure 12. Raman spectra of the lattice region in poly“ crystalline NH4ReO4 using 51453 Ar (II) laser excitation. 80 NH4 Re 04 Powder 77°K L I V | I 4'0 I 1120 200 Figure 12 Frequency (em-I) 81 Figure 13. Raman spectra of the lattice region in poly- crystalline ND4ReO4 using 51453 Ar(II) laser excitation. 82 NDAReO4 Powder I I 40 120 200 80 I I I l l I 40 120 200 Figure 13 . Frequency (cm’I) 83 Figure 14. Raman Spectra of the lat ice region in polym crystalline ND4IO4 using 48802 Ar(II) laser excitation. A N04 I04 Powder ( _ W k _ will) L 85 coaumufluxm momma AHHVH< Maven mcflms «oomemz ca mamusmempcsm Esficoeem mo muuommm smEmm .mH musmflm 86 comm .8H 683888 ATEuV >25:qu ooom _ _ oowp _ OJVP _ x.RR 3.25.. 63.412 87 .GOHumuHOxo Henna AHHVH¢ «mean msfims vommeoz ca mamucmEmUcnm EsflGOEEmoumusmU mo muuummm smfimm .®H munmflm .mH musmflm A783 xucoaqocm nun—nu nun—«N 00—3 on": 1..68.... 403.462 88 you onnm onNN coup cc: 8.. an 89 .soflumuwuxm momma AHHvud “mean can Nowme msflms voHsz CH manusmemos5m Eswcoafimoumusmo mo mnuommm cmamm .bH onsmflm 90 Dana owuu .bn musmwm A793 6:269."— Aja— 38 00:. 6636.. V0262 91 .Ame oucwuumwm.eoumm a mmom omem mmom mmvm m II amen In I: smmm m ma omoH «mam mmcm Hmem 8 mm mmmm mmvm ommm smam mmmm mcvm mmmm mmmm mem a ma m m smmm mama m mmHH I- ommm I: Hmmm I- mmmm m «a meme «mum m a noun ommm swam “sum smum mama mmmm I- mmom 4 Ha zoom 868mm mama gamma mass gamma mass gamma 508865866 vOanz vOvaz woodenz. voumvmz sowwsaom muuoiaam m , A mzvr .muumsosuuum cam moumooflnom on» Ca mamucoamussm o>wuumusmemm. vnz can +032 .8 dance 92 and combinations are labelled conventionally by the k_= 0 designations such as 2v2, although such labels are mis- leading since the phonon wave vector for these multiphonon transitions can range over the entire zone. The narrow half width of the two-phonon transitions and the small shift away from the predicted zero phonon frequency indi- cates that the internal ammonium modes show little disper- sion with the maximum in the phonon density of states away from k_= 0. These assignments agree with similar two- phonon bands in other ammonium salts (62,63,66—69). The isotopic ratios of the various fundamentals are compared to the valence force model predictions (58) in Table 8. The agreement is surprisingly good since we are not consid- ering isolated NH4+ ions, but distorted tetrahedrons in the solid. Symmetry assignments and motional character for the lattice modes are much more difficult since the isotopic ratios expected from a simple Born-von Karman model are not obeyed and correlation to the NaReO4 single crystal results is rather uncertain. Also, the various modes with the same symmetry certainly undergo mixing, resulting in modes with. mixed anion and cation character, giving an additional mechanism by which the isotopic ratios may be in error. The frequencies and assignments for the external modes are given in Tables 9 and 10 for the liquid nitrogen and room temperature results respectively. For the room temperature infrared modes no attempt is made to assign symmetries, 93 Table 8. .Isotopic ratios for NH4+ fundamentals compared to valence force model predictions. Symmetry R604 104 Theoretical v1 vD/VH 0.734 0.721 0.707 v2 vD/VH 0.719 0.720 0.707 -D D V V -V3, V4 fl 0.564 0.564 0.560 V3V4 Table 9. Raman active lattice modes in NH4+ and ND4+ per— rhenates and periodates near liquid nitrogen temperatures (cm’l). ReO4- Io4 Symmetry NH4(77°K) ND4(809K) NH‘(809K)a ND,(809K) 8g 49 50 65 64 39 68 69 90 86 .89 120 117 141 135 Ag 134 135 149 141 89 189 175 195 175 39 209 (194) 216 (195) E - - 241 213 9 A9 264 194 260 195 aSee Reference 37. T 94 able 10. Lattice modes in NH4+ and ND4+ perrhenates and periodates at 2940K (cm'l). .ReO4- a 104 Symmetry NH4 ND4 NH4' ND4 B9 48 46 64 58 :29 58 54 83 75 E9 >109 104 129 122 A .— _ ... .— 9 Ammonium .Modes 183 167 202 179 b 108 106 88 84 (A ,+ 3E ) 134 ~'140 — b (126,162) u u 170 169 190 188 aSee Reference 37. b See Reference 48. 95 since the optical branch modes are a combination of anion and cation motion. The infrared spectrum should contain two modes primarily of torsional character, one perrhenate motion and the other mainly ammonium character. If the infared results correlate with the Raman results only three modes should therefore be observable at room temperature, since the ammonium libration is not observed in the Raman spectra. The application of vibrational data to the determina- tion of transition temperatures relating to ammonium rota- tion in solids is extremely active at present (References 62—70 are a few representative examples). The results can be compared to other methods (65,70) of obtaining the transi- tion temperature, such as NMR line width studies, X-ray dif- fraction, or specific heat measurements.° The best indica- tion from IR and.Raman studies of free or restricted rota- tion is the presence or absence of the ammonium torsional vibration (v6) and combination bands such as v2 + v6 or 1).. + V6- In the ammonium ion salts studied here the presence of this librational mode at 77°K indicates that the ammonium group is not freely rotating near 779K but apparently undergoes a transition between 77°K and 3000K, where no evidence of the ammonium ion libration is present. The second librational mode, of E9 symmetry, was not observed in the perrhenate but was present in the periodate salt. This may indicate that the ammonium ion in ammonium 96 perrhenate is undergoing rotation about the Z axis at 779K, while still librating about a perpendicular axis. .Wmen warmed from 77°K the torsional mode in NH4ReO4 was observed to lose intensity rapidly and move toward lower frequencies; the spectra obtained at -500 and 25° are nearly identical with no evidence of the librational mode. Addi- tional discussion of the molecular motion in these com- pounds will be given in the next section in connection with the NQR results in NH4ReO4. B. Temperature Dependence 1. Normal Behavior: NaIO4 and MReO4 (M = Na, K, Ag) Before discussing the experimental results and method of analysis used for the compounds studied here a brief sur- vey of previous studies of the temperature dependence of NQR frequencies will be given since several of the conclu- sions to be drawn are better understood if earlier treat- ments are familiar. .Only inorganic salts showing normal temperature dependence of the frequencies will be discussed. ,The origin and mechanism producing positive temperature coefficients is covered in the following section. As shown earlier (Chapter I) a complete treatment of the NQR frequency shift with temperature also involves per- forming a separate pressure experiment to separate the volume effects. However, the required experiment is diffi- cult to perform, requiring special pressure bombs and presses, 97 and a complete treatment has been carried out in only a few cases (71a-d). In most cases approximations to both terms of the bracketed expression in Equation (41) are used. The approximation to the first term, (3L) 31(- v , dV 2 V,T has been discussed by Brown (72). This term represents the change in vibrational frequency upon volume and the effect of this change on the quadrupole frequency. .The variation in vibrational frequency with lattice size can usually be approximated by a linear function of temperature, V = V0 (1 - QT), (59) in which a is ordinarily an adjustable parameter but may be evaluated from IR-Raman data as discussed below, Va is the extrapolated frequency at T = 00K and v is the fre- quency at some temperature T. If the fixed frequency v2 in Equation (56) is replaced by the expression given above, Equation (59) provides an approximation to (av/amp,N given, in Equation (41). The remaining volume dependence is given T' which represents the effect of by the term (av/5V)V ,6, lattice expansion on the field gradient at the nucleus for which the temperature dependence is being calculated. In principle, this term could be calculated if data on the atomic positions as a function of temperature were available and,_more importantly, if an accurate procedure for calcu- lating field gradients in crystals was available. 98 Therefore, if a pressure versus frequency experiment is not carried out application of the two corrections given above should result in a better fit to experiment than using the Bayer term alone. In practice, calculation of the second term cannot be carried out due to a lack of the necessary X-ray data but some calculations have been made; Hewitt (73) performed a calculation for KNb03 based on thermal expansion data for BaTiO3, while Gutowsky and Wil- liams (74) calculated the volume dependence in NaClOa using only the change of unit cell parameters with temperature. The results in both cases were uncertain, due more to the approximate nature of the structural data available than to the approximations used in calculating the field gradient. Because of the problems in attempting to calculate the volume dependence of the field gradient properly, the usual procedure is to neglect this term and only use the correc- tion based on Equation (59). This involves varying both the vibrational frequency, v0, and its temperature varia- tion, a, until a best fit to experiment is obtained. In some cases the moment of inertia is included as an adjust- able parameter also. Use of these parameters along with v0, the rigid lattice NQR frequency, leads to an equation containing four adjustable parameters to be fit to a simple curve. This will be designated here as the adjusted Bayer theory and the resulting fit is usually quite good. It has, however, been applied in cases where some volume dependence of the field gradient is expected and therefore some 99 discrepancy between the adjusted Bayer theory and experi- ment is expected. Some compounds where this type of treat- ment has been carried out include LiNb03 (75) and a large number of chlorate and bromate salts (76-79). Since the periodates and perrhenates form ionic-type lattices the adjusted Bayer theory described above should not give close agreement to experiment over the whole tem- perature range studied. In order to reduce the number of the adustable parameters an infrared and Raman study was carried out to measure experimentally the vibrational frequencies and their temperature variations and the moments of inertia were calculated from the X-ray data, leaving only the rigid lattice quadrupole frequency v0 as an adjust— able parameter. The infrared and Raman results are sum- marized in the previous section except for the temperature variations of the lattice modes which are given in Table 11. Lattice translations do not average the field gradient tensor as the librational modes do and should not be in- cluded in the analysis. .Recalling the discussion of the Raman results for the ammonium ion presented in the last section, the four lowest lattice modes are primarily the Re04- motions. The A9 mode is assigned as a libration and the B9 as a translation on the basis of symmetry require- ments. The two Eg modes, intermediate in frequency between the Ag and Bg modes,are probably of mixed librational-translational character due to symmetry mixing of the states. Despite the mixed character the lower Eg 100 .ABo I Hvo> I > mcHEDmmdm a II 88H Amm.HV “mm II mmm II II II 89H Amv.Hv mmH m m 18m.8vm:mm.fiv mam Amm.mv mmH Amm.mv oHH Amm.mv mmH Amm.mv mmH 4 a Amm.mv mum Ams.mv mmH II mm Amm.mv mm AmH.Hv oaH Ammv.ov mmH m m Amm.mv mm Amm.mv mHH Amv.vv mm AmH.sv mm Amm.mv as AmH.mv om m m AHm.Hv as Amo.Hv mm Amm.mv mm Amm.mv mm Aomm.ov em AHv.HV om m AaIxocvoHRa :rxavvoaxa AHIxavvomxa meavvomxa AaImavaona AHImavvona AHIEOVo> AHIEOvc> AHIEOVo> AHIEOvo> AHIEOVo> AHIEOVo> unumafimm eon vonz 800mm< vommhm 600mm. a64.662 M .mmpoa mueuuma oumcmznumm can mumoofluom mo muocmcsmmmp ousumquEme .HH OHQMB 101 mode is more likely to be chiefly translational with the re- maining mode having more librational character. Because of their higher frequencies relative to the lattice modes, the internal vibrations are assumed to have a negligible effect on the averaging of the field gradient and will not be included in the analysis. Before discussing the appli- cation of these results to the calculation of the tempera- ture dependence of the NQR frequencies to be expected in these compounds, the experimental results will be presented. The experimental 1271 quadrupole frequencies in NaIO4 and 187Re quadrupole frequencies in NaReO4, KReO4, and AgReO4 are listed in Tables 12-15, at the experimental temperatures from 129K to 3000K. In both—NaIO4 and in the perrhenates the asymmetry parameter remained zero over the temperature range and only the (15/2 <-9 13/2) or the (13/2 <—> 11/2) transitions were followed, but not both. For the rhenium resonances the isotopic ratio 187Re/185Re remained constant so only the more abundant 187Re isotope resonance was followed. For NaReO4 and AgReO4 the higher frequency required to observe the resonance made necessary the use of a Dewar arrangement such that stable low temper- atures were very difficult to achieve and for these com- pounds fewer measurements are reported. It should be pointed out, however, that many more experimental measure- ments were made which could not be included because the rapidly changing temperature at the sample resulted in unacceptable errors in temperature measurement. The 102 Table 12. .Experimental fre uencies and temperatures for the (13/2 <—¢ 11 2) 187Re pure quadrupole reso- nance transition in NaReO4. Temperature (0K) Frequency (MHz) 30.4 46.3869 56.1 46.1382 59.3 46.0957 71.1 45.8242 93.4 45.6453 96.5 45.6184 203.1 44.0122 205.3 43.9873 206.7 43.9200 208.3 43.8918 208.6 43.9375 209.1 43.8877 209.5 43.9039 216.1 43.7088 226.8 43.7257 296.0 42.6060 103 Table 13. Experimental frequencies and temperatures for the 18 Re (13/2 <-9i1/2) pure quadrupole resonance transition in KReO4. Temperature Frequency Temperature Frequency (°K) (mm) (M (MHz) 16 27.9025 93.4 27.7920 18 27.8978 125.4 27.6247 23 27.9010 162.2 27.5056 30 27.8868 179.4 27.4554 36.6 27.8831 193.6 27.3650 38.6 27.8823 199.0 27.3465 42.8 27.8834 204.2 27.3111 46.3 27.8809 205.2 27.2777 49.0 27.8742 209.2 27.2508 52.0 27.8736 240.3 27.0104 54.3 27.8567 247.4 27.0870 59.0 27.8642 256.7 27.0269 63.5 27.8539 298.7 26.7750 66.4 27.8445 Points obtained with ENDOR Dewar.a 121 27.8233 200 27.3897 141 27.7091 212 27.3104 150 27.5971 228 27.2768 162 27.5425 238 27.2005 172 27.5060 248 27.1274 183 27.4578 263 26.9926 191 27.3897 268 27.0039 273 26.9500 298 26.9095 aObtained with an uncalibrated thermocouple. T Table 14. 104 Experimental frequencies and temperatures for the 187Re (13/2 <—> :1/2) pure quadrupole resonance transition in AgReO4. Temperature (0K) Frequency (MHz) 46.6» 63.6 76.2 86.0 95.0 96.5 98.5 120.5 131.5 252.4 275.4 283.5 39.3973 39.3851 39.3017 39.2579 39.0584 39.0710 39.0140 38.8742 38.7761 37.8706 37.7661 37.6475 Table 15. .Experimental frequencies and temperatures for the 1271 (15/2 <—> 13/2) pure quadrupole resonance transition in NaIO4. Temperature Frequency Temperature Frequency (°K) (MHz) (°I<) (MHz) 18 13.2850 200.7 13.0033 29 13.2852 201.2 13.0240 41.3 13.2790 194.4 13.0538 46.1 13.2804 212.3 12.9780 49.0 13.2796 223.0 12.9485 59.2 13.2671 233.1 12.9360 75.3 13.2598 250.4 12.8599 120.2 13.2288 251.5 12.8570 141.7 13.1579 273.0 12.8062 158.9 13.1125 273.2 12.7985 168.3 13.0975 300.2 12.7255 180.3 13.0586 105 temperature variation in the case of KReO4 was very thor- oughly studied and revealed anomalies, such as phase transi- tions to less symmetric phases or changes in slope of the temperature curve, so experimental points available for other simple perrhenate salts are not serious. The experi- mental errors, previously discussed in Chapter II, are 110K below.about 300K, and 10.10K above 800K; the frequency measurements have a relative accuracy of 15 kHz. A repre— sentative NQR signal is shown in Figure 18, with the center line indicated by an arrow. Usually the shape of the resonance line remained relatively constant over the range of temperatures so that the selection of center line was consistent. No problems were encountered with satura- tion of the signals at helium temperatures, although the resonance line was not observable in NaReO4 for about an hour after initial cool-down. This loss of signal was probably due to thermal strains induced by the rapid tem- perature changes. The room temperature resonance in KReO4 returned to full intensity quite slowly if the sample was abruptly warmed to room temperature; usually three to five days were required and in one instance, no signal was ob— servable the first two days after warming. If the samples were left in the helium Dewar to warm up gradully no loss of signal was encountered. Several different methods were used to calculate the temperature shifts to be compared with experiment. ~For applying the adjusted Bayer theory a computer program (TFIT) 106 .x on an voamx mcaaamummuumaom cw cmwwwwswmw e . I o Eds onMGOmoH waomsupmsv chum exams AN\HH Alv N\M+v osu m .mH unamwm 107 ~12 moonNN .mH wusmfim uri nhnodn 108 was written to calculate the temperature variation for any input values of lattice frequency, vibrational frequency temperature coefficient, and moment of inertia for up to four different vibrational modes. In the calculations in this thesis two different curves were calculated. The first assumed that the highest frequency ReO.- or 104- lattice mode of E9 symmetry is 100% librational (no sym- metry mixing) and this mode alone is used. Since there is an IR active Eu component the mode degeneracy used was four instead of two as implied by the E9 symmetry. The A9 mode is by symmetry the Z axis librational mode which has no effect on the EFG averaging and therefore is omitted. The second calculation assumes some degree of symmetry mixing between the E9 latice modes of the ReO4- and 104- ion and a mixed contribution from the two modes to the temperature dependence is calculated. The experimental and calculated curves for NaIO4 and sodium, potassium, and. silver perrhenate are shown in Figures 19-22. In each curve the solid line is drawn through the experimental points; the (--—-) curve is calculated assuming pure libraf tional character for the highest frequency 29 mode: the (—~—~) curve is calculated assuming partial librational character in both 29 modes. The interpretation of these results will be discussed after outlining the method of treating the data in which all the parameters of the ad- justed Bayer theory are allowed to vary. 109 .0608 mm mucosvunw smegma: up» How Houumumnu Hmcowumunwa musm museums AIIIIV 0>Hso ecu can .Hmuomumzo Hmcowamunfia RmN owoa aocmsvmum unused map paw Huuomumno HmGOflumuan Rmb once m mucmsvmum umunmflz mxu nufi3.mmooe H «OH. O3» 03» mo mcwxwfi muumaeam museumm Alzlmlo 0>Hsu may .Hmudmewummxm ma 0>Hsu pwaom use .vOHsz cm modmcommu mHomsHpmsU Hsmm may mo musmpsomoc musumuumemu ouumasuHmu new Heusoeflummxm .mH ousmfim 110 .mH wusmflm Cr; 0.5.301th ona onp Am\ma .Iv «\mav Hana vOHmz on Qu— «.m— (sz) Xausnbsq 111 .0008 mm musmswmnm ummann on» How Hmuumumnu HMCOHumunHH ousm mmfidmmm AIIIIV m>nsu mnu use .Hmuumumzu HmsoHumHQHH finm m©OE mocmswmum umwBOH may psmm Hmuomumnu HMCOHMMHQHH Rmb 060E m wocmsvmum amoann wnu suH3_mmer wood 03» man no mstHE muuofifihm moadmmm AIII;IQ o>Hso mnu .HmucoEHHmmxm wH 0>H50 UHHOm use .vommmz GH wonmcommu uHomsnvmsv umbmu may no oucmccmmoc mHDumHmmEou pwumHsuHmo was HmusuEHmexm .om mHamHm 112 .ON musmHm Axov scaocano... on“ - ‘ ' l ' l on— on 4 J 0.9. Am)“ Iv «\mi 3:3 vommmz 1 06V " " ' (sz) Aousnbsu 113 .0608 mm mucusvmum ummsmHs on» How Hmuomumzo HmcoHumunHH wusm moEdmmm AIIIIV 0>Hsu map was .Hmuomumnu HmcoHumHnHH finN woos wocosvmum ummBOH onu psmm Houomumno HmsoHumunHH $mb woos w musmsqunm umosmHn onu nuH3 mecca voom 03» ecu mo mstHE muumeemm moEsmmm AIIIIIQ 0>Hsu may .HmucmEHHmmxw mH 0>HSU GHHom was .eoomx GH musmsomou mHomsuomsw mmsmm on» mo mocoocwmww wusumummamu UoHMHSOHmu cam HmusmEHHmmxm .HN musmHm 114 ’ CC own HI AN\HH Alv «\MvamsmH comma .HN oHDmHh mcaumcanoh on— H CNN odu sz Aouanbaig 115 .0608 mm m000sv0um u00£mHn 05¢ How H0uomumsu HmsoHumHnHH 0000 0080000 AIIIIV 0>Hsu 0:0 6:0 .H0uumumno HmsoHuwHQHH RmN 0608 moc0sc0um 00030H 0gp 6cm H0uomnmao HmcoHumHQHH Run 0608 m mus0sv0nm u00£mH£ 0n» 50H? 00608 a 000m 030 0:0 60 mstHE muu088m0 0080000 Alzlle 0>Hsu 0:0 .Hmus08HHMQX0 0H 0>Hsu 6HH00 0:9 .vO0mmm 0H 00:0:000n 0Homsu6msw 0msmH 0:» no 0000680006 0usumn0m80u 60u0H50H00 6cm Hmuc08HH0mxm .NN 0H0mHm 116 ona AN\HH 41v m\m9v mmsma voommc .mm musmHm CC 0.30.30th 09 d I41 on Odo (ZHW) Aausnbsid 0.0" 117 As pointed out earlier one purpose of this study is to show that conventional treatments using the lattice fre- quencies and temperature variations as adjustable parameters can lead to a misleading agreement between theory and experi- ment. (In order to illustrate this point a conventional treatment of the compounds studied here was carried out to compare the parameterized librational frequencies with those obtained directly by experiment. For fitting the data a general non-linear least squares program (80), KINET, was used. Before fitting the experimental data an arti- ficial curve was constructed with simulated moments of inertia, vibrational frequencies, and temperature variations of the vibrational modes and the ability of a least-squares fit to reproduce the known simulated parameters was deter- mined. .The results of the simulation illustrate several prob- lems involved in fitting a curve with a function of the form used in the interpretation of experimental NQR fre- quency versus temperature curves (Equation 56). If both the moment of inertia AZ and the vibrational frequency wt in one of the summation terms in Equation (56) are allowed to vary, difficulties are encountered because they appear as a product and so are strongly correlated (105). The same problem with correlation is also present when the substitution of Equation (59) for the vibrational fre- quencies is made and both v0 and a: are allowed to vary. Fitting the constructed frequency versus temperature curve 118 using either of the above strongly correlated pairs gives excellent results; however, the final values of the param- eters in most instances were physically meaningless and far away from the actual values used in constructing the problem. In most cases one member of the parameter pro- duct would go to very large values while the other would go to extremely small values, the resulting product re- maining nearly constant. The second simulation tested the ability of the com- puter program to fit a constructed experimental curve with two separate terms in the adjusted Bayer summation (Equa- tions (56) and (57)). In this case the lower vibrational frequency was reproduced fairly well, but the second fre- quency was reproduced poorly, especially in cases where the second frequency was 30 to 50 cm”1 higher. ,In this case only one parameter in each term of the sum was allowed to vary in order to avoid correlation problems as discussed above. .In summary,therefore, attempting to fit experimental NQR frequency versus temperature plots using the adjusted Bayer theory must be carried out with care if the final parameter values are to correspond to physical reality. In the Scheelite-type perrhenates and periodates studied here agreement between experiment and theory over the entire range of temperatures studied can be obtained by such a procedure allowing only one frequency to vary. Be- cause of problems with strongly correlated parameters the moment of inertia for the lattice librational mode, which 119 averages the field gradient, was held constant: only the static lattice NQR frequency v0 and the libration fre- quency v2 were adjustable. The librational mode tempera- ture coefficient a usually used was the eXperimentally determined value. Attempts to improve the fit by varying 0 gave final values for v0 and a which were not phys— 2 ically meaningful. The values of Va and v which gave 2 the best fit of Equation (56) to the experimental data are listed in Table 16. The experimentally determined quantities vi , a, and In are also listed, along with the sum of the square of the residual frequency between calcu- lated and experimental data points. .In general this dif- ference was less than 25 kHz. Referring again to the results displayed in Figures 19- 22 several significant results obtained with the use of only one adjustable parameter, v0, in the adjusted Bayer theory are apparent. In the case of NaIO4, NaReO4, and —KReO4 the calculated curves begin to show deviations from the experimental curve below 1009K. This is precisely the effect expected from neglect of changes in the field gradi- ent directly due to lattice expansion, since one expects the lattice to begin showing significant expansion in the range 10-100°K. The extremely bad fit in the case of .NaReO4 implies that volume effects on the field gradient are of greater importance in this salt. In contrast, the agreement in the case of AgReO4 seems to indicate that neglect of the volume term may not be as serious here. 120 The extent to which the adjusted Bayer theory provides agreement with experiment might be expected to depend on the nature of the bonding in the crystal. However, the limited number of salts studied here has not provided any obvious correlation. Table 16. Final parameters obtained in applying the ad- justed Bayer theory. Adjustable Experimental Parameters 0 Constants Residual” compound (882) (szl) i::9:) IZ 2(Xi(exp)—xi(calc))2 Naneo4 47.136 63 0.492 18.55 3.28 x 10" «8604 28.150 85 1.14 22.80 5.48 x 10” AgReo4 39.895 84 2.00 18.72 4.33 x 10" Naro4 13.404 86 1.76 23.06 2.75 x 10‘2 An important deficiency in using the spectroscopically determined librational frequencies in the above method for fitting the experimental NOR frequency versus temperature curves is the neglect of phonon dispersion. The mechanism for averaging the field gradient involves phonons with wave vector ranging over the entire Brillioun zone and not restricted to the k_= 0, spectroscopically-active phonons. .Since the maximum in the phonon density of states usually doesn't occur at k_= 0 the choice of librational frequency to use in fitting the experimental curve is difficult. 121 However, in the compounds presented here the differences between the experimental vibrational frequencies and the average vibrational frequencies listed in Table 16 are too large to be accounted for solely by dispersion effects. In these compounds with ionic-type lattices neglecting field gradient changes due to lattice expansion or contrac- tion is not valid. 2. Anomalous Behavior: NH4ReO4 In contrast to the perrhenates just discussed, am- monium perrhenate exhibits a large positive temperature coefficient of the NQR frequencies indicating that the temperature dependent mechanism modulating the field gradi- ent is radically different in this case. Before discussing what appears to be the origin and nature of this mechanism the various proposals previously advanced to explain anomalous quadrupole temperature shifts will be discussed and their applicability to NH4ReO4 examined. A short but not comprehensive review of positive temperature coeffici- ents in NOR is also given. Positive temperature coefficients of nuclear quadrupole resonance signals have been observed in a relatively small number of compounds, mainly transition metal halides such as Tic14 and wc16 (81), TiBr4(82h Thc14 (83), and hexa- halometallate-type compounds which have been studied (28, 29,84). In all these compounds, measurements have been made only of the halogen NQR frequencies and not those of the 122 metal. Anomalous behavior in compounds not containing transition metals is much rarer, having been observed only in NH4I3 (85) to my knowledge. In the compounds containing transition metal complexes the transition metal invariably has d orbital vacancies, and the proposed mechanism causing the anomalous behavior is a destruction of the metal—halogen dw-pw bonds from the increased amplitude of the metal complex internal vibrations withincreasing tem— perature as discussed in Chapter I. Haas and Marram (26) attribute w-bond destruction through rehybridization to the bending fundamentals while Brown and Kent (28) attribute this to the bond stretching vibrations, showing that only the latter can be effective. Regardless of the type of fundamental causing the rehybridizations the mechanism predicts that all the isostructural compounds differing only in the cation should exhibit the same type of behavior. Since this is not the case for the perrhenate series this mechanism does not account for the present observations. A second mechanism, competing with the w-bonding mechanism just described, which also predicts positive NQR temperature coefficients, has been proposed by O'Leary (86). If the frequency of the torsional mode (wz) which, according to the conventional theories (Equation 56), averages the field gradient increases in frequency with increasing tem- perature at a fast enough rate, then the magnitude of the exponential term in Equation (56) will increase with in- creasing temperatures giving rise to a positive value for 123 the Bayer term, (g%) . Torsional modes with this type of P,N behavior are called soft librational modes and are usually observed on the high-temperature side of phase transitions. In cases where symmetry restrictions prohibit direct obser- vation of such a mode by Raman or infrared spectroscopy the NQR results can give previously unobtainable vibrational data. In the case of NH4ReO4 the librational mode is vibra- tionally active and since the infrared and Raman results show that the torsional mode in NH4ReO4 behaves normally, this mechanism is also not operable here. The magnitude of the temperature coefficients normally observed in cases where these two mechanisms are applicable preclude applying them to the NH4ReO4 data, but detailed discussion of this point will be deferred until after the experimental results are presented. The experimental rhenium quadrupole resonance frequen- cies measured at various temperatures are given in Table 17, with the results displayed graphically in Figure 23. The 185Re resonance is not shown since the isotopic ratio 185Re/187Re does not vary over the temperature range. It has been shown previously (11) that n - 0 over the range 257°K to 300°K and therefore only the (15/2 <—> 13/2) transition was followed. The disappearance of the NQR sig- nal at 245°K is not abrupt; the resonance gradually broadens and loses intensity as the temperature is de- creased. If an instrument of greater sensitivity were used the resonance definitely could be followed to lower tempera- tures. 124 Table 17. Experimental frequencies observed at various temperatures for the (15/2 <—>i3/2) transition of the 187Re pure quadrupole resonance spectrum. Temperature Frequency Temperature Frequency (0K) (882) (OK) (MHz) 245.4 32.0953 324.7 33.6283 246.4 32.1098 334.1 33.7560 248.4 32.1941 334.5 33.8153 250.9 32.2711 338.4 33.8274 256.2 32.4290 344.5 33.8905 262.0 32.5339 352.5 33.9648 265.7 32.6470 361.1 33.9937 273.9 32.8750 365.9 34.0160 274.6 32.8909 367.1 34.0428 278.7 32.9447 367.6 34.0294 283.3 33.0250 372.6 34.0574 294.2 33.2393 375.8 34.0699 296.8 33.2972 377.8 34.0758 308.2 33.4533 383.0 34.0837 313.6 33.5280 125 Figure 23. Experimental temperature dependence of the (:5/2 <—> 13/2) 187Re transition in NH4ReO4. 126 - 3315- AAHz f Frequency 3215 h l 250 , ) 350 Figure 23. Temperature ( K 127 In attempting to explain the experimentally observed behavior in NH4ReO4, the effect of the substitution of the ammonium ion for the simple cations should be considered, since in other respects the perrhenates with the Scheelite structure are quite similar. The strong temperature de- pendence of the field gradient at the rhenium nucleus sug— gests that changes in the ammonium ion motion may be responsible. Before going into detail regarding the mechanism of this effect, a somewhat analogous case will be presented. As previously mentioned the temperature be- havior of 1271 in NH4I3 (85) is anomalous, although the rubidium and cesium salts having the same crystal structure show completely normal behavior. .Kubo g£_31, (85) postu- late that this behavior may be caused by the onset of ammonium group rotation after which the rotating ammonium group simulates a simple alkali metal cation. No detailed model of the modulation of the NQR frequency by the change in ammonium motion was given. The NH413 results, particu- larly the magnitude of the temperature coefficient, will be compared to ammonium perrhenate below. The cfystal structure of NH4ReO4 is unfortunately not very reliable with regard to the oxygen positions since it was carried out quite a long time ago (87). However, recent redeterminations of the crystal structures of KReO4 (88) and.NaIO4 (89) have been carried out and some of the conclusions from these results are applicable to NH4ReO4. The question of main interest is the orientation of the 128 NH4 tetrahedron in the lattice since the hydrogen positions are not available from the crystal structure. The recent structure refinement of NaIO4 by Cruickshank (89) shows that the sodium atom has two tetrahedra of oxygen atoms enclosing it, four at a distance of 2.543 and the other four at 2.603. The same type of arrangement is present in KReO4, the corresponding Re-O bond distances being 2.773 and 2.883. The earlier determination of the oxygen posi- tions in Nan4 by Hazelwood (90) in 1938 gave Na-O separa- tions of 2.5733 and 2.5773 and indicated a much less dis- torted IO4_ tetrahedron than the work quoted above. The corresponding Re-O distances in NH4ReO4, using oxygen parameters from the old structure determination (87), are 2.843 and 2.873 indicating that the two sets of tetrahed- rally arranged oxygens are not equivalent. Since a struc- ture refinement could give as large a change as for NaIO4, the inequivalence and relative ordering of the two tetra- hedra may be open to question. If the X-ray results are taken to be qualitatively correct, however, a definite pre- ferred choice for the orientation of the ammonium ion 1 hydrogens can be made. The four H-atoms of the NH4+ group are naturally arranged in the correct shape to line up pointing toward the four closest tetrahedrally-arranged oxygen atoms at a distance of 2.843. The ordered state existing at very low temperatures, and possibly at 779K, would have all the ammonium ions in this favored configuration, which will be 129 denoted as Orientation I. At somewhat higher temperatures a second configuration, corresponding to alignment of the four hydrogen atoms toward the second group of oxygen atoms (Orientation II), should become energetically allowable and a transition of the order-disorder type should occur. Transitions between the two configurations would be achieved by some of the ammonium ions possessing enough energy to be in the continuum above the bound energy levels of the librating group and, therefore, undergo free rotation for a short time before losing energy and again becoming a librating group in either Orientation I or 11. Upon in- creasing the temperature a large number of the ammonium ions would tend to spend more time in the freely rotating state and the transition to a phase in which the ammonium ions are undergoing free rotation should be sharp. A model similar to this has been used to explain the phase transitions, NMR, and.Raman and infrared results in the ammonium halides (68,69,91). If the order-disorder transition occurred between 77°K and room temperature it should have been obserVable in the Raman results, but no changes in selection rules or large shifts of the ReO4- fundamentals from the other salts with the Scheelite structure were observed. Therefore, the be- havior of the ammonium fundamentals, and disappearance of the HH4 ion librational mode upon warming, tend to indicate a first-order transition (92) from bound to free rotation. This conclusion is consistent with the NQR results 130 independent of the exact mechanism which modifies the field gradient at the rhenium nucleus. In NH4ReO4 there are apparently four closest oxygen atoms arranged tetrahedrally around the ammonium ion and a second group of four oxygens slightly further away. In each of the ReO4- tetrahedra two of the oxygens are co- ordinated with hydrogens in Orientation I while the re- maining two are coordinated in Orientation II. Regardless of the mechanism, when the ammonium ion is aligned with the hydrogens in Orientation I the resulting field gradient at the rhenium is different than when the ammonium ion is in Orientation II. Therefore, if a field gradient exists in the lowest—temperature, completely ordered state it prob- ably has a large asymmetry parameter. Above the order- disorder transition the rapidly (on the NQR time scale) reorienting ammonium ions, from Orientation I to a mixed configuration of Orientations I and II, will cause a rapidly fluctuating field gradient at the rhenium nucleus, effectively making the resonance unobservable. .If a second transition to freely rotating ammonium ions occurs as the temperature is increased the rate of rotation will increase until, on the NQR time scale, an increasing number of am- monium ions begin to resemble alkali metal ions to the rhenium nuclei. The rhenium nuclei in the ReO4- tetra- hedra no longer having ammonium ions aligned toward their oxygen atoms produce an observable NQR signal which then increases in intensity as a greater fraction of the rotating ammonium ions resemble alkali metal cations. 131 The exact mechanism causing the observed positive tem- perature shift is probably a mixture of contributions from the change in covalent bonding of the Re—O bonds as the weak O--°H hydrogen bond breaks, and an electrostatic effect from the effective positive charge of the H+ atom being replaced by a NH4+ ion with the positive charge about 13 further away. This effective charge diSplacement would tend to decrease the positive contribution to the total field gradient from the ammonium ions in the lattice. Since the total field gradient is apparently negative (11), the above mechanism leads to an increasing field gradient with increasing temperature, as observed, and would have a much stronger effect on the field gradient than the mech- anisms discussed previously. This can be seen by a compar- ison of their NQR temperature coefficients. Table 18 lists a number of the larger positive temperature coefficients observed in other compounds for comparison with results for NH4ReO4. Note that the coefficients for two compounds, »NH4ReO4 and NH413, are an order of magnitude larger than ,those for the others. Because of the similarity between _ the temperature coefficients in NH4ReO4 and NH4I3 it is doubtful that the mechanism involved in modulating the field gradient in ammonium triiodide is direct hydrogen bonding, since no such possibility exists for ammonium perrhenate. However, if the N-H°'-I hydrogen bond is weak the effect on the field gradient might be about the same as in NH4ReO4. 132 Table 18. Positive temperature coefficients of NQR fre- quencies observed in NH4ReO4 and several other compounds. Compound %(g%0 x 103 0K.1 Reference NH4ReO4 1.0 This work NH4I3 1.29 85 KzReCla 0.0094 93 KzReBrG 0.022 93 (NH4)2ReBr6 0.0047 93 (NH4)2ReI6 0.022 93 wcl6 0.195 81 war. ‘ 0 .180 28 CsNbCls 0.130 28 0.107 CsWC16 0.080 28 .ngwcla 0.030 28 The NQR and Raman results are consistent with a model of NH4ReO4 in which the crystal is an ordered arrangement of librating ammonium ions at low temperatures and under- goes an order—disorder transition, apparently below 77°K, t0‘a disordered state consisting of librating ammonium ions in dynamic equilibrium between two different orientations. At some higher temperature (below 2450K) the ammonium ions begin rotating freely and, as the frequency of rotation for an appreciable fraction of the total number becomes greater 133 than ~106 Hz, an NQR signal becomes observable, although the frequency shifts strongly with temperature as the rate of rotation of the ammonium ion increases. Near 380°K the rate has increased to the point that the ammonium ions begin to resemble simple cations and the temperature vari- ation of the field gradient becomes normal. This model predicts an NQR signal with an appreciable asymmetry param- eter should be observable at liquid helium temperature if the librational motion of the ammonium ion does not per- turb the field gradient at the rhenium nucleus too strongly. C. Electric Field Gradient Calculations 1. Method of Calculation The attempts to calculate field gradients from X-ray diffraction data in this study were based on two relation- ships. The field gradient is related to the electron density (Equation (24)), which in turn can be expressed as an infinite Fourier series with the expansion coefficients related to the intensity of X-ray diffraction maxima (94, 97,98), p(x,y,z) = %- F(hkl) exp[-2wi(hx + ky + lz)), (60) CD 2 h,k,1 where V is the unit cell volume, .F(hkl) is the struc- ture factor, and p(x,y,z) is the electron density at some point in the crystal. The intensity of an X-ray diffraction maximum of order hkl is proportional to ‘|F(hkl)|2 so, in 134 principle, electron distributions in crystals may be cal- culated by obtaining |F(hkl)| from experiment. In practice, however, there are numerous problems associated with carrying out such a calculation, many of which are dis- cussed elsewhere (94,97,98). The problems directly re- lated to the field gradient calculation and its accuracy will be covered in the next section. The choice of a suitable crystal structure for use in calculating the field gradient was limited by two require- ments -- the structure determination had to be recent enough so that confidence could be placed in the observed structure factors and the compound had to be one for which experi- mental quadrupole resonance data were available. Addition- ally, it was hoped that the crystal structure would be a simple ionic-type lattice with few atoms per unit cell. Unfortunately, these requirements severely limit the number of choices, since structure determinations for most simple inorganic salts were carried out too long ago to be useful. The only compound for which field gradient calculations were actually carried out in this investigation was B-ICl; at room temperature there are two modifications of iodine monochloride, B-ICl is the metastable one. Electron density maps were also calculated for several other com- pounds but because of the unsatisfactory nature of the maps for field gradient calculations further work on these com- pounds was discontinued for reasons outlined below. 135 B-ICl is monoclinic, crystallizing in the space group le/c with eight molecules per crystallographic unit cell (95). The NQR results (106) show two inequivalent iodine atoms and two inequivalent chlorine atoms per unit cell, in agreement with the X-ray results. In the following discus- sion the two types of atoms are denoted as I(a), I(b), Cl(a), and Cl(b). The first step in the calculation of the field gradient tensors for B-ICl was to calculate the elec- tron density map using the observed structure factors from the crystal structure determination (95). The program used in evaluation of the Fourier sum was a general type (96) suitable for many different space groups. The program evaluates the sum in Equation (60) at a grid of points in the unit cell, then provides the electron density values in a matrix of numbers corresponding to slices in the unit cell. Options in the program allow evaluation of p(x,y,z) at either 30, 60, or 120 points along each axis leading to (30)3, (60)3, or (120)3 points in the unit cell where the density would be evaluated. The fineness of the grid chosen was dictated both by the amount of computer time available and, more importantly, by the experimental resolu- tion; dividing the cell into 120 parts along a particular axid usually exceeded the theoretical resolution for a par- ticular X-ray source and unit cell size. Because of sym- metry considerations, the grid points in only one-quarter of the unit cell needed to be evaluated using Equation (60), the remaining points being related by translations. 136 By drawing contour lines, corresponding to various fixed values of electron density, the center or maximum of electron density could be located for each atom. At vari~ ous distances away from each atomic position the electron density values dropped to the background level and the electron density of the atom was assumed to truncate at this point. Then, assuming that the nucleus was located at the maximum of electron density the field gradient from the surrounding atoms was calculated. The calculation was carried out using a modified pointwcharge method with each non-zero grid point representing a fractional electronic charge. The values of the electron density at the grid points around each atomic position were scaled such that the total charge on the iodine and chlorine atoms agreed with the degree of ionic character obtained from a Townes— Dailey treatment of the experimental quadrupole coupling constants of iodine and chlorine in ICl (19). Then, a point—charge-type field-gradient summation using these grid points was carried out using a modified version (LAT2U). of a program developed by Ryan (47) from the method of DeWette and Schacher (15,16). Input for this program was supplied by another computer program that calculated the additional grid points necessary to complete the unit cell, adjusted the fractional charges to give the correct ionic charges on the atoms, and punched the input deck for the lattice summation program. 137 The first problem encountered was the choice of atomic positions to use in the calculation. In B-ICl, as in most structure determinations, the atomic positions, based on the maxima in electron density (denoted in the following as the map origin), are not usually the final atomic positions; these are instead calculated using a least-squares procedure to minimize the difference between observed and calculated structure factors (referred to as the least-squares origin). As detailed below, the field gradient in B-ICl was extremely sensitive to the choice of origin. 2. Results and Problems Encountered The calculation for B-ICl involved a point-charge sum— mation with over 1000 points of fractional charge, resulting in an extremely lengthy calculation on the CDC 6500 computer (about one hour per run). The magnitude of the field gradi- ent computed at I(a) was very large due to an artifact in the calculation. .When the least-squares origin was employed, the origin was located asymmetrically between the grid points resulting in abnormally large contributions to the field gradient from the adjacent grid points. This problem was compensated for by removing the twelve grid points directly adjacent to the origin. When the map origin was used slight variations in the coordinates of the origin pro- duced large changes in the field gradients. The field gradients calculated using the map origin and using the least-squares origin were found to differ. Numerical 138 results for various choices of origin and various calcu- lation conditions are given in Table 19. One of the most serious deficiencies in the ICl calcu- lation is the unreliability in the field gradient introduced by not having the grid of points at which the electron density is calculated spaced such that the atomic position of 1(a) is coincident with a grid point. .However, the program used for calculating the density map in this work employs standard formulas (99) for p(x,y,z) for which the allowable values of x,y,z are n/30, n/60, or n/120 of a unit cell side. Shifting the unit cell origin in order to make a grid point coincident with a particular atomic position invalidates the standard formulas for elec- tron density, since the formula for a specific space group has been simplified by using symmetry relations of that particular space group. There is a possibility that trans- forming the unit cell to the triclinic system would allow any choice of origin but this was not attempted. Aside from these computational difficulties, there are some serious theoretical objections to expecting accurate results from this type of calculation, especially in the case of ICl. This calculation is based primarily on the equality between p(x,y,z) and the infinite Fourier sum- mation of Equation (60). Since the highest-order reflection observable experimentally is usually less than h,k, or 1 = 15 (12,0,0 for B-ICl), series truncation errors leading to inaccurate electron distributions occur. The results of 139 Table 19. Calculated electric field gradients in B-ICl. (in 2'3) Calculation Field Asymmetry Conditions Gradient Parameter (60.) (0) Field Gradient at 1(a)b Experimentala ~115.2 ~~~ Point Charge Model —0.043 0.21 Least-squares Origin -81106 0.033 Least-squares—12 points -380 0.34 Least—squares-all I(a) -22 0.22 Map Origin (.500,.118,.261)C -206 0.25 (.515, .118, .261) -971 °°“ (.500, .128, .261) -20634 °'° (.500, .118, .251) -206 0.25 Field Gradients at other Nuclei Cl(a)b: Least-squares Origin —10784 0.044 Cl(b)b: Least-squares Origin -2795 0.497 aReference 106. bAtomic positions from Reference 95: I(a)-(.5028, .1182, .2629), I(b)-(.8135, .0321, .1250), Cl(a)-(.2547, .2898, .3695), Cl(b)-(.8876, .3002, .1069). cFourier map position from Reference 95. 140 such truncation errors are clearly visible in the Fourier maps, manifested as a background electron density of ap— preciable magnitude. To partially eliminate the trunca- tion errors the field gradients were calculated using the actual Fourier map rather than directly substituting the Fourier summation from Equation (60) into the integral in Equation.(24). The second objection to this type of calculation con- cerns the thermal anisotropies. The contours of electron density drawn for the iodine and chlorine atoms in ICl show definite asymmetries, but these are almost certainly due to anisotropic thermal motion, rather than any effect at- tributable to chemical bonding. The contributions of the valence electrons to the density map are difficult to observe for several reasons. The outer valence electron shells contribute almost nothing to the total atomic scat- tering factor except for very low values of sin 9/1. In addition, competing effects from thermal motion must there- fore be completely removed before the difference electron density maps will show an accurate picture of chemical bonding. In ICl both the iodine and chlorine electron densities represent almost entirely *K and L shell elec— trons which are essentially spherically symmetrical and should yield an electric field gradient not greatly differ- ent from the contribution given by the point-charge model due to the neighboring nuclei. The anisotropic thermal motions would, however, lead to an additional contribution 141 to the field gradient at a given nucleus from the remaining atoms of the crystal. In the case of AlPO4, the additional polarization introduced by the anisotropic thermal motions was found (100) to be small. There is also a contribution to the field gradient at a given nucleus from the inner shell electrons of that atom. This would be essentially zero unless the anisotropic thermal motions in the crystal introduced a field gradient but this is unlikely since the electrons should follow the nuclei (Born-Oppenheimer approx- imation). In the calculations of Table 16, however, the largest contribution to the field gradient at the iodine nucleus appeared to arise from the electron distribution around that same nucleus. There are, then, three principal deficiencies in the calculations as presently carried out: (1) The contribu- tion of the valence electrons to the electric field gradient at the nuclei of a given molecule is not evaluated accurately since these electrons are ineffective in scattering the X- radiation and are not located precisely on the electron density maps. (2) The contribution of the inner shell electrons to the field gradient at the nucleus of an atom is very important but difficult to evaluate accurately since these electrons are smeared out anisotropically in the elec- tron density maps as a result of the anisotropic thermal motion of the atom. (3) The uncertainties in the field gradient resulting from having the grid points placed asym- metrically around the atomic positions in the electron 142 density map. As a result of these factors, the results for ICl are not reliable. Some other compounds, which might be less sensitive to these factors, were also tried. Cyanuric acid, C3N3H303, appeared to be the best choice available at present for correctly evaluating the effects of valence electrons since an extremely accurate X-ray structure determination has been carried out (101,102). Great care was taken in this X-ray investigation to obtain accurate structure factors and evidence of chemical bonding was obtained. Because of the low atomic numbers present the electron densities should reflect bonding character and field gradients calculated for this compound were expected to be more theoretically satisfying. Unfortunately, the electron density map which was calculated was quite complex and the number of grid points required to carry out a calculation was so large that the lattice sum would have taken too much computer time to properly evaluate. Other simple, or low atomic weight, compounds considered were Alzoaand CrClz. .In both the structure determination of CrClz (103) and the struc- ture refienment of A1203 (104), however, only a few struc— ture factors outside the hkO plane were reported and the resulting maps are effectively two-dimensional projections in the ab plane. In summary the calculation of field gradients using electron densities from.X-ray diffraction data might be possible if suitable X-ray data were available for a 143 favorable case. Such an ideal system would be a compound with component atoms of low atomic weight with a relatively simple structure. 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