Pfease Note: Not original copy.; Indistinct type on several pages. Filmed as re- celved. University Microfilms, Inc._ mymm m WYmm-mm as war: him an 1%!ng I 3333 23 Wit .Nm LATTICE DYNAMICS OF PERFECT AND IMPERFECT CRYSTALS OF NaCl STRUCTURE BY Sitaram S. Jaswal AN ABSTRACT OF A THESIS Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOC TOR OF PHILOSOPHY Department of Physics and Astronomy 1964 ABSTRACT LATTICE DYNAMICS OF PERFECT AND IMPERFECT CRYSTALS OF NaCl STRUCTURE by Sitaram S. Jaswal In the atomic theory of solids, both the equilibrium properties and the transport properties of crystals are influenced by vibrations of the crystal lattice into which the atoms are built. The primary concern of the present study is the computation and the application of eigenfrequencies and eigenvectors of normal vibrational modes resulting from the harmonic-approximation treatment of ionic crystals having the NaCl structure. The treatment takes into account short— range overlap forces and long-range Coulomb forces. First the dimensionless Coulomb coupling coefficients for the NaCl structure are evaluated for a moderately fine division of E-space (px/ZO, etc. ). Next eigenfrequencies and eigenvectors for NaCl and KCl are com- puted on both the rigid-ion model and a deformation-dipole model. Since the eigenvectors describe the actual motion of each ion, they have permitted us to classify the normal modes with respect to transversality or longitudinality. The eigenvectors and eigenvalues of the perfect lattice are useful, moreover, in various perturbation calculations for slightly imperfect lattices. We have used the values in a Green's-function computation for the frequencies of local modes and the corresponding amplitudes of impurity-atom vibrations that result from point-mass defects in NaCl and KCl. Finally our results for the frequencies of local modes due to U-centers (substitutional H" and D- ions) in NaCl and KCl are compared with experimental find- ings and with theoretical results by others. LATTICE DYNAMICS OF PERFECT AND IMPERFECT CRYSTALS OF NaCI STRUCTURE BY Sitaram S . Jaswal A THESIS Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Physics and Astronomy 1964 ACKNOWLEDGMENTS I am grateful to Professor D. J. Montgomery, who guided and advised me during the course of this work. I am thankful for the cooperation and assistance of Professor L. W. Von Tersch, Director of the Computer Laboratory, and his staff. Finally, Iwish to express my thanks to the Solid State Science Division, Air Force Office of Scientific Research, United States Air Force, and to the Metallurgy and Materials Branch, Division of Research, United States Atomic Energy Commission, for financial support during the investigation. I, ’1; ‘I, | ‘I, \l; J; I; ~l» \‘ l; '1 ‘1 s‘a \‘r \" )Is P pp )6 ,.\ I.‘ if >I‘ a" ¢.< >I‘ )'s >l‘ 1p ’1‘ II. ii TABLE OF CONTENTS CHAPTER Page I. INTRODUCTION . . . . . . . . ....... . . . . 1 II. THEORY .............. . . . . ..... 7 A. Perfect Lattice ...... . . . . . . . . . . 7 l. Rigid-Ion Model ..... . . . . . . 10 Z. Deformation-Dipole Model. . . . . . 16 B. Imperfect Lattice . . . . . . . . . . . . . . 19 III. COMPUTATION AND RESULTS. . . . . . . . . . . 22 A. Coulomb Coupling Coefficients. . . . . . . . 23 B. Eigenvectors and Eigenfrequencies of 23 Vibrations for the Perfect Lattice . . . . C. Impurity Modes ......... . ..... 41 IV. DISCUSSION 0 O O O O O ..... O O O O O O O O O O 68 REFERENCES . . .......... . . . . . . . . . . . . . 7O v LIST OF TABLES TABLE Page 1. Wave vectors for subdivision (px/ZO, py/ZO, pz/ZO) of first Brillouin zone and their weights. . . . . . . . 23 II. (a) Coulomb coupling coefficients of the form c (0(4)! 3474') 27 (b) Coulomb coupling coefficients of the form c (dd/N 31.) 33 III. Input data for calculations . . . . . . . . . . ..... 39 IV. NaCl lattice-vibration eigenvectors for wave propa- gation along the symmetry direction [10, 0, 0]/10 44 V. NaCl lattice-vibration eigenvectors for wave prOpaga- tion along the non—symmetry direction [9, 5, l]/10 . . 45 VI. Properties of lattice-vibration eigenvectors for NaCl onrigid-ionmodel................... 46 VII. Angular frequencies (in 1013 rad/sec) for impurity vibrations in NaCl and KCl 63 VIII. Coulomb coupling coefficients in matrix form for various wave vectors (px, py,pz) . . . . . . . . . . . 73 IX. Wave vectors for subdivision (p /10, p /10, p /10) of first Brillouin zone and their weight . . . .z. . . . 111 iv LIST OF FIGURES FIGURE 1. Frequency of local modes in NaCl as a function of €C1' on the basis of RI and DD models. . . . . . . . 2. Frequencies of local modes in KCl as a function of €C1’ on the basis of RI and DD models. . . . . . . . . Amplitude of vibration of impurity atom in NaCl, in terms of MC DC (f, 0) I 2, as a function of e Cl’ on the basis of and DD models ...... . . ..... . Amplitude of vibration of impurity atom in KCI, in terms of MC 'Y (f, 0)| 7‘, as a function of E‘Cl’ on thebasisof IandDDmodels. . . . . . . . . . . . . Page 64 65 66 67 APPENDIX A. LIST OF APPENDICES Page Table VIII - Coulomb coupling coefficients in matrixform............... ..... 73 Table IX - Wave vectors for subdivision (px/lO, py/IO, pz/IO) of first Brillouin zone and their weights...................... 111 vi CHAPTER I INTRODUC TION Just two centuries ago Lagrange (1) presented the,general theory of small oscillations of a dynamical system with a finite number of degrees of freedom (1762-65). Half a century ago Born and von Karman (2) applied the theory to the vibrations of the atoms that form a crystal lattice (1912), basing their treatment on quasi- elastic forces between nearest neighbors. * Subsequent develop- ments in the theory, insofar as they concern perfect ionic crystals in the harmonic approximation, are summarized in the section Perfect Lattices. The effect of changing the masses or the spring constants in an array of coupled particles was studied by Routh (4) in 1877 and by Rayleigh (5) in 1878. The subsequent extension of their ideas to disordered lattices, especially with respect to the effect of substi- tutional defects on infrared absorption, is described in the section Imperfect Lattices . Perfect Lattices To use the atomistic theory one needs to know the nature of forces between the particles of the system. The crystals simplest *In the same year Debye proposed his continuum theory for the vibrations of a solid. His model was so successful that little attention was paid to the work of Born and von Karman until 1935, when Blackman (3) made a systematic investigation of dispersion curves and frequency spectra of two- and three-dimensional lattices. He showed that sub- stantial deviations from Debye's theory should occur, especially at very low temperatures, and provided impetus for a renewed attack on lattice vibrations Xi_a_ the atomistic approach. 1 to deal with are ionic crystals, for which the interaction forces between ions were first given by Born (6). The central-force power- law model due to Born works fairly well for ionic crystals. Later an exponential version of short-range forces was given by Born and Mayer (7). For the Coulomb forces, which are of long range, direct sum- mation over all points is impossible. Born and Thompson (8) using a method developed by Ewald (9), suggested a way of transforming these sums into more rapidly convergent expressions. Thompson (10) gave the final expressions, but he made a slight error in the definition of coefficients. Broch (11) gave the sums for one-dimensional lattices. Lyddane and Herzfeld (12) extended Madelung's method (13). Their formulas are so complicated that they cannot be used for the whole frequency spectrum, but only for waves propagated along directions of symmetry. They took into account also the free-ion polarizability due to Pauling (14), which gave imaginary frequencies and hence an unstable lattice. Kellermann (15) used Ewald's method in a new form (16). This modification gave comparatively simple and rapidly converging expressions for the long-range Coulomb terms. Kellermann con- sidered nearest—neighbor short-range forces only. His work put the rigid-ion model in its final form. This model described the elastic constants and the specific—heat data pretty well. The theory is, however, inconsistent with the dielectric properties of the alkali halides, because it neglects the polarizability of the ions. Kellermann's method was extended to other ionic crystals of NaCl structure by Sayre and Beaver (17) and by Karo (18). Szigeti, in his study of dielectric properties of ionic crystals (19), derived relations connecting empirical static and optical-frequency dielectric constants to the compressibilities and restrahlen fre- * on an ion, instead quencies. He found that the apparent charge e of being an electronic charge e, is less than it (e* = 0.74e for NaCl). Since he took into account the electronic polarization, he attributed the discrepancy to distortion of ions due to overlap forces. Born and Huang (20) used this idea to determine the distortion dipoles due to the overlap forces between nearest neighbors. Instead of free-ion polarizabilities, Hardy (21) chose the crystal polarizabilities due to Tessman e_t a_l. (22) for the study of lattice dynamics of NaCl. He found the frequencies to be real, but the re- sults were worse than with the rigid-ion model. Therefore further modifications would be necessary. Szigeti's idea of the distortion of ions (19), as exploited by Born and Huang (20), was adapted by Hardy and Karo (23) for NaCl, and by Hardy (24) for KCl. For NaCl only negative ions were considered to be polarizable owing to short-range forces; good agreement was obtained between experiment and theory. For KCl, Hardy tried two models: deformation of negative ions only, and deformation of both kinds of ions. In the latter model he assumed the ratio of the dis- tortion dipoles of the two kinds of ions, m_(r)/m+(r), to be equal to the ratio of the square of the corresponding Zachariasen ionic radii, Rz_/RZ+. With this choice, though, the results were not so good as with the former model, which did give much improvement over the rigid- ion model. The connection between the polarization of the ions and the re- pulsive forces between them has been considered by Yarnashita and Kurosawa (25), by Dick and Overhauser (26), and by Hanlon and Lawson (27). The second and third pairs of authors have suggested a shell model for ions having closed- shell electron configurations. A shell model has been used by Cochran (28) to explain inelastic neutron scattering for NaI and Ge. The Chalk River group (29) first took the shell model in its simplest form, where only negative ions are polarizable and only nearest-neighbor short-range forces are con- sidered, in order to explain their neutron-scattering results for NaI. Later (30) they tried more complicated models involving the distortion of both kinds of ions, and including next-nearest-neighbor short-range interactions. They determined the parameters involved by fitting the theory to their neutron-scattering results. Tolpygo (31) took the deformations into account by expressing the perturbation energy as a quadratic function of nuclear displacements and atomic dipole moments, in a way somewhat similar to Hardy's. Maskevich and Tolpygo (31) gave wave-mechanical justification of this treatment by use of the tight-binding approximation. The Chalk River group have compared their simple shell model with the model of Hardy and of Tolpygo. Hardy's method neglects the short-range interaction between an ion and neighboring dipoles; Tolpygo's method neglects short—range dipole-dipole forces. In the present work the Coulomb coupling coefficients have been computed for a subdivision (px/ZO etc.) of k- space by summing series given by Kellermann,(15). Using Kellermann's subdivision of k-space as well as our own, we have computed the eigenfrequencies and the associated eigenvectors in NaCl and KCl on the rigid-ion model due to Kellermann, and on the deformation-dipole model due to Hardy. The eigenvectors have been used to classify the normal modes, and to calculate properties of local modes. Defects The earliest studies of defects in atomic-lattice vibrations are those of Lifshitz and his collaborators (32). Most of his work went unnoticed by subsequent workers. Montroll and his collaborators (33) carried out a parallel program at the University of Maryland. Other work in this area is that of Litzman (34). Dyson (35) did work on the frequency spectrum of a randomly-disordered linear chain. Emphasis in these calculations has been on choosing models simple enough that qualitative as well as quantitative answers to certain specific problems can be obtained. Most of the work that has been done is of qualitative nature. Owing to the large number of unknowns involved in the general disordered lattice, it is quite hard to give rigorous quantitative treatment of any realistic problem. Consequently, most of the work has been confined to isotopic impuri- ties where the only parameter that is changed is the mass, though a little work has been done on change of force constants (33). Some study has been made on thermodynamic properties of disordered lattices (33, 36-38). We are interested in the effect of substitutional defects on infrared absorption. Among the various changes originating from defects, the one that can be treated precisely is change in mass. Accordingly subsequent discussion is limited to this type of defect. Rosenstock e_t a}. (39) considered a single substitutional mass impurity in a linear monatomic chain with nearest-neighbor short— range forces only, and computed the frequency of the local or out-of- band mode. They showed, moreover, that this mode is optically active. Wallis gt a}. (40) considered one- and three-dimensional cases with nearest-neighbor central and non-central forces. Some analysis of optical absorption has been given by Maradudin (38). A general three-dimensional treatment of the problem has been outlined by Maradudin (38) and by Dawber and Elliott (41). They used a Green's—function method originally due to Lifshitz (32). Based on the theory outlined in references 38 and 41, we have computed the frequencies of local or out-of—band modes which lie above the longitudinal optical branch, as well as the corresponding ampli- tudes of vibration of the impurity atom, as functions of mass of the impurity in crystals of NaCl structure. We have compared our results with experimental findings (42) and with theoretical calcu- lations by others (39, 40). The present work is finot directly conclirned with dispersion curves of frequency versus wave numbers for materials studied. Such information is given for the rigid-ion model in reference 18, and for the deformation-dipole model in reference 43. CHAPTER II THEORY A. PERFECT LATTICE In the harmonic approximation (1) the equation of motion of a periodic lattice may be written as .- 2 zz' Q) MXUO‘(R) = z, 'le(xxt) U'ot'CX'): (1.18.) IN“ where M), is the mass of the X-th particle in the unit cell (X = 1, . . s, where s is the number of particles in the unit cell); ( g ) is the d—th component of the displacement from equilibrium Ha of the y-th particle in the Z-th cell (o(= 1,2,3; 1?: 1, . . . N, where N is the total number of cells in the crystal), and where 9.2.1.6511) [Wawa‘iwuwcéb 1, ,m gs being the potential energy of the system. With solutions of the form —| . o 110,613) = M; (10,00 exp E- wdt +277Lé-25 (£3 3 (1.2a) we get wzu“(y):;v Dea'GF/(X') u“, ()6), (1.3a) 7 where the modified dynamical matrix Dam! ( ZR—‘R' ) is given by -.L / DWI (of-x9 =-(M)< Mx’) 5?: géot’ git x €xpf—277LK-{L<(>€>—z(£:)]]. (1.4a) In general, I_) is hermitian, and by virtue of the geometric symmetry of the rock-salt structure it is real as well, and hence is symmetric. For non-trivial solutions of Eqn. (1. 3a) to exist, we must have (;_w:/ =0 (1.5a) For the rock-salt structure, this equation is of sixth degree in (4)2. Its roots give the eigenfrequencies of normal vibrational modes for a given wavevector If. These modes correspond to six branches of dispersion curves, which are labeled by j: , . . . 6. With 2 evaluated under the assumption of rigid-ion model, and nearest-neighbor short-range interactions, Kellermann (15) solved Eqn. (1.5a) for NaCl. As mentioned in the introduction, other authors (l7, 18) applied the same method to some other ionic crystals having the NaCl structure, and later workers (24, 29-31) made similar calculations with models more realistic than the rigid-ion one. For a given wavevector, corresponding to each branch we can find a vector 11 whose components satisfy the equation @ZQDWO, (De/f) = ga'Qd’GéX') Nd,(3e'/J-(‘), (1.6a) These equations determine w within a constant factor. The arbi- trariness in w can be removed by orthonormality conditions: * . A .. , E; nape/5;) “£10900 - (g , A * , _ Cf 1 ;%(XIJ)Wa'(X/f) "' dot/(£31, ('7a) This orthonormal set of 31's is the set of eigenvectors with which we are concerned here. In matrix form, equation (1.6a) can be written as he. =§efi °r Cigar—ea where the columns of S are the eigenvectors, and (£42 is a diagonal matrix with the eigenvalues as the diagonal elements. Since 2 is real and symmetric, it can be diagonalized by an orthogonal matrix S. The general motion of the lattice is given by a superposition of the elementary solutions (1. 2a): Leaf) = Ni: M: worm 5o Q do x is} 7\prEC@v(B)6+Z7TZ_/g 3g; (36)]. (1.8a) To compute the dynamical matrix we have used two models, the rigid-ion model (RI) due to Kellermann (15), the deformation- dipole (DD) model due to Hardy (24). 10 1. Rigid- ion model In this model the ions are treated as point-charges. The inter- action potential consists of a short-range central potential between the nearest neighbors, and a long-range Coulomb potential. The coupling coefficients, which are the elements of the dynamic- al matrix, are of two kinds: those associated with the motion of the ion under consideration, and those associated with the motion of the rest of the lattice. The coefficients of the first and second kind are given respectively by Edd’Nk‘J == ’2, gé‘dg()efl)e') (1.9a) [IR and (dot'ae 32') = g: CZiwé‘Exoex/szfllg oxygja, (1. 10a) Q o where T981“ 3 25. (Dd) —->£(D€’)o Coupling coefficients due to Coulomb potential Owing to the symmetry of the NaCl-structure, the contribution to eq. (1. 9a) due to the Coulomb terms is zero. The Coulomb part of eq. (1. 10a) is given C (dd'aep') = g: [92(8)< egg/l Ifx:/D/ auxa‘iwuare’io] expézm'A-zigp. 11a) II This series as it stands is notabsolutely convergent. It was evaluated by Kellermann with Ewald's formula for the Theta-function transformation , a 2 if? ; @905?an a!) +2 7741- Of] = All ;% ex/DEF7J?é/,+ft)z+2”5(éh+(1)7], (1.12a) where Va" is the volume of unit cell. This transformation makes the series quickly convergent, and maintains the cell neutrality. Next the summation and the differentiation ige eq. (1. 2112a) are interchanged. Then 1/ Ir] is replaced by 27T<> 2y Old. With this substitution in eq. (1. 11a), the Ewald transformation can be applied. From eq. (1. 12a) we note that the left-hand side convdrges rapidly for large values of J , whereas the right-hand side converges rapidly for small values of Cf”. Hence we break the integration into two parts of: 05f +5] , and use the apprOpriate integrand from eq. (1. 12a). After integration over J , differentiation is performed. Going through these mathematical details, Kellermann obtained the final form of the dimensionless coupling coefficients for the NaCl-structure as // 2 C(Xgfl) = %2C(xg//) =—G;y +HX§+,§—l7 J (£34 (1.13a) /2 Cay/2) =%z COW/2) = 6X3 - ng, (1.14a) where 12 = (Ax +x 633* 4- 4”; (n+5): ”lg/#7:“ 3)] 6;; =47: (II? 182%??? 3;) exp[—7§2/g,+_z)2]x C057(6x+4&_+/:2) H5: 2 ;[ {Mt-C fwgfl) %]cmrg._g {(E- —(2//7r) 6/222 + :VT(M) 3-[224- fig ear J2 2 {if 3e _, six/Egg) W0 / __ _. (5,92,) 2 442+ 651293 Here (h, hy ,h z) are sets of 61 ther all odd or all even integers, (j, fly, I) are sets of integers such that Z): Q = even, except the(0, Dy 0) set, which is excluded; and (mx, my, ,m:) are all sets of integers for which gm): = odd. The conditions on these integers arise from the NaCl structure as represented in Cartesian coordinates. These equations hold only for k 752. In this case, the ‘potential' satisfies Laplace's equation, which gives the condition C xxxx') + C(yygeae ') + C(zz )2 12'): O. 13 In performing the summation in equations (1. 13a) and (1. 14a), Kellermann took 0(2 1. We also took = 1 for our finer subdivision of _l_<_-space. For k = 2, the 'potential' satisfies Poisson's equation which, in case of cubic symmetry, gives c(dd3zx')=-477/3 , c.(o(oL>e)e)=47T/3 and c(otoL'ae)e)=c(ez>a')=0 The coupling coefficients corresponding to longitudinal modes are affected by the polarization on faces of the crystal perpendicular to the direction of motion of the ions; thus there arises an additional contribution of either +4 77 or -477 . Short- range repulsive forc e s The coupling coefficients in this case are given by FRI-310(52):?) = —; ;’ [QZUC/IRiI/D/aaueg) 3%!(362721‘ 15a) RW-‘RRQ: —; [a 27(/_J:et/)/aau(3<) 3“ “’(x’ )ZX expEzm'A 152,], <1- 16a) With nearest-neighbor interaction in the NaCl structure, RI: J Z [72‘ ” "(’2 J7: 2/6 a x H =- ’ ’X’ + X I , a 3 §17U —-2—*£ __ Xm’ 21.29] , (MM) (77303 14 where 1?": %¥)] _ 7 etc. -— e In Kellermann' s notation, ’__7:€z . II 82 where 7’, is the distance between nearest neighbors. , p a WW] = "£- 5; f; 2;, [A We; )2’ + 1 fl Ecchiflx‘a‘m" 12(55):] - (1. 18a) With near est-neighbor interactions, REXXII] = - (,4 +25) ; chyIZJ = O. (1.193) Similarly eq. (1. 16a) gives RCXH'Z> : o ,- "(Ta/ea R(x>e "g +Rfo¢o< 382?] where C5} '3 CE] +C[) etc. (1.25a) 2 . Deformation-Dipole model In addition to the terms considered in the rigiduion model, here the crystal polarizabilities and deformation dipoles due to short- range overlap forces are taken into account. We consider only the negative ion polarizable owing to overlap forces. Also, we consider only the distortions occurring between nearest neighbors. Let m(r) be the deformation dipole moment between a positive ion and a nega- tive ion. It is a function of their separation r, and the sign is taken as positive if the moment is directed from the negative towards the positive ion. To determine the deformation dipole moment to the first order in the displacement, the treatment is parallel to that of the short- range potential considered in part 1. Thus motion of the ion under consideration gives rise to the distortion term (3(32) =2(72’+2 Tana) , where Zée = [dmx(Y)/dy_7y and 23¢ = 072),.(Ya)/70 The distortion due to the motiorai of neighboring ions is given by Qfifix) = a [7,: C05 73% + 733 (Cos 77'5" +Cos 7736)] For the ion ( 3a ), the virtual work involving the field due to the dis- placement (_,_,L( Dog) is given by A W = -[ex~b(>e>J_£(a"e)-.L_L(e°e)—Z firs/.4: , “W o where E ( 3g ) is the field at ( 9.3 ). 17 The second term corresponds to the deformation induced on the nearest 0 a neighbor by the displacement of ( 2Q ). The field _E ( 3e ) is given by the lattice of dipoles: ’ ’ 'l/Z. I ~I B (,5) =[ex,u/, (>2) M2, + fi(9<’)+afl(><)ufl(5)MS/z _yz, . * -b(3<*’) ufl flab/Va] ex/JZEH’Lf'If] . (1.2b) I Here/”75(DQ) is the component of dipole moment resulting from the crystal polarizability. Hence the field is given by awr- —: o I —/ prfZ ”(AL-£21.] C (dot Sex')[ao,l(>< )Mx’ 23X :5 i“ 3%, 9:: +f4a’ (xQ/exi+é,§% 40,1(369— 5(X9Qq/wAZ-ffjfl. 3b) We can substitute this value of _E_ in the expression for AW . Then the Coulomb part of the force is given by 1372:) =-9AW/aue (3:) (1.4b) After going through various manipulations, Hardy put eq. (1.4b) in matrix form. Adding to the Coulomb part the short-range part, which is the same as that adopted by Kellermann, he arrived at the ~ D=-=_>S[h +1495 final form of the dynamical matrix: _ _..... e39: +Q+§QDX fiQQ—Ifl +§Qfig§j=x (1.51)) where 18 "X " pfdd'RX; + Cfaol’seae’}, ”(n =—b(92) 0;“: Canal + Qd(-32’) Cofw.’ (/_J 2229'), "C CgU’JIXD?’ , if = C(dol’aexb, l|>< I Iii-x 9 >3" )6 C; = 1 —- g4 UH U , :08? g 322’ -——(crystal polarizabilities). The distortion dipoles are computed from a relation given by Born and Huang (20): e*- e=2(7{: + 2 7.). (1.6b) * where the effective charge e is obtained from the second Szigeti ::< relation (19). If we write e 5 se, sis given by __ I 3: a), (60477 &)y2é2+2)(M€@)/22 “-79 with M, the reduced mass; Ea , the static dielectric constant; E” the high-frequency dielectric constant; and we: the infrared dis- per sion frequency. 19 B. IMPERFECT LATTICE We may write the equation of motion for an imperfect lattice as .__AM (£)LZ(£)+; 11¢ng (ya (36) (2.1) if we consider only substitutional defects. The first term on the right hand side contains the change in mass due to impurity, AM”, and the second term contains the change in force constants, A ¢°<°fi . If we express u“ ( f.) as the superposition of the amplitudes X of normal modes of imperfect lattice eq. (2.1) becomes 31: .x 1 Mf , 1 +2 De’d’g’QM,(XX) Xa (De/1) 2:},6 a! uiéQXwCXI/é’), (2.2) Md “(Xi') = _ AMx(j) (d2 otu’ 0;}, 5”"; Aéwgi’g’) where The normal modes of the imperfect lattice have been labeled by f, which takes 3sN values. Eq. (2.2) can be solved by a Green's—function method. The Green' 3 function for this equation is given by ,4" an goal/16531 “0+2, $.61? {Daw' i 0) ._—_ om” J22” 092”. (2.3) 20 If we expand gold, .1." by bilinear expansion in terms of the amplitudes of the normal modes of the perfect lattice, we get I /2- 1 (4.; a) =_N. __ , “1.0” 45) W: 1(2/ 4),, 301.1(sz ) (Mme/2 EN QJZ(B)— (AP-(71') ‘9 XP[2)TL'[ - (2; (fl) -_>$ (3):] (214) The amplitudes of the normal modes of the imperfect lattice are given x/f )= Z ? ) Xd( «,2 dot 3“) X Cow ’3?“ 390; .(38 /:§ ) (Z. 5) Eq. (2.5) gives the eigenvalue equation by loge, 93101161355; 63)C;(1 11631311”) “a“; 11 cg)?“ 501"]: 0.(2.6) If we consider a single isotopic impurity, then A ¢ . , is zero, I 1' and we have Cad (iin) = _ M21621 wcho‘a’ JMJ.‘ J; I; With €11 5(Mx‘A4)/MD< . With this value of Cod)!" eq. (2.6) reduces to (M,e E), 49230“, (323,1 3 ca) + 02.011 52 391/: o. (2.7) For a diatomic cubic crystal, eq. (2.7) gives us a triply- deg ene rate equation 21 1: € 2&4): /fl(x/J/f)/Z. (2.8) w’KfJ—wjd) If we apply the normalization condition to the amplitudes of impurity modes and use eq. (2. 5), the amplitude of the impurity atom is given by ‘F 2. Mxlch/aj = 4 / 31/ _, e: ___c_.)___(f) 21/021 1:299; 2“)me €212.19) Eq. (2.8), when solved for d 2(f), gives the frequencies of normal modes of the imperfect lattice. ’22 CHAPTER III COMPUTATION AND RESULTS A. COULOMB COUPLING COEFFICIENTS From eqs. (1. 13a) and (1. 14a) we have computed the Coulomb terms c (otd' 3g 1') with a finer subdivision of _l_<_- space (qxsz/ZO etc.) than Kellermann used. This finer subdivision gives 262 distinct points in the first zone, which by symmetry operations generate 8000 points in the first Brillouin zone. These points along with their weights are listed in Table I. The weight is determined by considering whether the point lies inside the zone, on the face, on the edge, or on the corner of the Brillouin zone. If the point lies inside the zone, the weight is given by the total number of permu- tations which are performed on the sets of p's generated by changing the sign of one of them at a time, two of them at a time, and three of them at a time (including, of course, the original set). For other cases, one has to take into account the neighboring cells. Computed dimensionless Coulomb coupling coefficients have been listed in Tables II (a) and II (b), and in matrix form in Table VIII (Appendix A). B. EIGENVECTORS AND EIGENFREQUENCIES OF VIBRATIONS FOR THE PERFECT LATTICE For NaCl and KCl we have made calculations on both the rigid- ion model (RI) and the deformation-dipole (DD) model. Most of the input data for these salts are given in Table III. 23 Table I Wavevectors for subdivision (px/ZO, Ry/ZO, pz/ZO) of first Brillouin zone and their weight. Wave Vector Weight Wave Vector Weight Px P), Pz PX Py Pz 0.0 0.0 0.0 l 8.0 8.0 4.0 24 2.0 0.0 0.0 6 8.0 6.0 6.0 24 2.0 2.0 0.0 12 8.0 8.0 6.0 24 2.0 2.0 2.0 8 8.0 8.0 8.0 8 4.0 0.0 0.0 ’ 6 10.0 0.0 0.0 4.0 2.0 0.0 24 10.0 2.0 0.0 24 4.0 4.0 0.0 12 10.0 4.0 0.0 24 4.0 2.0 2.0 24 10.0 6.0 0.0 24 4.0 4.0 2.0 24 10.0 8.0 0.0 24 4.0 4.0 4.0 8 10.0 10.0 0.0 12 6.0 0.0 0.0 10.0 2.0 2.0 24 6.0 2.0 0.0 24 10.0 4.0 2.0 48 6.0 4.0 0.0 24 10.0 6.0 2.0 48 6.0 6.0 0.0 12 10.0 8.0 2.0 48 6.0 2.0 2.0 24 10.0 10.0 2.0 24 6.0 4.0 2.0 48 10.0 4.0 4.0 24 6.0 6.0 2.0 24 10.0 6.0 4.0 48 6.0 4.0 4.0 24 10.0 8.0 4.0 48 6.0 6.0 4.0 24 10.0 10.0 4.0 24 6.0 6.0 6.0 8 10.0 6.0 6.0 24 8.0 0.0 0.0 6 10.0 8.0 6.0 48 8.0 2.0 0.0 24 10.0 10.0 6.0 24 8.0 4.0 0.0 24 10.0 8.0 8.0 24 8.0 6.0 0.0 24 10.0 10.0 8.0 24 8.0 8.0 0.0 12 10.0 10.0 10.0 74 8.0 2.0 2.0 24 12.0 0.0 0.0 6 8.0 4.0 2.0 48 12.0 2.0 0.0 24 8.0 6.0 2.0 48 12.0 4.0 0.0 24 8.0 8.0 2.0 24 12.0 6.0 0.0 24 8.0 4.0 4.0 24 12.0 8.0 0.0 24 8.0 6.0 4.0 48 12.0 1020 0.0 24 P 12.0 12.0 12.0 12.0 12.0 12.0 12.0 12.0 12.0 12.0 12.0 12.0 12.0 12.0 12.0 12.0 12.0 12.0 14.0 14.0 14.0 14.0 14.0 14.0 14.0 14.0 14.0 14.0 14.0 14.0 14.0 14.0 14.0 14.0 Wave Vector F? 12.0 2.0 4.0 6.0 8.0 10.0 12.0 4.0 6.0 8.0 10.0 12.0 6.0 8.0 10.0 12.0 8.0 10.0 .0.0 2.0 4.0 6.0 8.0 10.0 12.0 14.0 2.0 4.0 6.0 8.0 10.0 12.0 14.0 4.0 0.0 2.0 2.0 2.0 2.0 2.0 2.0 4.0 4.0 4.0 4.0 4.0 6.0 6.0 6.0 6.0 8.0 8.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 2.0 2.0 2.0 2.0 2.0 2.0 2.0 4.0 Weight 12 24 48 48 48 24 24 48 48 48 24 24 48 48 12 24 24 24 24 24 24 24 24 12 24 48 48 48 48 48 12 24 24 14.0 14.0 14.0 14.0 14.0 14.0 14.0 14.0 16.0 16.0 16.0 16.0 16.0 16.0 16.0 16.0 16.0 16.0 16.0 16.0 16.0 16.0 16.0 16.0 16.0 16.0 16.0 16.0 18.0 18.0 18.0 18.0 18.0 18.0 Wave Vector py 6.0 8.0 10.0 12.0 6.0 8.0 10.0 8.0 0.0 2.0 4.0 6.0 8.0 10.0 12.0 14.0 2.0 4.0 6.0 8.0 10.0 12.0 4.0 6.0 8.0 10.0 6.0 8.0 0.0 2.0 4.0 6.0 8.0 10.0 OOOOOOOOOOOOOOOOOOCOOOOOOOOOOOOOOO N Weight 48 48 48 24 24 48 24 12 24 24 24 24 24 24 24 48 48 48 48 24 24 48 48 24 24 24 24 24 24 24 24 Wave Vector P 18.0 18.0 18.0 18.0 18.0 18.0 18.0 18.0 18.0 18.0 20.0 20.0 20.0 20.0 20.0 20.0 20.0 20.0 20.0 20.0 20.0 20.0 1.0 3.0 3.0 3.0 5.0 5.0 5.0 5.0 5.0 5.0 7.0 py 12.0 2.0 4.0 6.0 8.0 10.0 4.0 6.0 8.0 6.0 0.0 2.0 4.0 6.0 8.0 10.0 2.0 4.0 6.0 8.0 4.0 6.0 1.0 1.0 3.0 3.0 1.0 3.0 5.0 3.0 5.0 5.0 1.0 0.0 2.0 2.0 2.0 2.0 2.0 4.0 4.0 4.0 6.0 0.0 0.0 0.0 0.0 0.0 0.0 2.0 2.0 2.0 2.0 4.0 4.0 1.0 1.0 1.0 3.0 1.0 1.0 1.0 3.0 3.0 5.0 1.0 Weight 24 48 48 48 24 24 48 24 12 12 12 12 12 12 24 24 16 12 16 24 24 24 48 24 24 24 24 25 7.0 7.0 7.0 7.0 7.0 7.0 7.0 7.0 7.0 9.0 9.0 9.0 9.0 9.0 9.0 9.0 9.0 9.0 9.0 9.0 9.0 9.0 9.0 9.0 11.0 11.0 11.0 11.0 11.0 11.0 11.0 11.0 11.0 Wave Vector py 3.0 5.0 7.0 3.0 5.0 7.0 5.0 7.0 7.0 1.0 3.0 5.0 7.0 9.0 3.0 5.0 7.0 9.0 5.0 7.0 9.0 7.0 9.0 9.0 1.0 3.0 5.0 7.0 9.0 1110 3.0 5.0 7.0 "U pa 9: p: r1 r1 H‘ r1 r1 H‘ m> \1 \1 U1 U1 U1 U) UJ-U2 U) H‘ P‘ P‘ #1 P‘ \1 U1 U1 0: U) U: #1 h‘ h‘ Weight 48 48 24 24 48 24 24 24 24 48 48 48 24 24 48 48 24 24 48 24 24 24 24 48 48 48 48 24 24 48 48 P 11.0 11.0 11.0 11.0 11.0 11.0 11.0 11.0 11.0 11.0 13.0 13.0 13.0 13.0 13.0 13.0 13.0 13.0 13.0 13.0 13.0 13.0 13.0 13.0 13.0 13.0 13.0 13.0 13.0 15.0 15.0 15.0 15.0 15.0 Wave Vector py 9.0 11.0 5.0 7.0 9.0 11.0 7.0 9.0 11.0 9.0 1.0 3.0 5.0 7.0 9.0 11.0 13.0 3.0 5.0 7.0 9.0 11.0' 13.0 5.0 7.0 9.0 11.0 7.0 9.0 1.0 3.0 5.0 7.0 9.0 3.0 3.0 5.0 5.0 5.0 5.0 7.0 7.0 7.0 9.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 3.0 3.0 3.0 3.0 3.0 3.0 5.0 5.0 5.0 5.0 7.0 7.0 1.0 1.0 1.0 1.0 1.0 Weight 26 15.0 15.0 15.0 15.0 15.0 15.0 15.0 15.0 15.0 15.0 15.0 17.0 17.0 17.0 17.0 17.0 17.0 17.0 17.0 17.0 17.0 17.0 17.0 19.0 19.0 19.0 19.0 19.0 19.0 19.0 19.0 19.0 Wave Vector FY 11. 0 13. 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OCDCDO 0 0‘0. 0 0 0‘00 0 oo OCDC)OOO(DC)O¢.9hJNth3N,‘ O p-o C O-‘O-O O 0 0(3 H O O c(xy11) -206635 ~205392 “300469 "302134 *300719 “209130 ‘301413 00000 ”101107 *200350 “206514 -209246 -208842 -205887 ”201002 ’100910 ’200026 “206167 “208966 ‘208677 ”205841 “201042 -109177 “205269 -208266 -208314 ‘205821 -204180 ‘207512 -208095 “207142 -04649 *103200 ’109754 -203641 -204784 ’203495 -200240 ‘102838 -109295 “203228 -204524 -203427 “108561 -202608 -204210 -202108 00000 ~07248 ‘103406 C(lel) -103812 *205398 ‘203814 -2.2616 ~202045 ’209130 “208317 00000 00000 00000 00000 00000 .0000 .0000 00000 ”100910 *100390 “09725 “.9111 '08635 *0 8500 ”08520 “109177 ’108105 “107154 -106505 -106434 -204183 '203316 ’202989 ‘207142 “04649 '04518 “04285 “.4043 ”03853 ‘03752 “03745 “102838 “102247 -101629 ~101169 “100975 -108561 -107840 ‘107401 “202108 00000 00000 00000 36 C(yzll) “103812 “1.7360 -C¢I707 —Z.40&5 ”£04161 ”5.7445 “3.0775 .0000 00000 OOOCK) 00090 00000 00003 00000 00000 ‘03217 -.6079 ‘08313 “09746 ‘100284 -09878 '08520 ”101528 -1.5842 “108689 -109859 “109213 -201941 -206142 ’208095 “301553 -00784 ‘02273 ’03542 ”04483 -.5021 “05102 “04695 “06600 “100313 “103100 -1.4733 *1.5041 ~106194 “200711 ~203489 “206750 00000 00000 00000 C(xy12) l 1 1 l 1 l 1 l 1 1 1 1 1 1 1 00925 02700 03878 02579 005C“ .8786 07475 00000 07238 02894 06039 06606 05153 02472 99272 .6871 .2232 05193 .5683 .4243 01641 .8580 00369 .2788 .3035 01596 .9198 .8955 .7433 .3866 .2911 oé31€38 .1679 . 046".) 01750 o}339 0971C 08216 09C3Q 08414 05335 OQCOO .--s,—- ~- v‘ I". ‘ ‘ a ... -..‘ .t, I 7 l" t ‘ 5‘. O '3 '_ C(x212) “02733 102700 oaajj .3737 m.n9¢9 .57flé 0300? 90030 0 000:) oOOCO 0000: .0000 o 0006 00300 OOCDQ 06571 C(y212) .— H .2733 .0275 00936 ".3654 u.- 0 0. (DC): b.J U .4383 02789 05200 000C¢ .OGQH .000; 00003 0000C 0033? 00033 00007 .0957 000$an O 'n‘ to bvaLDRJN- b'fid1b a“ m i~ . -; 1‘ ‘ .1.) } 0 Q. «)J fl 0 {J {U fl 0 :"x K. " '1 H \ "~ 9 ('1) I” o LP3)J 4 O m ’3h)b '\ “JP. ; I“ C - s}. 5’" . ' flu l , 3“, . (3 . J . ; ’ I.“ o ’3 fl ollfifl ~a1$7f ~.1597 3 c “00:“ A .- Iafi’ "3 L.._ i.“ J" «3737 _.a(n: .2987 «.aeaé "< ,OflDC)O(DC)OCDC)OOOCDC>OO<3C>Om<) u .:.. . .. . .. o o. . .. . .. . ..>. .. . N O b . .. . .. C)O(3CDC)O @CDO‘9C3030 O<3 €- 0 O O P. u.) 1C~JULJ~ O‘JCDO P 3.0 UEWLJUJU . . . . . (DC)O(3C)O h‘NTERJWFV . . C)O(DC)O(D ¢~b . . CvO 6.0 1.0 1.0 1.0 1.0 1.0 C(xyll) *1o7700 —l.9812 “1.9813 ~1.7998 -l.4741 ”.7154 “1.3252 *1.7539 *1.9693 *1.9765 “1.8023 —l.2954 *1.7136 ”1.9421 “1.9705 “1.6687 “1.9213 ~02744 “.7828 “1.1816 “1.4293 “1.5150 “1.4508 _.7668 —1.161d “1.4128 “1.5074 “1.1315 *1.3909 00000 w.3585 ~.6663 “.8860 -.9997 "1.0076 —.9217 “.3548 —.6604 ~.8901 ~.9959 #100073 4.6453 —.8658 —.9985 —.5590 “.0908 ~.2596 *.3934 ”.4762 a "‘ ".4090 ”F‘HHHh-l! C(lel) uh- I f v.- E f t _... «no u. ‘- ~— u..— .— 5. than ‘+ .0000 .0000 .0000 .QCOO .3300 .7154 .6908 .6597 .6326 .617! .6163 .2554 .2369 .1982 .1835 .6687 .6434 .2744 .2683 .2583 .2480 .2408 .2357 .7668 .7415 .7158 .7003 “1.1315 ~l.1053 .0000 .0000 .0000 .0000 .0000 .0000 .0000 . 3554cj . 3453 .3336 .3243 .3207 .6453 .6279 .6163 .8550 .0908 .0591 .0863 “.0536 37 C(yzll) w ‘_ u...- -- -1 l I -1 l C ‘2 I l .0000 .0000 .0000 .0000 .0000 .P945 .5635 .7842 .9382 .0104 ”.9885 .0869 -.5107 .8176 .9705 4.1261 . 572530 .0732 .2135 .3372 .4332 .4928 ".BCBS I Nani-owl t ) f i 2 .6249 o 9576.) .2729 .4542 .5077 _.0336 .0000 .0009 .0000 .0000 .0000 .0000 .3000 .9654 3:439 .7543 .9247 .0073 .0499 o ‘3 {‘2‘ 4‘. L). ‘1’; I) 1 ‘f1(‘:) .0712 ‘.2056 .3307 .4277 .4903 C(xy12) .9965 1.0456 .9677 .8049 .5028 .4173 .7495 .9423 .9855 .9053 .7463 .6324 .7667 .8081 .7821 .5474 .5302 .1614 .4517 .0552 .7498 .7380 .6475 .4054 .5003 .6638 .6447 .4541 .5003 .0000 .7071 .3743 .4751 .5027 .4681 .3915 .1965 .3545 . 4 ‘56 «([35 04722 .4363 .2976 .3715 c3528 Q?5}7 .Qfiifi .14Q8 .2111 .2423 .3396 C(x212) .0000 .0000 .0000 .0000 .0000 .4173 .3541 .2590 .1439 .0204 .1009 .6324 .4549 .2373 .0000 .5474 .2078 .1614 .1452 .1150 .C749 .0291 .0182 .4054 .3190 .2031 .0696 .4541 .2745 .0000 .0000 .9000 .0000 .0000 .0000 .0000 .1965 01561 .1197 .0525 .0000 .2976 .2101 .1009 .?517 .9516 .0464 .0365 .0233 .0050 C(y212) .0000 .0000 .0000 .0000 .0000 *.0517 “.1133 “.1884 -.2701 *o3422 *.3828 ”.2453 “.4085 .5721 ‘o7221 “.6520 *.9198 *.0171 ”.003,1 ~., .‘. (1 a. .‘l.’ .)l;‘. t ~.13UE *.1794 “.208? ~.l€51 “.2920 “.4274 ”.5947 $.51} w.7470 .0003 .0035 .0000 .0060 .0000 .0000 00000 “.0839 “.1731 “.2550 “.3511 ‘04767 “.3569 ~.5529 .7463 “.BRGC ““ 0 (3:34:30; I —-.O‘fi90 ~.1179 “.1673 “.2110 [alsll lllll l i I I! II II II ‘llul .II. III I 38 px py pz C(xyll) C(lel) C(yzll) C(xy12) C(x212) c(y212) ’0? 300 3.0 ~025b8 “0235C *0illd aidgfi 21597 —02108 2900 500 300 “03801 -02483 “-0‘3'71-3 01051} 010!“ ”03602 1900 700 300 ~04742 “02418 ~102604 02138 00630 *05118 1900 500 500 -03803 *03803 “105498 01444 01444 "06167 2000 00 00 00000 00000 00000 00000 00000 00300 2000 200 00 00000 00000 00000 00000 00000 00000 2000 400 00 00000 00000 00000 00000 00000 00000 2000 600 00 00000 00000 00000 00000 00000 00000 2000 800 00 00000 00000 00000 00000 00000 00000 2000 000 00 00000 00000 00000 00000 00000 00000 2000 200 200 00000 00000 ‘02790 00000 00000 ’00943 2000 400 200 00000 00000 “05385 00000 00000 ‘01925 3000 600 200 00000 00000 “07590 00000 00000 “02943 3000 800 200 00000 00000 “09216 00000 00000 "03915 4000 400 400 00000 00000 —}0Oa16 00000 00000 -03936 9000 600 400 00000 00000 ”104741 00000 00000 “06028 39 Table III. Input Data for Calculations Effective Lattice Crystal Polariz- Compressibility Charge Constant abilities (IO-lzdynes/cmz) >2 (10-8 cm) (10-24 (CHIP) B e /e rO 06' DC" NaCl 4.27 0.74 2.814 0.255 2.974 KCl 5.63 0.80 3.139 1.20 2.97 For the RI model, we need only fl and r0. For the DD model, we need as well e*/e, the effective charge ratio, and D(+ and 04’, the crystal polarizabilities of the two ions involved. The short-range ‘ coupling coefficients are computed with eqs. (1. 19a) and (1. 20a). To form the dynamical matrix, eq. (1. 25a) is used for the RI model, and eq. (1. 5b) for the DD model. The dynamical matrix has been taken in the following form: 1 {an} {xvii} flei? {nu} {xylzf §x212} §xy11} {W113 gyz11} {xylz} gyylzg gy2123 {ml 1} iyzll} {zzl 1} {ma} fyzlz} Ezle} {xxiz} w; {xz 12} We? {ma-2; {xzzzf ExylZ} gyyi z} {yzlz} {was gyyzzf {yzZZ} glez} {yziz} {zzlz} {xzzz} {yzzzg {zzzz} I Once we have the dynamical matrix on either model, our next step is to find the matrix g which diagonalises the dynamical matrix to give the diagonal “13”ng 7‘. Diagonalization has been performed on the CDC-3600 computer at Michigan State University, with an adaptation of 704-709 Fortran program number 664, 00-OP ID: F4 UCSD 1 EIGEN. This subroutine is based on the Jacobi method to 40 diagonalize real symmetric matrices. In this method the largest off-diagonal element is selected, and rotation is performed on the dynamical matrix so that that element becomes zero. This process is repeated successively till all the off-diagonal elements are smaller than a chosen limit. At the same time, the matrices which perform the rotation are multiplied together to give the final diagonaliz- ing matrix :_S_ . Thus in the end we get both the eigenvalues 6J2 and the eigenvector matrix g, with each column of § representing the eigenvector corresponding to its appropriate eigenvalue. We have done calculations with our subdivision of k-space as well as Kellermann's. His subdivision amounts to considering 48 points in the first zone in _k-space. These 48 points generate in all 1000 points by symmetry operations in the NaCl structure. In order to give examples of the eigenvectors and to illustrate their application in classification of normal modes, we give in this section the results for Kellermann's subdivision for the rigid-ion model. To see what typical eigenvectors look like, we have chosen two wavevectors for NaCl, one in a direction of symmetry and the other in a general direction, and have presented the corresponding eigen- vectors in Table IV and Table V, respectively. These eigenvectors, when normalized through division by (NMX )%-, give the amplitudes of vibration for the two kinds of ions in a given mode. Each column gives the eigenvectors for a given mode (whose frequency is indicated at the top), the first three elements correSponding to the Cartesian components of displacement of one kind of ion, and the last three of those of the other. For all the NaCl eigenvectors we have computed the ratio of the amplitudes of vibration for the two kinds of ions. We have calculated also the angle (Na, Cl) between the directions of motion for the two kinds of ions, and the angles (Na, _k) and (C1, _1_<_) between the directions 41 of motion for each kind of ion and the direction of propagation of the wave. The results are given in Table VI. For _l_<_ = 2, the ratio of the amplitudes of lighter to heavier atom is inversely proportional to their masses, in the optical branch; in general it increases with increasing 5. The corresponding ratio in the acoustical branch is equal to unity; it decreases with increasing _l_<_. Then, by continuity, we can classify the normal modes into various branches, and state their character as to longitudinality, transversality, or neither. The results for NaCl are shown in Table VI. We see that for _15 along [100], [110], and [111], the waves are purely transverse or longitudinal, and the direction of vibration of both kinds of ions is the same, as is apparent from the symmetry of the crystal structure. Also, in certain directions of somewhat lower symmetry, viz. , either when one of the components of wavevector is zero, or when two com- ponents are equal, there exists a pair of transverse waves, one optical and one acoustic. In all other cases, the waves are neither transverse nor longitudinal, and the directions of vibrations of two ions are different. C . Impurity Mode S A substitutional impurity in an otherwise perfect lattice changes the normal mode of vibration. The new modes are of two kinds, i_n_- band modes, and out-of-band or local modes. An in-band mode corresponds to a frequency which is shifted up or down from the frequency of perfect lattice by no more than the separation of con- secutive frequencies on either side. Out-of—band or local mode fre- quencies lie in gaps or regions forbidden in the perfect lattice. The impurity modes of concern here are the local modes which lie above the longitudinal optical branch, and which are optically active. 42 They have been observed experimentally by infrared absorption in alkali halides containing U-centers. We consider a substitutional impurity for a Cl- ion in NaCl and KCl. Examination of eq. (2.8) shows that it has solutions for 4J(f))£(JL, the largest frequency of the perfect lattice, for positive values of e Cl which are above a certain critical value é critical. To solve eq. (2.8) for these modes, one needs to know the eigenvectors and eigenfrequencies of the perfect lattice. These were obtained as indicated in part B. We have solved eq. (2. 8) numerically in the case of NaCl and KCl for the values of eCl lying between 0 and 1 with proper consideration of the weight of each point in k- space. Weights for Kellermann's subdivision of k-space are given in Table IX (Appendix B). We have solved this equation on both 10 and 20-fold subdivisions of k-space; the results are essentially the same. The frequencies of local mode _\_r_s_ é CI for NaCl and KCl are shown in Figs. 1 and 2 respectively. On the graphs are shown also the experimental findings (42) for U-centers, and results of theoretical calculations by others (39, 40) who treated U-centers as isotopic impurities. Knowledge of the frequency of the local mode has been used in eq. (2. 9) to find the amplitude of the impurity atom in that mode. Results for the amplitude of vibration of the impurity atom in a local mode, as a function of€c1, for NaCl and KCl are shown in Figs. 3 and 4 respectively. With respect to our own results, we note first from Figs. 1 and 2 that the RI model gives somewhat higher frequencies for the local modes than the DD model, as would be expected. Next we see that there are critical values of €Cl for local modes to occur above the longitudinal optical branch in both crystals. Table VII gives the results from our calculations for the frequencies of local modes, and their ratio, corresponding to 6C1 equal to 0. 972 and 0. 943, the _ . In...“ 43 values for H- and D-, respectively. It contains as well the experi— mental results for the absorption frequencies observed for H- and D- in NaCl and KCl. The ratio given by our calculations is about the same as that of Wallis and Maradudin (40), i. e. ,~,/2—. . It is seen that experimental values are far lower than the calculated values on both models and for both crystals. On the other hand, the experi- mental ratio, available only for KCl, does agree quite well with the theory. Amplitude of vibration of impurity atom, given by Figs. 3 and 4, increases with the decrease of mass of the impurity as expected. Since the DD model gives good results for the perfect lattice, the poor agreement between our results and the experimental findings for U-centers indicates that the U—center cannot be treated as a simple isotopic impurity. The changes in force constants, polari- zation, and effective charge are large enough to have significant effect on the frequencies of defect modes. Unfortunately it would be extremely complicated to take these changes into account. um” ‘ Table IV 44 NaCl lattice-vibration eigenvectors for wave prOpagation along the symmetry direction [10,0,0]/10 j 1 2 3 4 5 6 035 (1013/sec) 4.198 3.099 3.093 - 3.093 1.776 1.776 x -O.9284 -0.3716 0 0 0 0 Na+ y 0 0 -0.8435 0 0 -0.5372 2 0 0 0 -O.8435 -0.5372 0 x 0.3716 -0.9284 0 0 0 0 01 y 0 0 0.5372 0 0 -0.8435 2 0 o 0 0.5372 —O.8435 0 mm. 45 Table V7. NaCl lattice~vibration eigenvectors for wave propagation along the non—symmetry direction [9,5,l]/10 j ' 1 2 3 4 5 6 0351013/566) 3.596‘ ‘3.356 2.896 2.882 2.596 2 229—_ -0.6435 -0.7022_ -0.0052 -o.2421_ -0.1651 -0.0830_ Na+ -0.4145 0 0 0.7465 0.5206 0 ~0.6435 'f0.7022 0.0052 -0.2421 -0.1652 -0.0830 -0.0024 -0.0291 -O.664l -0.4035 0.5807 0.2902 01’ 0 0.1101 -0.4125 0 0 -0.9044 -o.0032 -0.0291 -0.6441 0.2902 0.4034 -0.5807 46 .Table 'VI. Properties of lattice-vibration eigenvectors for NaCl on rigid-ion model. Wavevector Eigenfreq. Angle (degrees) Na/Cl Hod. Class.* px py pz ,wjuo’isec) _(Na, 01) (Na, k) (01, k) Arnpl. Rat. 10 4 0 36.27 0 21.80 21.80 . 11.49 ’ 0 32.83 0 90 00 90.00 4.060 c 0 29.46 , 0 68.20 68.20 4.391 9 to 29.06 0 21.80 g 21.80 0.134 a ' 25.24 0 ' 90.00 90.00 0.380 c ' a 22.81 0 ' 68.20 68.20 0.351 p a 10 2 2 38.78 0 15.79 15.79 4.564 p 0 32.54 0 74.21 74.21 3.447 - 0 30.13 ' .0 15.79 15.79 0.338 a 29.36 0 90.00 90.00 ' 2.097 c 0 24.49 0 74.21 74.21 0.447 a 18.24 0 . 90.00 90.00 0.735 t a .9 .I‘ III‘ II I .6“ 'Uavcvector .PK 10 10 p Y 2 P 2 0 Eigenfreq. 40.32 31.34 30.59 30.55 20.16 19.58 41.98 30.99 30.93 30.93 17.76 l7.76‘ 35.96 33.56 28.96 "28.82 25.96 22.29 Angle (degreeS) 0 82.98 90.00 ‘ 89 98 '90.00 ‘90.00 90.00 11.31 90.00 11.31 ' 78.69 90.00 78.69 90.00 , 90.00 90.00 90.00 34.67 56.85 56.87 81.12.. .80.72 56.85 11.31 90.00 11.31 78.69 90.00 78.69 90.00 90.00 90.00 90.00 77.69 34.70 56.84 80.93 Na/Cl Mod. P.) 0. 313. 10. cl>j(1o*7sec) (Na, Cl) (Na, k) (6;, k) ' Ampl. Rat. .645 .192 .423 .181 .703 .707 .102 .497 .950 .950 791 .791 49 .009 .787 .863 .147 ' Class.* Wavevector P P K Y 9 3 3 9 3 1 9 1 1 Eigenfreq. 37.86 33.21 30.21. '27.50 26.58 18.75 38.96 32.10 .30.16 29.00 24.16 19.84 41.67 31.27 30.47 30.47 19.58 17.66 48 Angle (degrees) 21.01 65.04 57.03 30.95 7.621 28.66 73.33 57.28 12.06 28.00 2.728 48.61 56.53 0.922~q 16. 74. 40. 90. 75. 90. 87. 65. 75. 80. 80. 82 90. 37. 82. 90. 08 82 97 00 00 .838 49 58 86 25 46 .806 .17 00 97 88 00 4.926 40.14 16.06 90.00 73.49 90.00 4.252 66.87 11.45 47.39 84.22 81.99 0.077 49.22 90.00 18.55 83.80 90.00 Na/C] Mod. 10. L») P; w36109.66) (Na, 01) (Na, k) (Cl.k) Amp1.Rac. 35 £629 .256 .421 .107 .637 .986 .217 .771 .586 .426 .650 .642 .726 .785 Class.* 0 o a C O a t a o ,o a o a a o o t _ O a a t a Wavevector X Y P z pfiigenfreq. OJ j(10n7sec) 35.22 35.20 28.65 28.36 27.47 20.036. 39.72. 31.73 30.62 27.61 24.28 19.73 39.35 32.37 29.79 26.92 24.25 21.61 Angle (degrees) (Na, C1) 0 29.45 70.43 55.62 3.466 8.703 78.09 69.73 68.15 11.04 ' 7.961 50.38 68.55 20:49 I \ Na, R) 90.00 0.982 85.25 90.00 84.40 85.33 15.15 86.20 58.50 88.98 84.27 78.98 0.233 90.00 81.39 90.00 V 63.44 (01, k) 90.00 28.47 14.81 90.00 28.78 88.80 7.555 15.63 22.18 73.89 83.60 8.194 90.00 2.555 30.06 90.00 83.93 Na/Cl Nod. Amp). Rat. 75. 13. 0. 68 .816 .020 .391 .521 .926 .573 .050 .538 530 .188 .350 .793 .469 .460 .266 Class.* 1‘. O o a t a o a o o a o a a o L O a o t a a Wavevector 'Eigenfreq. UJj(10I2/sec) 41.77 30.91 29.62 29.17 22.99 17.52 42.56 31.07 29.99 28.15 19.21 16.92 43.83 30.69. 30.69 29.89 16.85 16.85 Angle (degrees) 50 (Na, Cl) 3.462 70.08 64.25 5.586 4.076 81.83 7.6.94 2.1.85 (Na, k) 4.503 72.90 51.55. 90.00 83.14 90.00 0.471 90.00 61.39 65.98 90.00 80.75 90.00 90.00 90.00 90.00 C cl. k) 7.964 37.02. 12.70 90.00 88.72 90.00 4.547 90.00 20.43 90.00 82.93 90.00 90.00 90.00 90.00 Na/Cl Mod. -Ampl. Rat. 2. to 0. 0. 724 .268 739 .769 Class.* 0 o a C O a t a o t o a o | t a a 43 o t O t 0 11 a t a t a Wavevector Eigenfreq. L"JjOOIZ/sec) 37.95 »33.29 30.25 27.39 26-66. 18.74 42.27 34.24 28.14 25.81 21.52 19.63 40.41 32.31 30.02 26.31 23.99 20.39 Angle (degrees) (Na, Cl) 21.32 70.16 57.94 39.85 89:99 90.00 90.00 90.00 90.00 90.00 7.310 59.77 53.20 64.45 64.02 37.36 kn 1 (Na, k) l\ to 84. 90. 18. 71 71 88. 82. .48 .00 15 00 11 .91 .91 98 81 .91 .27 .36 .12 .93 .55 (C1. k) (.11 b) 71 18. 48. 71. 71. 87. 0\ 28. 24. 80. 68. 89. .79 .71 .04 .00 .00 .93 09 52 93 97 61 Na/Cl Mod. Ampl. 19. 0. Rat. .764 .438 .273 .440 .094 .632- 73 268 .024 .339 .246 .105 .587 .969 .892 .207 Class.* \Javcvcctor PX {Y P 7 3 3 7 3 l 7 1 l Eigenfreq. wj (l 0'1/sec) 43.25 31.75 28.37 27.22 22.87 17.78 43.95 30.73 29.14 27.21 21.55 18.30 45.63 30.21 30.05 28.29. 17.66 15.98 52 Angle (degrees) Na/Cl Mod. (Na, 01) (Na, k) (Cl, k). Ampl. Rat. 2.898 10.44 7.539 2 983 21.93. 34.65 12.72 0.564 . 70.71 75.90 33.40 7.279 0 90.00 90 00 2.241 29.44 48.07 77.51 0.085 0 90.00 90.00 0.688 '2.669 5.195 7.084 2.252 7.707 85.81 86.96 2.373 41.62 36.55 5.404 0.889 49.09 82.62 51.66 2.002 13.97 88.17 88.46 0.514 4.836 82.04 84.54 0.670 1.787 2.577 4.364 2.092 26.72 79.85 73.43 1.998 0 90 00 90.00 1.855 27.07 15.65 11.42 ' 0.728 1.329 84.82 86.15 0.780 0 990.00 90.00 0.831 lass.* o .a o C 0 a t a o o a o a a O o t O a a C a Wavevector x y P 2 2 Eigenfreq. “j (10%...) 41.73 33.29 29.61 25.29 22.90 19.33 40.46 33.04 28.94 25.85 25.38 19.59 44.72 35.05 25.35 25.09 20.43 18.62 Angle (degrees) (Na, Cl) 17.71 61.31 ~49.79 10.24 7.856 20.98 62.39 37.69 . (Na, k) 16.57 77.42 70.07 90.00 57.56 90.00 90.00 90.00 90.00 8.237 30.90 80.29 90.00 61.40 90.00 Cl, k) 60.14 90.00. 67.80 90.00 90.00 90.00 90.00 0.382 9.919 17.89 90.00 90.00 Ampl. 5. Na/Cl Mod. Rat. 401 .560 .347 .862 .414 .399 .742 .030 .562 .306 .364 .985 .330 .41 .177 .078 .485 Class.* 0 a o t O a t a 4L 0 L O .8 a t 3 t O t a o a o t 0 a C a Wavevector‘ Eigenfreq. X Y 6 4 P . Z wj (l On/sec) 45.23 30.54 29.24 25.15 20.70 19.25 45.40 31.35' 27.71 24.93 21.29 19.34 47.67 28.67 28.28 . 27.43 19.37 15.53 54 Angle (degrees) (N5, 01) 2.353 53.03 54.68 36.65 40.34 19.86 1.490 23.04 39.82 . 8.484 0.635 37.18 34.10 1.645 _ (Na, k) ‘(Cl. k) 6.416 50.96 '70.72 85.74 75.57 80.70 2.464 90.00 23.22 85.65 90.00 78.93 5.556 90.00 75.06 27.52 89.42 ‘90.00 4. 54. 88. 79. 86. 90. 54. 90. 87 90. 67. O\ 87. 90. 02 58 03 .955 00 .188 52 00 .41 .190 00 76 .589 77 00 Na/Cl Mod. Amp] 0 r.) Rat. .326 .942 .699 .956 .424 .504 .043 .400 .835 .327 .642 .955 .875 .304 .830 .629 Class.* O a o o a a o C 0 a o t a a o ‘1'. O o a a C a Wavevector Eigenfreq. OJj(10n/sec) 48.11 30.41 28.31 26.25 16.47 16.22 48.74 30.10 30.10 25.93 14.24 14.24 45.22 36.42 23.97 23.97 19.30 19.30 Angle (degrees) 55 (Na, CI) 1.397 28.51 28.41 1 .' 724 (Na. 8) 3.150 90.00 78.30 14.55 90.00 83.03 90.00 90.00 90.00 ‘ 90.00 90.00 90.00 ’— (c1, k) Ln L\ 4. 90.00 73.19 13.86 90.00 84.76 90.00 90.00 90.00 90.00 90.00 90.00 Na/C1 Mod. Rat. .880' .875 .008 .794 .856 .762 .762 .831 .875 .875 Class.* 0 t .0 o a. t a a 46 , o t O t O L a t a t a it o 2 a t o t 0 C a t a ‘Wavevector Eigenfreq. b“j (10’7sec) '45.92 33.60 26.72_ 24.04 19.27 19.13 46.07 30.82 28.36 24.12 20.88 19.19 49.08 29.48 26.81 25.71 18.60 ' 16.09 56 Angle (degrees) (Na, C1) (Na, k) (C1, k) 1 4.137 _ 6.676 2.542 16.50 I 25.21 8.716 15.27 81.46 83.27 0 90.00 90.00 0 _ 90.00 90.00 ‘ 3.459. 74.64 79.09 1.903 ' 4'575, f 2.673 28.78 87.56 63.66 .28.04 8.049 19.99 0 90 00 90 00 3.429 73.76 77.19 0.726 4.521 3.795 8.670 14.38 5.706 0 90.00 90.00 11.66 - 83.13 85.21 2.068 80.81 82.88 NA/Cl Mod. Aupl. Rat. 2.935 0.551 2.599. 12.34 0.125 0.568 2.074 2.444 0.727 7.603 0.647 0.203 2.023 0.773 2.024 2.780 0.543 0.762 Class.* 0 a o t O 1: £1 a o o a t 0 a C a o a C ' O o a t a wavevector q a Ligenfre 50.08 29.77 26.04 25.25 17.45 15.74 51-43 30.23 28.28 23.35 15.68 10.77 48.59 32.17 24.97 24.97 17.66' 17.66 q. ‘93 (109566) 57 Angle (degrees) (Na, Cl) 0.628 4.104 43.19 40.72 4.820 3.104 0.280 3.133 0.629 (Na, 1() 3.483 88.97 74.93 27.33 88.05 83.74 2.341 90.00 86.58 1.020 88.30 90.00 90.00 90.00 89.97 90.00 (C1, k) 3.729 85.32 p... U! N O 87.62 84.99 2.621 90.00 89.71 2.340 88.93 90.00 90.00 89.97 90.00 Na/Cl Mod. Ampl. r—D p—A Rat. .805 .915 .070 .894 .712 .798 .737 .647 .673 .650 .650 .582 .582 Class.* 0- O o a a a o to o a 8 ca £0 46. CO to ta ta Wavevectors P P Y P 2 2 Eigenfreq. "8 j (1 O’z/sec) 50.66 28.29 27‘. 06 24.90 16.84 16.21 50.93 30.35 ~24.88 23.92 . 16.89 16.48 53.45 28.10 27.04 22.62 14.57 12.72 58 Angle (degrees) (Na, 01) _(Na, k) (01, k) 0.845 31.57 31.24 1.091 0.047 0.675 0.719 0.116 3.146 83.56 13.51 90.00 90.00 80.72 90.00 90. 00 90.00 90.00 2.761 90.00 87.00 2.547 88.02 90.00 2.301 64.87 17.73 90.00 90.00 81.81 90.00 90.00 90.00 90.00 2.807 90.00 87.68 1.828 87.90 90.00 Ampl. Na/Cl Mod. 1.861 1.935 0.834" 2.382 0.647 0.792 1.757 1.907 2.293 0.877 0.809 0.673 1.704 1.736 1.869 0.905 0.825 0.888 Rat. Class.* 0 o a t o t a a K, o t O t O 8 a t a t a o t O o a a- t a U: 9 0 p) in Eigenfreq. Angle (degrees) Na/CI Mod. Ldj (1012/5061) (Na, Cl) (Na, k) (Cl, k) Ampl. Rat. 53.93 0.338 1.94 2.287 1.662 29.65 0 90.00 90.00 1.700 27.31 “3.136 86.85 89.99 1.762 20.40 3.428 ~2.428 5.856 0.921 12.70 0.689 84.87 85.56 0.882 12.42 '0 90.00 90 00 0.907 54.38 0 0 0 1.642 29.40 0 90 00 90.00 1.642 29.40 0 90 00 90.00 1.642 18.98 '0 0 0 0.939 10.27 0 90.00 90.00 ‘ 0.939 10.27 0 90.00 90.00 0.939 53.34 0 0 0 1.751 26.47 0 90 00 90.00 1.867 26.47 0 90.00 90.00 1.867 24.78‘ 0 0 0 0.881 14.23 0 90.00 90.00 0.826 14.23 0, V 89.97 89.97 0.826 a wavevectors P P P x y z 3 3 1 Eigenfreq. wjflO’z/sec) 54.84 .9... 2.... 20.10 12.96 12.32 56.69 28.77- '28.28 . " 15.94 9.642 8.886 57.08 27.66 27.66 16.71 9.948 9.948 60 Angie (degrees) (Na, C1) 0.156 0.899 0.973 .0.214 .0.115 0.693 "0.783' 0.211 (Na. k) 1.176 89.20 90.00 3.233 90.00 86.02 1.426 90.00 88.28 2.578 86.80 90.00 90.00 90.00 -90.00 90.00 (Cl, k) 1.020 88.30 90.00 4.206 90.00 86.21 1.541 90.00 88.97 87.01 90.00 90.00 90.00 90.00 90.00 Na/Cl Mod. Ampl. Rat. 1.654 1.706 1.790 0.931 0.862 0.904 1.602 1.617 1.638 0.961 0.942 0.954 1.608 1.649, 0.959 0.935 0.935 Class.* 0 o C O a t a a o t o o a a t a .2 o t. O t 0 I; a t a L a Navevectors P P y z 2 0 0 0 1 1 Eigenfreq. j(10’7sec) 57.86 29.07 27.49 13.45 8.537 7.844 58.69 28.85 28.85 10.13 5.376 5.376 59.43 28.38 28.38 8.396 5.109 5.109 61 Angle (degrees) (Na, Cl) (Na, k) 90.00 90.00 90.00 90.00 90.00 90.00 90.00 90.00 90.00 89.99 89.96 90.00 (C1. k) 90.00 90.00 90.00 90.00 90.00 90.00 90.00 90.00 90.00 89.99 89.96 90.00 Na/Cl Mod. Ampl. Rat. 1.582 1.597 1.618 0.975 0.953 0.965 1.563 1.567 0.986 0.984 0.984 1.556 1.565 1.565 0.991 0.985 0.985 Class.* 2, o t O ‘C 0 1’. a t a C a 80 t O C O 1". .1 t a t a a. t O t O (,6 t a t a 62 Navevectors Eigenfre . Angle (degrees) Na/Cl Mod. Class.* <1 P P P wj (10’7sec) (Na, Cl) ' (Na, k) (C1, k) Ampl. Rat. 0 0 0 60.2 1.542 8. 0 28.61 - . 1.542 c 0 28.61 . 1.542 c o 0 1.000 -6 a 0 1.000 c ‘a 0 . 1.000 c 8 *Classification: 0, optical; a, acoustic; t, transverse; Z , longitudinal ON '30 Table VII. Angular frequencies (in 10 3 rad/sec) for impurity vibrations in NaCl and KCl. ' Calculated Eigenfrequencies U-Center Absorption NaCl KCI NaCl KCl Impurity . 8 RI DD RI DD Impur- , . Mass 601 00(1) 00(1) 00(1) 05(1) ity Ion w. 00. 1.009 0.972 15.9 15.177. 14.2 13.5 H" 10.52 9.36 2.015 ' 0.943 11.2 ,_ 10.6 10.1 9.4 0' - 6.73 1.42 _ 1.42 1.41 1.43 p 1.39 Ratio _. 20 9< 103551;" -———-THIS WORK: R1 '51=_TH'S WORK: 00 NaCl / -—-EXPTL: SCHAEFER (1960) —-THEOR: WALLIS ET AL.(|960) 10- —-— / I // I —————— — -—-;/-—(wg)=6.02xm‘3-—--§- --—- / RI 1 ‘ a/I . } 55...._—¢;::2:::' ________ .Q) = .8, §“_4.__:::' ( (2)061 9X16 ( 1 i 0.6 0.8 1.0 501 Fig. 1. Frequency of local modes in NaCl as a function of the basis of RI and DD models. 65 KCl 15 ~X10'38E0", --,--THlS WORK, H: R1 —-L THIS WORK,H: 00 _..—01an., H: WALLIS ET AL. (1960) --THEOR., H: ROSENSTOCK ET AL. (1980) ,0 :_:--THlS WORK, 0: RI _/ THIS WORK, 0: 00 EXPTL, H: SCHAEFER (1961) ~--- EXPTL.,.D: MlTSUlSHl ET AL. (1962) +—_——-—n.‘”u————~—~——_——-”—- ———_——’——_—_-—-—_—___—.~_ 51" 5”, .4 - ------------- ;; 7’—-——-————(w£)R- =458Xl03 -- | ---~ “figf’ . I “'*“"=: ““““““““ CU =3.87XIOB “- "-— ( 2100 !r‘ J J ! 1 05 0.7 0.9 6' Cl . . . a ’ ~ ., . ,. .r Y! "1 I ‘ .t‘.“v‘ .. .' O '- I' a Fig 2. Fre1u=neies of Local msdes in 131 as a Laaaeion or 6:31, On .5 the basis or RF mi in models 66 '50 L 1 1 1 NaCl 40 —- 4 N___ 301- ‘ 5.. 3: 2S 5 2 20- - 10 =- - . O 0.6 LO .6 Cl. Fig. 3. Amplitude of vibration of impurity atom in NaCl, in terms of MCl(X(f, O)‘ 2, as a function of 6C1’ on the basis of RI and DD models . so - - 4o - -~ N— ES. 2 20 -— .. 105- _. o 05 Fig. 4. Amplitude of vibration of impurity atom in K61, in terms of MCI IX“, O)|2, as a function of 661’ or; the basis of R1 and D3 models. 68 CHAPTER IV DISCUSSION We have seen that for wavevectors directed along axes of symmetry in the perfect rock- salt structure, the lattice vibration waves are either purely transverse or purely longitudinal. For directions of somewhat lower symmetry, viz. , when either one component of the wavevector is zero or two of them are equal, there is‘a pair of transverse waves. But for all other wavevectors , the waves are neither transverse nor longitudinal. We have seen also that when isotopic impurities are introduced in the rock-salt lattice, local modes will appear above the optical branches for mass differences beyond a certain critical value. From earlier theoretical calculations based on simplified models, it appeared as if U-centers in NaCl and KCI behave quite like isotopic impurities. But evidently it is inadequate to treat them as simple isotopic impurities harmonically coupled to the lattice, without con- sidering changes in force constants, polarization, and effective charge. Unfortunately there is no way to introduce these quantities explicitly, One can, however, take the frequency of the local mode as a known! parameter, and make some estimate of the change in short-range force constants between nearest neighbors. One can assume that r0 remains unchanged, and hence that the first derivative of the short-range potential is unchanged from that given by eq. (1 . 23a). If we assume that there is partial compensation among some of these quantities, and that changes in short-range potential absorb some of them, the only parameter which changes is the quantity A given by eq. (1. 24a). With these simplifying assumptions, it is possible to 69 I write a Coax, ( )5 ”I ) which would include change in mass and in A. 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Rev. 129, 2024 (1963). 0.0 2.0 2.0 4.0 COULOMB COUPLING COEFFICIENTS IN 0.0 0.0 «8.3776 .0000 .0000 8.3776 .0000 .0000 .0 .0 «8.2846 .0000 .0000 8.5351 .0000 .0000 2.0 .0 -2.047O ~6.2246 .0000 2.1740 6.0909 .0000 2.0 2.0 .0000 -4.I335 ~4.1335 .0000 3.9950 3.9950 .0 .0 ~8.0i33 .C000 .0000 8.9937 .0000 .0000 2.0 .0 «5.5427 ”4.9053 .0000 6.4048 4.6486 .0000 4.0 00 ~I.8981 ~6.0331 .0000 2.4151 5.5403 .0000 Appendix A .0000 4.1888 .0000 .0000 “4.1888 00000 .0000 4.1423 .0000 .0000 *4.2676 .0000 -6.2246 ”2.0470 .0000 6.0909 2.1740 00000 -4.1335 .0000 -4.1335 3.9950 .0000 3.9950 00000 4.0066 00000 .0000 ’4.4969 .0000 *4.9052 1.5897 .0000 4.6486 ~1.8228 00000 ‘6.0331 “1.8981 .0000 5.5403 2.4151 .0000 73 TABLE .0000 .0000 4.1888 .0000 .0000 *4.1888 .0000 .0000 4.1423 .0000 .0000 *4 o 2676 .0000 .0000 4.0940 .0000 .0000 -4.3479 ~4.1335 -4.1335 .0000 3.9950 3.9950 .0000 .0000 .0000 4.0066 .0000 .0000 “4 . 4969 .0000 .0000 3.9530 .0000 .0000 “4.5820 .0000 .0000 3.7963 .0000 .0000 w4.8303 VIII 8.3776 .0000 .0000 ~8.3776 .0000 .0000 8.5351 .0000 .0000 “8.2846 .0000 .0000 2.1740 6.0909 .0000 ”2.0470 ~6.2246 .0000 .0000 3.9950 3.9950 .0000 ~4.1335 ~4.1335 8.9937 .0000 .0000 -8.0132 .0000 .0000 6.4048 4.6486 .0000 *5.5427 -4.9052 .0000 2.4151 5.5403 .0000 ”1.8981 -6.0331 .0000 MATRIX FORM .0000 *401888 .0000 .0000 4.1888 .0000 .0000 "4.2676 .0000 .0000 4.1423 .0000 6.0909 2.1740 .0000 ~6.2246 —2.0470 .0000 3.9950 .0000 3.9950 44.1335 .0000 -4.1335 .0000 -4.4969 .0000 .0000 4.0066 .0000 4.6486 -1.8228 .0000 -4.9052 1.5897 .0000 5.5403 2.4151 .0000 ~6.0331 -l.8981 .0000 .0000 .0000 ”4.1888 .0000 .0000 4.1888 .0000 .0000 -4.2676 .0000 .0000 4.1423 .0000 .0000 ~4.3479 .0000 .0000 4.0940 3.9950 3.9950 .0000 —4.l335 ~4.1335 .0000 .0000 .0000 ~4.4969 .0000 .0000 4.0066 .0000 .0000 -4.5820 .0000 .0000 3.9530 .0000 .0000 -4.8303 .0000 .0000 3.7963 74 s . -5 1 i i? ~ '9 e (3.7 I§C§ “" v - 7 '7”’ ’ ’1“ '?1 1 {7 0 37'\' J? . fi 3 7’03 ~54 a. f'f 733.3 . 956‘...) - Q; o i: 5: 27:3 :L g L’ 0 .5 Cw? . 325.2 3.. :- . 8 962 “9.0739 *2.0489 1.9639 3.5079 1.8962 “2.3261 4.6522 3.6079 3.8079 -.-3.':;}115”.5 ~400739 "4.0739 3.8079 *2.3261 1.8962 ~4.0739 1.9559 “2.0489 3.8079 1.8962 ‘2.3261 ‘4.0739 ~290489 1.9559 4.0 4.0 4.0 “1.2499 *5.3477 “2.6969 1.6350 4.8365 2.4030 ~5.3477 *1.2499 *2.6969 4.8365 1.6350 2.4030 “2.6969 “2.6969 2.4998 2.4030 2.4030 “3.2701 1.6353 4.8365 2.4030 “1.2499 —5a3477 “2.6969 4.8365 1.6350 2.4030 "5.3477 *1.2499 ’2.6969 2.4030 2.4030 “3.2701 “2.6969 ~2.6969 2.4998 4.3 4.0 4.0 .0000 -3.9892 "3.9892 .0000 3.4234 3.4234 ~3.9892 .0000 *3.Q892 3.4234 .0000 3.4234 "3.98923, ~3.9392 .0000 3.42334 3.42351 .0000 00000 3.4234 3.4234 .0000 ~3c95‘392 -‘ .13.".5‘t‘f9‘2 3.4234 .0000 3.4234 “3.9892 .0000 - .3.’:.45':$'_.32 3.4234 3.4234 .0000 ‘3.9892 “3.989? .0600 6.0 .0 .0 “7.5855 .0000 .0000 9.7125 .0000 .0000 .3000 3.7928 .0000 .0000 ~4.8563 .0000 .0000 00000 3.7928 .0000 .0000 ‘9.8563 9.7125 .0000 .0000 “7.5855 .0000 .0000 .0000 -4.8563 .0000 .0000 3.7928 .0000 .00 I .0000 “4.8563 .0000 .0000 3.7928 6.0 2.3 .0 *L’.366{3 “3.5774 .0000 8.3840 3.2191 .0000 “3.5774 2.6362 .0000 \.2191 "3.4349 .0000 .0000 00000 3.7306 .0000 .0000 “4.9492 8.3840 3.2191 .0000 “6.36158 23.5774; .-’Ji’j0£) 3.2191 “3.4349 .0000 “3.5774 2.6362 .0000 .0000 .0000 "4.9492 .0000 .0000 3.7303 0.0 4.0 .0 *3.8684 "5.4057 .0000 5.5654 4.7174 0&0? ~5.4057 .3196 .0000 4.717“ ‘03452 '"NV .0000 00000 3.5487 .0000 .0000 5.5654 4.7174 .0000 *3.8684 ~5.4067 9.7174 ".3452 .0000 ”5.4057 .3196 00000 00000 ”5.2203 .0000 .0000 0 0 6.0 .0 *1.6304 ~5.66d7 .0000 2.8235 4.7062 *5.6687 “1.6304 .0000 4.7062 3.6235 .0000 .0000 3.2607 .0000 .0000 2.8235 4.7062 .0000 “1.6304 ~5.6687 4.7002 2.8235 .0000 ~'5.6687 ~1.6304 .0000 00000 “5.6470 .0000 .0000 6.0 0.0 m . D 8.C 2.0 2.0 ~5.3783 —3.2441 -3.2441 7.2825 2.8722 2.8722 .0 2.0 ~3.2918 —5.0100 -2.5376 4.8659 4.2950 2.1256 .0 2.0 -1.3530 -5.3649 ~1.8744 2.4071 4.3640 1.3973 .0 4.0 -1.9985 -4.1145 ~4.1145 3.2131 3.3207 3.3207 6.0 4.0 -.6987 -4.6331 -3.1961 1.3453 3.5179 2.2739 6.0 6.0 .0000 ~3.8120 —3.8120 .0000 2.5098 2.5098 .0 .0 -7.0381 .0000 .0000 10.6263 .0000 .0000 ~3.2441 2.6891 “1.1128 2.8722 *3.6412 .9371 -5.0100 .5374 -1.7308 4.2950 -.7111 1.3919 ~5.3649 ‘1.3530 -1.8744 4.3640 2.4071 1.3973 ~4.1145 .9993 -2.8187 3.3207 -1.6065 2.1643 -4.6331 -.6987 *3.1961 3.5179 1.3453 2.2739 -3.8120 .0000 -3.8120 2.5098 .0000 2.5098 .0000 3.5190 .0000 .0000 “5.3131 .0000 75 —3.2441 ‘1.1128 2.6891 2.8722 .9371 ~3.6412 -2.5376 -1.7308 2.7545 2.1256 1.3919 -4.1546 —1.8744 -l.8744 2.7061 1.3973 1.3973 —4.8142 —4.1145 -2.8187 .9993 3.3207 2.1643 -1.6065 —3.l961 -3.1961 1.3975 2.2739 2.2739 ~2.6907 -3.8120 -3.8120 .0000 2.5098 2.5098 .0000 .0000 .0000 3.5190 .0000 .0000 ~5.3131 7.2825 2.8722 2.8722 ~5.3783 -3.2441 -3.2441 4.8659 4.2950 2.1256 ‘3.2918 -5.0100 “2.5376 2.4071 4.3640 1.3973 “1.3530 -5.3649 -1.8744 3.2131 3.3207 3.3207 ‘1.9985 -4.1145 -4.1145 1.3453 3.5179 2.2739 -.6987 -4.6331 -3.1961 .0000 2.5098 2.5098 .0000 ”3.8120 ‘3.8120 10.6263 .0000 .0000 -7.0381 .0000 .0000 2.8722 —3.6412 .9371 ~3.2441 2.6891 -1.1128 4.2950 ”.7111 1.3919 '-5.0100 .5374 ‘1.7308 4.3640 2.4071 1.3973 ~5.3649 -1.3530 -l.8744 3.3207 41.6065 2.1643 -4.1145 .9993 -2.8187 3.5179 1.3453 2.2739 -4.6331 -.6987 -3.1961 2.5098 .0000 2.5098 -3.8120 .0000 ~3.8120 .0000 -5.3131 .0000 .0000 3.5190 .0000 2.8722 .9371 -3.6412 -3.2441 -1.1128 2.6891 2.1256 1.3919 -4.1548 “2.5376 “1.7308 2.7545 1.3973 1.3973 ~4.8142 ‘1.8744 ‘1.8744 2.7061 3.3207 2.1643 -1.6065 -4.1145 —2.8187 .9993 2.2739 2.2739 -2.6907 -3.1961 -3.1961 1.3975 2.5098 2.5098 .0000 -3.8120 -3.8120 .0000 .0000 .0000 -5.3131 .0000 .0000 3.5190 ”‘1 . 3.0 (I) . 0 CD 9 C) 8.0 4.0 .0 "4.6463 ~4.4649 .0000 7.8509 3.6403 .COQO é.-: .c "2.7837 ~5.1673 .0000 5.5382 4.0132 .0000 -‘ ~2.5277 «R . ”39377 9.0941 2.0821 2.0821 4.0 2.0 ~4.2049 ~4.2489 ~2ol641 7.2992 3.3913 1 .615685 5.0 2.0 -2.5199 ~4.9694 *1.7613 5.1487 3.7672 1.1844 '2.6801 2.8838 .0000 2.2513 “4.4036 .0000 ”4.4649 1.4155 .0000 3.6403 "2.1335 .0000 “5.1673 ~.1057 .0000 4.0132 .6548 .0000 “5.0866 “1.2238 .0000 3.7007 3.4018 .0000 -2.5277 2.8524 ".6898 2.0821 “4 .5471 .4837 -4.2489 1.4655 “1.1744 3.3913 “2.3533 .7738 ~4.9694 .0037 “1.4013 3.7672 .3848 .8374 76 .0000 .0000 3.4457 .0000 .0000 -5.4162 .0000 .0000 3.2308 .0000 .0000 ”5.7174 .0000 .0000 2.8894 .0000 .0000 “6.1929 .0000 .0000 2.4476 .0000 .0000 ~6.8037 "2.5277 ".8898 2.8524 2.0821 .4837 "4.5471 “2.1641 “1.1744 2.7394 1.6685 .7756 -4 . 9459 ”1.7613 “1.4013 2.5162 1.1844 .8374 ~5.5335 9.8198 2.2513 .0000 “6.3295 “2.5801 .0000 7.8509 3.6403 .0000 *4.6463 *4 . 464 C) .0000 5.5382 4.0132 .0000 ”2.7837 “5.1673 .0000 3.4018 3.7007 .0000 “1.2238 “5.0866 .0000 9.0941 2.0821 2.0821 “5.7047 ~2.5277 *2.5277 7.2992 3.3913 1.6685 “4.2049 ”4.2489 "2.1641 5.1487 3.7672 1.1844 ”2.5199 "4 . 9694 ”1.7613 2.2513 “4.4036 .0000 ~2.6807 2.8838 .0000 3.6403 -2.1335 .0000 ”4.4649 1.4155 .0000 4.0132 .6548 .0000 -5.1673 ”.1057 .0000 3.7007 3.4018 .0000 ~5.0866 ~1.2238 .0000 2.0821 *4.5471 .4837 *295277 2.8524 -.6898 3.3913 ~2.3533 .7756 «4.2489 1.4655 ~1.1744 3.7672 n 384.“ .8374 ”4.9694 .0037 "1.4313 .0000 .0000 *5.4162 .0000 .0000 3.4457 .0000 .0000 -5.7174 .0000 .0000 3.2308 .0000 .0000 ”6.1929 .0000 .0000 2.8894 .0000 .0000 “6.8037 .0000 .0000 2.4476 2.0821 .4837 —4.E’)471 —2.5277 -.éd98 2.8524 1 . 6685‘» .7756 “4.9459 *2.1641 "1.1744 2.7394 1.1844 .8374 ~5.533: ”1.7613 ”1.4013 2.5162 Qmmr-u 8.0 8.0 8.0 8.0 8.0 2.0 ~1.0898 -4.9395 -1.4288 3.1215 3.4931 .7428 4.0 4.0 -3.1189 —3.7201 -3.7201 5.8903 2.7640 2.7640 6.0 4.0 ~1.8550 ~4g4710 —3.1116 4.1151 3.1253 1.9944 8.0 4.0 *.7498 -4.5611 -2.5914 2.3505 2.9333 1.2572 6.0 6.0 -1.0559 —3.8729 —3.8729 2.7287 2.2912 2.2912 8.0 6.0 -.3433 -4.o932 -3.3421 1.2524 2.1650 1.4399 8.0 8.0 .0000 —3.6695 ~3.6695 .0000 1.3291 1.3291 -4.9395 “1.0898 ‘1.4288 3.4931 3.1215 .7428 -3.7201 1.5595 -2.0391 2.7640 -2.9452 1.2666 -4.4710 .2627 ~2.4896 3.1253 -.3462 1.3955 -4.5611 -.7498 -2.5914 2.9333 2.3505 1.2572 -3.8729 .5279 -3.1266 2.2912 -1.3644 1.5725 -4.0932 -.3433 -3.3421 2.1650 1.2524 1.4399 -3.6695 .0000 -3.6695 1.3291 .0000 1.3291 77 -l.4288 -l.4288 2.1795 .7428 .7428 -6.2430 “3.7201 ~2.0391 1.5595 2.7640 1.2666 ”2.9452 -3.1116 -2.4896 1.5923 1.9944 1.3955 ~3.7689 -2.5914 -2.5914 1.4996 1.2572 1.2572 *4.7009 -3.8729 -3.1266 .5279 2.2912 1.5725 ‘1.3644 -3.3421 -3.3421 .6867 1.4399 1.4399 ”2.5049 -3.6695 -3.6695 .0000 1.3291 1.3291 .0000 3.1215 3.4931 .7428 -1.0898 -4.9395 -l.4288 5.8903 2.7640 2.7640 43.1189 ~3.7201 -3.7201 4.1151 3.1253 1.9944 *1.8550 “4.4710 -3.1116 2.3505 2.9333 1.2572 -.7498 -4.5611 *2.5914 2.7287 2.2912 2.2912 -1.0559 -3.8729 -3.8729 1.2524 2.1650 1.4399 ”.3433 -4.0932 ~3.3421 .0000 1.3291 1.3291 .0000 ”3.6695 -3.6695 3.4931 3.1215 .7428 -4.9395 -l.0898 -1.4288 2.7640 -2.9452 1.2666 -3.7201 1.5595 ~2.0391 3.1253 -.3462 1.3955 ~4g4710 .2627 -2.4896 2.9333 2.3505 1.2572 -4.5611 -.7498 -2.5914 2.2912 -1.3644 1.5725 ~3.8729 .5279 -3.1266 2.1650 1.2524 1.4399 -4go932 *.3433 -3.3421 1.3291 .0000 1.3291 -3.6695 .0000 -3.6695 .7428 .7428 -6.2430 ”1.4288 -1.4288 2.1795 2.7640 1.2666 -2.9452 -3.7201 -2.0391 1.5595 1.9944 1.3955 -3.7689 -3.1116 -2.4896 1.5923 1.2572 1.2572 -4.7009 -2.5914 -2.5914 1.4996 2.2912 1.5725 -1.3644 -3.8729 -3.1266 .5279 1.4399 1.4399 -2.5049 -3.3421 -3.3421 .6867 1.3291 1.3291 .0000 '3.6695 “3.6695 .0000 t ' f '0‘ 3 ..._I‘ C .." .7- 'j “ 1., 1 ';J .0000 .0000 11.6497 .0000 .0000 u 1 ,\ f. 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Q 5- m 0 0 17.0 17.0 17.0 7.0 5.0 “1.7292 «2.2608 “1.7840 9.9527 .9339 .5218 9.0 5.0 ~.8407 ~2.421O “1.7401 8.7259 .8414 .1786 7.0 7.0 ~1.0537 ~2.2108 —2.2108 8.9538 .5335 .5335 1.0 1.0 “4.4649 “.2744 “.2744. 14.5462 .1614 .1614 3.0 1.0 ~4.0679 “.7828 “.2683 14.0854 .4517 .1452 5.0 1.0 “3.3405 “1.1816 “.2583 13.2360 .6562 .1150 7.0 1.0 «2.3945 “1.4293 “.2480 12.1208 .7498 .0749 “2.2608 1.0070 “2.0711 .9039 “3.5319 “.4846 -2.4210 .8399 -2.3489 .8414 “1.2129 «.6821 “2.2108 .5268 “2.6750 .5335 “4.4769 “.7603 «.2744 2.2324 ~-.O732 .1614 -7.2631 —.0171 “.7828 2.0875 -.2138 .4517 “8.5072 —.0534 “1.1816 1.8513 —.3372 .6562 -5.0661 ~.0942 “1.4293 1.6074 “.4332 .7498 “3.1109 “.1382 107 “1.7840 “2.0711 .7222 .5218 “.4846 “6.4208 ~1.7401 ~2.3aao .0000 .1786 —.6821 -7.5130 “2.21C8 “2.6750 .5268 .5335 “.7603 ~4.4769 “.2744 “.0738 2.2324 .1614 “.0171 “7.2731 “.2683 “.2138 1.9884 .1452 “.0534 “7.5781 “.2583 —.3372 1.4892 .1150 —.0942 —8.1698 “.2480 “.4332 .7871 .0749 “.1382 “9.0099 9.9527 .9039 .5218 “1.7292 “2.2608 “1.7840 8.7259 .8414 .1786 “.8407 “2.4210 “1.7401 8.9538 .5335 .5335 “1.0537 -2.2108 “2.2108 14.5462 .1614 .1614 “4.4649 “.2744 “.2744 14.0854 .4517 .1452 “4.0679 “.7828 “.2683 13.2360 .6562 .1150 “3.3405 “1.1816 “.2583 12.1208 .7498 .0749 “2.3945 “1.4293 “.2480 .9039 “3.5319 “.4846 “2.2608 1.0070 “2.0711 .8414 “1.2129 “.6821 “2.4210 .8399 “2.3489 .5335 “4.4769 “.7603 -2.2108 .5268 ~2.6750 .1614 “7.2731 “.0171 “.2744 2.2324 “.0732 .4517 -6.5072 -.0534 “.7828 2.0875 -.2138 .6562 “5.0661 “.0942 “1.1816 1.8513 “.3372 .7498 “3.1109 “.1382 “1.4293 1.6074 “.4332 .5218 “.4846 “6.4208 “1.7840 “2.0711 .7222 .1786 “.6821 “7.5130 “1.7401 “2.3489 .0000 .5335 “.7603 “4.4769 “2.2108 “2.6750 .5268 .1614 “.0171 “7.2731 “.2744 “.0732 2.2324 .1452 “.0534 “7.5781 “.2683 “.2138 1.9804 .1150 “.0942 “8.1698 “.2583 “.3372 1.4892 .0749 “.1382 “9.0099 “.2480 “.4332 .7871 9.0 1.0 “1.3568 ~1.5150 “.2408 10.8820 .7380 .0291 17.011.0 1.0 “.3434 -1.4508 “.2387 9.6546 .6475 “.0182 17.C 3.0 3.0 “3.6931 “.7668 “.7668 13.6445 .4054 .4054 17.0 5.0 3.0 -3.0054 -1.1618 “.7412 12.8303 .5860 .3190 17.0 7.0 3.0 ~2.1097 -1.4128 “.7158 11.7585 .6638 .2031 17.0 9.0 3.0 -1.1252 “1.5074 “.7003 10.5641 .6447 .0696 17.0 5.0 5.0 ~2.3895 -1.1315 “1.1315 12.0786 .4541 .4541 3.4 ‘- J O L) “1.5150 1.4362 -.4928 .7380 “.8441 “.1794 -1.4508 1.3854 —.5085 .6475 1.5151 -.2082 “.7668 1.8465 “.6249 .4054 “6.8222 -.1661 -1.1618 1.6315 —.9876 .5860 -5.3962 -.2920 “1.4128 1.4184 “1.2729 .6638 -3.4531 —.4274 “1.5074 1.2876 “1.4542 .6447 “1.1875 —.5547 “1.1315 1.1947 “1.5677 .4541 “6.0393 “.5113 108 “.2408 “.4928 “.0794 .0291 “.1794 “10.0379 -.2387 -.5085 -1.0420 -.0182 -.2082 ~11.1697 “.7668 “.6249 1.8465 .4054 “.1661 “6.8222 —.7412 -.9876 1.3739 .3190 -.2920 —7.4341 -.7158 -1.2729 .6913 .2031 -.4274 -8.3054 “.7003 “1.4542 “.1624 .0696 “.5547 “9.3767 “1.1315 “1.5677 1.1947 .4541 “.5113 “6.0393 10.8820 .7380 .0291 “1.3568 “1.5150 “.2408 9.6546 .6475 “.0182 “.3434 “1.4508 “.2387 13.6445 .4054 .4054 “3.6931 “.7668 “.7668 12.8303 .5860 .3190 “3.0054 “1.1618 ~.7412 11.7585 .6638 .2031 “2.1097 “1.4128 “.7158 10.5641 .6447 .0696 “1.1252 “1.5074 “.7003 12.0786 .4541 .4541 -2.3895 “1.1315 -1.1315 .7380 “.8441 “.1794 “1.5150 1.4362 “.4928 .6475 1.5151 “.2082 “1.4508 1.3854 “.5085 .4054 -6.8222 -.1661 “.7668 1.8465 “.6249 .5860 —5.3962 “.2920 -1.1618 1.6315 “.9876 .6638 -3.4531 “.4274 -1.4128 1.4184 -1.2729 .6447 -1.1875 “.5547 “1.5074 1.2876 “1.4542 .4541 -6.0393 “.5113 -1.1315 1.1947 —1.5677 .0291 “.1794 “10.0379 “.2408 “.4928 “.0794 -.0182 “.2082 ~11.1697 “.2387 “.5085 -1.0420 .4054 -.1661 -6.8222 “.7668 “.6249 1.8465 .3190 -.2920 “7.4341 “.7412 “.9876 1.3739 .2031 “.4274 “8.3054 “.7158 “1.2729 .6913 .0696 “.5547 “9.3767 “.7003 “1.4542 “.1624 .4541 “.5113 -6.0393 -1.1315 -1.5677 1.1947 17. O 19.0 19.0 (4 0 I O 1?).0 19.3 7.0 5.0 “1.5849 “1.3909 “1.1053 11.0841 .5003 .2745 1.0 1.0 —4.2642 “.0908 “.0908 14.8893 .0516 .0516 3.0 1.0 “3.8912 “.2596 “.0891 14.4654 .1448 .0464 5.0 1.0 ~3.2016 “.3934 “.0863 13.6789 .2111 .0365 7.0 1.0 “2.2927 “.4782 “.0836 12.6364 .2423 .0233 9.0 1.0 “1.2799 “.5096 “.0820 11.4662 .2396 .0080 3.0 3.0 “3.5359 “.2552 “.2552 14.0584 .1297 .1297 “1.3909 1.0345 “2.0336 .5003 “4.1234 “.7470 “.0998 2.1321 “.0712 .0516 “7.4446 “.0226 “.2596 2.0212 “.2086 .1448 “6.7070 “.0690 “.3934 1.8424 “.3307 .2111 “5.3113 “.1179 -.4782 1.6645 -.4277 .2423 “3.4028 “.1673 —.5096 1.5550 -.4903 .2396 “1.1710 -.2110 -.2552 1.7680 -.6113 .1297 “7.0292 “.2108 109 “1.1053 “2.0336 .5504 .2745 “.7470 “6.9607 “.0908 “.0712 2.1321 .0516 “.0226 “7.4446 “.0891 “.2086 1.8701 .0464 “.0690 “7.7584 “.0863 “.3307 1.3592 .0365 “.1179 “8.3676 “.0836 “.4277 .6283 .0233 “.1673 -9.2337 “.0820 “.4903 “.2750 .0080 “.2110 10.2952 “.2552 “.6113 1.7680 .1297 “.2108 “7.0292 11.0841 .5003 .2745 “1.5849 -1.3909 -1.1053 14.8893 .0516 .0516 “4.2642 “.0908 “.0908 14.4654 .1448 .0464 -3.8912 “.2596 -.0891 13.6789 .2111 .0365 “3.2016 “.3934 “.0863 12.6364 .2423 .0233 “2.2927 “.4782 “00836 11.4662 .2396 .0080 “1.2799 “.5096 “.0820 14.0584 .1297 .1297 “3.5359 “.2552 “.2552 .5003 “4.1234 “.7470 “1.3909 1.0345 “2.0336 .0516 “7.4446 “.0226 “.0908 2.1321 “.0712 .1448 “6.7070 “.0690 “.2596 2.0212 “.2086 .2111 “5.3113 “.1179 “.3934 1.8424 “.3307 .2423 “3.4028 “.1673 “.4782 1.6645 “.4277 .2396 “1.1710 “.2110 “.5096 1.5550 “.4903 .1297 “7.0292 “.2108 “.2552 1.7680 “.6113 .2745 “.7470 “6.9607 “1.1053 “2.0336 .5504 .0516 “.0226 “7.4446 “.0908 “.0712 2.1321 .0464 “.0690 “7.7584 “.0891 “.2086 1.8701 .0365 “.1179 “8.3676 “.0863 “.3307 1.3592 .0233 “.1673 “9.2337 “.0836 “.4277 .6283 .0080 “.2110 “10.2952 “.0820 “.4903 “.2750 .1297 “.2108 —7.0292 “.2552 “.6113 1.7680 L37 19.0 5.0 3.0 “2.8783 “.3881 “.2483 13.3023 .1880 .1014 19.0 7.0 3.0 “2.0105 “.4742 “.2418 12.2986 .2138 .0630 19.0 5.0 5.0 “2.2791 “.3803 “.3803 12.6013 .1444 .1444 “.3881 1.6069 “.9713 .1880 “5.6462 “.3602 “.4742 1.4564 “1.2604 .2138 “3.7476 “.5118 “.3803 1.1395 “1.5498 .1444 “6.3007 “.6167 110 “.2483 “.9713 1.2714 .1014 “.3602 “7.6561 “.2418 “1.2604 .5541 .0630 “.5118 “8.5509 “.3803 “1.5498 1.1395 .1444 “.6167 “6.3007 13.3023 .1880 .1014 “2.8783 “.3881 “.2483 12.2986 .2138 .0630 “2.0105 “.4742 “.2418 12.6013 .1444 .1444 “2.2791 “.3803 “.3803 .1880 “5.6462 “.3602 “.3881 1.6069 “.9713 .2138 «3.7476 -.5118 ~.4742 1.4564 ‘1.2604 .1444 “6.3007 “.6167 “.3803 1.1395 “1.5498 .1014 “.3602 “7.6561 “.2483 “.9713 1.2714 .0630 “.5118 “8.5509 “.2418 “1.2604 .5541 .1444 “.6167 “6.3007 “.3803 “1.5498 1.1395 “Ala! A 111 Appendix B 'Table’IX Z /10, py/IO, P /10) of first Brillouin zone and their weights. X Wavevectors for subdivision (p WEIGHT WAVEVECTOR WEIGHT WAVEVECTOR 24 24 24 12 12 12 10 10 10 10 24 12 48 24 24 24 48 24 24 48 24 24 24 24 24 12 24 24 12 24 48 24 24 24 48 24 24 12 24 48 12 r: ”an - V .- . H.i1|'..!lpl\.a. IIDILi EEufv’sv L I. . ..