Mahanty, J

Sharma, Ram Kishore

English

IACS Institutional Repository

DISCRETE FREQUENCIES IN A LATTICE PERTURBED BY ISOTOPE DEFECTSJ. MAHANTY a n d  KAM ICISHORE SHARMAV llV S K 'S  D lO ’AKTAlliN'J’. UMlVKJIM'J V, ( 'jIANJJIU M ill-.‘I\iwll 2 1 , 1DG2 )14ABSTR4GT. An invi\siiifutinn is inaili’ <if Uii' j^kum’hs ol giMunutioa of diHoirlt' vibra- iion fri'qiKuini'b ouisiclr lJu* ullimi'd inujiK’ju'y liumb nl a l.iUicp Avlien li is pi'i'llU'lii'cl by tlu' oluuigi) of llin muss of an atom, A i nli'im n for mojialomu’ lailicuH in dosciibod wliu-li rimbloH one to jiii'du’l. wliollu'r in tlio ]jort-url)i'd of Uin liiltnr, Iho cb.scJ'cU' fVrqui'ncy gojU'mLod from n,jiy braiicb is ijidcpi'julmt of fbo otfior Inimclu'S. Kmally, flic diHLTi'>U' froqiu'nf'ii's in Uio caso of Ni'vi'iid linoni lal.liir modi'fs al■»^  ovaliuiti d1N T K O i)  U (f T 11) NThe problem of f^enertition of (h,sovetc vibrjitiontil frequontueH due to irtoiopc rleJceis m n liittiee bus becMi Kluduid by LilHhitz (Hl-tihi, 1013b, 1014, 1056), unci by Moiitroll and J^ oti s (I0b5, 1050). The gcnioi-al tediniqucs developed by them have been iqfplied mostly to ti liittieo model m whieli there is no eouj)hng between the various eom[>oiients of the displacements of the atoms. Uecontly, Nardolli (1000) has studied the problem in the case of cubic lattices. The object of this paper is lo discuss a few lattice models in wJiich tJiere is coupling between the ibs])Iacements; hut still it is [lossiblc to evaluate the discrete frequencies due to ail isotope defect exactly.T I\ ir. S K C 11 Ja A K t: Q U A T T 0 N F () l i  T II F 111 iS U l i  E T E F l i  t: Q LT F N C yWe will adopt here the notation of Lifshitz (1043a) with shglit ihauges, The lattice under consideration lias N unit cells in each of Avhich there arc p atoms, the mass of the .s‘-th atom being ni,. Tlic coordinate of the unit cell is given hy the vector ty, Avhose dimension equals the diiiiensjon of the lattice. Periodic boundary conditions AVill be used throngliout.The eigenfrequencies of the unperturbed lattice are th(^  roots ol the equation| i - A o ^ l - 0  (1)Avhero the dynamical matrix A is detcmiiiied from the equations of motion of the atoms in the lattice. Kach element of A has six indices, three for the row and127128 J .  Mahanty awl Ram Kishore Sharmathroe for the column. Tho following equation gives the correspondence between this notation and that of Born and Huang (1954);y'nism.r ' >(2)The index j  l efcrs it> the component of tho djsplacement of tho atoms.In the presence of an isotojv  ^ defect of mass m / a,t the s-th position in tho ry-tli unit cell, the eigenlVecpiencms are the vool.s of the equationl / d G V l A l  ---0 (3)wlicre C/((d) -- {A aiid /\ is the pej-tiirliation m,atj-ix. In this case Aa diagonal niatrix wliose elements are given bywhoi'o 6 , ^ 1 -  .(4)In a throe dimensional latt ice, Ibr examjile, there aviII lie only three nonvaiiish-Aiiig elements in A> diagonal elements cJiara.ctorised by tlwindices ry, <s‘, and the three values of y1’he determinant in equation (!3) eaii be ri'-dnoed to tlie form1 -hc,co^ty„(.s, L I .y, 1) 11 y,2) s, | y,3)I -s, 1) 1H - I  y,2) e,oiV;„(y,2 | s,3)i I .s-d) I ,s-2) lH-r.,o)‘^ (;„(.sv3 |y,3)... (5)Tlio evaluation of the elements of (5 matrix whiiih oceur in equation (5) ean bo (lone by the bilinear formula for the matrix elements of a function of an operator.The eigenvectors of A span a space of 3pN diiiumsions, and eonstitute an orthog(Jual set. (The degenei’atc eigenvectors can be orthogomibscd in tho usual way). For convenience we will use the symbol | A’, r >  foi’ a typical normalised eigenvectorof A, wliei’e k can take up N  values inside the unit coll of the reciprocal lattice, of volume (27r)“/iV, and r can take up 3yi values.Each eigenvector of d  has 3yjJV components. We will denote a typical oompo- iient by the symbol | A:, r; g', y,y>.  These eompoiioiits arc known to be of the form (Lifshitz, 1943a)1 r; q, .v, j >  ^   ^ Qt^ ’W  ff ) (6)Discrete Frequencies in a Lattice Perturbed by Isotope^ etc, 1 2 9where Q/> {^k) is the (s, j)-th coiuponent of the r-th oigeiweetoi* of a Her-mitean lyiatrix aj^  which is related to A by the equationa I y) =  ^ i  I y) k. q ) (7)The eigenvalues ol are the 'Ap branch (scpiared) frequencies to, (/*) loi‘ a given value ol Ic. The branch index r can take up 3p values, in each branch there arc N  frequencies coiTcsponding to the N  values of fc.liy the bilinear formula, the nijiti ix element of G isClq-q>{^J\«\y)</i:, r, q\ ti\ y  \k,r\ f/, .s, j >  k,r t^0r^k)-~to^_  I V  f d} k^Q/' {^h)QS '^'-i\k) exp [2ni Ic. {g-^q')](27t)“ *r J fOr“(/<:)W® (8)In this, tlie suinniation over k has been replaced by an integration, V* is the volume of the unit cell of tlie jecijU'ocaJ lattice*.In ))articular, the elements of 0  that occur in equiition (5)aro given by Go(*. U « . J ) -  ^ 2- -  • • ■Actual comiiutation of the discrete frequencies has been done in those lattice models in which there is no coupling between the various components of the dis­placements of the atoms, since the evaluation of the matrix elements of G is easy for such models. We will now prove a. theoicm thiit if a ccT'tain ciltcrion is satisfied in a lattice model, an isoto]»e defoed perturbs each froquency branchseparately, and evaluation of the matrix elements of G is soincA\hat sinqjler even in the presence of coupling between the dispiacemeiit components.Only inonatomie lattices can satisfy this criterion. We null evaluate the dis- ci'ete frequeniiioH in certain linear lattici  ^ models satisfying this criterion, and it will be shown that when this criterion is satisfied, the range of interaction between♦111 tins equation the vector k is normalised, in  sueli a way th a t the volume V * equals(2tt)w.2130 J» Mahanty and Ram Kishore Sharmaiiic atoms IS (j1 no (joiiscquenuo, as I'ar as the solvability of the problem is concerned,T I T  1C C n i T E l l I O N  K O l t  T I T E  U N  M  J X  I N  O  O K  T H E  J MC H T  U K  n A T  I O N  J N  D I K K E K E N T  H K  A N  (J H  1C NJjct US consider a monatomic lattice model {p — 1), in vdiich each atom hastliT('e (hi^recs of freedom, and in udiich the elements A satisfy the equation-*]"{'} I J/ ) ~  1j/ ) ■■■ (^ 0)where a (j'j") is a, nuinboj' independent of q.The dynainicaJ inatnees (!orTi;spoiiding to the various branches are obtained by a similarity ti'anslbrmationP a P - ^ ^ b  ... (11)whore il is a matrix ol the formBX  0  O b  Y 6  b  6  7j(la)Here X, Y, and Z arc N x N  submatrices.In Ihis model it is easily shown that the elements of P  are Pg/_Qw(.  ^ l_y ) — /J(j 1^  )andX ^ x GY  =  yC _Z ^ z G(13)(14)where x, y, ;ind z arc pure numbers and G is a N x N  cyclic matrix, x, y and z are, in fact, the eigenvalues of tlie 3 x 3  matrix a  whose elements are <x{j' Using equation (11) it can be shoAvn that the 3 x 3  matrix fi w^ hose elements are /j(j'|j") diagonulises a by tlie similarity transformationa Ji-~'X 0 o'o y 0_0 0 3.(15)Tho woliknown symnicliy properties ol\4 recpiire ilml a must lie a syiimiotrie matrix, so that fi must be an rirtliogonal matrix satisfying tlie condition.^ W |j') -  ('«)jThe matrix for isotope defects i*,an bo wTitten in the formDiscrete Frequencies in a Lattice Perturbed by Isolojie, etc. 131A "-S 0 0b  h 0  b b s(17)wJicre S IS n N \ N  diagonal subinatrix. Using ccpiation ( 10 ) and the faid- thatS commutes with each of the submatrices of P (wnich iirc diagonal), it is easily shown that for this modelP A P-' -  A -  ( 1 « )Therelore, if we perform a similarity transformation A^ uth respei!!- to P  on equation (8), we obtain\ T+G(x)S\ |/+(7(y/)§| 0where all matrices are now N xN,nrn\d { x )0[y) =  [z/U-/o)2]-iG{z) -  [=C'™fc>2)-i( 19)... (20)From equation (19) it is apparent that in Ji, monatomic lattice satisfying the criterion of equation (1 0 ), the effect of isotope defects on each frequency branch is independent of the other branches,With a single isotope defect, equation (19) reduces to the three equations1 +f dVc _  ^(271)3 “... (2 1 a)cto® f __ Q(27T)3^  ^to\(ifc) —(0®... (2 1 h)eti)3 f d^ h _ ... (2 1 c)(27t)“ J, a%(k)~bi‘132 J . Mahauty and Ram Kishore SharmaAn. exmn.pl(iAs a simple example of a physical lattice model satisfying the critonon of equation (LO), we consider tlie case of a. linear chain with nearest neighbour inter­actions and three degrees of freedom per atom Ff exp (^ Ci)f), exp (uo )^, and exp (i(i)t) denote the two i-ransversi* and one longitudinal eomiionents respec- tivel}'  ^ of the ?i.-th atom from cqiulilirium, the equations of motion can bo Amtteii * in the form... (22)Here y.j is tlie coupling bet wiien y/ and components, and is the coupling between u and y/i or between v and components respectively.Tlie matrix a in this case is of the form7 i 7a 7am m m 1a •— 72 7 i 7_am m m7a 7:i 7 i__ m m m —and the N x N  matrix 0  is of the form2 - J 0 0 ... 0 - 1- J  2 - 1 0  ... 0 0C - 2 - 1 0 0(23)(25)The three eigenvalues of a, which we denote as before by x, y, and z, are easily obtained and are given by the equations» =  ( r i -r n )Discrete Frequencies in a Lattice Pert/m'bed by Isotope, etc. 138y 7i) -1- A/(ri 1-7l -r4)"+ «7;i“]J_2m [(7 1 + 7 2 -1-74 ) -V (7 i  H-7a-74)“ |-«7a^ T(25)The eigenlVequeiK’ieH in the vnrions blanches as functions of /.:(wl\uh is a. one (liniensjonal vector in this ca-so) is obtamod iiy talciiij^  the Foniicr transforni oFB  in accordance with the formula of equation (7);— 2.'t( I ■ COS k)- -  2 :i/(J-cos/^)(.)“# )  -  2s(l -<‘os/0 ... (2(5)Wc will discuss the eejuation foj‘ the discrete frequency ^^ enora.torl from one of the branches only. Gt|mition (2 1 a) ran be written in the following form by substituting for the denominator of the integrand IVoin equation (2 b);S!;r dk1 1  f ___  ______ ^  027T J 2;r(l —COS/c)-to“(27)For discrete, frequmuaes Ave are interested in the region c.i- >  Avhere— hr, the maximum frequency of that brancli. In this region erjuation (27) reduces to the form1 - ... (28)A Holiition of this equation for m >  0 .^ 2/ exist only for positive e, i.e., for a lighter isotope defect, and this solution is•»«0 =  ~ y —  'V/l-'C-... (29)The treatment of the other two secular equations proceeds along identical lines.Thus, in this example, a single lighter isotope defect generatiis three discrete frequencies, one from each of the acoustic: branches. However, for small c the physical observable discrete frequency is the one, which is above the maximum frequeiuy of the stiffest branch, since the other two discrete freipioncios will lie submerged among the quasicontmiinm of frequencies of the stiffest branch.A L [ N E  A TR, L A T T I C E  W I  T  I f L O N  (f  I t  A N Cl E  1 N T  E R  A ( V r i  O N SIn a linear lattice with long range interactions, the liquations of motion can be wi'itton in the form (with infinite number of atoms)(30)134 J .  Mahanty and Ram Kishore Sharwuu"ith two HiniiliU' ec(u,‘UionH for the (»thor two comj)oneiitH, Fforo js thocoupling c'oriKtaiit In'twccMi tiu; /^-diKplaccTimiit of an fitoin and th<* ?;-(lisj)laccnicnt of ilK ^j-th nojglil)onj‘,A ])l)yHifiallv roasonablc aasumpfion to make on the properties of the eoiipling of)nstants is lhat they all depend on the distanee of separatifm of the interacting atoms in acuordanee with the same law. In that- case, one can write(III)where the function f{'p) suitably describes the dependence of the eouphng cons­tants on the distance of separation of the atoms.This model satisfies the criterion of cipiation (1 0 ). The matrix a will have the same form as in ('quafion (23), if we use the follosing equivalences ,y{or) __ y('f 'J)_ ~y i U j W )  „  y(W^U)  „  y(»,aa _  y(M',«) :The mati'ix (J  is uoav of the form:y ,;2S/(^-)P - / ( I ) - m- / ( I ) 2 S/(P)p - / ( ] )0  - - / (2 ) - / ( I ) 2 S /(p )i>-/(:>) - / (2 ) -./■(I)(32)Tho eigonfi'oqncnoics of the various br-mches are<0 ,^(1 :) -  [ £  f ( p ) -  £  f(p) c.osplc ] ... (33)L jj-=i 31= 1 J■with two similar equations for the other two branches. The maximum frequencyof the branch is evidently 4a: S /(2 ^ + l ) j  . As before, we will evaluatethe discrete frequency generated from one of the branches only. The secular equation is(IkI  2 * [s {/(;> )-/(;))co B p * :}j-o i*=  0 ... (34)The integral in equation (34) can bo done exactly for several assiinicd forms We give belov' tlie results for the discrete frequency arising outside the band in a few eases.Discrete Frequencies in a Lattice Perturbed by Isotope^ etc, 1350) J{p) ^  exp {-p(T).The integral in ecjuation (34) can be done exactly for w and in terms of a, variable 0 — cosli-^w/w^ ,^ ? discrete frequency is the I'oot of the equationdoth (35)Jt IS easy to sliow graphically and oth(iVwise that for vabuis of <, in the range 0 <  r <  1  a solution of tins transcemlental equation must exist, and hence a discrete frcipiency must arise. For very small values of (, the discrete frequency is... (36)(ii) f{p) -- (p)-2^; r Integer.The series oe.cumug in the denominator- of the integrand in equation (34) can be summed exactly in accordance witli the formulae (Jeffreys and Jeffreys, 1956),S (p)-2'‘ COS^ /C =-- ---- {4:77  ^ V^ iTand s  (p)-“'- =  a*>-)p 1where P.^ f and are respectively the Bernoulli xiolynomial and the Bernoulli number of order 2 r respectively, and (^2 r) is the Riemann Zeta function. (^2r) is related to h^ j. through the cijuationm  -( - 1 )^ - [4n'^ Y hrThe hist few Bernoulli polynomials are ^ Pt{x) ^  i  (:c>-2 r '+ * - ) ;1\{X) : 1720 \ 2 2 /The firwi Icmv IhirnouJli muiihors are136 J .  Mahanty and Ram Kishore Sharma' “ =  12= ' “ = - 4 =I30240 ’ 1200(500 ■The equation loi tlie diseiete ticqueiiey for r =  1 is ( witJi — --Oixi,r  , J Isiii-  ^ --W - 1  V ^A solution of this equation will always exist lor 0 <  1 , and for c -disuretc frequency is. (.‘17)1 , theFor r — 2 , the e({uation for the discrete frequency is>/ 1 V> I"V4 I , V . . 1 1, —— -  ^ siii“  ^ - =In this case also, a solulion will always exist for 0  <  t <  1 , and for the discrete fiequency is given hy— ~ ( > + '.T ).. (38).. (39) c < < l.. (40)C O N C h U 8 1 0  NIf the oonditi(m described in equation (10) does not hold for any given lattice model, the pert urbation due to th(5 isotojie defect mixes up the va,rious frequency bran(;hes, so that it may not be possililc to find a one to one correspondence between, the disiirotc frequencies and the unperturlied frequency branches. The models discussed here arc one dimensional, but there is no essential complication in the case of two and three dimensional crystals as long as they satisfy the criterion of equation (1 0 ). For three dimensional crystals the evaluation of the integrals in equation (2 () (;an lie done numerically if one knows the freijuency distribution function of the unperturbed lattice. Whether any actual crystals can be ade­quately represented by a model satistying the criterion of equation (1 0 ) must remain an open question.We have seen that the difference hetvveon the diacroto frequency and the mnxiraimi frequency of any branch depends on c^ . This property is related to the exist<mce of the square root singularity in the fi'oqiieney distribution function of a linear latti(;e. A aiinple proof of the fact that for small c one can expect the discrete frequency to depend on 6- irrespective of the details of the law of inter­action between the atoms in a linear lattice is given in the Appendix.Discrete frequencies in a Lattice Perturbed by Isotope, etc. 137A PPEN D IXWhen e is very small, a solution of equation (34) would exist only if w is slightly greater than LotP=iF{k) hrt-s a maximum a t k =  tt, and evidently=  2xF{n).In the neighbourhood of this maximiiin wo can expand F{k) in a power scries in k' — k—n. Roinombering that most of the contribution to the integral comes from the neighbourhood of k' 0, we can write the integral in equation (34) in the formdk' 7T 1 ^xF"{n)We thus got the following approximate secular equation for the discrete frequency for small e;1 - 0 - 712\/ xF"{n) V 0)2—W®a:jr^The solution of this equation isA C K  N O W L E D G M E N 1'One of the authors (R.K.S.) is indebted to the authorities of the Punjab University for the award of a research scholarship.138 J . Mahanty and Ram Kishore SharmaE  fi P E l i  E N C E SJnffroyN, I f .  mid Ji'ffrrya, B, S , J9.10, M(‘,thod.s o f MatlioJimtioul Ph y sifa , (Catnbi'jdgo Univf'rHiiy p. 988.hifHfiity,. I M., 194311, J .  Phifs. IJPSR , 7, 21o.LiMiiU, T. M., 1943b, J .  Fln/s. USSR., 1, 249.[.ifaliit/-, ].M , 1944, PhjR. USSR , 8, 89. 4, 71G.ATonI.roll, 1C, W., and PottK, K B , 19r)f), Rhys, Rev., 100, r»2.‘).Mojilroll, p:. \V. atid PoUh, R. B.. 1996, Pln/s Jipv., 102, 72.Nai’dolli, (jr V., 1960, DyiuimicH orOubic Oryaial Latiifoa Peitui'bndby a Local Chango of Maaa, (Tech Hop., Inalitutjcj of Phyaica, Univeraify of Milan).

Discrete Frequencies in a Lattice Perturbed by Isotope Defects

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Discrete Frequencies in a Lattice Perturbed by Isotope Defects

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