The frequency shift and lifetime of a localized vibration mode

ANNALS

OF PHYSICS:

The

30, 371-410

Frequency

(1964)

Shift and Vibration A. A.

Westinghouse

Research

Lifetime of a Localized * Mode

MARADUDIN

Laboratories,

Pittsburgh,

Pennsylvania

A localized vibration mode due to a light impurity atom in a crystal is an exact eigenstate of the crystal when the latter is studied in the harmonic approximation. When the anharmonic nature of the interatomic forces is taken into account, it is found that the frequency of the localized mode suffers a complex shift from its value in the harmonic approximation. The real part of this shift describes an actual change in the frequency of this mode, while the imaginary part is conventionally related to the reciprocal of the lifetime of this mode. In this paper expressions are obtained for the real and imaginary parts of the complex shift of the frequency of a localized vibration mode associated with an isotopic impurity in an arbitrary cubic Bravais host crystal. Cubic and quartic anharmonic terms in the expansion of the crystal potential energy in powers of the atomic displacements are retained in the present calculation. Our results are exact to the lowest nonvanishing order in the anharmonic force constants. To display some of the qualitative features of the formal results, the expressions obtained are evaluated in an approximate fashion in t#he high temperature limit for the special case that the host crystal is a monatomic linear chain. I. INTRODUCTION

The localized vibration modes’ associated wit’h a sufficiently light impurity atom in a crystal are exact eigenstates of the crystal Hamiltonian when only quadratic terms are retained in the expansion of the crystal potential energy in powers of t’he displacement,s of t’he atoms from their equilibrium positions, and are consequent’ly infinitely long lived. However, if one goes beyond this harmonic approximation tJo the crystal Hamiltonian, and retains the third, fourth, . . . , order terms in the atomic displacements in the expansion of the crystal potential energy, the exact eigenstates of the harmonic approximation are no longer exact eigenstates of t’he anharmonic crystal. The normal coordinate transformation which expresses the Hamilt80nian of the crystal in the harmonic approximation * This research was partially supported by the Director for Materials Sciences and was technically Scientific Research under Contract AF 49(638)-1145. I See, for example, ref. 1. 371

Advanced monitored

Research Projects by the Air Force

Agency, Office of

372

MARADUDIK

as the sum of independent harmonic oscillator Hamiltonians, each of which describes an independent normal mode of vibration and has its own characteristic frequency, in an anharmonic crystal leads to a coupling between t,he formerly independent normal modes. This coupling, which allows the exchange of energy between the normal modes of the harmonic crystal, among other effects imparts a complex shift to the frequency of each of t)hese modes. The real part, of t,his shift represents an act,ual change in the frequency of the harmonic normal mode, while the imaginary part of the shift is conventionally int’erpreted by saying that it is half of t,he reciprocal of the lifetime of the mode, or it is defined in an alternative, operational, way as half t,he width of the peak in some absorpt.ion or scattering experiment in which, for one reason or another, the dynamical properties of that mode are singled out by the nature of the experiment. These frequency shifts and widths are temperature dependent. In particular, t,he frequency of a localized mode suffers such a complex shift. The first calculation of the lifetime of a localized vibrat,ion mode was carried out by Klemens (Z), who used perturbation t.heory. He found that. the lifetime at low temperatures is of the order of 100 vibrational periods, and decreases with increasing temperat,ures. At about the same time Krivoglaz ($?a) obtained formal expressions for the shift and width of a localized mode due to cubic anharmonic interactions, but he did not make any quantitative estimates of these quanDities. Subsequent,ly, Maradudin (5) carried out a similar approximate calculation by examining the poles of t’he phonon propagator for an anharmonic Bravais crystal containing an isotopic impurity. His result,s for the lifetime of a localized mode at the absolute zero of temperature were somewhat longer than those of Klemens, and showed that the lifetime becomes infinite as the frequency of the localized mode approaches twice the maximum frequency of the harmonic crystal. This last result is a consequence of the fact that. only cubic anharmanic terms were kept in the anharmonic cont’ribution to the crystal potential energy. It was also shown (4) that the lifetime of the localized mode at a temperature of the order of the Debye t,emperature could be shorter by an order of magnitude than it is at the absolute zero of temperature. More recently Visscher (5) has presented a calculation of the lifetime of a localized mode which differs considerably from those mentioned previously. Visscher assumed t,hat the only anharmonic interatomic forces act between the impurity at’om and its nearest neighbors, alt’hough this is not an essential part of his method of calculation. Assuming a truncat’ed cubic anharmonic Hamiltonian as a perturbation, he was able to write the Hamilt.onian for an isotopic impurit’y in an anharmanic crystal in a form equivalent to t,he Hamiltonian which appears in the theory of the unstable V-particle in the Lee model of a field theory (6) and by exploiting this similarity he was able to solve the reduced problem exactly. Visscher’s approach seems to be limited t,o the absolute zero of t’emperahure, but

LOCALIZED

MODE

LIFETIME

373

his result for the localized mode lifetime is compatible with the results of Klemens and Naradudin. Other calculations of the lifetime of a localized mode have been reported by Kagan and Iosilevskii (?‘) and by hlozer (8), but the details of these calculat8ionsand their results are unknown to the author. The shift and width of localized vibrat,ion modes, have been measured as functions of temperature in the infrared opt#ical absorpt’ion spectra of ionic crystals containing small concentrations of hydride or deutcride ions as substit.utional impurities (9). It has also been suggested by several aut)hors (10) t’hat, t,hese effects should also be observed in t,he one phonon crosssection for the absorption of gamma rays by nuclei bound in a crystal. It. was therefore t,hought t,o be wort.hwhile to present, a calculation of not only the widt.h of a localized mode but, of its frequency shift as well, without, making some of the approximations and rcstrict’ive assumptions present’ in earlier treat’ments of this problem. In this paper me present a formal calculation of both the frequency shift and widt,h of a localized vibration mode associated with a sufliciently light isotope impurity at,om in a cubic Bravais crystal. The calculation is presented in the form of a calculation of the localized mode contribution to the cross section for the absorption of a gamma ray photon by the impurity nucleus which recoils and emits or absorbs one quantum of vibrat,ional energy. This approach was chosen because it’ leads to an operational definit.ion of the phonon width and shift.. The expressionswe obt,ain for the widt,h and shift. are exact to the lowest. nonvanishing order in t.he anharmonic force constants for an arbitrary cubic Bravais host crystal. Our formal results for the width and shift, are rather complicated expressionswhose numerical evaluation would not, be easy at, the present time. Therefore, in order to estimak t’he order of magnitude of t)he shift. and width, and t’o exhibit the qua1itat.k features of these results, their dependence on the frequency of the localized mode and on the anharmonic force const,ams, they are evaluated numerically in t,he high temperature limit for the special case of a monatomic linear chain. Our starting point, is the Hamiltonian for a Bravais crystal containing a single mass defect which, with no loss of generalit’y, we can assumet,o be at the origin of our coordinate system: H = Ho + HD + H,

(1.1)

where (1.2)

374

MARADUDIPU‘

In Eqs. (1.2)-( 1.4) u,(Z) and p,(Z) are the a-Cartesian components of the displacement and conjugate momentum operators for the Zth atom in the crystal. M is the mass of one of t,he host atoms and M’ is t,he mass of the impurity atom. The coefficients emb(ZZ’), @aar(ZZ’Z”), . . . , are the second order, t’hird order, . . . > at’omic force constants of the crystal, respectively. In this work we retain only the cubic and quartic anharmonic terms in t’he anharmonic Hamiltonian Ha , since they give contributions to the frequency shift which are of comparable magnitude, to the order to which we work. The nth order anharmonic term in t,he expansion of the potential energy is smaller t’han the harmonic term by approximately a factor (~/a~)“-*, where u is a root, mean square displacement amplit#ude and aa is the lattice parameter (11). This fact gives us a convenient way to order the various anharmonic contributions to any physical quantity of interest. We can write HA formally as

where 7 is an order parameter which can be set equal to unity at the end of the calculation. If we retain terms in HA up to some particular (even) order in 7, we are assured of obtaining a consistent result if we retain terms to the same order in 7 in our final answer. In what follows we work only to 0( v2). Our first step is to carry out a normal coordinate transformation which diagonalizes the total harmonic Hamiltonian Ho + HD . We write this transformation formally as (1.5) p,(Z)

= f (u>‘;’

T

(~,)~‘~Bh”‘(l)(a,

- a,+>.

(1.6)

In these equations a,+ and a, are phonon creation and destruction operators for the normal mode s of the harmonic Hamiltonian Ho + Ho . ws is the frequency of the sth normal mode. The index s takes the values 1, 2, . . . , 3N. Ml is the mass of the Zth atom. It equals M for every atom except for the one at 1 = 0, where it equals 111’. The expansion coefficient B&“‘(Z) is the (1, 0~) element of the eigenvector of the dynamical matrix of the harmonic crystal corresponding to the eigenvalue u,.~, z

D&‘)BB”‘(Z’)

= w:B?(Z)

s = 1, 2, 3, ... , 3N.

(1.7)

LOCALIZED

375

MODE LIFETIME

The elements of the dynamical matrix D are defined by

(1.8) where the {@J&U’)} are the second order atomic force constants which appear in the unperturbed part of the harmonic Hamilt80nian Eq. (1.2). The matrix D is real and symmetric in the index pairs (t)

and (‘d’>. The eigenvector com-

ponents B’,“‘(I) are real and can be made to satisfy the orthonormality closure conditions z

B:‘(z)Bh”“(l)

c BI;“(Z)BB”‘(Z’) s

and

= &,I

(1.9a)

= l&3621’.

(1.9b)

If we substitute the expansions given by Eqs. (1.5) and (1.6) into the expression for Ho + HD given by Eqs. (1.2) an (1.3) and make use of Eqs. (1.7)(1.9)) we obtain for the harmonic part of the Hamiltonian (1.10)

Hh = Ho + Hn = c fio,(a,+a, + $4). Applying tribution

the same normal coordinate transformation to the anharmonic conto the Hamiltonian, Eq. (l-4), we obtain

where the coefficients b’(sls~so)and V(ss2s3s~)are given explicitly

by

(1.12b)

They are completely symmetric in the indices s1, sz , sS, . . . . Because they appear naturally in this problem, in writing Eq. (1.11) we have introduced the operat’ors A, which are defined by A, = a, + a,’ = A,+.

(1.13)

These operators sat#isfythe commutation relaCons

[As,

A,!1= 0,

(1.14)

376

NARADUDIN

which follow from t,he commut,ation relations sat,isfied by the operators us and a,+, jl.15) From the Hamilt,onian Hh , Eq. ( 1.10), and the commutation relations ( 1.15) we can obt,ain the results (a,+a,l)h = Bssms (w&h

= &,l(% + 1)

(a&,~),~ = (a,+&

(1.16)

= 0,

where the brackets (. . .)h denote a thermodynamic average carried out in the canonical ensemble defined by the harmonic Hamiltonian HP,. The function n, is t,he mean number of phonons in t,he sth normal mode at temperat)ure T, n, = [exp /3fiw, - 11-l II.

THE

CROSS AN

SECTION IMPURITY

FOR THE NUCLEUS

/3 = (kT)-‘.

ABSORPTION BOUND IN

OF GAMMA A CRYSTAL

(1.17) RAYS

BY

Although one can think of the lifet.ime of a vibration mode of a crystal without reference to any experiment by which it may be determined, it seemsmore satisfactory to discussthis quantity wit,hin an experimental context, since ultimately it is only t,hrough the results of some experiment that the concept of a phonon lifetime acquires a concrete rather than a purely abstract significance. We have chosen, therefore, to calculate the localized mode contribution t.o the one phonon cross section for the absorption of a gamma ray by an impurit,y nucleus bound in a crystal. This is t’he cross section for those processesin which the nucleus absorbs the gamma ray and recoils, exciting or taking up one quantum of vibrational energy. We will find that the localized mode cont’ribution to this cross section is a broadened peak, whose width at half maximum can be interpreted with t,he aid of the uncertainty principle as the inverse of the lifetime of this mode. Since what would actuaIly be measuredin a solid state resonance absorption experiment of t,he type just described is t,he line shape and hence the line width, one can argue that to describe the experimental result, in t,erms of the anthropomorphic concept of a phonon lifetime is completely unnecessary. However, the concept has a picturesque appeal and we retain it for this reason. The crosssection for the absorption of a gamma ray of energy A’ by a nucleus bound in a crystal can be written in the form (1%‘) m q,(E)

=

?,5 co Y

I

m

dte

iwt--rlil

(e-i~.u’“‘t’eiK’U’l’O’)

(2.1)

LOCALIZED

MODE

377

LIFETIME

where u,, is the resonance absorption cross section for the absorbing u(Z, t) is the displacement of the impurity nucleus from its equilibrium at t,inle 1, K is t’he wave vector of the gamma ray, and we have put,

fiw = E - Eo

y = r/2fi.

nucleus, posit’iou

(\2.2)

In I$. (2.2) & is the energy difference between t’he final and initial nuclear stat’es of the absorbing nucleus, and r is the natural width of the excit’cd staic of the nucleus. The angular bracket#s in Eq. (2.1) denote an average with respect) to the canonical enselnble described by t.he Hamiltonian HI, + HA . With no loss of generality we can assume that, t,he impurity nucleus is sit,uated at. t.he origin of our coordinat,e system. Our first task is to simplify the correlatiotl function appearing in Eq. (2.1). It, is couvenieIlt t,o do this by the use of curnulant,s (13, 14). If z and y arc two nonconunuting random variables, t’he average (e-“c’) can be written as (e-g)

= (o,e-x+fJ) = exp $;

(2.3) lOA-2

+ y)“‘)e

where 0, is an operator which orders all powers of z t’o the left of all powers of g. The subscript c on the angular brackets denotes the curnulant average of the enclosed quantity. The cunmlant, average of the nth power of a single random variable, is defined by the generating function (16) (2.4) so t,hat for the first few cunlulant

averages we obtain

lx>, = lx) (x2), = (2) - (ly

(2.5)

(x3), = (x3) - 3(x")(x) + 2($. .. . Equation

(2.4) can be writ&n

conveniently

as

(et”) = exp (e’” If we write

I), .

(2.6)

out the first’ few terms in the exponent on the right side of Ey. (2.3),

(e-“e”> = exp i -(x>

+ (y) + ?4[(2”> - 2(w)

+ (!/“>I

- $&g2 - 2(x)(y) + (y)'] + . . .\,

(2.7)

we see that, two kinds of t’erm occur. In the first category are the terms which x and 1/ appear independently of each other. These are of the form

in

378

MARADUDIN (,~‘)“yx”2)“2

. . .

and (y”‘)“‘(y”“)“”

. . . . In the second category are the terms in which x and y appear together in the same term such as (zy) or (z)(y). It is clear that the terms in the first category arise from the product (eC)(eY) = exp (e-” We can therefore

write

l)C exp (e’ - l)C .

(2.8 1

the desired average as

(e-zey = (e-“)(ey> Ve”) (e-W9

(2.9)

= (e-“)(e’) exp(O,(e-”

-

l)(e”

-

l)), .

If we identify zr as i~.u(Z, t) and y as zk.u(Z, 0), and make use of the time independence of the crystal Hamiltonian, we can write the correlation function appearing in Eq. (2.1) as (e-i”.u(z’t)eix.u~z’o~ ) = eczM exp (Ot(e--iw’u(ztt) - l)(eiK’ucE’o) - l))c (2.10) where factor

Ot orders all powers

of u( 1, t) to the left of all powers e-2M

=

, (eiK.uu,o))

of u( I, 0). The

12

(2.11)

is the Debye-Wailer factor for the impurity nucleus. We will not consider it further here since it does not affect the shape of the absorption cross section in any significant way. The leading contribution to the one phonon absorption cross section is obtained if we retain only the first order terms in the correlation function in the exponent on the right side of Eq. (2.10), (e-iK.u(z,t)eiK.u(z,o)

) g e-2M exp ((~c.u(Z, t)~.u(Z, -

0))

(Kd.l(Z, t))(K*U(z,

(2.12) 0))).

The one phonon absorption cross section is obtained if we substitute Eq. (2.12) into Eq. (2.1) and expand the second exponential in Eq. (2.12) in powers of its argument. The term linear in the argument gives the desired cross section, (2.13)

In obtaining Eq. (2.13), in addition to neglecting anharmonic effects of all but the lowest order, we have neglected the three, five, seven, . . . , phonon processes in which the localized modes are excited and de-excited in all possible ways such that the net energy exchange between the impurity resonant nucleus

LOCALIZED

MODE

379

LIFETIME

and the host crystal is one quantum of t,he vibrational energy in one of the localized modes. Because the amplitudes of the atoms vibrating in a localized mode are of 0( 1) rather than of 0( K”‘) as is the case for the wavelike modes, where N is the total number of atoms in the cry&al, these multiphonon contributions sum up in the harmonic approximation to give Bessel function factors which would appear on the right side of Eq. (2.13).” The nature of the modification of t’he Bessel function fact’ors in the presence of anharmonic forces has not yet been det’ermined rigorously. However, a crude calculation based on the assumpCon that, t’he main effect of anharmonicity is to replace the simple time dependence of the st,h normal mode, ei”“’ and eiw8’ by c~~“-~“” and e--zw8t-rS’t’ respcctively, shows t’hat to a good approximation the Bessel function factors can be replaced by unity.3 We can simplify somewhat, the expression for o?‘(e) given by Eq. (2.13 j. While it is true that in the simultaneous presence of both a mass defect and anharmonic terms in the crystal potential energy the average (K.u(Z)) does not vanish in general (18)) it does vanish in the case of a Bravais crystal in which the index 1 refers to t’he site of the impurity atom itself. This is a consequence of the fact that the impurity is situated at a center of inversion symmetry. In addition, since t$he widt,h of the peak which represents t#he localized mode contribution to the one phonon absorption cross sect’ion is several orders of magnitude larger than the width y which appears under the int,egral sign in Eq. (2.13), we can neglect the latter with the introduction of a negligible error int’o our final result. Our start’ing point is t’herefore the following expression for t,he one phonon absorption cross sect’ion us(l)(E)

= a uoyeP

where we have explicitly indicated origin of our coordinate system. If we substit’ute int,o Eq. (2.14) by Eq. (1.5) it takes the form

SI dteiwt(y..u(O, that t’he impurity

t)K.u(O,O)),

(2.14

nucleus is located at t,he

the normal coordinate

decomposition

given

We denote by I?,,~(u~ the function 2 See, for example, ref. 16. 3 Dr. M. A. Krivoglaz has informed basis of similar arguments. See ref.

me that 17.

he has come

j

to the same

conclusion

on the

380

MARADUDIN

F&(W)= 1, dte’“t(A,(t)A,~(0)),

(2.16)

and turn to it)s det,ermination. III.

THE

DETERMINATION

OF

F,,/(w)

If we write out explicitly the Heisenberg operator appearing in the correlation an d write t’he thermodynamic average as a trace over the function (AS(L)A eigenstates of t’he Hamiltonian H, we can carry out the integration over t in Eq. (2.16) with the result that F.&(w)

= g c e-pEm(m ( A, 1 n)(n 1Ad ( m) x s(w + + (23, - En,). mlL

To calculate FS8’(~) we introduce

the imaginary

G,,)(u) = (TA,(u)A,,(O)),

time correlation

function (3.2)

-pSU$p

where T is the time ordering operator which orders a product of operators right t’o left in order of increasing arguments. A,(u) is defined by As(u) The function

G,,!(u)

= euHA,(0)evuH.

has the periodicity G&u

+ P) =

and we expand it in a Fourier Gszl(u) The Fourier coefficient

from (3.3)

property

G,,!(u)

-/3Su%O

(3.4)

series,

= Lx~m g(ss’; iwL)eih”’

2d

(3..5)

wL=pk’

g( ss; iwl) is given by

We construct a function g(ss’; z) of the complex variable coefficient g(ss’; iwl) by replacing iwl by x, g(ss’;z)

(3.1)

= J- C ePEm (m (A pnz mn

x from

( n)(n ( A,< I m) X ,l,~~~~E~, 172

,‘n

the Fourier

2.

(3.7)

The discontinuity of g(ss’; z) across the branch cut it possesses along the real x axis is simply related to F,,r (w), g(ss’;w)

= ?&

(g(ss’; w + i0) - g(ss’; w - iO)J

e@hw= 2apn

1

Pa-,/(-w).

(3.8)

We therefore obtain the result, that

F&(W) = - l yybhd g(ss’; - WI.

(3.9)

LOCALIZED

MODE

381

LIFETIME

To obtain the Fourier coefficient g(ss’; LX) we employ (19). We find t’hat ii. is the solution of the Dyson equation g(ss’; iwl)

= &,~a(s; io1) + a(s; icol) c

s1

standard

@(SSl ; iw~)g(sls’;

methods

iWl)

(3.10)

where n( s; iwl) is the free phonou propagator,

(3.11) and ~(ss’; iwl) is the proper self energy matrix. We will see below that the Icading contribut,ion to ~(ss’; iwl‘, is of 0 (17’). As long as we are inkrested in a solut,ion of t,he Dyson equation which is correct only to O(q’), we can proceed in the following manner. We iterat,e I he Dyson equatiou g(ss’;

iwz) = 6,,a(s;

ioz) + a(s; iwz)@(ss’;

c u(s; iWl)@(SSl ; &)U(Sl 8I

+

iw&L(s’;

; iwz)cP(s&

iwz) iwz)u(s’;

(3.12) icol) + . . . .

Since the nondiagonal elements g( ss’. , iwl) (s # s’) are at least of 0( 17’)) as can be seen directly from Eq. (3.101, we focus our attention on the diagonal term. These arc given by (3.13)

We rI(s;

now

introduce

a function

lI(s;

ion) by the expansion

icol) = cp(ss; LIZ) + C’ 6(SSl ; iWl)U(Sl ; iw)6(w; 81 + C’ (P(SSl ; icdZ)U(Sl ; iWl)6qSlSY ; iwz)u(s2 llsg

iw) (3.11)

; iwz)B(w;

index s1 , sz , . . , can

where t,he princes on the sums mean that no sumnation equal s. In terms of this function Eq. (3.13) becomes g(ss;iw~)

= a(s; ioz) + a(s; icdl) {rI(s; + r13(s; iw~)c?(s; = u-‘(s;iw~)

if&) +

iwz) + rI”(s;

iw)u(s;

iWl)

. * . )u(s; iwz)

1 - rI(s;iwz)

1 =- et& cfi &Ji2+ Wr2 - (2C&//3fi)II(s;i4 Since ~(ss;

iwz) + . .

(3.15)

*

iwl) is already of 0( q2) it, is a consistent

approxirna~,ion

to replace

382 II(s;

MARADUDIN

iw~) in Eq. (3.15) by the leading approximation

to @(ss; iwl),

1 g(ss; iwz) s ZEf pn wz2 + ws2 - @w8/pfi)@fss;

(3.16)

iwz) -

If we define a function of the complex variable z, which we denote by @(s; z), by replacing iwl by z in @(ss; iwt), and introduce in turn the functions A,(w) and r,(w) by -(l/@~)@(s; then combining

wf

i0) = A,(w)

F iI’,(

(3.17)

Eqs. (3.9) and (3.16) we obtain the approximate 2w,r,(w) - ws2 - 2w,As(w)]”

result (3.18)

+ 4w,2ra2(w) ’

In writing this expression we have anticipated one of the results of the next section, that A,(w) is an even function of w while l?,(w) is an odd function of W. We can rewrite Eq. (3.18) as

where

Q*(w)= w*[l +s!$J’ = w, + A,(w)

(3.20a)

1 A&> + . . . .

- z-

(3.20b)

WS

We define the center of the line described by Eq. (3.19) by the equation (3.21)

WC = as(wc>. To the lowest

order in the anharmonicity

the solution of this equation is (3.22)

we = w,s + A,(w,). If r,(w) is small and slowly varying proximate Fssj(w) further as F&w)

iZ &,I

of we we can ap-

N82r8(Wo) 1 - 2e-phoo cw- w,)2+ Nsw(wc> +

where

for w in the neighborhood

N82r8bd 2,E’h% i cw+ wc)2+ xw(4

(3.23)

>

LOCALIZED

MODE

383

LIFETIME

(b) FIG. FIG. FIG.

1.

The 2. The

1

FIG.

2

first order diagram which contributes to the proper self energy to O($). second order diagrams which contribute to the proper serf energy to O($). N, =

2% 2w,[l

(3.24)

- c&‘(a)1*

To lowest order in the anharmonicity we can replace N,I’,(w,) by I’,, We t#hus see that A,(ws) gives the shift of the frequency of the sth normal mode from its value in the harmonic approximation, w8 , while 2I’,(w, j is the width at half maximum of the resonance peak contribut,ed by the sth normal mode to the onephonon absorption cross section. The lifetime T of the localized mode is therefore given by

(3.25)

l/T* = m,(w,),

where s refers to the localized mode. The contributions to 6(ss; iwl) of O(q”) are associated with the diagrams shown in Figs. 1 and 2. The explicit expressions for these contributions are 6”‘GW P)(ss;

iwz) = -12PC

c v(sss~s1)~(s~ 81 21

; iL!Jl,)

iwz) = 18/3”~ c V(ss,s*)V(s~s~s)a(s~ 5182 11

(3.26a)

; iw,,) (3.26b)

x a(s2 ; iwz - iwz,) dzb)(ss;

iwz) = 18”zsz

T

V(sss,)V(s~s~s~hz(s~ ; 0) (3.26~)

x a(ss ; iwz,). Since ~“‘(ss; iwl) and @(2b)(~~; iwl) are independent of I, t,hey contribute only to the shift of the frequency of the sth mode, and not to its width. The latter contribution vanishes for a perfect Bravais cryst’al. It does not vanish in general for a Bravais crystal containing a mass defect, but we will show in the next section that at temperatures high enough for the classical limit to hold (or equivalently, in the limit as fi -+ 0), the contribution to the shift from P(*~)(ss; z&) vanishes. Combining Eqs. (3.17) and (3.26) we obt’ain the results

A?(w) =$V(

Swd(2%,

+ 1)

(3.27a)

384

MeRADUDPi

Ai’“’

=

-g

c

V2(sas2)

(n,,

SlSB

Lw + (%

-

+

%)

+ w.t+

ws,Lr

=

-;g

r:‘“‘(w)

=

g

c

V(

SlS!2

(0 -

1 wsl + wsz)p -

[ (w -

?7hyls2)

A:‘“‘(w)

1)

ns2 +

l

(2n,,

ss1s2) ((n,,

+

n,,

+

+

wsz)p

ws:-

(w + ws,‘-

1

1)

ws.?) - P

1)

l)[S(w

(3.2713)

(3.27~) -

ws, -

- S(W + w,, + wag)] + (nsl - n,,)b(w

w,,)

+ us, - us,)

(3.28)

- s(w - a, + 411. These expressions for t-he shift and (half) width of the sth phonon are exact t’o 0( 7’). However, as they stand they are not well suited for the numerical evaluation of these quantities since they involve t’he eigenfrequencies and eigenvectors of the normal modes of t,he perturbed crystal in a complicated way. In t’he next’ section we transform Eqs. (3.26) in such a way that only the frequencies and eigenvectors of the unperturbed crystal appear in our final expressions. IV.

EXPRESSIONS

FOR

THE

SHIFT MODE

A. We consider first) the term ~‘“(ss;

iwl)

AND

WIDTH

OF

A LOCALIZED

which is given by

after we substitut’e the explicit expressions for B(ssslsl) and a(a ; iwl, ) into Eq. (3.26a). To simplify this expression and t’he ones we will come to below, we now make use of Born’s theorem (,%)), which we can state in the form of t,he following equation, c B%)f(w,2)B,i”‘(I’) 8

= U(D)lk”;

(4.2)

where D is the dynamical matrix defined by Eq. (1.8). Through the use of this theorem we can eliminate the unknown eigenfrequencies w, and the unknown eigenvectors {B?‘(Z)) in favor of quantities we know. Substit,uting Eq. (4.2) into Eq. (4.1) we obtain

LOCALIZED

MODE

At, this stage, let us do two things. amplitudes

and let us introduce

a nlat,rix U(z’)

Let us introduce

explicitly

the reduced

by

$1 + D)-’ With these two substitutions

385

LIFETIME

(4.5 1

= M1’2U(z”)M”2.

Eq. (4.3) becomes

(4.6) This result is convenient because the amplitude C&8’(1) for a perturbed mode, in particular for a localized mode, is related to the amplitude of t’he impurity atom by4 C&“‘(Z) = dfu?F

c.4.7 )

G&Z; ws”)C~“‘(O),

where G,pjZ; w’) is the Green’s funct’iou for the unperturbed the Fourier series expansion (21)

host’ cryst,al. It has

(4.8) where wj(k) is t,he freyueucy of t,he normal mode of the perfect by the mave vector k and phonon branch index j, while e(kj) unit polarization vector. We see from Ey. (4.7) the amplitude of the impurit’y atom the solutSion of the homogeneous set of three equations iu the Ck”‘CO>(ci = 2, y, 21, C:‘(O)

= t!\lw,‘~

G&O;

crystal described is the associat’ed itself is given by three unknowns

ws”)C;“‘(O).

In the special case of a cubic Rravais crystal C,p(O; w8’) is isotropic, haBG(o)$‘), and t.he amplit’udes C?‘(O) are t’he solut’ions of 4 See, for

example,

ref.

21.

(4.9) G,o(O; w,‘)

=

386

MARADUDIN

C’(0) c$‘(o)

c:‘(o)

i

=

dIIw,*G( OS*) 0

0 dblw.?G(w.?)

0

ii

0

0 0

(4.10)

tMw 8‘G(w 8“)

There are clearly three, degenerate, localized modes possible in a cubic Bravais crystal containing a mass defect. Any three mutually perpendicular vectors can be chosen for the vectors C’“‘(O), subject to the appropriate normalization, which for the localized modes is given by Eq. (4.17) below. We will denote the amplitudes of the three degenerate localized modes by C”’ (0), C”‘(O), and Ct3’(0), and the frequency of these modes, which we denote by wo , is the solution of 1 = &w:G(w;) =U,~(ll’;

6W02

x2) can also be expanded in Fourier

UaB(ZZ’; x”) = 5 kF, e,(kj)F(kj; where the coefficient F(kj;

kj”;

(4.11)

3N g wo* -lwj”(k).

F(kj;

kj”;

k’j’;

series,

z2)es(k’jl)

X ~2aik~x~z~+2rik’~x~~‘~ (4.12)

x2) is found in the Appendix

to be

1 A(k + k’)6+ z”) = __ NM z* + wj2(k) 2

1

+ AT&f (x2 + wj*(k))(z* K(z2)

= $

g

e(kj).e(k’j’) 1 - EZ*K(Z’) *

+ wjt2(k’)) z2

+ ii,(k) *

(4.13)

(4.14)

In Eq. (4.13) A(k) equals unity if k is a translation vector of the reciprocal lattice and vanishes otherwise. When we substitute Eqs. (4.7), (4.8), and (4.12) into Eq. (4.6), it becomes

e,(k~jde&h)

xc

klknkska

C(2

jljzj,j,

Ws

-

w;,(kd)b.e*

A&l -

+ kz + kg + kt)

(4.15)

wj2JW

where @(k& ; ktj2 ; k3j3 ; k&) is the Fourier transformed quartic anharmonic force constant of Born and Huang (5%). We can simplify this expression a bit if we assumethat. the host crystal is a cubic Bravais crystal. We look at the sums over kI , . . . , kh . It is not difficult to

LOCALIZED

MODE

387

LIFETIME

show that, if we carry out a real orthogonal transformation k-space which takes the crystal into itself, the functions *t(k),

A(kl

+ kz + k3 + k4)Ww.l

of coordinate

; kzj, ; k&

axes in

; k&),

and F(kj; k’j’; z”) are left unchanged, while the factor e,(kl&)e,(k&) transforms as the product I&,& . The presence of t,hree mut,ually perpendicular mirror planes among t.he symmetry elements of a cubic crystal has the consequence that the sum over kl , * . . , k4 in Eq. (4.15) vanishes unless P = v. The equivaience of t.he x-, y-, and z-coordinate axes has the consequence that’ the sum has the same value whether it is e,(k&)eZ(k&), e,(k&)e,(k,j,), or e,(kljl)e,(k&) that appears in the summand. We accordingly take 46 of the sum of the three terms which arise when we replace e,(kljl)e,(k&) by d,,e,(kljl)e,(kzjo) and let’ p take on the values X, y, x. In this way we obt’ain

dl’(ss;iWl)= xc

ktkikaka

c,r

jljzjzj,

e&h).4k2j2) *s

-

&(kd)bs2

-

w;,(W)

A(kl + kz + ka + k4) (4'16)

x @(kljl ; kzj2; kzj.3;k4jdF(k& ; k& ; wn'). This is a convenient result’ because the value of the squared amplitude impurity atom in the localized mode has been calculated (25, 24))

F (Cs”‘(0))”= 6”Mwo2B( 1

wo2)

of the

(4.17a) (4.17b)

Because the {C(‘) (0)} (s = 1, 2, 3) are the normalized eigenvectors of a 3 X 3 real, symmetric matrix, it follows t,hat they must also satisfy the closure condition

With this result we obtain finally t’hat for any of the three degenerate localized modes (S = 1, 2, 3)

-(w&-

w;,(kl))(wo2 - w;,(k,)) A(kl + k2 + k3 + k4)

(4.19)

388

MARADUDIN

It is only in the high temperature limit that this expression takes a comparatively simple form. This is due to the following circumst’ance. The temperature dependence of ~‘“(ss; &) is contained in the factor F(k& ; k& ; w,‘) through the summation variable wn = 27rn/@fi. For all n except n = 0, high temperatures mean small /3 and hence large wn . If we therefore separate out t,he R = 0 term from t,he sum over n on t’he right side of Eq. (4.19)) we can expand t#he remaining terms in inverse powers of an2 to obt,ain sums over n that can be evaluated analytically. From Eq. (4.13) we therefote obtain the result that in the high temperature limit

+-

c 1-e

If we retain only the leading term ~‘“(ss; iwl) we obtain the result t’hat

e(kj).e(ky)

N2M

1

+ o(T-4)

in the high temperature

expansion

d”(ss;io2)=

of

(4.21)

+ O(T2). B. We consider next, the contribution

~@)(ss;

i&l). This is given by

(4.22)

w;,: & If we now make use of Born’s t’his expression can be rewritten

tlheorem and the definition as

BS”2’( m”) (J4,,,)1’2 . of the matrix

U(z”),

(4.23)

LOCALIZED

MODE

Wheu we subst,itmute into this expressiou the Fourier matrix elenieuts that appear in it, it becomes

o,(kde,(k&)

' (co; - w;,(k,))(wt

- w:,(kz))

389

LIFETIME

expansions

of the various

A(kl + kz + k,)A(k, + ks + ks) (4.24)

If we now make use of the assumed cubic symmetry with Eq. (4.17 j we can simplify this result to

of the host crystal

togethel,

e(k&).e(kzjd __ A(kl + k, + kdA(ka + ks + ksj x (a: - w;,(kJ)(c$ - o;,(kd)

We can simplify

this expression

eveu more. From Eq. (4.18) me see that

F(kj;k'j';O) If, moreover,

1 A(k + k')hjjf = m w?(k) '

(4.20)

we use t’he fact that the coefficient

A(k + k' + k")+(kj;k'j';k"j") vanishes and the only the trihutiou

(4.25)

(4.27)

when auy t,wo of the wave vectors appearing in it’ are equal aud opposite correspoudiug j’s are equaJ5 we see from Eys. (4.13) and (4.25) that second term ou the right side of Eg. (4.13) gives a nonvanishiug couto P(“~’ (ss; &) w h en it is substituted into Eq. (4.25) for F(ksjb

We therefore

obtain finally

5 See, for example, ref. 25.

; k6.76 ; tin’).

390

’

MARADUDIN

(002 - w;,(kd)(w:

- c&M)

A(kl

+ kz + k&(-k3

+ ks + k6) (4.28)

e(k&J . e($jd Ew,2K(Wn2) In the high temperature

d2*yss;iWl) = ’

(w:

x cP(k$

-

limit this expression reduces to

n 1m&

1

e(kA) . e(k&) &(kl))(W: - &(k2)) ; kzjz ; k&)@(-k3j3

A(k,

+ kz + k,)A(-k, ’

; ksjs ; k8$)

+ ks + k)

e(k.&) - @&J w:,(ks)

(4.29)

+ o( T-4)

We see from this result that the leading contribution t’o @(2b)(~~; iw~) in the high temperature limit is the first quantum correction to the classical result, since the latter is identically zero in this case. C. We come finally to the contribution @(2a)(ss; iiwl). If we substitute explicit expressions for V(SISSS~) and a(s; iwl) into Eq. (3.2613) we obtain

x aaa,( ZE’E”)(Px,,(mm’m”)

An application

of Born’s

By(r) _ (Met)“*

BJ8l)(m’) 1 Wn2 + w4, (M?d)1’2

the

(4.30)

theorem yields the result (4.31)

X ~PEIB-r(ZZ’Z~)(Phav(mm’mN)U~ir(Z’m’; wn2)U,x(ZHm; We now utilize the Fourier

expansions

of the various

(wt -

wn)‘).

matrix element,s to obtain

LOCALIZED

’

(us’ -

&(kl))(a8”

-

X @(kljl

&(kz))

MODE

A&l

391

LIFETIME

+ k3 + kdA(k2 ; k4j4 ; k&JF(k&

; k3j3 ; k&Mk&

x F&-s

+ kt, + ks)

(4.32)

; k4j4 ; u),~)

; k6jtt ; (w - d>.

Finally, making use of the cubic symmetry of the perfect host crystal, we obtain for t#helocalized mode

A@, + kr+kdA(k:! + k4+ k-s)(4.33) x (co: - w:,(t))(d - &(kz)) To complete our calculation of cP’*~’ (ss; &l) we use the fact that F(kj; kj”; x’) possesses the spectral represent’ation (4.34) which is derived in the Appendix, where an explicit expression for the spectral density f(kj; k’j’; V) is presented. If we substitute Eq. (4.34) into Eq. (4.33) we can carry out the sum over n directly to obtain

e(tjd

’

. e(k&>

(co: - w;,(t))(woz

- o:,(W)

A&l

+ k3 + kdA(kz

+ k4 + ks)

(4.35)

x f(ksj, ; k&j ; v2) e(v1) + eb2) Vl + v2 - iw1’ where we have put e(x) = >@fi coth >@hx.

(4.36)

Combining Eqs. (4.19)) (4.28)) and (4.35)) and recalling the definit.ions, Eq.

392

MARADUDIN

(3.17), we obtain finally for the frequency the localized mode (s = 1, 2, 3) &(u)

=

nluokT 12NB(w,?)

x A(kl x a( -k&

&

klzSkb

&,

(u;

+ kz + kz)A(-ks

; k&

; k&d

-&$ 33

12 B(w,y

s--oodn s-1

x A(kl

'b2

c

c

c

c

klk2k3

j,isi,

knksks

jljsi6

+ k3 + b)A(k,

=

x(

r--

s-m

&(k2))

(4.37a)

_

;;~;;);~~2k!2)w?

(k2))

; k% ; kd

; k4.h ; vdf(k6.G

; k6j6 ; ~2)

Mud + e(vdl x (VI + v2- w)p dv1 -1 dvz c s klk2k3

e(kjd ae(k2j2) uo* - w;,(k))(uo2

of

32

m

kT M’wo 12 B(&)

cuo2

+ k4 + kd@(kljl

x c@(k2j2 ; k4j4 ; ksjG)f(k&

r,(o)

~;$$)j(e:b2~)

width

.e(kd EWn2K(Wn2)

x1-

m

-

shift and (half)

+ ks + k6)@(k,j, ; kzj2 ; k&) 1 ( un2 + WS5(k5))(Wn2 + ws6(k6))

e(k&)

-- IcT M2uo

dependeut

- w$(kz))

Ah

c

jljzj,

c

k4k5k6

c

jdjsj6

+ ka + kdA(kz

+ k4 + k6)

(4.3713)

X @(kljl

; k3j3 ; k&)+(k2jz ; k4j4 ; k6js)f(k3j3 ; k4j4 ; VI) x f(kej,; k6.G ; v2)bbd + eb2)l~h + v2 ~1. If we set w = wO in Eq. (4.37) we obtain what. are conventionally called the localized mode shift and (half) width. However, it must be kept in mind t’hat it is in fact A, ( CO)and l’,(o) which appear in the absorption cross section, Eq. (3.19). If we note that in the expressions for A,(o) and l’,(o) for the localized mode the reference to the mode occurs only through the presence of its frequency, we see that we can combine Eqs. (2.15), (2.16) and (3.19), wit,h the help of Eq. (4.18)) to write for the localized mode eontribut,ion to the one phonon absorpt,ion cross se&ion -2M

(4.38)

where s now refers to anyone of the degenerate localized modes. The reeu.lt given

LOCALIZED

MODE

393

LIFETINE

by Eq. (4.38) has the same form as the corresponding expression for a mass defect’ in a harmonic cubic Bravais crystal (24), where O,(w) plays t’he role of the harmonic localized mode frequency and J?,(w) plays the role of the (half) width of the excited nuclear st,ate of the resonant nucleus. V. EVALUATION

OF

THE

SHIFT AND IN A LINEAR

WIDTH CHAIN

OF

A LOCALIZED

MODE

The expressions we have obtained in the preceding sect,ion for A,( U) and I’,(w) are exact t,o t,he lowest’ nonvanishing order in the anharmonic force constants. Their evaluation for some particular model of a Bravais crystal undoubtedly would require the use of a high speed computer. Alt#hough calculations of this general type have been carried out successfully (13, 1.4, 5X), t.he expressions given by Eq. (4.37) are suflicient.ly more complicated t.han those which have been studied t,o date that it seems unlikely t’hat their exact (numerical) evaluation will be effected in the near future. In order to obt’ain at least a qualitabive feeling for the dependence of A,(U) and T’,(W) on w and on t,he anharmonic force constanm, we carry out in this section an evaluat,ion of these functions in the classical limit for an isotope defect in a monat~omic linear chain with nearest neighbor interact,ions between atoms. The calculat.ion is carried out. in the classical limit, so t.hat we need not, consider the contribution to A,(w) from CY(‘~)(ss; iwl), and because the expressions for the contributions to A,(w) and I’,(w) from cP@~)(ss; z&) take much simpler forms in this limit t’han at’ lower temperatures. The choice of a monatomic linear chain for the host cryst)al enables us t,o carry out most of the required int,egrals analytically rather than numerically. Despit’e the simplicity of our cry&al model, t’he calculat’ions are not, completely trivial. The Fourier transformed anharmonic force constants for our model are (27)

@(klkzka) Nhkzhh)

= +&

= s2 ( 2i)4e-

. aks T,473 (2i)3e-(si’N)(k1+k?+k3) sin 7rkl -N sin -N sin -N (ai/.V)(k,+k,+k3+k*)

Tkl

. sin -

N

lrka

sin -?rh sin - sin -2, N N

(5.1) (5.2)

where g and h are the third and fourth derivatives of the interat’omic potential function evaluated at the nearest neighbor separation, respectively, while M is the atomic mass. N is the number of atoms in the cryst,al, and lcl , k, , . . . , are integers which take on values between ( -N/2) + 1 and N/2. The unpert’urbed normal mode frequencies are given by Wk = WL 1sin (nlc/N)

j)

( 5.3)

where wL = (4f/M) I” is the maximum frequency, in terms of the harmonic force constant f. Suppressing factors of 3 which have their origin in the assumed cubic sym-

394

MARADUDIN

metry of the host crystal obvious notation

employed

in the preceding section, we can write,

wokT 1 @(kl - klkz - kd --x = 4JqwoZ) p ).@) (oo2 - 4,>“4, a For the present model we have (1)

in an (5.4)

A%)

2 wo2= -1”

(5.5)

$’

while (5.6) Substituting Eq. (5.3) into Eq. (5.6), and replacing tegration according to

summation

over k by in-

rk &y+ (5.7)

we obtain sl2

B(Wo2)= AI0 1rWL2

sin26 ”

(x2 - sin2 4)” (5.8)

1 =- 1 GIL22X(22 - 1)3’2 ’ where we have put x = wo/wL . Combining Eqs. (5.2), (5.3), (5.4), and (5.6) we obtain the result n/2 A:“( U> xkT 16 h 1 sin’ 41 ,-E--pd’2 (x2 - sin* &)” 4B(wo2) ?T2 (MWL2)2 cfJL2 WL kT h =--xx. 4 f” Turning now to A?( temperature limit

x A(kl

(5.9)

w), we obtain from Eq. (4.37) the result that in the high

+ k, + kdA(kz

x f(kak, ; vdf(hke ; vd LVlV2 where

in the present

context

A(k)

(5.10)

+ k4 + ke)Wcxkaks)+,(kzk4kd

equals unity

Vl + v2 (Vl

+

v2

if k = 0, +N,

-

o)p

~t2N,

'

. . . and

LOCALIZED

vanishes otherwise.

lMODE

39.5

LIFETIME

We have the representation A(k)

ND c

= ;

e2risk’Na

(5.11)

s-(N/2)+1

To cont,inue past t,his point we make t.he following approximation. We expand t’he spectral density f(kk’; V) in powers of c and retain only the leading term. This approximation corresponds to treating the localized mode exactly, but assuming that t,he continuum modes are unaffected by the presence of the impurity atom. It can be improved systematically by ret,aining higher order terms in E in the expansion of bhe spectral density but’ we will not do so here. Wit!h this single approximat,ion we can carry out an exaci- evaluat,ion of A.,(w) and I’,(w). From Eq. (A.38) we find that f(kk’;

v) = NG

A(k

+

k’) ‘(’

-

@‘) 2-ks(v

+ wk)

(5.12) s(v - wo>- s(v + 00) + 2Azkf B(WO*)(wo2 - Wk2)(W$- w,$) * We substitute Eq. (5.12) into Eq. (fi.10) to obt,ain cc kT M2~o A(2a)(W) = _ dVl s 4 B(W02)s--m s-a klhks k&&e x A(kl + k3 + ks)A(kz + k4 + kg) @(k1kskd@(k2keks) (003 - 4J(wo2 - 4z,> 1 A(k, + kMk5 + kc) Vl + v2 XL VlV2

.[&I

-

W3)8(V2

-

(Vl

+

W5)

-

v2

-

6(t’1

w),

4w3 W5

NZA!P

-

w3)8(v2

+

W5)

-

&I

+

w3)8b2

-

w5)

+ 6(Vl+ w3)8(v2 + w5)l+ NM J- A(k3 + k4)2N2M wo 2~3 (5.13)

1

B(wo”)(wo” -

6(Vl

-

W3)6(V2

+

WO)

-

w52)(wo2

6(V1

+

_

w62)

ml

-

WO)

u3)6(VZ

-

+

W3)6(V2

‘%Vl

-

+

wo)

W&(V2

al)1

t

1 1 A(ks + ks) wo + xm 2N2M B(wo2)(wo2- w3ywo2 - w42) 2-a x

MY1

-

wo)3(vn

-

w5)

-

6(Vl

-

wo)6(v2

+

w5)

- 6(Vl + wo)6(vz+ w5)+ 6(Vl + wo)fdvz+ 41 + &@ x [6(Vl- wo)6(vz- wo)- 6(Vl - wo)6(v2+ WI>- av1 + wo)& - wo) 1 +

b+

WNQ

+

41

B2(W*)(Wo2 =

The integrations over

v1

_

v2

-

wa2)(Wo2

-

w52)(wo2

+ A:ia’(w) + A:?‘(w)

-

w62)

+ A?“(W). can be performed immediately.

A:?‘(w)

and

w32)(wo2

>

396

MARADUDIK

If we use the representabion

given by Eq. (5.11) together with

(k)p we can rewrite

the expression

= fs[

clle-‘tsinat,

(5.14)

for A:“,” (CO) as

2

A:?‘( CO) = - 16kTwo (M;L2)5 & &

gs(x) cos 2+

r

cos 2% ,

I--

xl’ - wJ,dtsin wtcos

w3 t

cos w6 t

1

(5.15) ,

where the funct,ion g8(x) is defined by cos(2~sk/N) sin”(7&/N)’ (x2 - sin2(7&/N))2

(5.16)

’

so that gs(x)

= (-l)“[x

- -\/x2 -

112’“‘[1 + 2 1S j x&?*

-

11.

(5.17)

It decreases exponentially with increasing 1s 1 and wit’h increasing 2. Replacing the sums over k, and k5 by integrals in Eq. (5.15) we are led to the result (28) A!?(W)

= - y =

wg; 2 s=--oD 5 gs(x)[6,0 -

w lm sin

wtJi,(w,t)dt]

-(l-is)

where P”(x)

and B,(x)

are the Legendre functions -v,

a, (x) _ 7r1’2 r(v

Y

+ 1) L

2y+lIyv + yg 2””

F

2’2;

(5.18a)

w > 2wL

(5.18b)

1 < x < 1

(5.19a)

z > 1

(5.19b)

defined by

l-x I + 1; 1; _____ 2 v + 1 v+ 2

0 < w < 2wL

-

v+?.l2’2

and F(a, b; c; x) is Gauss’ hypergeometric function. In what follows we will not be interested in the region w > 2wL because, as we will short’ly see, rS ( W) vanishes identically in this frequency range. Therefore, setting w = w. we obtain as the contribution to the shift in the local mode frequency

LOCALIZED

Proceeding

in t#he same way,

MODE

397

LIFETIME

we obtaiu the following

expression

for A$?‘( w),

cltwt w0t 1 sin

cos

cos

(5.21)

w3t

where we have intmroduced the function fy(x)

=

i

c

.

e(n=ik’N)(s-l’2)

Thy

‘ln z N k [e” - sin?(&/N)] r/2 2i sin(2s - 1>1$ sin C$ & -I7r -0 x2 - sin”+ = 1 (--I)” ; dFl

(2. - ~FTi)2s-1

= ; i-g1

(x -

~x,ly+l

(5.22) SZl s = -/CT].

We see from these results that’ j’q(r) is an odd funct)ion of s about the point s = +$. That is, we seet’hat j; = -f. , fZ = -j’L , ja = -je2 , etc. The sum over k3 in Eq. (5.21) is readily carried out’, A:,““‘(w) = - y

w L clt sin wt cos cd0tJ2(s,-,s.I,(cdLt)

1

(5.23)

.

Setting w = w. , we obtain for the contribution t,o the localized mode frequency shift6

We have used the fact that Jzn(-r) = LZn(x), where rz is an integer. It is easy to see, by a relabeling of dunmy sumnlatlion variables, that’ the contribution to the frequency shift giveu by A$@(wo) equals that given by A~:“‘(,u,,), 6 See ref.

28, p. 166.

398

MARADUDIN

(5.25)

A$‘=‘(w,) = A!?‘&,).

We consider next the contribution to the localized mode frequency shift given by A!:“( 0). This can be written as A:?‘(W)

= 4kT

g”

c

f” (z)ffkd

WL6BT(W02)(MwI,2)3 s1s.J 81

4woff

w2.

(5.26)

Because f*(x) is an odd function of s about the point s = 45, the contribution A::“‘(W) to the shift of the frequency of the localized mode, vanishes. Thus processes in which the localized mode phonon decays into two localized mode phonons do not contribute to the frequency shift. We come finally to the expression for the (half) width I’,(o). If we take into account the remarks at the end of the appendix, we can write I’,(w) in the approximation we are employing in this section as

(5.27)

(5.28)

X $11

dtemiut cos wd cos wd,

where we have used Eq. (5.11) and the representation (5.29)

in going from Eq. (5.27) to Eq. (5.28). Carrying out the integrations kB we obtain the result’

over kz and

(5.30a) 0 < w < 2WL

= 0 7 See ref.

w >i2WL ) 28, p. 65.

(5.30b)

LOCllLIZED

MODE

399

LIFETIME

where Q”(X) is the Legendre function of t’he second kind, defined for real arguments in t’he int’erval ( - 1, 1) by (29) Qv(x)

= ;&

-v,

v + 1; 1; ‘+) (5.31) -v,v+l;l;--

1+x 2

.

In Lhc present case we have that

Qe,s,-i(x) =;F

-2ls/++

+;;I;+.

(5.32)

The vanishing of I’,(o) for w > 20~ is a reflection of the fact that energy cannot be conserved in the decay of the localized mode into two continuum modes when &is condition is satisfied. Setting w = w. , we obtain for t’he (half) width of the localized vibration mode

- WL = ~$&-$ r,(w)

gs(x)Qz,s,+ (1 - ;x2).

1 < x < 2

(5.33)

Tables of the Legendre functions P,(s) and Q”(Z) for nonintegral values of v and for x in the interval (- 1, 1) do not seem to exist. We therefore proceeded to generat,e t’hese functions in the following way. Py(n) and Q”(X) satisfy the recurrence relat’ions (SO) (v + l)P&(Z) (v +

= (2, + l)zP,(z)

l)Qrt~(x)

= (2,

- VP,-l(X)

+ 1)x&v(x) - V&,-~(X).

(5.34a) (5.34b)

Therefore, if we know P-l,z(z) and Pl,2(x), we can generate all the Legendre functions required for the evaluation of A%‘E(cc,,).Similarly, knowing Q--1,2(~) and Q112(5) we can compute I’,( wg) . We begin by noting t,hat. (31) P-1,2(x)

= F

(

- k,f;

Q-1,2(x) = z"F (;,;;

l;L$) 1; T)

= zK(dq) = K(,jF),

(5.3ja) (5.35b)

where K(k) is the complete elliptic integral of the first kind. The following tion of contiguity for the hypergeometric function,* (a *See

ref.

P)F(a, 31,

p.

P; Y; xl:= 32.

aF(a! + 1, P; Y; x) - PF(cq P + 1; 7;:~:)

rela(5.36)

400

with CT= -45,

MARADUDIN

0 = 34, y = 1, together with the fact that (31) F( -g,

where E(k) is t’he complete establish the result that

$5; 1; 2) = (2/7r)E(&, ellipt’ic integral

(5.37)

of Dhe second kind, enables us to

(5.38a)

(5.3813)

The results given by Eqs. (5.34), (5.35)) and (5.38) enable us to ealcuIate all the Legendre functions we require. The sums in the expressions for Aifa’(uo), ALia’(wo), and I’$(wO) converge very rapidly. In no case was it necessary t,o go beyond / s 1 = 5. We have calculated A~2a’(wo) and I’,( CO,,)as functions of z for IZ:lying between 1 and 2. The results are plotted in Figs. 3 and 4, respectively. We have added a second horizontal scale to show the value of t,he ratio of the impurity atom mass to the mass of t’he atom it replaces, M//M, corresponding to a particular value of A?“(uO) or Ins. Fram Fig. 3 we seethat the contribution to A:““‘(w,) from A%“ (w,,) is of the sameorder of magnitude as the contribution from A62”‘(wo). It should be recalled that the latter contribut,ion is associated with the decay of the localized mode into two continuum modes, while the former contribution is associated with the decay of the localized mode into a localized mode and a continuum mode. Energy is not conserved in this last process, so that there is no analogous contribution to the half widt’h of the localized mode, which is plotted in Fig. 4. To obtain a rough estimate of the magnitudes of the shift and width we evaluat’ed the coefficients kTh/4f2 and kTg2/4f3 using the values off, g, and h which we have determined previously for a model of lead (32). The use of force constants determined for a three dimensional cryst’al model in a calculation based on a linear chain means t.hat the results can have only a qualitative significance. In this way we obt,ain the results

LOCALIZED

MODE

401

LIFETIME

where C-,, is the limiting high temperature value of the equivalent Debye temperature. It equals 143.4% for our model of lead (3.2). Wit.h the aid of these results we have computed A,(oo‘i and T,(wO) as functions of ZE,and the results are presented in Table I. The shift is seen to increase with increasing t.emperature. This is due in the present case t,o the fact that’ the quartic anharmonic X 2.0

1.2 I A (2a) sl %’ OL

1.5

1.4 I kT / T;3

1.6 I

1.8 I

2.0 I

2 / MC

/ 1.0

/

/

4-

-

/

/’ /I /I // 1’ /I /I /I

7

-1.0

-1.5 0.5

0.4

0.3

0.2

0.134

M’IM FIG. 3. A plot of A?’ (wo), quency of a localized vibration and of t,he ratio of the mass of culation is carried out in the chain.

the cubic anharmonic contribution to mode, as a function of the frequency the impurity atom to that of the atom high temperature limit for an isotopic

the shift in the freof the localized mode it replaces. The calimpurity in a linear

402

MARADUDIN

FIG. 4. A plot of I’,(u,), the complex shift in the frequency quency and of the ratio of the The calculation is carried out linear chain.

cubic anharmonic contribution to the imaginary part of the of a localized mode, as a function of the localized mode fremass of the impurity atom to that of the atom it replaces. in the high temperature limit for an isotopic impurity in a

18

TABLE x

&(w)/(~/@,)~L rsb)/(~/@,bL

1.2

1.4

1.6

1.8

1.95

0.0219 0.1344

0.0715 0.1424

0.1172 0.1187

0.1480 0.0701

0.1599 0.0194

a Values of the real and imaginary parts of the complex shift in the frequency of a localized mode as functions of the local mode frequency in the high temperature limit. The calculations are based on a linear chain whose force constants have been chosen to represent lead.

contribution dominates the cubic contribution. The experimental results of Schaefer (9) showsthat of the ionic crystals studied by him it is only in KI that the frequency of the localized mode (caused by a U-center) shifts upward with increasing temperature. In the other ionic crystals studied by him, NaCl, NaBr, KCl, KBr, RbCl, RbBr, RbI, the shift is downward with increasing temperature. These results suggest that the sign of the shift probably depends rather sensitively on the interatomic forces, and a more realistic model than has been employed here is required in any given caseto reproduce the shift accurately. There does not seemto be any experimental evidence at the present time which bears on the predicted increase of A,(W) with x at fixed temperature. The increase in

LOCALIZED

MODE

LIFETIME

403

I’s(wO) with increasing temperature at fixed x is confirmed by the experiments of Schaefer (9). It should be kept in mind, however, that in any case our results can have only a qualitative significance for the interpretation of t’he experimental data of Schaefer. The frequencies of t’he localized modes studied by him are more than twice the maximum frequency of the host crystal, so that these modes can decay into continuum modes only through four phonon processes, while we have studied only t,he decay through three phonon processes. The values of I’,(w,,) presented in Table I are somewhat larger than those obtained in previous calculations of this function for three-dimensional crystals (.2,S, 5). While this may be due in part to the fact that the density of final states for the decay of a localized mode into t,wo cont8inuum modes can have a qualitat.ively different character for a linear chain compared with a three-dimensional crystal, it is most likely due to our use of values off, g, and h det’ermined for a three-dimensional cryst’al in a one-dimensional model. Another way of choosing values for these constants may lead to results more closely in agreement with the results of the three-dimensional calculation. In conclusion, we should like to point out that grant,ing the approximation (5.12) the remainder of the analysis in t.his section is exact. That is to say, the wave vector and frequency conservation conditions which appear in the expressions for A,(wo) and l?s(wO) have been taken into account exactly. This is a consequence of our use of a linear chain as the host crystal and of the assumption of the high temperature limit, and as such has a rather special character. APPENDIX

In this appendix we study someproperties of the matrix element (~~1+ D);::w where D is the 3N X 3N dynamical matrix for a Bravais crystal containing a mass defect. With the use of Born’s theorem we may express this element as

(A.11 The matrix U(z’) which we define by (~~1+ D)-’

= M1’2U(z”)M”2,

L4.2)

where M is the (diagonal) matrix of the massesof the atoms in the pert#urbed crystal, therefore has elements (A.3)

404

MARADUDIN

In the preceding expressions z is not restricted to be a real variable, complex. We can now write a spectral representation for U=B(11’:2’) :

it may be

(A.4) where the spectral density ~~~(21’;v) is given explicitly by eYm%~‘) I&&

[*(

v - w> - s(v + dl.

(A.5)

I?rom Eq. (A.5) we see that ~~(11’; v) is an odd function of v, so that we can rewrite Eq. (A.4) as (-4.6) It follows from Eq. (A.6) that the spectral density u&U’; sgn vu,,(ZZ’; y) =

V) is given by

iy,,(ZZ’; -y2 - i0) - Ua)9(ZZ’; -v* + io>. 2ir3

(A.7)

To det’ermine the spectral density u,b(ZZ’; Y) we must first determine U,,(EZ’; a”). We begin by rewriting the matrix 2’1 + D as a21+ D = M-“2(x2M

+ @)M-“’

(A-8)

where @ is the force constant matrix for the perturbed crystal. Let us now write the matrix in brackets on t,he right side of Eq. (A.8) as t)he sum of the contribution corresponding t,o the perfect crystal and t,he contribution describing the defect, Mx’ + @ = L + 6L

(A.91

where L = M,$ + @o 6L = (M - Mo)z2 + (@ - a,,).

(A.lOa) (A.lOb)

If we denote the matrix inverse t,o L by 9, $j = L-l, then combining Eqs. (A.2), (AS),

(A.9) and (A.111 we find that

u = (L + 6L)-l, and is the solution of the equation

(A.ll)

(A.12)

LOCALIZED

MODE

405

LIFETIME

u = s - gBLU. We now determine $j and U. The matrix equation

(A.13)

element $jmb(ZZ’) is the solution of the

& La4z”)s7p4w

= &x&l, .

(A.14)

We expand $L~(ZZ’) and 6,&rLI as $&II’)

= g e~(kj)F(kj)e,(kj)e’“~k”x’~‘-x’~“~

(A.lTia)

= ; g ea(kj)es(kj)e2rzk’(x(tJ-r(“))

(A.lSb)

&&,

where the fact that $jos(ZZ’) depends on 1 and 2’ only through their difference is a consequence of the fact that the perfect crystal possesses translational invariance. If we use the fact that L&11) then the equation for F(kj) Mz.’ z

= Af&&,t

+ aq!! (ZZ’),

(A.16)

becomes

eu(kj)e,(kj)e’“‘k”“‘~~-x’~‘~‘~(kj) + M GF

~,,(k)e,(kj)e,(kj)e’~ak”x’~‘-x’~”’~(kj) = $ g

(A.17)

e,(kj)ea(kj)e”“‘k’(X!l’-“(l’“.

In Eq. (A.17) D,@(k) is the Fourier transformed dynamical matrix for the perfect crystal, i.e., it is t,he matrix whose eigenvalues are the squared frequencies W:(k) and whose eigenvectors are the { e(kj) \ : F D,,dk)edkjJ The element, D,,(k)

= wj’(k)e,(kjj.

is given explicitly

(A.18)

by (A.19)

where 1 = Z -

I’. Combining

Eqs. (A.17)

FUG)

= N$

and (A.18) we obtain finally that, z2 + ]w.(v fL

7

(A.20)

so that e,(kjh(kj)

e?aik.(x(l~--x(l’))

(A.21)

406

MARADUDIN

The matrix element G,o(ZZ’; 02) introduced $xs(ll’; 2’) by

in Eq. (4.7) of the text is related to

G,p(ZZ’; w’) = -$&(ZZ’; We obtain U&II’; series as U&U’;

-w”).

2”) in much the same way. We expand it in a double Fourier

z”) = )$ k& e,(kj)F(kj;

kj”; z2)ea(ky)

c2*i’k’x”‘+k”x’2”‘.

(A.22)

The use of a double Fourier series is necessitated by the fact that bhe crystal containing a mass defect no longer possesses translational invariance. The expression for CC&W’) for a mass defect at the origin of an arbitrary Bravais crystal is

(A.23)

6L,8(ZZ’) = -EM&&&o where

When this result and the expansions (A.21) and (A.22) are substituted into the equation for U, Eq. (A.13), we find that the coefficient F(kj; k’j’; x2) satisfies the equation 1 A(k + k')&,r F(kj; kj”; x2) = NM z2+ wj'(k)

(A.25)

+ $ ?

dWe,(hs) F(hs; kj"; 2"). 3 z2 + wj2(k)

In terms of the matrix K whose elements are given by

(A.26) we can write the solution to Eq. (A.25) in the form

F(kj;k'j';z2)

= - 1 A(k + k')& NM z2+ wf(k)

2

x NyM x2 +

1wj2(k) 22 + iz,ck,j

(A.27) 5 e,(kj)(I

- ~z2K(s2))L&(kj’l).

In the particular case of a cubic Bravais host crystal the matrix K(z’) becomes diagonal and isotropic,

(A.28)

LOCALIZED

and the expressionrfor F(kj; WC

k’i;

MODE

407

LIFETIME

kj”; z’) simplifies to

1 A(k + k’)6jjr x”> = N-M 22 + w,2(k) I 2

(A.29) 1

1 x2 + c,$(k) x2 + c+(k)

+ h&

e(kj).e(k’j’> 1 - ez2K(zT) ’

In what follows we will consider only the case of a cubic Bravais host cry&al. If we write a spectral representation for F(kj; k’j’ ; z’) , $‘(kj;

k’j’; z”) = i:

,&jck;;k;;

‘)

(A.30)

where sgn vf’(kj; kj”; v) = ki

[F(kj;

kj”;

- v* -

i0) -

then the spectral density u,~(ll’;

(A.31) F(kj;

k’j’;

--?

+ iO)l,

v) is given by

u,~(Zl’; V) = & kF, e,(kj)f(kj;

kj”;

v)eb(kj”)

~2?ri~k~x~~~~k’~x~z’~~.(A.32)

However, in the application of the results of this appendix in the text of this paper, it is t#he spectral density f(kj; k’j’; v) that is required, and we turn tjo its determination. Combining Eqs. (A.29) and (A.31) we obtain the result f(kj;

k’j’; v) = N& A(k + k’)&jt

sgn v6(v2 - @j’(k))

2

e(kj) .e(k’j’> [1 + sgm v NYM [l - wy&(v2)]2 + 7T%2v4G02(v2) x

6(v2 - co;(k)) (v” - c+(k’))p

EV2G( v2)]

+ 6(v* - w;t(k’)) b’ - dk))p (A.33)

+ ev2Go(y2)[(v” - wF(k)),‘cvz - w(?i,(k))P

- a26(v2- $(k))8(vZ

6(v2- wo’)

+ sgnv wo2e(kj)-e&j”) N2M

- &(k’))

B(ao2) (~0’ - ~j2(k))(~02

where we have introduced t#he functions

- u$ (k’))

’

408

MARADUDIN

GO(v2) = 3+ z 6(v2- q%d) GO(v2) = &

& (g

(ASa)

1 _ Wj*(k))p’

(A.34b)

Go( Y”) is recognized to be the distribution function for the squares of the normal mode frequencies of the perfect, host crystal, while Go( v”) is its Hilbert transform. In writ*ing Eq. (A.34) we have denot)ed the localized mode frequency by w. . In the special case of a monatomic linear chain, which is of interest for t,he discussion in Section V of the text, we have that Go(v2)

=

;

(y2(wL2

!

VP <

y2))1/2

(A.35)

G(y2)= 0 =

We therefore sgn vf(kk’;

WL2

v*< WL2

(&qv2

_’

v*> WL2.

wL2))W

obtain the result] that

v) = N&A(”

+ k’)6(u2

-

Q*) 2 v

lV‘lv1

1

(A.36)

t

7r‘wYio‘(v‘)

(L(v’

-

Wil)P

(v2

-

wb> Wk2)P

1+w*Go(v*) v*
.

[

(v2

_

wnJ(v2

_

w;,)p

-

w0”

T26b2

-

w!mv*

-

wit>

6(v2- 00”)

N*M B(wo2)(wo2- wk*)(wo*- w$) *

v*> WL2

The last term on the right hand side of Eq. (A.33) and of Eq. (A.371 can be omitted in calculations of the phonon width. The retention of this term corresponds to including processesin which the localized mode decays into two localized modes, or into a localized mode and into a band mode. However, it is not difficult to see that such processesdo not conserve energy, and therefore cannot contribute to the width in this order of perturbation theory. In a more accurate theory in which the free phonon propagator a(s; iwl) appearing in Eqs. (3.26) is replaced by a dressed propagator, the d-functions in the last term on the right side of Eqs. (A.33) and (b.37) are replaced by finite, broadened peaks. In this case it, is possiblefor the localized mode to decay into t’wo localized modesor into

LOCALIZED

MODE LIFETIME

409

a localized mode and into a continuunl mode. In this casethe function r,(w) as a function of w would have a resonance character in the neighborhood of w = 2~ . ACKNOWLEDGMENT

Part of the work described in this paper was carried out while the author was a visitor at the Laboratory of Atomic and Solid State Physics at Cornell University, under a grant from the IJnion Carbide Corporation. He would like to thank Prof. J. A. Krumhansl for extending to him the hospitality of his Laboratory and for providing the conditions which made this work possible. He would also like to thank Brenda J. Kagle for help in carrying out some of the numerical calculations reported here.

RECEIVED:

Alay 8,19M REFERENCES

1. 2. 2a. 3.

4. 5. 6. 7. 8. 9. 10.

11. 12. 13. 14. 16. 16. 17. 18. 19. 20. 21. 22. $3. 24.

E. W. MO~VTROLL AND I<. B. POTTS, Phys. Rev. 100, 525 (1955). P. G. KLEMENS, Phys. Rev. 122, 443 (1961). M. A. KRIV~GL~Z, Soviet Phys.-JETP 13, 397 (1961). A. A. MAR.~DUDIN, in “Astrophysics and the Many-Body Problem,” p. 107. Benjamin, New York, 196% A. A. M.~R.~DUDIN, unpublished work. W. M. VISSCHER, Phys. Rev. 134, A965 (1965). 1:. GL.\SER .IND G. K.ILLEN, i\:&. Phys. 2, 706 (1956). Yu. K~c.~N AND Y-4. IOSILEVSKII, Soviet Phys.-JETP 1’7, 195 (1963). B. MOZER, &ll. Am. Phys. Sot. Ser II 8, 593 (1963). G. SCHAEFER, J. Phys. Che,m. Solids12,233 (19GO); B. FRITZ, J. Phys. Chem. Solids 23, 375 (1962). S. \:. M.\LEEV, Soviet Phys.-JEEP 12,617 (19Gl); W.M.VISSCHER,~~ Proc. Second M&sbauer Conf., Paris, 1961. Wiley, New York, 1962; B. M~ZER END G. H. VINEY~~RD, Bull. Am. Phys. Sot. Ser. II 6, 135 (1961). 1,. V~\N HOVE, N. M. HUGENHOLTZ, AND L. I’. HOWL~ND, “Quantum Theory of Manyp. 11. Benjamin, New York, 1902. Particle Systems,” R. L. M&SBSUER, 2. Physik 161, 124 (1958). A. A. M.~R.\DUDIN AND A. E. FEIN, Phys. Rev. 128,2589 (1962). \I. AMBEG.\BKAR, J. CON~.\Y, .\NI) G. BAYM, Proc. 1963 Intern. Conf. Lattice Dynamics. Pergamon Press, New York, 1964. “The Advanced Theory of Statistics,” Chap. 3. Griffin, M. G. KENDALL .~ND A. STUIRT, London, 1958. B. KACFM.~N ,\ND H. J. LIPKIN, ilnn. Phys. (N. I’.) 18, 294 (1962). M. A. KRIVOGLAZ, Zh. Eksperim. i. Teor. Fiz. 46, 637 (1964). J. MELNG.~ILIS, private communication. J. M. LUTTINGER AND J. C. WORD, Phys. lieu. 118,1417 (1960); see also ref. 13. M. BORN, Kepts. Progr. Phys. 9, 294 (1942-43). A. A. M.~R.~DVDIN, P. A. FLINN, A\~~ S. L. RUBY, Phys. Rev. 126,9 (1962). M. BORN .~ND K. Hu.~NG, “Dynamical Theory of Crystal Lat,tices,” p. 305. Oxford Univ. Press, Oxford, 1954. P. G. D.IWBER AND R. J. ELLIOTT, Proc. Roy. Sot. A273,222 (1963). A. A. M~R:\DUDIN, Westinghouse Research Laboratories Scientific Paper No. 63-129103-P9 (1963).

410

MARADUDIN

26. R. E. PEIERLS, “Quantum Theory of Solids,” p. 37, second footnote. Oxford Univ. Press, Oxford, 1955. 26. R. A. COWLEY, Proc. 1963 Intern. Conf. Lattice Dynamics. Pergamon Press, New York, 1964. 27. A. A. MARADUDIN, P. A. FLINN, AND R. A. COLDWELL-HORSFALL, Ann. Phys. (N. Y.) 16, 337 (1961). 28. F. OBERHETTINGER, “Tabellen zur Fourier Transformation,” p. 169. Springer, Berlin, 1957. 29. E. W. HOBSON, “The Theory of Spherical and Ellipsoidal Harmonics,” p. 229. Cambridge Univ. Press, Cambridge, 1931. SO. A. ERDELYI, W. MAGNUS, F. OBERHETTINGER, AND F. G. TRICOMI, “Higher Transcendental Functions,” Vol. I, pp. 161-162. McGraw-Hill, New York, 1953. 81. c. SNOW, “Hypergeometric and Legendre Functions with Applications to Integral Equations of Potential Theory,” p. 3. National Bureau of Standards Applied Mathematics Series-19. U. S. Government Printing Office, Washington, D. C., 1952. 32. A. A, MARADUDIN AND P. A. FLINN, Phys. Rev. 129, 2529 (1963).