a theory of the lattice vibration of anharmonic solids § 1. introduction

Supplement of the Progress of Theoretical Physics, No. 45, 1970

A Theory of the Lattice Vibration of Anharmonic Solids

Shozo T AKENO

Research Institute for Fundamental Physics

Kyoto University, Kyoto

The lattice vibration of anharmonic or "non-ideal" solids is studied using a method of double-time Green's function, without assuming the smallness of the anharmonicity of atomic vibrations from the outset- To encompass disordered and amorphous solids and molecular systems as well, a system under consideration is taken to be arbitrary in structure and in composition. In treating equations satisfied by Green's functions a decoupling approximation is employed, which leads to a self-consistent phonon theory. A set of hierarchy equations are thereby obtained which give renorroalized eigenfrequencies of phonons in terms of "effective interatomic or intermolecular potentials" or "effective force constants" in an implicit manner. In contrast to almost all current self-consistent phonon theories in which attention is focused to the case of quantum crystals, the theory is designed to reformulate theory of lattice dynamics and therefore to study the anharmonic vibrational properties of solids in general. In solving the hierarcy equations the first and the second order approximations are employed which give renorroalized harmonic phonons and renorroalized anharmonic phonons with threephonon scattering processes taken into account, respectively. A detailed discussion is given of the properties of effective potentials. It is shown that the effect of the anharmonicity of atomic vibrations is to modulate a bare potential in a manner similar to the random thermal modulation of X rays or neutrons in solids, thus giving rise to a decrease in the minimum value and an increase in the minimum position of the effective potential as the anharmonicity increases; its asymptotic expression reduces to a mean intermolecular potential energy in liquids or in imperfect gases. As applications and implications of results obtained here, a brief discussion is given of some general properties of effective potential, the dynamical instability or the melting of solids, phonons in liquids, anharmonic vibrational properties of phonon impurity modes in crystals, impurity- or defect-induced anharmonic vibrations in solids, etc.

§ 1. Introduction

137

The traditional theory of lattice dynamics initiated by Born and von Karman more than fifty years ago, has been considered to be one of the most firmly rooted concepts in the solid state physics.1)-a) The fundamental assumption used in this theory is that the displacements of atoms in a lattice from their equilibrium positions are small compared to interatomic spacing, and therefore that the truncation of expansion terms of interatomic or intermolecular potentials in powers of the displacements at first few anharmonic terms (usually the third and the fourth orders in powers of the displacements) is legitimate and these can then be treated as small perturbations.4 ) However, it has recently been recognized that such an apparently reasonable assumption cannot be used from the outset for quantum crystals such as solid helium and

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138 S. Takeno

some of the other rare gas crystals5'' 6 '·*> and solid molecular hydrogen as well.7> For example, it has been shown that conventional harmonic approximations which form the basis set for the perturbation expansion do not exist for solid helium.8'·**> A similar situation exists in solid hydrogens, but to lesser extent.9> Another important class of solids for which the anharmonicity of atomic vibrations plays an essential role is ferroelectrics. Furthermore, it can generally be assumed that a similar situation holds for almost all types of solids near their melting temperatures.

To study the vibrational properties of highly anharmonic solids, some kind of renormalized perturbation expansion for treating anharmonic vibrations is required. A naive form of such a method has already been devised by Hooton10' in the nineteen-fifties, following the work of Bom.11'' 12' It is, however, only recently that more general and elegant renormalized perturbation theories based on the modem techniques of many body theory have appeared, which enable us to make extensive studies of the physical properties of quantum crystals. These theories, presented in a variety of forms, are mainly classified into two types, one having a single-particle picture and the other collective picture, as called by Werthamer.5>

In the former case the Schrodinger equation for an individual atomic motion is studied using an approximation similar to the temperature-dependent Hartree self-consistent field method.13' The phonons are then obtained as the normal modes of response of a system to an externally applied disturbance from equilibrium. This line of approach has been made by Brenig/4' by Fredkin and Werthamer/5' by Meissner/6 ' by Gillis and Werthamer17' and by de Wette, Nosanow and Werthamer.18' Its application to problems of ferroelectrics has been made by Miller and K wok.19'

In the latter treatment the low-lying excitation spectrum of a solid is regarded as a collection of lattice displacement waves or phonons. Its leading approximation, usually called self-consistent harmonic phonon scheme, has been derived many times by a number of authors. Its earliest and modest derivation is probably due to Hooton/0' whose work has sometimes been overlooked. Several approaches based on the variation principle to define effective harmonic Hamiltonian for anharmonic crystals have been made by Boccara and Sharma/0 '

by Gillis, Werthamer and Koehler21' and by Koehler.22' The merit of these elegant methods is that without lengthy calculation we can arrive at results equivalent to renormalized-harmonic-phonon approximations. Most recently, Werthamer has devised a more general functional variation method.23' While, Choquard has made extensive studies of the vibrational properties of anharmonic

*> The references 5) and 6) are review articles in which extensive lists of papers on quantum crystals are given.

*i:l The eigenfrequencies of phonons in the harmonic approximation are all imaginary over the whole Brillouin zone.

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A Theory of the Lattice Vibration of Anharmonic Solids 139

crystals utilizing the teclmique of renonnalized perturbation method.24> Green's function methods have also been formulated by Horner5> and by Plakida and Siklos/6> who have applied the method of Dominicis and Martin27> and that of Bogoliubov and Tyablikov,28> respectively. With these methods, we can proceed from the renonnalized-harmonic-phonon approximations in a straightforward manner and arrive at a result of any desired degree of accuracy. However, as is often the case with exact methods, its implementation leads to more involved calculations in general than does the use of the variational principle. A different method from the above ones has also been presented by Fixman using a technique employed in polymer theory. 29> Applications of renonnalized perturbation theoretic methods have been made by Ranninger30> and also by Gotze and Michel31> to study the transport properties of quantum crystals. Kugler has applied a variation method to problems of the Wigner electron lattice.32> Noolandi and Kranendonk have studied the vibrational properties of solid hydrogen.7>

Almost all previous self-consistent phonon theories, developed considerably in recent few years, have been formulated to make extensive studies of the vibrational and the thermodynamical properties of quantum crystals, in which atomic vibrations are highly anharmonic due to large zero-point motion of atoms. It appears, however, that, in addition to this rather specific problem, the physical contents of the theory can be applied to much general and wider class of problems, such as the vibrational properties of anharmonic solids in general, (softening of solids upon heating, for example), the melting of solids, phonons in liquids near their freezing temperatures, electron-phonon interaction in solids at high temperatures, the vibration of polymers, etc. Several recent works have also shown that the anharmonicity of vibrations in solids may be more wide-spread than is generally realized.33>-aB> Moreover, anharmonic vibrations seem to be generally important for virtual localized or localized modes in impure crystals due to the large excursion of impurity atoms. There have been some experimental indications which support this conjecture. 39>·40> Theoretical studies of the vibrational properties of quantum crystals containing isotopic impurities have been made very recently by Jones41> and by Vanna.42>

It is the purpose of this paper to study the properties of anharmonic solids without assuming the smallness of the anharmonicity of atomic vibrations from the outset. In view of the general importance of anharmonic vibrations in solids, a theory to be developed here is designed to reformulate theory of lattice dynamics from the viewpoint of self-consistent phonon theory. No particular attention is paid to the case of quantum crystals which have been the main object of current self-consistent phonon theories.5>.G> The principal concern of this paper is the anharmonic properties of solids, crystalline (pure or impure) or noncrystalline, in general. Attention is also given to their bearing on physics of liquids and imperfect gases. So, aside from some

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140 S. Takeno

general results, problems to be touched upon in this paper are effective interatomic or intermolecular potential or effective force constants, the dynamical instability or the melting of solids, phonons in liquids, anharmonic properties of impurity modes in crystals, impurity- or defect-induced anharmonic vibrations in solids, etc.

The program of this paper is as follows. In the next section the formulation of the problem is made using the method of double-time Green's function28l

to study the physical properties of phonons in anharmonic solids. A similar analytical method has been used previously by Plakida and Siklos.26 l In treating a set of hierarchy equations inherent in double-time Green's function method, a decoupling approximation procedure is employed which naturally leads to a self-consistent treatment for studying the vibration of anharmonic solids. In §3 general expressions for effective potentials are obtained, which are average values of bare interatomic potentials over the thermal motion of atoms in solids. In order to reduce these to tractable forms, a method of cumulant averages is employed.43 l In §4 it is shown that the first-order approximation for a self-consistent phonon theory developed in §2 yields renormalized harmonic phonons. Several sum rules convenient for studying the vibrational properties of anharmonic solids are obtained. In §5 we make use of a higher order approximation procedure to take into account (renormalized) phonon(renormalized) phonon interactions. In §6 a brief discussion is given on each of the problems mentioned previously. The last section is devoted to a brief summary of the results obtained in this paper.

§2. Formulation

We study the dynamical properties of a system of N atoms (or ions) under the assumption that the Born-Oppenheimer approximation1l is well satisfied. To encompass disordered and amorphous systems, we shall suppose our system to be arbitrary in structure and in composition. Let r. and P. be the position vector and the momentum of an atom which is numbered by index n in the system. To simplify notations in this paper, the index n and the Cartesian component a are subsumed in a composite index x, whenever appropriate; The a component A.cx of a vector A. associated with then atom is abbreviated as A(x). The Hamiltonian which describes the motion of atoms in the system is generally taken to be of the form

.9£= S{P(x) 2/2M(x)} + VN(r1rz···rN)- SF(x)r(x), (2·1) s s

where M(x) ==M. is the mass of the n atom, VN== VN( {r}) == VN(r1rz···rN) is the N-body potential which represents interactions of all atoms in the system, and F(x) is an external force which acts on the n atom. The basic assumption used in this paper is that atoms in our system constitute a solid,

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crystalline, noncrystalline or amorphous. This means that the free energy has a minimum at a set of atomic equilibrium positions. Thus, we decompose r(x) into two terms:

r(x) =(r(x))+u(x) ==R(x) +u(x), (2·2)

where R(x) is a mean or equilibrium position of r(x), u(x) is the displacement of the n atom from its equilibrium position,*> and the angular bracket denotes a canonical or thermal average. It is assumed that the solid under consideration is in thermal equilibrium with a heat bath. The equilibrim position R(x) is determined by the condition that a net force acting on a given atom at its equilibrium position is zero, namely**>

id(P(x))/dt=f(x)(VN)-F(x) =0, (2·3)

where f(x) is an abbreviation of 8/8R.rx.. When the solid is assumed to be in equilibrium with an externally applied stress arx.rx.'' we get the equation4>

arx.rx.,= (1/V)2J(8(VN)/fJR.rx.)R.a, (2·4) . which, for a uniform external pressure P, reduces to

P= ( -1/3V)2Jf(x)(VN)R(x). (2·5) z

Here V is the volume of the solid. Equation (2 · 5) can be used either to calculate the equilibrium distance of atoms in the solid for a fixed pressure or to calculate the pressure for a particular choice of atomic configurations.

We study atom-displacement waves or phonons in the solid, taking a set of displacements {u(x)} and momenta {P(x)} as dynamical variables. We therefore expand the potential VN in powers of the u's:

VN( {r}) =exp{2Ju(x.)f(x.)} VN( {R}) • "" =VN({R})+2J(1/n!) 2J K.(XtXz"·x.)u(xt)u(xz)···u(x.), (2·6)

n=l Z1S'2···S'n

where

(2·7)

is the n-th order force constants, in which the symbol [ ] o means that the bracketed expression is evaluated with all atomic coordinates at the equilibrium positions. It is easily seen that the force constants K.(xtXz"·x.) (n= 1, 2, 3, · · ·) are symmetric functions of Xt, xz, · · ·, x., namely

*> It is to be remarked that in the case of anharmonic solids the equilibrium positions {R} do not coincide with "rest positions" determined from the minima of the potential energy even when no external force is applied to them.

**' We use units with h=l throughout this paper.

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142 S. Takeno

To take into account the anharmonicity of atomic vibrations in full detail, we introduce a retarded or advanced double-time Green's function composed of a pair of operators A and B: 28 l

((A(t); B(t')))=+iO[±(t-t')]([A(t),B(t')]), (2·9)

with its Fourier transform

((A; B)).,= (1/2n) r~(A(t); B(t'))) exp [i(l)(t-t')] d(t-t'), (2·10)

where A(t) =exp(iHt)A exp( -iHt) is the Heisenberg operator for A, O(t) is Heaviside's step function. We are now concerned with a double-time Green's function G.(xx', t-t') =((u(x, t); u(x', t'))) composed of displacement-displacement correlation functions of atoms in the system. With the use of the commutation relations

[u(x),u(x')]=[P(x),P(x')]=O and [u(x),P(x')]=itl(xx'), (2·11)

an equation of motion satisfied by G. is written as

idG.(xx', t-t')/dt= (i/M(x))((P(x, t); u(x', t'))). (2·12)

Differentiating again the above equation with respect to t, we get

-M(x)d2G.(xx', t-t') /dt2=tl(t-t')tl(xx') ~

+ ~(1/n!) { ~ K.(xx2···x.)((u(x2, t) ···u(x., t); u(x', t') ))+ ··· n=2 xz···X,.

Taking into account the symmetry property of the K's, we can reduce the above equation to

-M(x)d2 G.(xx', t-t') /dt2 =B(t-t')tl(xx') ~

+ ~(1/n !) ~ K.+t(XXtXz···x.)((u(xt, t)u(xz, t) ···u(x., t); u(x', t') )). n=l Zt-S2'''%JI

(2·14)

An equation obeyed by G.(xx', (I)) =((u(x); u(x') ))., is therefore given by

M(x)(I)2 G.(xx', (I))= (1/2n)B(xx') ~

+ ~(1/n !) ~ K.+t(XXtXz···x.)((u(xt)u(xz) ···u(x.); u(x') ))., . n=l Z!Zz·•·Z,.

(2·15)

Retaining only the term with n=1 is equivalent to the harmonic approximation traditionally used in lattice dynamics.

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For later discussion it is more convenient to introduce the mass-reduced coordinates and momenta:

u(x) =M(x)-112 q(x) and P(x) =M(x) 112p(x). (2-16)

The commutation relations obeyed by these have the same form as Eq. (2 -11) :

[q(x),q(x')] = [p(x),p(x')] =0 and [q(x),p(x')] =ilJ(xx'). (2·17)

The Hamiltonian of the system is then rewritten as

-.j{,= (1/2)2Jp(x) 2 + 2J(1/n!) 2J Dn(X1X2'"X.)q(xi)q(x2) "·q(xn) x rh=l .:r1z2 ... .:r.

- 2Jf(x)q(x) +constant terms, (2·18) "

where f(x) =F(x)M(x)-112 and

Dn(X1X2'"Xn) =Kn(X1X2'"Xn) / [M(x1)M(x2) ... M(xn)] 112. (2·19)

The mass-reduced force constants D.Cx1X2'"Xn) is also a symmetric function of x1,X2, "',Xn, satisfying the same symmetry relation as Eq. (2·8). It is then easily seen that an equation satisfied by the Fourier transform G(xx', ro) ==G(xx') of a Green's function G(xx' t-t') ==.((q(x, t); q(x', t'))) takes the form

ro2G(xx', ro) = (1/2n)D(xx')

-+ 2J(1/n!) 2J Dn+1(xx1x2·--x.)((q(x1)q(x2)· .. q(x.); q(x'))).,. 8=1 X!Z2•••Z11

(2·20) A relationship between G.(xx', ro) and G(xx', ro) is

G.(xx', ro) = [M(x)M(x')] - 112 G(xx', ro). (2·21)

As it must, the second term in Eq. (2 · 20) or (2 ·15) contains higherorder Green's functions in the form of an infinite series. The conventional method here is to truncate the series at n=2 or n=3 at most and to treat anharmonic terms thus retained as small perturbations. One of the most natural procedure to arrive at a renormalized perturbation theory using a method of double-time Green's function is to make use of the following decoupling approximation*>

+ 2J( II q(x,) )((q(x,)q(x1); q(x'))) i<J "'""if)

+ 2J (II q(xJ))((q(x,)q(x1)q(x.); q(x')))+ .. ·, i<J<k l(""'iik)

(2·22)

*' We hereafter omit the subscript co for the Fourier-transformed Green's function, whenever appropriate.

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144 S. Tak:eno

which is in consonant with Wick's theorem of contraction of operators employed in field theory.44> Combining Eq. (2 · 22) with Eq. (2 · 20), we get

aiG(xx', ro) = (1/2n)tl(xx') ~

+~(1/n!) ~ 9J•+1(xx1xz···x.)((q(x1)q(xz)···q(x.);q(x'))), (2·23) 11=1 Z1Z2···Z,.

where

ffJ.(x1xz· .. x.) =0.(x1xz···x.) / [M(x1)M(xz) ···M(x.)] 112,

in which

(2·24)

(2·25)

is the n-th order "effective force constant" derived from an "effective or an average potential" ( VN>· In obtaining the above results, we have made use of the following relations

~ . ~(1/n!) ~ D.+1(xx1xz···x.)~( II q(x,)>((q(x;); q(x'))) n=l St.%2""Xn i=l J(~i)

= ~9Jz(xy1)((q(y1); q(x'))), (2·26a) 71

~

~(1/n!) ~ D.+1Cxx1xz···x.)~( II q(x.)>((q(x;)q(x1); q(x'))) n-1 s,sa···Sn I<J k(-foij)

= (1/2!)~9Jz(XY1Yz)((q(y1)q(yz); q(x'))), (2·26b) ~

~(1/n!) ~ D.+1Cxx1xz···x.) ~ ( II q(x;)>((q(x;)q(x1)q(xk); q(x'))) n-1 XlSg•••Xn i<J<k /(~jjk)

= (1/3!) ~ 9Ja(XY1YzYa)((q(y1)q(yz)q(ya); q(x'))), (2·26c) J't>'2.13

etc.

These can be obtained in a straightforward manner using the symmetry property of the D's. We see that Eq. (2 · 23) is formally equivalent to Eq. (2 · 20) provided the "effective force constants" ffJ.(x1xz···x.) are replaced by the "bare force constants" D.(x1xz···x.).

At this stage it becomes clear why the phrase "self-consistent" has been used. The phonon eigenfrequencies are determined by the effective force constants which depend on the correlation functions of the displacements (CFD) of atoms in the solid and CFD itself depends on the phonon frequencies. It is easily seen that truncation of the series on the right-hand side of Eq. (2 · 23) at n = 1 gives renormalized harmonic phonon frequencies, while truncations at n=2 and n=3 describe three and four (renorrnalized) phonon scattering processes, respectively. It should further be noted that in the renormalized harmonic approximation only even numbers of derivatives of the potential enter, while with the truncation procedure at n = 2 we can include odd derivatives. The physical meaning of the effective force constant may be understood as follows. An atom in an anharmonic solid, when moving through

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a large excursion, experiences anharmonic forces, which depend on how the other atoms are moving. A field seen by a given atom is thus smeared out or undergoes a random thermal modulation of the other atoms. Thus, it is seen that in anharmonic or "real" solids the physically meaningful quantities are effective potentials or effective force constants rather than bare potentials or bare force constants themselves.

§3. Effective potential

We study in this section the properties of the effective potential ( VN> Its explicit form is given by

N

(VN( {r}) >= (exp(::Eu.·P" .) > VN( {R} ). (3·1) #=l

It is now convenient to introduce the Fourier transform of VN:

or

( VN( {r}) >== (27t)~aN~ ··· ~d{k} CV N( {k} )exp(i~1k.· R.)

= (27t)-aN~···~d{k} VN({k})(exp(i~1k.·r.)>. (3·2)'

( {k} = (kl, k2, ···, k.))

Here the function ( exp (i ::E:-1 k. · r .) > is the structure factor of the atoms in the solid. By virtue of Eq. (3 · 2), the effective potential can be expressed in the form

where

(VN( {r} )>= ~ ···~d{R'} VN( {R+R'} )PN( {R'})

= ~···~d{R'} VN( {R'} )PN( {R-R'} ),

(d{R} = (dR1dR2 .. ·dRN))

PN( {R}) = (27t)-aN~ ···~d{k}(exp(iiik.·u.) >exp( -i"fdk.·R.)

(3·3)

(3·3)'

(3·4)

is the probability distribution function of the displacements of all the atoms in the solid, in which < exp (i ::E:-1 k. · u.) > is the characteristic function or the moment generating function45> of PN( {R} ). Such an exponential function appears directly in the theory of diffuse scattering of X rays or thermal neutrons by the thermal vibration of atoms in crystals.46> Defining the Fourier transform CV N( {k}) of < VN( {r}) > by Eq. (3 · 2)', we get

N

CV N( {k}) = (exp(i::Ek. · u.) > VN( {k}). (3·5) •-1

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146 S. Takeno

The factor < exp (i ~:=1 u. · k.)) can be understood as a measure of the random thermal motion of atoms in the solid which modulates the bare potential.

The evaluation of the average of the exponential functions appearing in Eqs. (3·1), (3·4) and (3·5) can be effected most conveniently by introducing the method of cumulant average. Following the method suggested by Kubo,48>

we obtain SN co SN

<exp(~u(x1)fi'(x1))) VN( {R}) =exp [~(1/ml)< {~u(x1)fi'(x1)} '").] 1=1 .. =2 1=1

X VN( {R}) (3·6) and

N - N <exp(i~k.·u.))=exp [~(i'" /m !)< {~k.·u.}'").],

n=l m=2 •=1 (3·6)'

where the angular bracket with subscript c denotes the cumulant average. The conventional approximation for treating Eq. (3·6) or (3·6)' is to

truncate the series in the exponent at the first term m = 2, which is equivalent to approximating the distribution function PN to be Gaussian in 3N-dimensional space. Equations (3·6) and (3·6)' then reduce to

SN

<exp(~u(xl)fi'(x,))) VN( {R}) 1=1

=expa~<u(x;)u(x1))fi' (x;)fi'(x1)} VN( {R} ), ij

(3·7)

N

<exp(i~k.·u.)) =exp{ -~~<u(x1)u(x;) )k(x1)k(x1) }. •=1 ij

(3· 7)'

Insertion of Eq. (3·7)' into Eq. (3·4) gives

PN( {R}) = (2n)-3N12 (detAN)-112 exp{ -~~A-1 (ij)R(x1)R(x1)}, (3·8) ij

where

(3·9)

is a 3N-dimensional dispersion or correlation matrix. With this approximation the effective N-body potential takes the form

<VN( {r} ))= (2n)-3N12 (detAN)-112~ ···~d{R'} VN( {R+ R'})

Xexp{ -~~A!l(ij)R(x1)'R(x1)'} ij

= (2n)-sN/2 (detAN)-112~ ... ~d {R'} VN( {R'})

Xexp[ -~~Ai/(ij) {R(x;) -R(x;)'} {R(x1) -R(x;)'} ]. (3·10) ij

We give here an example of self-consistent treatments, which in the renormalized harmonic approximation, proceeds as follows; The displacementdisplacement correlation function <u(x)u(x')) is evaluated from a solution

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for a Green's function ((u(x'); u(x))) using the relation28l

< ( ) ( '))= 2~~ Im[G(x'x, ro-iO)]d U XU X (o ) 1 ro,

-~ exp pro -(3·11)

where Im [A] denotes the imaginary parts of A and {1 = 1/ k8 T, k8 and T being the Boltzmann constant and the absolute temperature, respectively. Equation (2 · 23) with the first term in the series only, (3 ·10) and (3 ·11), together with the definitions of Eqs. (2·24) and (2·25), represent an implicit non-linear equation for the renormalized phonons such that their frequencies depend on their own thermal population.

It is to be noted that in the Gaussian approximation for PN no shortrange correlation effects for atoms arising from strong repulsive forces at their small mutual separations is taken into account; Here the penetration of the atoms into the core region of their comoving neighbours, though small but not vanishing, takes place. Physically, of course, the probability function PN must be sharply diminished for those configuration in which the potential develops a strong repulsion at small separations of the atoms where the electronic core of the atoms start to overlap. This point, particularly important for solid helium for example, will be touched upon very briefly in later discussion.

The above discussion for the effective potential is too general, which is rather inconvenient to obtain explicit forms for the effective potential. For this reason, we assume, as usual, the potential function VN to be pair-wise additive and spherically symmetric, namely

(3·12)

It is then a straightforward matter to specialize the result obtained above to this case. The effective spherically symmetric pair potential is given by

(3·13)

where ro is the instantaneous position of an atom whose mean position Ro==(ro) is taken to be the origin of the coordinate. As is the case for the N-body potential, Eq. (3 ·13) can be expressed in terms of the probability distribution function P1 as follows :

(Vl(r.-ro))= ~dR'V1(R.+ R')P1(R') = ~dR'V1(R')P1(R.-R'), (3·14)

where

(3·15)

m which

u.o=u.-uo (3·16)

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148 S. Takeno

is the relative displacement of a pair of atoms in the solid. Let us introduce, as before, the Fourier transform of the pair potential:

(3 ·17)

and

( v;_(r.-ro) )== (2n)-3~dkCVt(k)exp(ik· R.)

= (2n)-3~dk v;_(k) (exp{ik· (r.-ro)} ). (3·17)'

Defining, as before, the Fourier transform CVt(k) of (V1(r.-r0)) by the above equation, we get

CVt(k) =(exp(ik·u.o)) v;_(k). (3·18)

A cumulant expansion for the exponential function (exp (ik · u.o)) can be made in exactly the same way as before. We obtain

co

(exp(ik· u.o)) =exp{2:; (i'" /m !) ( (k·u.o)'").}, m=2

which, when truncating at the first term (m=2), reduces to

(exp(ik· u.o)) =exp [ -~ {( (k·u0) 2) + ( (k· u.) 2)}]

exp[!{((k·uo) (k·u.))+((k·u.) (k·uo))} ].

(3·19)

(3· 20)

In the above equation the first factor exp[ -H((k·uo) 2)+((k·u.) 2)}] is the familiar Debye-Waller factor associated with the zero-phonon process while the second exponential factor describes multi-phonon processes. Within the Gaussian approximation the effective pair potential is given by

( v;_ (r.-ro)) = (2n)-812 (detAt(nO) )-112~dR'v;.(R.+ R')

(3·21)

= (2n)-312 (detAt (nO) )-112~dR'Vt(R')

Xexp{ -~Z::A11 (n0) (R.-R')a(R.-R')a}, (3·21)' ail

where (3·22)

is the three-dimensional matrix composed of the pair correlation function of the relative displacement u.-uo.

We now give a brief discussion of the effect of short-range correlations which have not been taken into account in the Gaussian approximation. Most of theoretical studies on this problem have been done for helium with attention primarily paid to the zero-temperature ground state of the solid helium.47>-49>

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The usual approach has been to adopt a variational method rather than a Green's function technique and so to express a trial ground-state wavefunction ..P.Cr1r2···rN) in the Jastrow form: 50 l

where C is a constant and K is taken as an adjustable parameter. It is here understood that without the Jastrow factor a variation with respect to the tensor coefficients A1(mn) yields a renormalized harmonic approximation. The calculation of the energy expectation value using Eq. (3 · 23) has been made extensively by Nosanow and coworkers51l and by others52l'53 l using cluster expansion methods. The program of Nosanow et al. is first to use Eq. (3 · 23) in an energy expectation value truncated to include only two-body clusters, and then to use a variational procedure. The result thus obtained is that within certain approximations we still obtain renormalized phonons, but with a modified potential:

V1(r)-.. W1(r) =h(r) 2{ V1(r)- (ti}/2M)f72lnh(r) }.

(h(r) ==exp( -KV1(r))

(3·24)

Here, M is the atomic mass of helium and the f72lnh(r) term represents the cost in kinetic energy of having the overall wavefunctions vary rapidly in f(r). While, Meissner have suggested that for the evaluation of the effective harmonic force constants we should use a formula similar to Eq. (3 ·14) using an "exact" form for P1 (R) in which both short-range and long-range correlations are incorporated. It appears that the interrelationship between the results of Nosanow et al. and that of Meissner is not yet clear. We do not dwell upon this specific problem, so no further discussion is given on this problem. We merely mention here that in the spirit of Green's function method we can in principle follow a method suggested by Meissner.

We close this section with a discussion on the relation between the effective potential obtained here and a mean total potential energy of an imperfect gas or a liquid. In the limit of small relative displacement of atoms in the solids, of course, effective potential reduces to bare potential, which is surely an upper bound of the effective potential. Our next concern is an asymptotic expression for the effective potential in the limit of large atomic excursions. In this case the probability distribution P1 (R), for example, can be taken to be constant, namely

P1(R) =const=1/V. (3·25)

Equation (3 · 25) means that the distribution of atoms in the system is uniform. Inserting this into Eq. (3 ·14), we obtain

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150 S. Takeno

(Vl(r-ro))= (1/V) ~ V,.(R)dR. (3·26)

The physical meaning of this equation is made clear by considering the average value of the total potential. Combining of Eq. (3 · 26) with Eq. (3 ·12) gives

(VN( {r} ))= (N(N-1) /2V) ~ V1(R)dR, (3·27)

which is just a mean potential energy of an imperfect gas in first order perturbation theory.54> Equation (3 · 27) has previously been derived by Choquard as an asymptotic limit of integral ( 3 · 17)' for large A1 (nO).

In Eq. (3·25), no correlation between any pair of atoms is taken into account. We can generalize Eq. (3 · 27) to a more general situation. To do this, we remember that the formulation of the present paper has been made under the assumption that an eqilibrium position exists for every atom in the system. If we lift this assumption, the probability distribution function P1 (R- R') in Eq. (3 ·14) may be replaced by P1 (R'), which can be identified with a two-body correlation function of atoms in the system. With the use of the same procedure as before the mean value of the total potential energy is thus obtained as follows:

(3·28)

which is obviously a generalization of Eq. (3 · 27). If the pair potential and the two-body correlation function is spherically symmetric, Eq. (3 · 28) is rewritten as

(VN( {r} ))=2rcN(N-1) ~ V,.(R)g(R)R2dR, (3·29)

where g(R) is a pair correlation function. Equation (3·28) or (3·29) has been used for an approximate expression for a mean potential energy of a liquid or an imperfect gas.54>·55> Equation (3 · 26) can be considered as a lower bound of the effective potential.

The results obtained in this section for the properties of the effective potential have a number of applications to various problems in the vibrational properties of solids and physics of liquids as well, some of which will be discussed in §6.

§4. Renormalized harmonic approximation

In this section we employ the lowest-order approximation m treating Eq. (2·23), truncating the series at n=1, namely

(J)2G(xx') = (1/2rc)6(xx') + ~[J}z(xx")G(x"x'). (4·1) ""

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With this approximation the Green's function G constitutes a closed set of equations by themselves. We therefore obtain renormalized phonons without damping. An eigenvalue equation associated with Eq. ( 4 ·1) is given by

al,P.(x)- ~.fDz(xx'),P.(x') =0. (4·2) zl

Let c.o(A) and ,Y.(x, ..l) be the eigenvalues and the eigenfunctions of this equation characterized by a set of quanties ..l which takes on 3N values. Here, the ket I ..l) is said to be the representation in which the operator or matrix .fD2 = (I!J2(xx')) becomes diagonal. We impose the orthonormality and the closure condition for the ,Y.'s:

~,P.(x, ..l),Y.(x, l) * =lJ(Al), z (4·3) ~ ,P. (x, ..l) ,P. (x', ..l) * = lJ (xx').

A

Then, the Green's function G(x, x') can be expressed as follows

G(xx') = (1/2n) g(xx'), (4·4)

where

g(xx') ==.g(xx', c.o) = ~ ,P.(x, ..l),Y.(x', ..l) * A c.o-c.o2(A) (4·5)

is the (renormalized) phonon Green's function. The Green's function G.(xx', c.o) is therefore given by

G.(xx', c.o) = (1/2n) [M(x)M(x')] - 112g(xx'). (4· 6)

We give an explicit procedure for the self-consistent method within the framework of the renormalized harmonic approximation, under the assumption that the eigenvalue problem given by Eq. ( 4 · 2) has been solved. From Eq. (3·11) the displacement-displacement correlation function (u(x')u(x)) is expressed in terms of g (xx') as follows

(u(x')u(x))= (1/n [M(x)M(x') )112) r~ lm[g(xx', c.o-iO)] dc.o. (4· 7) J-~ exp(/1c.o) -1

Inserting Eq. ( 4 · 4) into this, we obtain

(u(x')u(x))

= (M(x)M(x') )-112 2j (1/2c.o(A) )coth(J1c.o(A) /2),Y.(x, ..l),Y.(x', ..l) * ( 4· 8) A

or

(q(x')q(x) )= 2j(1/2c.o(A) )coth(J1c.o(A) /2),Y.(x, ..l),Y.(x', ..l) *. (4·8)' A

In the asymptotic limit T"::;phc.oM/k8 (c.oM is the maximum value of c.o(A), approximately identified with De bye cut-off frequency) and T-o, where the

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152 S. Takeno

function (w(A) /2)coth(Sw(A) /2) reduces to k8 T(classical) and w(A) /2(quantal), respectively, Eq. ( 4 · 8) reduces to

{ [ko T/ (M(x)M(x')) 112] ~w(A)-2 '1/r(x, .A.)'I[r(x', A.)* ( 4· 9a)

(u(x')u(x))= J..

(M(x)M(x'))-112 ~w(A)-1 'l/r(x, .A.)'I[r(x', A.)*. (4·9b) }..

Equations (4·1) or (4·2), (2·24), (2·25), (3·21) or (3·21)', (4·8) or (4·8)' should be treated simultaneousely.

A useful sum rule can be obtained by considering the sum ~zz,fP2(xx') X(u(x)u(x')). With the aid of Eqs. (4·8)', (4·2) and (4·3) the sum is

calculated to be*l

S==~tPz(xx')(u(x)u(x') )= ~9J2(xx')(q(x)q(x') )= ~E(A, T), (4·10) - - }.. where

E(A, T) = (w(A) /2)coth(Sw(.A.) /2) (4·11)

1s the mean or average energy of the .A.-mode phonon at temperature T. Equation (4·10), when considering high and low temperature limits, reduces to

{3NkoT s-~w(A) /2.

}..

(4·12a)

(4·12b)

An alternative expression for the sum rule can be obtained by observing the fact that for the pair-wise potential we have

(4·13)

Using this relation, we get

2 ~ ca< V';_) /8At (xx')) At (xx') = ~ E(A, T). (4·14) - }..

This sum rule has been obtained by Choquard.24l We shall now specialize the results obtained in this section to the case

of a pure crystal composed of identical atoms with mass M, assuming the crystal structure to be primitive for the sake of simplicity. The eigenfunction then takes the form

'l/r (x, A.) =='1/ra (n, A.) = N-112 exp (ik · R.) ea (k, t1) (4·15)

with

~ea (k, a)ea (k, a')* =B(aa'), ~ ea (k, t1 )ea' (k, a)* =B(aa'), (4·16) a a

where k is the wave vector, a denotes the branches of phonon modes, one is

*l Equation (4·10) shows that the Virial theorem is satisfied.

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longitudinal and the other two are transverse, and e(k, o) is the polarization vector. An explicit expression for Eq. ( 4 · 2) is then given by

Wu(k) 2 ea (k, a)

= (1/M)h h f1 afla,( Vl(r.-ro) ){1-exp( -ik·R.)}ea,(k, o), (4·17) a' •("TO)

where we have made use of the relation

hKaa,(nn') =0. (4·18) •'

Equation ( 4 ·17) has been obtained by a number of authors. The parameter A1 (n0) entering into Eqs. (3·21) is determined by the equation

Al(nO)a.a

= (2/MN) h {cothSwu(k) /2} {1-cos(k· R.) }ea(k, a)e.a(k, o) *. ( 4·19) ku

It is instructive to obtain an alternative form for Eq. ( 4 ·17) by introducing the Fourier transform of the pair potential. For the present case it is more convenient to rewrite Eq. (3 ·17) as

~(R.) = (1/N)hVl(k)exp(ik·R.). (4·20) k

Putting this into Eq. ( 4 ·17) and using Eq. (3 ·17)', we get

wu(k) 2 e(k, a)= (1/M) hq(q·e(k, o)) ~(q) (1/N)h [exp{i(q-k) ·R.} • •

-exp(iq· R.)] (exp{iq· (u.-uo)} ). (4· 21)

With the aid of the orthonormality condition for the e's, we can also obtain the following sum rule:

Mhwu(k) 2 = hifV1(q) (1/N)h [exp{i(q-k) ·R.} 0" • •

-exp(iq·R.)] (exp{iq· (u.-uo)} ). (4·22)

In treating the modulation factor (exp{iq· (u.-uo)} ), here we confine ourselves to using the Gaussian approximation. Then, in view of Eqs. (3·20), ( 4 · 21) and ( 4 · 22) are rewritten as

wu(k) 2 e(k, o) = (1/M) hq(q· e(k, a)) ~(q)exp(- (q·u0) 2)S(k, q) ( 4· 23) •

and

Mhwu(k) 2 = hq2 ~(q)exp(- (q·uo) 2)S(k, q) (4·24) " .

respectively, where

S(k, q) = (1/N)h [exp{i(q-k) ·R.} . -exp(iq· R.)] exp( (q· uo) (q· u.) ). (4·25)

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154 S. Takeno

The conventional procedure for treating the factor exp( ( q · uo) ( q · u.)) 1s to expand it as a power series:

exp( (q· uo) (q· u.))

= 1 +((q·uo) (q·u.))+i((q·uo) (q·u.))2 + ···, (4·26)

with the observation that the first and the second terms describe zero-phonon and one-phonon processes, respectively while all the other terms characterize multi-phonon processes. It is seen that there is a close correspondence between the manner of the random thermal modulation of phonons in anharmonic crystals and that of X rays or thermal neutrons in crystals.

As is the case for harmonic solids, Eq. ( 4 ·17) has been used extensively; with a modification due to a hard-core effect, for the calculation of phonon dispersion curves of quantum solids. Equation ( 4 · 21) with the average of the exponential factor replaced by unity56' has often been used for the evaluation of phonon dispersion curves in metals. *'· 57>-sa> It will be shown in §6 that the results obtained above are useful for studying the dynamic stability of anharmonic solids within the framework of the renormalized harmonic approximation.

§5. Higher-order approximation

In order to take into account phonon scattering processes, we proceed from the renormalized harmonic approximation, retaining the terms with n = 1 and 2 on the right-hand side of Eq. (2·23). This approximation is equivalent to taking into account three (renormalized) phonon scattering processes. We then obtain

aiG(xx') = (1/2TC)8(xx') + ":E!JJ2(xx'')G(:r!'x') zll

+! ";E. !JJ2 (xx" :r!") G2 (x" x"', x'), (5·1) zllzlll

where G2Cxx",x')-==((q(x)q(x"); q(:r!))). Now it is a straightforward matter, with the aid of Eqs. (2 ·17) and (2 ·18), to obtain an equation of motion for the new Green's function G2(x:r!',x'; t-t')==((q(x,t)q(:r!',t); q(:r!,t'))). It is written as

-d2 G2 (xx'', x'; t-t') I dt2 = - f(x)G(:r!' :r!; t-t')-f(x") G(xx', t -t')

-2((p(x, t)p(x", t); p(:r!, t'))) ~

+";E. (1/n!) ";E. D.+l Cxx1x2···x.) ((q(:r!', t)q(xl, t)q(x2, t) ···q(x., t); q(x', t'))) 11=1 Z}%z···Z11

~

+";E. (1/n!) ";E. D.+l (x"x1x2·· ·x.)((q(x, t)q(xl, t)q(x2, t) ···q(x., t) ; q(:r!, t') )). n=l Z!Zz··•Zn

(5·2)

*' A self-consistent treatment of anharmonic phonons in metals is worth separate discussion, and it is omitted in this paper.

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Applying the Fourier transformation, we get

oiG(xx",:x!) = -f(x)G(x"x') -f(:x!')G(x:x!) -2((p(x)p(x"); q(x'))) .. + 2j(1/n!) 2j Dn+t(XXtXz .. ·x.)((q(:x!')q(xt)q(x2) ···q(x.); q(x')))

H=l Zl%z•••z11 .. + 2j (1/n!) 2j D.+l (:x!'xtx2· ··x,.) ((q(x)q(xt)q(xz) ···q(x.); q(:x!) )}.

#=l .s'!Zz•••%11

(5·3)

Equation (5·2) (or (5·3)) yields a new Green's function G,(x:x!',:x!; t-t') ==((p(x, t)p(:x!', t); q(:x!, t'))) (or its Fourier transform G,(x:x!', :x!; (J)) =G,(xx", :x!) =((p(x)p(:x!'); q(:x!) ))) to be studied. Using the same procedure as before, we can express an equation of motion for this in the form

dG, (x:x!', x'; t- t') / dt = f(x") ((p (x, t) ; q (:x!, t'))) + f(x) ((p (:x!', t) ; q (:x!, t'))) .. - 2j(1/n !) 2j D.+t(x"XtXz···x.)((p(x, t)q(xt, t)q(xz, t) ···q(x., t); q(x', t')))

•=1 z1zz ... z,. .. - 2j (1/n !) 2j D.+t(XXtXz···x.)((q(xt, t)q(xz, t) ···q(x., t)p(x", t); q(:x!, t') )),

H=l z 1zz•••Z11

(5·4)

where use has been made of the relation <p(x))=O. A Fourier-transformed equation of motion for this is

i(J)G,(xx", x') =f(:x!')((p(x); q(:x!) ))+ f(x)((p(x"); q(x'))) .. + 2j(1/n !) 2j D.+t(X1XtXz···x.)((p(x)q(xt)q(xz) ···q(x.); q(:x!)))

n=l ZIXZ···z8 .. + 2j (1/n!) 2j D•+t(XXtXz···x.)((q(xt)q(xz) ···q(x.)p(x"); q(x') )).

n=l Zizz··•za

(5·5)

It is seen that except ((p(x, t); q(:x!, t'))) a pair of new types of Green's functions ((p(x, t)q(xt. t)q(xz, t) ···q(x., t); q(x', t'))) and ((q(xt, t)q(xz, t) ··· q(x., t)p(x", t); q(x', t'))) or their Fourier transforms appear further.

Before obtaining equations of motion for these, we employ a decoupling approximation for treating the right-hand side of Eqs. (5·3) and (5·5), obtaining

• ((q(x)q(xt)q(xz) ···q(x.); q(:x!) ))= < IIq(x;) )((q(x); q(x')))

i=l

+ 2j<q(x) II q(x1) )((q(x;); q(:x!))) + 2j< II q(x1) )((q(x)q(x;); q(x'))) i Kf-il i J<+il

+2j<q(x) II q(xk))((q(x;)q(x1); q(x')))+···, (5·6) i<J k(+ii)

n

((p(x)q(xt)q(x2)···q(x.); q(:x!)))=<IIq(x;))((p(x); q(x'))) i=l

+ 2j<p(x) II q(x1))((q(x;); q(:x!)))+ 2j< II q(x1))((p(x)q(x;); q(x'))) j j( oFi) i<J j( +il

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156 S. Takeno

(5·7a)

• ((q(xt)q(xz)···q(x.)p(x''); q(x')))=(llq(x;))((p(x''); q(x')))

i=l

+ 2X rr q(xj)p(x'') )((q(x;); q(x') ))+ 2X rr q(xj) )((q(x;)p(x''); q(x'))) ; J("'ei) 1<1 j("'el)

+ 2X rr q(xk)p(x'') )((q(x;)q(xJ; q(x') ))+ .... I<J k("'eii)

(5·7b)

The approximation used above, in its sprit, is eqivalent to Eq. (2 · 22). A set of equations to be studied further on within the approximation used in this section are ((p(x); q(x'))), ((q(x)p(x''); q(x'))) and ((p(x)q(x''); q(x'))). The first one of these, ((p(x); q(x'))), is related to G(xx') by the equation

((p(x); q(x')))= -iwG(xx'). (5·8)

With the aid of the same straightforward procedure as before, equations of motion for the remaining two are written as

d((q(x, t)p(x", t); q(x', t') ))/dt=f(x")G(xx'; t-t') +Gp(xx", x'; t-t')

-- ~ (1/n !) ~ D.+t(x''xtXz"·x.)((q(x, t)q(xb t)q(xz, t) .. ·q(x., t); q(x', t') )), n=l Xtzz···Xn

(5·9)

d((p(x, t)q(x'', t); q(x', t') ))/dt= f(x)G(x"x'; t-t') + Gp(xx", x'; t-t')

-- ~ (1/n !) ~ .Dn+t(XXtXz"·x.)((q(x", t)q(xt, t)q(xz, t) .. ·q(x., t); q(x', t'))). 8=1 ZtS'z···Z,.

(5·10)

Fourier-transformed equations corresponding to these are

-iw((q(x)p(x"); q(x') ))= f(x'')G(xx') +GP(xx'', x') -- ~(1/n!) ~ D.+t(x''xtXz"·x.)((q(x)q(xt)q(xz) .. ·q(x.); q(x'))), n=l ZtSz···Z·

(5·11)

-iw((p(x)q(x''); q(x')))=J(x)G(xx'') +GP(xx'',x')

-- ~(1/n !) ~ D.+t(XXtXz"·x.)((q(x")q(xt)q(xz) ... q(x.); q(x') )). •=1 ztzz···z"

(5·12)

We shall now use the same procedure as that employed previously in proceeding from Eq. (2·20) to Eq. (2·26). A set of relationships thus obtained are

-~(1/n!) ~ D.+t(x''xtXz"'X.)(q(xt)q(x2) .. ·q(x.))((q(x); q(x'))) fJ=l ZtZz···S,.

(5·13a)

-~(1/n!) ~ D.+1(x''XtX2'"Xn)~(q(x) rr q(xj))((q(x;); q(x'))) n=l z 1zz···z,. i j(~=i)

= ~_q);(x''y1; x)G(y1x'), (5·13b) ¥1

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A Theory of the Lattice Vibration of Anharmonic Solids 157 ~

~(1/n!) ~ Dn+t(x''xtXz···x.)~( IT q(x1))((q(x)q(x,); q(x'))) #=l Z!Xz•".l'* i J(~i)

= ~.fDz(x''yt)Gz(Xyt, x'), (5 ·13c) ,, ~(1/n!) ~ Dn+t(x''xtXz···x.)~(q(x) IT q(x.))((q(x,)q(x1); q(x'))) •=1 .<1.<2···.<0 I<J lt(-/oli)

= (1/2!)~.fl)g(X11YtYz; x)Gz(YtYz,x'), (5·13d) .11Y2

where

and

.fD;(x''yt; x) =P"(x'')P"(yt)(VNq(x)),

9JK(x''ytyz; x) =P"(x'')P"(yt)P"(yz)(VNq(x)),

~ n

~(1/n!) ~ Dn+t(x''xtXz···x.)(ITq(x,))((p(x); q(x'))) n=l Zl.S:Z•••%11 i=l

=.fDt(x)((p(x); q(x'))),

~(1/n!) ~ Dn+t(X11XtXz···x.)~(p(x) IT q(x1))((q(x,); q(x'))) ft=l X}X2•••Z,. i j(~i)

(5·14a)

(5·14b)

(5·15a)

=~~(x; x"yt)G(ytx'), (5·15b) ,, -~(1/n!) ~ Dn+t(XXtXz···x.)~( IT q(x1)p(x''))((q(x,); q(x'))) n=l x1zz .. ·s• i j(~i)

= ~~(xy1; x'')G(ytx'), (5·15c) ,, -~(1/n!) ~ D.+l(x"xtxz···x.)~( IT q(x1))((p(x)q(x1); q(x'))) •=1 _.,_., ... _.. I<J j(~l)

= ~.fDz(x''yt)((p(x)q(yt); q(x') )), (5·15d) J'l

~

~(1/n!) ~ Dn+t(XXtXz···x.)~( II q(x1))((q(x1)p(x''); q(x'))) •=1 .<t.<2···.<o I<J j( ... l)

= ~IfJz(Xyt)((q(yt)p(x''); q(x'))), (5·15e) J'l

~

~(1/n!) ~ Dn+t(x''xtxz···x.)~(p(x) II q(x.))((q(x,)q(x1); q(x'))) 11=1 z1zz•••%• i<J k(~ij)

= (1/2!)~IJJ:(x; x''YtYz)Gz(YtYz,x'), (5·15£) .11.12

~

~(1/n!) ~ D.+t(XXtXz···x.)~( IT q(x.)p(x''))((q(x1)q(x1); q(x'))) IJ=l Z!ZZ•••.I'n i<J k(~ij)

= (1/2 !)~IJJ:(xytyz; x'')Gz(YtYz, x'), (5·15g) J'lYl

where

~(x; x''yt) =P"(x")P"(yt)(p(x) VN),

~(xy1; x") =P"(x)P"(yt)(VNp(x")),

IJJ:(x; x''YtYz) =P"(x")P"(yt)P"(yz)(p(x) VN),

IfJ:(XYtYz; x") =P" (x)P" (yt)P" (yz)( VNp(x'') ).

(5·16a)

(5·16b)

(5·16c)

(5·16d)

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158 S. Takeno

Insertion the relations (5·13) and (5·15) into Eqs. (5·3), (5·5), (5·11) and (5 ·12), we can rewrite these equations as

oiGz(x:r!', x') = -2G;(xx", x') + 2j{.fD;(xy1; x'') +.fD;(x"yt; x) }G(y1x') Y1

+ 2J {_q:)z(xyl)Gz(x''yl, x') +mz(x''yt)Gz(xyl, x')} .1'1

+i2J{_q:):(xy1yz; x'') + _q:)g(x"ylyz; x)} GzCY1Yz, x'), (5·17)

+ 2J{_q:)z(xy1)((q(yt)p(x''); q(x')))+_q:)z(x''yt)((p(x)q(y1); q(x')))} .1'1

+!2J{!l)C(xy1yz; x'') +!DC(x; X 11Y1Yz)}GzCY1Yz, x'), (5·18)

-i(J)((q(x)p(x''); q(x')))=G;(x:r!', x')- 2j_q:)j(x''y1; x)G(ytx') .1'1

(5·19)

-i(J)((p(x)q(x''); q(x') ))= G;(x:r!', x')- 2j_q:)j(xy1; x'')G(ytx') J'1

- 2J_q:)z(Xyt)Gz(x"yt, x') -!2J~(XYtYz; x")Gz(YtYz, x'). (5·20) j't .YtYZ

Here, all the terms containing the external force f have been cancelled out exactly with the terms involving first derivative of the effective potentials (See Eq. (2·3)). It is seen that in the above set of equations there arise "new type of force constants" defined by Eqs. (5 ·14) and (5 ·16) which are derivatives of the correlation functions of the coordinates or momenta and the bare potential.

Equations (5·17)-(5·20), in conjunction with Eq. (5·1), give the eigenfrequencies of renormalized phonons with phonon-phonon interactions taken into account. A detailed study of solutions of this simultaneous equation deserves separate discussion, so it is omitted here. As is expected from the results obtained in §4, the above set of equations must yield, within certain approximations, self-consistent renormalized anharmonic phonons characterized by the third order effective force constants ma' which, when ma is replaced by the bare force constants Da, reduce to anharmonic phonons obtainable using conventional perturbation method. To arrive at this result, it is necessary to obtain relationship between the force constants m. and m: and 9Y. (n=2, 3). This can be achieved by employing further a decoupling approximation for m: and m~. We illustrate this procedure for g)z(x"yt; x) and g)~(x; x"y1), separately. It is expressed as follows:

-g};(x"y1; x) = 2J(1/n !) 2J D.+z(x''ytXtXz···x.)<q(x)q(xl)q(xz) ···q(x.)) ft=l ZtXz•••Z• -=2J(1/n!) 2J D.+z(X11YtX1Xz···x.)2J<q(x)q(x;))< II q(xi)):

n=l ZtXz··•X,. i j(~i)

= 2Jg)a(X11YtYz)<q(x)q(yz)) (5· 21a) .)'2

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and

gyg(x; x"y,) = 2:}(1/n!) 2J D.+z(x'y,x,xz···x.)(p(x)q(x,)q(xz) ... q(x.)) n=l XtXz···x.

~

=2:}(1/n!) 2J D.+z(x"y,x,xz···x.)2J(p(x)q(x;))( II q(x1)) n=l XtZz···s8 i i(~i)

= 2J§}a(:x!'y,yz)(p(x)q(yz) ). (5· 21b) ,. Using the same method, we are lead to the results

With the aid of Eqs. (5·21), Eqs. (5·17)--(5·20) reduce to

ro2 Gz (xx", x') = - 2Gp(x:x!', x') + 2J {.fDa (xy,yz) (q(yz)q(:x!')) .VtY2

+ §Ja(x"y,yz)(q(yz)q(x))} G(y,x')

+ 2J {.fDz(xy,)Gz(:x!'y,, :x!) + §Jz(x"y,)Gz(xy,, x')}, ,,

+ §Ja(x"y,y2) (p(x)q(yz) )} G(y,x')

(5· 21c)

(5·21d)

(5·21e)

(5·21£)

(5·22)

+ 2J{§}z(xy,)((q(y,)p(:x!'); q(x')))+.fDz(x"y,)((p(x)q(y,); q(x')))}, , (5·23)

iro((q(x)p(:x!'); q(x'))) = - Gp(XX11, x') + 2J§Ja(x"y,yz)(q(yz)q(x) )G(y,:x!) J'IJ'Z

+ 2J.fDz(x'y,)Gz(xy,, x'), (5· 24) ,,

+ 2JSDz(xy,)Gz(x"y,, :x!), (5·25) n

where we have neglected, for the sake of simplicity, all the terms containing higher order effective constants SD4. Insertion of Eqs. (5·24) and (5·25) into (5·23) then gives

ro2 Gp(xx", x') = 2:} {SD2(xy,)Gp(y1x", x') + SDz(x"y,)Gp(xy,, x')} Y!

- 2:} {.fD2(xy,).fD2(x"y2) + SD2(x"y,)SD2(xy2)} G2(y,y2, x') Y1Y2

- 2J {§Ja(xy,y2)C'(y2x") + SDa(x"y,y2)C"(xy2)} G(y,:x!) YtYz

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160

where

S. Takeno

C(xx!) =(q(x)q(x')), C'(xx') =iw(q(x)p(x')),

C"(xx!) =iw(p(x)q(x')). (5·27)

Equations (5·1), (5·22) and (5·26) are a set of equations to be solved within the approximation used above. With the use of the first order approximation employed in §4 for Eq. (5 · 27), Eq. (5 · 26) can be simplified further. From Eqs. (3·11) and (5·8), we then obtain the relation

C'(xx') =- (w/2)8(xx') and C"(xx") = (w/2)8(xx'). (5·28)

With this procedure Eq. (5 · 26) reduces to

w2 Gp(xx", x')- h {9Jz(xyt)GP(YtX11, x') + 9Jz(x"yt)Gp(Xyt, x')} Yl

- h {9Jz(Xyt)9Ja(X11YtYz) +9Jz(x"yt)9Js(x'YtYz) }C(ysyt)G(yz, x'). Y1Y2YS

(5·29)

It is now convenient to introduce the A-representation for which 9J2 ,

namely the Green's function G in the renormalized harmonic approximation, becomes diagonal. Let the A·representation of the Green's functions G, G2 , Gp, 9Js be G(A, l), Gz(;.J.", l), Gp(U", l) and 9Ja(..l, l'l), respectively. In terms of these, Eqs. (5·1), (5·22) and (5·29) can be expressed in the form

(w2 -w(A) 2)G(;., l) = (1/2n)8(U') +! h 9Js(;., l'l")Gz(l'l", l), (5· 30) )t,/1)..1/f

(w2 -w(A) 2 -w(l') 2)Gz(U", l) = -2Gp(U", l)

+ h {C(l')SJJs(A, l"l) +C(;.)9Ja(l', l"l)} G(t", t), (5· 31) ),..f/1

(w2 -w(A) 2 -w(;.'') 2)Gp(U", l) = -2w(A) 2 w(l') 2 Gz(U", l)

-C(J.)w(;.) 2 h 9Js(l', l";.')G(l", l) -C(l')w(l') 2 h SJJa(;., l";.')G(l", l), )..Ill ).,/11

(5·32) where

C(A) =coth(Sw(;.) /2) /2w(;.) = {2N(A, T) + 1} /2w(;.), (5·33)

in which

N(A, T) = {exp(Sw(;.)) -1} -t. (5·34)

A solution of Eqs. (5·30) ~(5·32) for the diagonal part of the Green's function G(;., ..l) can be expressed in terms of the self-energy ll(A, ..l) as follows:

(5·35)

where

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2(;., ..l)

=a> (A) ~ 1 CVs(A, l'l") 12{ N(l') +,,NO"') ;f; 1 N(A") + N(A"') + 1 Jo.ll),.fll a>-a>(A ) -a>(A ) a>+a>(t') +a>(l")

N(t') - N(A"') N(A")-N(t") } + a>+a>(t')-a>(t") + a>-a>(t')+a>(l") ' (5·36)

in which

CVs(A, l'l") =flJs(A, l'l") / [&o(..l)a>(l')a>(l")] 112• (5·37)

The result obtained here can be considered as a self-consistent anharmonic phonon theory. In fact, Eq. (5 · 36) has exactly the same form as that obtained previously using conventional perturbation theoretic method for the case of perfect crystals60 >'61> and also that of imperfect crystals62> provided the effective third order force constants flJa are replaced by the bare third order force constants Da. It is to be noted that in order to proceed from Eq. (5 ·1) to (5 · 36) a number of approximations and simplifications have been used. It is important to obtain explicit results for self-consistent anharmonic phonons which are free from the above restriction. A study of this problem is, however, omitted in this paper.

§6. Applications

In this section we apply the results obtained in the preceding sections to various problems. Several anharmonic properties of solids and implications of the obtained results are discussed from the viewpoint of the self-consistent phonon theory.

(1) Explicit forms for the effective potential using specific model potentials

We are concerned with an explicit form for the effective potential, assuming the bare potential to be known. We do this for the case of pair-wise additive and spherical potentials. For this purpose we assume, for the sake of simplicity, that the correlation matrix A1(nO) given by Eq. (3 · 22) is of the form

(6·1)

In fact, this relation holds exactly for a cubic Bravais lattice.3> Equations (3 · 21) and (3 · 21') then reduce to

(Yt(r.-ro))==(Yt(r.o))= (2n..l(n0))-312~dR'Vt(R + R')exp{- R'2 /2.-l(nO)}

(6·2) and

(Vl(r.0))= (2n..l(n0))-312~dR'Vt(R')exp{- (R'-R) 2/2..l(n0)}, (6·2)'

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162 S. Takeno

respectively. For practical calculations Eq. ( 6 · 2)' is more convenient, which, for the case of spherical potentials, reduces to

(Vt(r.o))= (1/Tt112R)(r { [ {2A(nO)P12x+R] V1C {2A(n0)} 112x+R)e-s'dx -R/{2.\(.0)}112 -r [ {2A(nO)P12x-R] V1( {2A(nO)P12x-R)}e-s'dxJ. (6·3)

R/2.\(•0)} 112

The evaluation of Eq. (6 · 3) is made here for the Morse potential and the Lennard-Janes potential, respectively.

(A) Case of the Morse potential The bare potenial takes the form

V,.(R) = Uo[exp{ -2a(R-Ro)} -2 exp{ -a(R-Ro)}] (6·4)

with a minimum value - Uo at R = Ro, where Uo, Ro and a are constants. Inserting this into Eq. ( 6 · 3), we obtain

( Vt(r-ro))/Uo==C{lt(R, A) /Uo

=A(R, a, A) -2A(R, a/2, A) +A( -R, a, A) -2A( -R, a/2, A), (6·5)

where A= (1/3)((uR-uo) 2) and

A(R,a,A)

= {1- (2aA/R)}exp{ -2a(R-Ro) + 2a2A}7t-112 Erfc((2A) 112a- (2A)-112R),

(6·6) in which

Erfc(x) = r exp( -t2)dt= (7t112/2) -Erf(x)==(7t112/2)- ~:exp( -t2)dt

(6·7)

is the Gauss error function. Equation ( 6 · 6) is slightly different from the result obtained by Choquard.24 l It is easily seen from the properties of this function that for small A112 / Ro the main contribution to the effective potential comes from the first two terms in Eq. (6·5). An approximate expression for Eq. (6·5) for this case can be obtained by using the following asymptotic expressions for Erfc(x) and Erfc( -x) for large x: 63 l

~

Erfc(x) =exp( -x2 ) :L;( -1)" (2n)! /22"+1n! x 2"+1 (6·8) n=O

and

Erfc( -x) =Tt112 -exp( -x2 ) :L;( -1)" (2n)! /22"+1n! x 2"+1. (6· 8)' n=O

(x>O) Using these equations, we get

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CV1(R, ..<)/Uo=exp{ -2a(R-R0)} {1- (2a..</R)}exp(2a2A)

-2 exp{ -a(R-Ro)} {1- (a..</R)}exp(a2..</2) ~

-n112 exp(-R 2 /2..<)~ [ ( -1)" (2n)! /2"+112 n!] (.,{112 /R) 2"+1 H=l

X {exp(2aRo) [{1-(2..<a/R)}-2"-{1+(2..<a/R)}-2"]

-2 exp(aRo) [ {1- (Aa/R)} -2"- {1 + (..<a/R)} -2"]}. (6·9)

For small ..<112 /Ro the last term, expressed in the form of a series, can be neglected as compared with the first two terms. We then obtain

CV1(R, ..<)/Uo=exp{ -2a(R-Ro)} {1- (2a..</R)}exp(2a2..<)

-2 exp{ -a(R-Ro)} {1- (a..</R)}exp(a2..</2).

This result has been obtained by Plakida and Siklos.26 J

(B) Case of the Lennard-Janes type potential

(6·9)'

We take a generalized Lennard-Janes type potential to be of the form

Vt(R) == U0 {(m/(n-m)) (Ro/R)"- (n/(n-m)) (Ro/R)m}, (6·10)

(n>m)

where Uo and Ro are constants and n and m are integers. As is the case for the Morse potential, this has a minimum value - Uo at R = Ro. Here, there exist some difficulties in the calculation of the effective potential. If both of n and m are even, an effective potential can be expressed in the form

q; (R ..<)/U. _ 1 [ n ( Ro )m \~ exp( -x2 )dx 1 ' o- n112 n-m R J-~ [ {(2A) 112x/R} -1] m 1

m ( Ro )" \ ~ exp( -x2)dx J n-m R J-~[{(2..<) 112 x/R}-1]" 1 ·

(6·11)

However, if either of n and m are odd, no such a simple expression can be obtained, so a discussion of this case is omitted. For the evaluation of the above integral the following relation is exploited

(6·12)

It is here understood that the principal value of the integral is to be taken. We then obtain

CV1 (R, ..<)/Uo= n~m (~or- n:!_m (~or + 1 i_;CA112/Ro) 2"(Ro)2"{m(Ro)"C2P+n-2)!

n-m p=1 2"p! R R (n-2)!

-n( ~0 rc2~~~;)~)! }· (n,m:even) (6·13)

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164 S. Takeno

The second term represents the effect of random thermal modulation on the bare potential.

The effective potentials (6 · 6) and (6 ·13) have been calculated numerically as a function of x=R/ Ro for S==J.112 I Ro = 0.00 (bare potential)' 0.01, 0.10, 0.30, and the result is shown in Figs. 1 and 2. In Fig. 2 plot of curve for S=0.30 is omitted because the series (6 ·13) is not convergent. This may probably be due to the inadequacy of the use of the Gaussin approximation for the evaluation of the effective potential for large {j. From the numerical result obtained above, we see that the effect of the anharmonicity of atomic vibrations is to give rise to a decrease in the minimum value and an increase in the minimum position of the effective potential as the anharmonicity increases. This result is of course intimately connected with the thermal expansion and the softening of solids upon heating, which may eventually lead to the melting of solids. It is also shown that the present result is consistent with the result obtained at the end of §3 for the general properties of the effective potential.

In order to keep the paper not too lengthy, the following problems are only briefly touched upon as implications of the results obtained here. Detailed discussions of some of these will be made elsewhere.

cr;,(R,A.)/Uo /3=0.10

0.5 ~rO.OO....-? /3=0.30 1.0.01

3.0 R!R. 0.0 1-.l..--'-----'--IH'---:-':::--L.__L_.J__j_--=-'-~'---..1..._--L__L_L_ 0 1.0 2.0

-0.5

-1.0

Fig. 1. Effective potential for the Morse potential. The curves for "'-''2/R=O.OO and 0.01 are nearly coincident.

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cr;,(R,A.)jU,

0.5 /3 =0.00

~./3~0.01

/3=0.!0 _/

RIR0

Fig. 2. Effective potential for the Lennard-Jones 12-6 potential (m=6 and n=12).

(2) Some general properties of e.ffectit•e potentials

The above numerical results for the effective potential have been obtained by approximating the probability distribution function P to be Gaussian and at the same time the pair potential to be of central type. We can obtain certain general properties of effective potential which are free from the above restrictions. It can be shown, first of all, that qualitative features of a shift in the minimum position and a decrease in the minimum value of the effective potential can also be obtained from Eq. (3 ·14) by assuming a plausible form for P1(R). Another important effect of the anharmonicity is that it has a tendency to make an effective potential more isotropic than the bare potential itself, thus giving rise to a decrease of effective noncentral force constants or of shearing stresses. This may be particularly important for studying the dynamical stability of crystals whose intermolecular potential is highly anisotropic, such as covalent crystals (silicon, germanium, etc.). It is also important to study the other anharmonic properties of solids, such as the Griineisen constants, the temperature dependence of elastic constants from the properties of effective potentials.

(3) Instability and melting of solids

Physically, the difference between a solid and a liquid is that the solid has elastic resistance against shearing stress, while the liquid has not. Thus,

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166 S. Takeno

It IS not too much to say that a theory of melting of solids should contain an investigation of the shear instability or the instability of transverse phonon modes. According to the results obtained before, qualitative features of the melting can be understood as being due to the smearing of effective potentials in anharmonic solids. A direct insight into this problem can be gained by studying Eq. ( 4 · 23) or ( 4 · 24) in more detail. For this purpose we employ the lowest order approximation, retaining only the first term on the righthand side of Eq. (4·26). A more detailed discussion using the higher order approximation will be given elsewhere. With the use of this approximation Eq. ( 4 · 23) reduces to

(J}u(k) 2 e(k, a)= (N/M) ~(k)k(k·e(k, a))exp{ -((k·u0) 2)}

+(N/M) ~ [~(k+K.)(k+K.){(k+K.)·e(k,a)} K.(-¥J)

Xexp[ -( {(k+K.) ·uo} 2)]- Vl(K.)K.(K.·e(k, a))exp{ -((K.·u0) 2)}],

(6·14)

where K. are the reciprocal lattice vectors. As is the case for harmonic crystals, our observation here is that the first and the second terms on the right-hand side of the above equation lead to longitudinal and transverse phonons, respectively. It is seen that the transverse phonons undergo much larger thermal modulations than do the longitudinal phonons due to the presence of the reciprocal lattice vectors K. which appear in the Debye-Waller factor. Thus, the squared-eigenfrequencies of the transverse phonons decrease faster than those of the longitudinal phonons as the anharmonicity increases, which may eventually lead to the dynamical instability of solids. A quantitative study of this problem requires the calculation of phonon frequencies as a function of mean square displacement of an atom in the solid, however it is omitted here. At this stage it is to be noted that stress correlation functions, rather than phonon eigenfrequencies themelves, are physically more meaningful quantities with which we study the instability of solids. It appears that the above result is in agreement with the computer experiment of Dickey and Paskin using molecular dynamic techniques.64> Very recently, Meissner has published a paper on the instability of highly anharmonic solids.65>

( 4) High-frequency phonons in classical liquids

When considering atomic motions in classical liquids, we can distinguish two possible regimes, collisionless or statistical, depending on whether we look at short- or long-time behaviors of atomic motions. Let -r be some time scale characterizing atom-atom collisions in the liquids, which may be of the order of 10-11-10-12 sec. We may then consider that the physical properties of liquids in the limits t4/;,-r and t';;Pr have some similarities to those of solids and imperfect gases, respectively. The conventional approach to physics of classical liquids is from the side of statistical limit. Recently, there have been

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several experimental indications that collective oscillations in classical liquids look something like phonons in amorphous solids.66 l Here, the self-consistent phonon theory developed in this paper is applied to amorphous solids, with the observation that it may equally be applicable to high-frequency phonons in liquids provided the anharmonicity of atomic vibrations is fully taken into account. It is shown that if we neglect the damping of phonons arising from the structural disorder, the squared-eigenfrequencies of phonons in a liquid can be determined, within the framework of the renormalized harmonic approximation, by the secular equation

detloh1(a8)- (p/M) ~dR[P'al7aCVl(R, II) ]g(R) {1-exp( -ik· R)}l =0,

(6·15)

where p is the density of the liquid, M is the atomic mass, and g(R) is a pair correlation function of atoms in the liquid. If we replace the effective potential by the bare potential, the above result reduces to those obtained by Zwanzig,67l by Hubbard and Beeby,68 l and by others. Numerical calculations

10

5 5

• •• • • x-x· •

X

2

(a)

• • • • •

• • .... :• ... ••

2

(b)

Fig. 3. Phonon dispersion curves in liquid argon. Figures 3(a) and 3(b) correspond to longitudinal and transverse modes, respectively. The solid circles give the results of the machine calculations of Rahman, and the crosses give the experimental results of Skold and Larsson.

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168 S. Takeno

of Eq. (5·16) have been made for liquid argon at 84.4°K with p=2.113 X 10-3 / A 3, using the harmonic approximation. In doing this use is made of the result obtained by Khan for the pair correlation function,69l and the parameters appearing in the expression for the Lennard-Janes model potential V1(R) =4E{(a/R) 12 - (a/R) 6 } are taken to be E/kB=119.7°K and a=3.407 A. The calculated phonon dispersion curves are compared with the results of machine calculations of Rahman70 l and those of the experimental results of Skold and Larsson.71 l It is seen that fairly good agreement with the experiments is obtained for the longitudinal phonons, however the frequencies of the transverse phonons obtained here are generally greater than those by Rahman. In their recent work Chung and Yip72 l and Singwi, Skold and Tosi73l have calculated the frequencies of high frequency phonons in liquid argon. Although the methods used therein are different from that employed here, the obtained results are similar to the present one. A detailed discussion on the derivation of Eq. (6 ·15) and the properties of phonons obtainable using the method described here will be given elsewhere.

(5) Anharmonic properties of resonant or localized modes in crystals

Several experiments on the infrared absorption spectra of crystals containing impurities have revealed remarkable anharmonic effects which are associated with resonant or localized phonon modes. Their manifestation is a rapid temperature dependence of the integrated absorption coefficients40 l and the appearance of phonon side bands.74l Here, we wish to call attention to the fact that a dynamical softening of force constants may occur in the vicinity of an impurity in a crystal which is capable of giving rise to resonant or localized modes. This is due to the large excursion of the impurity atom, which, as shown previously, gives rise to a decrease of effective force constants. We now give a number of experimental indications which lend support to this theoretical consideration, using the result obtained in a previous paper by the author.75l·*l To illustrate the physical situation explicitly, localized modes, gap localized modes and in-band resonant modes in alkali-halide crystals are considered. In Table I the force constant ratios K' /K and the observed frequency of impurity mode mo are given for each sample. Here K' and K are, respectively, the force constant between an impurity and its neighbouring host atoms and that between a pair of host atoms. The numerical values of K' /K were estimated there using a specific crystal model of simple cubic lattice with interactions between nearest neighbours only. The use of this "exactly analytically soluble model" is due to the general observation that this crystal model, although physically somewhat unrealistic, retains all the

*' The calculation of the force constant ratios was done there using the experimental data of Schaefer (reference 39)) and that of Sievers (See, for example, Lectures on Elementary Excitations and Their Interactions in Solids, NATO Advanced Study Institute, Cortina, Italy, 1966).

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qualitative features of more realistic models. From this Table, we see that for all the samples except the case of KI: es+ the force constant ratio K' I K is much less than unity. This result can be considered to be more than accidental and attributed to a remarkable anharmonicity which occurs in the vicinity of an impurity. A detailed discussion of this problem will also be given elsewhere.

(6) Impurity- or defect-induced anharmonic vibrations in solids

Here, the result obtained above for the anharmonic properties of impurity modes in crystals is extended to more general situations. It has recently been shown by several authors that the anharmonicity of vibrations in solids may be more wide-spread than is generally realized. 33l-38l Generally speaking, the anharmonicity of atomic vibrations in solids depends on the degree of excursion of atoms, which is in turn characterized by the mean square displacement of the atoms. As is seen from Eq. (4·8), which can be applied to any type of solids, crystalline or noncrystalline, the mean

Table Ia.

Sample

NaBr:H-

NaCl:I-r KBr:I-r KCl:I-r

KI:I-r RbBr:H-

RbCl:H-

Rbi:I-r

Table lb.

Sample

KI:Cr

KI:Cs•

KI:C~

KI:NO.-

Case of localized modes.

roo (cm-1)

496 563 445 500 379 425

475 362

Case of gap modes.

roo (em-')

77

74 84 81

71

80

K'/K

0.79 0.62 0.53 0.57 0.42

0.58 0.49 0.39

K'/K

0.38 0.81

1.50

0.31 0.40 0.56

Table lc. Case of resonant modes.

Sample

KI:Ag• NaCl:Ag• KBr:Li•• KBr:Li•7

NaCl:Li•

33.5 38.5 17.4 52.2 17.9

16.3 44.0

K'/K

0.36 0.33 0.10 0.58 0.005 0:005 0.026

square displacement is intimately connected with the localization of eigenfunctions of phonon modes; In a crystal with N atoms the displacements due to a in-band or out-of-band localized mode is 0(1) while the displacements due to a nonlocalized mode is O(N-112). This situation is even more pronounced for a low-frequency in-band localized mode due to the presence of the factor ro(A) in the denominator of Eq. (4·8).

To show more explicitly the importance of low-frequency resonant modes, the root mean square displacement of Li+ impurity ion in KBr crystal is estimated using the data in Table Ic as a typical example.*l For this purpose use is made of the following resulf6l for the mean square displacement of an

*l A detailed discussion on the infrared absorption of Li-doped KBr crystals has been made by Sievers and Takeno (see, A. ]. Sievers and S. Takeno, Phys. Rev. 140 (1965), A1030).

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170 S. Takeno

impurity in a crystal:

(u2) = li, coth(/3wo/2) /21M'-Ml Wo. (6·16)

Here M is the mass of a host atom which an impurity with mass M' replaces and wo is the eigenfrequency of the resonant mode. Equation (6 ·16) can be used if the spectral function associated with the mean square displacement of the impurity has a sufficiently sharp maximum at w=w0 • Putting M=39, M' = 6 and 7 for u+G and Li+7, respectively and T= 1.5°K which is the temperature at which the experiments was done, we get

< I {8.8 X 10-9 em

u2)12= 9.5X 10-9 em

for u+s

for Li+7.

The lattice constant a of KBr crystal 1s a=3.3A. Thus, we get

(6·17)

which shows that the root mean square displacement of Li+ ion in KBr cyrstal is almost one-third of its interatomic spacing, a situation very similar to the case of solid helium.5J,GJ Thus, it is seen that the vibration of lithium ion in KBr crystal is very anharmonic, which may account for various anomalous properties of lithium ions in alkali halide crystals.

We may conclude that an impurity or a defect in a crystal lattice which gives rise to localized or resonant modes can induce anharmonic vibrations in its vicinity. This may be generalized as follows; any type of in-band or outof-band localized mode in a solid is likely to induce anharmonic vibrations. It is understood here that in the above surface phonon modes or defect modes due to an extended defect are also included. Impurity-induced anharmonic vibrations discussed here may have important implications on various problems in solid state physics, such as the diffusion of impurity or vacancy in solids, the thermal conductivity of insulating solids, defectons in quantum solids,77l

etc. A detailed discussion of these problem is omitted in this paper.

§7. Concluding remarks

In this paper a theory of the anharmonic lattice vibration of solids has been developed. Although the anharmonic nature of solids has been recognized for a long time, conventional theory of lattice dynamics since the work of Born and von Karman has limited its discussion to harmonic or nearly harmonic solids or "ideal solids", in which the effect of the anharmonicity, when taken into account, is regarded as small perturbations. It has been shown that a self-consistent treatment of the atomic vibration in solids makes it possible to study the vibrational properties of anharmonic solids or real or "non-ideal" solids that become unstable and eventually undergo transition to liquids as

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the anharmonicity increases. Results obtainable using such a method may be used to reformulate theory of lattice dynamics as well as to study the vibrational properties of "soft" or "hot" solids. In this paper the formulation of the problem is designed to give results which may account for the anharmonic properties of solids in general, crystalline or non-crystalline, whereas in almost all current self-consistent phonon theories particular attention has been focussed primarily on the case of quantum crystals. The results obtained in this paper have been applied to various problems in §6. A by-product of the obtained results here is that the self-consistent method may be useful to study the interrelationship between the localization of phonon modes in disordered systems and anharmonic vibrations. It may also be used as an approach to physics of liquids from the side of solid state physics.

Acknowledgements

The author would like to express his sincere thanks to Prof. H. Yukawa for drawing his attention to physics of liquids.

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a theory of the lattice vibration of anharmonic solids § 1. introduction

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