A Computational method in the spectral analysis of the harmonic oscillations of a crystal lattice

Harmonic oscillationsof a crystal lattice

163

11. VLADIMIROV, V. S., Mathematical problems of the one-velocity theory of particle transport. Tr. hfatem. in-taAkod. NaukSSSR, 61,196l. 12. SOBOLEVSKII, P. E., On equations of the parabolic type in Banach space. Tr. Mask. matem. ob-va, 10, 297-350,196l.

A COMPUTATIONAL METHOD IN THE SPECTRAL ANALYSIS OF THE HARMONIC OSCILLATIONS OF A CRYSTAL LATTICE* F. P. ONUFRIEVA Odessa (Received 16 May 1974) A METHOD is developed for establishing the frequency distribution function from its specified moments, using the apparatus of orthogonal polynomials. An operator generalization of a wellknown theorem of M. G. Krein devoted to this problem is given. The proposed approach is used to compute the spectral density of the harmonic oscillations of a facecentred cubic lattice. The density constructed is used to calculate the thermodynamic functions of the lattice and to construct tables of them.

1. Introduction Among the methods of constructing the spectral density of the normal oscillations of a threedimensional crystal lattice p ( CO’)developed up to the present time, the most used and simplest is Montroll’s method of moments [ 1,2] . However, a drawback of this method is the fact that the spectral densities of various orders of approximation may oscillate strongly with respect to one another and even become negative with respect to the exact density on some segments of o. In this paper a new version of the reconstruction of the spectral density from its known moments @L’ Sk=

J

ClPp

(cd)

do2

( or. is the maximum frequency of the oscillations), is developed, which is also fairly simple, but in our.opinion is free from the deficiencies of Montroll’s method. The procedure for computing the actual moments in the harmonic approximation is developed in detail, and the necessary calculations are performed for various types of lattices in various approximations 12-41. This approach is based on general propositions of the classical moment problem. From the mathematical point of view the formulation of the problem considered is as follows. We know the r t 1 first moments {Sk}OTof the distribution function o(t) , increasing in a finite interval of the real axis [u, b] : sk=

J tkdt3(t); (1

it is required to describe the distribution functions o ,.(t)- with specified r + 1 first moments, the points of increase of which belong to the same interval [a. b] . As is known from classical theory 151, *Zh. vjchisl. Mat. mat. Fiz., 15.5, 1262-1275,

1975.

F. P. Chufrieva

164

in the case of a finite number of moments the problem of moments is indeterminate and the functions o, (t) sought form some set. The properties of this set are described in a fundamental paper [6] for o (t) , specified in the interval (- 00, m), and in [7] for the case where the interval of specification of the function is finite and fured. The main results of these papers are as follows. Let the finite sequence of moments {Sk}0zs of the distribution function o (t) considered be specified. For the strictly positive sequence {&} OZnwe construct by a well-known method the orthonormal polynomials D,(t) , k=O, 1, . . . , n, and the polynomials Ck(t) conjugate to them (see

PI): a{D,,(t)a(t)}=GmR,

Gt(z)=@

a(t)--Dk(z)

1

t_

z

>

The polynomials L+(t) are defined explicitly by the formula

Dk (q

=

s, s,. . . Sk I Sl s, . . . Sk,, . . . . . . . . (A,Ar-J’iz s,_, s, . . . S?,;_, 1

t...

s, s;...s, 1

Ak

S1 Sa .

=

,S,

th’

. .

Sk+1

*---.**

Sc+r . . .S 2k

.

(1)

They possess the following properties: 1) they have exactly k real roots which are situated within the interval (a, b); 2) they preserve a constant sign outside (a, b), namely: for tab

&(l)>O,

for Ku

D,(t)>0

for even k,

D,(t)<0

for odd

(2) k;

3) the roots of adjacent polynomials Dk (t) and D,, , (t) strictly alternate. We denote the set of all the solutions of the moment problem by I’[ a, b; &,, I. The following theorem holds. Theorem 1 (see [6]) For a sequence {S,} ,,2’1 strictly positive in the interval (- 00, m) the formula

-m

s-

do(t) _ c,,+i(2) +Q (z) cn(2) z-t D,+, (z) +Q(z)Dn (z)

establishes a one-to-one correspondence between the solutions o=X[ -7, problem and the functions 51(z) , representable in the form

(3) 00; S,,,] of the moment

(4)

Harmonic oscilkhons of a crystal lattice

where a(Q)

165

is a real number, and t (t) is a non-decreasing function.

In the theorem from [7] cited below we require the quasi-orthogonal polynomials vanishing at one of the ends of the interval [a, b] ,

II’ D,(a) Dk(t)

B(t)=(-l)*ll;+i;t; k+i

a

Qk(t)

=

Dk (t) Dk+, (t> II Dk,,

(b)

Dk

(b)

II

and the polynomials conjugate to them ck(t>

,,,t,=ll;+i;;;

11, Dk

k+l

(a)

;k;“b: k

11.

Note that a,(t) and h(t) also have exactly k zeros, situated inside the interval (a, b) and strictly alternating, the (k t I)-th zero coincides with the corresponding end of the interval. Theorem 2 (see [7] ) For a sequence {&} 02” strictly positive in [a, b] the formula

sb

do(t)

-a

a

Z-i?

P,(z)+o(z)Pn(z)

established a one-to-one correspondence between the solutionso= V[ a, b; S,,] problem and the functions w(z) representable in the form

0

(5)

Qn(Z)+d)Q?&)

(2)= (b-2)

a(t) Jbt-_z

of the moment

(6)

a

where 5 (t) is a non-decreasing function. The class of functions representable in the form (6) is denoted by S[u, b] , and the functions representable in the form (4), for which a(Q) =O, by R[ -00, m]. In [7] it is mentioned that both results can be generalized operationally in the theory of the extension of bounded Hermitian operators, and the way in which this generalization can be performed is indicated. The content of section 2 of this paper constitutes a new proof of Theorem 2 in this way. In section 3 the results obtained are used to calculate the distribution function of a face-centred cubic lattice (f.c.c.l.) in the harmonic phonon approximation taking into account the interaction of only the nearest neighbours. We immediately notice that the choice of this particlar approximation for the calculation is dictated merely by the existence of the exact solution [8] (by the root selection method), required for comparison. By the program constructed we can calculate the approximate spectral density for various types of lattices to various approximations (also taking into account the interactions of successive neighbours, in the quasiharmonic approximation etc) based on moments calculated to a high degree of accuracy in [3]. Section 4 gives detailed tables of the reduced thermodynamic functions calculated to the given approximation based on the construction of the spectral density in the interval of reduced temperatures 0=0.05+2.70. We mention that the method described in [9], where the apparatus of the theory of orthogonal polynomials is also used, is close in spirit to the method of constructing the spectral density developed

F. P. Onufrieva

166

in this paper. This method is probably to be preferred when the calculation of the moments of the spectral density is difficult, for example, in the case of a crystal with defects. But in the case where the moments are known our approach is more simple and natural. Moreover, it is obviously more flexible, since the selection of a particular function w(z) enables spectral densities with the correct behaviour at the desired critical points to be chosen.

2, An operator approach to the proof of the theorems of R. New&ma and M. G. Krein Let {Sk}orn be a sequence of moments of the ~st~bution function, increasing in the general case in an inftite interval of the real axis. We denote by L,, m=l, 2, . , . , the fmitedimensional Hilbert space of polynomials of degree not exceeding m, with the scalar product m

(cp? $1L,

=

z

m

&+k’Ckbk’,

q

(t) =

k.k'w.0

z Cktk,

9

(t) =

2

R=O

bktk.

k-0

ln the halfspace L,cL n+i we defme an operator A, of multiplication by t: (A,rp) (t) =tcg (t). It (4 =rp, $)= (rp, A,+@),cp, $E&. Let A be any self-conjugate extension of the operator A, with an outlet into the space H, satisfying the condition (A,cp, I$)= (Arp, sp) for any polynom~s c~EL, and @s!&. Also, let E(r)_be the spectral function of the operatorA, e(t)=1 be avector fromL,+1, and J?,(A) =(A--zf)-‘, Im z=O. the resolvent of the operator A. Then the function o(t) = (E(t) e, e) , connected with the resolvent R,(A) by the equation is obvious that the operator A, possesses the property

J

do(t)

4,

__,=(R,We,

is one of hand,if operator equation

-

(7)

the solutions of the moment problem considered: omV[ --oc, 00;. &,I. On the other , then the space La2 is an extension of the space L,,l , and the o=F[- 00, w; SSn] of m~tip~cation by f in L, 2 is an extension of the operator A,, and hence the following holds

:

e.7 J tktpdo(t) = (Anke,AnPe),

063,

paz.

-00’

Therefore, the problem of describing all the solutions of the given moment problem V [ -Q), 00; S,,) reduces to the problem of describing the family of functions (R,(A) e, e) for all selfconjugate extensions A of the operator A,. We denote by Pn and P,,I the orthoprojectors from H onto the subspaces i;, c G+ f dY* and L,lcH respectively. Since the subspace P,,iA,L,cL,+, is one-dimensional; then by the fund~ent~ theorem of Riesz on the general form of a linear functional (see, for example, [l O]), for any extension A of the operator A, a vector qEL, and a unit vector kL,+,@L, exist such that

A~&?4n~Ln+(~, q)h,

A IL,I=R31 Lnl+(-, h)q.

‘3

Therefore, every self-conjugate extension A of the operator A, is of the form

(9)

Harmonic oscillationsof a crystal lattice

167

where the operator A, O=P,,A,,P, is uniquely defmed by the given sequence of moments (Sk} Z”, and Q=P,,LA,P,A is an arbitrary self-conjugate operator. From (9) as a result of several calculations we obtain a formula for the resolvent of the operator A : R,(d)=R~(An”)P,,+R~(Q)P,II+I1-(R~(A~,0)q,q)(R~(Q)h,h)l-i X{[--6, Rz*(A.O)q)+(R(A.O)q, q) (., R’(Q)h) IRz(Q)h +[-(., Rz’(QM)+(Rz(Q)k h) (s, Rz*(Ak’)q) ]&*(A2)qf (the asterisk is the sign of the Hermitian conjugate), and from it the form of the function which interests us:

(h(d)e, e)=(Rz(&“)e,

e)+A-‘(2)

(R,(Q)h,

h) (R,(A,,O)e, q)

X 0% (A,B) 4, e),

(10)

Wd=~-UWL”)g,

q)

(Rz(Q)ht h).

We determine the quantities occurring on the right side of (10). Since Q is an arbitrary selfconjugate operator in the space L ,‘-, the scalarproduct (R,(Q) h, h) is also an arbitrary function m, -1. The rern~~g functions are associated with the operator A,0 of i;Z(z)oftheclassR[and must therefore be explicitly expressed in terms of the moments (SA}0”‘. We will fmd the corresponding expressions, first establishing the explicit form of the operator A,0 and the vector q. It follows from the defmition of the operator A,0 that its action in the space (A.?, h) h. Having chosen as the unit-vector h the orthogonal L, is as follows: d,“v=A,cp~lyno~~ Dnfl (t) , of degree n + 1, we obtain for Aa0

(A.Ocp) (O=@(t)-c,

q(t)= &,tk. (+ 1“*LL+&), n-t*

(11)

RIO

On the other hand, Eq. (8) implies that

(AC-A,)

60=- (cp,Q)h.

(12)

It follows from a comparison of Eqs. (11) and (12) that (cp, q) =c~(AJA~+~)‘~. By the last formula, Q(t) _=An(A~+~A~~~) -‘BD,_t (t) . We now consider the general form of the functions

U&(A.O)cp) (t) ==P(t),

I%,.

(13)

Applying to both sides of (I 3) the operator (A ,2-- 21), taking into account (1 l), we obtain the following equation for p(t):

(14) Since PEL,

and is hence a ~lyno~~

of degree n, it necessarily follows from Eq. (14) that

cpw+cn (+-n+i 1‘k+1(z) =o and consequently that

F. P. Onufrieva

168 Inthecasescp(t)=e(t)

and cp(t)=q(t)

ofinteresttouswehave

From Eqs. (15), taking into account the properties of orthogonal polynomials, in particular the equation (Dk( t) , t”‘) = ( AJAA-~) %,w, and the definition of the polynomials conjugate to them, we obtain for the scalar products required the following expressions:

Substitu~g

Eq. (16) into Eq. (10) and using the identity [5]

c,+,(z)D,(z)-D,+,(z)C*(z)=b~

-1

,

(17)

we fmally obtain

Equation (18) contains polynomials of degree n + 1 whose construction uses the sequence of the moments are known only up to the ‘Ln-theclair, moments {Sk}2n+i 0 II But since by hooters the (2n + I)& moment can be expressed arbitrarily, which, as is obvious from Eqs. (1), is equivalent to the addition of an arbitrary real constant OLto the function Q(z). Introducing the factor b,-’ into the function 62(z) ; we obtain Eq. (3), connecting the aggregate of solutions of the moment problem ok V[ - oo,*oo;S;, ] with the class of functions of the type

Therefore, Theorem 1 is proved. We now require that the unknown function grow only on the given interval [a, b] , For the set V[a, b; &,] not to be empty it isnecessary that the spectrum of the operator A,* belong to the segment [a, bl. If this condition is satisfied, then, obviously, the problem considered reduces to fmding all the extensions of A whose spectra belong to [a, b] . Therefore, all the scalar products in Eq. (IO), constructed on the operators &(A,“) and R, ((8 , are regular outside [a, b] . It is obvious that the necessary and s~~cient condition of regularity of the left side of (10) subject to the condition of regularity of all the functions on the right side is the non-vanishing of A (z) in the same interval: A (z) ZO for zb. The functions occurring in A(z) are representable in the form

crystal lattice

Harmonic oscillations

169

from which the following properties of them follow: 1) regularity outside [a, b] , whence non-negativeness in the interval (-m, positiveness in the interval (b, w ) ;

u) and non-

2) monotonic increase outside [a, b] ; 3) vanishing at z=*m. From the above it follows that A (z)30

for ZGZ and s>b,

A(z) :=*ce= 1.

(19)

The following inequalities are a strengthening of inequality (19):

bW)

l-j -Q(z) t-a

>O

for ztU,

1-

b d?-l(t) -Q(z)>0 ‘s t-b

for

z>b.

When Eqs. (16) are used they assume the form

1

1+

as(a)

b,3 D”+i(a> if1

for zta,

i-2(2)=-0

(20)

D,(b) !A(z) >o bn3 Dn+i (b)

for z>b.

On the other hand, the validity of these same inequalities for z> b and z
o(z)= x

[

If

t-1) “+‘&+I (b) = Dn+t(~1

(_1)

,,+,

Dn+i(b) +Dn(b) 52(z) b,-’

(21)

Dn+,(~)+D,(+2(~)b,-~

which satisfies the following conditions, following from the properties of the orthogonal polynomials, the function Q(z) and the inequalities (20): 1) regularity and non-negativeness in the intervals (-00, a) and (b, 00) ; 2) monotonic growth in these intervals, which follows from the monotonic growth of Q (z) of the positiveness of the determinant of the linear-fractional transformation (21):

in view

(-,)‘I,’

1

Dn (b)

&+i

Dn (a)

Dn+i (a)

o(z)i_*~=(-l)~+’

(b) I

= t-1)

Dn+i(b) > Dn+,W * o

n+rDn W&+i

(a)

F. P. Unujrieva

170

Conditions 1) and 2) for this function are equivalent to the statement (see, for example, [ 1 1] ), that o(z) belongs to the classS[a, b]. Condition 3) imposes a constraint on the normalization: b

saft)-(--l)“+’

a

Dn+i(a) (b) D

,

(22)

zt1

We notice in passing that conditions 2) and 3) imply that

o

tzj

>

Dw (b)

D,,i(a)

(--QS+’

for s% (23)

o(z) < Dnt: (b) D,+,(a) (-i)n+i

for zab.

We have established that if the arbitrary function Q (z) belongs to the class R [a, b] , then the function o (z) connected with it by the relation (21) belongs to the class S[n, b] . The converse also holds, that is, if o (z) ES [a, b] , then the function connected with it by formula (21) Q(z)er$[a, b).Itiseasytoverifythisifwerewrite(21)intheform

a(z)= [ o(z)--(-I)%+$

D,ti (a) b,z3 Dn+1(b) ] (-l)“f’D,(b)--w(z)D,(a)

Dnti(a)

*

(24)

From condition I) for the function o (z) taking into account the properties of the orthogonal (CO) foreven(odd)n and polynomials (2) we obtain, firstly, (-1) n+iD,(b)-o(z)D,(u)>O secondly,

Dn+i(4 (-1)“+‘D,(b)--o(z)D,(a)

“*

The first implies the regularity outside [a, b] of the function 5t (z) , defmed by Eq. (24), and from the second, on the basis of (23), its non-negativeness in (-T, a) and non-positiveness in (b, m) s Properties 2) and 3) of the function &l(z) obviously follow from conditions 2) and 3), which the function o (z) satisfies. What has been proved implies the admissibility of the substitution of the function r;Z(z) , defmed by formula (24), in formula (18). Making this substitution we immediately obtain Eq. (5), connecting the aggregate of functions of the distribution c(t), whose first moments {Sk)_.,,2nequaI the corresponding moments of the given ~t~bution function, with the functions 0 (2) of the class S [ a, b 1. We notice that although the qua&orthogonal polynomials occurring in the formula are defmed by using D,,+, (t) , the indeterminateness in the (2n + I)-th moment for them is inessential, as is easily verif=d directly, and consequently, in Krein’s formula the (2n t I)-th moment occurs inessentially. However, here o (z) must be an arbitrarily numbered function of the class S[a, b] : But the use of the no~~ation (22) for ce (z) selects from the aggregate of solutions V[ca, b; &,f all the solutions possessing also the correct (2n + l)-th moment, that is, the functions o=V[a, b:Sz,,il. Therefore, it has been proved that the aggregate of solutions Via, b; & ] is described by Eq. (5) in both cases: m = 2n and m = 2n + 1: the difference between them being realized by a different normalization of w(z), arbitrary in the fust case and defmite in the second.

Harmonic oscillationsof a crystal lattice

171

3. Construction of the approximate spectral density of the normal oscillations of a f.c.c.1. The app~cation of Theorem 2 for calculating the spectral density of the squared frequencies of the normal oscillations of a crystal lattice is based on the following formulas connecting the density with the dynamic phonon Green’s function Gap(e-e’, co’) for e-e’=0 (see, for example, [13): e&zp (AZ)dh2 02_h2 , 0

G(02)=~G.,(0,02)=~ cc

~(~z)=~limImG(~2-~~) v

Considering that in this case the function a(t), occurring in Eq. (5) is an absolutely continuous function, do(t) =p ( t) dt, and ~troduc~g the notation kt(t)=limRew(t-ie), 860

~(t)=IimImo(t--ie), e&O

we obtain from Eq. (5) the following relations:

The identity ( 1’7)implies the identity

P,(t)Qn(t)-P,(t)Q,(t)=Q,(b)b,-*. and hence

pnft)L3_

n

Qrm

[~11(t>+Cl(t)~*(t)12+[~(t)~n(t)12

Equation (25) is the initial formula for the subsequent calculations. We again emphasize that in the case of an arbitrary no~~ation of w (z) it describes the aggregatefrla, b; &,I, and in the case of a normalization by (22) it describes the aggregate V Ea, b; SznfL 1. From the mathematical point of view the problem formulated may be regarded as solved, the choice of the function o (z) here remains arbitrary. However, from the physical point of view the different densities of this aggregate are not equivalent in the sense of a satisfactory description of the real spectral density of the normal oscillations of a threedimensional lattice. It may be attempted, by imposing certain constraints on w (z) , to select those functions (or that function) of this aggregate, which would possess characteristic features of the latter, known in advance. It is a question of such properties of it as piecewise continuity and positiveness on a spectrum, the bounded (known for various types of lattice) number of critical points, and the radical nature of the dependence on c at the edges of the spectrum. In our view the most successful in this sense would be the o (z) , for which u (t) -0. This point of view can be explained by the following semi-intuitive considerations. As is obvious from (29, for a smooth u(t) the nature of

F. P. Onufrieva

172

the behaviour of the approximate spectral density (singular points, maxima, minima) is determined by the nature of the behaviour of the denominator of (25). In the case where u(t) =O, the zeros of the two positive terms of the denominator are displaced in the interval [ 0, tL ] in accordance with the displacement of the zeros of the polynomials gn (t) and Qn (t) , which excludes the occurrence of singular points of the pole type within the spectrum;%oreover, the position of the maxima and minima of the denominator, and hence also of pn (t) , is also determined by the values of the quasi-orthogonal polynomials determined by the given moments. Accordingly, these maxhna and minima have a natural origin, since the moments in fact carry information about the actual spectral density. On the other hand, for p(t) ZO it is possible for false maxima and minima to occur whose positions depend on the arbitrary function p(t). In the class S[ a, b I, to which w (z) , belongs, the only function with a zero imaginary part is the following:

0

(2) =iK

(gy=- f (t-z)((:) (t_a)]‘”

(26)

(K is a normalizing factor). It must be mentioned that with these merits the approximate spectral density corresponding to the function o(z), defmed by formula (26), possesses an inherent deficiency, namely: it is inversely proportional to the square root oft at the edges of the spectrum. However, it is shown by calculations that these spurious features occur in very narrow intervals close to the edges of the spectrum, after them the correct radical law is restored. There also exist functions w(z) giving the approximate spectral densities with the correct radical behaviour close to the edges of the spectrum. One of them is e~(z)=&+K~{ H (b-z)

(b-z)

: {K,6@4)fK,[ J

n

+i[ (b-z)

(z-u)

I’“}

(t-a)/@-i)]“‘}&

(27)

t-z

However, it possesses all the enumerated deficiencies associated with functions with p (t) PO. This is confumed by a calculation made for illustrative purposes whose results for m = 9 are shown in Fig. 1.

1

40 -

I

I

6.i 0.2 0.3 6.4 0.5 0.6 0.7 0.8 0.9,

FIG. 1. Distribution functions: - actual; - - - with correct behaviour close fo the edges of the spectrum.

Harmonic oscillations of a crystntluttice

173

FIG. 2. Approbate d~t~bution functions: - actual; the remainder correspond to formula (26): - - - form=&-v--form=9,...form=I5. All the remaining calculations, both of the spectral density of a f.c.c.l., and also of its thermodynamic functions, were based on the o(z) defined by formula (26). Here the initial information used was the moments of the dimensionless spectral density (defmed in the interval (0, 1) of the dimensionless frequencies ~=w’/u~’ ) of a f.c.c.1. with the central interaction of the closest neighbours calculated in the harmonic approximation with great accuracy in [4]. The graphs of the approximate spectral densities of the frequencies of normal oscillations (g(o) ==2@P(@YJt constructed from 9,10 and 16 (including the zeroth) fust moments of the exact density, are shown in Fig. 2. Comp~~on with the data of [8] showed excellent a~eement already for m = 8, which improves when the number of moments is increased. The spurious features close to the edges of the spectrum are not visible in Fig. 2, since not very narrow intervals occur: t~0.000001 and DO.99 for m = 8, which are further compressed as m increases. It is necessary to stipulate that when the number of moments used is increased their accuracy must also be increased. The inadequacy for large m of nine places for performing the calculations, apparently reveals itself as a small peak at lower frequencies for m = 15. We conclude this section by mentioning that the comparison carried out of the moments {Sk) #In”of the constructed spectral densities pn (t) with the moments of the actual density demonstrated their a~eement in both cases (26) and (27), with an accuracy practically specified for the numerical integration. Moreover, in case (26) they are identical with reasonable accuracy (to tenths of a percent) for subsequent moments also. The latter fact is a confrrmation of the closeness of the corresponding density to the actual density of the f.c.c.1.

4. Thermodynamic functions of a face-centred cubic lattice in the harmonic approximation On the basis of the spectral densities constructed the reduced ~e~~yn~ic functions of the f.c.c.1. were calculated: the free energy F*, thespecific heat at constant volume C,*, the

174

F. P. Onufieva

internal energy E *, the entropy S * and the mean-square displacement (U”) as functions of the reduced temperature based on the formulas, well-known in lattice dynamics (for example, [l] ), l

F*=8S)ln2sh(JJ

’ t’” C,*= JGcschZ

p(t)&,

0

(;)

p(t)&,

0

E’ = jtl., cth ( -gI’!?) p(t)&

[$cth

S*=)

(&-ln(?sh;)]p(t)dt,

II

‘= j,,,

($,$&it.

0

On integration it appeared to us to be natural to extrapolate continuously the correct radical behaviour of the spectral density integrated in intervals where spurious features occurred. Generally speaking, in the numerical calculation this did not introduce appreciable changes in the thermodynamic functions (except for (U’)*, where the variation <2% ), but it seems more correct theoretically. Table 1 shows the reduced thermodynamic functions constructed from 15 moments. (The results for various values of m differ by l- 1.5%.) The actual thermodynamic functions in a calculation on one particle were associated with the reduced simple relations: C,=3kC,*, F=3h,F’, E=3Ao,E*/2, S=3kS’,
0.05 . 10 .15 .2C) .25 .30 .35 .40 .45 .50 .55 .RO .70 .80 .90 1.00 . IO .2C) .30 .40 .51J .60 .7O .80 .90 2.00 .10 .20 .30 .40 .50 .60 .70

0.00855 .(I969 .263 .428 .558 .655 .726 .778 .818 .847 .8iO .889 .915 .933

.945

.954 .961 .966 .970 .973 .976 .978 .98U .982 .983 .984 .985 .986 .986 .987 .988 .988 .988

E*

s*

0.678

0.00195 .0285 .O982 .197 .307 .417

.682 .7w

.735 .785 .845

.915 .990 1.070 1;:;

.326 . xl7

.692 .879 2.069 .260

.456 .650 ,844 3.039

.235

.428 .625 .822 4.018 .215 .412 .610

.807

5.004

.202 .400

.524 .625 .719

.808

.890 .970 1.105 .229 .239 .439 .530 .614 .691

.763

.830 .893 .952 1.009 .0062 .112 .I60

.206 .250 .293 .333

.372

.410

F*

(

c*> *

0.339

1.608

.335 .327

.741 ,889 2.076 .289 .520

.338

.314 .296 .273 .245 .211 .172 .130

.085

-0.019 .I35

.263 .402 .551

.708

.874 -1.049 .228 .415

.607 .805

-2.009 .218 .432 .650 .873 -3.100

.330 .565 .803

.644

.766

3.024 .287 .556

.830

4.386 .951 5.522 6.097 .676 7.257

.840

8.424 9.010 .596 10.183 .771 11.360 .948 12.538 13.127 .718 14.307 .898 15.489 16.080

Harmonic osci~~tio~s of @crystallattice

175

When using this table for specific substances possessing a f.c.c.l. it is only necessary to know the maximum oscillation frequency oL , which can be taken from the corresponding experiments or calculated to some approximation. For example, we performed a calculation of the specific heat C, for solidified krypton - Table 2 TABLE 2

fi IO 15 20 25 30

0.0627 .238 .462 .ti25 .731 ,798

0.0623 .237 .46d .623 .727 .792

35 40 4.5 31) .55 6U

.845 a877 .892 .915 ,929 ,939

.835 .866 .882 ,896 .9ij7 .915

Here we used the experimental value 0,=9.2. X1O-*2 set -’ [ 121. For comparison Table 2 gives the experimental data on the specific heat C, frdm [13], with which a compa~son is advisable only up to T1:50”+60°, since the further course of C, as one approaches the tempera~re of the phase transition, naturally cannot be described within the limitations of the assumptions made about the idealness of the crystal lattice and the sufficiency of the harmonic approximation. The author sincerely thanks V. M. Adamyan, without whose aid and supervision the present paper could not have appeared, and A. E. Glauberman for his interest. Translated

by

J. Berry

REFERENCES 1.

MARADUDIN, A., ~ONTROLL, E. and WEISS, G., Theory of lattice dynamics in the h~rno~~c ap~~ximation (DiMmicheskaya teoriya ~ist~cheskoi reshetki v g~rno~che~orn prib~hen~). “Mir”, Moscow, 1965.

2.

MONTROLL, E. W., Frequency spectrum of crystalline solids. J. Chem. Phyr, 10,4,219-229,1942; MONTROLL, E. W. and PEASLER, D., Frequency spectrum of crystalline solids 3. Body-centred cubic lattices.J. C’hem. Phyr, 12,3,98-106, 1944.

3.

ISENBERG, C., Moment calculations in lattice dynamics, 2. J. Phyr, C4,164-173,

4.

ISENBERG, C., Moment calculations in lattice dynamics 1. For lattice with nearest neighbor interactions. Phys. Rev., 132,2427-2433,1963.

5.

AKHIEZER, N. I., The ckicai 1961.

6.

~V~NL~NA, R., Asymptot~he en~icklungen beschriinkter funk~~n und das Stielt~he momenten* problem. Ann. Acad. Sci. Fertnieae, A18,4-68,1922; Uber beschriinkte analytische funktionen. Ann. Acad, Sci Fennicae, A32,3-X,1929.

I.

KREIN, M. G., Description of aJl the solutions of the truncated power moment problem and some operational questions, in: Mathematical investigations (Matem. issledovaniya), 2, section 2,114-132, Izd-vo Kishinevsk. un-ta, Kishinev, 1967.

8.

LEIGHTON, R., The vibrational spectrum and specific heat of a face centered cubic crystal. Rev. Mod. Phys., 24 16%174,1948.

9.

PERSADA, V. I., On the thermodynamic quantities of an ideal crystal lattice in the harmonic approximation. Zh. e&g. teor. fiz,., 53,605-614,1967; PERSADA, V. I. and AFANAS’EV, V. N., On an analytic method of representing the frequency distriiution functions of the oscillations of an ideal crystal lattice. Zh. e&p. teor. fizz., 58,135-144,197O.

1971.

moment problem (Klassicheskaya problema momentov), Fizmatgiz, Moscow,

176

L. S. Klabukova

10. YOSIDA, K., Functional analysis (Funktsional’nyi

an&z), “Mu”, Moscow, 1967.

11. KREIN, M. G. and NUDEL’MAN, A. A., Markov’s moment problem and extreme1 problems (Problema momentov Markova i ekstremal’nye zadachi). “Nauka”, Moscow, 1973. 12. BROWN, I. S. and HORTON, G. H., Model potentials and the dispersion law in solid krypton. Phyr. Rev. Letters, 18,647-649,1967. 13. LOSEE, D. L. and SIMMONS, R. O., Thermal-expansion Phys Rev., 172,944-957,1968.

measurements and thermodynamics

of solid krypton.

THE CORRECTNESS OF BOUNDARY VALUE PROBLEMS IN THE THEORY OF MOMENTLESS THIN ELASTIC SHELLS OF POSITIVE CURVATURE AND THEIR SOLUTION BY THE MESH METHOD* L. S. KLABUKOVA

Moscow (Received 7 December 1973) THREE types of boundary value problems of the theory of momentless thin elastic shells of positive curvature are considered, and for each problem sufficient conditions for its correctness are indicated. Using inequalities proving the stability of the problems, we demonstrate the possibility of solving them by the mesh method. A difference system of equations is obtained by minimization of a quadratic functional (of the least-squares functional type) in a class of functions defmed on a net.

Introduction We will retain the definitions and notation of [I] . We denote by S the mean surface of the shell and assume that the surface S has the positive curvature K. Let the coordinate system x1, x2 be introduced on the surface S. We denote the radius vector of the point (z’, x2) by r=r (z’, x2), and we denote by r1 and r2 the basis vectors:

r,=adad,

rz=ada2.

We introduce the conjugate basis vectors r1 and r2 defined by the conditions r”re=&3,

a, j3=1,2

( rarg is the scalar product). denote by n=n (z’, 5’) the normal unit vector to the surface S at the point (x’, x2). The vectors rl, r2, n constitute the fundamental basis, and rl, r2, n constitute the conjugate basis. We

We write

*Zh. vTchis1.Mat. mat. Fiz., l&5,1276-1288,

1975.