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Thermodynamics of mesoscopic soft modes in strongly disturbed lattices
Physica A 166 (1990) 229-262 North-Holland
THERMODYNAMICS OF MESOSCOPIC SOFT MODES IN STRONGLY DISTURBED LATTICES Max WAGNER
and Theodor
MOUGIOS
Institut fiir Theoretische Physik, Universitiit Stuttgart, Pfaffenwaldring Fed. Rep. Germany
57, D-7000 Stuttgart 80,
Received 3 November 1989 Revised manuscript received 13 February 1990
A generalized Lifshitz procedure is developed, which allows the evaluation of frequency densities in mesoscopically disturbed harmonic lattices. Archetypical examples are calculated explicitly. In particular it is shown that lattice softening at mesoscopically distant sites produces a low-frequency enhancement of the increase in mode density, which may be ascribed to the oscillation of mesoscopically large masses. Possible relevance for the anomalous low-temperature behaviour of amorphous systems is discussed.
1. Introduction The unusual thermodynamic and thermal-conductivity properties of amorphous solids, first found by Zeller and Pohl 19 years ago [20], have stimulated a great amount of experimental and theoretical activity [6,15]. While the experimental situation is far advanced, the theoretical understanding of these phenomena is still not very satisfactory. The tunneling model of Anderson, Halperin and Varma [l] and of Phillips [14] has been of great phenomenological help for the explanation of the T1 behaviour of the specific heat at low temperatures as well as of the T2 behaviour of thermal conductivity, although it is not yet established whether it can account for the “plateau” both in thermal conductivity and in acoustic attenuation. For this problem, phonon scattering beyond the one-quantum regime must be accounted for, and in this region a Green function technique beyond Hartree-Fock may be useful [7]. Although the microscopic footing of the tunneling model is still unresolved, it may well preserve an important future role in the microscopic explanation of the phenomena at very low temperatures. On the other hand, the recent inelastic neutron scattering experiments of Buchenau and his collaborators [3,4] have given a new impetus to theoretical study, since they seem to provide a hint to the microscopic nature of the 0378-4371/90/$03.50 0
1990 - Elsevier Science Publishers B.V. (North-Holland)
230
M. Wagner and T. Mougios
tunneling
centers.
harmonic
soft modes.
the acoustic measurements vectors
Moreover,
phonon
Their
I Mesoscopic
they
in silica glasses
of the additional
emphasize
contribution
contributions
soft modes in disturbed lattices
(Debye)
importance
to the specific in the region
also incorporate
soft modes,
the
heat
of additional already
exceeds
above 2.5 K [3]. These
information
on the amplitude
from which one may deduce
that they are
coupled rotational oscillations of SiO, tetrahedra. Another piece of evidence for the importance of “mesoscopic” modes for the low-temperature thermodynamics of glasses is given by Pohl [16], who showed that pressed granulary systems of crystallites display the same specific heat anomaly as glasses. In view of this background, it would seem appropriate to introduce the concept of “mesoscopic” soft modes embedded in a quasicontinuum of more or less undisturbed acoustic modes. In the present study, the conventional Green function formalism of lattice dynamics is modified in such a manner that the softening of intrinsic mesoscopic motions of a given reference lattice can be handled in the spirit of the original Lifshitz technique [9-111. In the Lifshitz philosophy, the disturbed Green function is traced back to the undisturbed one by means of the solution of a matrix equation, the rank of which is given by the number of degrees of freedom involved in the disturbance. In lattice dynamical problems [2,5, 131, this technique has been applied to problems where the disturbance involves one or a few degrees of freedom in Cartesian space. In the present study, the idea is to start from a fixed mesoscopic combination of Cartesian coordinates, which defines a single mesoscopic degree of freedom, but may involve many Cartesian coordinates. If this object is known to turn soft, the dominant effective disturbance at the lattice will only involve this collective coordinate and its Cartesian partner coordinates which describe the coupling to the surroundings. The latter may also be combined to a singular mesoscopic mode. In this manner, a disturbance of low rank is established, and after an appropriate transcription a Lifshitz procedure is applicable. Moreover, since mesoscopic soft modes almost behave as individual oscillators, it would seem advantageous for the calculations to handle them as foreign degrees of freedom attached to an almost undisturbed reference lattice. To consider
this question,
another
modification
of the Lifshitz
procedure
is
required, which accounts for the additional degrees of freedom. Such an extension of the formalism has been given by one of us [17] earlier and is adapted here to the mesoscopic mode case.
2. Lattice dynamical Let us consider tonian
preliminaries
an oscillatory
system
which
is characterized
by the Hamil-
M. Wagner and T. Mougios
I Mesoscopic
soft modes in disturbed lattices
H=~~P:,+gium,xmx”, where X, are mass-reduced Cartesian coordinates (X,,, = Mz2x,). eigenvectors T(K) = [T,(K)] of the “dynamical” matrix urn,, %JLz(‘d
constitute
(1)
mn
m
c n
231
=
02,
a complete
‘L(K)
(2)
7
orthonormal
Then the
set
‘?;(K>‘?&‘> = kc, 3
(3)
F r);(K)‘L(K)= %zn
(4)
;
7
which may be employed
to define normal coordinates
Q, >
(5)
Q2,= cm ‘7;(K) x,n 3
(6)
X, = c ‘I,(K)
I(
which diagonalize
the Hamiltonian (7)
We note the sum rule
(8) We introduce
the Zubarev
Green function (GF) [21]
(9) which satisfies the equation
of motion
232
M. Wagner and T. Mougios
/ Mesoscopic
soft modes
in disturbed
lattices
G,,, (El = & %z, z (E2%,,- ~nz,,> = ;
‘%,@W2L - u,z.,),
(10)
where the latter equality is due to G being a symmetric matrix. Using the Plemelj formula l/(x + is) = P(l lx) + &S(x), we may interrelate the imaginary part of the GF with the density of oscillatory frequencies, G,,(w
+
is) - G,,(w
- ic)
= +i c 77,(K) n,*(K) $ ]a(W - & ) - a(W + 0, )] K K =&
c 77,(K) ‘-/,*(K)]a(O - 0,) + a(W + a,)]. K
Applying the orthonormality oscillatory frequencies
(11)
relation (3), we deduce from this the density of
p(w) = 2io z [Gmm(m + is) - G,,(o = -4w Im 2 G,,(o m
+ ic) ,
-is)] for 6~20,
(14
and the sum rule cc
I
dwp(w)=N=-41m
i
i
0
0
dwwCG,,(w+is). m
(13)
The latter two formulae are of enormous utility for the calculation of thermodynamic properties, as will be seen later. If an oscillatory system may be conceived as a deviation from an “undisturbed” one, the dynamics of which is known, H=HO+W,
(14)
w=;cwmnxmxn )
mn
(15)
such that the “disturbance” W only embodies a small number of coordinates, we may apply the Lifshitz formalism [lo] to calculate the disturbed GF from the undisturbed one. The equation of motion (10) may then be rewritten in the form
M. Wagner and T. Mougios
(E’Z -
I Mesoscopic
U”) G(E) = & Z +wG(E)
soft modes in disturbed lattices
233
,
(16)
where Z denotes the unity matrix Z,,,, = 6,“. By using Go(E) =
& (E*Z-
U”)-’
(17)
eq. (16) may be formally solved G(E) = Go(E) + 2rr Go(E) wG(E) ,
(18)
or likewise, since G(E) is symmetric, G(E) = Go(E) + 27r G(E) wG’(E)
.
(19)
Upon inserting (19) in (18), we end up with the compact formula G(E) = Go(E) + 2~ Go(E) wG’(E)
+ (~IT)*G’(E)
wG(E)
wG'(E)
. (20)
This expression most clearly evidences the very essence of Lifshitz’ formalism, the statement of which is that the full GF G(E) can be written down, if G(E) is known in the “small space” of the disturbance w. This can be deduced by considering the last term on the right-hand side of (20), which in the product wG(E) w just effects the projection of G(E) onto the “small space”, wG(E) w = w g(E) w,
(21)
where g(E) is that part of G(E) which pertains to the space of w. It is the reduction (21) which establishes the main computational option of the Lifshitz formalism. By using this expression, the “small space” version of eq. (20) assumes a closed form g(E) = go(E) + 2ng”(E)
wg’(E) + (24* go(E) wg(E)
wg’(E),
(22)
or, finally, g(E)
= [I - 2q”(E)
WI-’ go(E)
.
(23)
Inserting (23) in (21), and this then in (20), the overall expression for G(E)
234
M. Wagner and T. Mougios
I Mesoscopic
soft modes in disturbed lattices
reads G(E) = Go(E) + 2nG”(E)
w[Z - 2rrg”(E) WI-’ Go(E) ,
(24)
which attaches the G(E) parts outside the “small space” to those inside.
3. Mesoscopic coordinates It has to be noted, however, that the full efficiency of the Lifshitz procedure becomes effective only if one realizes that the true rank of w may be much smaller than the number of Cartesian coordinates involved in the disturbance. To exploit this, we introduce a set {S(y)} of orthonormal coordinates,
w=CaY)x,. m such that the disturbance
y=l,2,.
2
(25)
(15) displays the form
We emphasize that the number of Cartesian coordinates involved, may be much larger than the number R of effective “disturbance coordinates” S( 7). Evidently, by inserting (25) in (26) and identifying it with (15) we have (27)
For the sake of argument, we add to the given set of unit vectors {mm(y)}, constituting the disturbance coordinates (25), fictitious vectors with y = R + 1 . . 9 N, such that a complete orthornormal system in the full space is esiablished, $ c+:(Y) Cm(Y’) = &
$4xY)
(284
9
(28b)
Cn(Y) = a,, .
Then all matrices may be projected
onto this vector set, e.g.
M. Wagner and T. Mougios
I Mesoscopic
6%‘)
soft modes in disturbed lattices
T&(K))
(2
n
Ron*
%w)
235
T
(29)
etc. We note that Wvr, = 0 for y > R or y ’ > R. The Lifshitz equation (18) then assumes the form G&5)
= G;,(E)
+ 2lr 5 ~:,.(E)w,..,...~,...,.(E) Y“Y“’
)
(31)
which is se_en to be a closed system in the “small space” (1 s y s R),in which we write G = g”, g”(E) = g”O(E) + 2lrg”O(E) r@(E) )
(32)
which may be solved, since R is a small number,
g”O(E).
k?(E) = ]I - 2lrgO(E)w]-’ We insert this in the projected
(33)
form of (20),
G(E) = GO(E) + 27FG0(E)[G +27&&7(E) ;;I GO(E) = GO(E) +2&O(E)
W[Z + 27F[Z- 2$O(E)
k?-‘g”O(E) 61 GO(E) (34)
or GO(E) = GO(E) + 2&O(E)
W[Z - 27rg”O(E) WI-’ GO(E),
(35)
and returning to Cartesian space again by means of (28, 29), we finally end up with G,,(E)
= 5 U,(Y) &(E)
6Xr’)
YY'
=
G:,(E) + 257 2 G:,@) m’n’
x[Z - 27rg”‘(E) E],;.v,*.(~‘~)
5 vm,(y) cyy YY’ Y”
GO,.,(E) .
(36)
236
M. Wagner and T. Mougios
I Mesoscopic
soft modes in disturbed lattices
This formula reveals that the additional fictitious vectors u(y) for y > R have only played the role of an artifice. In the final result (21) they do not longer appear. Formula (33) will be our central equation for the evaluation of thermodynamical quantities in strongly disturbed lattices. In particular, applying (12 and 13) we derive the alteration in the density of oscillatory frequencies [2,5,131 Ap(o) = -4w Im c [Gmm(w + i&) - GL,(w
+ ia)]
R =
-8,~
Im c GL,,,. (W + ic) c u,,(y) mm’ YY’ n’ Y”
X[Z - 2rrg”‘(w + is) @],,$a,*,(r”)
GYy,
Gi,,(w
+ i&)
(37)
and the sum rule
I
dw Ahp(w) = 0.
(38)
4. Mass disturbances
Up to this point, we have assumed that mass-reduced coordinates are used throughout, which of course implies that also mass defects appear in the form of disturbances in the potential energy (see expression (15)). However, if mass defects are present, it sometimes is preferable to retain the mass-reduced Cartesian coordinates of the undisturbed system (X, = (Mi)1’2~,). This amounts to the definition of a modified GF G”(E) such that
G,,(E) =
G;,(E) (3)“’ . (3) 1’2
Multiplying (M,lM;)“2
(39)
n
m
(10) from the left by (M,lMi)-1’2 we find the equation of motion
-u”,,)G”,.,(E)=~6,,+E2(1-~)Gf.(E), T tE2%m,
and from
m
the right by
(40)
where U;,(E)
=
(g)‘;’ urn, (!$,“’ . m
n
(41)
M. Wagner and T. Mougios
I Mesoscopic
Eq. (40) again displays the structural =E2am~mn, WZZ”
soft modes in disturbed lattices
237
form of (16), with
,,=I-$,
(42)
m
hence we may proceed similarly up to formulae (35) and (36). We now may interpret y as the index of those degrees of freedom, the masses of which are disturbed (y = 1, . . . , R, ay # 0) and u~( y ) = S,,. Then the analogue of (36) reads G;,(E)
= G;,(E)
+ 21rE~ 5 G;,(E) YY’
a,[Z - 2ng”(E)
w”];;.G;.,(E)
.
(43) This procedure has been followed by many workers in the field [13] and dates back to Lifshitz himself. Its main virtue lies in the fact that for each Cartesian degree of freedom which is subjected to a mass alteration, only a single coordinate makes its appearance in the “disturbed” space. However, for the calculation of thermodynamical quantities it is less optimal, since the return from G” to G (see eq. (39)) implies some inexpedience in the formulae. On the other hand, we may as well remain in the legitimate mass-reduced formulation and exemplify the utility of a mesoscopic coordinate description for a massdefect type disturbance. Let us assume that the oscillatory coordinate m = 1 displays a mass defect (M, # Mi), whereas all others have the old masses. Then the deviation from the undisturbed Hamiltonian generally reads kv=;~[(~)1’2(~)“2-1]u&~x,x,, m n
(44)
and specifically, using p,,, = Mf, JM,,, - 1, v,,, = (MII,/M,)1’2
w= &qJ;lx;
+ $vlx, 2
uymxm.
rnfl
- 1,
(45)
At a first glance, this would appear to be a complicated form, since many Cartesian degrees of freedom are involved. But as a matter of fact only two “mesoscopic”degrees of freedom are needed. We introduce the unit vectors o-(l) = (l,O, 0,. . . ) ) a(2) = u-‘(0,
lJL, u;,,
(46) ...)
UL, . . .)
)
(47)
238
M. Wagner and T. Mougios
I Mesoscopic
soft modes in disturbed lattices
with the abbreviation: u=
( c (u:,)‘)1/7 ,
(48)
iTI#l
and their corresponding
coordinates
S(l) = x1 9 S(2) = u-l
c
(49) ULX,
.
(50)
??I#1
Then (45) assumes the form W= &_QJ;, S(1)2 + V,US(l) S(2) ) and the matrix quantities
6 and g”” (see expressions
(51) (23) and (27)),
(52)
(53) where (see (30))
(54)
Inserting this in (36), we are left with a 2 x 2 matrix diagonalization problem, which is easily handled; nevertheless, we will not pursue this further.
5. Extrinsic degrees of freedom
In the preceding sections, we have assumed that the disturbance of the reference lattice does not alter the number of degrees of freedom. There are two aspects, however, which motivate the study of reference lattices with
M. Wagner and T. Mougios
I Mesoscopic
soft modes in disturbed lattices
239
additional (“extrinsic”) degrees of freedom. On the one hand, there actually may exist such foreign degrees of freedom, e.g. if molecular defects are installed in the reference lattice. On the other hand, the disturbance itself may be such that some mesoscopic motion of the lattice behaves almost as an independent mode. Then the calculation may be simplified considerably if this motion is taken as an additional degree of freedom, provided the coupling to the background lattice is adjusted accordingly. The latter concept has been first adopted by Krumhansl and Matthews [8]. Extrinsic degrees of freedom may be handled by a method which has been developed by one of us (M.W.) many years ago [17] as an extension of the We modify this procedure slightly to fit our original Lifshitz procedure. purpose. We write the Hamiltonian in the form
where
(56) (57) H’“’
;
2
mr
(X,,,K,,X,
+ X,K,,X,J
,
JL, = Km
3
(58)
and we introduce the notation that the indices {m, n} are always confined to the intrinsic degrees of freedom and {r, s} to the extrinsic ones. The extended set of GF equations then reads ;
(E*Lc - U:‘,.)G,d~) = & 4,,, + c K,,,, G,,(E) , r
c @*a,,, - ~?)G,~,(~)
= x K,,,, G,,,(E)
7 (E*L - ~%LW
= & 4,
r’
?
m
,
+ ; K,,G,,(E) ,
(E*s,,, - U::.)G,.,(E) = T K,n,G,,(~)~
(59) (60)
(61) (62)
Since we tacitly assume that the number of foreign degrees of freedom {r} is small, it is easy to achieve the inversion of the matrix (E2Z - IV’“‘), and we therefore introduce the “extrinsic” GF
M. Wagner and T. Mougios
240
GE’ =
where
&
[E2Z’“’
Zlz’ = S,,. Then
_
U”‘],’
/ Mesoscopic
soft modes
in disturbed
lattices
(63)
)
eq. (60) transmutes
into
c
G,,(E) = 2,~sm G::‘(E)K,,G,,(E) , which we insert
(64)
in (59),
T (E*L, - ~imJ%,(E)
+2~ c LG:: rs
where the intrinsic dynamical matrix WE!. In this manner we have returned effective intrinsic disturbance
k) G,,,(E) >
W
(65)
has been separated into UEi = UtA + to an equation of the type (16) with an
where the formalism is traced back to the one given in sections 2 and 3. This, however, only refers to the intrinsic part G,,(E) of the total GF. Since the density of frequencies now also incorporates the foreign degrees of freedom we also need the complementary part G,,(E) of the GF. This is easily found from eqs (61) and (62). Defining G;‘, =
G,,
and remembering
=
&(E*Z
- U’i’),;
that by definition
c
(m,nfr,s) m, n # r, s we have from
G,,(E) = 27rnr G:;(E) K,,G,,(E) and inserted
(67)
,
(62)
(68)
in (61)
or G,,(E)
= $
{E2Z”’ - [U’“’ + ~TKG”‘(E)
K]},’
.
(69b)
M. Wagner and T. Mougios
I Mesoscopic
241
soft modes in disturbed lattices
The solution of this equation requires the knowledge of G”‘(E) (see expression (67)), which is found by writing UCi) = U” + wCi)and applying the original Lifshitz formalism of section 2. From expression (69b), in analogy with expression (12)) we deduce the modified extrinsic density of states p,,(o)
= -4w Im c G,,(o + ic) .
(70)
I
6. Green function thermodynamics The internal energy of an oscillatory system with frequencies
{a,}
is given
by 1 u=$&coth(tpO,J, K
‘=k,T’
(71)
-
from which we derive the specific heat (72) As mentioned earlier, the key formulae for a Green function description are the spectral density formulae (12) and (37). Specifically, the excess thermodynamic quantities are given as
AU = U - U” = ; j- dw Ap(o)w coth( $w)
,
(73)
m
’
ACeC-C’=
dw Ap(o)
4k,T?
0
O2 , sinh2( i PO)
where Ap(o) is given by expression (37). Inserting in this expression definition (9) of the undistrubed GF, we have (E = o t ia) Ap(o)
=
-$
Im
c
r*
(y,,)
n’
and employing
the
‘~2K!~~!;!*
mm’n KX’ x
(74)
z
Y”
‘7%‘) 77:tK’)* E2
-
(@r)’
the orthonormality
’
relation (3),
(75)
M. Wagner and T. Mougios
242
A&)=-$Im
/ Mesoscopic
c l “2’ (E2 - (Q’)’
soft modes in disturbed lattices
i [‘L&(K)* a,,(Y)] yy’ Y”
x$,,[z-27Tg""(E) ti$$[~7,*~(~') ?$(K)] . Now, by noticing
that
g”&(E)= &
(see definition
c b%“‘)
m’n’
(76)
(30))
d~(K)l[d*(K)*‘=m,(Y)l
(77)
3
(L’jl)’
E* -
K
we may write
b%“‘) 77~~(K)h%‘)* ~,n,(‘Y)l [(o
K
+
1 L
g”,,
= -2W do
ie)2 - (fl,“)‘]’
(w
+
iE) ’
y”y
(78) and upon
inserting
in (76) we arrive
Ap(w) = 2 Im 5
“,,,,[I
- BITT”’
at
$&,(E),
WI,&, 2
E=w+i.s.
YY’
Y”
(79) If W does not depend
Ap(w) = 2 Im &
on w, this may be simplified
$
{ln[Z-
2ng"'(E) G]},,
In passing, we note that the latter strongly by using the trace-determinant Tr{ln by means
to
expression relation
A} = ln{det A} ,
formally
may
be simplified
of matrices
(81)
of which we have
Ap(w) = 2 Im &
ln[Z - 27~g”‘(E) W] .
(84
This transcription sometimes may be advantageous. We now insert expression (79) in expressions (73) and (74). Then we get for the excess internal energy
M. Wagner and T. Mougios
I Mesoscopic
243
soft modes in disturbed lattices
m
AU=U-U’=Im
J^
dw wcoth($w)
i
~,,,.[Z-27~g”~(E)
CS];;.
YY’ Y v
0
E=w+iE,
(83)
and for the excess specific heat AC=C-Co =yIm
1
dw sinh;; I
2k,T
x
$
pw) ;, G;vy,]Z- 27~g”‘(E) @I],;~
0
Y”
g”,“,,,(E)
E=U+ie.
,
(84)
In applications of these formulae, it would seem that the computation turns very complicated if the number of unit vectors o(y) (see (2.5)), which constitute the space of the disturbance, is large, since the diagonalization of matrix $[I - ~IT~“‘(E)v?]-’ dg”‘(E) /dw is required. This, however, need not be the case, since the defect may be governed by a symmetry group. Then a(y) would be irreducible base vectors to this group,
where r is the multiplicity index of the irreducible representation Z in the “small space” of the disturbance, r = 1, . . . , R(T), whereas j denotes the degeneracy index of Z, j = 1, . . . , fr . Then Ap(o) decomposes into irreducible parts,
b(w) =
c fr MC
~1,
(86)
I-
R(r)
Ap(Z; w) = 2 Im c
rr’ r”
E=w+is,
w”,,(Zj)[Z - 2ng”‘(Z’j; E) W(I’j)]~~,, &
&(rj;
E)
(87)
being independent of the degeneracy index j. In this way, the remaining matrix diagonalization sub-problems are of rank Z?(T). Particularly, if an irreducible representation Z only appears once (Z?(T) = l), there is no diagonalization problem whatsoever.
244
M. Wagner and T. Mougios
7. Three elucidative
1 Mesoscopic
soft modes in disturbed lattices
case studies (intrinsic softening)
In this section, we apply the formalism to a singular chain of N atoms with periodic boundary conditions, X,,,,, = X,, and nearest neighbour spring interaction,
Ho= ; j, P:,+ ;$
i
(x, -x,,,+,>’ ,
(88)
m-l
with the Bloch eigenvectors 77,(Kj
=
~-112
ei27rKmlN
, K
=
0,
+I,
k2, . ..,?$N-l,+$N,
where N has been taken as an even number. The undisturbed then read 0: = 0, sin(rKlN)
)
n,
= (4flM)“2
(89)
eigenfrequencies
)
(90)
and it is worth recording the sum rules
(91)
The Green function pertaining
to this linear lattice model,
i2nr(m-n)lN G:,(E)
=
kN
c I(
e
E2 _
(@>2
=
eJ-9
(92)
3
can be given in a closed form [19], ilm-nlq
G;,(E)=
-&
e D
sincp
for O
’
and w>L!,
,
(93)
and where w is expressed via the quantity cp, w = 0, sin( $9) ,
osn,:
O~(p~~,
wS.n,:
cp=n+iz,
For 0, - E < w < 0, + E a separate
O
consideration
is needed.
(94) But we may
M. Wagner and T. Mougios
/ Mesoscopic
soft modes in disturbed lattices
245
discard this analysis, since we may determine the p(w) behaviour in this region via sum rules. We now consider several defect cases. 7.1. Single spring constant defect ,=lf’-f
* 7
(4
The perturbation a(1) = $
(1, -l,O, 1,
S(1) = 5
(95)
may be described by a single-unit vector (see (25))
m.
with coordinate
- X2)’ *
2, 3,
. . . ) 0) ) . . . ,N ,
(96)
(see (25)) (X, - X,) .
(97)
Then
w=
$wS(1)2
The projection
)
+_2f’-f
(98)
M
(30) is now of the form
g”‘(E) = G;, - GL,
(99)
or
for w > 0, , where expression (93) has been inserted. We note that g”“(o + is) is not yet specified in an &-surrounding around w = 0, and displays a singular behaviour there. This is known since the works of Lifshitz and Kosewich [ll] and Mahanty et al. [12]. But we need not specify this behaviour, because we can determine the behaviour of Ap(w) in this
M. Wagner and T. Mougios
246
I Mesoscopic
soft modes in disturbed lattices
region by means of the sum rule (38). Inserting g”O(E) and W in (79), (83) and (84), we further note that the region o > 0, (by way of the infinitesimal imaginary part of g”‘(E) in this region) only plays a role if the denominator 1 - 2ng”‘(E)G displays a pole in this region. This would indicate then a localized mode, but it only appears if W > 0 (f’ >f). We refrain from discussing this case, since it has often been discussed in the literature (see e.g. the books of Maradudin et al. [13] and Dederichs et al. [5]). Henceforth, we only regard the “softening” case W < 0 (f’ < f). Then formula (79) assumes the form (see fig. la)
Ap(w) = -2 9... Im
=-_ 1 f’ f-f’
(l-x2)l/2,(~)~+(l’
non,f f
forx=w/,f&, = -
;qw2 - 0;)
O~L~GLZ~-E,
foran-s
)
(101a) (101b)
where the latter formula has been found via sum rule (38). Specifically in the low-frequency region we find
(102)
which yields for the excess specific heat (74), (84) the low-temperature ($@‘,)a
result
‘)
AC=
which is in the extremely AC=
f
f’
f -f’ f
low-temperature
regime ( f p0,(
f ‘lf) 9 1)
TkB k,T 3
fl,
(104a) )
whereas for somewhat higher temperatures,
provided
f’ 4 f,
M. Wagner and T. Mougios
I Mesoscopic soft modes in disturbed lattices
3-
72 52. * c?
247
.I -
f’=O.lf
_---- -___
f’=-JSf f’ = 0.9 f
a‘.I’
x=w/
I I1 .: .I .’ I :.I,I ,I i: : I, .’: .,’
n,
Fig. 1. (a) Excessive frequency density in a linear chain with nearest neighbour springs f, when a single spring is softened (f’ < f). Solid line: f’ = 0. lf, dashed line: f’ = OSf, dotted line: f’ = 0.9f. (b) Excessive specific heat, when a single spring is softened (f’
M. Wagner and T. Mougios
248
I Mesoscopic
soft modes
in disturbed
lattices
AC= +k,,
(104b)
which is the Dulong-Petit limit. The total numerical behaviour employing the exact formula (lOl), is depicted in fig. lb. 7.2. Softening of two neighbouring
w= ;
f%f
[(X_,
of AC(T),
springs
- X0)’ + (X0-
Xl)“] .
(105)
Now we have two reduced Cartesian coordinates X_, - X0 and X0 - X, involved in the disturbance, but since the defect exhibits inversion symmetry, the coordinates uniquely split into an odd and an even symmetry coordinate. Hence we may proceed along the lines indicated in (85), (86) and (87). We introduce the symmetry vectors (u = “ungerade” = odd, g = “gerade” = even), a(u) =
$
(. . . ,
m.
o,-1,2,-l,o
. . . ) -2,
-l,O,
(106)
1,2 )...)
o,-l,O,
4d=$C..,
)... ),
1,o )... ),
with symmetry coordinates S(u) = $
(-X-r
+2x,
- X,) )
(107) %)=&(x,-x_,).
Expression
(105) is then converted
w= &+A) s(u)2+
&c(g)
into
s(g)*
qu)
= 3
)
f%f
)
qg>
= f+f.
The projections of the Green functions onto the irreducible base vectors (106) read (E = o + is) g”‘(u; E) = ;[6Gi,(E)
- 8Gy,,(E)
2 =s
(
+ 2Gy,_,(E)]
-1 - 2x2 - 2i V&
>
,
x=o/n,<1,
(109)
M. Wagner and T. Mougios
I Mesoscopic
soft modes in disturbed lattices
249
iok E) = G:,(E)- G:,_,(E) (-1 + 2x2 - 2ix Vi?)
= --$
,
x=w/On,
(110)
D
where formula (89) has been used. From (108), (109) and (110) inserted in expression (87)) we find the excessive densities of state Ap(u; W) = 2 Im G(u)[l - 27rg”‘(u; w + is) C(U)]-’ -& g”‘(u; w +is)
=--
2 f-f’
f
7FoD
forx=o/aD,
[3(f)-41x*}
’
OSx
Ap( g; 0) = 2 Im $u)[l
(111)
- 2rg”‘( g; w + ic) W(U)]-’ -& i”( g; w + is)
=- 2 f-f’ ToD f forx=w/OD,
OSx
(112)
The behaviour of these two excessive frequency densities is depicted in figs. 2a and 2b. Expressions (111) and (112) are used for the computation of the excessive specific heat given in figs. 3a and 3b. In the low-frequency region and for f’ 4 f formulae (111) and (112) simplify to
(113)
forx=olR,41,
f’*f,
(114)
250
M. Wagner and T. Mougios
/ Mesoscopic
yielding the very-low temperature AC(u; T) = $
$
soft modes in disturbed lattices
excess of specific heat (i/30,
s 1)
k,(y)‘,
(115)
D
2n f T) = 3 7
Wg;
k,T k, F
(116)
,
whereas for somewhat higher temperatures AC(u; T) =
k,
AC(g; T) =
$k,
( 4 @I,(
f ‘lf) G 1 < i pa,),
,
we find (117)
.
(118)
7.3. Spring softening at distant sites (mesoscopic modes)
w=~f~f[(x2-xl)2+(x,+3-xr+2)2],
r=l,2
)....
(119)
r thus counts the number of undisturbed springs between the two disturbed ones. Again there are two Cartesian coordinates (X, - X, and X,+3 - Xr+2) involved in the disturbance, and there is inversion symmetry around site i(r + 4), whence we may proceed similarly as in section 7.2. We introduce the symmetry parity-ordered vectors a(u)=i(
l,-l,o
m:
1,
dg>
=
2,3
)...)
0,-l,
,...,
0,
l,o )...)
;w,
1,
r+Z,r+3,
0,-l,
0 )...)
O),
(120a)
. . . . N,
1,
0, . . 30) ,
(120b)
with symmetry coordinates S(u)
=
t(x,
S(g)
=
ic-Xl
Expression
-
x2
-
x,+2
+
K-+3)
(119) then transmutes
w&qu)=qg)=2f+f.
(121a) (121b)
+X,-X+2+X,+,).
w= +w[s(u)2+ s(gy]
>
)
into
(122) (123)
M. Wagner and T. Mougios
I Mesoscopic
soft modes in disturbed lattices
251
The projections of the Green functions onto the irreducible base vectors (120a) and (120b) read now
go@; El = Gi,(E)- G?,(E) + $G~,,(E) - G:+,,,(E)
+ $G;+,,,(E)
, (124)
g”“(g;E) = G:,(E)
-
G;,(E)- 4G:,,(E)+ G:+,,,(E)- $G;+,,,(E). (125)
Inserting (93) we find for arbitrary
-$
g”“(u;E) = -
r
{1+tg(tcp)sin[(r+l)cp]+2itg(+cp)sin2[~(r+l)cp]}, D (126)
8”‘(g; E) = - --$
{l-tg(fcp)sin[(r+l)cp]+2itg($cp)cos*[$(r+l)~]}. D
(127) For r = 1 this reads
g”“(u;E) = --$ (- 1 - 4x2 + 8x4 - Six3 Vi-?)
,
D
(128)
x = 6J/.nD.
For r = 2:
g”‘(u;E) = --&
2ix3(9 - 24x2 + 16x4)
-1-6x2+32x4-32x6-
g”“(
g;
E) = L
ma:,
)
vi-7
D
(
_1 +
6x2
_
32x4
+
32x6
_
%x(1 - 9x2 + 24x4 - 16x6) e
x=6_l/.nD,
whereas for
g”‘(u;E)
r =
’
(129) 4, the expressions
= -& D
(
already are of a somewhat
lengthy form:
-1 - 10x2 + 160x4 - 672x6 + 1024x8 - 512x”
- 2ix3(25 - 200x2 + 560x4 - 640x6 + 256x8) Vi=?
2
252
M. Wagner and T. Mougios
1 = 2 non,
I Mesoscopic
soft modes
in disturbed
lattices
-1 + 10x2 - 160x4 + 672x6 - 1024x8 + 512x”
2ix( 1 - 2.5x2 + 200x4 - 560x6 + 640x* - 256x”) >7
Vi? x=c.d/f2D.
(130)
Inserting expressions (123) and (128) into (130), we find the respective densities of states Ap(u; w) and Ap( g; o) and therefrom the excessive specific heat expressions AC(u; T) and AC( g; T). The numerical results are depicted in figs. 2a, 2b, 3a and 3b. We first analyse the sharp peaks in the intermediate and upper frequency domain, e.g. the one pertaining to r = 1 in fig. 2a. In this case there is one simple “hard” spring f between the two softened springs f’. Thus, in the limiting case f’ G f, the two masses between the two soft spring sites will display an internal “molecular” mode of the approximate frequency 0 = (2f/M)“2 = 0,/V?. It is this singular (localized) mode which generates the sharp peaking at x = 0.71 in fig. 2a. In the same manner the other sharp maxima find their explanation. For I = 4 the molecular unit between the soft springs has two even and two odd internal vibrations, which make their appearance in figs. 2a and 2b.
___-__
-5 L1 !i
-10 0.00
a
0.50
0.25 x=w/
0.75 n,
1.00
M,
Wagner
and
Mougios
T.
I Mesoscopic
soft
modesin distwbed
25-
, I'
i
I I
0'
I
20 15 -
r=O
_- ___---_ r=l r=2 _-----
I I
I
I I II I, I II ,I 1; I II
I :I, !I /: ! :I
I’ ____________-__;_I_________________j_I: - - - - _:_ i _‘,__ _ “’_ _ _.- “._ _ _,l’ _-,y__’ ---__ --_ _..,Ilb, \ \ : \ .I !,:; ,8
-5-10 0.00
r=4
I
;I I! I’I :I
f'= 0.1f
253
lattices
b
0.75
0.50
0.25
x=w/
1.00
R,
1
0.0
C
0.1
0.3
0.2 x=w/
0.5
0.4
i-l,
Fig. 2. (a) Excessive density of frequencies in a linear chain. Softening of two with r undisturbed springs in between. Solid line: r = 0, dashed line: r = 1, dash-dotted line: r = 4. Even mode case. (b) Excessive density of frequencies in a linear chain. Softening of two springs undisturbed springs in between. Solid line: r = 0, dashed line: r = 1, dotted line: line r = 4. Odd mode case. (c) Dependence of the softening of the odd mode excessive frequency density undisturbed springs between the two undisturbed ones (f’ = O.lf). Solid line: r = 1, dotted line: r = 2, dash-dotted line: r = 4.
springs dotted
(f' = O.lf)
line:
r = 2,
(f’ = O.lf) with r r = 2, dash-dotted: on the number r of r = 0, dashed line:
M. Wagner and T. Mougios
I Mesoscopic
soft modes in disturbed lattices
1.00
f'=O.lf 0.75 -
r=O _____---- r=l
r” ;
OSO-
r=2
3 a
___---
-----_____ 0.00 0.00
a
0.25
0.50 T/
r=4
--‘-----_-____,_ 0.75
1.00
0,
f'=O.lf r=O
---------
r= 1 r=2
__----
r=4
0.25
/
‘b
0.25
0.50 T/
0.75
0,
Fig. 3. (a) Excessive specific heat for a linear chain with softening of two springs undisturbed springs in between. Solid line: r = 0, dashed line: r = 1, dotted line: line: r = 4. Even mode case. (b) Excessive specific heat for a linear chain with softening of two springs undisturbed springs in between. Solid line: r = 0, dashed line: r = 1, dotted line: line: r = 4. Odd mode case.
(f’ = O.lf) with r r = 2, dash-dotted (f’ = O.lf) with r r = 2, dash-dotted
M. Wagner and T. Mougios
I Mesoscopic
255
soft modes in disturbed lattices
However, in our context the more important feature of these figures is the low-frequency enhancement of the mode density, which in the odd mode case (fig. 2b) is a manifestation of the center-of-mass motion of the “mesoscopic” unit between the soft spring sites. Accordingly, the position of the maximum shifts to lower frequencies with increasing number of atoms belonging to the mesoscopic unit. Astonishingly, there also is a low-frequency enhancement in the even mode density (fig. 2a), which does not seem to have a simple qualitative perspective. In the low-frequency region and for f’ 6 f, the respective excess density formulae derived from (123) to (130) simplify to 3 f_I.X2 - 2(r + 1)X4 Ap(u; w) = -$
f
(r + 1)’ D
(131)
f_l - 2(r + 1)x2 2 + 4(r + 1)4X6 ’
>
f
x=wmD,
r=0,1,2,4.
(132)
From this formulae we deduce that the odd mode softening has a pronounced dependence on the number of atoms between the disturbed springs, whereas for the even modes it is less salient. For Ap(u; w), the prefactor to the low-frequency slope increases in proportion to (r + l)‘, whereas the position of the maximum decreases in proportion to (r + 1)-“2, but the maximal height increasing again in proportion to (r + 1) (see fig. 2~). Taken together, these features establish the tendency to increase the low-temperature specific heat with ascending number (r + 1) of atoms between the disturbed springs. Quantitatively we derive this from (131) if we insert it in (74). For pL!,~l,
f’%f:
AC(u; T) = $
f
(r + l)Zk,( z)2,
for r = 0, 1,2,4 .
D
(133)
For p~,(f’lf)~l~p0,: AC(u; T) = k, . Similarly, inserting (132) in (74), we find the low-temperature the even modes.
(134) behaviour
for
M. Wagner and T. Mougios
256
/ Mesoscopic
soft modes in disturbed lattices
For pJ& % 1, f’ +f: 2n f T) = 3 7
AC(g;
For @I,(
f ‘lf) G
(135)
.
14 POD:
AC(g; T) = ;k,
8. Extrinsic
k,T k,n,
(136)
.
soft mode
To get insight into this problem we again study a one-dimensional model of prototype nature. This is depicted in fig. 4. The Hamiltonian of this model is given by H
=
H(i)
+ H(‘)
+ ~(“1
(137)
2
where (138) H’“’ = ; (P,’ + L?,“Q,‘) ,
(139) (140)
(141)
n3=2$ s
We now apply the formalism of section 5 with the specifications (see eq. (66)) w(i) f, mn = z %&L,,
+ L,)
)
Fig. 4. Prototype model of an extrinsic degree of freedom coupled to a lattice.
(142)
M. Wagner and T. Mougios
I Mesoscopic
soft modes in disturbed lattices
L Km
@e’=~ ss
=
-
(MM,)‘/2
G%l,,
+
1
The effective intrinsic disturbance
1 +
We introduce a,(u)
z
(143)
%I-,),
E=w+i~,
2~ E2 - fl,2 ’
251
(144)
(66) acquires the form
0; E2
_
a2
s
Pm,pk--p
+
%-,%,
+
4ws~rn>P
+
%I-ii
,I) * (145)
the symmetry vectors (see (25))
= -$
m,(g) = $
(Sm,, + Sm._&)
(“ungerade”),
(146)
(S,,, - S,,-P)
(“gerade”)
(147)
,
by means of which the formalism of section 6 becomes applicable. Thus, the intrinsic change of the frequency density is given by expression (87), where r= u or r = g. Introducing the projections (cr*IG’Icr) and (~T*~w@~~)(u)(see eqs. (29) (30)) (erf) w”(u, E) = WE,;) + wF _IL _- 2f (eff) kC(g) = wf,f;) - w,,_,
E2 M E2 - fl,z ’
(148)
)
(149)
_
- zL
g”“‘(u) = G;,‘,(E)
+ G’,,‘,(E)
= &
A,(E),
(150)
g”‘“‘(g) = GE;(E)
- G;:,(E)
= &
B,(E),
(151)
we get
ApCi)(u, 0) = 2 Im w(u, E)[l - 21r g”(‘)(u, w + ie) W(u, E)]-’ X &
g”‘O’(u, 0 +i~)
E2 - 0; - (f,IM)E2A,(E)
’
(152)
258
M. Wagner and T. Mougios
Ap”‘( g, W) = $ k Im = -$
I-
Im &
I Mesoscopic
soft modes in disturbed lattices
(JYM)B,(E) ln[l -
(hIM)B,(E)]
.
(153)
As explained at the end of section 5, this density has to be supplemented by the extrinsic contribution ensuing from the extrinsic GF (70), which in our example reads G,,(E) = & [ E2 -
0,Z(l +21r ;
where g”“‘(u, E) is the “statically”
g”“‘(u,E)
=
,c.CT;(U)G;‘,(E)
= [l - 24’)(u,
g”(‘)(u, E))] -’ ,
disturbed
(154)
GF defined by (67),
un(u)
E) F?,‘~‘(u)]-’ g”“‘(u, E)
A,(E) =- 1 2~ 1 - (f,IM)A,(E)
(155)
’
where Gci’(u) = f,lM has been used for the “static” intrinsic disturbance. this we find 1 I- (L/M) A,(E) GSS(E) = % E* - 0; - (f,/M)E*A,(E)
From
(156)
’
which by means of (70) yields the modified extrinsic density of states p,(w) = -F
Im
1 - (f,IM)A,(E) E* - 0; - ( fsIM)E2A,(E)
Combining this with expression Apci)(u, o) + p,(o) = -a
’
(157)
(152), we finally arrive at (E = w + ia) Im -& ln[E2 - 0: - (f,IM)E2A,(E)].
(158)
This is drawn in fig. 5, where MS = (2~ + l)M and& = f/l0 has been chosen. It is therefore suggestive to compare this figure with the corresponding one of an intrinsic softening (see fig. 2b), identifying r with /_L.Then in both cases, the peak in the excessive mode density progresses towards lower frequencies with increasing effective mass of the soft mode. However, in the extrinsic case
M. Wagner and T. Mougios
I Mesoscopic
soft modes in disturbed lattices
259
25
-5 -
0.00
0.25
0.50
1.00
0.75
x=w/R,
Fig. 5. Modified frequency density of a singular oscillator (0,’ = 2f,lM) coupled to a chain of oscillators (M, f) at sites t/~ with springs f, (see fig. 4). f, = O.lf, MS = (2~ + l)M. Solid line: p = 1, dashed line: p = 2, dotted line: /.L= 3, dash-dotted line: y = 4.
(fig. 5) the peak position for I_L= 1 is already considerably lower than for r = 1 in the intrinsic case (fig. 2b), and the further progression is towards higher peaks for higher p-values (fig. 5), in contrast to almost unaltered heights for higher r-values (fig. 2b). The high-frequency parts of both figures naturally differ in a qualitative manner. The slight wavy nature of the curves in the upper frequency regime of fig. 5 is due to the wavy nature of the sin(2pq) and cos*( ~CJJ)terms in the projected GF A,(E) (see below expression (159)). In fig. 6 a fixed distance p = 2 is chosen for a variety of foreign mass MS values and the total excessive density of states is drawn. Expressions (153) and (158) may be analytically exploited, if we employ formula (89) for the evaluation of the quantities A,(E) and B,(E),
A,(E)
= $
&
hG-w)
-
co~*b-41y
(159)
bw)I,
(160)
D
B,(E)=-+
&
[~WG-w~
+
sin*(
D
which have the low-frequency A,(E)
= -$
(4P[l D
behaviour
(X = w/L?,)
- 5(4P2 - QX2] - i]2 - (‘<
- ‘)‘*I)
,
(161)
260
M. Wagner and T. Mougios
1 Mesoscopic soft modes in distwbed lattices
40
, f, = 0.1 f, p = 2
30 .
20
-
-
I’\ ,’ ‘1
M,=M
--------
_ MscZM
______
0.1
0.0
0.2
0.3 x=w/
0.4
0.5
n,
Fig. 6. Total excessive density of states due to a singular oscillator (of = 2f,lM,) coupled to a chain of oscillators (M, f) at sites p = 22 with springs f, = O.lf. Solid line: MS = M, dashed line: MS = 2M, dotted line: MS = 4M, dash-dotted line: M, = SM.
B,(E)
= -$
{441.[1- :(4p*
- 1)x*] - i8p2x}
.
(162)
D
From this, inserted in eqs. (153) and (158) respectively,
we get (x = o/O,)
(163)
(164)
9. Summary
and remarks
This work has been stimulated by the physical question in what way the softening of atomistic elastic properties (spring constants) of condensed quasiharmonic systems influence the low-frequency mode density, and consequently the low-temperature specific heat. This question seems to be an urgent one in
M. Wagner and T. Mougios
I Mesoscopic
soft modes in disturbed lattices
261
view of the experimentally suggested (neutron scattering) low-frequency enhancement of harmonic modes in glasses on the one hand, and of the suggested strongly anharmonic models (tunneling, etc.) for the unusual low-temperature properties of glasses on the other. The work has also been initiated by the background concept of mesoscopic soft modes, i.e. the concept of a soft cooperative motion of a mesoscopic number of atoms. We have developed an extended Green-function-Lifshitz procedure, which, after introduction of appropriate mesoscopic disturbance vectors, allows for a utilization of the computational options of the original Lifshitz technique, which had been restricted to a small number of Cartesian coordinates. The formalism covers both the cases of an intrinsic disturbance (no additional degrees of freedom) as well as that of an extrinsic disturbance (additional degrees of freedom). The formalism is applied to archetypical model systems. Specifically, we have made several case studies for linear lattices disturbed at mesoscopically distant sites (spring softening). It is found that there always is a low-frequency peaked enhancement of mode density. For a fixed elastic alteration, the peak position descends to lower frequencies with ascending number of atoms between the disturbed sites, showing that the most effective soft modes are those of mesoscopic nature. We further have calculated the behaviour of a foreign mesoscopic mass MS (aMlattice ) which is harmonically coupled to a linear lattice with various distances between the coupling sites. It is found that the arrangement somewhat simulates the low-frequency mode density of a corresponding intrinsic disturbance. Here it is also found that the position of the low-frequency density peak descends with increasng mass MS, if M, is chosen to increase in proportion to the coupling distance CL.However, in detail there are pronounced qualitative differences between the considered intrinsic and extrinsic cases, such that a simulation of the intrinsic behaviour by an extrinsic mode must be done with care. We have not presented here the calculation of mesoscopic disturbances in three-dimensional lattices. This calculation will be given elsewhere. In this context, mesoscopic modes of librational character would be of particular interest, since modes of this nature have been found experimentally (Buchenau). The presented formalism is also well fit to handle these modes, although the computational effort is considerably higher and less lucid.
Acknowledgement
We have profited from discussions with G. Zavt, simplifications of some of our formulae.
which have lead to
262
M. Wagner and T. Mougios
I Mesoscopic
soft modes in disturbed lattices
References [l] [2] [3] [4] [5] [6] [7] [8] [9] [lo] [ll] [12] [13]
[14] [15] [16] [17] [18] [19] [20] [21]
P.W. Anderson, B.I. Halperin and C.M. Varma, Phil. Mag 25 (1972) 1. H. Bottger, Principlees of the Theory of Lattice Dynamics (Physik-Verlag, Weinheim, 1983). U. Buchenau, Physica Scripta T19 (1987) 350. U. Buchenau, H.M. Zhou, N. Nucker, K.S. Gilroy and W.A. Phillips, Phys. Rev. Lett. 60 (1988) 1318. P.H. Dederichs and R. Zeller, in: Point Defects in Metals II, Springer Tracts in Modem Physics, vol. 87 (Springer, Berlin, 1980). S. Hunklinger and W. Arnold, in: Physical Acoustics, vol. 12, W.P. Mason and R.N. Thurston, eds. (Academic Press, New York, 1976) p. 155. W. Junker and M. Wagner, J. Phys: Condens. Matter 1 (1989) 5065. J.A. Krumhansl and A.D. Matthews, Phys. Rev. 166 (1968) 856. I.M. Lifshitz, JETP 17 (1947) 1017. I.M. Lifshitz, Nuovo Cimento Suppl. 10 3 (1956) 716. I.M. Lifshitz and A.M. Kosevich, Rep. Prog. Phys. 29 (1966) 217. J. Mahanty and A.A. Maradudin, Prog. Theor. Phys. 20 (1958) 369. A.A. Maradudin, E.W. Montroll, C.H. Weiss and I.P. Ipatova, Theory of Lattice Dynamics in the Harmonic Approximation, Solid State Physics, Suppl. 3, 2nd ed. (Academic Press, New York, 1971). W.A. Phillips, J. Low Temp. Phys. 7 (1972) 351. Properties (Springer, Berlin, W.A. Phillips, ed., Amorphous Solids, Low-Temperature Heidelberg, New York, 1981). R.O. Pohl, see his article in ref. [15]. M. Wagner, Phys. Rev. 131 (1963) 2520. M. Wagner, Phys. Rev. A 133 (1964) 750. R.F. Wallis and A.A. Maradudin, Prog. Theor. Phys. 24 (1960) 1055. R.C. Zeller and R.O. Pohl, Phys. Rev. B 4 (1971) 2029. D.N. Zubarev, Sov. Phys. Usp. 3 (1960) 320.
THERMODYNAMICS OF MESOSCOPIC SOFT MODES IN STRONGLY DISTURBED LATTICES Max WAGNER
and Theodor
MOUGIOS
Institut fiir Theoretische Physik, Universitiit Stuttgart, Pfaffenwaldring Fed. Rep. Germany
57, D-7000 Stuttgart 80,
Received 3 November 1989 Revised manuscript received 13 February 1990
A generalized Lifshitz procedure is developed, which allows the evaluation of frequency densities in mesoscopically disturbed harmonic lattices. Archetypical examples are calculated explicitly. In particular it is shown that lattice softening at mesoscopically distant sites produces a low-frequency enhancement of the increase in mode density, which may be ascribed to the oscillation of mesoscopically large masses. Possible relevance for the anomalous low-temperature behaviour of amorphous systems is discussed.
1. Introduction The unusual thermodynamic and thermal-conductivity properties of amorphous solids, first found by Zeller and Pohl 19 years ago [20], have stimulated a great amount of experimental and theoretical activity [6,15]. While the experimental situation is far advanced, the theoretical understanding of these phenomena is still not very satisfactory. The tunneling model of Anderson, Halperin and Varma [l] and of Phillips [14] has been of great phenomenological help for the explanation of the T1 behaviour of the specific heat at low temperatures as well as of the T2 behaviour of thermal conductivity, although it is not yet established whether it can account for the “plateau” both in thermal conductivity and in acoustic attenuation. For this problem, phonon scattering beyond the one-quantum regime must be accounted for, and in this region a Green function technique beyond Hartree-Fock may be useful [7]. Although the microscopic footing of the tunneling model is still unresolved, it may well preserve an important future role in the microscopic explanation of the phenomena at very low temperatures. On the other hand, the recent inelastic neutron scattering experiments of Buchenau and his collaborators [3,4] have given a new impetus to theoretical study, since they seem to provide a hint to the microscopic nature of the 0378-4371/90/$03.50 0
1990 - Elsevier Science Publishers B.V. (North-Holland)
230
M. Wagner and T. Mougios
tunneling
centers.
harmonic
soft modes.
the acoustic measurements vectors
Moreover,
phonon
Their
I Mesoscopic
they
in silica glasses
of the additional
emphasize
contribution
contributions
soft modes in disturbed lattices
(Debye)
importance
to the specific in the region
also incorporate
soft modes,
the
heat
of additional already
exceeds
above 2.5 K [3]. These
information
on the amplitude
from which one may deduce
that they are
coupled rotational oscillations of SiO, tetrahedra. Another piece of evidence for the importance of “mesoscopic” modes for the low-temperature thermodynamics of glasses is given by Pohl [16], who showed that pressed granulary systems of crystallites display the same specific heat anomaly as glasses. In view of this background, it would seem appropriate to introduce the concept of “mesoscopic” soft modes embedded in a quasicontinuum of more or less undisturbed acoustic modes. In the present study, the conventional Green function formalism of lattice dynamics is modified in such a manner that the softening of intrinsic mesoscopic motions of a given reference lattice can be handled in the spirit of the original Lifshitz technique [9-111. In the Lifshitz philosophy, the disturbed Green function is traced back to the undisturbed one by means of the solution of a matrix equation, the rank of which is given by the number of degrees of freedom involved in the disturbance. In lattice dynamical problems [2,5, 131, this technique has been applied to problems where the disturbance involves one or a few degrees of freedom in Cartesian space. In the present study, the idea is to start from a fixed mesoscopic combination of Cartesian coordinates, which defines a single mesoscopic degree of freedom, but may involve many Cartesian coordinates. If this object is known to turn soft, the dominant effective disturbance at the lattice will only involve this collective coordinate and its Cartesian partner coordinates which describe the coupling to the surroundings. The latter may also be combined to a singular mesoscopic mode. In this manner, a disturbance of low rank is established, and after an appropriate transcription a Lifshitz procedure is applicable. Moreover, since mesoscopic soft modes almost behave as individual oscillators, it would seem advantageous for the calculations to handle them as foreign degrees of freedom attached to an almost undisturbed reference lattice. To consider
this question,
another
modification
of the Lifshitz
procedure
is
required, which accounts for the additional degrees of freedom. Such an extension of the formalism has been given by one of us [17] earlier and is adapted here to the mesoscopic mode case.
2. Lattice dynamical Let us consider tonian
preliminaries
an oscillatory
system
which
is characterized
by the Hamil-
M. Wagner and T. Mougios
I Mesoscopic
soft modes in disturbed lattices
H=~~P:,+gium,xmx”, where X, are mass-reduced Cartesian coordinates (X,,, = Mz2x,). eigenvectors T(K) = [T,(K)] of the “dynamical” matrix urn,, %JLz(‘d
constitute
(1)
mn
m
c n
231
=
02,
a complete
‘L(K)
(2)
7
orthonormal
Then the
set
‘?;(K>‘?&‘> = kc, 3
(3)
F r);(K)‘L(K)= %zn
(4)
;
7
which may be employed
to define normal coordinates
Q, >
(5)
Q2,= cm ‘7;(K) x,n 3
(6)
X, = c ‘I,(K)
I(
which diagonalize
the Hamiltonian (7)
We note the sum rule
(8) We introduce
the Zubarev
Green function (GF) [21]
(9) which satisfies the equation
of motion
232
M. Wagner and T. Mougios
/ Mesoscopic
soft modes
in disturbed
lattices
G,,, (El = & %z, z (E2%,,- ~nz,,> = ;
‘%,@W2L - u,z.,),
(10)
where the latter equality is due to G being a symmetric matrix. Using the Plemelj formula l/(x + is) = P(l lx) + &S(x), we may interrelate the imaginary part of the GF with the density of oscillatory frequencies, G,,(w
+
is) - G,,(w
- ic)
= +i c 77,(K) n,*(K) $ ]a(W - & ) - a(W + 0, )] K K =&
c 77,(K) ‘-/,*(K)]a(O - 0,) + a(W + a,)]. K
Applying the orthonormality oscillatory frequencies
(11)
relation (3), we deduce from this the density of
p(w) = 2io z [Gmm(m + is) - G,,(o = -4w Im 2 G,,(o m
+ ic) ,
-is)] for 6~20,
(14
and the sum rule cc
I
dwp(w)=N=-41m
i
i
0
0
dwwCG,,(w+is). m
(13)
The latter two formulae are of enormous utility for the calculation of thermodynamic properties, as will be seen later. If an oscillatory system may be conceived as a deviation from an “undisturbed” one, the dynamics of which is known, H=HO+W,
(14)
w=;cwmnxmxn )
mn
(15)
such that the “disturbance” W only embodies a small number of coordinates, we may apply the Lifshitz formalism [lo] to calculate the disturbed GF from the undisturbed one. The equation of motion (10) may then be rewritten in the form
M. Wagner and T. Mougios
(E’Z -
I Mesoscopic
U”) G(E) = & Z +wG(E)
soft modes in disturbed lattices
233
,
(16)
where Z denotes the unity matrix Z,,,, = 6,“. By using Go(E) =
& (E*Z-
U”)-’
(17)
eq. (16) may be formally solved G(E) = Go(E) + 2rr Go(E) wG(E) ,
(18)
or likewise, since G(E) is symmetric, G(E) = Go(E) + 27r G(E) wG’(E)
.
(19)
Upon inserting (19) in (18), we end up with the compact formula G(E) = Go(E) + 2~ Go(E) wG’(E)
+ (~IT)*G’(E)
wG(E)
wG'(E)
. (20)
This expression most clearly evidences the very essence of Lifshitz’ formalism, the statement of which is that the full GF G(E) can be written down, if G(E) is known in the “small space” of the disturbance w. This can be deduced by considering the last term on the right-hand side of (20), which in the product wG(E) w just effects the projection of G(E) onto the “small space”, wG(E) w = w g(E) w,
(21)
where g(E) is that part of G(E) which pertains to the space of w. It is the reduction (21) which establishes the main computational option of the Lifshitz formalism. By using this expression, the “small space” version of eq. (20) assumes a closed form g(E) = go(E) + 2ng”(E)
wg’(E) + (24* go(E) wg(E)
wg’(E),
(22)
or, finally, g(E)
= [I - 2q”(E)
WI-’ go(E)
.
(23)
Inserting (23) in (21), and this then in (20), the overall expression for G(E)
234
M. Wagner and T. Mougios
I Mesoscopic
soft modes in disturbed lattices
reads G(E) = Go(E) + 2nG”(E)
w[Z - 2rrg”(E) WI-’ Go(E) ,
(24)
which attaches the G(E) parts outside the “small space” to those inside.
3. Mesoscopic coordinates It has to be noted, however, that the full efficiency of the Lifshitz procedure becomes effective only if one realizes that the true rank of w may be much smaller than the number of Cartesian coordinates involved in the disturbance. To exploit this, we introduce a set {S(y)} of orthonormal coordinates,
w=CaY)x,. m such that the disturbance
y=l,2,.
2
(25)
(15) displays the form
We emphasize that the number of Cartesian coordinates involved, may be much larger than the number R of effective “disturbance coordinates” S( 7). Evidently, by inserting (25) in (26) and identifying it with (15) we have (27)
For the sake of argument, we add to the given set of unit vectors {mm(y)}, constituting the disturbance coordinates (25), fictitious vectors with y = R + 1 . . 9 N, such that a complete orthornormal system in the full space is esiablished, $ c+:(Y) Cm(Y’) = &
$4xY)
(284
9
(28b)
Cn(Y) = a,, .
Then all matrices may be projected
onto this vector set, e.g.
M. Wagner and T. Mougios
I Mesoscopic
6%‘)
soft modes in disturbed lattices
T&(K))
(2
n
Ron*
%w)
235
T
(29)
etc. We note that Wvr, = 0 for y > R or y ’ > R. The Lifshitz equation (18) then assumes the form G&5)
= G;,(E)
+ 2lr 5 ~:,.(E)w,..,...~,...,.(E) Y“Y“’
)
(31)
which is se_en to be a closed system in the “small space” (1 s y s R),in which we write G = g”, g”(E) = g”O(E) + 2lrg”O(E) r@(E) )
(32)
which may be solved, since R is a small number,
g”O(E).
k?(E) = ]I - 2lrgO(E)w]-’ We insert this in the projected
(33)
form of (20),
G(E) = GO(E) + 27FG0(E)[G +27&&7(E) ;;I GO(E) = GO(E) +2&O(E)
W[Z + 27F[Z- 2$O(E)
k?-‘g”O(E) 61 GO(E) (34)
or GO(E) = GO(E) + 2&O(E)
W[Z - 27rg”O(E) WI-’ GO(E),
(35)
and returning to Cartesian space again by means of (28, 29), we finally end up with G,,(E)
= 5 U,(Y) &(E)
6Xr’)
YY'
=
G:,(E) + 257 2 G:,@) m’n’
x[Z - 27rg”‘(E) E],;.v,*.(~‘~)
5 vm,(y) cyy YY’ Y”
GO,.,(E) .
(36)
236
M. Wagner and T. Mougios
I Mesoscopic
soft modes in disturbed lattices
This formula reveals that the additional fictitious vectors u(y) for y > R have only played the role of an artifice. In the final result (21) they do not longer appear. Formula (33) will be our central equation for the evaluation of thermodynamical quantities in strongly disturbed lattices. In particular, applying (12 and 13) we derive the alteration in the density of oscillatory frequencies [2,5,131 Ap(o) = -4w Im c [Gmm(w + i&) - GL,(w
+ ia)]
R =
-8,~
Im c GL,,,. (W + ic) c u,,(y) mm’ YY’ n’ Y”
X[Z - 2rrg”‘(w + is) @],,$a,*,(r”)
GYy,
Gi,,(w
+ i&)
(37)
and the sum rule
I
dw Ahp(w) = 0.
(38)
4. Mass disturbances
Up to this point, we have assumed that mass-reduced coordinates are used throughout, which of course implies that also mass defects appear in the form of disturbances in the potential energy (see expression (15)). However, if mass defects are present, it sometimes is preferable to retain the mass-reduced Cartesian coordinates of the undisturbed system (X, = (Mi)1’2~,). This amounts to the definition of a modified GF G”(E) such that
G,,(E) =
G;,(E) (3)“’ . (3) 1’2
Multiplying (M,lM;)“2
(39)
n
m
(10) from the left by (M,lMi)-1’2 we find the equation of motion
-u”,,)G”,.,(E)=~6,,+E2(1-~)Gf.(E), T tE2%m,
and from
m
the right by
(40)
where U;,(E)
=
(g)‘;’ urn, (!$,“’ . m
n
(41)
M. Wagner and T. Mougios
I Mesoscopic
Eq. (40) again displays the structural =E2am~mn, WZZ”
soft modes in disturbed lattices
237
form of (16), with
,,=I-$,
(42)
m
hence we may proceed similarly up to formulae (35) and (36). We now may interpret y as the index of those degrees of freedom, the masses of which are disturbed (y = 1, . . . , R, ay # 0) and u~( y ) = S,,. Then the analogue of (36) reads G;,(E)
= G;,(E)
+ 21rE~ 5 G;,(E) YY’
a,[Z - 2ng”(E)
w”];;.G;.,(E)
.
(43) This procedure has been followed by many workers in the field [13] and dates back to Lifshitz himself. Its main virtue lies in the fact that for each Cartesian degree of freedom which is subjected to a mass alteration, only a single coordinate makes its appearance in the “disturbed” space. However, for the calculation of thermodynamical quantities it is less optimal, since the return from G” to G (see eq. (39)) implies some inexpedience in the formulae. On the other hand, we may as well remain in the legitimate mass-reduced formulation and exemplify the utility of a mesoscopic coordinate description for a massdefect type disturbance. Let us assume that the oscillatory coordinate m = 1 displays a mass defect (M, # Mi), whereas all others have the old masses. Then the deviation from the undisturbed Hamiltonian generally reads kv=;~[(~)1’2(~)“2-1]u&~x,x,, m n
(44)
and specifically, using p,,, = Mf, JM,,, - 1, v,,, = (MII,/M,)1’2
w= &qJ;lx;
+ $vlx, 2
uymxm.
rnfl
- 1,
(45)
At a first glance, this would appear to be a complicated form, since many Cartesian degrees of freedom are involved. But as a matter of fact only two “mesoscopic”degrees of freedom are needed. We introduce the unit vectors o-(l) = (l,O, 0,. . . ) ) a(2) = u-‘(0,
lJL, u;,,
(46) ...)
UL, . . .)
)
(47)
238
M. Wagner and T. Mougios
I Mesoscopic
soft modes in disturbed lattices
with the abbreviation: u=
( c (u:,)‘)1/7 ,
(48)
iTI#l
and their corresponding
coordinates
S(l) = x1 9 S(2) = u-l
c
(49) ULX,
.
(50)
??I#1
Then (45) assumes the form W= &_QJ;, S(1)2 + V,US(l) S(2) ) and the matrix quantities
6 and g”” (see expressions
(51) (23) and (27)),
(52)
(53) where (see (30))
(54)
Inserting this in (36), we are left with a 2 x 2 matrix diagonalization problem, which is easily handled; nevertheless, we will not pursue this further.
5. Extrinsic degrees of freedom
In the preceding sections, we have assumed that the disturbance of the reference lattice does not alter the number of degrees of freedom. There are two aspects, however, which motivate the study of reference lattices with
M. Wagner and T. Mougios
I Mesoscopic
soft modes in disturbed lattices
239
additional (“extrinsic”) degrees of freedom. On the one hand, there actually may exist such foreign degrees of freedom, e.g. if molecular defects are installed in the reference lattice. On the other hand, the disturbance itself may be such that some mesoscopic motion of the lattice behaves almost as an independent mode. Then the calculation may be simplified considerably if this motion is taken as an additional degree of freedom, provided the coupling to the background lattice is adjusted accordingly. The latter concept has been first adopted by Krumhansl and Matthews [8]. Extrinsic degrees of freedom may be handled by a method which has been developed by one of us (M.W.) many years ago [17] as an extension of the We modify this procedure slightly to fit our original Lifshitz procedure. purpose. We write the Hamiltonian in the form
where
(56) (57) H’“’
;
2
mr
(X,,,K,,X,
+ X,K,,X,J
,
JL, = Km
3
(58)
and we introduce the notation that the indices {m, n} are always confined to the intrinsic degrees of freedom and {r, s} to the extrinsic ones. The extended set of GF equations then reads ;
(E*Lc - U:‘,.)G,d~) = & 4,,, + c K,,,, G,,(E) , r
c @*a,,, - ~?)G,~,(~)
= x K,,,, G,,,(E)
7 (E*L - ~%LW
= & 4,
r’
?
m
,
+ ; K,,G,,(E) ,
(E*s,,, - U::.)G,.,(E) = T K,n,G,,(~)~
(59) (60)
(61) (62)
Since we tacitly assume that the number of foreign degrees of freedom {r} is small, it is easy to achieve the inversion of the matrix (E2Z - IV’“‘), and we therefore introduce the “extrinsic” GF
M. Wagner and T. Mougios
240
GE’ =
where
&
[E2Z’“’
Zlz’ = S,,. Then
_
U”‘],’
/ Mesoscopic
soft modes
in disturbed
lattices
(63)
)
eq. (60) transmutes
into
c
G,,(E) = 2,~sm G::‘(E)K,,G,,(E) , which we insert
(64)
in (59),
T (E*L, - ~imJ%,(E)
+2~ c LG:: rs
where the intrinsic dynamical matrix WE!. In this manner we have returned effective intrinsic disturbance
k) G,,,(E) >
W
(65)
has been separated into UEi = UtA + to an equation of the type (16) with an
where the formalism is traced back to the one given in sections 2 and 3. This, however, only refers to the intrinsic part G,,(E) of the total GF. Since the density of frequencies now also incorporates the foreign degrees of freedom we also need the complementary part G,,(E) of the GF. This is easily found from eqs (61) and (62). Defining G;‘, =
G,,
and remembering
=
&(E*Z
- U’i’),;
that by definition
c
(m,nfr,s) m, n # r, s we have from
G,,(E) = 27rnr G:;(E) K,,G,,(E) and inserted
(67)
,
(62)
(68)
in (61)
or G,,(E)
= $
{E2Z”’ - [U’“’ + ~TKG”‘(E)
K]},’
.
(69b)
M. Wagner and T. Mougios
I Mesoscopic
241
soft modes in disturbed lattices
The solution of this equation requires the knowledge of G”‘(E) (see expression (67)), which is found by writing UCi) = U” + wCi)and applying the original Lifshitz formalism of section 2. From expression (69b), in analogy with expression (12)) we deduce the modified extrinsic density of states p,,(o)
= -4w Im c G,,(o + ic) .
(70)
I
6. Green function thermodynamics The internal energy of an oscillatory system with frequencies
{a,}
is given
by 1 u=$&coth(tpO,J, K
‘=k,T’
(71)
-
from which we derive the specific heat (72) As mentioned earlier, the key formulae for a Green function description are the spectral density formulae (12) and (37). Specifically, the excess thermodynamic quantities are given as
AU = U - U” = ; j- dw Ap(o)w coth( $w)
,
(73)
m
’
ACeC-C’=
dw Ap(o)
4k,T?
0
O2 , sinh2( i PO)
where Ap(o) is given by expression (37). Inserting in this expression definition (9) of the undistrubed GF, we have (E = o t ia) Ap(o)
=
-$
Im
c
r*
(y,,)
n’
and employing
the
‘~2K!~~!;!*
mm’n KX’ x
(74)
z
Y”
‘7%‘) 77:tK’)* E2
-
(@r)’
the orthonormality
’
relation (3),
(75)
M. Wagner and T. Mougios
242
A&)=-$Im
/ Mesoscopic
c l “2’ (E2 - (Q’)’
soft modes in disturbed lattices
i [‘L&(K)* a,,(Y)] yy’ Y”
x$,,[z-27Tg""(E) ti$$[~7,*~(~') ?$(K)] . Now, by noticing
that
g”&(E)= &
(see definition
c b%“‘)
m’n’
(76)
(30))
d~(K)l[d*(K)*‘=m,(Y)l
(77)
3
(L’jl)’
E* -
K
we may write
b%“‘) 77~~(K)h%‘)* ~,n,(‘Y)l [(o
K
+
1 L
g”,,
= -2W do
ie)2 - (fl,“)‘]’
(w
+
iE) ’
y”y
(78) and upon
inserting
in (76) we arrive
Ap(w) = 2 Im 5
“,,,,[I
- BITT”’
at
$&,(E),
WI,&, 2
E=w+i.s.
YY’
Y”
(79) If W does not depend
Ap(w) = 2 Im &
on w, this may be simplified
$
{ln[Z-
2ng"'(E) G]},,
In passing, we note that the latter strongly by using the trace-determinant Tr{ln by means
to
expression relation
A} = ln{det A} ,
formally
may
be simplified
of matrices
(81)
of which we have
Ap(w) = 2 Im &
ln[Z - 27~g”‘(E) W] .
(84
This transcription sometimes may be advantageous. We now insert expression (79) in expressions (73) and (74). Then we get for the excess internal energy
M. Wagner and T. Mougios
I Mesoscopic
243
soft modes in disturbed lattices
m
AU=U-U’=Im
J^
dw wcoth($w)
i
~,,,.[Z-27~g”~(E)
CS];;.
YY’ Y v
0
E=w+iE,
(83)
and for the excess specific heat AC=C-Co =yIm
1
dw sinh;; I
2k,T
x
$
pw) ;, G;vy,]Z- 27~g”‘(E) @I],;~
0
Y”
g”,“,,,(E)
E=U+ie.
,
(84)
In applications of these formulae, it would seem that the computation turns very complicated if the number of unit vectors o(y) (see (2.5)), which constitute the space of the disturbance, is large, since the diagonalization of matrix $[I - ~IT~“‘(E)v?]-’ dg”‘(E) /dw is required. This, however, need not be the case, since the defect may be governed by a symmetry group. Then a(y) would be irreducible base vectors to this group,
where r is the multiplicity index of the irreducible representation Z in the “small space” of the disturbance, r = 1, . . . , R(T), whereas j denotes the degeneracy index of Z, j = 1, . . . , fr . Then Ap(o) decomposes into irreducible parts,
b(w) =
c fr MC
~1,
(86)
I-
R(r)
Ap(Z; w) = 2 Im c
rr’ r”
E=w+is,
w”,,(Zj)[Z - 2ng”‘(Z’j; E) W(I’j)]~~,, &
&(rj;
E)
(87)
being independent of the degeneracy index j. In this way, the remaining matrix diagonalization sub-problems are of rank Z?(T). Particularly, if an irreducible representation Z only appears once (Z?(T) = l), there is no diagonalization problem whatsoever.
244
M. Wagner and T. Mougios
7. Three elucidative
1 Mesoscopic
soft modes in disturbed lattices
case studies (intrinsic softening)
In this section, we apply the formalism to a singular chain of N atoms with periodic boundary conditions, X,,,,, = X,, and nearest neighbour spring interaction,
Ho= ; j, P:,+ ;$
i
(x, -x,,,+,>’ ,
(88)
m-l
with the Bloch eigenvectors 77,(Kj
=
~-112
ei27rKmlN
, K
=
0,
+I,
k2, . ..,?$N-l,+$N,
where N has been taken as an even number. The undisturbed then read 0: = 0, sin(rKlN)
)
n,
= (4flM)“2
(89)
eigenfrequencies
)
(90)
and it is worth recording the sum rules
(91)
The Green function pertaining
to this linear lattice model,
i2nr(m-n)lN G:,(E)
=
kN
c I(
e
E2 _
(@>2
=
eJ-9
(92)
3
can be given in a closed form [19], ilm-nlq
G;,(E)=
-&
e D
sincp
for O
’
and w>L!,
,
(93)
and where w is expressed via the quantity cp, w = 0, sin( $9) ,
osn,:
O~(p~~,
wS.n,:
cp=n+iz,
For 0, - E < w < 0, + E a separate
O
consideration
is needed.
(94) But we may
M. Wagner and T. Mougios
/ Mesoscopic
soft modes in disturbed lattices
245
discard this analysis, since we may determine the p(w) behaviour in this region via sum rules. We now consider several defect cases. 7.1. Single spring constant defect ,=lf’-f
* 7
(4
The perturbation a(1) = $
(1, -l,O, 1,
S(1) = 5
(95)
may be described by a single-unit vector (see (25))
m.
with coordinate
- X2)’ *
2, 3,
. . . ) 0) ) . . . ,N ,
(96)
(see (25)) (X, - X,) .
(97)
Then
w=
$wS(1)2
The projection
)
+_2f’-f
(98)
M
(30) is now of the form
g”‘(E) = G;, - GL,
(99)
or
for w > 0, , where expression (93) has been inserted. We note that g”“(o + is) is not yet specified in an &-surrounding around w = 0, and displays a singular behaviour there. This is known since the works of Lifshitz and Kosewich [ll] and Mahanty et al. [12]. But we need not specify this behaviour, because we can determine the behaviour of Ap(w) in this
M. Wagner and T. Mougios
246
I Mesoscopic
soft modes in disturbed lattices
region by means of the sum rule (38). Inserting g”O(E) and W in (79), (83) and (84), we further note that the region o > 0, (by way of the infinitesimal imaginary part of g”‘(E) in this region) only plays a role if the denominator 1 - 2ng”‘(E)G displays a pole in this region. This would indicate then a localized mode, but it only appears if W > 0 (f’ >f). We refrain from discussing this case, since it has often been discussed in the literature (see e.g. the books of Maradudin et al. [13] and Dederichs et al. [5]). Henceforth, we only regard the “softening” case W < 0 (f’ < f). Then formula (79) assumes the form (see fig. la)
Ap(w) = -2 9... Im
=-_ 1 f’ f-f’
(l-x2)l/2,(~)~+(l’
non,f f
forx=w/,f&, = -
;qw2 - 0;)
O~L~GLZ~-E,
foran-s
)
(101a) (101b)
where the latter formula has been found via sum rule (38). Specifically in the low-frequency region we find
(102)
which yields for the excess specific heat (74), (84) the low-temperature ($@‘,)a
result
‘)
AC=
which is in the extremely AC=
f
f’
f -f’ f
low-temperature
regime ( f p0,(
f ‘lf) 9 1)
TkB k,T 3
fl,
(104a) )
whereas for somewhat higher temperatures,
provided
f’ 4 f,
M. Wagner and T. Mougios
I Mesoscopic soft modes in disturbed lattices
3-
72 52. * c?
247
.I -
f’=O.lf
_---- -___
f’=-JSf f’ = 0.9 f
a‘.I’
x=w/
I I1 .: .I .’ I :.I,I ,I i: : I, .’: .,’
n,
Fig. 1. (a) Excessive frequency density in a linear chain with nearest neighbour springs f, when a single spring is softened (f’ < f). Solid line: f’ = 0. lf, dashed line: f’ = OSf, dotted line: f’ = 0.9f. (b) Excessive specific heat, when a single spring is softened (f’
M. Wagner and T. Mougios
248
I Mesoscopic
soft modes
in disturbed
lattices
AC= +k,,
(104b)
which is the Dulong-Petit limit. The total numerical behaviour employing the exact formula (lOl), is depicted in fig. lb. 7.2. Softening of two neighbouring
w= ;
f%f
[(X_,
of AC(T),
springs
- X0)’ + (X0-
Xl)“] .
(105)
Now we have two reduced Cartesian coordinates X_, - X0 and X0 - X, involved in the disturbance, but since the defect exhibits inversion symmetry, the coordinates uniquely split into an odd and an even symmetry coordinate. Hence we may proceed along the lines indicated in (85), (86) and (87). We introduce the symmetry vectors (u = “ungerade” = odd, g = “gerade” = even), a(u) =
$
(. . . ,
m.
o,-1,2,-l,o
. . . ) -2,
-l,O,
(106)
1,2 )...)
o,-l,O,
4d=$C..,
)... ),
1,o )... ),
with symmetry coordinates S(u) = $
(-X-r
+2x,
- X,) )
(107) %)=&(x,-x_,).
Expression
(105) is then converted
w= &+A) s(u)2+
&c(g)
into
s(g)*
qu)
= 3
)
f%f
)
qg>
= f+f.
The projections of the Green functions onto the irreducible base vectors (106) read (E = o + is) g”‘(u; E) = ;[6Gi,(E)
- 8Gy,,(E)
2 =s
(
+ 2Gy,_,(E)]
-1 - 2x2 - 2i V&
>
,
x=o/n,<1,
(109)
M. Wagner and T. Mougios
I Mesoscopic
soft modes in disturbed lattices
249
iok E) = G:,(E)- G:,_,(E) (-1 + 2x2 - 2ix Vi?)
= --$
,
x=w/On,
(110)
D
where formula (89) has been used. From (108), (109) and (110) inserted in expression (87)) we find the excessive densities of state Ap(u; W) = 2 Im G(u)[l - 27rg”‘(u; w + is) C(U)]-’ -& g”‘(u; w +is)
=--
2 f-f’
f
7FoD
forx=o/aD,
[3(f)-41x*}
’
OSx
Ap( g; 0) = 2 Im $u)[l
(111)
- 2rg”‘( g; w + ic) W(U)]-’ -& i”( g; w + is)
=- 2 f-f’ ToD f forx=w/OD,
OSx
(112)
The behaviour of these two excessive frequency densities is depicted in figs. 2a and 2b. Expressions (111) and (112) are used for the computation of the excessive specific heat given in figs. 3a and 3b. In the low-frequency region and for f’ 4 f formulae (111) and (112) simplify to
(113)
forx=olR,41,
f’*f,
(114)
250
M. Wagner and T. Mougios
/ Mesoscopic
yielding the very-low temperature AC(u; T) = $
$
soft modes in disturbed lattices
excess of specific heat (i/30,
s 1)
k,(y)‘,
(115)
D
2n f T) = 3 7
Wg;
k,T k, F
(116)
,
whereas for somewhat higher temperatures AC(u; T) =
k,
AC(g; T) =
$k,
( 4 @I,(
f ‘lf) G 1 < i pa,),
,
we find (117)
.
(118)
7.3. Spring softening at distant sites (mesoscopic modes)
w=~f~f[(x2-xl)2+(x,+3-xr+2)2],
r=l,2
)....
(119)
r thus counts the number of undisturbed springs between the two disturbed ones. Again there are two Cartesian coordinates (X, - X, and X,+3 - Xr+2) involved in the disturbance, and there is inversion symmetry around site i(r + 4), whence we may proceed similarly as in section 7.2. We introduce the symmetry parity-ordered vectors a(u)=i(
l,-l,o
m:
1,
dg>
=
2,3
)...)
0,-l,
,...,
0,
l,o )...)
;w,
1,
r+Z,r+3,
0,-l,
0 )...)
O),
(120a)
. . . . N,
1,
0, . . 30) ,
(120b)
with symmetry coordinates S(u)
=
t(x,
S(g)
=
ic-Xl
Expression
-
x2
-
x,+2
+
K-+3)
(119) then transmutes
w&qu)=qg)=2f+f.
(121a) (121b)
+X,-X+2+X,+,).
w= +w[s(u)2+ s(gy]
>
)
into
(122) (123)
M. Wagner and T. Mougios
I Mesoscopic
soft modes in disturbed lattices
251
The projections of the Green functions onto the irreducible base vectors (120a) and (120b) read now
go@; El = Gi,(E)- G?,(E) + $G~,,(E) - G:+,,,(E)
+ $G;+,,,(E)
, (124)
g”“(g;E) = G:,(E)
-
G;,(E)- 4G:,,(E)+ G:+,,,(E)- $G;+,,,(E). (125)
Inserting (93) we find for arbitrary
-$
g”“(u;E) = -
r
{1+tg(tcp)sin[(r+l)cp]+2itg(+cp)sin2[~(r+l)cp]}, D (126)
8”‘(g; E) = - --$
{l-tg(fcp)sin[(r+l)cp]+2itg($cp)cos*[$(r+l)~]}. D
(127) For r = 1 this reads
g”“(u;E) = --$ (- 1 - 4x2 + 8x4 - Six3 Vi-?)
,
D
(128)
x = 6J/.nD.
For r = 2:
g”‘(u;E) = --&
2ix3(9 - 24x2 + 16x4)
-1-6x2+32x4-32x6-
g”“(
g;
E) = L
ma:,
)
vi-7
D
(
_1 +
6x2
_
32x4
+
32x6
_
%x(1 - 9x2 + 24x4 - 16x6) e
x=6_l/.nD,
whereas for
g”‘(u;E)
r =
’
(129) 4, the expressions
= -& D
(
already are of a somewhat
lengthy form:
-1 - 10x2 + 160x4 - 672x6 + 1024x8 - 512x”
- 2ix3(25 - 200x2 + 560x4 - 640x6 + 256x8) Vi=?
2
252
M. Wagner and T. Mougios
1 = 2 non,
I Mesoscopic
soft modes
in disturbed
lattices
-1 + 10x2 - 160x4 + 672x6 - 1024x8 + 512x”
2ix( 1 - 2.5x2 + 200x4 - 560x6 + 640x* - 256x”) >7
Vi? x=c.d/f2D.
(130)
Inserting expressions (123) and (128) into (130), we find the respective densities of states Ap(u; w) and Ap( g; o) and therefrom the excessive specific heat expressions AC(u; T) and AC( g; T). The numerical results are depicted in figs. 2a, 2b, 3a and 3b. We first analyse the sharp peaks in the intermediate and upper frequency domain, e.g. the one pertaining to r = 1 in fig. 2a. In this case there is one simple “hard” spring f between the two softened springs f’. Thus, in the limiting case f’ G f, the two masses between the two soft spring sites will display an internal “molecular” mode of the approximate frequency 0 = (2f/M)“2 = 0,/V?. It is this singular (localized) mode which generates the sharp peaking at x = 0.71 in fig. 2a. In the same manner the other sharp maxima find their explanation. For I = 4 the molecular unit between the soft springs has two even and two odd internal vibrations, which make their appearance in figs. 2a and 2b.
___-__
-5 L1 !i
-10 0.00
a
0.50
0.25 x=w/
0.75 n,
1.00
M,
Wagner
and
Mougios
T.
I Mesoscopic
soft
modesin distwbed
25-
, I'
i
I I
0'
I
20 15 -
r=O
_- ___---_ r=l r=2 _-----
I I
I
I I II I, I II ,I 1; I II
I :I, !I /: ! :I
I’ ____________-__;_I_________________j_I: - - - - _:_ i _‘,__ _ “’_ _ _.- “._ _ _,l’ _-,y__’ ---__ --_ _..,Ilb, \ \ : \ .I !,:; ,8
-5-10 0.00
r=4
I
;I I! I’I :I
f'= 0.1f
253
lattices
b
0.75
0.50
0.25
x=w/
1.00
R,
1
0.0
C
0.1
0.3
0.2 x=w/
0.5
0.4
i-l,
Fig. 2. (a) Excessive density of frequencies in a linear chain. Softening of two with r undisturbed springs in between. Solid line: r = 0, dashed line: r = 1, dash-dotted line: r = 4. Even mode case. (b) Excessive density of frequencies in a linear chain. Softening of two springs undisturbed springs in between. Solid line: r = 0, dashed line: r = 1, dotted line: line r = 4. Odd mode case. (c) Dependence of the softening of the odd mode excessive frequency density undisturbed springs between the two undisturbed ones (f’ = O.lf). Solid line: r = 1, dotted line: r = 2, dash-dotted line: r = 4.
springs dotted
(f' = O.lf)
line:
r = 2,
(f’ = O.lf) with r r = 2, dash-dotted: on the number r of r = 0, dashed line:
M. Wagner and T. Mougios
I Mesoscopic
soft modes in disturbed lattices
1.00
f'=O.lf 0.75 -
r=O _____---- r=l
r” ;
OSO-
r=2
3 a
___---
-----_____ 0.00 0.00
a
0.25
0.50 T/
r=4
--‘-----_-____,_ 0.75
1.00
0,
f'=O.lf r=O
---------
r= 1 r=2
__----
r=4
0.25
/
‘b
0.25
0.50 T/
0.75
0,
Fig. 3. (a) Excessive specific heat for a linear chain with softening of two springs undisturbed springs in between. Solid line: r = 0, dashed line: r = 1, dotted line: line: r = 4. Even mode case. (b) Excessive specific heat for a linear chain with softening of two springs undisturbed springs in between. Solid line: r = 0, dashed line: r = 1, dotted line: line: r = 4. Odd mode case.
(f’ = O.lf) with r r = 2, dash-dotted (f’ = O.lf) with r r = 2, dash-dotted
M. Wagner and T. Mougios
I Mesoscopic
255
soft modes in disturbed lattices
However, in our context the more important feature of these figures is the low-frequency enhancement of the mode density, which in the odd mode case (fig. 2b) is a manifestation of the center-of-mass motion of the “mesoscopic” unit between the soft spring sites. Accordingly, the position of the maximum shifts to lower frequencies with increasing number of atoms belonging to the mesoscopic unit. Astonishingly, there also is a low-frequency enhancement in the even mode density (fig. 2a), which does not seem to have a simple qualitative perspective. In the low-frequency region and for f’ 6 f, the respective excess density formulae derived from (123) to (130) simplify to 3 f_I.X2 - 2(r + 1)X4 Ap(u; w) = -$
f
(r + 1)’ D
(131)
f_l - 2(r + 1)x2 2 + 4(r + 1)4X6 ’
>
f
x=wmD,
r=0,1,2,4.
(132)
From this formulae we deduce that the odd mode softening has a pronounced dependence on the number of atoms between the disturbed springs, whereas for the even modes it is less salient. For Ap(u; w), the prefactor to the low-frequency slope increases in proportion to (r + l)‘, whereas the position of the maximum decreases in proportion to (r + 1)-“2, but the maximal height increasing again in proportion to (r + 1) (see fig. 2~). Taken together, these features establish the tendency to increase the low-temperature specific heat with ascending number (r + 1) of atoms between the disturbed springs. Quantitatively we derive this from (131) if we insert it in (74). For pL!,~l,
f’%f:
AC(u; T) = $
f
(r + l)Zk,( z)2,
for r = 0, 1,2,4 .
D
(133)
For p~,(f’lf)~l~p0,: AC(u; T) = k, . Similarly, inserting (132) in (74), we find the low-temperature the even modes.
(134) behaviour
for
M. Wagner and T. Mougios
256
/ Mesoscopic
soft modes in disturbed lattices
For pJ& % 1, f’ +f: 2n f T) = 3 7
AC(g;
For @I,(
f ‘lf) G
(135)
.
14 POD:
AC(g; T) = ;k,
8. Extrinsic
k,T k,n,
(136)
.
soft mode
To get insight into this problem we again study a one-dimensional model of prototype nature. This is depicted in fig. 4. The Hamiltonian of this model is given by H
=
H(i)
+ H(‘)
+ ~(“1
(137)
2
where (138) H’“’ = ; (P,’ + L?,“Q,‘) ,
(139) (140)
(141)
n3=2$ s
We now apply the formalism of section 5 with the specifications (see eq. (66)) w(i) f, mn = z %&L,,
+ L,)
)
Fig. 4. Prototype model of an extrinsic degree of freedom coupled to a lattice.
(142)
M. Wagner and T. Mougios
I Mesoscopic
soft modes in disturbed lattices
L Km
@e’=~ ss
=
-
(MM,)‘/2
G%l,,
+
1
The effective intrinsic disturbance
1 +
We introduce a,(u)
z
(143)
%I-,),
E=w+i~,
2~ E2 - fl,2 ’
251
(144)
(66) acquires the form
0; E2
_
a2
s
Pm,pk--p
+
%-,%,
+
4ws~rn>P
+
%I-ii
,I) * (145)
the symmetry vectors (see (25))
= -$
m,(g) = $
(Sm,, + Sm._&)
(“ungerade”),
(146)
(S,,, - S,,-P)
(“gerade”)
(147)
,
by means of which the formalism of section 6 becomes applicable. Thus, the intrinsic change of the frequency density is given by expression (87), where r= u or r = g. Introducing the projections (cr*IG’Icr) and (~T*~w@~~)(u)(see eqs. (29) (30)) (erf) w”(u, E) = WE,;) + wF _IL _- 2f (eff) kC(g) = wf,f;) - w,,_,
E2 M E2 - fl,z ’
(148)
)
(149)
_
- zL
g”“‘(u) = G;,‘,(E)
+ G’,,‘,(E)
= &
A,(E),
(150)
g”‘“‘(g) = GE;(E)
- G;:,(E)
= &
B,(E),
(151)
we get
ApCi)(u, 0) = 2 Im w(u, E)[l - 21r g”(‘)(u, w + ie) W(u, E)]-’ X &
g”‘O’(u, 0 +i~)
E2 - 0; - (f,IM)E2A,(E)
’
(152)
258
M. Wagner and T. Mougios
Ap”‘( g, W) = $ k Im = -$
I-
Im &
I Mesoscopic
soft modes in disturbed lattices
(JYM)B,(E) ln[l -
(hIM)B,(E)]
.
(153)
As explained at the end of section 5, this density has to be supplemented by the extrinsic contribution ensuing from the extrinsic GF (70), which in our example reads G,,(E) = & [ E2 -
0,Z(l +21r ;
where g”“‘(u, E) is the “statically”
g”“‘(u,E)
=
,c.CT;(U)G;‘,(E)
= [l - 24’)(u,
g”(‘)(u, E))] -’ ,
disturbed
(154)
GF defined by (67),
un(u)
E) F?,‘~‘(u)]-’ g”“‘(u, E)
A,(E) =- 1 2~ 1 - (f,IM)A,(E)
(155)
’
where Gci’(u) = f,lM has been used for the “static” intrinsic disturbance. this we find 1 I- (L/M) A,(E) GSS(E) = % E* - 0; - (f,/M)E*A,(E)
From
(156)
’
which by means of (70) yields the modified extrinsic density of states p,(w) = -F
Im
1 - (f,IM)A,(E) E* - 0; - ( fsIM)E2A,(E)
Combining this with expression Apci)(u, o) + p,(o) = -a
’
(157)
(152), we finally arrive at (E = w + ia) Im -& ln[E2 - 0: - (f,IM)E2A,(E)].
(158)
This is drawn in fig. 5, where MS = (2~ + l)M and& = f/l0 has been chosen. It is therefore suggestive to compare this figure with the corresponding one of an intrinsic softening (see fig. 2b), identifying r with /_L.Then in both cases, the peak in the excessive mode density progresses towards lower frequencies with increasing effective mass of the soft mode. However, in the extrinsic case
M. Wagner and T. Mougios
I Mesoscopic
soft modes in disturbed lattices
259
25
-5 -
0.00
0.25
0.50
1.00
0.75
x=w/R,
Fig. 5. Modified frequency density of a singular oscillator (0,’ = 2f,lM) coupled to a chain of oscillators (M, f) at sites t/~ with springs f, (see fig. 4). f, = O.lf, MS = (2~ + l)M. Solid line: p = 1, dashed line: p = 2, dotted line: /.L= 3, dash-dotted line: y = 4.
(fig. 5) the peak position for I_L= 1 is already considerably lower than for r = 1 in the intrinsic case (fig. 2b), and the further progression is towards higher peaks for higher p-values (fig. 5), in contrast to almost unaltered heights for higher r-values (fig. 2b). The high-frequency parts of both figures naturally differ in a qualitative manner. The slight wavy nature of the curves in the upper frequency regime of fig. 5 is due to the wavy nature of the sin(2pq) and cos*( ~CJJ)terms in the projected GF A,(E) (see below expression (159)). In fig. 6 a fixed distance p = 2 is chosen for a variety of foreign mass MS values and the total excessive density of states is drawn. Expressions (153) and (158) may be analytically exploited, if we employ formula (89) for the evaluation of the quantities A,(E) and B,(E),
A,(E)
= $
&
hG-w)
-
co~*b-41y
(159)
bw)I,
(160)
D
B,(E)=-+
&
[~WG-w~
+
sin*(
D
which have the low-frequency A,(E)
= -$
(4P[l D
behaviour
(X = w/L?,)
- 5(4P2 - QX2] - i]2 - (‘<
- ‘)‘*I)
,
(161)
260
M. Wagner and T. Mougios
1 Mesoscopic soft modes in distwbed lattices
40
, f, = 0.1 f, p = 2
30 .
20
-
-
I’\ ,’ ‘1
M,=M
--------
_ MscZM
______
0.1
0.0
0.2
0.3 x=w/
0.4
0.5
n,
Fig. 6. Total excessive density of states due to a singular oscillator (of = 2f,lM,) coupled to a chain of oscillators (M, f) at sites p = 22 with springs f, = O.lf. Solid line: MS = M, dashed line: MS = 2M, dotted line: MS = 4M, dash-dotted line: M, = SM.
B,(E)
= -$
{441.[1- :(4p*
- 1)x*] - i8p2x}
.
(162)
D
From this, inserted in eqs. (153) and (158) respectively,
we get (x = o/O,)
(163)
(164)
9. Summary
and remarks
This work has been stimulated by the physical question in what way the softening of atomistic elastic properties (spring constants) of condensed quasiharmonic systems influence the low-frequency mode density, and consequently the low-temperature specific heat. This question seems to be an urgent one in
M. Wagner and T. Mougios
I Mesoscopic
soft modes in disturbed lattices
261
view of the experimentally suggested (neutron scattering) low-frequency enhancement of harmonic modes in glasses on the one hand, and of the suggested strongly anharmonic models (tunneling, etc.) for the unusual low-temperature properties of glasses on the other. The work has also been initiated by the background concept of mesoscopic soft modes, i.e. the concept of a soft cooperative motion of a mesoscopic number of atoms. We have developed an extended Green-function-Lifshitz procedure, which, after introduction of appropriate mesoscopic disturbance vectors, allows for a utilization of the computational options of the original Lifshitz technique, which had been restricted to a small number of Cartesian coordinates. The formalism covers both the cases of an intrinsic disturbance (no additional degrees of freedom) as well as that of an extrinsic disturbance (additional degrees of freedom). The formalism is applied to archetypical model systems. Specifically, we have made several case studies for linear lattices disturbed at mesoscopically distant sites (spring softening). It is found that there always is a low-frequency peaked enhancement of mode density. For a fixed elastic alteration, the peak position descends to lower frequencies with ascending number of atoms between the disturbed sites, showing that the most effective soft modes are those of mesoscopic nature. We further have calculated the behaviour of a foreign mesoscopic mass MS (aMlattice ) which is harmonically coupled to a linear lattice with various distances between the coupling sites. It is found that the arrangement somewhat simulates the low-frequency mode density of a corresponding intrinsic disturbance. Here it is also found that the position of the low-frequency density peak descends with increasng mass MS, if M, is chosen to increase in proportion to the coupling distance CL.However, in detail there are pronounced qualitative differences between the considered intrinsic and extrinsic cases, such that a simulation of the intrinsic behaviour by an extrinsic mode must be done with care. We have not presented here the calculation of mesoscopic disturbances in three-dimensional lattices. This calculation will be given elsewhere. In this context, mesoscopic modes of librational character would be of particular interest, since modes of this nature have been found experimentally (Buchenau). The presented formalism is also well fit to handle these modes, although the computational effort is considerably higher and less lucid.
Acknowledgement
We have profited from discussions with G. Zavt, simplifications of some of our formulae.
which have lead to
262
M. Wagner and T. Mougios
I Mesoscopic
soft modes in disturbed lattices
References [l] [2] [3] [4] [5] [6] [7] [8] [9] [lo] [ll] [12] [13]
[14] [15] [16] [17] [18] [19] [20] [21]
P.W. Anderson, B.I. Halperin and C.M. Varma, Phil. Mag 25 (1972) 1. H. Bottger, Principlees of the Theory of Lattice Dynamics (Physik-Verlag, Weinheim, 1983). U. Buchenau, Physica Scripta T19 (1987) 350. U. Buchenau, H.M. Zhou, N. Nucker, K.S. Gilroy and W.A. Phillips, Phys. Rev. Lett. 60 (1988) 1318. P.H. Dederichs and R. Zeller, in: Point Defects in Metals II, Springer Tracts in Modem Physics, vol. 87 (Springer, Berlin, 1980). S. Hunklinger and W. Arnold, in: Physical Acoustics, vol. 12, W.P. Mason and R.N. Thurston, eds. (Academic Press, New York, 1976) p. 155. W. Junker and M. Wagner, J. Phys: Condens. Matter 1 (1989) 5065. J.A. Krumhansl and A.D. Matthews, Phys. Rev. 166 (1968) 856. I.M. Lifshitz, JETP 17 (1947) 1017. I.M. Lifshitz, Nuovo Cimento Suppl. 10 3 (1956) 716. I.M. Lifshitz and A.M. Kosevich, Rep. Prog. Phys. 29 (1966) 217. J. Mahanty and A.A. Maradudin, Prog. Theor. Phys. 20 (1958) 369. A.A. Maradudin, E.W. Montroll, C.H. Weiss and I.P. Ipatova, Theory of Lattice Dynamics in the Harmonic Approximation, Solid State Physics, Suppl. 3, 2nd ed. (Academic Press, New York, 1971). W.A. Phillips, J. Low Temp. Phys. 7 (1972) 351. Properties (Springer, Berlin, W.A. Phillips, ed., Amorphous Solids, Low-Temperature Heidelberg, New York, 1981). R.O. Pohl, see his article in ref. [15]. M. Wagner, Phys. Rev. 131 (1963) 2520. M. Wagner, Phys. Rev. A 133 (1964) 750. R.F. Wallis and A.A. Maradudin, Prog. Theor. Phys. 24 (1960) 1055. R.C. Zeller and R.O. Pohl, Phys. Rev. B 4 (1971) 2029. D.N. Zubarev, Sov. Phys. Usp. 3 (1960) 320.