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On the lifetime of phonons in one-dimensional crystals
PHYSICS
Volume 2, number 6
LETTERS
ON THE LIFETIME IN ONE-DIMENSIONAL
OF
15 October !962
PHONONS CRYSTALS
A. A. MARADUDIN Westinghouse Research Laboratories, Pittsburgh 35, Pennsylvania Received 17 September 1962
In this note we report on an exact calculation of the lifetime of a phonon in a linear chain, due to cubic anharmonic terms in the crystal’s potential energy. The result is found to be independent of the wavelength of the phonon, and decreases as T ml with increasing temperature in the high-temperature limit. The problem of evaluating the lifetime of a phonon which is coupled to other phonons in a crystal through the anharmonic terms in the crystal’s potential energy is of interest for several reasons. One of these is that an ex erimental determination of phonon lifetimes is now possible by means of neutron spectrometry 1~2‘5. If the energy distribution of neutrons scattered coherently by one-phonon processes is measured for a given scattering direction, it is found to consist of three peaks (in the case of a Bravais crystal), centered at the energies of the phonons responsible for the scattering processes. The widths of these peaks at half maximum can be identified with the reciprocals of the lifetimes of the corresponding phonons. Theoretical expressions for the phonon widths have recently been obtained by several authors S-5). The evaluation of these expressions even for simple models of crystals presents formidable computational problems 5,6). It is our purpose in this note to present an exact calculation of the phonon lifetime for one special, non-trivial, model. The model we study is a monatomic linear chain with nearest neighbour interactions between atoms. In particular, we consider the high-temperature limit of the expression for the lifetime. Our model, while non-trivial, is also non-physical. However, in the absence of any exact result for realistic crystal models, we feel that the existence of even one exact result, even if it is for a onedimensional crystal, is worth presenting, since it can serve as a check on the approximation methods which must be used in dealing with any more realistic models. The expression for the inverse lifetime of a phonon in a Bravais crystal in the high-temperature limit, obtained from eq. (5. llb) of ref. 5), and correct to the lowest non-vanishing order in the anharmonic force constants is
x
(6(w(kj) + w(kljl)
+ u&$2))
+ GW)
+ 6(w(ki) + w(k&)
- 4kl j,) + 4&4)
-
w(k2j2))
+ 6(w(kA - wVq.il) - w(k2_$)3 . (1)
In this expression w(kj) is the frequency of the normal mode described by the wave vector k and polarisation index j; A(k) equals unity if k vanishes, modulo a translation vector of the reciprocal lattice, and is zero otherwise; N is the number of unit cells in the crystal; and a(-kj; kl jl; kZj2) is the Fourier transform of the cubic anharmonic force constants 7). The allowed values of the wave vector k are uniformly and densely distributed throughout a unit cell qf the reciprocal lattice. In the case of a linear chain the polarisation indices iji] are suppressed, and eq. (1) reduces to 1
nk@
?k
N
1 klk2
A(-k+kl+k2)
I@(-k;kl;k2)i2 ug
$2 1 L”h2
1 d(‘Gk+wkl+“/
‘2
) + 6(c~k+yq~~)
+ where the Cki f are integers which take on the values -iN+l, 298
-:N+2,.
6(cL’/z-ql+q~2)+ Cwy-u1~~)1 . . ,$N-1, :N,
and 8)
, (2)
Volum% 2, number 6
PHYSICS LE TTERS
15 October 1962 1
o~h
= eL I s i n - ~ l
,
(e~(_k;kl;k2)l 2
o~L
=
6452
(3)
,
w21 w22
is the nearest-neighbour harmonic force constant and 5 is the nearest-neighbour cubic anharmonic force constant. If we make use of the representations
½N
1
Z
e 2nisk/N
(5)
A(k) = ~ s=-½N+I 5(o~) = ~-~
e ic°t dt
(s)
,
we can rewrite eq. (2) as
kBT62
TcosJ-2 k ; °° .eos s
t ~ Z
oo
2.sk cos u~kt)2
k
~
(7) "
We now pass to the limit as N -- ~o and replace summation over k by integration according to --~--"cp, In this way we obtain ~ ' 1 ( ¢ ) = TkB ~ -T~ o52~ 2 ( ¢ )
~~o
cos2s~o
S='°o
k
"~ ,
(8)
d~0.
f o dtcos~o(¢)t ( 2 f ~-n 2 ck°c°s2s~c°s(o~Ltsin ~o)) 2 -oo
"
(9)
0
If we make use of the results that 9) 1.
(o;Lt sin
d~ cos 2s¢ cos
~o) = J2s(~L t)
(i0)
o and I0)
J2s(OOLt) =-~
J4s(2O~Lt cos
(11)
x) dx ,
0
oo
we find that
kBT52 !oo ~-1(¢) = 2. ~ t°21~) s=-oo ~ cos 2s~ oo dt cos
~.
oo(¢)t o
J4sl2O~LtCOS x)
(12)
(ix.
We can c a r r y out the sum over s if we use the following expansions 11) COS (z sin ~o) = ~-
n=.oo
J2n(z) COS 2n~,
COS (z COS ~0) =
(-1) n
J2n(Z)
cos 2n~0 ,
(13)
n=-oo
with the result kBT52 ~--l(g))=-~--~
oj2(~o) f .oo
dt COS 00(~o)t ~ " dx {cos
(2oJLt cos
x sin ½~o)+ cos
(2ooLt cos
x cos ½~o)] • (14)
O
The integral over t is given by eq. (6), so that
7"1(~°) =~kBT
-~52W2(~°)f½"o dx (5(COS½~0+cosx)X +)½~0 5(cos s i½~0n cos
+ 5(sin ½~.0+cosx) e o +s x½~ 5(sin ) . ) c o½~so-
- .
(15)
We can assume, with no loss of generality, that ¢ is in the interval (0,½~). Then only the second and fourth terms in the integrand on the right side of eq. (15) contribute to the integral, and we obtain finally 299
Volume 2, number 6
PHYSICS
LETTERS
kBT r-l(¢P)-
2
15Octobbr 1962
62 ~3 COL "
(16)
T h e r e s u l t e x p r e s s e d b y eq. (16) s h o w s u s t h a t a t high t e m p e r a t u r e s t h e phonon l i f e t i m e i s t h e s a m e f o r a l l p h o n o n s and v a r i e s i n v e r s e l y w i t h t h e t e m p e r a t u r e . T h a t we o b t a i n a f i n i t e , n o n - v a n i s h i n g r e s u l t f o r t h e phonon l i f e t i m e in a l i n e a r c h a i n m a y be s u r p r i s i n g a t f i r s t g l a n c e , s i n c e i t m i g h t b e a r g u e d t h a t t h e n u m b e r of p o i n t s in k 1 a n d k 2 s p a c e f o r w h i c h t h e w a v e v e c t o r a n d f r e q u e n c y c o n s e r v a t i o n c o n d i t i o n s in eq. (2) a r e s i m u l t a n e o u s l y s a t i s f i e d i s s o s m a l l t h a t r k "1 v a n i s h e s in t h e l i m i t a s N - 0% T h i s a r g u m e n t , h o w e v e r , c a n n o t be t a k e n v e r y s e r i o u s l y . T a k e n l i t e r a l l y , i t w o u l d i m p l y t h a t T ' l ( k j ) s h o u l d v a n i s h in t h e t h r e e - d i m e n s i o n a l c a s e a s w e l l , s i n c e t h e s u m s ( i n t e g r a l s in t h e l i m i t a s N - oo) in eq. (1) a r e c a r r i e d out o v e r a t w o - d i m e n s i o n a l s u r f a c e in w a v e v e c t o r s p a c e a n d h e n c e g i v e a r e s u l t w h i c h i s p r o p o r t i o n a l to N2/3. S i n c e t h e r e i s a f a c t o r of N - 1 m u l t i p l y i n g t h e s u m s o v e r k 1 and k 2 , r ' l ( k j ) a c c o r d i n g to t h i s a r g u m e n t would v a n i s h a s N-l/3 in the l i m i t a s N - oo. T h a t one o b t a i n s a n o n - v a n i s h i n g r e s u l t f o r r "1 in both t h e t h r e e - d i m e n s i o n a l and o n e - d i m e n s i o n a l c a s e s s e e m s to b e d u e to t h e f a c t t h a t in a c t u a l c a l c u l a t i o n s , t h e 6 - f u n c t i o n w h i c h e x p r e s s e s t h e f r e q u e n c y c o n s e r v a t i o n c o n d i t i o n s m u s t n o t b e i n t e r p r e t e d a s r e s t r i c t i n g t h e i n t e g r a t i o n s o v e r k 1 a n d k 2 (or k 1 a n d k 2 in o n e d i m e n s i o n ) to b e c a r r i e d out o v e r a r i g o r o u s l y t w o - d i m e n s i o n a l s u r f a c e in w a v e v e c t o r s p a c e . I n s t e a d , t h e i n t e g r a t i o n s a r e c a r r i e d out t h r o u g h o u t a s h e l l of f i n i t e v o l u m e w h i c h e n c l o s e s t h e s u r f a c e d e f i n e d b y t h e s i m u l t a n e o u s s a t i s f a c t i o n of t h e w a v e v e c t o r and f r e q u e n c y c o n s e r v a t i o n c o n d i t i o n s . T h i s s t a t e m e n t i s i m p l i e d b y t h e f a c t t h a t t h e 5 - f u n c t i o n s in e q s . (1) and (2) a r i s e e i t h e r in t h e f o r m 5(x) = l i m
1
¢
(17)
e-O+ ~ x 2 + ¢2
(time-independent formalism),
or 6(x)
sin xt
=
(18)
lira - t.~oo fiX
( t i m e - d e p e n d e n t f o r m a l i s m ) . If t h e r e p r e s e n t a t i o n s (12) o r (18) a r e u s e d in t h e s u m s o v e r w a v e v e c t o r s b e f o r e t h e i n d i c a t e d l i m i t s a r e t a k e n , and t h e l i m i t s a r e t a k e n l a s t p h y s i c a l l y r e a s o n a b l e r e s u l t s a r e o b t a i n e d . O u r u s e of t h e F o u r i e r i n t e g r a l r e p r e s e n t a t i o n f o r t h e 5 - f u n c t i o n , eq. (6), h a s t h e e f f e c t of s m e a r i n g out t h e p o r t i o n of k - s p a c e w h i c h c o n t r i b u t e s to t h e s u m (2) into a f i n i t e i n t e r v a l r a t h e r than a d i s c r e t e s e t of p o i n t s , in t h e l i m i t a s N -- oo. W e c o n c l u d e b y p r e s e n t i n g an a l t e r n a t i v e , but in s o m e w a y s l e s s u s e f u l , e x p r e s s i o n f o r ~-'1(~0). C o m b i n i n g e q s . (9), (10), a n d 12) cO
j" dtcos,o
(coL ) :~d~L 1 Q2ls[_½ (1 -½ ~LL 2) ,
0 < co< coL ,
(19)
o
=0
,
otherwise ,
where Qn(x) is a Legendre function of the second kind, we obtain oO
kBT 62 CO2(cP)
T'I(~) - 4~ 73
~
COL s=-oo
Q2
i (1 - ½ co2(~)) cos 2s~. [s l-~ coL
(20)
It i s a p l e a s u r e to a c k n o w l e d g e t h e h o s p i t a l i t y of t h e B r o o k h a v e n N a t i o n a l L a b o r a t o r y d u r i n g t h e s u m m e r of 1962 when t h e w o r k r e p o r t e d in t h i s n o t e w a s c a r r i e d out.
References 1) B.N. Brockhouse et al., Inelastic scattering of neutrons in solids and liquids {International Atomic E n e r gy Agency, Vienna, 1961) p. 113. 2) K . E . Larsson, U. Dahlborg and S. Holmryd, Arkiv Fysik 17 (1960 } 369. 3) V.N.Kashcheev and M.A.Krivoglaz, Soviet Physics Solid State 3 (1961) 1107. 4) J . J . J . K o k k e d e e , Physica 28 (1962) 374. 5) A . A . Maradudin and A. E. Fein, Westinghouse R e s e a r c h Laboratories Scientific paper 62-129-103-P8, 1962. 6) A.A. Maradudin, A. E. Fein and G. H. Vineyard, to be published. 300
7) M. Born and K. Huang, Dynamical theory of crystal lattices (Oxford University P r e s s , Oxford, 1954) p. 217. 8) A.A. Maradudin, P . A . Flinn and R. A. ColdwellHorsfall, Ann. Phys. 15 (1961) 337. 9) W. Grobner and N. Hofreiter, Integraltafel, Vo]. 2 (Springer Verlag, Vienna and Innsbruck, 1949). 10) E.g.: E . T . W h i t t a k e r and G.N.Watson, Modern analysis (Cambridge University P r e s s , Cambridge, 1952) p. 380, example 16. 11) Ibid. p. 379, example 1. 12) F. Oberhettinger, TabeHen zur F o u r i e r T r a n s f o r mation (Springer Verlag, Berlin, 1957).
Volume 2, number 6
LETTERS
ON THE LIFETIME IN ONE-DIMENSIONAL
OF
15 October !962
PHONONS CRYSTALS
A. A. MARADUDIN Westinghouse Research Laboratories, Pittsburgh 35, Pennsylvania Received 17 September 1962
In this note we report on an exact calculation of the lifetime of a phonon in a linear chain, due to cubic anharmonic terms in the crystal’s potential energy. The result is found to be independent of the wavelength of the phonon, and decreases as T ml with increasing temperature in the high-temperature limit. The problem of evaluating the lifetime of a phonon which is coupled to other phonons in a crystal through the anharmonic terms in the crystal’s potential energy is of interest for several reasons. One of these is that an ex erimental determination of phonon lifetimes is now possible by means of neutron spectrometry 1~2‘5. If the energy distribution of neutrons scattered coherently by one-phonon processes is measured for a given scattering direction, it is found to consist of three peaks (in the case of a Bravais crystal), centered at the energies of the phonons responsible for the scattering processes. The widths of these peaks at half maximum can be identified with the reciprocals of the lifetimes of the corresponding phonons. Theoretical expressions for the phonon widths have recently been obtained by several authors S-5). The evaluation of these expressions even for simple models of crystals presents formidable computational problems 5,6). It is our purpose in this note to present an exact calculation of the phonon lifetime for one special, non-trivial, model. The model we study is a monatomic linear chain with nearest neighbour interactions between atoms. In particular, we consider the high-temperature limit of the expression for the lifetime. Our model, while non-trivial, is also non-physical. However, in the absence of any exact result for realistic crystal models, we feel that the existence of even one exact result, even if it is for a onedimensional crystal, is worth presenting, since it can serve as a check on the approximation methods which must be used in dealing with any more realistic models. The expression for the inverse lifetime of a phonon in a Bravais crystal in the high-temperature limit, obtained from eq. (5. llb) of ref. 5), and correct to the lowest non-vanishing order in the anharmonic force constants is
x
(6(w(kj) + w(kljl)
+ u&$2))
+ GW)
+ 6(w(ki) + w(k&)
- 4kl j,) + 4&4)
-
w(k2j2))
+ 6(w(kA - wVq.il) - w(k2_$)3 . (1)
In this expression w(kj) is the frequency of the normal mode described by the wave vector k and polarisation index j; A(k) equals unity if k vanishes, modulo a translation vector of the reciprocal lattice, and is zero otherwise; N is the number of unit cells in the crystal; and a(-kj; kl jl; kZj2) is the Fourier transform of the cubic anharmonic force constants 7). The allowed values of the wave vector k are uniformly and densely distributed throughout a unit cell qf the reciprocal lattice. In the case of a linear chain the polarisation indices iji] are suppressed, and eq. (1) reduces to 1
nk@
?k
N
1 klk2
A(-k+kl+k2)
I@(-k;kl;k2)i2 ug
$2 1 L”h2
1 d(‘Gk+wkl+“/
‘2
) + 6(c~k+yq~~)
+ where the Cki f are integers which take on the values -iN+l, 298
-:N+2,.
6(cL’/z-ql+q~2)+ Cwy-u1~~)1 . . ,$N-1, :N,
and 8)
, (2)
Volum% 2, number 6
PHYSICS LE TTERS
15 October 1962 1
o~h
= eL I s i n - ~ l
,
(e~(_k;kl;k2)l 2
o~L
=
6452
(3)
,
w21 w22
is the nearest-neighbour harmonic force constant and 5 is the nearest-neighbour cubic anharmonic force constant. If we make use of the representations
½N
1
Z
e 2nisk/N
(5)
A(k) = ~ s=-½N+I 5(o~) = ~-~
e ic°t dt
(s)
,
we can rewrite eq. (2) as
kBT62
TcosJ-2 k ; °° .eos s
t ~ Z
oo
2.sk cos u~kt)2
k
~
(7) "
We now pass to the limit as N -- ~o and replace summation over k by integration according to --~--"cp, In this way we obtain ~ ' 1 ( ¢ ) = TkB ~ -T~ o52~ 2 ( ¢ )
~~o
cos2s~o
S='°o
k
"~ ,
(8)
d~0.
f o dtcos~o(¢)t ( 2 f ~-n 2 ck°c°s2s~c°s(o~Ltsin ~o)) 2 -oo
"
(9)
0
If we make use of the results that 9) 1.
(o;Lt sin
d~ cos 2s¢ cos
~o) = J2s(~L t)
(i0)
o and I0)
J2s(OOLt) =-~
J4s(2O~Lt cos
(11)
x) dx ,
0
oo
we find that
kBT52 !oo ~-1(¢) = 2. ~ t°21~) s=-oo ~ cos 2s~ oo dt cos
~.
oo(¢)t o
J4sl2O~LtCOS x)
(12)
(ix.
We can c a r r y out the sum over s if we use the following expansions 11) COS (z sin ~o) = ~-
n=.oo
J2n(z) COS 2n~,
COS (z COS ~0) =
(-1) n
J2n(Z)
cos 2n~0 ,
(13)
n=-oo
with the result kBT52 ~--l(g))=-~--~
oj2(~o) f .oo
dt COS 00(~o)t ~ " dx {cos
(2oJLt cos
x sin ½~o)+ cos
(2ooLt cos
x cos ½~o)] • (14)
O
The integral over t is given by eq. (6), so that
7"1(~°) =~kBT
-~52W2(~°)f½"o dx (5(COS½~0+cosx)X +)½~0 5(cos s i½~0n cos
+ 5(sin ½~.0+cosx) e o +s x½~ 5(sin ) . ) c o½~so-
- .
(15)
We can assume, with no loss of generality, that ¢ is in the interval (0,½~). Then only the second and fourth terms in the integrand on the right side of eq. (15) contribute to the integral, and we obtain finally 299
Volume 2, number 6
PHYSICS
LETTERS
kBT r-l(¢P)-
2
15Octobbr 1962
62 ~3 COL "
(16)
T h e r e s u l t e x p r e s s e d b y eq. (16) s h o w s u s t h a t a t high t e m p e r a t u r e s t h e phonon l i f e t i m e i s t h e s a m e f o r a l l p h o n o n s and v a r i e s i n v e r s e l y w i t h t h e t e m p e r a t u r e . T h a t we o b t a i n a f i n i t e , n o n - v a n i s h i n g r e s u l t f o r t h e phonon l i f e t i m e in a l i n e a r c h a i n m a y be s u r p r i s i n g a t f i r s t g l a n c e , s i n c e i t m i g h t b e a r g u e d t h a t t h e n u m b e r of p o i n t s in k 1 a n d k 2 s p a c e f o r w h i c h t h e w a v e v e c t o r a n d f r e q u e n c y c o n s e r v a t i o n c o n d i t i o n s in eq. (2) a r e s i m u l t a n e o u s l y s a t i s f i e d i s s o s m a l l t h a t r k "1 v a n i s h e s in t h e l i m i t a s N - 0% T h i s a r g u m e n t , h o w e v e r , c a n n o t be t a k e n v e r y s e r i o u s l y . T a k e n l i t e r a l l y , i t w o u l d i m p l y t h a t T ' l ( k j ) s h o u l d v a n i s h in t h e t h r e e - d i m e n s i o n a l c a s e a s w e l l , s i n c e t h e s u m s ( i n t e g r a l s in t h e l i m i t a s N - oo) in eq. (1) a r e c a r r i e d out o v e r a t w o - d i m e n s i o n a l s u r f a c e in w a v e v e c t o r s p a c e a n d h e n c e g i v e a r e s u l t w h i c h i s p r o p o r t i o n a l to N2/3. S i n c e t h e r e i s a f a c t o r of N - 1 m u l t i p l y i n g t h e s u m s o v e r k 1 and k 2 , r ' l ( k j ) a c c o r d i n g to t h i s a r g u m e n t would v a n i s h a s N-l/3 in the l i m i t a s N - oo. T h a t one o b t a i n s a n o n - v a n i s h i n g r e s u l t f o r r "1 in both t h e t h r e e - d i m e n s i o n a l and o n e - d i m e n s i o n a l c a s e s s e e m s to b e d u e to t h e f a c t t h a t in a c t u a l c a l c u l a t i o n s , t h e 6 - f u n c t i o n w h i c h e x p r e s s e s t h e f r e q u e n c y c o n s e r v a t i o n c o n d i t i o n s m u s t n o t b e i n t e r p r e t e d a s r e s t r i c t i n g t h e i n t e g r a t i o n s o v e r k 1 a n d k 2 (or k 1 a n d k 2 in o n e d i m e n s i o n ) to b e c a r r i e d out o v e r a r i g o r o u s l y t w o - d i m e n s i o n a l s u r f a c e in w a v e v e c t o r s p a c e . I n s t e a d , t h e i n t e g r a t i o n s a r e c a r r i e d out t h r o u g h o u t a s h e l l of f i n i t e v o l u m e w h i c h e n c l o s e s t h e s u r f a c e d e f i n e d b y t h e s i m u l t a n e o u s s a t i s f a c t i o n of t h e w a v e v e c t o r and f r e q u e n c y c o n s e r v a t i o n c o n d i t i o n s . T h i s s t a t e m e n t i s i m p l i e d b y t h e f a c t t h a t t h e 5 - f u n c t i o n s in e q s . (1) and (2) a r i s e e i t h e r in t h e f o r m 5(x) = l i m
1
¢
(17)
e-O+ ~ x 2 + ¢2
(time-independent formalism),
or 6(x)
sin xt
=
(18)
lira - t.~oo fiX
( t i m e - d e p e n d e n t f o r m a l i s m ) . If t h e r e p r e s e n t a t i o n s (12) o r (18) a r e u s e d in t h e s u m s o v e r w a v e v e c t o r s b e f o r e t h e i n d i c a t e d l i m i t s a r e t a k e n , and t h e l i m i t s a r e t a k e n l a s t p h y s i c a l l y r e a s o n a b l e r e s u l t s a r e o b t a i n e d . O u r u s e of t h e F o u r i e r i n t e g r a l r e p r e s e n t a t i o n f o r t h e 5 - f u n c t i o n , eq. (6), h a s t h e e f f e c t of s m e a r i n g out t h e p o r t i o n of k - s p a c e w h i c h c o n t r i b u t e s to t h e s u m (2) into a f i n i t e i n t e r v a l r a t h e r than a d i s c r e t e s e t of p o i n t s , in t h e l i m i t a s N -- oo. W e c o n c l u d e b y p r e s e n t i n g an a l t e r n a t i v e , but in s o m e w a y s l e s s u s e f u l , e x p r e s s i o n f o r ~-'1(~0). C o m b i n i n g e q s . (9), (10), a n d 12) cO
j" dtcos,o
(coL ) :~d~L 1 Q2ls[_½ (1 -½ ~LL 2) ,
0 < co< coL ,
(19)
o
=0
,
otherwise ,
where Qn(x) is a Legendre function of the second kind, we obtain oO
kBT 62 CO2(cP)
T'I(~) - 4~ 73
~
COL s=-oo
Q2
i (1 - ½ co2(~)) cos 2s~. [s l-~ coL
(20)
It i s a p l e a s u r e to a c k n o w l e d g e t h e h o s p i t a l i t y of t h e B r o o k h a v e n N a t i o n a l L a b o r a t o r y d u r i n g t h e s u m m e r of 1962 when t h e w o r k r e p o r t e d in t h i s n o t e w a s c a r r i e d out.
References 1) B.N. Brockhouse et al., Inelastic scattering of neutrons in solids and liquids {International Atomic E n e r gy Agency, Vienna, 1961) p. 113. 2) K . E . Larsson, U. Dahlborg and S. Holmryd, Arkiv Fysik 17 (1960 } 369. 3) V.N.Kashcheev and M.A.Krivoglaz, Soviet Physics Solid State 3 (1961) 1107. 4) J . J . J . K o k k e d e e , Physica 28 (1962) 374. 5) A . A . Maradudin and A. E. Fein, Westinghouse R e s e a r c h Laboratories Scientific paper 62-129-103-P8, 1962. 6) A.A. Maradudin, A. E. Fein and G. H. Vineyard, to be published. 300
7) M. Born and K. Huang, Dynamical theory of crystal lattices (Oxford University P r e s s , Oxford, 1954) p. 217. 8) A.A. Maradudin, P . A . Flinn and R. A. ColdwellHorsfall, Ann. Phys. 15 (1961) 337. 9) W. Grobner and N. Hofreiter, Integraltafel, Vo]. 2 (Springer Verlag, Vienna and Innsbruck, 1949). 10) E.g.: E . T . W h i t t a k e r and G.N.Watson, Modern analysis (Cambridge University P r e s s , Cambridge, 1952) p. 380, example 16. 11) Ibid. p. 379, example 1. 12) F. Oberhettinger, TabeHen zur F o u r i e r T r a n s f o r mation (Springer Verlag, Berlin, 1957).