9 Computer Simulation of Collective Modes in Solids
Downloaded via UNIV OF CALIFORNIA SANTA BARBARA on July 11, 2018 at 12:18:46 (UTC). See https://pubs.acs.org/sharingguidelines for options on how to legitimately share published articles.
M. L. KLEIN Chemistry Division, National Research Council of Canada, Ottawa, Canada K1A 0R6
The molecular dynamics technique i s now widely used t o c a l c u l a t e the timedependent p r o p e r t i e s of many body systems. This paper reviews the a p p l i c a t i o n o f this computer s i m u l a t i o n technique t o the c a l c u l a t i o n o f the normal modes o f v i b r a t i o n o f s o l i d s and t h e i r l i f e t i m e s . Examples will be r e p r e s e n t a t i v e o f most c l a s s e s o f solid found i n nature, namely the i n e r t gases, i o n i c c r y s t a l s , metals, a l l o y s , and molecular s o l i d s . Normal modes o f v i b r a t i o n o f s o l i d s are r o u t i n e l y s t u d i e d experimentally u s i n g i n e l a s t i c s c a t t e r i n g o f neutrons or photons ( 1 ) . A l s o the theory o f the dynamics o f c r y s t a l s is now w e l l developed and even anharmonic e f f e c t s are u s u a l l y i n c o r p o r a t e d i n t o t h e o r e t i c a l models ( 1 ) . T h i s being the case one may wonder why computer s i m u l a t i o n is a t all necessary? W e l l , neutron s c a t t e r i n g i s r a t h e r c o s t l y so that although t h i s technique is in p r i n c i p a l very powerful it is l i m i t e d i n t h i s sense and light s c a t t e r i n g i s mostly r e s t r i c t e d t o small wave v e c t o r s . The s i m u l a t i o n method, on the other hand, i s r e a l l y a compliment t o the r e a l experiments s i n c e i t allows one t o compare approximate t h e o r e t i c a l approaches t o the c r y s t a l dynamics with exact r e s u l t s for a given i n t e r m o l e c u l a r f o r c e model, subject o n l y to the l i m i t a t i o n that c l a s s i c a l s t a t i s t i c a l mechanics a p p l i e s . F o r t u n a t e l y , f o r most s o l i d s , t h i s l a t t e r r e s t r i c t i o n i s not a severe l i m i t a t i o n . Moreover, i n the computer s i m u l a t i o n s t h e r e i s an unambiguous d i s t i n c t i o n between coherent and incoherent s c a t t e r i n g , and between one-phonon and multiphonon e f f e c t s , a s i t u a t i o n which does not always p e r t a i n to r e a l experiments. We have used the molecular dynamics (MD) method o r i g i n a l l y pioneered by Alder and Wainwright (2) f o r hard spheres and by Rahman ( 3 ) f o r continuous potentials. The c o l l e c t i v e modes (phonons) we wish t o study are r e l a t e d t o the spectrum o f d e n s i t y f l u c t u a t i o n s i n the s o l i d . Such motions are obtained by i n t e g r a t i n g the c l a s s i c a l equations of motion (using a time step o f 10"~ to 1 0 ~ sees) f o r a model system composed o f approximately 1 0 p a r t i c l e s , i n t e r a c t i n g v i a some assumed f o r c e law, with the p e r i o d i c boundary c o n d i t i o n being used t o simulate an i n f i n i t e system. These boundary 11+
1 5
3
0-8412-0463-2/78/47-086-094$05.00/0 © 1978 American Chemical Society Lykos; Computer Modeling of Matter ACS Symposium Series; American Chemical Society: Washington, DC, 1978.
9.
KLEIN
Collective Modes in Solids
95
c o n d i t i o n s "quantize" the allowed wave v e c t o r s t h a t can "be studied. The d e t a i l s o f the phase space t r a j e c t o r i e s are stored so that at a l a t e r time s t a t i s t i c a l averages can be performed. This i n essence i s the method. I t has been widely discussed i n the l i t e r a t u r e and w i l l not be d e s c r i b e d i n d e t a i l here. The
Dynamical S t r u c t u r e Factor S(Q,a))
I t i s w e l l known that apart from d u l l f a c t o r s the coherent s c a t t e r i n g o f neutrons from l i q u i d s and s o l i d s i s determined by the Van Hove f u n c t i o n (dynamical s t r u c t u r e f a c t o r )
S(Q,(o) =
dte
i a ) t
F(Q,t)
where F(Q,t) i s the intermediate s c a t t e r i n g f u n c t i o n . This i n t u r n i s r e l a t e d t o the time c o r r e l a t i o n f u n c t i o n o f the d e n s i t y operator
p (t) = I y i=l
e ^
Q
{
t
)
where N i s the number o f p a r t i c l e s i n the system whose p o s i t i o n at time t are given by r ( t ) . The momentum and energy t r a n s f e r r e d to the system i n the s c a t t e r i n g process are Q and w r e s p e c t i v e l y . In d e t a i l F(Q,t) = ( l / N ) < p ( t ) p Q
(0)> ,
where the angular bracket denotes a s t a t i s t i c a l average which w i l l be evaluated by the c l a s s i c a l molecular dynamics method (MD). The f i r s t c a l c u l a t i o n s o f t h i s type were c a r r i e d out f o r l i q u i d r a r e gases near the t r i p l e p o i n t (h) . Since then a v a r i e t y o f f l u i d s have been s t u d i e d i n c l u d i n g l i q u i d metals (5.), l i q u i d n i t r o g e n (6), the c l a s s i c a l one-component plasma (7.)» molten s a l t s (8) and water (9.). Because l i q u i d s are i s o t r o p i c S(g,a)) depends only on |g| whereas f o r s o l i d s t h i s i s not t r u e . A f u r t h e r d i s t i n c t i o n a r i s e s because f o r many s o l i d s under a wide v a r i e t y o f s t a t e c o n d i t i o n s the c o n s t i t u e n t p a r t i c l e s execute small amplitude v i b r a t i o n s about w e l l defined e q u i l i b r i u m p o s i t i o n s (noteworthy exceptions are p l a s t i c c r y s t a l s ) . If this s i t u a t i o n p e r t a i n s i t i s p o s s i b l e t o express r ^ ( t ) = R$+u£(t) the instantaneous p o s i t i o n o f p a r t i c l e £ i n terms o f i t s mean p o s i tion and i t s time dependent displacement uj^(t) . I f U£ i s i n some sense small with respect t o R^ one can develop F(§,t) as a power s e r i e s i n §*u. This i s the s o - c a l l e d phonon expansion (10)
Lykos; Computer Modeling of Matter ACS Symposium Series; American Chemical Society: Washington, DC, 1978.
96
COMPUTER
F(Q,t) = Fo + F + F . x
n t
MODELING
OF MATTER
+F + 2
where the successive terms d e s c r i b e the e l a s t i c (zero-phonon) s c a t t e r i n g one-phonon i n e l a s t i c s c a t t e r i n g , the i n t e r f e r e n c e between one and two phonon s c a t t e r i n g , and the two-phonon s c a t t e r i n g , e t c . For a c l a s s i c a l s o l i d these i n d i v i d u a l c o r r e l a t i o n f u n c t i o n s can be evaluated u s i n g the MD method as can the f u l l F(Q,t). From a t e c h n i c a l p o i n t o f view one can proceed as f o l l o w s . One can evaluate F(Q,t) f o r a given Q and then take the F o u r i e r transform. A l t e r n a t i v e l y , one can proceed v i a the F o u r i e r Laplace transform o f the d e n s i t y operator PQ(CO) u s i n g the formula T
T
N S(Q,o)) = l i m j e T
i u ) t
?
3
~* ° o
= lim|p
p (t)dt j ~
(w)\ / 2
e~
i ( A ) t
' p _ ( t ' )dt ?
J^
o X
Both methods have been used i n the l i t e r a t u r e (11). Other methods i n v o l v i n g p e r t u r b a t i o n s o f t r a j e c t o r i e s have a l s o been used t o study c o l l e c t i v e modes i n s o l i d s (12). The one-phonon approximation t o the dynamical s t r u c t u r e f a c t o r S i can be w r i t t e n (13) N S!(
