C o l l e c t i v e E x c i t a t i o n s i n L i q u i d C o l l e c t i v e E x c i t a t i o n s i n L i q u i d S i Fig. 1. Selected S( Q, ω ) spectra normalized t o S ( Q ) . E x p e r i m e n t a l d a t a a r e g i v e n b y circles with error bars, and thick solid lines represent fits of the DHO model convoluted w i t h t h e r e s o l u t i o n f u n c t i o n ( d a s h e d l i n e ) . Do t- da sh ed li ne s sh ow th e be st co nv ol ut ed f i t s f o r t h e q u a s i e l a s t i c l i n e s u s i n g a Lorentzian in comparison to the quasi-Voigt fits (thin solid lines, shifted from the data). T h e s t r u c t u r a l l y s i m p l e s t ‘ s e m i c o n d u c t o r ’ , S i , i s m a n u f a c t u r e d b y g r o w i n g t h e c r y s t a l d i r e c t l y f r o m the ‘ metallic’ melt. This interplay between a metallic disordered phase and the semiconducting c r y s t a l l i n e s t a t e h a s s t i m u l a t e d m u c h t h e o r e t i c a l and experimental interest in the static and dynamic p r o p e r t i e s o f t h i s s y s t e m . F o r e x a m p l e , t h e influence of covalent bonds on the dynamics of the m e t a l l i c m e l t w a s i n v e s t i g a t e d i n a n e a r l y f i r s t - p r i n c i p l e s m o l e c u l a r - d y n a m i c s s i m u l a t i o n b y t h e o r i g i n a t o r s [ 1 ] . H o w e v e r , t h e e x p e r i m e n t a l inv esti gat ion of the micr osc opi c dyn ami cs has yet b e e n h i n d e r e d b y t h e f a c t t h a t t h e c o l l e c t i v e longitudinal modes in liquid Si are out of reach of thermal neutrons due to the high sound velocity (~ 4 , 0 0 0 m s -1 ) a n d t h e k i n e m a t i c r e s t r i c t i o n s o f t h i s t e c h n i q u e . H i g h - r e s o l u t i o n i n e l a s t i c X - r a y sc at te ri ng ( IX S ) is an ot he r te ch ni qu e th at pe rm it s t h e s t u d y o f Q d e p e n d e n c e o f e x c i t a t i o n s i n t h e meV range, but in contrast to neutron scattering, it h a s n o k i n e m a t i c r e s t r i c t i o n s a n d t h e s c a t t e r e d r a d i a t i o n i s e n t i r e l y c o h e r e n t w i t h i n t h e e n e r g y r a n g e o f i n t e r e s t . C o m b i n e d w i t h a suit able high - te mp er at ur e sa mp le en vi ro nm en t, we we re ab le to m e a s u r e f o r t h e f i r s t t i m e t h e d y n a m i c s c a t t e r i n g law S ( Q, ) of liquid Si [2]. The experiments were carried out at beamline B L 3 5 X U u s i n g a h o r i z o n t a l I X S s p e c t r o m e t e r [ 3 ] (energy resolution: ~1.8 meV FWHM at 21.8 keV ). T h e h o t s a m p l e ( T = 1 7 3 3 K ) w a s l o c a t e d i n a sapphire container, which was a slight modification of the so-called Tamura-type cell [4]. It was placed in a vessel equipped with continuous Be windows [5] capab le of cover ing scatt ering angle s betwe en 0 and 25 . Figure 1 shows selected spectra normalized to t h e r e s p e c t i v e i n t e n s i t y . A l s o g i v e n i s a t y p i c a l e x a m p l e o f t h e r e s o l u t i o n f u n c t i o n ( d a s h e d l i n e ) . Th e da ta cl ea rl y pr ov e th e ex is te nc e of lo ng it ud in al co ll ec ti ve sh or t wa ve le ng th mo de s, wh ic h ap pe ar ω 0 . 3 5 0 . 3 0 0 . 2 5 0 . 2 0 0 . 1 5 0 . 1 0 0 . 0 5 0 . 0 0 1 8 . 9 1 1 . 2 7 . 4 4 . 3 2 5 . 0 2 . 0 2 9 . 6 - 4 0 - 2 0 0 2 0 4 0 l i q u i d S i 1 7 3 3 K Q ( n m - 1 ) ω ( m e V ) S ( Q , ω ) / S ( Q ) 28 as pea ks or sho uld ers in the low er Q ran ge. For the res olu tio n cor rec tio n, a mod el S ( Q, ) fun cti on c o n v o l u t e d w i t h t h e r e s o l u t i o n f u n c t i o n w a s f i t t e d to the data. For this model, we approximated the central line by a Lorentzian at lower Q values or by a p s e u d o - V o i g t f u n c t i o n a t h i g h e r Q v a l u e s ( s e e t e x t b e l o w ) , a n d t h e i n e l a s t i c c o n t r i b u t i o n b y a damped harmonic oscillator ( DHO ). Using the fitted re su lt s (t hi ck so li d li ne s) , th e ex ci ta ti on en er gy ω c and the line width Γ Q were determined as shown in Fig. 2 . The dashed line represents the dispersion of hy dr od yn am ic so un d, an d it s sl op e is gi ve n by t h e b u l k a d i a b a t i c s o u n d v e l o c i t y v s = 3 9 5 2 m s -1 . T h e f r e q u e n c i e s o f t h e s h o r t w a v e l e n g t h m o d e s i n c r e a s e n o t i c e a b l y f a s t e r ( ~ 1 7 % ) w i t h Q t h a n p r e d i c t e d b y c l a s s i c a l h y d r o d y n a m i c s . T h i s s o - c a l l e d ‘ p o s i t i v e ’ d i s p e r s i o n w a s a l r e a d y f o u n d earlier in liquid alkali metals and also in liquid Hg. Fig. 2. Dispersion relation (circles) and line width (triangles) of the collective modes in liquid Si. The collective modes are highly damped at higher Q values compared to those in liquid alkali metals. The usual choice, in which a quasielastic line is modelled by a Lorentzian, is not suitable for liquid Si beyond Q = 20 nm -1 . Instead we used a quasi- V o i g t f u n c t i o n , i . e . , a l i n e a r c o m b i n a t i o n o f a G a u s s i a n a n d a L o r e n t z i a n . D o t - d a s h e d l i n e s i n F i g . 1 s h o w c o n v o l u t e d f i t s u s i n g a L o r e n t z i a n i n co mp ar is on to th e qu as i- Vo ig t fi ts (t hi n so li d li ne , t h e s a m e a s t h e t h i c k o n e s ) . T h e q u a s i - V o i g t function fits the data well, whereas the deviation of t h e L o r e n t z i a n f i t s i s c o n s i d e r a b l e . C i r c l e s a n d triangles in Fig. 3 represent the line width Γ 0 s and t h e G a u s s i a n f r a c t i o n c o f t h e q u a s i e l a s t i c l i n e s . T h e G a u s s i a n c o n t r i b u t i o n b e c o m e s n o t i c e a b l y important at about 20 nm -1 and reaches about 50% a t Q ~ 3 0 n m - 1 . A r o u n d t h e Q v a l u e w h e r e t h e m a x i m u m i n S ( Q ) i s l o c a t e d , a m i n i m u m i n Γ 0 ω 29 5 0 4 0 3 0 2 0 1 0 0 2 0 1 5 1 0 5 0 2 0 1 0 0 Q ( n m - 1 ) ω c ( m e V ) Γ 0 ( m e V ) l i q u i d S i 1 7 3 3 K v s = 3 9 5 2 m / s F i g . 3 . Q d e p e n d e n c e o f t h e q u a s i e l a s t i c l i n e w i d t h Γ 0 ( c i r c l e s ) . W i t h increasing Q, a Gaussian component (open circles) is needed in addition to the Lorentzian (full circles) to model the central line. Triangles give the Gaussian fraction, c. The arrow indicates the Q position of the first maximum in S(Q). I n s e t : S ( Q ) d e t e r m i n e d f r o m t h e z e r o f r e q u e n c y m o m e n t o f t h e p r e s e n t experiment (squares) together with the result from elastic X-ray scattering [6]. (usu ally Γ 0L ) is expe cted , whic h is the well -kno wn de Gennes narrowing. However it is worth noting that the minimum of Γ 0L is found at about 2 2 . 5 n m - 1 , while the maximum in S ( Q ) is located at 27.3 nm -1 (arrow in Fig. 3 , see also S ( Q ) given in the inset). Be si de s th e da mp ed ph on on mo de s an d th e [ 2 ] S . H o s o k a w a , W . - C . P i l g r i m , Y . K a w a k i t a , K . O h s h i m a , S . T a k e d a , D . I s h i k a w a , S . T s u t s u i , Y . Tanaka and A.Q.R. Baron, submitted in Phys. Rev. Lett. [3] A.Q.R. Baron et al. , J. Phys. Chem. Solids 61 (2000) 461. [ 4 ] K . T a m u r a e t a l . , R e v . S c i . I n s t r u m . 7 0 ( 1 9 9 9 ) 1 4 4 . [ 5 ] S . H o s o k a w a a n d W . - C . P i l g r i m , R e v . S c i . Instrum. 72 (2001) 1721. [6] Y. Waseda and K. Suzuki, Z. Phys. B 20 (1975) 339. Shinya Hosokawa a , Wolf-Christian Pilgrim a and Yukinobu Kawakita b (a) Philipps Universit ät Marburg, Germany (b) Kyushu University E-mail: hosokawa @ mailer.uni-marburg.de anoma lies in the quasi elast ic line, howev er, direc t e v i d e n c e f o r t h e e x i s t e n c e o f c o v a l e n t b o n d s i n liquid Si ( e.g. localized modes) was not observed. A d e t a i l e d a n a l y s i s u s i n g e . g . a m o d e - c o u p l i n g theory would be useful for a further understanding of the present results, and is now in progress. References [1] I. Stich, R. Car and M. Parrinello, Phys. Rev. B 44 (1991) 4262. 30 1 . 5 1 . 0 0 . 5 0 . 0 4 0 3 0 2 0 1 0 0 Q ( n m - 1 ) S ( Q ) 1 6 1 4 1 2 1 0 8 6 4 2 0 3 5 3 0 2 5 2 0 1 5 1 0 5 0 1 . 0 0 . 5 0 . 0 c Q ( n m - 1 ) l i q u i d S i 1 7 3 3 K Γ 0 ( m e V )
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