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NUCLEAR-NUCLEAR CORRELATIONS IN LIQUID
RUBIDIUM
P. Egelstaff, J. Suck, W. Gläser, R. Mcpherson, A. Teitsma
To cite this version:
JOURNAL DE PHYSIQUE CoZZoque C8, suppZ6ment au n08, Tome 41, aogt 1980, page
Cg-222
NUCLEAR-NUCLEAR CORRELATIONS I N L I Q U I D RUBIDIUM
*
P.A. Egelstaff, J.B. Suck
,
W.laser**,
R. McPherson and A . TeitsmaUniversity of GueZph, GueZph, Ontario N I G 2W1, Canada. " ~ n s t i t u t Laue-Langevin, 38042 GrenobZe, F ~ a n c e . **Physik Department, T . U . Miinchen, R. F. A.
Abstract.- The neutron diffr ction patterns of liquid rubidium have been measured for momentum
0-B
transfers between 0.4-2.4 A for twelve states between 328-473 K and 10-1200 atmospheres using the IN4 spectrometer at the Institut Laue-Langevin. These states were chosen to give isothermal, isochoric and isobaric data.
In this paper we present the experimental results and the isothermal pressure derivatives of the liquid structure factor. The latter may be predicted from a model in which the electron fluid is assumed to control the position 02 the ions, so that when it is compressed all inter-ion distances vary as (density)-lI3. The experimental derivatives are close to the predictions of the model, and some interpretations of this observation are-given.
I n t r o d u c t i o n
The i o n - i o n s t r u c t u r e o f a l i q u i d metal has been t h e s u b j e c t o f many papers i n recent years [I]. However the comparisons o f pseudopotential models w i t h the data have o f t e n seemed a r t i f i c i a l , because t h e f i t t o a s i n g l e s t a t e i s an inadequate t e s t o f t h e model. To overcome t h i s problem we have meas- ured t h e s t r u c t u r e f a c t o r f o r l i q u i d rubidium f o r 1 2 s t a t e s chosen so t h a t isothermal, i s o c h o r i c and i s o - b a r i c paths may be followed. I n t h i s paper we des- c r i b e the experimental method, t h e a n a l y s i s o f t h e d a t a and the r e s u l t s . F i n a l l y we compare t h e i s o - thermal data w i t h t h e " u n i f o r m f l u i d model" t o t e s t the d e n s i t y dependence of t h e pseudopotential [21. Other comparisons w i l l be made i n la t e r p u b l i c a t i o n s . T h e o r e t i c a l Notes
measure t h e i n t e n s i t y
I ( % )
f o r t h e s c a t t e r - i n g o f monochromatic neutrons i n l i q u i d r u b i d i u m through an angle 0, and t h i s q u a n t i t y w i l l be r e - l a t e d ( l a t e r ) t o the d i f f e r e n t i a l cross s e c t i o n ( d o / d ~ ) f o r s i n g l e s c a t t e r i n g . I n terms of t h e s t r u c t u r e f a c t o r S(q) f o r rubidium n u c l e i t h e d i f -and h i s t h e neutron wavelength):
where acoh i s t h e coherent cross s e c t i o n and us i s t h e t o t a l s c a t t e r i n g cross s e c t i o n of rubidium, and P(0) i s t h e "Placzek c o r r e c t i o n " [3] f o r t h e i n e l - a s t i c i t y o f t h e s c a t t e r i n g . The s t r u c t u r e f a c t o r i s r e l a t e d t o the r a d i a l d i s t r i b u t i o n f u n c t i o n by
We a r e i n t e r e s t e d i n t h e v a r i a t i o n of S(q) o r g ( r ) from s t a t e t o s t a t e and t h e e x t e n t t o which t h i s may be p r e d i c t e d t h e o r e t i c a l l y [4]. I f t h e spacing o f the i o n s i s c o n t r o l l e d by k f and t h i s i s p r o p o r t i o n a l t o p 1 l 3 then we may expect t h a t t h e d i s t a n c e s c a l e i n g ( r ) i s p r o p o r t i o n a l t o
$I3.
I n t h i s case t h e d e n s i t y dependence o f S(q) may be expressed by t h e formula:We s h a l l compare t h e p r e d i c t i o n s o f t h i s simple formula t o our experimental r e s u l t s , and make com- p a r i s o n s w i t h more s o p h i s t i c a t e d models i n l a t e r p u b l i c a t i o n s . For example i n a system i n which t h e f e r e n t i a l cross s e c t i o n i s (where q = 41rSin(8/2fi
p o t e n t i a l energy i s a f u n c t i o n o f t h e n u c l e a r p o s i - t i o n s , the d e n s i t y d e r i v a t i v e o f g ( r ) may be r e l a t e d
t o t h e t r i p l e t c o r r e l a t i o n f u n c t i o n (e.g. r e f . 6). Because we may t h i n k o f an a l k a l i metal as a two f l u i d system o f e l e c t r o n s and ions t h i s r e l a t i o n m a y be v a l i d on1 y approximate1 y. Consequent1 y we s h a l l make comparisons i n v o l v i n g t h r e e body c o r r e l a t i o n s e l sewhere.
The Sample
The sample c o n s i s t e d o f
1
i q u i d r u b i d i um (99.99% pure) contained i n a c y l i n d r i c a l t i t a n i u m a l l o y vessel. The a x i s o f t h e c y l i n d e r was h o r i - z o n t a l and heaters placed a t each end o f the sample r e g i o n c o n t r o l l e d t h e temperature t o t h e d e s i r e d value. Because o f t h e c o n f i g u r a t i o n o f t h e i n s t - rument, t h e a x i s o f the c o n t a i n e r was turned t o t h e v e r t i c a l a t 8 cm from t h e sample region. L i q u i d r u b i d i u m w i t h a s u i t a b l e temperature gradient, con- t r o l l e d by heaters, extended up t h e v e r t i c a l s e c t i o n and was i n c o n t a c t w i t h s i l i c o n e o i l a t a temperat- u r e o f 80 C. The s i l i c o n e o i 1 was connected through pressure tubing, a t room temperature, t o a manually operated pump. T h i s system was t e s t e d a t 2000 a t - mospheres using a sample a t 250 C, before conducting these experiments a t maximum c o n d i t i o n s of 1250 a t s and 200 C. The r a d i a l expansion o f the sample r e g i o n was n e g l i g i b l e because the r e d u c t i o n of d i a - meter due t o a x i a l e x t e n s i o n almost balanced t h e increase i n r a d i a l dimensions. States were chosen t o meet t h e c o n d i t i o n s l i s t e d i n t h e i n t r o d u c t i o n and a r e given i n t a b l e 1.Experimental Method and Data Reduction
The experiment was performed u s i n g t h e thermal neutron t i m e - o f - f l i g h t spectrometer IN4 a t t h e High Flux Reactor o f t h e I n s t i t u t e Laue-Langevin i n Grenoble. The i n c i d e n t energy Eo from t h e two r o - t a t i n g p y r o l y t i c g r a p h i t e monochromators was 12.6 meV. S i x t y t h r e e spectra were recorded on separate
Table 1. Summary o f Experiments.
S t a t e P P
x
l o z 2
atoms/cc T OC ~ / m ~12 1.047 200 125.1
*
Temperature r a i s e d 100 t o a v o i d t h e phase boundary d e t e c t o r s f o r s c a t t e r i n g angles 0 between 9 and 84 degrees, covering momentum t r a n s f e r s q between 0.4 and 3.3 Our o b j e c t was t o measure b o t h t h e dynamic and s t a t i c s t r u c t u r e f a c t o r s (S(q,o) and S ( q ) ) over a l i m i t e d range o f q and w f o r many states. I n t h i s paper we present o n l y t h e d a t a on t h e s t a t i c s t r u c t u r e f a c t o r .f4easurements were made f o r a standard s p i r a l of vanadium f o i l (0.025 mm t h i c k ) having approxirjlately t h e same dimensions and s c a t t e r i n g t h e same f r a c t i o n o f the beam as t h e sample. F u r t h e r measurements were made w i t h t h e empty c o n t a i n e r a t each o f t h e o p e r a t i n g temperatures shown i n t a b l e 1. Then t h e l i q u i d rubidium was siphoned i n t o t h e c o n t a i n e r and covered w i t h s i l i c o n e o i l , so t h a t data c o u l d be taken f o r each o f t h e s t a t e s shown i n t a b l e 1. For each s t a t e t h e c o u n t i n g t i m e was about 24 hours.
c8-224 JOUKNAL DE PHYSIQUE
Fig. 1. Observed s t r u c t u r e f a c t o r s f o r s t a t e s 5 t o 8 c o n t a i n e r , and were normal i s e d t o the vanadium read- ings. Normal "PI aczek c o r r e c t i o n s "
[ 3
j were appl i e d t o take account of t h e i n e l a s t i c i t y o f t h e s c a t t e r - ing. M u l t i p l e s c a t t e r i n g was c a l c u l a t e d f o r a bare sample o f l i q u i d rubidium, and was subtracted. We assume t h a t the a d d i t i o n a l m u l t i p l e s c a t t e r i n g due t o t h e c o n t a i n e r i s (approximately) independent o f angle, and can be found by n o r l n a l i s i n g t o t h e known v a l u e o f S(q=O) = pkT%
through s u b t r a c t i o n o f a constant q u a n t i t y .As an example, t h e f o u r s t r u c t u r e f a c t o r s taken a t 398 K a r e shown i n Fig. 1. I n t h e r e g i o n of t h e
second maximum of the s t r u c t u r e f a c t o r t h e data a r e i n a c c u r a t e because t h e Debye-Scherrer peaks,
stemming from t h e t h i c k w a l l e d pressure c e l l , g i v e r i s e t o l a r g e s t a t i s t i c a l e r r o r s and s p o i l the model [63 from which t h e a b s o r p t i o n f a c t o r s a r e c a l c u l - ated. This r e g i o n i s o m i t t e d from t h e present p l o t s
0
which extend o n l y t o 2.4 A-I, b u t some data on S(q,w) f o r q > 2.4 w i l l be presented elsewhere as t h i s e f f e c t o n l y occurs f o r w % 0. As t h e d e n s i t y i s changed the peak p o s i t i o n s h i f t s s l i g h t l y , b u t otherwise very l i t t l e change i s seen.
Comparison lJi t h t h e Uniform F l u i d Model
The d e n s i t y d e r i v a t i v e o f S(q) was found by the f o l l o w i n g equation, f o r two s t a t e s l a b e l l e d 1 and 2 a t t h e same temperature:
A S / A P / ~ ~ ~ ~ = tS(q9p1)
-
S ( q , ~ ~ ) l/
(pl-
P ~ ) (4) and t h e d e r i v a t i v e w i t h r e s p e c t t o q, f o r adjacent p o i n t s q, and qb, was found f o r the same mean s t a t eEquation ( 4 ) i s the l e f t - h a n d s i d e o f equation ( 3 ) w h i l e the r i g h t hand s i d e i s t h e q u a n t i t y :
I n f i g u r e 2 we show a s e t o f e i g h t comparisons o f equation ( 4 )
-
c i r c l e s-
w i t h equation (6)-
smooth l i n e . These correspond t o t h e means o f a l l adjacent isothermal s t a t e s 1 is t e d i n t a b l e I , e x c l u d i n g s t a t e 4. The o v e r a l l agreement i s s a t i s f a c t o r y ,i n d i c a t i n g t h a t t o a good approximation t h e i n t e r - n u c l e a r distances vary as p-
.
I f s i m i l a r exper-behavior of aS/ap demonstrates t h a t t h e " f r e e volume" i n l i q u i d rubidium i s c o n t r o l l e d by t h e e l e c t r o n f l u i d . I n c o n s i d e r i n g these data i t should be remembered t h a t the shape o f t h e p r i n c i p a l peak o f S(q) i s determined by the r e l a t i v e p o s i t i o n i n g o f successive o s c i l l a t i o n s i n g ( r ) , and t h a t the range o f t h e i n t e r a c t i o n s being s t u d i e d i s set by
0
t h e maximum value o f q t o be 3 A.
The small d i f f e r e n c e s between the model and t h e data r e q u i r e a more d e t a i l e d a n a l y s i s than can be given here.
References
1. " L i q u i d Metals 1976", Ed. R. Evans and 0. A. Greenwood, The I n s t i t u t e o f Physics Conf. Series No. 30
2. Egelstaff, P. A. and Wang, S. S. (1972) Can. J. Phys.
50,
24613. Enderby, J. E. (1968) I n "Physics of Simple L i q u i d s " , Ed. H. Temperly, H. Rowlinson and G Rushbrooke, North Holland (Amsterdam)
4. Gubbins, K. E., Gray, C. G. and E g e l s t a f f , P. A. (1978) Molec. Phys.
35,
315; E g e l s t a f f , P. A. (1973) p. 13. I n "The P r o p e r t i e s o f L i q u i d Metals" Ed. S. Takeuchi, T a y l o r and Francis L t d (London)5. Howells, U. 5 . (1978) I n s t i t u t Laue-Langevin Report No. 77H046T
6. Kendig, A. P. and Pings, C. J. (1968) J. Sppl. Phys.
36,
16927. Winfield, D. and E g e l s t a f f , P. A. (1973) Can. J. Phys.
