NEUTRON SCATTERING ON LIQUID ARGON AT SEVERAL DENSITIES AND TRIPLET CORRELATION FUNCTIONS

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NEUTRON SCATTERING ON LIQUID ARGON AT SEVERAL DENSITIES AND TRIPLET

CORRELATION FUNCTIONS

P. Verkerk

To cite this version:

P. Verkerk. NEUTRON SCATTERING ON LIQUID ARGON AT SEVERAL DENSITIES AND

TRIPLET CORRELATION FUNCTIONS. Journal de Physique Colloques, 1985, 46 (C9), pp.C9-17-

C9-22. �10.1051/jphyscol:1985902�. �jpa-00225254�

JOURNAL DE PHYSIQUE

Colloque C9, supplément au n012, Tome 46, décembre 1985 page ~ 9 - 1 7

NEUTRON SCATTERING ON LIQUID ARGON AT SEVERAL DENSITIES AND TRIPLET CORRELATION FUNCTIONS

P . Verkerk

I n t e r u n i v e r s i t a i r Reactor Instituut,2629 JB Delft, The NetherZands

Abstract

-

Recently two s e t s of t h e m l neutron scattering tiirie-of-flight measurements on liquid argon a t 120 K were published. The c d i n e d results provide the s t a t i c and the dynamic structure factor a t £ive densities. The f i r s t and second isothermal density derivatives of the s t a t i c structure factor have evaluateà. The derivatives have been capared with the uni- form f l u i d m d e l and with a s i q l e m d e l for g based on the h-bond e p m i o n by Abe. In both cases the aqreemnt is unsatis2actory. The f i r s t density derivative of the dynamic structure factor has been evaluated and agrees qualitatively with c a l d a t i o n s by G r o o n i l , Gubbins and Dufty.

I f non-additive three particle interaction p o t m t i a l s cannotbe neglecteà, as i n high-density noble gases, t r i p l e t distribution functions are important e.g. for the calculation of thermdynamic quantities /1/. Moreover, knmledge of the t r i p l e t dis- tribution function enables one t o calculate the pair distribution function through an equation of the BBGY hierarchy in the tirne-independent case /2/, and through an equation derived f r m the BEGKY hierarchy i n the tirne-dependent case /3/. Approxima- tions f o r the t r i p l e t distribution functions can be tested e x p r h m ~ t a l l y , because the isothermal pressure derivative of the pair distribution function is related t o an integral over space of the t r i p l e t distribution function /4,5/. The pair distri- bution function can be measured by neutron scattering for instance.

Recently two neutron scattering experiments on liquid argon a t 120

K

were performed t o determine the dynamic structure factor S(k,w) a t several pressures. One experi- mtwas performedwith the rotating-crystal tim-of-flight s p e c t r m t e r RKS 1 a t the 2 MW reactor of the Interuniversitair Reactor Instituut, Delft, a t three pres- sures /6,7/ (see Table 1). The second experiment w a s performed with the rotating- crystal time-of-flight ç p e c t r c i ~ t e r IN-4 a t the High Flux Reactor of the I n s t i t u t Laue-Langevin a t four pressures /8/ (Table 1). F r m Table 1 it can be seen that the consistency of the two expriments can be checked a t 20 and a t 270 bar, and that S(k,w) data are available a t 120 K and £ive different pressures. The s t a t i c s t r u e ture factor S(k) can be obtained by i n m a t i o n of S(k,w) over w.

F i r s t the basic theory of t r i p l e t distribution functions i s reviwed. Next cqer- imental results for the f i r s t and second density (pressure) derivative of S(k) are presented and c a p r e d with predictions f r m two simple models. Finally the f i r s t density derivative of S(k,w) is presented. A s f a r as we knavI, approximations t h a t r e l a t e the tiirie-dependent t r i p l e t correlation function t o the pressure derivative

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1985902

C9-18 J O U R N A L D E PHYSIQUE

of S (k,w) are not y e t available.

Table 1. S u n a r y of meaçurementç a t 120 K used i n this p a p r .

(bar) (mu-3)

IN-4 (ILL) 20.1 17.6 0.202

As Schofield /4/ has shown, the t r i p l e t density correlation function 93

(f ,s)

for a system in which the potential energy is a function of p a r t i c l e posi- tions only, i s related t o the isother- mal pressure derivative of the pair correlation function g (r) :

Here p is the p a r t i c l e nunker density,

kg

Boltzmann's constant, T the temperature, P the pressure and S(o) the value a t k=a of the s t a t i c structure factor. Egelstaff /5/ has given a gerieralisation of eq.1 for tine-dependent density correlation func- tions. For instance the pressure derivative of the tirne-dependent Van Have total correlation function G ( r , t ) can be written as:

The density

derivative can simply be calculated using %Ta/aP=S(o)a/ap. The s t a t i c and time-dep=ndent correlation functions are defined as:

m e

spis

the position of p a r t i c l e i a t time t.

A usefui approach is t o approxhmte g3 i n the so-called h-bond m i o n /9/:

+ +

g3(?,Q)=g(r)g(s)g( l ~ b l ) e x p ~ - r ( r ~ s , I r - s l :P) 1 * (3a) + + + +

T (r, s,

1 :-zl;

h(w)h(r-w)h(w-s)

+

higher

order

t- in hl (3b) with h (r)=g(r)-1. Expnding the expnential i n eq. 3a, Egelstaff /1/ writes g3 a s an

expansion i n h. Using only the s h p l e s t tenns the r e s u l t i s

Eq.4 is equivalent to the approximation ac(k)/ap=o, with c ( k ) the Fourier trançfom of the d i r e c t correlation function, related t o S (k) by c (k) = (s(k)-l)/p S (k)

(H. Fredrikze, private camminication). Haynset e t a l . /IO/ obtain excellent r e s u l t s taking i n t o acoount the f i r s t t w o tenns i n eq.3b. The c a l d a t i o n s neceçsary t o obtain aS(k)/ap £rom this approximation are rather c a p l i c a t e d and have not y e t been perfonned f o r the data i n this paper.

An alternative expression for the isothermal density derivative of S(k) c m be derived f m the a s s q t i o n that the only effect of appïying pressure to the f luid,

Similarly the second derivative i n t h i s so-called uniforin f l u i d m d e l is:

So f a r only Egelstaff has given a few mdels f o r the-dependent t r i p l e t correlation functions /5/. 'keçe d e l s are straightforward yeneralizations ~Eaoproximationsfor 93. Conse-luently the result3 are only aoplicable t o t!e pressure derivative of the d i s t i n c t part Gd of the Van Hove function, and cannot be _applied to the derivative of the total correlation h c t i o n , w h i c h is the Fourier transfonn of S(k,w).

III

-

^RESULTS

The two neutron scattering experhxts mentioned i n the introduction have been des- cribed i n ref.7-8. It has also been d m n s t r a t e d that the two experiments are con- s i s t e n t /7/. The range of wavenmdxrs k covered by both experiments is 4.2-22.2 nn-'.

I n this chapter the four -4 meaçurawnts are used, canbined with the IlKS 1 meaçurawnt a t 844 bar (Table 1). The s t a t i c structure factor S(k) was obtained frcm S(k,w) by n-ical integration as described in refs.7 and 8. The results f o r 20 and 844 bar a r e given i n Figs.1 and 2.

1-Density derivatives of S (k)

.

In order to determine the f i r s t and second density derivatives of S (k)

,

a p o l y m i a l i n p was f i t t e d to the £ive experimental data points f o r S(k) a t each value of k within the range 4.2 to 22.2 nm-1. The degree of the polyncanial w a s chosen to be

the lowest one that yielded a correlation of a t l e a s t 99% be.tiween the data and the values predicted by the regression. This was achieved by a polyncanial of degree 1 o r 2 except f o r one k value (18.6 nn-l) . The resulting f i r s t derivative is given i n Fig. 3 for the lmst and for the highest density i n the range covered by the experimmts ( p = 17.6 and 21.6 n ~ t - ~ ) .

The g 3 mode1 (eq.4) is unsa- tisfactory f o r both densities.

0 5

a4 S ( k )

0.3

0.2

0.1

O O 25 20 bar (apen circïes) and

k(nm-'1 844 bar (closed circles)

.

I I I I r

+

-

4

-

O O 0

Q o .

O O O

- -

O O

O * 0

O

( 2 0

- 0 . _O

-

^-

~~o~~~~~~~ O

.

⁰

-

*..*** ^{O .}

-

+..o. o ~ O

.

0 0 0 0 0 0 0 0 0 ~ ~ ~ 0 0 * * *

.

.OUO.. O.****

1

5 10 15 20

w

The same mode1 has also been w e dby Egelstaff /1/

with neutron diffraction m e a s u r m t s on liquid neon

2.0 /11/ and by Winfield and Egelstaff with neutron dif-

fraction data on krypton near its c r i t i c a l point /12/. Both

l.5 cases lead to similar conclu- sions

as

the present case.

Haymet and Rice /15/ canpred the second order h-bond

ex-

pançion of T (eq.3b) with the

'"

neon data obtained by de Graaf and Mozer /Il/. They find qualitative a q r m t , but sans systematic differen- ces in the density derivative of g ( r ) .

Fig. 1

-

Ekperimental S(k) of 0 liquid argon a t 120 K and a t

JOURNAL

DE

PHYSIQUE

Fig. 2- Experimental S (k,w) of liquid argon a t 120K and a t 20 ( l e f t ) and 844 bar (right).

In f i g . 4 the scaled density de- rivatives of S(k) i n l i w d neon and i n liquid argon a r e cmpared.

Although the t h e m d y m m i c states correspnd, there is a small but signif icant systematic differ- ence. Therefore it would be in- teresting to make a direct cm- parison of the second-order h-bond e x p m i o n with the pres- ent data. This should be done i n k-space, since the k-range of the exprh-ental data is too r e s t r i c M to a l l m a Fourier transfom to r-space.

The second density derivative of S (k)

,

w h i c h vanishes w i t h i n the exgerimental uncertabty except a t a few k-values on the slope of the f i r s t peak, is sham i n Fig. 5. W e n t l y , i n the densi- t y r a n g e p = 17.6

-

21.6 mn-3 the unifonn f l u i d mode1 can describe neither the f i r s t nor t h e second density derivative of S (k) (see Figs. 3 and 5 ) . Egelstaff e t a l . /13/ report t h a t eq. 5a is in satisfactory agreement with neu- tron s m t t e r i n g data on liquid

-0.2

O 5 IO l5 k(nm-') 20 25 bar(b). Dashed 1ine:eq. 4; solid f i e : eq. 5; dots: expriment.

1 1

1

I I

' 5 !

I '.,'

,

I

Fig. 3

-

~sothermal density deri- vative of S (k) i n l i w d argon a t 120 K a t 20 &(a) and a t 844

A 1

+++ ^{Fig. 4} derivative of S (k) i n liquid

^-

^{1 ~ 0density ~ 1} arqon a t 120 K and

p

=

18.52 mi'-3 (dots) and in liquid neon a t 35 K and p = 33.18 n ~ n - ~ (solid line).

O The derivative of S (k) in

neon has been approximated by the difference quotient and has been s d e d by means of

-0.1 -

-

= 1.21, w i t h a a m u r e

of the atan diameter.

O 5 IO 15 20 25

kinm'')

rubidium, which they ascribe t o the presence of the electron

fluia. Egelstaff and Wang /14/

-1 ^{, , ,} 3'

¹

1

calculate the second derivative

-

^{4 $}

..'

¹

of S (k) f m the exprimental data for neon of ref. 11 and

caanpare the result with eq. 5b. _-6 m e n t l y the second deriva-

tives

in liquid

neon

(Fig. 2 ^O 5 IO ¹⁵ 20 25

of ref. 14)

and

in liquid argon k(nm")

(Fig. 5) a t comesponding ther- mdynamic states have o p p s i t e

si-. Consquently neon appears to deviate even mre frcm the uniforrn fluidthanargon.

2

-

Dençity derivative of S(k,w)

.

The f i r s t àensity derivative of S(k,o) has been determj.ned using a cubic mth spline as fundion of p, that takes into account the estimated errors of the data, SmxkWng is obtained by minimizing the average of the secand derivative, which w i l l

be

biased as a consquence. Therefore we w i l l lirnit ourselves here to the

f i r s t derivative. (Fitting a plynanial in p yields rather oscillatory and unsatis- facto- results for the density derivative). The density derivative appearç t o

be

hardly density m d e n t w i t h i n the k- and p-range studied here. The result for p = 19.5 m-3 is given i n Fig. 6 together with S(k,w) and is qualitatively very sirnilar to the results calculated by GKxrirr e t a l . /16/, using kinetic theory for liquid argon near its t r i p l e point.

Fig. 5

-

Experinwtal second +H isothevmal density derivative of ₂ S (k) i n liquid argon a t 120 K bZç(t) a n d a t 2 7 O b a r ( d o t s ) c a p a r e d bp2

lT

with the uniformfluid mode1 (

eq. 5b a t 20 bar (da&& line) and a t 844 bar (solid l i n e ) .

The f i r s t and the second dençity derivatives of the s t a t i c structure factor i n liquid argon a t 120 K have been detennined

over

a density range extending fran the vapour- liquid coexistence region to the vicinity of solidification. Liquid argon does not behave as a uniformly canpressible fluid in t h i s density range, in m t r a s t to liquid rubidium. The f i r s t and especially the second density derivative of S(k) are a p p r e ciable different for liquid argon and liquid neon a t correspondirg teqeratures and densities. The data presented here appear to be accwate enough for a meaningful mn- parison with the second-order h-hond expansion. The f i r s t density derivative of

-

-

0 . .

I

i, :1

JOURNAL DE PHYSIQUE

Fig. 6

-

S (k,w) (solid line) and its i s o t h e m a l pressure derivative (error bars) i n l i q u i d argon a t 120 K and 270 bar.

S(k,w) has a l s o been calculated and t h e r e s u l t c m be used t o test approximtions of the t h e d e p e n d e n t t r i p l e t correlation function.

REFERENCES

/l/ P.A. Egelstaff in: Ann.Rev.Phys.Qlem., H. Eyring, ed. (Annual Review Inc., Pal0 Alto, California, 1973)p. 159.

/2/ J.A. Barker

and

D. Henderson, Rev.Mod.Phys. (1976) 587.

/3/ G.F. Mazenko, Phys. Rev. AI (1973) 209.

/4/ P. Schofield, P m . Phys.

Soc.

(1966) 149.

/5/ P.A. Egelstaff in: Pmc. Symp. Neutron I n e l a s t i c Scattering 1972 (IAEA, Vienna, 1972) p. 383.

/6/

P.

Verkerk in: P m . Symp. Neutron I n e l a s t i c Scattering, Vienna 1977 (IAFA, Vienna, 1978) p. 53.

/7/ P. Verkerk, Ph. D. Thesis, University of T e c h n o l ~ , Delft, 1985.

/8/ A.A. van W e l l , P. Verkerk, L.A. de Graaf, J.-B. Suck and J.R.D. Copley, Phys.

Rev. (1985) 3391.

/9/ R. Abe, Progr. Theor. Phys.

2

(1959) 421.

/10/ A.D.J. Haymet, S.A. Rice and W.G. Madden, J. Chan. Phys. 74 (1981) 3033.

/il/ L.A.

de

Graaf and B. Mozer, J. Chem. Phys. (1971) 4967.

/12/ D.J. Winfield and P.A. Egelstaff, Can. J. Phys.

2

(1973) 1965.

/13/ P.A. Egelstaff, J.-B. Suck, W. G l a s e r , R. Mcpherson and A. Teitsma, J. de Phys.

0

(1980) C8-222.

/14/ P.A. Egelstaff and S.S. Wang, Can. J. Phys. (1972) 684.

/15/ A.D.J. Haymet

and

S.A. Rice, J. Chan. Phys. (1982) 661.

/16/ L. Grocme, K. Gubbins and J. Dufty, Phys. Rev. A g (1976) 437.