Electron charge distribution in liquid sodium

Journal of Non-CrystaJline Solids 117/118 (1990) 80-83 North-Holland

80

ELECTRON CHARGE DISTRIBUTION IN LIQUID SODIUM S h i n ' i c h i TAKEDA, Sh~zi HARADA*, Shigeru TAMAKI** and Yoshio WASEDA+ College of General Education, Kyushu University, Ropponmatsu, Fukuoka 810, JAPAN * Faculty of Engineering, Niigata University, Niigata 950-21, JAPAN * * Department of Physics, Faculty of Science, Niigata University, Niigata 950-21, JAPAN + Research i n s t i t u t e of Mineral Dressing and Metallurgy, Tohoku University, Sendai 980, JAPAN The ion-electron c o r r e l a t i o n has been evaluated from a small difference in the structure factors by X-ray and neutron d i f f r a c t i o n in l i q u i d sodium and a q u a n t i t a t i v e evaluation of the electron charge d i s t r i b u t i o n in l i q u i d Na has also been derived from the ion-electron c o r r e l a t i o n function. I. INTRODUCTION I t is well known that neutron scattering

agreement with those proposed by Pauling 7,

gives the structure f a c t o r among the nuclei

Such successful results prompt us to do some

centered at an ion, whereas X-ray scattering

f u r t h e r i n v e s t i g a t i o n on the d e r i v a t i o n of

is caused by both ions and valence electrons.

e l e c t r o n - i o n c o r r e l a t i o n function in simple

Egelstaff et a l . 1 pointed out the p o s s i b i l i t y

l i q u i d metals such as l i q u i d Na.

that three types of c o r r e l a t i o n ( e l e c t r o n - i o n ,

In t h i s paper we present the results of

e l e c t r o n - e l e c t r o n , anci ion-ion c o r r e l a t i o n s )

newly measured structural data of neutron

in l i q u i d metals can be separated by the three

d i f f r a c t i o n for l i q u i d Na, and to provide the

d i f f e r e n t measurements such as X-ray, neutron,

electron charge d i s t r i b u t i o n in this l i q u i d .

and electron d i f f r a c t i o n s , and several e f f o r t s

2. EXPERIMENTALAND RESULTS

have been devoted to separate these three c o r r e l a t i o n s from the d i f f r a c t i o n data. 2'3

The sample in t h i s measurement was sealed in a thin aluminum tube and Figs. l ( a ) and

On the other hand, there have been many

l ( b ) show the observed structure factors of

t h e o r e t i c a l works on the l i q u i d metals, which

l i q u i d Na by neutron and X-ray d i f f r a c t i o n s .

regarded as a binary mixture or a strongly

The uncertainty in the present structural

coupled plasma composed of electrons and ions,

information can be estimated as follows.

and also many t h e o r e t i c a l works on the p a r t i a l

Neutron d i f f r a c t i o n : The accumulated i n t e n s i t y

c o r r e l a t i o n functions among ions and electrons

counts are of the order of 4.0xlO 4 around the

in l i q u i d metals. 4'5

f i r s t peak region, and those other region are

Under these circumstances, so far, we

of the order of 1.5xlO4. The accumulated

presented the e l e c t r o n - i o n c o r r e l a t i o n s of

i n t e n s i t y counts of empty aluminum tube is

several l i q u i d polyvalent metals such as Bi,

about O.2xlO4 except f o r the Bragg angle.

Sn, Zn, Pb, T1 and Ga6, using the results of

Combining the d e t a i l e d discussion f o r

X-ray and neutron d i f f r a c t i o n measurements

systematic analysis in neutron d i f f r a c t i o n of

with high accuracy incorporating with the

l i q u i d metals by North et al 8, the t o t a l

e l e c t r o n - e l e c t r o n c o r r e l a t i o n function

experimental uncertainties in the structure

t h e o r e t i c a l l y calculated. The ionic r a d i i f o r

f a c t o r SN(Q) were estimated to be almost 0.6%

these elements evaluated from the e l e c t r o n - i o n

X-ray d i f f r a c t i o n : The accumulated counts by

c o r r e l a t i o n functions are found to be in good

the X-ray d i f f r a c t i o n methods, varying from

0022-3093/90/$03.50 (~) Elsevier Science Publishers B.V. (North-Holland)

S. Takeda et al./ Electron charge distribution in liquid sodium e s s e n t i a l agreement is encouraging very much,

2xlO 5 at low angles to 7xlO 4 at high angles, were to hold counting s t a t i s t i c s

81

approximately

although there are some d i f f e r e n c e s in d e t a i l .

uniform. The t o t a l experimental u n c e r t a i n t y in

Na

the s t r u c t u r e f a c t o r of l i q u i d Na does not (a)

exceed over 1.0%. As e a s i l y seen from the Figs, a small but c l e a r d i f f e r e n c e can be well appreciated around the low Q region. The q u a n t i t y of [Sx(Q)-SN(Q)] c a l c u l a t e d from two s t r u c t u r e f a c t o r s are shown in Fig. 2 and Xray d i f f r a c t i o n

(b) c

~

:~"~--"~'--

data were independently

obtained by d i f f e r e n t workers 9 ' I 0 and the

Q (l/A) FIGURE 2 [Sy(Q)-SN(Q)] f o r l i q u i d Na. X-ray d i f f r a c t i o n da~a are taken from the reference 9 f o r (a) and I0 f o r (b).

No

2

(a)

.... Neutron(110"C) 3. THEORY AND ANALYSIS Since the theory of d e r i v a t i o n f o r the i o n - e l e c t r o n c o r r e l a t i o n f u n c t i o n in l i q u i d metals has been reported in d e t a i l in the previous papers, 5

only the e s s e n t i a l points

are given below.

o~

2

3

4

5

6

-~

-~

~-

lo

e (~/A)

Let us consider a l i q u i d metal composed of N ions and zN-valence e l e c t r o n s .

I f we use

the s o - c a l l e d Ashcroft-Langreth type p a r t i a l s t r u c t u r e f a c t o r s of p a r t i c l e s as ions and e l e c t r o n s , the coherent X-ray s c a t t e r i n g intensity, NQ

2

(b)

l~°h(Q),

is expressed as f o l l o w s ;

l~°h(Q) = N[f~(Q)Sii(Q) + 2 z l / 2 f i ( Q ) S i e ( Q )

..... Neutron(110"C) -- X-ray (105"C)

+ zSee(Q)] (I) where f i ( Q ) is the form f a c t o r of the ion and z the number of valence electrons per atom. S i i ( Q ) can be replaced by SN(Q), because the nucleus is located at the center of ion. On the other hand, the conventional

0

1

2

3

4

5

Q (l/A)

6

?

8

9

10

FIGURE 1 Structure f a c t o r s of l i q u i d Na. Dotted curve is neutron and s o l i d curve X-ray d i f f r a c t i o n . X-ray d i f f r a c t i o n data is taken from the referrence 9 f o r (a) and I0 f o r (b).

expression f o r the s t r u c t u r e f a c t o r obtained by X-ray d i f f r a c t i o n

method, Sx(Q), is

expressed in the f o l l o w i n g d e f i n i t i o n , Sx(Q) =

l~°h(Q) Nf~(Q)

(2)

S. Takeda et al. / Electron charge distribution in liquid sodium

82

where fa(Q) is the form factor of free atom.

shown in Figs. 1 and 2, and the atomic form

Therefore the following equation can be obtained.

factor fa(Q) is in accord with the ionic form

1 [ f2(Q) SX(Q)_f2(Q) SN(Q) Sie(Q)~2z½fi (Q) -zSee(Q) ] (3) I t is noteworthy to mention that the f i r s t

Therefore, the information for Sie(Q)

factor f i ( Q ) beyond Q=5.6A- I

12 for sodium.

presently obtained is enough to discuss the p a r t i a l pair correlation functions.

term of Eq. (3), f2(Q)Sx(Q), is just the observed coherent X-ray scattering i n t e n s i t y per atom.

The Q-dependence of fi(Q) may be NQ

close to the form factor of z-th valent ion

H-T

with atomic number Zo A usual representation of f i ( Q ) is given by the Hartree-Fock method. The electron-electron correlation in metals have long been studied 4 and the pair v

correlation function gee(r), corresponding to the Fourier transform of See(q), is well established for various r a d i i of conduction electrons.

The Eq. (3) enables us to evaluate

the p a r t i a l structure factor of ion-electron 0

pairs Sie(Q ) from measured structural

1

2

3

4 O (I/A)

5

6

7

8

information of Sx(Q) and SN(Q), when coupled with the theoretical values of See(Q). We calculated See(Q) for radius of conduction

FIGURE 3 lon-electron structure factor of l i q u i d Na.

electron rs:2o14 under the Utsumi-lchimaru schemeI 1 N(3

The p a r t i a l pair correlation function between ions and electrons, g i e ( r ) , is

I ri rA rB r~ q

(a) H - T

obtained from Sie(Q) by the Fourier transformation using the following equation, gie(r) = 1 + ~

IoZ-~Sie(e)Q s i n ( e r ) d e

(5) where po=zN/VM and VM being the molar volume and N the Avogadoro's number. 4. RESULTS AND DISCUSSION The p a r t i a l structure factor of ion-electron pairs, Sie(Q ), is evaluated from Eq. (3) and shown in Fig 3. The v e r t i c a l lines indicate the experimental uncertainty due to the counting s t a t i s t i c a l errors of i n t e n s i t y . The difference in the structure factors between Sx(Q) and SN(Q) disappear beyond 5.5A - I as

0

1

2

3

4

5 r (A)

6

7

8

9

10

FIGURE 4 lon-electron pair correlation functions in l i q u i d Na. (a):X-ray data is cited from the reference 8, (b):X-ray data is cited from the reference 9. r : : minimum point, r ^ : f i r s t maximum point, rR: minimum position between the neighbou~s, rl:distance of the nearest neighbour lons,

S. Takeda et al. / Electron charge distribution in liquid sodium

The p a r t i a l p a i r c o r r e l a t i o n functions, g i e ( r ) , are obtained from Sie(Q) and the results are shown in Fig. 4.

As shown in Fig.

4, the essential agreement is found in these two p r o f i l e of g i e ( r ) and the sharp change of g i e ( r ) in l i q u i d Na occurs at about 1.05 A in both results.

I t may correspond to the ionic

radius of Na+, although i t is s l i g h t l y larger than the value proposed by Pauling 7, However i t is noteworthy that g i e ( r ) increases sharply less than r i, which is d i f f e r e n t behaviour from those of l i q u i d polyvalent metals such as Zn and Sn. The present authors maintain the

83

behaviour is consistent with a nearly free electron picture. This is a d i s t i n c t feature compared with other l i q u i d polyvalent metal. According to the ordinary ion-ion d i s t r i b u t i o n and the curve of g i e ( r ) , the schematic representation for the electron d i s t r i b u t i o n around the ions can be evaluated to be about 0.7. This implies that about two t h i r d number of electrons loosely d i s t r i b u t e around Na+ ion and the remainders e x i s t in the interstitial

region among the ions.

ACKNOWLEDGMENTS The authors express t h e i r thanks to Dr. J.

view that t h i s is caused by the penetration of

Chihara, Japan Atomic Energy Research

electron charge from out side to the inside of

I n s t i t u t e , f or his t h e o r e t i c a l comments and

the ion core. Eq. (3) may not be, more or

discussion on t h i s work.

less, enough to describe completely the electron and ion d i s t r i b u t i o n s in l i q u i d metals p a r t i c u l a r l y f o r metals where the

REFERENCES I. P.A. Egelstaff, N.H. March and N.C. McGill, Can. J. Phys. 52 (1974) 1651.

electron charge d i s t r i b u t e s both outside and inside of the ion core. The f i r s t

peak at 1.65A in g i e ( r ) means the

electron charge d i s t r i b u t i o n around Na+ ion located at the o r i g i n . The second peak at about 3.10A indicates the electron cloud around the nearest neighbour ions. Since the nearest neighbour ions coordinate around the distance of r=3.70A from the ordinary ion-ion d i s t r i b u t i o n function, the distance of the first

peak in g i e ( r ) , r=l.65A, corresponds to

the middle region among nearest neighbours, and the height of the corresponding peak is not so high than 1.0. A ft e r a careful consideration to above results, i t is suggested that the electrons more than the average density are d i s t r i b u t e d over the intermediate region among the nearest neighbour ions. In other words, t h i s implies that the valence electron charges in l i q u i d sodium are not bound at a certain sodium ion, but rather spread out over the intermediate region among the nearest neighbours. Such

2. P.J. Dobson, J. Phys. C 1978 L1 L295. 3. H. Olbrich, H. Ruppersberg and S. Steeb, Z. Naturforsch. 38a (1983) 1328. 4. S. Ichimaru, Rev. Mod. Phys. 54 (1984) 1017. 5. J. Chihara, J. Phys. F, 17 (1987) 295. 6. S. Takeda, S. Tamaki, Y. Waseda, and S. Harada, J. Phys. Soc. Jpn. 55 (1986) 184; Z. Phys. Chem. 157 (1988) 45~TT. 7. L. Pauling, The Nature of the Chemical Bonds (Cornell Univ. Press, Ithaca, New York, 1939). 8. D.M. North, J.E. Enderby and P.A. Egelstaff, J. Phys. C ! (1968) 1075. 9. M.J. Huijben and W. van der Lugt, Acta. Cryst. A3___55(1979) 431. I0. Y. Waseda, The structure of Non-Crystalline Materials (McGraw-Hill, New York, 1980). I I . K. Utsumi and S. Ichimaru, Phys. Rev. B22 (1980) 5203. 12. D. Cromer and J.T. Waber: Acta Cryst. 18 (1965) 104.