Electron-ion correlation in liquid tellurium

] O U R N A L OF

Journal of Non-Crystalline Solids 156-158 (1993) 683-686 North-Holland

~I.1.11~ ~I.I1W AIVII-VA%l Ulllltll~! lJ

Electron-ion correlation in liquid tellurium Shin'ichi T a k e d a a, Masanori Inui and Yoshio W a s e d a d

a,

Shigeru Tamaki b, Kenji M a r u y a m a c

a Department of Physics, College of General Education, Kyushu University, 4-2-1 Ropponmatsu, Chuo-ku, Fukuoka 810, Japan b Department of Physics, Faculty of Science, Niigata University, Niigata 950-21, Japan c Graduate School of Science and Technology, Niigata University, Niigata 950-21, Japan d Institute for Advanced Materials Processing, Tohoku University, Sendai 980, Japan

The structure factor of liquid Te at 470°C has been obtained by X-ray and neutron diffraction measurements. A remarkable and clear difference in these structure factors has been found and the ion-electron correlation function in liquid Te has been evaluated from this difference. The valence electron charge distribution function around a central ion has maxima at 0.82 and 1.69 ,~, corresponding to the lone pair electrons and covalent bonded electrons around a central Te ion.

1. Introduction

Liquid Te has been extensively investigated by many workers, because of the anomalous temperature dependence of its electronic [1,2] and thermodynamic properties [3]. This anomalous temperature dependence has been considered as a gradual transition from a non-metallic to a metallic state, and many investigations of the local structure of liquid Te [4-8] have been carried out. Such a transition was discussed within the concept of a change of the local bonding character of Te at a microscopic level [9,10]. It is interesting to study how the bonding electrons behave at such a non-metal to metal transition. Liquid metals are a binary mixture of rigid ions and conduction electrons, giving a strongly coupled plasma. Based on this idea, the structure of liquid metals can be described explicitly in terms of three types of correlations, namely, electron-electron, electron-ion and ion-ion pairs. The three types of correlations in liquid metals can be separated by three different measureCorrespondence to: Professor S. Takeda, Department of Physics, College of General Education, Kyushu University, 4-2-1 Ropponmatsu, Chuo-ku, Fukuoka 810, Japan. Tel: + 8192 771 4161, ext. 281. Telefax: +81-92 724 0790.

ments such as X-ray, neutron, and electron diffraction. As is well known, neutrons are scattered by nuclei at the center of ions, while X-rays are scattered by both the bound electrons of all ions and the valence electrons distributed among the ions. These different scattering mechanisms yield a small, but distinct difference in the observed structural data of liquid metals. [11,12]. An effort has been made by several workers [13,14] to estimate the electron charge distribution in liquid metals. In this paper, we present newly measured structural data from X-ray and neutron diffraction for liquid Te, and discuss the electron charge distribution around the Te ions.

2. Experimental details and results

Neutron scattering measurements for liquid Te were carried out at 470°C using the two-axis diffractometer of the Institute of Solid State Physics, The University of Tokyo at JRR-3. The sample was sealed in a quartz tube. After correcting the measured intensity for absorption [15], inelastic [16] and multiple scattering [17], we obtain the structure factor, SN(Q). The experimental uncertainty in the present experiment can be

0022-3093/93/$06.00 © 1993 - Elsevier Science Publishers B.V. All rights reserved

684

S. Takeda et aL / Electron-ion correlation in liquid Te

evaluated as follows. The uncertainty in the wave-number AQ/Q, is less than 8% even in the low-Q region near Q = 2 ,~-1. The accumulated intensity counts are of the order of 5.0 × 104 around the first peak region and of the order of 2.0 × 104 in other regions, while those of the empty cell are about 0.2 × 104. The total experimental uncertainties in the neutron structure factor, SN(Q), of liquid Te are estimated to be less than 1.0% and then, for example, the error bars in the SN(Q) values are of the order of 0.02 around the first peak. The X-ray diffraction measurement for liquid Te was carried out at the same temperature as the neutron diffraction measurement. The accumulated counts vary from 4 x 104 at low angles to 4 × 103 at high angles, so that the counting statistics are approximately uniform. The total experimental uncertainty in the structure factor of liquid Te does not exceed 1.0%. This follows from a detailed discussion of the source of systematic errors in X-ray diffraction experiments. Thus the uncertainty is of the order of 0.02 in the value of Sx(Q) near the first peak. In both cases, S(Q ~ O) is determined by thermodynamic measurements of density, sound velocity and specific heat. Figure 1 shows the structure factor of liquid Te at 470°C determined by neutron and X-ray diffraction. The systematic differences are clearly

/ 21

Te

i

470"C

-- X-roy .... N D

S N

_

g021 % O,

,~'.

~,...."

o -'..'..".

......

.,.) ~-.,,.,~.,,....,¢ . . . .

recognizable, although the qualitative coincidence in both structural profiles, is retained. The quantity [Sx(Q) -SN(Q)] obtained from the two structure factors is also given at the bottom of fig. 1.

3. Analysis If we use the Ashcroft-Langreth type partial structure factors of a binary mixture composed of ions and electrons, the total X-ray scattering intensity for Q ~ 0 is expressed as

l x ( O ) / N = [fi2(a)sii(O) + 2f~fi(Q)Sie(a ) +zSee(Q)] + (Z-z)S~nc(o),

where fi(Q) is the form factor of the ion, Z the atomic number, and z the number of valence electrons per atom. Sii(Q), Si~(Q) and Se~(Q) are the partial structure factors of ion-ion, ion-electron and electron-electron pairs, respectively and s~nc(Q) is the incoherent (Compton) scattering factor of the ions. The static structure of the valence electrons, See(Q ) is given by

See(Q) =

]p(Q) 12sii(Q) z

+ s e(o) et"~,q) ,~

(2)

where p(Q) is the Fourier transform of the electron charge distribution around an ion in the liquid metal and S~°)(Q) is the structure factor of uniformly distributed electrons. Since (Z-z)S~nC(Q) is the Compton scattering factor of ions, it may be calculated and then eliminated properly from the measured intensity data. The structure factor of uniformly distributed electrons, S~_°)(Q), is known to make a negligible contribution to the coherent X-ray scattering. Hence, we obtain the coherent X-ray scattering intensity, i~oh x (Q), (which can be determined from measured intensity data) in the form

I~°h(Q)/N= [ fi(Q) + p(Q)] 2Sii(Q) =fi(Q)zSii(Q) + 2fzfi(Q)Sie(Q ).

-0.2 Fig. l. Structure factors of liquid Te by X-ray (solid curve) and neutron diffration (ND, closed circles) at 470°C, and the difference between the two curves, [ S x ( Q ) - SN(Q)].

(1)

(3)

Except for very light elements such as Li and Be, the term See is negligibly small compared with the first and second terms. This is also

S. Takeda et al.

/ Electron-ion correlation in liquid Te

quantitatively confirmed for several liquid metals [141. p(Q) is useful for discussing the local charge density around an ion, while Si~(Q) in eq. (3) is convenient to describe the electron distribution for an intermediate region, say, a few times larger than the atomic distance. The partial structure factor for ion-ion pairs is well known to be equal to the coherent structure factor determined by neutron diffraction, S i i ( Q ) = SN(Q) =l~h(Q)/Nb 2, where b is the scattering length of neutrons and i~Oh(Q) is the coherent neutron scattering intensity. Therefore, electron-ion correlation Si~(Q) is given by 1 S i e ( a ) = 2v~-fi(Q) [ I ~ O h ( a ) _ f i 2 ( O ) S N ( O ) ]

(4) or

p(Q) = v~Sie(Q)/SN(Q).

(5)

The theoretical ionic form factor, fi(Q), is usually calculated by the Hartree-Fock method, and is also obtainable from the X-ray scattering intensity measurements for the ionic crystals. Therefore, Si~(Q) can readily be converted into p(Q) using eq. (5), or vice versa. The Q dependence of fi(Q) for the Te 4+ ion is calculated by the Hartree-Fock method. Therefore eqs. (4) and (5) enable us to evaluate the partial structure factor of ion-electron pairs Sie(Q) and p(Q) from measured structural information of Sx(Q) and SN(Q). The partial pair correlation function between ions and electrons, gi¢(r), and charge density around a Te 4+ ion, p(r), can be obtained from Sie(Q) and p(Q) by the Fourier transformation, 1

=

g i e ( r ) = 1 + 2,rr2p0---------~f 0

z-1/2si~(Q)a

sin

Or dQ,

685

4. Discussion The partial structure factor for ion-electron pairs, Sie(Q) , for liquid Te evaluated from eq. (4) is given in fig. 2. The vertical lines in this figure indicate the experimental uncertainty due to the statistical counting error of the X-ray and neutron diffraction measurements. Therefore, the information on Sie(Q) obtained here is sufficient to discuss the partial pair correlation functions, although the information on Sie(Q) is limited to amax = 8.7 .~ -1 . The partial pair correlation function for ions and electrons, gie(r), is obtained from Sie(Q). The charge distribution around a Te ion, 4-rrr 2Pogie(r) has a firStoPeak at 0.82 ,~ and a second peak around 1.69 A. A theoretical study of crystalline Te by Joannopoulos [18] shows that the charge density maxima of two non-bonding pelectrons (lone pair) and of two bonding p-electrons (a-bond) around a central atom are located at distances of 0.85 and 1.45 A, respectively. Hence, the first peak in 4"rrr2pogi~(r) corresponds to the charge distribution of the lone pair electrons around the Te 4÷ ion and the second peak shows the electron charge distribution of the bonding stage in liquid Te. To see the electron configuration around an ion more clearly, it may be useful to deduce the electron charge distribution, 4~r2p(r), in real space around a central ion. p(r) can be obtained from the charge density distribution in Q space, p(Q). The resultant 4~rr2p(r) is shown in fig. 3. The dotted line in the figure represents the charge o

Te ~70"C

S

(6) p( r )

1 2.rr2por

fo=p(Q)

sin

Qr dQ,

(7)

where P0 = zN/VM (I'M is the molar volume and N is Avogadoro's number).

-~

~

~

~

~

g

~

~

~

Q (l/A)

Fig. 2. Ion-electron pair correlation function in liquid Te.

10

686

S. Takeda et al. / Electron-ion correlation in liquid Te

5. Conclusions In conclusion, we have obtained f u n d a m e n t a l information on the valence electron charge distribution in liquid Te by performing n e u t r o n and X-ray diffraction measurements. 0

r (A) Fig. 3. Valence electron charge density distribution of liquid Te in real space. The dotted line denotes the charge density distribution of 5p electrons in a free atom. density distribution of 5p electrons in a free Te atom. 4 ~ r r e p ( r ) has maxima at 0.82 and 1.69 ,& and the coordination n u m b e r s are 0.7 and 3.3, respectively. Liquid Te has an anomalous t e m p e r a t u r e dep e n d e n c e of the electrical conductivity and this anomaly has b e e n considered as a gradual transition f r o m a non-metallic to a metallic state. T h e transition is related to the c h a n g e of the local b o n d i n g character f r o m twofold to threefold coordination. T h e s e results show that liquid Te is c o m p o s e d of about 35% twofold and 65% threefold c o o r d i n a t e d atoms at 470°C, and this is consistent with the result o b t a i n e d f r o m the measu r e m e n t o f the specific heat [10]. Since the ordinary i o n - i o n radial distribution function o f liquid Te obtained by n e u t r o n diffraction shows 2.7 ions within a distance of 2.90 A as the nearest neighbour ions, this is also consistent with o u r results. A tentative interpretation is that the n u m b e r of the b o n d i n g p-electrons increases from 2 to 2.7 on melting. If the three electrons a m o n g four 5p electrons f o r m b o n d i n g orbitals, one electron must occupy an anti-bonding orbital. This may cause the b o n d distance to be stretched to some extent in the liquid state c o m p a r e d with the solid state and, as a result, liquid Te may have metallic properties.

T h e authors are grateful to Professor A. A z u m a for providing the wave-function of an isolated Te ion. T h e authors also express their thanks to T h e Institute of Solid State Physics, T h e University of Tokyo, for providing the facilities o f the diffractometer 4 G in J R R - 3 .

References [1] J.C. Perron, Adv. Phys. 16 (1967) 657. [2] J.A. Gardner and M. Cutler, Phys. Rev. B20 (1979) 529. [3] H. Thurn and J. Ruska, J. Non-Cryst. Solids 22 (1976) 331. [4] Y. Waseda and S. Tamaki, Z. Natufforsch. 30a (1975) 1655. [5] S. Takeda, S. Tamaki and Y. Waseda, J. Phys. Soc. Jpn. 53 (1984) 3830. [6] A. Mennele, R. Bellissent and A.M. Flank, Europhys. Lett. 4 (187) 705. [7] J.E. Enderby and A.C. Barnes, Rep. Prog. Phys. 53 (1990) 85. [8] K. Tamura, M. Inui, M. Yao, H. Endo, S, Hosokawa, H. Hoshino, Y. Katayama and K. Maruyama, J. Phys.: Condens. Matter 3 (1991) 7495. [9] B. Cabane and J. Friedel, J. Phys. (Paris) 32 (1971) 73. [10] S. Takeda, H. Okazaki and S. Tamaki, J. Phys. C15 (1982) 5203. [11] P.A. Egelstaff, N.H. March and N.C. McGill, Can. J. Phys. 52 (174) 1651. [12] J. Chihara, J. Phys. F17 (1987) 295. [13] C. Petrillo and F. Sacchetti, J. Phys. F16 (1986) L283. [14] S. Takeda, S. Harada, S. Tamaki and Y. Waseda, J. Phys. Soc. Jpn. 60 (1991) 2241. [15] H.H. Paalman and C.J, Pingth, J. Appl. Phys. 33 (1962) 2635. [16] D.M. North, J.E. Enderby and P.A. Egelstaff, J. Phys. C1 (168) 2635. [17] V.F. Sears, Adv. Phys. 24 (1975) 1. [18] J.D. Joannopoulos, The Physics of Selenium and Tellurium, ed. E. Gerlach and P. Grosse (Springer, Berlin, 1979) p. 2.