Electron–ion correlation in divalent liquid metals by diffraction method

Journal of Physics and Chemistry of Solids 60 (1999) 1553–1556

Electron–ion correlation in divalent liquid metals by diffraction method S. Takeda a,*, Y. Kawakita a, M. Kanehira a, S. Tamaki b, Y. Waseda c a

Department of Physics, Faculty of Science, Kyushu University, Ropponmatsu, Fukuoka 810-8560, Japan b Department of Physics, Faculty of Science, Niigata University, Niigata 950-2181, Japan c Institute for Advanced Materials Processing, Tohoku University, Tohoku 980-0812, Japan

Abstract The structure factor of liquid Zn has newly been obtained by neutron diffraction above melting temperature with sufficient accuracy by a powder diffractometer with multi-detector system, HERMES, which is installed at T1-3 beam line in JRR-3M. ˚ 21 and connected with The observed structure factor of liquid Zn has a very clear profile in the low Q region less than Q ˆ 4.5 A the structural data obtained previously. The structure factor of liquid Zn by X-ray diffraction has an asymmetric peak profile, while that by neutron diffraction does not. There has been detected a small but clear difference around the first peak region. Electron-ion correlation function and electron charge distribution around an ion have been obtained. The relation between the liquid structure and the electron charge distribution around an ion for liquid Zn will be discussed as compared with the simple liquid metals. q 1999 Elsevier Science Ltd. All rights reserved. Keywords: Liquid metals; C. X-ray diffraction

1. Introduction Liquid metals can be viewed as a binary mixtures between ions and conduction electrons moving nearly freely through an assembly of ions, and many of their physical properties are linked to the corresponding correlation functions of ion–ion, ion–electron and electron–electron. As is well known, neutrons are scattered by nuclei at the center of ions, therefore, the structure factor measured by neutron diffraction gives the pair correlation function among ions. In contrast, X-ray are scattered by electrons, which are the bound electrons of all ions and valence electrons distributed among the ions. They are sensitive to both ionic parts and electronic liquid parts and X-ray scattering intensity includes three types correlations, viz, ion–ion, electron– electron and electron–ion parts. The three types of correlations in liquid metals can be separated by the three different measurements such as X-ray, neutron and electron diffrac-

tion [1]. These different scattering mechanisms give a small, but certain difference among the observed structure factors. Several efforts [2–7] have been devoted to separate the three type partial structures and to evaluate the electron–ion and electron–electron correlations in the liquid metals. Recent experiments with sufficiently accurate results of Xray and neutron diffraction for liquid metals enable us to extract the electron–ion correlation on non-simple liquid metals. In the serial works for simple liquid metals, we reported the electron–ion correlation and electron–ion pair distribution [3–5], and some workers reported the theoretical results for simple liquid metals [5]. The main purpose of this article is to report newly measured structural data of neutron diffraction for liquid Zn and provide the electron– ion correlation in the metallic liquid. Then we provide the information on the charge distribution of valence electron around Zn 21 ion in liquid Zn from the X-ray and neutron diffraction data. 2. Neutron diffraction of liquid Zn

* Corresponding author. Tel.: 1 81-92-762-4728; fax: 1 81-92762-4728/4841. E-mail address: [email protected] (S. Takeda)

The neutron scattering measurements for liquid Zn was carried out at 4708C using powder diffractometer with a

0022-3697/99/$ - see front matter q 1999 Elsevier Science Ltd. All rights reserved. PII: S0022-369 7(99)00171-7

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S. Takeda et al. / Journal of Physics and Chemistry of Solids 60 (1999) 1553–1556

atom, fa(Q), as follows: Sx …Q† ˆ Ixcoh …Q†=Nfa …Q†2

Fig. 1. Structure factors of liquid Zn by neutron (solid curve) and Xray diffraction (broken curve). Also given is the difference between the two curves, SX(Q) 2 SN(Q).

multi detector, HERMES [8], of The Tohoku University and also the two-axis diffractometer, TAS-4G, of The Institute for Solid State Physics, The University of Tokyo at JRR-3 M. The samples were sealed in thin quartz tubes of inner diameter 8.0 mm with 0.3 mm wall thickness under a vacuum of 10 25 Pa. To eliminate the contamination of the Bragg reflections from the furnaces, they were coated with GdO powder which is an absorber for neutron. The sample and the heaters were put into a stainless steel chamber with an Al window under a He atmosphere. The used wave˚ for HERMES length, l , of the neutron beam was 1.83 A ˚ for TAS-4G. After the correction of the and 1.05 A measured intensity for absorption, inelastic and multiple scattering, we have the structure factor, SN(Q). The structure factor of SN(Q) by neutron diffraction is obtained from the coherently scattered intensity of neutron, Ixcoh …Q†; as folows; SN …Q† ˆ INcoh …Q†=Nb2

…1†

where N is number of atoms irradiated by neutron beam and b is scattering length. The accumulated intensity of counts are of the order of 2.0 × 10 4 around the first peak region and of the order of 1.5 × 10 4 in other regions, while those of empty cell are about 4.5 × 10 3. So the total experimental uncertainty in the structure factors are estimated to be less than ^1.0%. The structure factor Sx(Q) for liquid Zn by X-ray diffraction was obtained by one of the present authors [9] and their uncertainties are given below, for convenience of discussion in this work. The accumulated counts by X-ray diffraction, varying from 2 × 10 5 at low angles to 7 × 10 4 at high angles, were to hold counting statistics approximately uniform. The conventional structure factor of Sx(Q) by X-ray diffraction for liquid metals can be expressed by the coherent scattered intensity of X-ray, Ixcoh …Q†; and the form factor of a free

…2†

The normalization procedure to obtain Sx(Q) from the measured intensity was carried out based on Krogh–Moe– Norman method and fa(Q) and the incoherent (Compton) scattering factor were compiled from International Tables for X-ray Crystallography. The total experimental uncertainty in the structure factor of liquid Zn at 4708C does not exceed 1.0%. In both cases S(Q ! 0) was determined by the thermodynamic measurements of density, sound velocity and specific heat. Fig. 1 shows the structure factors of liquid Zn determined by neutron and X-ray diffraction. The systematic differences are well appreciated, although the qualitative coincidence in the structural profile is almost retained. The quantity of [Sx(Q) 2 SN(Q)] evaluated from two structure factors is also given in the bottom of Fig. 1, which may help to see the well appreciated difference and oscillating profile in [Sx(Q) 2 SN(Q)]. 3. Analysis X-rays are scattered by the electrons, and they are scattered by both ionic part and electronic liquid part. If we use the Ashcroft–Langreth type partial structure factors [10] of binary mixture composed of ions ( ˆ i) and electrons ( ˆ e), the total X-ray scattering intensity except for Q ˆ 0 is expressed from the combination of Ashcroft–Langreth partial structure forms as follows: IX …Q†=N ˆ ‰fi …Q†2 Sii …Q† 1 2z1=2 fi …Q†Sie …Q† 1 zSee …Q† 1 …Z 2 z†Sinc i …Q†

…3†

where subscripts, i and e, represents ion and electron, respectively, fi(Q) is the form factor of the ion, Z the atomic number and z the number of valence electrons per atom. Sii(Q), Sie(Q) and See(Q) are the partial structures of ion– ion, ion–electron and electron–electron correlations and Sinc i …Q† is the incoherent (Compton) scattering factor of ions due to bound electrons. The static structure factor of valence electrons, See(Q), is composed of two parts of local charge density bound to the ionic core and the uniformly distributed part. See(Q)and Sie(Q) in liquid metal are also given by the following forms: See …Q† ˆ ur…Q†u2 Sii …Q†=z 1 S…0† ee …Q†

…4†

Sie …Q† ˆ r…Q†Sii …Q†=z1=2

…5†

where r (Q) is the Fourier transform of the local charge density function of valence electrons and S…0† ee …Q† is the structure factor of uniformly distributed electrons. Both …Z 2 …0† z†Sinc i …Q† and zSee …Q† in Eq. (3) are incoherent parts, and they are eliminated from the measured intensity data on obtaining the coherent X-ray scattering intensity, IXcoh …Q†:

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Fig. 2. Charge density function, r (Q) (solid curve), obtained from back-transformed spectrum for liquid Zn and an isolated Zn atom (dotted curve).

Fig. 4. Charge density function, r (r), of liquid Zn (solid and broken curve) and that of an isolated Zn atom (dotted curve). r1; distance of the nearest neighbour ions.

This term can be rewritten in the following form:

mediate region as a few times larger than the atomic distance. It may be worthwhile to mention that no approximation can be applied the present derivation of electron–ion correlation and charge density function.

IXcoh …Q†=N ˆ ‰fi …Q† 1 r…Q†Š2 Sii …Q†:

…6†

However, the structure factor of SN(Q) is discussed as Eq. (1). Since Sii(Q) is equal to SN(Q), the local charge density distribution in the momentum space, r (Q), can be obtained by Eqs. (6) and (1), as follows:

r…Q† ˆ

‰b2 INcoh …Q†=IXcoh …Q†Š1=2

2 fi …Q†:

…7†

r (Q) in Eq. (7) is useful for discussing the local charge density around an ion, while Sie(Q) in Eq. (5) is convenient for discussing the electron charge distribution for an inter-

Fig. 3. Ion–electron structure factor, Sie(Q), obtained from structural data.

4. Discussion Eq. (7) enables us to evaluate the local charge density, r (Q), from the measured structural data of IXcoh …Q† and INcoh …Q†: Since r (Q) has uncertainty due to the relatively small contribution of the electron–ion correlation to the total correlation in the liquid metals, the profile of r (Q) was assumed to be a smooth function and the average values at each Q. Then, we obtained the r (Q) by the back Fourier transform technique. Fig. 2 shows the resulting local charge density function, r (Q), together with those of isolated Zn atom obtained from [fa(Q) 2 fi(Q)]. As shown in the figure the large value can be ˚ 21, which is close to the value of 2kF of found around 3.1 A Zn. The corresponding Sie(Q) is shown in Fig. 3. Fig. 4 shows the local charge density distribution functions, r (r), together with the charge density for isolated atom. As shown in the figure, the curve of r (r) crosses zero ˚ and a sharp increase toward the center of an around 1.3 A ˚ . Although this value is close ion can be seen less than 0.74 A to the ionic radius, the accurate spectra of r (Q) in the high Q region will be required for the further understanding of the charge distribution in the very small r region less than ˚ , which corresponds to the ionic radius and also 0.74 A near the first maximum position of charge density of an isolated atom. However, the structural information outside ˚ should be valid and useful for the region larger than 0.74 A

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5. Conclusion In conclusion we have obtained the structure factor for liquid Zn from newly measured neutron diffraction spectrum. By utilizing the difference in structure factors obtained from neutron and X-ray diffraction spectra, we have deduced the electron charge distribution around Zn 21 ion and also metallic form factor of liquid Zn.

Acknowledgements

Fig. 5. Form factor of liquid Zn (solid curve) and dotted curve shows that of an isolated Zn 21 ion.

understanding the charge distribution outside the ionic core region. The charge density function in Fig. 4 shows that the electron charge screens the ionic charge rather closely contact with the ions and the maximum point exists rather ˚ , which is rather close to the nearest Zn ion. distant at 2.2 A In case of simple liquid metals as Na, Mg and Al, the electron charge distribution has maximum value around the mid point between the nearest neighbour ions to screen the central ionic charge. However, the charge distribution of liquid Zn takes minimum value around the mid point among the nearest neighbour ions. The charge distribution of liquid Zn is different from the simple liquid metal case. It seems not easy to separate the charge distribution clearly from the central ionic charge in case of liquid Zn. The first term of Eq. (7) indicate the metallic form factor, fM(Q), and Fig. 5 shows the obtained metallic form factor (solid and broken curve) and ionic form factor (dotted curve) of liquid Zn. The obtained metallic form factor of liquid Zn indicates the interesting feature around 2kF region.

The authors express their thanks to the Institute for Material Research, Tohoku University, for providing us the fascilities for the neutron diffraction measurement, HERMES, in JRR-3M and also Dr Miki for his helpful collaboration on this experiments. The authors are grateful to the Ministry of Education, Science and Culture for Financial support of Grant-in-Aid.

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