Structure of Liquid Gallium and Rubidium by Pulsed Neutron Diffraction Using Electron Linac

By Kenji Suzuki*,Masakatsu Misawa* and Yoshiaki Fukushima*

The static structure factors of liquid gallium and rubidium were measured at several tem- peratures above their melting points using the time of flight neutron diffractometer installed on the 300MeV Tohoku University electron linac as a pulsed neutron source. The characteristic oscillation of the structure factor of liquid gallium in a high momentum transfer region has been shown to be well understood in terms of a diatomic molecule-like atomic association with the bond length of 2.69Å. It has been discussed, however, that the subsidiary maximum on the high momentum transfer side of the first peak in the structure factor of liquid gallium may appear due to the second and subsequent peaks in the pair correlation function to shift to a larger distance in comparison with those of simple liquid metals such as alkali metals.

(Received October 9, 1974)

sentative simple metallic liquid and gallium as Ⅰ. Introduction a liquid metal with anomalous structure factor. The time of flight (T-O-F) neutron diffraction based on the electron linear accelerator as Ⅱ. Experimental a pulsed neutron source is known to provide particularly useful information on the structure For a structural study of the liquidus, all the factor of molecular liquids in the high mo- measurements of static structure factor were mentum transfer region(1)(2). On the other made using the L0≫L T-O-F neutron dif- hand, the precise measurement even at the fractometer(5) installed at the 300-MeV Tohoku momentum transfer of about Q=10Å-1 is University electron linear accelerator(6). The quite difficult in the conventional neutron schematic diagram of the diffractometer is diffraction experiment using a monochro- shown in Fig. 1 and its essential feature is briefly matized neutron beam from the nuclear described below. reactor. Neutrons scattered from sample are counted The high momentum transfer structure factor simultaneously at four scattering angles, 2θ= of liquid metals is necessary not only to perform 15, 30, 60 and 150° by He-3 counters (Reuter an accurate Fourier transformation into a pair correlation function but also to identify a compound-like atomic association in liquid alloys(3) and derive the steepness of the repulsive core in the pair potential of liquid metals(4)

. From such a point of view, the T-O-F neutron diffraction experiment was carried out to measure the structure factor over a wide momentum transfer range for the liquid state of metallic gallium and rubidium at several tem- peratures. Rubidium was chosen as a repre-

* The Research Institute for Iron , Steel and Other Metals, Tohoku University, Sendai 980, Japan. Fig. 1 Schematic diagram of the diffractometer.

Trans. JIM 1975 vol. 16 298 Kenji Suzuki, Masakatsu Misawa and Yoshiaki Fukushima

Stokes, 10atm. He-3, 1 inch diameter). The flight path length of incident neutron (L0) is 4.40m. Those of scattered neutron (L) are fixed at 0.44m for the scattering angle 2θ=15° and 0.42m for 2θ=30, 60 and 150°, respectively. The accelerator was operated at an electron acceleration energy of 250MeV with a peak beam current of 60mA and at a pulse duration of 3μsec with a pulse repetition frequency of 100pps. The liquid sample was sealed in vacuum in a silica tube of 0.30mm wall thickness and 10mm inner diameter. The electric furnace to heat the sample was specially designed, as shown in Fig. 2, for heating elements not to be located in the way of neutron beams. The sample tem- peratures were kept at 40,455 and 920℃ for liquid gallium, and 55 and 370℃ for liquid rubidium during the measurements. Incoherent scattering from vanadium was used as a standard for the calibration of the observed intensity. The detailed procedures of the corrections for absorption, multiple scattering, silica tube vessel and so on have been Fig. 3 Relative contributions from the sample with given in the previous papers(5)(7) as well as the vessel (S), empty vessel (Cell), vanadium rod (V), discussion of resolution and performance of and background (BG) to the observed intensity in this T-O-F neutron diffractometer. Figure 3 the case of liquid gallium at 40℃. shows the relative contributions from the sample with a vessel, an empty vessel, a vana- intensity in the case of liquid gallium measured dium rod and the background to the observed at the temperature of 40℃ with a scattering angle of 2θ=15°. The wavelength range of pulsed neutron used in this experiment is from 0.5 to 2.0Å, which covers the momentum

transfer range from 0.8 to 3.2Å-1 for 2θ=15°, from 1.7 to 6.6Å-1 for 2θ=30°, from 3.2 to 12.7Å-1 for 28=60° and from 6.0 to 33.6Å-1 for 2θ=150°.

Ⅲ. High Momentum Transfer Structure Factor

The structure factors S(Q) were obtained over a wide range of momentum transfer from

Q=1 to 30Å-1. However, the amplitudes of oscillation in S(Q) are so small in the momentum transfer range higher than about 20 Å-1 for liquid galiium and 10Å-1 for liquid rubidium that they are buried in the overall experimental error. The observed structure Fig. 2 Electric heating furnace. factors of liquid gallium ranged up to Q=20 Structure of Liquid Gallium and Rubidium by Pulsed Neutron Diffraction Using Electron Linac 299

ment is quite good in both the gallium and rubidium liquids near the melting point. The oscillation in S(Q) for liquid gallium is more pronounced compared with that of liquid rubidium. Rapid damping of oscillation in S(Q) for liquid rubidium comes from a large atomic diameter and a soft repulsive core in the pair potential(4). On the other hand, the strong oscillation in S(Q) for liquid gallium may be due to both hard repulsive core and persistence of an atomic association like a molecule in the liquid state. The existence of a Ga2-molecule in liquid gallium near the melting point has been often speculated, for example, by NMR studies(13), because the crystal structure of α-gallium is isomorphous with that of solid bromine which is composed from Br2-molecules as a con- stituting unit. If we assume that liquid gallium may be Fig. 4 Observed structure factors S(Q) of liquid regarded as a diatomic molecular liquid, the gallium. observed structure factor S(Q) can be divided into two parts as follows (14): (1) In the right hand side of eq. (1) the first term F1(Q) is the form factor for a single molecule which gives intramolecular structural information and the second term corresponds to inter- molecular information where F2(Q) is the orientational correlation function between molecules and Sc(Q) is the structure factor for molecular centers. In the high Q region, the observed structure factor S(Q) tends asymptotically to the form factor for a single molecule F1(Q), that is, (2)

because the correlation of molecular centers becomes lost. F1(Q) for a diatomic molecule is Fig. 5 Observed structure factors S(Q) of liquid written as rubidium. (3)

Å-1 are shown in Fig. 4, together with that of where r is the internuclear separation in a amorphous gallium by Ichikawa(8). Figure 5 single molecule. also shows the observed structure factors of The dashed line in Fig. 6 is the form factor liquid rubidium ranged up to 10Å-1. The F1(Q) calculated by eq. (3) for r=2.44Å agreement between the present results by the corresponding to the atomic separation in a linac experiment and other experimental gallium atom-pair existing in α-gallium crystal data(9)～(12) by the conventional reactor experi- (Fig. 7), while the dotted line in Fig. 6 is 300 Kenji Suzuki, Masakatsu Misawa and Yoshiaki Fukushima

than small compared with that of β-gallium crystal as a high pressure phase.

Ⅳ. Pair Correlation Function g(r)

The observed structure factor S(Q) is Fourier transformed to give the pair correlation function g(r) by the following expression:

(4) Fig. 6 Comparison of observed S(Q) (full line) with where ρ0 is the mean number density in the calculated form factor F1(Q) (dashed line for r= 2.44Å and dotted line for r=2.69Å respectively) liquid. The integration of eq. (4) should be in liquid gallium. made over all the range of Q from zero to infinite. However, the numerical calculation of calculated by using r=2.69Å which is the Fourier transformation eq. (4) has to be nearest neighbour atomic distance in an atom- truncated at a finite value of Qmaxbecause of a chaih in β-galHum crystal (Fig. 7). The position limited range of Q over which S(Q) can ex- of the subsidiary maximum on the high Q side perimentally be observed. A considerable of the first peak in S(Q) is found to coincide amount of error is known to be introduced into with the first peak of F1(Q) in either of the two g(r) when the integration of eq. (4) is truncated models. However, the phase of oscillation of at a low value of Q, even if a sufficiently ac- S(Q) for liquid gallium in the high Q region is curate S(Q) is available. Fortunately the in good agreement with that of F1(Q) for the T-O-F neutron diffraction method using the

β-phase crystal model. It is interesting to men- electron linac as a pulsed neutron source is very tion that the density of liquid gallium is larger effective in measuring the structure factor S(Q) than that of α-gallium crystal and close rather for liquids in high Q region and therefore makes

Fig. 7 Crystal structures and phase diagram of metallic gallium. Structure of Liquid Gallium and Rubidium by Pulsed Neutron Diffraction Using Electron Linac 301 it possible to explore experimentally the famous "truncation effect" in the numerical calculation of eq. (4). We used the sum-rule, putting r=0 in eq. (4),

(5) to find an optimum value of Qmaxas the upper limit of integration of eq. (4). The value of the left hand side of eq. (5) oscillates around the mean number density ρ0 and tends to it with increasing Qmax. The amplitude of the oscillation becomes, as shown in Fig. 8, less than 10 % of ρ0 and reaches the minimum at Qmax=17 -1

Å . All the pair correlation functions g(r) obtained here (Figs. 9 and 10) were calculated by truncating the Fourier integration of the corresponding S(Q) at the optimum values of Qmax,which were determined by the condition that the amplitude of oscillation of eq. (5) becomes the minimum at the position of Qmax. Figure 11 shows how the spurious ripples appearing in g(r) due to the truncation effect does deform the profile of g(r) in accordance with the choice of Qmaxvalue. Although the Fig. 9 Pair correlation functions g(r) of liquid gallium. position of the first peak maximum may be fixed When truncating at Qmax=8Å-1, the shape and height is not established until

Qmax>12Å-1. The spurious ripples are negligibly small in g(r) obtained by truncating at

Qmax=17Å-1. As shown in Fig. 9, a fine oscillation is

Fig. 8 Number density of liquid gallium at 40℃ calculated by sum-rule, eq. (5). ρE means the number Fig. 10 Pair correlation functions g(r) of liquid density obtained experimentally. rubidium. 302 Kenji Suzuki, Masakatsu Misawa and Yoshiaki Fukushima

validity of the static approximation(16). How- ever, the useful methods of correction for the static approximation have been proposed by Placzek(17) and Ascarelli and Caglioti(18). Ascarelli and Caglioti have demonstrated that the subsidiary maximum on the high Q side of the first peak for liquid gallium is pronounced in the quasi-elastic diffraction pattern. In our T-O-F neutron diffraction measurement the position of the subsidiary maximum just corresponds to the elastic scattering of neutron with wavelength λ=0.5Å, at scattering angle 2θ=15°, λ=1Å at 2θ=30° and λ=2Å at 2θ=60°, respectively. Therefore, the dependence of the apparent total scattering cross section on neutron energy could be explored simultaneously. The resolution width of Q at 2θ=15° counter system is wider than the range of Q where the Fig. 11 Truncation effect of Fourier transformation, subsidiary maximum appears. In the case of eq. (4), in the case of liquid gallium at 40℃. the 2θ=60° counter system the resolution width is reasonably fine but the energy integration superimposed on g(r) for liquid gallium at 40℃ in the range from the first minimum to the second peak maximum. Such an oscillation does not occur within the range smaller than atomic diameter, which means that this fine oscillation is not the spurious ghost due to truncating errors but may reflect an alternative configuration for local atomic ordering in liquid gallium near the melting point. A computer simulation(15) for a dense fluid of Lennard-Jones disks, in fact, shows that the similar additional structure appears in the range between the first minimum and the second peak maximum of g(r). In amorphous gallium this behavior becomes so pronounced that the second maximum of g(r) is completely divided into three parts as shown in Fig. 9 (by Ichikawa).

Ⅴ. Subsidiary Maximnm in S(Q) of Liquid Gallium

To get an accurate S(Q) from neutron scattering experiments, measurements have to be done in the condition that the static approximation does hold. Use of incident neutron with the Fig. 12 Subsidiary maximum on high Q side of first peak in S(Q) of liquid gallium of 40℃ at scattering energy of more than 100meV should be an angle 2θ=15° (… ○ …), 30° (-●-) and 60° empirically suitable choice to realize the (---△---). Structure of Liquid Gallium and Rubidium by Pulsed Neutron Diffraction Using Electron Linac 303 satisfies only partially the constant-Q condition. Hence the data by the 2θ=30° counter system was used as the profile of S(Q) around the subsidiary maximum. As shown in Fig. 12, the two curves corresponding to a higher neutron energy coincides well with each other, which is likely to guarantee the validity of the static approximation in the experimental condition of this work. In con- trast, the curve for λ=2Å neutron has an emphasized shoulder indicating the elastic scattering which may be predominant in this energy region. The mean number densities and atomic radii of liquid gallium and aluminum are known to be exactly the same near the melting point, respectively(19). Therefore, a comparison of the diffraction data for the liquid gallium and aluminum is quite instructive, because liquid aluminum has a really simple diffraction pattern without any subsidiary maxima. As shown in Fig. 13(a), the position of the first peak in g(r) for liquid gallium and rubidium coincides, while the subsequent peaks for

Fig. 14 Simulation on reproducing a subsidiary maximum in S(Q) due to modification of g(r) for liquid aluminum. Dashed lines are the S(Q) and g(r) observed experimentally for liquid aluminum.

liquid gallium is shifted to the larger r region compared with those of liquid aluminum. On the other hand, Fig. 13(b) shows that the high Q behavior of S(Q) closely resembles each other but a large difference comes from the different shapes of the first peak in S(Q) for liquid gallium and rubidium. Therefore, the subsidiary maximum in S(Q) for liquid gallium may be expected to be due to the shift of the second peak and the subsequent ones in the g(r) to larger r region. In fact, the subsidiary maximum can be artificially re- produced, as shown in Fig. 14, by modifying the g(r) for liquid aluminum in the fashion described above. Figure 15 shows that such a behavior of g(r) may generally be found in the other liquid metals having the asymmetric first peak or the Fig. 13 (a) Pair correlation functions g(r) of liquid aluminum and gallium. (b) Structure factors subsidiary maximum on the high Q side of the S(Q) of liquid aluminum and gallium. first peak in S(Q). Further investigation based 304 Kenji Suzuki, Masakatsu Misawa and Yoshiaki Fukushima

REFERENCES (1) M. Misawa, Y. Fukushima, K. Suzuki and S. Takeuchi: Physics Letters A, 45A (1973), 273. (2) J. G. Powles: Molecular Physics, 26 (1973), 1325. (3) S. Takeuchi, K. Suzuki, F. Itoh, K. Kai, M. Misawa and K. Murakami: The Properties of Liquid Metals (Proc. 2nd Intern. Conf.), Ed. by S. Takeuchi, Taylor and Francis Ltd., London, (1973), p. 69. (4) Y. Waseda and K. Suzuki: ibid., p. 37. (5) M. Misawa, K. Kai, K. Suzuki and S. Takeuchi: Res. Reports Lab. Nucl. Sci., Tohoku Univ., 5 [2](1972), 73. (6) M. Kimura, T. Kubota, N. Niimura, M. Oyamada, A. Ozora, K. Sato, F. Takahashi, S. Takeda, S. Tomiyoshi, Y. Yamada and N. Watanabe: Proceedings of the Joint Meeting on Pulsed Neutrons and Their Utilization, Euratom- Japan Atomic Energy Society, Ispra, Sept., 17- 18, (1971), (1973), p. 75. (7) M. Misawa, Y. Fukushima, K. Suzuki and S. Takeuchi: Res. Reports Nucl. Sci., Tohoku Univ., 6[1] (1973),88. (8) T. Ichikawa: Phys. Status Solidi, (a), 19 (1973), Fig. 15 Comparison of pair correlation functions 347. g(r) between liquid metals with and without sub- (9) A. H. Narten: J. Chem. Phys., 56 (1972), 1185. sidiary maximum in S(Q). r1 is the position of main (10) D. I. Page, D. H. Sanderson and C. G. Windsor: peak in g(r). J. Phys. C, 6 (1973), 212. (11) N. S. Gingrich and L. Heaton: J. Chem. Phys., 34 (1961), 873. on more sophisticated triplet and higher cor- (12) J. E. Enderby, The Properties of Liquid Metals relation functions is necessary to understand (Proc. 2nd Intern. Conf.), Ed. by S. Takeuchi, the total behavior of the subsidiary maximum Taylor and Francis Ltd., London, (1973), p. 3. in anomalous structure factors for some kinds (13) K. Suzuki and O. Uemura: J. Phys. Chem. Solids, 32 (1971), 180. of liquid metals. (14) P. A. Egelstaff, D. I. Page and J. G. Powles: Molecular Phys., 20 (1971), 881. Acknowledgments (15) P. L. Fehder, J. Chem. Phys., 52 (1970), 791. The authors would like to thank Professor (16) J. E. Enderby: Physics of Simple Liquids, Ed. by S. Takeuchi for his continuous encouragement H. N. V. Temperley et al., North-Holland, (1968), and the members of Laboratory of Nuclear p. 611. (17) G. Placzek: Phys. Rev., 86 (1952), 377. Science, Tohoku University for the operation of (18) P. Ascarelli and Caglioti: Nuovo Cimento, 43B the electron linac. (1966), 375. (19) Y. Waseda and K. Suzuki: Sci. Rep. RITU, A-24 (1973), 139.