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NEUTRONOGRAPHY OF LIQUIDS
Using neutron diffraction, Chamberlain studied [1] the structure of liquid metals (lead and bismuth) and liquid sulfur. He used a special optical method based on the fact that the differential cross section for neutron scattering by atoms is a function of the scattering parameter \(\dfrac{\sin \Theta}{\lambda}\), where \(\Theta\) is the scattering angle. Consequently, in determining this function (and this is the aim of the experiment) one may use not monochromatic neutron beams, but beams with a certain range of wavelengths, though with a definite value of \(\dfrac{\sin \Theta}{\lambda}\); this sharply increases the luminosity of the method. The diffraction apparatus shown in Fig. 1 differs from the usual one in that the incident beam is not parallel but homocentric, and the reflecting atomic planes of the monochromator crystal are perpendicular to the outer face (the flat crystal works in transmission).
The monochromator selects from the incident neutron beam with definite parameters and focuses them on the central slit, which makes it possible, in the main, to get rid of the background of incoherent scattering.
The collimator slit, the monochromator crystal, and the central slit remain in fixed positions during operation. The scattering specimen can rotate about a vertical axis and move along the central beam. In this way the parameters \(\varphi\) and \(\gamma\) indicated in Fig. 1 are varied.
For neutrons passing along the central beam, the Bragg–Wulff condition is valid:
\[ \lambda = 2d \sin \Theta_0 \]
(for \(n = 1\)) and, consequently,
\[ \frac{\sin \Theta}{\lambda} = \frac{\sin \Theta}{2d \sin \Theta_0}, \]
where \(\lambda\) is the wavelength of the incident neutrons, \(d\) is the interplanar spacing of the crystal, and the angles \(\Theta\) and \(\Theta_0\) are indicated in the drawing.
For neutrons not passing along the central beam, the parameters \(\varphi\) and \(\gamma\) must be chosen so that for them as well \(\dfrac{\sin \Theta}{\lambda}\) has the same ...
same value. The choice of \(\varphi\) and \(\gamma\) can be made analytically from the conditions
\[ \left(\frac{dx}{d\alpha}\right)_{\alpha=0}=0;\qquad \left(\frac{d^2x}{d\alpha^2}\right)_{\alpha=0}=0,\quad \text{where } x=\frac{\sin\Theta}{\lambda}. \]
However, the analytical form of these equations is very complicated and solving them is very laborious. Therefore the solution was carried out graphically.
The collimator slit was installed inside the boiler lining in a graphite column. The monochromator crystal consisted of 4 natural crystals of \(\mathrm{CaF}_2\), mounted on a support with individual adjustment. The working surface of the crystal was \(150 \times 150\) mm, and its thickness was 6 mm. The crystals were ground along a natural cubic face and were set to reflect from the plane (200). The monochromator crystal was set so that \(\Theta_0 = 15^\circ\). This is the largest angle for which reflection of the second order can be neglected.
The scattering sample and the neutron counter are mounted on a carriage, which can move along the central beam, thereby setting the value \(\gamma\) (the distance from the sample to the counter is constant). In turn, the counter rotates about the scattering sample. The scattering liquid is placed in a thin flat chamber with thin (1 mm) aluminum walls. The chamber can also rotate about a vertical axis. Such a device makes it possible (by selecting the position of the carriage and the angle \(\varphi\)) to achieve constancy of the factor \(\dfrac{\sin\Theta}{\lambda}\) for the entire beam of scattered neutrons. In the course of measurement errors could arise as a result of scattering by air and by the walls of the chamber, and also as a result of multiple scattering (the latter was always corrected for). In addition, for different \(\varphi\) the angle of scattering is not the same. Therefore the measurements were made with: 1) the shutter closed (the background of fast neutrons and cosmic radiation was measured); 2) the shutter open without the chamber (scattering by air was measured);
Fig. 1. Apparatus for a neutronographic study of liquids.
Labels in the figure: collimator slit; graphite column; boiler lining; monochromator crystal; shutter; cadmium protective screen; central slit; scattering sample; movable carriage; counter; rails; \(\dfrac{p}{p}=\gamma\).
3) with an empty chamber (the scattering by the walls of the chamber was measured); 4) with a chamber filled with the liquid under investigation (the scattering of the sample was measured), and, finally, 5) with a piece of Plexiglas inserted in place of the chamber (first, to monitor the intensity of the neutron beam, which could vary with time; second, to take account of the correction for different solid angles; and third, to monitor the efficiency of the counter).
As a result of such measurements, dependences of the differential scattering cross section (per 1 atom in unit solid angle) \(\frac{d\sigma}{d\omega}\) on \(\frac{\sin \theta}{\lambda}\) were obtained for liquid sulfur, lead, and bismuth (the curves for sulfur and lead are given in Figs. 2 and 3). The vertical scale was obtained from the normalization condition for \(\frac{d\sigma}{d\omega}\). The measurements were carried out twice (solid and dashed lines).
Fig. 2. Neutronogram of liquid sulfur. Below, for comparison, is shown the X-ray pattern of liquid sulfur (without observing the vertical scale).
Fig. 3. Neutronogram of liquid lead.
As was to be expected, the curves obtained are analogous to those known for X-rays (without allowance for the atomic factor). The background of incoherent scattering was determined by measuring the scattering intensity at small angles, since, as a result of intermolecular interference, coherent scattering is absent at small angles. This background was then subtracted from the total intensity at all angles, since the incoherent scattering of neutrons is almost isotropic. (For sulfur this operation could not be performed, for, owing to the small radius of the sulfur atoms, the first maximum of coherent scattering lies in the region of small angles.)
The equation connecting the distribution of the atomic density \(\rho(R)\) in a liquid at various distances \(R\) from a fixed atom with the differential scattering cross section \(\frac{d\sigma}{d\omega}\), obtained
Chernik and Prins² for X-rays can easily be transformed for the case of neutron scattering. In final form it is as follows:
\[ R[\rho(R)-\rho_0]=8\int_{0}^{\infty} dx\,x \left(\frac{4\pi}{\sigma_s}\cdot\frac{d\sigma}{d\omega}-1\right) \sin(4\pi xR), \]
where \(\sigma_s\) is the scattering cross section of one bound atom, \(\rho_0\) is the mean atomic density, and \(x=\dfrac{\sin\Theta}{\lambda}\).
This formula is derived on the assumption that the atoms of the liquid are identical and that the spin of the neutrons does not affect the scattering process.
The results of calculating the atomic density for liquid lead are shown in Fig. 4; for comparison, the curve of the mean density is also given. From this and analogous curves the data for liquid sulfur, lead, and bismuth shown in the table were obtained. Some data for these elements obtained in their X-ray investigation in the solid state are also given there.
Fig. 4. Atomic density in liquid lead at definite distances from a fixed atom.
In lead the nearest neighbors (12 in all) are not sharply separated from the remaining atoms. Glocker and Hendus³, in studying the diffraction of X-rays in liquid lead, found the presence of 8 nearest neighbors and 4 atoms at somewhat larger distances. A complete solution of this question can be obtained by the method described, by studying diffraction at large values of \(\dfrac{\sin\Theta}{\lambda}\). In other respects the agreement with Glocker and Hendus is very close.
Sauerwald and Teske⁴ assumed that metals having a dense packing in the solid state retain it upon melting. It should be expected that the volume during melting changes insignificantly. The results obtained with a specimen of liquid lead confirm this assumption. This assumption is also supported by the fact that the diffraction maxima of the neutronogram of liquid lead are located at the same values of \(\dfrac{\sin\Theta}{\lambda}\) that can be calculated theoretically from the structure of solid lead. On this basis it is possible to index the neutronogram.
The preservation of the positions of the maxima of coherent scattering, the preservation of the minimum distance between atoms, and the preservation of the coordination number of sulfur upon its melting permit one to suppose that the atoms tend to form groups with a configuration analogous to the structure of solid sulfur. The diffraction maxima of liquid sulfur are also located at those positions that can be found for the solid state. The data for sulfur are in complete agreement
FROM CURRENT LITERATURE
with the work of Gingrich^5, who studied hard and liquid sulfur by X-ray diffraction.
In bismuth it is more difficult to count the number of nearest neighbors, since they cannot be well separated from atoms at greater distances. However, it is evident that at a distance of 3.1 Å there are 8 atoms. On melting, the coordination number of bismuth increases. This means that its structure becomes denser. This can explain the decrease in the volume of bismuth upon melting. Bismuth shows no tendency to form crystal-like groups (as sulfur does).
Table
| Element | Crystal lattice in the solid state | Number of nearest neighbors in liquid state | Number of nearest neighbors in solid state | Distances between nearest neighbors in liquid state, Å | Distances between nearest neighbors in solid state, Å | Distances with excess atomic density, $\rho>\rho_0$, Å | Distances with deficient atomic density, $\rho<\rho_0$, Å |
|---|---|---|---|---|---|---|---|
| Sulfur | Rhombic | 2 | 2* | 2.1 | 2.12* | 2.1 4 5 |
2.6 6 |
| Lead | Face-centered cubic | 12 $\pm 1$ |
12* | 3.4 | 3.49* | 3.4 6.4 |
4.7 8 |
| Bismuth | Rhombohedral | 8 | 3* | 3.1 | 3.10* | 3.1 6.7 |
5.3 7.7 |
Note. Figures marked with an asterisk are taken from other sources.
From the curves it is also evident that at distances exceeding 6–8 atomic distances all $R$ are equiprobable. Thus, it may be asserted that in liquid sulfur, lead, and bismuth an ordered arrangement of atoms is preserved in small regions.
The works described have shown the possibility of applying neutronography to the study of the structure of liquid and gaseous samples. As to the merits of neutronography, the following must be said. Different elements have different ratios of the cross sections of coherent and total scattering $\left(\dfrac{\sigma_{\text{coh.}}}{\sigma_{\text{tot.}}}=\delta\right)$. Neutronography is successfully applicable in the case of large $\delta$ (for example, deuterium has $\delta=0.57$). In this case neutronography gives a much clearer picture than X-ray structural analysis (see Fig. 2). The situation is worse for small $\delta$ (for hydrogen, for example, $\delta=0.025$), since diffuse scattering masks the coherent scattering. Further, some heavy elements
have a small transverse cross section for neutron absorption, then, since the coefficient of absorption of X-rays is very large for them. Therefore, in solving problems of structural analysis of both solid and liquid and gaseous substances, a reasonable combination of both methods is necessary.
R. Ozerov
REFERENCES
- O. Chamberlain, Phys. Rev. 77, 305 (1950).
- F. Zernike, J. Prins, Zeits. f. Physik 41, 184 (1927).
- R. Clocker, H. Hendus, Ann. d. Physik 43, 513 (1943).
- F. Sauerwald, W. Teske, Zeits. anorg. und. alg. Chemie 210, 247 (1933).
- N. S. Cingrich, J. Chem. Phys. 8, 29 (1940).