CRYSTALLOGRAPHY

I. I. Yamzin, Yu. Z. Nozik, and Academician N. V. Belov

NEUTRON-DIFFRACTION STUDY OF THE CUBIC MODIFICATION OF PbF₂

The cubic (high-temperature) modification of PbF₂ is assigned to the structural type CaF₂ (¹). However, an X-ray diffraction study is difficult in this case because of the large difference between the atomic amplitudes of fluorine and lead \((Z_\mathrm{F}=9,\ Z_\mathrm{Pb}=82)\), and this conclusion is based mainly on crystallochemical analogies. In neutron diffraction the situation is more favorable, since the amplitudes of coherent nuclear scattering are \(b_\mathrm{F}=+0.55\cdot 10^{-12}\ \text{cm}\) and \(b_\mathrm{Pb}=+0.96\cdot 10^{-12}\ \text{cm}\) (²).

We carried out a neutron-diffraction study of cubic PbF₂, single crystals of which were grown in the laboratory of optical single crystals of the Institute of Crystallography, Academy of Sciences of the USSR.

Using a neutron diffractometer for the study of single crystals (³) and applying the method described in (⁴), we obtained, in all, 32 reflections from planes of the zones \([100]\) and \([110]\). The diffraction pattern was recorded automatically on the chart tape of an electronic potentiometer during continuous slow rotation of the specimen and of the neutron detector (with preservation of the Bragg condition) in angular intervals encompassing the calculated positions of the diffraction maxima.

Table 1

Measured and calculated (without allowance for the temperature factor) intensities of reflections

To determine the diffraction intensities, the areas lying under the recorded count-rate curve were measured, with subtraction of the background level. The intensities measured in this way were reduced to the intensity of the primary neutron beam, which was monitored by a monitor...

counter. We estimate the relative error in measuring the intensities to be \(\pm 5\%\) \(^{(4)}\).

Table 1 gives the measured intensities and the values of \(L\Phi\) and \(L\Phi^2\) (reduced to an absolute scale \(^{(5)}\)), calculated under the assumption of the \(\mathrm{CaF}_2\) structure with the coordinates of the basis atoms \(\mathrm{Pb}(0,0,0)\), \(\mathrm{F}_1(1/4,1/4,1/4)\), \(\mathrm{F}_2(3/4,3/4,3/4)\). Here \(L=\dfrac{1}{\sin^2\vartheta}\) is the Lorentz factor, and the structure amplitude \(\Phi_{hkl}\) for the given space group \(Fm3m\) has the form

\[ \Phi_{hkl} = \begin{cases} f_{\mathrm{Pb}} + 2f_{\mathrm{F}} & \text{for } h+k+l = 4n,\\ f_{\mathrm{Pb}} & \text{for } h+k+l = 2n+1,\\ f_{\mathrm{Pb}} - 2f_{\mathrm{F}} & \text{for } h+k+l = 4n+2. \end{cases} \]

From consideration of these results it may be concluded that cubic \(\mathrm{PbF}_2\) does indeed have a fluorite-type structure. The close agreement of the experimental intensity values with the quantities \(L\Phi\) indicates the predominantly dynamic character of the scattering, as was to be expected for the \(\mathrm{PbF}_2\) specimen used in the study (a sphere, \(d=8\) mm). Indeed, the discrepancy coefficient calculated on the assumption of a dynamic character of the scattering proves to be \(20.3\%\) (without allowance for the temperature correction!). If, however, a kinematic character of the scattering is assumed, the corresponding coefficient exceeds the above value by more than a factor of 2.

It is well known that X-ray analysis of crystal structures containing elements whose atomic numbers differ by approximately a factor of 10 is very difficult. The use of the neutron-diffraction method eliminates the difficulties associated with the poor detectability of a light atom in the presence of a heavy one. The contribution of F atoms to the intensities of reflections with a sum of indices that is a multiple of \(4n\) even exceeds the contribution of the much heavier Pb atoms.

Institute of Crystallography
Academy of Sciences of the USSR

Received
20 I 1961

CITED LITERATURE

1. Strukturber., 1, 1931, pp. 148–150.
2. G. Bacon, Neutron Diffraction, Foreign Literature Publishing House, 1957, p. 34.
3. I. I. Yamzin, Crystallography, 4, 3, 423 (1959).
4. I. I. Yamzin, Yu. Z. Nozik, Crystallography, 6, 3 (1961).
5. M. A. Porai-Koshits, Practical Course in X-ray Structural Analysis, Moscow State University Press, part 2, 1960, p. 166.