ON THE TRANSITION IN STRUCTURAL AND SPECTROSCOPIC STUDIES FROM kX UNITS TO Å UNITS

I

At the basis of structural studies lies, as is known, the Wulff–Bragg equation

\[ n\lambda = 2d_{hkl}\sin\vartheta . \tag{1} \]

In calculations using this equation one employs the wavelengths of X-rays and, consequently, obtains values of interplanar distances expressed in special radiographic or spectroscopic units—“X units.”

The X unit was at one time chosen equal to \(1/3.02904 \times 10^3\) of the value of the interplanar distance between the cleavage planes of calcite \(d_{100}\), which was taken to be \(3.02904\ \text{Å}\) at \(18^\circ\text{C}\), as the average of three measurements published by Siegbahn in 1919[^1] (\(3.02903,\ 3.02907\), and \(3.02902\)), i.e., 30 years ago.

Thus \(1\ \mathrm{X}\) was taken to be equal to \(10^{-11}\ \text{cm}\), and \(1000\ \mathrm{X}\), or 1 “kilo-X” (\(1\ \mathrm{kX}\)), was taken to be equal to \(10^{-8}\ \text{cm}\), i.e., \(1\ \text{Å}\).

Therefore, throughout the structural and roentgenographic literature up to 1945 and even up to September 1947, 1 kX was called 1 Å.

In recent years the atomic constants have undergone a fundamental revision, including Avogadro’s number ($6.023\cdot 10^{23}$ instead of $6.06\times 10^{23}$), for example according to the summary of DuMond and Cohen[^2] (January, 1948), based on Birge’s work.

Already by 1945 it had become finally evident that 1 kX is not exactly equal to 1 Å, and that, at the present level of our knowledge, to convert 1 kX into 1 Å one should use a conversion coefficient, which we shall denote by $\gamma$. In 1945 $\gamma$ was to be taken as equal to $1.00203_4$.

This discrepancy in the magnitude of 1 kX and 1 Å leads us to the necessity of introducing corrections into the wavelength values of characteristic X-ray radiation. For example, $\lambda$ for Cu $K_{\alpha_1}$ should be taken as not $1.537395$ Å, as has been done up to now and as formed the basis of the corresponding calculations of interplanar distances by the Wulff–Bragg formula, but $1.537395$ kX, or $1.537395\cdot\gamma$ Å. Meanwhile, for several years the question of the final value of the conversion coefficient officially adopted by roentgenographers remained open.

In September 1947 the American Society of X-ray and Electron Diffraction published a note by E. Armstrong Wood[^3] and a report by Bragg that, as a result of consultations of English roentgenographers with the American Society of X-ray Diffraction, as well as with Siegbahn, the highest coefficient $\gamma$ is henceforth accepted as equal to $1.00202\pm 0.003\%$. In the same report by Bragg a table of changed wavelengths is given, with the note that henceforth in all roentgenographic works the wavelengths adopted for calculation should be specified exactly. We reproduce this table below. (The wavelengths are given in Å.)

As Bragg indicates, $K_{\alpha}$ represents the mean value of $K_{\alpha_1}$ and $K_{\alpha_2}$*).

*) In calculating $K_{\alpha}$ it was assumed that the weight of $K_{\alpha_1}$ is twice as large as that of $K_{\alpha_2}$; for example, for Cu

\[ K_{\alpha}=\frac{2\cdot 1.54050+1.54434}{3}=1.5418. \]

From what has been set forth above it follows that, since from 1912 to 1945 and even up to 1947 the units kX were erroneously equated in the literature with units \(\mathring{\mathrm A}\), the values of interplanar and interatomic distances published in articles and in Strukturberichte should be regarded as expressed not in \(\mathring{\mathrm A}\), but in kX, and for an exact conversion to \(\mathring{\mathrm A}\) one should use the conversion coefficient \(\gamma\), introducing a correction of about \(0.2\%\).

II

As is known, for calculating the X-ray density of a substance one uses the equation

\[ \sigma_x=\frac{\Sigma A}{Nv}\ \text{g/cm}^3, \tag{2} \]

where \(\Sigma A\) is the sum of the atomic weights of all atoms entering into the unit cell, \(N\) is Avogadro’s number, and \(v\) is the volume of the unit cell, expressed in cu. cm. In the case of identical atoms

\[ \sigma_x=\frac{nA}{Nv}\ \text{g/cm}^3, \tag{3} \]

where \(n\) is the number of atoms in the unit cell. Substituting the modern value of Avogadro’s number, we obtain the simple relation

\[ \sigma_x=\frac{\Sigma A}{6.023\cdot 10^{23}\cdot v\cdot 10^{-24}}. \tag{4} \]

Whence

\[ \sigma_x=1.66020\,\Sigma A/v\ \text{g/cm}, \tag{5} \]

where \(v\) is the volume of the unit cell, expressed in \(\mathring{\mathrm A}^3\). This final formula, without derivation, is given in Bragg’s article.

It is evident that, in order to convert the volume of the unit cell expressed in \((\mathrm{kX})^3\) into \(\mathring{\mathrm A}^3\), the volume in kiloxes must be multiplied by the conversion factor, which we shall denote by \(\omega\).

\[ \omega=(1.00202)^3=1.00607. \tag{6} \]

Since the majority of structural data available in the literature, in accordance with what was set forth above, are expressed in kiloxes, it seems useful to us to use, instead of formula (5), the formula

\[ \sigma_x=1.65018\,\Sigma A/v\ \text{g/cm}^3, \tag{7} \]

where \(v\) is the volume of the unit cell, expressed in \(\mathrm{kX}^3\).

This equation is obtained from (5) by dividing the numerator and denominator of the fraction by the conversion coefficient \(\omega\).

B. F. Ormont

CITED LITERATURE

1. Siegbahn, Phil. Mag. 37, 611 (1919); ann. d. phys. 59, 56 (1919).
2. Dumond a. Cohen, Reviews of Mod. Phys. 20, 82 (1948).
3. E. Armstrong Wood, Phys. Rev. 72, 436 (1947).
4. W. L. Bragg, Phys. Rev. 72, 437 (1947).