PRECISION METHOD FOR MEASURING THE ENERGY OF $\gamma$-RAYS

In recent years DuMond and co-workers$^{1—4}$ have developed a new modification of the diffraction method for measuring the energy of $\gamma$-rays.

As is known, the study of the diffraction of $\gamma$-rays by crystals is, historically, one of the first methods of $\gamma$-spectroscopy. However, this method rather soon ceased to be used, since it required the use of very intense narrow beams of $\gamma$-rays; moreover, as the energy of the $\gamma$-quanta increased, in order to obtain satisfactory resolving power it was necessary to reduce sharply the luminosity of the apparatus. This circumstance is easy to understand if one bears in mind that, in the diffraction analysis of $\gamma$-radiation by the rotating-crystal method, the Bragg reflection condition must be satisfied:

\[ 2d \sin \vartheta = n\lambda, \tag{1} \]

where $d$ is the crystal-lattice constant, $\lambda$ is the wavelength of the $\gamma$-radiation, $\vartheta$ is the glancing angle, and $n$ is the order of the maximum (usually one works with the first order, i.e. $n = 1$). If $\Delta\lambda$ is the smallest difference between the wavelengths of two $\gamma$-lines that are still distinctly resolved by the instrument, then from (1) it follows that, for small $\vartheta$,

\[ \frac{\Delta\lambda}{\lambda} \sim \frac{\Delta\vartheta}{\vartheta}. \]

Since with increasing energy of the $\gamma$-rays $\lambda$ decreases, the angle $\vartheta$ corresponding to $n = 1$ will also decrease, and in order to preserve constancy one has to reduce the beam aperture $\Delta\vartheta$. For example, for $d \sim 1 \text{ Å}$, $\lambda \sim 10^{-2}\text{ Å}$ ($h\nu \sim 10^6 \text{ eV}$), $\vartheta$ will be $\sim 2^\circ$, and to achieve

\[ \frac{\Delta\lambda}{\lambda} \sim 1\% \]

$\Delta\vartheta$ must be only $0.02^\circ$.

The rotating-crystal method for the analysis of $\gamma$-radiation was first used by Rutherford and Andrade (1914). Subsequently, by a number of investigators—

method was improved, and the best result was considered to be the measurement of the wavelength of one of the $\gamma$-lines of Ra $(B+C)$, $\lambda=1.6\cdot10^{-2}\ \text{\AA}$ ($h\nu=770\ \mathrm{KeV}$), with an accuracy of up to 7%. A new instrument, constructed by DuMond and collaborators, makes it possible to measure wavelengths down to $7\cdot10^{-5}\ \text{\AA}$ ($h\nu=1.75\times10^{8}\ \mathrm{eV}$), and moreover with an accuracy 100 times exceeding the accuracy of the best measurements by the rotating-crystal method. Such accuracy surpasses that of all methods known to date for determining $\gamma$-ray energies, including methods associated with the use of magnetic $\beta$-spectrographs (for example, by measuring the energy of conversion electrons). The table gives the results of measurements by DuMond and collaborators of the energies of the $\gamma$-lines of Xe$^{131}$, formed as a result of the $\beta$-decay of J$^{131}$ (half-life 8.0 days). A comparison of the results of DuMond et al. with the data of the best $\beta$-spectrographic works of Metzger and Deutsch$^{5}$ and Owen, Moe, and Cook$^{6}$ shows that the accuracy attained by these authors is 20–30 times lower than the accuracy of the measurements of DuMond and collaborators. In general, the greatest accuracy in measuring the energies of $\gamma$-lines with the aid of modern magnetic $\beta$-spectrographs does not exceed 0.1%, whereas the diffraction measurements of DuMond and collaborators give errors of the order of 0.01%. From the data presented it is clear that we are dealing with a precision method of $\gamma$-spectroscopy.

Energies of the $\gamma$-lines of Xe$^{131}$

The basic scheme of the new instrument is shown in the figure. The crystal is bent along an arc of a circle of radius $R$, and the extensions of the lines forming the atomic planes converge at $S_0$. The source of $\gamma$-rays is placed at $S$, and the diffracted rays occupy the region $S'$. Thus the spectrometer operates in “transmitted light.” In studying hard X-rays, when the radiation source is the anticathode of a tube, it is convenient to place it at $S'$ and to register the diffracted rays at $S$. Since the angles $\delta_1$, $\delta_2$, $\delta_3$, etc. are equal to one another (as corresponding angles resting on the same arc), it is obvious that the finite width of the beam of $\gamma$-rays emerging from $S$ will in no way affect the width of the diffraction maximum. The diffracted rays, after passing through a lead collimator, are registered by a counter. For a given $\gamma$-ray energy and a fixed position of the collimator together with the counter, the Bragg condition will be fulfilled only for a certain position of the radiation source on the circumference. The maximum of the counter readings for $\gamma$-radiation of different wavelengths will therefore be observed at different positions of the source $S$. The possibility of applying the electrical method of recording diffracted $\gamma$-radiation in the described instru-

in the instrument is due to its considerable dispersion and in itself constitutes an advantage in comparison with the photographic method previously used, which required the use of much more intense beams or the carrying out of long exposures.

The calculation shows that the resolving power of the spectrograph is connected exclusively with errors in the manufacture of the crystal, namely with the deviation of the surfaces bounding it from an arc of a circle. If this deviation is small, then the relation holds

\[ \frac{\Delta\lambda}{\lambda}\simeq \frac{\cos\vartheta(1-\cos\vartheta)} {\cos(\alpha+\vartheta)}, \tag{2} \]

where \(\alpha\) is the “aperture” angle (see the drawing). In the case of hard \(\gamma\)-rays \(\vartheta<\alpha\ll 1\), and (2) becomes

\[ \frac{\Delta\lambda}{\lambda}\simeq \frac{1}{2}\alpha^2. \tag{3} \]

Thus, the resolving power of the spectrograph in the region of sufficiently short wavelengths does not depend on the wavelength, in contrast to the situation in many spectrometers based on the principle of a rotating crystal.

The concrete data of the instrument constructed by DuMond and his collaborators are as follows: \(R=2\) m, crystal dimensions \(80\) mm \(\times 70\) mm \(\times 1\) mm, \(\alpha\simeq 0.02\) radian, dispersion at short wavelengths \(1.186\cdot 10^{-3}\) Å/mm.

In one of the works cited above, a detailed description is given of the technology for manufacturing bent crystals.

I. S. Shapiro

CITED LITERATURE

1. J. W. M. Du Mond, Rev. Sci. Instr. 18, 626 (1947).
2. J. W. M. Du Mond, D. A. Lind, E. R. Cohen, Rev. Sci. Instr. 18, 617 (1947).
3. J. W. M. Du Mond, D. A. Lind, Watson, Phys. Rev. 73, 1392 (1948).
4. J. W. M. Du Mond, D. Lind, J. Brown, D. Klein, D. Müller, Phys. Rev. 75, No. 10 (1949).
5. F. Metzger, M. Deutsch, Phys. Rev. 74, 1640 (1948).
6. G. E. Owen, D. Moe, C. S. Cook, Phys. Rev. 74, 1879 (1948).