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X-RAY DIFFRACTION BY PARTICLES OF 0.3 MICRON SIZE
Specific effects of X-ray scattering at small angles (up to \(2^\circ\)) have been discussed in detail in this journal[^1]. Calculation and experiment have shown that near the primary beam there is a distribution of the intensity of the scattered radiation similar to that which occurs in the diffraction of light by an aperture, if only the scattering system consists of individual particles ranging in size from several tens to several hundreds of angstroms. With the exception of one case[^2], investigators have found only a broadening of the primary beam, but not a diffraction ring.
The figure shows a pattern, obtained in a recently performed study[^3], of X-ray scattering at small angles. In the photograph
distinct diffraction rings of very high orders are clearly visible (on top down—exposures of 10, 30, and 20 h).
The scattering object is latex (Dow latex 580G), used in electron microscopy as a magnification standard.^4 The use of latex for this purpose is due to the exceptional uniformity of its sizes and to the spherical shape of its particles.
The diffraction patterns shown in the figure were obtained for the purpose of an objective determination of the size of the latex particles. It appears that both the phenomenon itself and its application are described in the cited work for the first time.
The wide use of measurements of sizes with the aid of the electron microscope makes the exact determination of the diameter of the spherical particles adopted as a standard very important. Many works have been devoted to measuring the sizes of latex particles by various methods. At the same time, the figures obtained are not in agreement, but fluctuate between 2500 and 3100 Å.
In the work being reviewed, the particle diameter was established with a probable error of less than ±1%; it was found to be 2780 Å.
The scattering theory for the case of interest to us gives the following expression for the scattering intensity as a function of the scattering angle:
\[ I = MN^2\Phi^2(u)\left\{1+P\left\{5\,\frac{\sin 2u}{2u}-6\Phi(2u)\right\}\right\}, \]
where \(I\) is the intensity, \(M\) is the number of particles in the specimen, \(N\) is the number of electrons in a particle,
\[ \Phi(u)=\frac{3}{u^3}(\sin u-u\cos u),\qquad u=\pi\varepsilon\frac{D}{\lambda}; \]
\(\varepsilon\) is the scattering angle, \(D\) is the diameter of the sphere, \(\lambda\) is the wavelength of the x-ray beam, and \(P\) is the packing coefficient of the spherical particles. The intensity curve consists of a large number of maxima close to one another, the distance between which must be of the order of four units in the value of \(u\), which corresponds to approximately two or three angular minutes.
In order to resolve such closely spaced diffraction rings, the experiment was carried out with the softest of the available radiations (\(K_{\alpha}\)-line of chromium, \(\lambda=2.286\) Å, monochromatization by \(V_2O_5\)) in a vacuum camera 2 m long. The scattering specimen was placed approximately in the middle of the camera.
The diffraction rings were observed for values of \(u\) from 15 to 38. From the formula given above, the abscissae of the maxima \(u\) were calculated; from the x-ray photographs the values of \(\varepsilon\) for these maxima were determined, and then from each maximum the value
\[ D=\frac{\lambda u}{\pi\varepsilon}. \]
was calculated. The scatter of the values did not exceed ±0.6%.
The success of the described experiment in resolving diffraction rings and in accurately determining particle size shows that a new method has been found, suitable for determining the sizes of viruses and other homogeneous particles and molecules with diameters of less than 0.3 micron.
A. K.
References
- E. A. Porai-Koshits, UFN 39, 573 (1949).
- K. L. Yudowitch, J. Appl. Phys. 20, 174 (1949).
- K. L. Yudowitch, J. Appl. Phys. 22, 214 (1951).
- See, for example, UFN 39, 142 (1949).