K. S. Shifrin. Light Scattering in a Turbid Medium. State Publishing House of Technical-Theoretical Literature, Moscow–Leningrad, 1951, 288 pp., print run 4000 copies. Price 10 rubles 85 kopecks.

The optics of turbid media has long since become an extensive and independent branch of physics. Interest in it has grown especially in recent years, when a whole series of practically important problems—such as radiolocation, questions of visibility, the thermal balance of the atmosphere, the physics of colloids, and many others—have brought to the fore the need to know the laws of scattering and propagation of light in turbid media. Meanwhile, despite the abundance of works devoted to questions of light scattering, a number not only of secondary but also of fundamental points still remain unexplained, either theoretically or experimentally. The explanation for this should be sought in the exceptional complexity and diversity both of the phenomenon of light scattering itself and of the mathematical and experimental methods that must be brought to bear in studying it. This confusion and the underdevelopment of a number of basic problems has manifested itself, in particular, in the almost complete absence of monographic literature, as a result of which problems in the optics of turbid media have remained in large measure inaccessible to broad circles of physicists. This is especially true of the problem of scattering by large inhomogeneities (particles), on which monographic literature was altogether lacking.

The book under review is intended precisely to fill this gap. It should at once be noted, however, that the content of the book is considerably narrower than its title. The author confines himself exclusively to questions of scattering (and, in part, absorption) of electromagnetic waves by extraneous inclusions (particles). Questions of molecular (fluctuation) scattering (including critical opalescence) and many other matters lie outside the scope of the book. A series of questions of light propagation in turbid media—multiple scattering, the transfer equation, and so on—also remain outside it.

Having declined to consider all these diverse questions and having limited himself to the relatively narrow field of the elementary act of scattering (more precisely, diffraction) by spherical (chiefly) particles, the author has acted quite properly. The significance of the book consists chiefly in the fact that it is the first monograph devoted to this important phenomenon. It is especially valuable that the author has not only put in order the existing literary material, eliminating a number of substantial disagreements and errors, but has also introduced much that is new, supplementing it with his own investigations, which have made a considerable contribution to bringing clarity and completeness to the problem under consideration. One can only regret that the author’s own interests have pushed him toward an even greater, and in our opinion unjustified, narrowing of the subject. In essence, the monograph is devoted only to the theory of light scattering by individual particles (as its title should have been). Experimental data are cited very sparingly, for the purpose of illustrating or substantiating calculations; otherwise the author confines himself to bibliographic references, often lapsing into a simple enumeration (for example, pp. 142–144 and 182–183) and not always even explaining what is actually contained in the cited reference (for example, references to the works of G. I. Pokrovsky on p. 221).

Let us turn to the content of the book.

In Chapter I (pp. 9–37) the general formulation of the problem is given, and the optical properties of a substance essential for its formulation are briefly reported (the properties of metals are described in somewhat greater detail and, in particular, those of water, because of the importance of scattering by hydrosols for meteorology, which chiefly attracts the author’s attention). Of undoubted interest is the analysis of the dimensionless parameters characteri-

scattering a turbid medium and predetermining the method of approaching the study of the phenomena of scattering and propagation of light in such a medium (§ 3). However, the presentation here (as in a number of other places in the book, for example pp. 139—140, 151—153) is excessively declarative in character and insufficiently convincing. The author’s assertion (pp. 20—21) is puzzling: that in the case when both the sizes of the particles and the distances between them are smaller than the wavelength, the usual division of the problem of scattering into two independent parts—analysis of the elementary act of scattering and allowance for multiple scattering—has no place. The author overlooks the fact that in this case the concept of the elementary act of scattering may be referred not to an individual scattering particle, but to an element of volume whose dimensions are larger than the wavelength.

Chapter II (pp. 38—57) is devoted to the successive solution of the problem of diffraction by a sphere and to the derivation of the coefficients of attenuation, scattering, and absorption. The author rightly sees the need for a detailed analysis of these questions, which have repeatedly been elucidated in the literature, in the fact that in the existing accounts (Mie, Born, Frank and Mises, Stratton, and others) there is a considerable lack of coordination, which seriously hinders practical use of the resulting formulas.

The treatment of partial waves and of a number of questions necessary for carrying out practical calculations (asymptotic formulas, various forms of notation), given in Chapter III (pp. 59—80), is very successful.

Chapter IV (pp. 81—98) examines scattering by small particles. Here the author focuses the reader’s attention on the well-known Rayleigh law, on the limits of its applicability and deviations from this law in the case of a large refractive index of the material of the particle, and also on the substantial difference in the dependence of the absorption and scattering coefficients on wavelength. Especially significant is the detailed analysis of the peculiar resonant phenomena that occur at large (but finite) values of the refractive index, as well as the analysis of scattering indicatrices.

While dwelling in detail on theoretical considerations, however, the author completely bypasses the question of the extent to which these considerations are confirmed by experiment. The material known in this field is available, and ignoring it is not justified by anything.

Chapter V (pp. 99—136) discusses scattering by large (transparent) particles, the phenomenon of scattering being considered here from the standpoint of geometrical optics as the result of successive reflections and refractions of rays at the boundary of the scattering sphere. The questions of the intensity of the refracted and reflected rays and their polarization as functions of the scattering angle (indicatrix) are considered in detail. At the same time, a clear interpretation is given of the so-called “negative” polarization observed in scattering by large particles and long “intriguing” many investigators. Here, too, a detailed table is given of the distribution of energy among the various orders of the reflected and refracted rays as a function of the scattering angle. As a special case, the theory of rainbows of various orders is included here. Scattering by completely opaque particles is considered separately (§ 6). It is shown that the scattering indicatrix in the case of completely reflecting particles is spherically symmetric, while in the case of completely absorbing particles it is elongated forward. In this case, absorption of light by particles leads only to a weakening of the scattered light (a decrease in the scattering coefficient), without affecting the attenuation coefficient, which (for large particles) does not depend on the physical properties of the particle. The author completely bypasses the important question of semitransparent particles.

Chapter VI (pp. 137—169) presents the results of the rigorous theory for completely absorbing and reflecting spherical particles of any size.

dimensions. First the author considers the dependence of the attenuation coefficient on wavelength and the relation between the attenuation coefficient and the scattering amplitude in the direction of the primary ray. The considerations developed by the author concerning this relation, for all their naturalness, are not very convincing and require substantially more solid justification. A feeling of complete dissatisfaction is left by pp. 142–144, where the author dismisses, in a few general phrases, the important and difficult question of the scattering and absorption of semitransparent particles.

The discussion of the asymptotic behavior of the attenuation coefficient (§ 2) is very substantial; here the author explains in detail the nature of the discrepancy between the results of rigorous theory and those of a theory based on considerations of geometrical optics (by the multiplier, equal to 2). The meaning of this discrepancy (well known also in the phenomena of the so-called diffraction scattering of neutrons), which consists in the fact that as the dimensions of the scattering particle increase, diffraction phenomena not only do not disappear, but the intensity of the diffracted light grows in proportion to the square of the particle radius, is revealed by the author quite fully.

However, the illustrative considerations based on Babinet’s principle (pp. 150–152) suffer from excessive vagueness and do not sufficiently reveal the physical essence of the phenomenon, which thus remains masked by mathematical formulas—an circumstance characteristic also of a number of other places in the book (for example, to the highest degree, the interesting § 4 of the same chapter).

The analysis of the asymptotic behavior of the rigorous solution of the problem of diffraction by a sphere also leads the author to an explanation of the frequently observed halo phenomenon, which for a long time remained unexplained.

Further, the author investigates the behavior of the scattering indicatrix and the polarization of scattered light as functions of particle size and wavelength.

The author’s results on the modification of the laws of emission for small particles (§ 4) are very interesting. Here the author shows that the dependence of the total energy emitted by a particle per unit time on temperature does not obey the Stefan–Boltzmann law and changes with the properties of the particles.

Thus, for small particles the emitted energy is proportional to the fifth power of the temperature, while at low temperatures for conducting particles (in connection with the dependence of electrical conductivity on temperature) it is proportional to the first power of the temperature. At the same time the constant in Wien’s displacement law also changes.

In Chapter VII (pp. 170–227) the author turns to the consideration of transparent spherical particles of arbitrary size. First, the dependence of the attenuation coefficient on particle size and wavelength is analyzed on the basis of tabulated data from exact formulas, and experimental data of geophysical interest are briefly described. In this respect § 1 sharply falls out of the style of the whole book, but here too the experimental material is insufficiently systematized and is presented, essentially, apart from the theory.

The term “free path length of radiation,” introduced by the author on p. 178 and indicating the distance over which the intensity of a ray decreases by a factor of \(e\), is clearly unsuccessful. It would be more appropriate to call this quantity, for example, the mean depth of penetration of the radiation.

Then (§ 2) the author proceeds to the consideration of the geometrical optics of the sphere as a limiting case of diffraction. Along the way, an explanation is given of the phenomena of coronas (forward diffraction) and glories (backward diffraction), as well as rainbows.

Finally, an approximate technique is given for calculating the indicatrix for large particles, which is extremely important for meteorological applications. Here too the author departs from his “rule” and gives experimental data.

Chapter VIII (pp. 228–271) is devoted to an exposition of the relaxation method developed by the author for scattering by inhomogeneities, fundamentally different from the method of solving the diffraction problem and suitable for particles whose properties deviate little from the properties of the surrounding medium. This method is based on the solution of an integral equation analogous to the equation underlying the theory of light propagation in continuous media and relating the intensity of the electric field to the dipole moments of the molecules forming the particle. The solution is carried out by the method of successive approximations, which already in the second approximation gives good agreement with the rigorous theory.

The advantage of the method is the simplicity of the resulting formulas, which makes it possible to consider scattering not only by spherical particles, but also by particles of other shapes (ellipsoidal). The chapter concludes with an extremely important paragraph containing an analysis of the question of coherent scattering of light by a system of particles.

At the end there is appended a table of angular functions, computed by the author, corresponding to the facilitation of the solution of practical problems.

The book is supplied with an extensive bibliography containing 128 titles. An undoubted merit is that the author has illuminated, with the necessary fullness, the role of domestic science in the development of questions of the scattering of light by large particles. Here he may be reproached only for the fact that, when speaking of many experimental works, he too often confines himself to a simple mention or outlines them with excessive schematism.

From what has been said it is clear that the book under review rather fully summarizes the present state of the theory of scattering of electromagnetic waves by inhomogeneous inclusions and, what is especially important, systematizes this theory, creating the possibility not only of comparatively simple acquaintance with it, but also of its practical application and further development, which until now has been substantially hindered, as has already been noted, by the diversity present in the numerous original investigations. Clearly taking the latter circumstance into account, the author has taken care to present the material in the form most convenient for its practical use in calculations, and he has fully succeeded in this. The author’s own contribution to the field of physics under consideration has given the book freshness and, in places, a polemical tone. It has also conditioned a certain narrowness in the range of questions touched upon by the author. Dwelling in detail on, apparently, problems of atmospheric optics especially close to him, the author pays insufficient attention to other, no less important questions (for example, colloids), and essentially bypasses some questions altogether (for example, backscattering, which is the basis of radiolocation methods for investigating clouds and precipitation).

Serious objections are aroused by the character of the exposition. Instead of emphasizing the obscurities and difficulties, and thereby revealing the direction of further investigations, the author in most cases cautiously skirts them, striving to create an impression of complete “well-being.”

The book is written in clear language. However, excessive repetitions (for example, concerning the error made by a number of authors in calculating the intensity of scattered light) and a certain diffuseness of the exposition deprive the book of that wholeness which is implicit in the plan of its construction, and create an impression of looseness and incompleteness. This impression is aided by

Bibliography

…exists, as does the manner of handling experimental material already noted above. The author apparently does not have a sufficiently clear sense of his readership and therefore not infrequently permits sharp violations of proportion. An example is the detailed, three-page discussion (pp. 34–36) of an absolutely black body, which does not go beyond elementary truths.

The book’s shortcomings are, in many respects, of course, due to the fact that it is the first attempt at a monograph on this range of questions. They do not prevent the conclusion that we have before us a necessary and useful book, for which the Soviet reader will be grateful both to the author and to the publishing house.

G. V. Rozenberg