The Theory of Heat Radiation (1914) — part 1 of 12

ASTRONOMY DEPT ,

THE THEORY OF HEAT RADIATION

PLANCK AND MASIUS

Li

THE THEORY

OF

HEAT RADIATION

BY

DR. MAX PLANCK

PROFESSOR OF THEORETICAL PHYSICS IN THE UNIVERSITY OF BERLIN

AUTHORISEDJTRANSLATION BY

MORTON JVUSIUS, M. A., Ph. D. (Leipzig)

INSTRUCTOR IN PHYSICS IN THE WORCESTER POLYTECHNIC INSTITUTE

WITH 7 ILLUSTRATIONS

PHILADELPHIA

P. BLAKISTON'S SON & CO.

1012 WALNUT STREET

SEP 29

ASTRONOMY DEFT;

COPYRIGHT, 1914, BY P. BLAKISTON'S SON & Co.

THE. MAPLH' PRESS -YORK- PA

£

TRANSLATOR'S PREFACE

The present volume is a translation of the second edition of Professor Planck's WAERMESTRAHLUNG (1913). The profoundly original ideas introduced by Planck in the endeavor to reconcile the electromagnetic theory of radiation with experimental facts have proven to be of the greatest importance in many parts of physics. Probably no single book since the appearance of Clerk Maxwell's ELECTRICITY AND MAGNETISM has had a deeper influence on the development of physical theories. The great majority of English-speaking physicists are, of course, able to read the work in the language in which it was written, but I believe that many will welcome the opportunity offered by a translation to study the ideas set forth by Planck without the difficulties that frequently arise in attempting to follow a new and somewhat difficult line of reasoning in a foreign language.

Recent developments of physical theories have placed the quan tum of action in the foreground of interest. Questions regarding the bearing of the quantum theory on the law of equipartition of energy, its application to the theory of specific heats and to photoelectric effects, attempts to form some concrete idea of the physical significance of the quantum, that is, to devise a "model" for it, have created within the last few years a large and ever increasing literature. Professor Planck has, however, in this book confined himself exclusively to radiation phenomena and it has seemed to me probable that a brief resume of this literature might prove useful to the reader who wishes to pursue the subject further. I have, therefore, with Professor Planck's permission, given in an appendix a list of the most important papers on the subjects treated of in this book and others closely related to them. I have also added a short note on one or two derivations of formulae where the treatment in the book seemed too brief or to present some difficulties.

vi TRANSLATOR'S PREFACE

In preparing the translation I have been under obligation for advice and helpful suggestions to several friends and colleagues and especially to Professor A. W. Duff who has read the manu script and the galley proof.

MORTON MASIUS. WORCESTER, MASS., February, 1914.

PREFACE TO SECOND EDITION

Recent advances in physical research have, on the whole, been favorable to the special theory outlined in this book, in particular to the hypothesis of an elementary quantity of action. My radi ation formula especially has so far stood all tests satisfactorily, including even the refined systematic measurements which have been carried out in the Physikalisch-technische Reichsanstalt at Charlottenburg during the last year. Probably the most direct support for the fundamental idea of the hypothesis of quanta is supplied by the values of the elementary quanta of matter and electricity derived from it. When, twelve years ago, I made my first calculation of the value of the elementary electric charge and found it to be 4.69-10"10 electrostatic units, the value of this quantity deduced by J. J. Thomson from his ingenious experiments on the condensation of water vapor on gas ions, namely 6.5-10~10 was quite generally regarded as the most reliable value. This value exceeds the one given by me by 38 per cent. Meanwhile the experimental methods, improved in an admirable way by the labors of E. Rutherford, E. Regener, J. Perrin, R. A. Millikan, The Svedberg and others, have without exception decided in favor of the value deduced from the theory of radiation which lies between the values of Perrin and Millikan.

To the two mutually independent confirmations mentioned, there has been added, as a further strong support of the hypothe sis of quanta, the heat theorem which has been in the meantime announced by W. Nernst, and which seems to point unmistakably to the fact that, not only the processes of radiation, but also the molecular processes take place in accordance with certain ele mentary quanta of a definite finite magnitude. For the hypoth esis of quanta as well as the heat theorem of Nernst may be re duced to the simple proposition that the thermodynamic proba bility (Sec. 120) of a physical state is a definite integral number, or, what amounts to the same thing, that the entropy of a state has a quite definite, positive value, which, as a minimum, becomes

vii

viii PREFACE TO SECOND EDITION

zero, while in contrast therewith the entropy may, according to the classical thermodynamics, decrease without limit to minus infinity. For the present, I would consider this proposition as the very quintessence of the hypothesis of quanta.

In spite of the satisfactory agreement of the results mentioned with one another as well as with experiment, the ideas from which they originated have met with wide interest but, so far as I am able to judge, with little general acceptance, the reason probably being that the hypothesis of quanta has not as yet been satis factorily completed. While many physicists, through conserva tism, reject the ideas developed by me, or, at any rate, maintain an expectant attitude, a few authors have attacked them for the opposite reason, namely, as being inadequate, and have felt com pelled to supplement them by assumptions of a still more radical nature, for example, by the assumption that any radiant energy whatever, even though it travel freely in a vacuum, consists of indivisible quanta or cells. Since nothing probably is a greater drawback to the successful development of a new hypothesis than overstepping its boundaries, I have always stood for making as close a connection between the hypothesis of quanta and the classical dynamics as possible, and for not stepping outside of the boundaries of the latter until the experimental facts leave no other course open. I have attempted to keep to this standpoint in the revision of this treatise necessary for a new edition.

The main fault of the original treatment was that it began with the classical electrodynamical laws of emission and absorption, whereas later on it became evident that, in order to meet the demand of experimental measurements, the assumption of finite energy elements must be introduced, an assumption which is in direct contradiction to the fundamental ideas of classical electro dynamics. It is true that this inconsistency is greatly reduced by the fact that, in reality, only mean values of energy are taken from classical electrodynamics, while, for the statistical calcula tion, the real values are used; nevertheless the treatment must, on the whole, have left the reader with the unsatisfactory feeling that it was not clearly to be seen, which of the assumptions made in the beginning could, and which could not, be finally retained.

In contrast thereto I have now attempted to treat the subject from the very outset in such a way that none of the laws stated

PREFACE TO SECOND EDITION ix

need, later on, be restricted or modified. This presents the advantage that the theory, so far as it is treated here, shows no contradiction in itself, though certainly I do not mean that it does not seem to call for improvements in many respects, as regards both its internal structure and its external form. To treat of the numerous applications, many of them, very important, which the hypothesis of quanta has already found in other parts of physics, I have not regarded as part of my task, still less to discuss all differing opinions.

Thus, while the new edition of this book may not claim to bring the theory of heat radiation to a conclusion that is satis factory in all respects, this deficiency will not be of decisive importance in judging the theory. For any one who would make his attitude concerning the hypothesis of quanta depend on whether the significance of the quantum of action for the ele mentary physical processes is made clear in every respect or may be demonstrated by some simple dynamical model, misunder stands, I believe, the character and the meaning of the hy pothesis of quanta. It is impossible to express a really new principle in terms of a model following old laws. And, as re gards the final formulation of the hypothesis, we should not forget that, from the classical point of view, the physics of the atom really has always remained a very obscure, inacces sible region, into which the introduction of the elementary quantum of action promises to throw some light.

Hence it follows from the nature of the case that it will require painstaking experimental and theoretical work for many years to come to make gradual advances in the new field. Any one who, at present, devotes his efforts to the hypothesis of quanta, must, for the time being, be content with the knowledge that the fruits of the labor spent will probably be gathered by a future generation.

THE AUTHOR. BERLIN, November, 1912.

PREFACE TO FIRST EDITION

In this book the main contents of the lectures which I gave at the University of Berlin during the winter semester 1906-07 are presented. My original intention was merely to put together in a connected account the results of my own investigations, begun ten years ago, on the theory of heat radiation; it soon be came evident, however, that it was desirable to include also the foundation of this theory in the treatment, starting with Kirch- hoff s Law on emitting and absorbing power; and so I attempted to write a treatise which should also be capable of serving as an introduction to the study of the entire theory of radiant heat on a consistent thermodynamic basis. Accordingly the treatment starts from the simple known experimental laws of optics and advances, by gradual extension and by the addition of the results of electrodynamics and thermodynamics, to the problems of the spectral distribution of energy and of irreversibility. In doing this I have deviated frequently from the customary methods of treatment, wherever the matter presented or considerations regarding the form of presentation seemed to call for it, especially in deriving Kirchhoff's laws, in calculating Maxwell's radiation pressure, in deriving Wien's displacement law, and in generalizing it for radiations of any spectral distribution of energy whatever.

I have at the proper place introduced the results of my own investigations into the treatment. A list of these has been added at the end of the book to facilitate comparison and examination as regards special details.

I wish, however, to emphasize here what has been stated more fully in the last paragraph of this book, namely, that the theory thus developed does not by any means claim to be perfect or complete, although I believe that it points out a possible way of accounting for the processes of radiant energy from the same point of view as for the processes of molecular motion.

XI

TABLE OF CONTENTS

PART I FUNDAMENTAL FACTS AND DEFINITIONS

CHAPTER PAGE

I. General Introduction 1

II. Radiation at Thermodynamic Equilibrium. Kirchhoff's Law. Black Radiation 22

PART II

DEDUCTIONS FROM ELECTRODYNAMICS AND THERMODYNAMICS

, I. Maxwell's Radiation Pressure 49

II. Stefan-Boltzmann Law of Radiation 59

III. Wien's Displacement Law 69

IV. Radiation of any Arbitrary Spectral Distribution of Energy. Entropy and Temperature of Monochromatic Radiation. . . 87

V. Electrodynamical Processes in a Stationary Field of Radiation . . 103

PART III ENTROPY AND PROBABILITY

I. Fundamental Definitions and Laws. Hypothesis of Quanta . . 113 II. Ideal Monatomic Gases 127

III. Ideal Linear Oscillators 135

IV. Direct Calculation of the Entropy in the Case of Thermodynamic Equilibrium 144

PART IV

A SYSTEM OF OSCILLATORS IN A STATIONARY FIELD OF

RADIATION

I. The Elementary Dynamical Law for the Vibrations of an Ideal

Oscillator. Hypothesis of Emission of Quanta 151

II. Absorbed Energy 155

III. Emitted Energy. Stationary State 161

IV. The Law of the Normal Distribution of Energy. Elementary Quanta of Matter and of Electricity 167

xiii

xiv TABLE OF CONTENTS

PART V IRREVERSIBLE RADIATION PROCESSES

' I. Fields of Radiation in General 189

II. One Oscillator in the Field of Radiation 196

III. A System of Oscillators 200

IV. Conservation of Energy and Increase of Entropy. Conclusion . . 205 List of Papers on Heat Radiation and the Hypothesis of Quanta

by the Author 216

Appendices 218

Errata . . 225

PART I FUNDAMENTAL FACTS AND DEFINITIONS

RADIATION OF HEAT

CHAPTER I GENERAL INTRODUCTION

1. Heat may be propagated in a stationary medium in two entirely different ways, namely, by conduction and by radiation. Conduction of heat depends on the temperature of the medium in which it takes place, or more strictly speaking, on the non- uniform distribution of the temperature in space, as measured by the temperature gradient. In a region where the temperature of the medium is the same at all points there is no trace of heat conduction.

Radiation of heat, however, is in itself entirely independent of the temperature of the medium through which it passes. It is possible, for example, to concentrate the solar rays at a focus by passing them through a converging lens of ice, the latter remaining at a constant temperature of 0°, and so to ignite an inflammable body. Generally speaking, radiation is a far more complicated phenomenon than conduction of heat. The reason for this is that the state of the radiation at a%given instant and at a given point of the medium cannot be represented, as can the flow of heat by conduction, by a single vector (that is, a single directed quantity). All heat rays which at a given instant pass through the same point of the medium are perfectly independent of one another, and in order to specify completely the state of the radiation the intensity of radiation must be known in all the directions, infinite in number, which pass through the point in question; for this purpose two opposite directions must be considered as distinct, because the radiation in one of them is quite independent of the radiation in the other.

1

2 FUNDAMENTAL FACTS AND DEFINITIONS

2. Putting aside for the present any special theory of heat radiation, we shall state for our further use a law supported by a large number of experimental facts. This law is that, so far as their physical properties are concerned, heat rays are identical with light rays of the same wave length. The term "heat radia tion," then, will be applied to all physical phenomena of the same nature as light rays. Every light ray is simultaneously a heat ray. We shall also, for the sake of brevity, occasionally speak of the " color" of a heat ray in order to denote its wave length or period. As a further consequence of this law we shall apply to the radiation of heat all the well-known laws of experi mental optics, especially those of reflection and refraction, as well as those relating to the propagation of light. Only the phenomena of diffraction, so far at least as they take place in space of considerable dimensions, we shall exclude on account of their rather complicated nature. We are therefore obliged to introduce right at the start a certain restriction with respect to the size of the parts of space to be considered. Throughout the following discussion it will be assumed that the linear dimensions of all parts of space considered, as well as the radii of curvature of all surfaces under consideration, are large compared with the wave lengths of the rays considered. With this assumption we may, without appreciable error, entirely neglect the influence of diffraction caused by the bounding surfaces, and everywhere apply the ordinary laws of reflection and refraction of light. To sum up: We distinguish once for all between two kinds of lengths of entirely different orders of magnitude — dimensions of bodies and wave lengths. Moreover, even the differentials of the former, i.e., elements of length, area and volume, will be regarded as large compared with the corresponding powers of wave lengths. The greater, therefore, the wave length of the rays we wish to consider, the larger must be the parts of space considered. But, inasmuch as there is no other restriction on our choice of size of the parts of space to be considered, this assumption will not give rise to any particular difficulty.

3. Even more essential for the whole theory of heat radiation than the distinction between large and small lengths, is the distinction between long and short intervals of time. For the definition of intensity of a heat ray, as being the energy trans-

GENERAL INTRODUCTION 3

mitted by the ray per unit time, implies the assumption that the unit of time chosen is large compared with the period of vibration corresponding to the color of the ray. If this were not so, obvi ously the value of the intensity of the radiation would, in general, depend upon the particular phase of vibration at which the measurement of the 'energy of the ray was begun, and the inten sity of a ray of constant period and amplitude would not be inde pendent of the initial phase, unless by chance the unit of time were an integral multiple of the period. To avoid this difficulty, we are obliged to postulate quite generally that the unit of time, or rather that element of time used in defining the intensity, even if it appear in the form of a differential, must be large compared with the period of all colors contained in the ray in question.

The last statement leads to an important conclusion as to radiation of variable intensity. If, using an acoustic analogy, we speak of " beats" in the case of intensities undergoing peri odic changes, the "unit" of time required for a definition of the instantaneous intensity of radiation must necessarily be small compared with the period of the beats. Now, since from the previous statement, our unit must be large compared with a period of vibration, it follows that the period of the beats must be large compared with that of a vibration. Without this restriction it would be impossible to distinguish properly between "beats" and simple "vibrations." Similarly, in the general case of an arbitrarily variable intensity of radiation, the vibrations must take place very rapidly as compared with the relatively slower changes in intensity. These statements imply, of course, a certain far-reaching restriction as to the generality of the radiation phenomena to be considered.

It might be added that a very similar and equally essential restriction is made in the kinetic theory of gases by dividing the motions of a chemically simple gas into two classes: visible, coarse, or molar, and invisible, fine, or molecular. For, since the velocity of a single molecule is a perfectly unambiguous quantity, this distinction cannot be drawn unless the assumption be made that the velocity-components of the molecules contained in suffi ciently small volumes have certain mean values, independent of the size of the volumes. This in general need not by any means be the case. If such a mean value, including the value zero, does not

4 FUNDAMENTAL FACTS AND DEFINITIONS

exist, the distinction between motion of the gas as a whole and random undirected heat motion cannot be made.

Turning now to the investigation of the laws in accordance with which the phenomena of radiation take place in a medium sup posed to be at rest, the problem may be approached in two ways: We must either select a certain point in space and investigate the different rays passing through this one point as time goes on, or we must select one distinct ray and inquire into its history, that is, into the way in which it was created, propagated, and finally destroyed. For the following discussion, it will be advisable to start with the second method of treatment and to consider first the three processes just mentioned.

4. Emission. — The creation of a heat ray is generally denoted by the word emission. According to the principle of the conserva tion of energy, emission always takes place at the expense of other forms of energy (heat,1 chemical or electric energy, etc.) and hence it follows that only material particles, not geometrical volumes or surfaces, can emit heat rays. It is true that for the sake of brevity we frequently speak of the surface of a body as radiating heat to the surroundings, but this form of expression does not imply that the surface actually emits heat rays. Strictly speaking, the surface of a body never emits rays, but rather it allows part of the rays coming from the interior to pass through. The other part is reflected inward and according as the fraction transmitted is larger or smaller the surface seems to emit more or less intense radiations.

We shall now consider the interior of an emitting substance assumed to be physically homogeneous, and in it we shall select any volume-element dr of not too small size. Then the energy which is emitted by radiation in unit time by all particles in this volume-element will be proportional to dr. Should we attempt a closer analysis of the process of emission and resolve it into its elements, we should undoubtedly meet very complicated con ditions, for then it would be necessary to consider elements of space of such small size that it would no longer be admissible to think of the substance as homogeneous, and we would have to allow for the atomic constitution. Hence the finite quantity

1 Here as in the following the German "Korperwarme" will be rendered simply as "heat." (Tr.)

GENERAL INTRODUCTION 5

obtained by dividing the radiation emitted by a volume-element dr by this element dr is to be considered only as a certain mean value. Nevertheless, we shall as a rule be able to treat the phe nomenon of emission as if all points of the volume-element dr took part in the emission in a uniform manner, thereby greatly simplifying our calculation. Every point of dr will then be the vertex of a pencil of rays diverging in all directions. Such a pencil coming from one single point of course does not represent a finite amount of energy, because a finite amount is emitted only by a finite though possibly small volume, not by a single point.

We shall next assume our substance to be isotropic. Hence the radiation of the volume-element dr is emitted uniformly in all directions of space. Draw a cone in an arbitrary direction, having any point of the radiating element as vertex, and describe around the vertex as center a sphere of unit radius. This sphere intersects the cone in what is known as the solid angle of the cone, and from the isotropy of the medium it follows that the radiation in any such conical element will be proportional to its solid angle. This holds for cones of any size. If we take the solid angle as in finitely small and of size dtt we may speak of the radiation emitted in a certain direction, but always in the sense that for the emis sion of a finite amount of energy an infinite number of directions are necessary and these form a finite solid angle.

5. The distribution of energy in the radiation is in general quite arbitrary; that is, the different colors of a certain radiation may have quite different intensities. The color of a ray in experi mental physics is usually denoted by its wave length, because this quantity is measured directly. For the theoretical treatment, however, it is usually preferable to use the frequency v instead, since the characteristic of color is not so much the wave length, which changes from one medium to another, as the frequency, which remains unchanged in a light or heat ray passing through stationary media. We shall, therefore, hereafter denote a cer tain color by the corresponding value of v, and a certain interval of color by the limits of the interval v and /, where vr> v. The radiation lying in a certain interval of color divided by the magni tude v'-v of the interval, we shall call the mean radiation in the interval v to v '. We shall then assume that if, keeping v constant,

6 FUNDAMENTAL FACTS AND DEFINITIONS

we take the interval v'-v sufficiently small and denote it by dv the value of the mean radiation approaches a definite limiting value, independent of the size of dv, and this we shall briefly call the " radiation of frequency v." To produce a finite intensity of radiation, the frequency interval, though perhaps small, must also be finite.

We have finally to allow for the polarization of the emitted radiation. Since the medium was assumed to be isotropic the emitted rays are unpolarized. Hence every ray has just twice the intensity of one of its plane polarized components, which could, e.g., be obtained by passing the ray through a Nicol's prism.

6. Summing up everything said so far, we may equate the total energy in a range of frequency from v to v--dv emitted in the time dt in the direction of the conical element d ft by a volume element dr to

dt'dT-dtt'dv'2*,. (1)

The finite quantity ev is called the coefficient of emission of the medium for the frequency v. It is a positive function of v and refers to a plane polarized ray of definite color and direction. The total emission of the volume-element dr may be obtained from this by integrating over all directions and all frequencies. Since €„ is independent of the direction, and since the integral over all conical elements dtt is 4,w, we get:

CO

dt-dr.Sw I <,dv. t (2)

7. The coefficient of emission e depends, not only on the fre quency v, but also on the condition of the emitting substance contained in the volume-element dr, and, generally speaking, in a very complicated way, according to the physical and chemical processes which take place in the elements of time and volume in question. But the empirical law that the emission of any volume- element depends entirely on what takes place inside of this ele ment holds true in all cases (Prevost's principle). A body A at 100° C. emits toward a body B at 0° C. exactly the same amount of radiation as toward an equally large and similarly situated body B' at 1000° C. The fact that the body A is cooled

GENERAL INTRODUCTION 7

by B and heated by B' is due entirely to the fact that B is a weaker, B' a stronger emitter than A.

We shall now introduce the further simplifying assumption that the physical and chemical condition of the emitting sub stance depends on but a single variable, namely, on its absolute temperature T. A necessary consequence of this is that the coefficient of emission e depends, apart from the frequency v and the nature of the medium, only on the temperature T. The last statement excludes from our consideration a number of radiation phenomena, such as fluorescence, phosphorescence, electrical and chemical luminosity, to which E. Wiedemann has given the common name " phenomena of luminescence." We shall deal with pure " temperature radiation" exclusively.

A special case of temperature radiation is the case of the chemical nature of the emitting substance being invariable. In this case the emission takes place entirely at the expense of the heat of the body. Nevertheless, it is possible, according to what has been said, to have temperature radiation while chemical changes are taking place, provided the chemical condition is com?- pletely determined by the temperature.

8. Propagation. — The propagation of the radiation in a medium assumed to be homogeneous, isotropic, and at rest takes place in straight lines and with the same velocity in all directions, diffrac tion phenomena being entirely excluded. Yet, in general, each ray suffers during its propagation a certain weakening, because a certain fraction of its energy is continuously deviated from its original direction and scattered in all directions. This phenome non of " scattering," which means neither a creation nor a destruction of radiant energy but simply a change in distribution, takes place, generally speaking, in all media differing from an absolute vacuum, even in substances which are perfectly pure chemically.1 The cause of this is that no substance is homogene ous in the absolute sense of the word. The smallest elements of space always exhibit some discontinuities on account of their atomic structure. Small impurities, as, for instance, particles of dust, increase the influence of scattering without, however, appre ciably affecting its general character. Hence, so-called "turbid"

1 See, e.g., Lobry de Bruyn and L. K. Wolff, Rec. des Trav. Chim. des Pays-Bas 23, p. 155, 1904.

8 FUNDAMENTAL FACTS AND DEFINITIONS

media, i.e., such as contain foreign particles, may be quite prop erly regarded as optically homogeneous,1 provided only that the linear dimensions of the foreign particles as well as the distances of neighboring particles are sufficiently small compared with the wave lengths of the rays considered. As regards optical phenom ena, then, there is no fundamental distinction between chemically pure substances and the turbid media just described. No space is optically void in the absolute sense except a vacuum. Hence a chemically pure substance may be spoken of as a vacuum made turbid by the presence of molecules.

A typical example of scattering is offered by the behavior of sunlight in the atmosphere. When, with a clear sky, the sun stands in the zenith, only about two-thirds of the direct radiation of the sun reaches the surface of the earth. The remainder is intercepted by the atmosphere, being partly absorbed and changed into heat of the air, partly, however, scattered and changed into diffuse skylight. This phenomenon is produced probably not so much by the particles suspended in the atmos phere as by the air molecules themselves.

Whether the scattering depends on reflection, on diffraction, or on a resonance effect on the molecules or particles is a point that we may leave entirely aside. We only take account of the fact that every ray on its path through any medium loses a certain fraction of its intensity. For a very small distance, s, this frac- tio'n is proportional to s, say

fts (3)

where the positive quantity ft is independent of the intensity of radiation and is called the " coefficient of scattering" of the me dium. Inasmuch as the medium is assumed to be isotropic, ft is also independent of the direction of propagation and polariza tion of the ray. It depends, however, as indicated by the subscript v, not only on the physical and chemical constitution of the body but also to a very marked degree on the frequency. For certain values of v, ft may be so large that the straight-line propagation of the rays is virtually destroyed. For other values of Vj however, ft may become so small that the scattering can

1 To restrict the word homogeneous to its absolute sense would mean that it could not be applied to any material substance.

GENERAL INTRODUCTION 9

be entirely neglected. For generality we shall assume a mean value of ft. In the cases of most importance ft increases quite appreciably as v increases, i.e., the scattering is noticeably larger for rays of shorter wave length;1 hence the blue color of diffuse skylight.

The scattered radiation energy is propagated from the place where the scattering occurs in a way similar to that in which the emitted energy is propagated from the place of emission, since it travels in all directions in space. It does not, however, have the same intensity in all directions, and moreover is polarized in some special directions, depending to a large extent on the direction of the original ray. We need not, however, enter into any further discussion of these questions.

9. While the phenomenon of scattering means a continuous modification in the interior of the medium, a discontinuous change in both the direction and the intensity of a ray occurs when it reaches the boundary of a medium and meets the surface of a second medium. The latter, like the former, will be assumed to be homogeneous and isotropic. In this case, the ray is in general partly reflected and partly transmitted. The reflection and refraction may be " regular," there being a single reflected ray according to the simple law of reflection and a single trans mitted ray, according to Snell's law of refraction, or, they may be "diffuse," which means that from the point of incidence on the surface the radiation spreads out into the two media with intensi ties that are different in different directions. We accordingly describe the surface of the second medium as "smooth" or "rough" respectively. Diffuse reflection occurring at a rough surface should be carefully distinguished from reflection at a smooth surface of a turbid medium. In both cases part of the incident ray goes back to the first medium as diffuse radiation. But in the first case the scattering occurs on the surface, in the second in more or less thick layers entirely inside of the second medium.

10. When a smooth surface completely reflects all incident rays, as is approximately the case with many metallic surfaces, it is termed "reflecting." When a rough surface reflects all incident rays completely and uniformly in all directions, it is

i Lord Rayleigh, Phil. Mag., 47, p. 379, 1899.

10 FUNDAMENTAL FACTS AND DEFINITIONS

called " white." The other extreme, namely, complete trans mission of all incident rays through the surface never occurs with smooth surfaces, at least if the two contiguous media are at all optically different. A rough surface having the property of completely transmitting the incident radiation is described as " black."