NUMBER ¢ AUGUST 15, 1948 On the Classical Equations of Motion of Point Charges Peer Havas Department of Physics, Lehigh University, Bthchem, Pennsylvania, (Received February 9, 1948) Recent attempts to obtain the force of radiative reaction in the classical equations of motion of point charges have proceeded from two diferent viewpoints. Each of these has to introduce one basic assumption in addition to ‘Maxwell's equations, namely the conservation law for the electromagnetic energy-momentum tensor in field theory and the relation between the Lorentz force and the ‘momentum of the particle in action-at-a-dstance theory. In previous fld-theoreical derivations the Lorentz-Dirac equations including radiation damping are obtained only ‘fone takes the field produced by the particle to be the re- ‘tarded field; but equations of motion without the damping. term are obtained if one uses half the sum of retarded and 1. INTRODUCTION NE of the major problems of classical electrodynamics has been to account for the force experienced by a charge as a result of its motion, The first attempt at a solution due ‘to Lorentz! was based on a model of an extended charge and attributed the force of radiative reaction to the action of one part of the particle ‘on another. It appeared as the first term in a series in powers of the radius of the particle, and all higher terms depended upon the charge distribution assumed. This, and the fact that it appears to be very difficult to fit finite-sized elementary particles into the schemes of rela tivity and of quantum mechanics makes it desirable to treat these particles as mathematical points? In Lorentz’ theory, however, infinities appear in the equations ifthe radius goes to zero. Recent attempts to obtain the force of radia tive reaction have proceeded from two different viewpoints. One is that of field theory, which considers the otal field at all points in space to be the fundamental physical quantity and the point charges as singularities of the field. The other is that of action at a distance, which considers only the forces exerted on a charge by other charges to be physically meaningful. ‘The field theoretical point of view was first THEA Lorentz, Collected Papers (ML. Niof, The Hague, 1930), Va, pp. 281 and $43. Also The Theory of Electrons (Teutiner; Leipe, 1908) pp 49 and 153. (SE rate Zeta Biya 3,318 (1925) advanced fields. On the other hand, the theory of action at a distance as developed by Wheeler and Feynman was able to obtain che radiation damping using fields sym- ‘metric in time. Ie is noted that the need for the exclusive ‘ise of retarded fields arote only in the fel-theoretieal derivations for the one-particle problem. The considera- tons of Wheeler and Feynman on the total fld due to all particles of the universe are, however, applicable to eld theory in the symmetric form as well as to actionat-a- distance theory. The acceptance of their condition of, “complete absorption” again leads to the radiation damping term in the equations of motion of the many-body problem. Some implications of this result are discussed applied successfully to this problem by Dirac’ who showed that the equations of motion are suggested by the conservation law for the electro- ‘magnetic energy-momentum tensor. He clearly stated it as follows: “The usual derivation of the stress-tensor is valid only for continuous charge distributions and we are here using it for point charges. This involves adopting as a fundamental assumption the point of view that energy and momentum are localized in the field jin accordance with Maxwell's and Poynting’s ideas." Using this conservation law and Max- well’s equations he obtained Lorentz’ equations of motion, but whereas these equations were considered by Lorentz to be only approximate, Dirac concluded that ‘there is good reason for believing them exact, within the limits of the classical theory.”” Another field theoretical derivation of these equations was given by Infeld and Wallace,‘ who succeeded in linking it with the general method of obtaining equations of motion in general relativity of Einstein, Infeld and Hoffmann.® ‘The main interest of their paper for the present. purpose is that they showed explicitly that, for the case of a single particle, the Lorentz-Dirac equations including radiation damping are ob- YP. A. M. Dirac, Prot, Roy. Soc. A167, 148 (1938). In the flowing quoted ay D. "Le tnfeld and P. H. Wallace, Phys. Rey. 87,797 (1940) Inthe following quoted as IW. VA Blasteins Es Tnfeld and B, Hoffmann, Ann, Math 39, 68 (1938). 456CLASSICAL EQUATIONS FOR POINT CHARGES tained only if one takes the field produced by the particle tobe the retarded feld; but equations of motion without the damping term are obtained if one uses half the sum of retarded and advanced fields. ‘The first derivation of the force of radiative reaction on the basis of action at a distance is due to Wheeler and Feynman.® As itis necessary in any action-ata-distance theory to introduce an assumption on the basic equations of motion, they had to assume that the Lorentz force acting on a particle equals its rate of change of momen- tum. On the other hand, they were able to obtain the Lorentz-Dirac equations taking the forces on the charges as determined by half the sum of the retarded and the advanced field. It was of considerable importance to show that the fundamental law of fore is symmetric with respect to past and future, and to settle the question first raised in 1909, in which “Ritz considers the limitation to retarded potentials as one of the foundations of the second law of thermodynamics, while Einstein believes that the irreversibility of radiation depends exclu- sively on considerations of probability."7 It appeared that the theory of action at a distance was preferable to the point of view of field theory, which seemed incapable of explaining the radiation reaction using fields symmetric in time. However, we shall show that this is actually not the case. The need for the exclusive use of retarded fields for the explanation of the radia- tive reaction arose only in the field-theoretical derivations for the one-particle problem. ‘The considerations of WF on the total field de to al particles are, however, applicable to field theory as well as to the theory of action at a distance ‘The acceptance of their condition of “complete absorption" does not yield any new results in field theory if the retarded field alone is used, tout in the case of half-advanced, half-retarded fields it does provide the radiation damping term in the equations of motion. Therefore, subject to this condition, one can oain the Lorens-Dirac fquations in both theories starting with fields 1,8, Wiel and. P. Feyaman, Rev. Mod, Phys 17, 187 (945). In the following a Ba dada Eiotehs Pys Zets 0,323 (1909). 437 symmetric in time, in spite of the fundamentally different underlying physical ideas.* UL. FIELD THEORY: THE FORMULATION OF INFELD AND WALLACE We shall first outline the field theoretical derivation of the equations of motion due to Infeld and Wallace. We write down the con- servation laws for the electromagnetic energy- ‘momentum tensor Tn =Tnas aw 21) Toma Tone (aw 2.22) where Tn =Maxwell stress tensor =FrasFss—FraoFao H1Aige PsP $1 /28maF Pon (IW 2.3) Poynting vector= FyaF ix (uw 24) Energy density =1/2F.F, +1/2F Fr, (IW 2.3) Fu=E, and Pax=tqnoll where B, and H, are the electric and magnetic field respectively, Sqn is the Kronecker symbol, and eae is the permutation symbol (Levi-Civita tensor dens- ity). Latin letters run from 1 to 3 and repetition ‘of an index implies summation over this range. “wn and “o” denote partial derivatives with respect to the coordinates x* and time, respec- tively. The velocity of light is taken as unity. The conservation laws break down only at points occupied by a singularity. We shall choose a Lorentz frame of reference in which the point charge is instantaneously at rest at the origin at some moment f. Then if we take any surface enclosing the singularity it can be shown that the four surface integrals Stortvaanris —aw2.10) fotwtveonas awa. Tip awn”. tay ain ata distance and field theory, WE refer ‘only to the formal adie raion eects seh ataan niente here Seeger de seach ire mies Mas Scene rhe ene Nerctaeteenontysncey ore eat ealenca Sere ipeeine ise eine? Pet Pa ae ccgarheed eerie ESS iS ta ote458 PETER are independent of the shape and size of the surface chosen, and hence can depend only on {quantities characterizing the singularity, in par- ticular the coordinates of the singularity and their time derivatives, Here \" are the direction cosines of the normal to the surface of integra- tion Yu Va— in Vem e (aw 2.14) (aw 2.15) where Yq, Jo are arbitrary harmonic functions and Yq, Fe are solutions of the Poisson equation which’do not contain in their development with respect tor an harmonic funetion of the type 1/. ‘The motion of the singularity is determined by the assignment of consistent values to the four surface integrals. We ean assume that they are equal to zero and shift the task of determining the motion to the choice of the arbitrary har- monic functions. Calling the coordinates of the singularity 9"(0), and indicating derivatives with. respect to time by dots, we have as the simplest vial choice? vi and obtain then for the equations of motion =m.li"/n) main —1/he f Tan Ye dMER. (IW 2.16) We note that up to this stage it did not matter if retarded or advanced felds were used. We are only concerned with this problem when wwe evaluate the integral over Ty. ‘The integrals over Ye,» can be shown to be zero. For the fells determining Tn we have t0 take the total fekds aw La), aw 12) the electromagnetic scalar potential, Fan tne omtentPan Far tne Yam tent ae where eis Ti has been shown in IW that this choice corresponds to the choice of an energy-momentim tepor for matter snd that the arbitrariness can be removed by an appeal the eneal rity theory. The same argument bole of the derivation nD Equations of marion based on the Sesion of B, but using a diferent choice of the abe trary functions, have ben sugested by C. J. Elica, Brog. Camb, Phi, Soc: 42,218 (1516) Teappeae rom the hve iat ne se euton ate pte enue ith the general theory HAVAS and 7 the vector potential of the field of the point charge itself, and wiFy» is the external field (including the fields of other point charges). If we take for 7, half the sum of the retarded and advanced potentials, we obtain for the equations of motion mei =e elm (aw 2.17) where uw, is the external electric field (the fields of the other point charges also being taken as 1/2 (retarded +advanced) field) evaluated at the position of the singularity at the moment #. These ate just the usual equations of motion without radiation reaction, except for the special ‘form of the felds of the other charges. If we choose for 7, retarded potentials only wwe obtain main —e alin + 2/309" (IW 2.18) where we have to take in wy the contribution due to other charges as retarded fields only ‘These equations do contain the radiation rea tion. The above equations held in a special coordi- nate system only. We consider a four-space with coordinates 2 and signature + — — —, Greek letters taking the values 0,1, 2, 3 where x* is the time coordinate and x','x, x are the space coordinates. The vector ois defined as the four-dimensional velocity vector of the point charge. Accents indicate differentiation with respect to arc length in space-time. Then it may be shown that in the first case (from now on referred to as the symmetric case) the equations of motion become For (IW 2.19) and in the second case (from now on referred to as the retarded case) mf? oF +2/300"* +2/3e%* (IW 2.20) where ae? is evaluated at the world point of the singularity and contains symmetric contri- butions from the other charges in the symmetric case and retarded contributions only in the retarded case." Equations (IW 2.20) are just the Lorentz-Dirac equations. ‘This distinction in the meaning of may" i the two sets ‘of equations has not been made explicit in IW, ae that was not concerned with aryapplication of thowe ‘abalone, but it obvious from their Gerivaton. maCLASSICAL EQUATIONS FOR POINT For future reference we shall make the dis- tinction between the meaning of waF,* in Eqs. (IW 2.19) and (IW 2.20) explicit by writing it as waPP= E 1/2 FH” + aah AO) + FY (1) in the symmetric case and oF Z vel Ey @ in the retarded case. Here ya FiM and yay” are the retarded and advanced fields of the k-th charge, and .F,* is a solution of Maxwell's ‘equations for empty space, which of course is not, necessarily the same in the retarded and in the symmetric case. We obtain, therefore, for the equations of motion mapa = 06 EAA FA + aay EAP) teas eet (3) for the symmetric case, and acaba EPPO Hee Erte OY papsecee$2/Sececrae for the retarded case. All #'s are evaluated at the world point of the a-th singularity, whose rest mass we have denoted by m, now instead of ‘ma its velocity vector by 144, and its charge by e,. Dirac’s results are entirely equivalent to those of Infeld and Wallace. As Dirac states, however, of Eq. (4) that it has been “obtained in a theory which is fundamentally symmetrical between retarded and advanced potentials”, we shall show in the Appendix that actually he did not use any symmetry relations in his derivation of the above ‘equations, but retarded fields only. MIL ACTION AT A DISTANCE: THE WHEELER- FEYNMAN THEORY In the theory of action at a distance as developed by Fokker" and Wheeler and Feyn- man’ the equations of motion of the a-th charge are assumed to be! mata eaP 04" ©) WA. D. Fokker, ets, Physik $8, 86 (1929), We are got entre herewith cer formulations actin a **Hlere and inthe following we shall change the notation ‘of WE ightly to conform tothe one employed above. CHARGES 459 where the right-hand side is the Lorentz force acting on the particle, and F is the field LUA P+ say), (WF 38) ‘Therefore, we have malt ee EAA FO + ao PM}! O) ‘The expression (WF 38) can be broken down into three parts Em FO 1/2 wa —1/2 wae) ~ E12 Fay FO), (WF 39) ‘The second term of this will contribute to the force an expression Cal /2raFoM© —1/2uae PM)" ‘This reduces, according to Dirac, to the form 2/3eaate! —04"ty)04? (WE 41) which can also be written 2/Seoa"*-+04!099), ‘The third term has no singularities anywhere and is, therefore, a solution of Maxwell's equations for empty space, which we shall call /F to distinguish it from the empty-space solutions introduced in (1) and (2), or E 1/2 mF a sar P) o ‘Then we obtain for the final equations of motion rmesE mbes $2/eaMoa! + for the case which WF call “ tion”. The case which they call “complete absorption" is characterized by E (oF — sav F®) =O (everywhere). Using t “7 While (WF 37) was obtained in WF from the relation 3 (12 iF 41/2 sayF)=0 (outside the absorber) (WE 37) relation, * we obtain for the equations {is only (WWF 37) which i used inthe equations of motion A GWE 3) te a division of the universe into, Dart inside” the absorber and’a part “outside” it it Sopears irreconcilable. with any” current comologial460 PETER of motion mand ta E oh! +2/3e2(ou"+2%00). (WE 42) IV, FIELD THEORY AND THE CONSIDERATIONS ‘OF WHEELER AND FEYNMAN Following Infeld and Wallace we obtained two different sets of equations of motion for the retarded and the symmetric case (Eqs. (3) and (4) respectively. If we considered the equations significant for the case of a single particle, we would obtain a force of radiative reaction in the retarded case only. However, itis clear that we can only compare those equations with experi- ‘ment which take account of the existence of a large number of particles in the universe (which may or may not lead to the same conclusions as the simpler equations for the one-particle case). We shall show that we can take over the considerations of Wheeler and Feynman on the field of all particles in the universe into field theory and we shall then obtain the radiative reaction also én the symmetric case just as in the theory of action at a distance. Except for its last term, Eq. (3) of feld theory is of the same form as Eq. (6), the starting point of the theory of action at a’ distance. Its first term on the right-hand side is just the expression (WF 38) and we can break it down into three different fields exactly as WF have done, for none of their arguments (once Eq. WF 38 is accepted) involves any distinction between field theory and action at a distance. Therefore, we we obtain finally, corresponding to (WF 44), ca mF PV + 2/3004 040) telFet Feet (8) where the sum in the last term is due to two fields each of which is a solution of Maxwell's equations for empty space and, therefore, still a solution for empty space. If we accept (WF 37), we obtain corresponding theory, while (WF. 37) holds everywhere and ia therefore secon wd ana maghe even bea consequence eons: ‘geal considerations, Alaa (WF 33) does not have fo hold EGHESH o be tre, Tae makes ore platable at sreentainpytotaks OVE 31) asan additonal aesmption BRC Ros: HAVAS to (WF 42) tata! 60 Esa A i $2/eao"-+ 0.0) tee Fee. (9) Equation (4) (the retarded case) is already of the form of Eq. (8) (except that the term involving F if absent), and nothing new is obtained if one takes into account all particles the universe We have considered in outline the derivation of the equations of motion from the point of view of field theory first in order to show that, starting from the total field, one may obtain the Eqs. (3) or (4) which apparently do not involve the total field, but only the “external” one, and are therefore of the same form as in action at a distance. This, then, enabled us to show that the application of the WF considerations to the symmetric case yields the radiation damping term which appears in the retarded case already without these considerations. However, we could have seen without any calculation ‘that under the so-called “complete absorption” conditions any results of field theory ‘must be the same in the symmetric and in the retarded (and also the advanced) case. For the condition (WF 37) can also be written Eh = Tae =F 1/2 PO +1/2 sae), (10) and these are the fields (due to sources) which have to be inserted into the energy-momentum tensor at the start of any field-theoretical calcu- lation in the retarded, advanced and symmetric cases, respectively. As these are equal, it is obvious that the equations of motion must also be the same in all cases.¥¢ While this demonstrates the equality of the equations of motion arrived at in the various cases, the explicit form of the equations must be obtained by a calculation such as those of Dirac or Infeld and Wallace. It should be noted that in the theory of action ‘at a distance, contrary to field theory, Eqs. (10) or (WF 37) do not lead to the same equations of “he ton (10) was sgzested by A. insti, Phys Zit 0,18 C00), Cl aaa seleneCLASSICAL EQUATIONS FOR POINT CHARGES 461 TAME I Basic assumptions Fields used Retarded Equations obtained ® Additional assumption None Final equations © ‘Terms common to all fal equations Additional terms (to be multiplied by eat) * Field theory Action ata distance “Maxwells equations Conservation laws forthe electro- smametic energy-momentusn tensor Lorente Eq. (5) awa) Symmetric Symmetric ® © None (WF 37) None. (WE 37) ° OWE 44) WF 42) eBing $2) eae) a ae None motion in all cases. As it is not the total field which enters the starting equations (8), the final equations (WF 42) ean be obtained in the sym- metric case only. If one does not assume condition (WF 37), we have from (7) E ea FE (/2 FED aay) 4 F =D seP+2,F. (a) Therefore, the total fields in the three cases differ only by solutions of Maxwell's equ for empty space, and the equations of motion will only. differ in the terms involving such solutions (ef. Eqs. 4 and 8). In short, the mathematical reason for the similarity of (4) and (8) is simply that two solutions of the inhomogeneous wave equation can at most differ by a solution of the homo- sgencous one. DISCUSSION We shall now inquire into the relationship between the five different sets of equations of motion: Eqs. (4), (8) and (9) from field theory and (WF 42) and (WF 44) from action at a distance. ‘To facilitate the comparison we summarize the assumptions and results in Table I. ‘The last row shows that these equations differ mathematically only in fields which represent ‘empty-space solutions. However, physically the fields .F and ,F do not have the same significance. JF is a solution of the homogeneous wave equa- tion, which arises only in field theory and may be due wholly, partly, or not at all to sources; but at present field theory is unable to analyze it any further. ;F on the other hand is uniquely determined by the sources according to relation (@, which states that it is half the difference of the retarded and advanced fields of all particles. As shown in section III, the term ¢.0."/F,* con- tains a part which just cancels the radiation damping term 2/3ez(o4"*+04"4). ‘Therefore, the similarity of Eqs. (8) and (WF 44) to (4), (9) and (WF 42) is purely formal and (8) and (WF 44) actually do not describe radiating icles (ef. the discussion of “‘incomplete ab- "in WF). If we adopt the assumption frequently used in field theory that .F=0, Eqs. (4) and (9) reduce to (WF 42). However, it should be kept in mind ‘that while the solutions of the equations of the symmetric case (9) and (WF 42) are subject to the restriction (WF 37), those of the retarded case (4) are not. Therefore, we (and also WF) have not shown the complete equivalence of the retarded and symmetric cases. Keeping the assumption .'=0, we have, on the other hand, complete equivalence of Eq. (9) and (WF 42) of field theory and action-at-a- distance theory, respectively, describing the mo- tion of radiating charges. These identical equa- tions have been obtained from two basically different starting points, while using in common Maxwell's equations, fields symmetric in time, and the condition (WF 37).462 PETER In both theories we have to introduce one additional assumption, namely, either the con- servation law for the electromagnetic energy= momentum tensor, or the relation between Lorentz force and momentum. The advantage of action at a distance is the plausibility of the physical idea of reducing everything to the interaction of particles. But this is balanced by the difficulties connected with the conservation" and transmission of energy and momentum. Therefore, none of the viewpoints appears to be preferable to the other from considerations of simp! Clearly a direct verification of the fundamental assumptions of the two viewpoints is impossible, as one cannot observe a field without a test charge, the effect of which, however, would have to be included in the total field. ‘The only experimentally verifiable conclusions are pre- cisely the equations of motion. Comparing these equations, we see that there is mo effect which would require the point of view of action at a distance. On the other hand, the demonstration of a non-vanishing .F would show an effect which can only be explained by field theory Such a demonstration, while possible in princi ple, appears to be impossible in practice, a would amount to finding whether the observed field is “only” due to the retarded fields of all the charges in the universe, or whether there is still another part. ‘Therefore, as far as the symmetric case is concerned, there do not seem to be any com- pelling reasons at present to prefer either of the ‘two points of view. It appears possible, however, that the application of these viewpoints to general relativity or to quantum mechanics will provide such reasons. ‘The author wishes to thank Professors J. A. Wheeler and R, P. Feynman for a stimulating discussion, and Professor R. J. Emrich for his valuable criticism of the manuscript. APPENDIX ‘We want to show that Dirac’s formulation of the field theory for the one-particle problem leads to the same conclusions as that of Infeld and Wallace, namely that one does obtain the term of radiative reaction only in the retarded, HAVAS but not in the symmetric case, ‘tual field as"® saP ra tial where iF is the incident field, a solution of the homogeneous wave equation, as is ue, which is defined by (D8) wePal oul (D9) ‘The difference bP = oy, PF = in (D10) a = a —aaeP wm is called the field of radiation in D. The equations ‘of motion are obtained by substituting the actual field (D 8) into the energy-momentum tensor. After a rather long calculation, which, however, wolves only the use of Maxwell's equations, his stress-tensor and (D 8), Dirac obtains the result ot (D2) where wP-1/UvaF+uivF), — (D 13) J=1/Moa Pav) inF from which we get (see (WF 41) above) mo! —2/38(0"*-+ 00H) =00" Fi? (D 24) If we had used the field of what we called above symmetric case, we would have had sesF 1/2 + uaF) +inP. It can be easily seen, following Dirac’s caleula- tions, that the introduction of this expression ‘would still have led to (D 22) with the definition (13) or with Sauk and, therefore, we would have obtained as our equations of motion mle meo? WF, do not include the term of radiative Here, and in the following, we shall change the nota- tion of B slightly to-conform to the one employed above ‘Alo, as we shall only have to follow the argument of D ‘without any detailed caleutatons, we shailomie sub and ereipa for conveience wietever there w no dager S97 reaule has alzo been obtained b ieee, Rev, Mods Piya oy 147 GBH) fae b= Syn waPHOTO-VoLTS For the case of several particles, analogous considerations would lead to our Eqs. (4) and (3) respectively. ‘Therefore Dirac’s derivation is in accord wit the one of Infeld and Wallace outlined previ- ously. However, we noted that Dirac himself calls his theory fundamentally symmetrical be- tween retarded and advanced potentials. The contradiction seems to come from the fact that Dirac considers his theory symmetrical due to the apparent symmetry of the use of retarded and advanced fields in definitions (D 8) and (D9) (and the analogous definitions (D 38) and (D 39) for the many-body problem). Ic EFFECT 463 However, it should be noted that in the derivation by Dirac outlined above, he is only using (D 8) (respectively D 38). ‘The definition (D9) never enters into any of which lead to his equations of mi used in a purely formal manner to introduce the notion of aM (see (D 10)) into some of the ‘equations; but the results themselves are inde- pendent of it "The purely formal character of any definition of the radiation Bild can be sce hwo Trom the fact that in WE {he term “radiation field” i teed for just 1/2 mF of D, stain Sy a dingo otros 4 he ony ROESa of cifpon chartbe “mations VOLUME 74, NUMBER ¢ august 1s, The Photo-Voltaic Effect K, Lanovee ‘Signal Corps Engineering Laboratories, Fort Monmouth, New Jersey (Received March 19, 148) ‘The Schothy-Mott theory of the barrier layer rectification is extended with respect to the action of light absorbed in the barrier layer. The essential physical assumptions to be used are as follows: (a) The barrier layer is a boundary layer of a semiconductor with a reduced ‘density of mobile charges (cther electrons or holes"); (b) both positive and negative mobile ‘charges are released by light; (€) the recombination within the barrier layer is negligible; ‘and, (€) the electrons and "holes" have the same properties whether released by light or by thermal agitation. Thus an “equation of state” connecting. photowoltage, photo-currea, Tight intensity, wave-length, external resistance, etc, i derived. Among others the regularities of short circuit exerent, open circuit voltage, photo-characteristc, dark characteristic (barrier layer rectification), power output, and spectral distribution of the quantum yield are involved. 1 INTRODUCTION HE, most successful theory in the explana- tion of the barrier rectification is the theory of W. Schottky? and N. F. Mott.! In his discus- sion of the action of light in a barrier layer of the nature described by the above theory, N. F. Mott has already succeeded in explaining the sign of the photo-voltage* In this paper we generalize the fundamental assumptions of W. Schottky and N. F. Mott and derive a comprehensive formula for the barrier layer photo-effect.‘ The establishment of one formula aS dy nl 30 9 Zt Sige i act oe vere a oy. Soe London AUT, 27 (1950) tN’ Fi Mot’ Pros Roy Soe: London UTLE area bas bees conmnnisied fs © previous note by the author in Optik 1, tse 3 (1946) for the manifold of barrier layer photo-cells varying in peculiarities is possible since the peculiarities of minor importance enter only into the parameters of the formula, which remain ‘open to a further discussion in special cases. We shall treat explicitly the photo-effect in semiconductors only for the case where the mobile carriers of charge are electrons (n-type semiconductors) and shall state the corre- sponding results for semiconductors with “hole” conductance (p-type semiconductors).* TL BASIC CONCEPTS ABOUT THE BARRIER LAYER In principle both barrier layer rectifiers and barrier layer photo-cells consists of a semicon- K, Lehover, Zeit. f. Naturforsch. 2,398 (1947).