No. 6] Proc. Jpn. Acad., Ser. B 93 (2017) 399
How the Klein–Nishina formula was derived: Based on the Sangokan Nishina Source Materials
† By Yuji YAZAKI
(Communicated by Makoto KOBAYASHI, M.J.A.)
Abstract: In 1928, Klein and Nishina investigated Compton scattering based on the Dirac equation just proposed in the same year, and derived the Klein–Nishina formula for the scattering cross section of a photon. At that time the Dirac equation had the following unsettled conceptual questions: the negative energy states, its four-component wave functions, and the spin states of an electron. Hence, during their investigation struggles, they encountered various difficulties. In this article, we describe their struggles to derive the formula using the “Sangokan Nishina Source Materials” retained in the the Nishina Memorial Foundation.
Keywords: Klein–Nishina formula, Dirac equation, semi-classical treatment, field quantization, negative energy states
including Compton scattering should be treated 1. Introduction quantum mechanically. As for Compton scattering, In Compton scattering, the scattering of an X- he obtained energy-momentum conservation for the ray by an electron, the wavelength of the scattered Compton scattering, but did not show the intensity X-ray varies with the scattering angle. In 1923, using distribution of the scattered wave. In 1927, Gordon4) the light quantum theory for X-rays, A.H. Compton1) and Klein5) independently developed the so-called explained the wavelength shift upon scattering by Klein–Gordon equation to describe the behavior of using the conservation principle of energy and an electron to include relativistic effects in Compton momentum of the light quantum and electron scattering. The energy and momentum relation of an system. This analysis showed that the Compton electron used by Dirac3) was equivalent to that of the effect was one of the important experimental facts Klein–Gordon equation. confirming the quantum theory of light. Soon after Dirac presented his relativistic In his analysis concerning the angular distribu- electron theory,6),7) O. Klein and Y. Nishina suc- tion of the intensity of the scattered wave, Compton ceeded to derive the famous Klein–Nishina formu- used relativistic theory and the Doppler effect. In la,8),9) calculating the intensity distribution of the 1926, Breit2) discussed this problem using the scattered wave in the Compton scattering based on correspondence principle in the old quantum theory, the Dirac equation. The Klein–Nishina formula has and obtained the same result as what Dirac3) and been firmly accepted and widely used, even now. subsequently Gordon4) obtained independently, using When Klein and Nishina started to attack this quantum mechanics. problem, they intended in part to confirm the They deduced the energy and momentum validity of the Dirac equation, as stated in Introduc- conservation law on the light-quantum and electron tion of Ref. 8. Recalling that time, Bohr wrote to system quantum mechanically, which Compton Nishina in 193410),11) that “the striking confirmation hypothesized, and derived a formula giving the which this formula has obtained became soon the angular distribution of the intensity of the scattered main support for the correctness of Dirac’s theory wave. In 1927, Klein5) discussed how the interaction when it was apparently confronted with so many between an electron and an electromagnetic field grave difficulties.” Ekspong discussed in Ref. 12, that the experiments conducted to confirm the Klein–
† Nishina formula were to suggest the existence of then Correspondence should be addressed: Y. Yazaki, Ooaza- Toyosato 5918-298, Koumimachi, Minamisakugun, Nagano 384- unknown phenomena, namely the pair production 1103, Japan (e-mail: [email protected]). and annihilation of positive and negative electrons. doi: 10.2183/pjab.93.025 ©2017 The Japan Academy 400 Y. YAZAKI [Vol. 93,
The following arguments in this article are based consider a system consisting only electrons without on the Sangokan Nishina Source, preserved by the any electromagnetic field): Nishina Memorial Foundation. In Section 2 through 1) The system is to be described by a set of Section 5, we survey Klein–Nishina’s theory and canonical variables, J and w([w, J] F ih). consider the process they undertook when deriving 2) The Hamiltonian, H, of this system is to be their formula. In Section 6 and Section 7, we look expressed by a function of J only, namely H(J). back their efforts to solve the most difficult problem 3) The coordinate x and momentum p of this for them, namely how to set the final states of the system are both to be expanded in the form i,w i,w electron after scattering, and consider why they x F ∑,C,(J)e and p F ∑,D,(J)e , where adopted the semi-classical method in treating electro- C,(J) and D,(J) are amplitudes and ,Bs are magnetic waves. Finally, in Section 8, we discuss the integers. equivalence between Klein–Nishina’s semi-classical The term ei,w is related to the transition of the method and the quantum field theoretical method. system from the state JB to the state JB ! ,h. The frequency of radiation due to this transition is 8 F – ’ 2. Works preceding to Klein Nishina s theory [H(JB) ! H(JB ! ,h)]/h. Then, the intensity of the In this section we give the energy and momen- emitted radiation observed in the direction perpen- tum conservation law on the Compton scattering dicular to the x-axis is obtained by substituting presented by Compton, himself, and the equation C,(JB) for the amplitude of the displacement of the of the frequency shift accompanying the scattering. charged particle in the formula for the intensity of The energy and momentum conservation laws were dipole radiation, namely regarded as being a basic hypothesis in Compton’s1) e2 04 and Breit’s2) papers. In Section 2 through Section 5 ¼ j ð 0Þj2 ½ I 3 2 C J ; 2-3 we adopt the notations used by Klein and Nishina8): 8 c r h denotes Planck’s constant divided by 2: and 8 where r denotes the distance between the position of denotes frequency multiplied by 2:, while m and !e the charged particle and the observation point. This are the mass and charge of an electron and c is the is the fundamental assumption of Dirac’s theory. velocity of light in a vacuum. This indicates that the interaction between a charged The energy and momentum of a light quantum particle and an electromagnetic field should be having frequency 8 and propagating direction n are handled by replacing quantities in the classical given by h8 and (h8/c)n. Suppose the frequency and theory with their quantum mechanical alternatives, propagating direction of a light quantum before and as was done in the semi-classical treatment of the after the scattering are 8, n and 8B, nB, respectively, electromagnetic field used by Gordon and Klein, as and the momenta of an electron before and after the shown later. scattering are 0 and p, respectively, then the energy We apply the above-mentioned general theory and momentum conservation laws are given by to an electron in an incident electromagnetic wave. pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ 2 ¼ 0 þ 2 2 þ 2 4 Taking into account only the oscillating term, we can h mc h c p m c ; fi 0 0 apply the de nition of the periodic system to the ðh =cÞn ¼ðh =cÞn þ p ½2-1 electron considered here. Using his proper mathe- The frequency of the X-ray scattered in the direction matical insight, Dirac succeeded to construct a set of angle 3 measured from the incident direction of canonical variables, J and w, which describe becomes scattering with the original variables xi, pi, t, and E. The transition from JB to JB ! h corresponds to a 0 h ¼ ; ¼ ½2-2 scattering process; in addition he showed that 8B F 1 þ ð1 cos Þ mc2 [H(JB) ! H(JB ! h)]/h, "J F!h and "J2 F "J3 F 0, 3) 2.1 Dirac’s theory. Dirac treated Compton where J2 and J3 were remaining variables commut- scattering based on quantum mechanics for the first able with J corresponding to the energy and time, using his q-number algebraic method. As the momentum conservation laws. Then the intensity first step, he developed a general theory of a quantum of the scattered wave can be obtained using Eq. [2-3] mechanical multi-periodic system. as For simplicity, we take here a system in which e4 sin2 the degree of freedom is unity. The periodic system is I ¼ I ; ½2-4 r2m2c4 0 ½ þ ð Þ 3 defined by the following properties 1), 2) and 3) (we 1 1 cos No. 6] How the Klein–Nishina formula was derived: Based on the Sangokan Nishina Source Materials 401
where I0 is the intensity of the initial wave, 3 is the (b) Charge density and current density. We define scattering angle, and ? is the angle between the A as a solution of Eq. [2-5] and H as its complex propagating direction of the scattered wave and the conjugate. We can then express the charge density, ;, direction of the electric field accompanying the and the current density, J,as incident wave. ie @ @’ 2ie 2.2 Gordon’s theory. We now explain ¼ ’ þ V’ ; c2 @t @t h Gordon’s theory in some detail, since his work has a e 2i e close relation to Klein–Nishina’s work. Gordon J ¼ ’ grad grad ’ A’ ½2-6 treated the scattering of an electromagnetic wave i h c by an electron based on the correspondence principle, Since these relations satisfy the equation of continu- which means that at first writing down the classical ity, ∂;/∂t D div J F 0, we can interpret ; and J as equations of the interaction between an electron and being the charge density and the current density, an electromagnetic wave, and then replacing the respectively. Hereafter, the letter J denotes the quantities, such as the current density or the charge current density, though in sub-section 2.1, the letter density, with the corresponding quantum mechanical J denotes a canonical variable. expressions. Gordon’s theory can be characterized (c) Electromagnetic potential produced by an by compromising the use of classical and quantum electron. We conjecture that the electromagnetic mechanical theories, which came to be called a “semi- potential produced by an electron can be obtained by classical treatment” later.11),12) Gordon discussed the inserting ; and J into the expression of the retarded dynamics of an electron in an electromagnetic field potential in classical electromagnetic theory, using purely classical theory and quantum-mechani- Z ½J Z ½ 1 t R t R cal theory in parallel; he confirmed that the results A0 ¼ c dr;V0 ¼ c dr; ½2-7 based on quantum-mechanical theory for an electro- c R R magnetic field produced by an electron coincided with where R denotes the distance between the volume those based on classical theory in the limit of Planck’s element dr and the observation point. This is the constant, where h ! 0. Gordon called this treatment fundamental hypothesis of the semi-classical treat- as the “correspondence principle”, and took it as a ment. basis for the validity of the semi-classical treatment, (d) Compton scattering. We assume here that while Klein5) called the semi-classical treatment, the incident wave is a linearly polarized plane itself, as the “correspondence principle method”. wave, and denote its electromagnetic potential as (a) Klein–Gordon equation. In classical theory, )i F ai cos 8(t ! nr/c), where i F 0, 1, 2, 3; )0 F V, the basic equations of an electron in an electro- )k F Ak. magnetic potential A and V are given by the The solution of the Klein–Gordon Equation [2-5] relativistic Hamilton–Jacobis’ equation, can be obtained in the perturbed form of the solution p @ 2 X3 @ 2 of a free electron with momentum , 1 W W e 2 2 eV þ þ Ak þ m c ¼ 0 2 @ @ i ðp r EtÞ c t k¼1 t c ðpÞ¼CðpÞeh 1 þ½quantity of OðaiÞ Following the procedure that Schrödinger took when n r he derived his non-relativistic wave equation, we sin t ; replace @W and @W in the Hamilton–Jacobis’ equation c @xk @t by where E denotes the energy ofp theffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi free electron p ¼ 2 2 þ 2 4 @W h @ @W @ with momentum , namely E c p m c . The ! ; ! ih coefficient C(p) is the normalization factor, deter- @x i @x @t @t k k mined by the requirement of the correspondence and operate it on the wave function, A. principle in the h ! 0 limit, resulting in normal- We then obtain the following Klein–Gordon ization of the /-function type. Multiplying A(p)by equation: the weight factor, z(p) (real quantity), and super- posing them, we can write down the general solution 1 @ 2 X3 h @ e 2 ih þ eV þ þ A þ m2c2 as 2 @ @ k Z c t k¼1 i xk c ¼ ðpÞ ðpÞ ðpÞ p ¼ 0 ½2-5 z C d : 402 Y. YAZAKI [Vol. 93,
When * is inserted into the expression of ; and J the original authors’ notation). The wave function A [2-6], the electromagnetic potential of the scattered has four components and is expressed as a 4–1 wave can be obtained using Equation [2-7] as matrix. By denoting the Hermitian conjugate wave Z A H ; 1 function of as , the charge density and the 0 7) ¼ < ½quantity of OðaiÞ current density J are given by i r 0 i ðP P0Þ r0 i 0ðt rÞ 0 0 ¼ J ¼ < ½ zðpÞzðp Þeh c dpdp dr ; e’ ; ec’ 1 ; KN-3 where which satisfy the equation of continuity, ∂;/∂t D 0 div J F 0. E þ h E h E þ h 0 ¼ ; P ¼ p þ n n0; (b) Free electron. The Dirac equation for a free h c c electron is obtained by setting A F V F 0 in [KN-1]; E0 P0 ¼ p0 n0: and the solution is written in the following form: c i ðEt ðprÞÞ ðpÞ¼vðpÞe h ; ½KN-4 Here, the equation of 8B gives the energy conservation 0 law of the photon and electron system, and integra- where v(p)isa4–1 matrix independent on r and t, tion over rB gives /(P ! PB), which reduces to the and satisfies the equation momentum conservation law. Taking the weight E factor z(p) and z(pB) into account, we obtain the þ ð The perturbed solution is written down in the form of given by H(p, pB) F rot A(p, pB) F i8B/cA(p, pB) # nB. the solution of a free electron with momentum p, Assuming that the initial momentum, p, of the modified by the incident radiation field as electron is zero, we define H(0, pB) F H0. The terms i ðt ðnrÞ=cÞ i ðt ðnrÞ=cÞ g(pB) and f(p) in [KN-36] contain the terms < and ðpÞ¼½1 þ gðpÞe þ g ðpÞe 0ðpÞ; ;1<, as shown in [KN-25], and [KN-36] has such a ½KN-23 structure that g(pB) and f(p) multiplied by ;1< are where g ðpÞ does not mean the complex conjugate of placed between u(p) on left and v(pB) on right. g(p), but merely means the coefficient of e!i8(t!(nr)/c). Using the arithmetic rule on < and the equation Rewriting the Dirac equation in the second-order that each of u(p)andv(pB) satisfies, we can express equation and inserting the form of [KN-23] into the each term in [KN-36] in terms of d F u(p)v(pB), wave function, A, in the rewritten Dirac equation, s F u(p) 4 0 3 0 clearly different from the result of Dirac’s and e 0 2 H 2 ¼ þ C2 2ðn CÞ ; Gordon’s theories. Therefore, experiments on this 0 m2c4r2 0 can confirm the validity of Klein–Nishina theory and, ½ KN-58 moreover, of the Dirac equation. For this purpose, where C denotes the amplitude of the electric field of using the experimental conditions at that time, the incident wave and nB denotes the scattering Nishina calculated the intensity distribution of the direction. Setting the scattering angle as 3 and the doubly scattered wave with the scattering direction angle between the scattering direction and the being normal to each other on the result of the Klein– direction of the polarization of the incident wave as Nishina work. ?, and taking into account [2-2], 8B F 8/[1 D ,(1 ! Here, we follow Nishina’s calculation. First we cos 3)], we obtain the intensity, I, of the scattered rewrite the expression for the magnetic field of the wave: scattered wave [KN-44] in Klein–Nishina’s paper8); we separate the scattered wave into two mutually e4 sin2 I ¼ I independent elliptically polarized components in the 0 m2c4r2 ð þ ð ÞÞ3 "#1 1 cos following way. In this section we use the same ð Þ2 equation numbers as those used in Nishina’s paper13): þ 2 1 cos ½ 1 2 KN-59 0 r 0 r ð þ ð ÞÞ i ðt þ 1Þ i ðt þ 2Þ 2 sin 1 1 cos H 0 /ðA iBÞe c ðD þ iCÞe c þ c:c:; ½N-2 Here, I0 denotes the intensity of the incident wave, 2 (c/4:)2C ; the factor 2 comes from the electric field of where /1 and /2 are arbitrary phases, as in section 3, the incident wave, being written as C[ei8B(t!(nr)/c) D expressing the final states of the electron, ! 8B ! nr þ 0 0 0 0 e i (t ( )/c)]. Comparing [KN-59] with [2-4], we see v ðp Þei 1 þ v ðp Þei 2 . Here, A, B, C and D that the difference between Klein–Nishina’s theory denote vector functions, though in other places A and Dirac’s or Gordon’s theories is in the second denotes the vector potential. The electric field of the term added in the brackets of [KN-59]. By multi- scattered wave, E0 F H0 # nB, can be written as 8B 8 0 0 plying both sides of [KN-59] by (I/h )/(I0/h ), we 0 0 i ðt rþ Þ 0 0 i ðt rþ Þ E /ðA iB Þe c 1 ðD þ iC Þe c 2 obtain the differential cross section of photons as 0 þ c:c: ½N-3 d 1 e4 0 2 0 ¼ þ ðn0e Þ2 Then, the magnetic field of the scattered wave for an 2 4 0 2 0 d 2 m c elliptically polarized incident wave can be obtained This result is obtained after summing over the on the basis of [KN-44] by setting the electric field E electron spin and over the polarization of the of the incident wave as 11),12) scattered photon. This indicates that the corre- ð ðnrÞÞ E ¼ðC þ C Þ i t c þ c c ½ spondence principle method used to derive [KN-59] a i b e : : N-5 implicitly included the above-mentioned procedure. The intensity of the scattered wave depends on the spin direction, l, of the electron in the initial state. ’ 4. Nishina s individual paper This is different from that of a linearly polarized Immediately after joint papers with Klein, incident wave. If we average over the direction l, Nishina by himself wrote subsequent papers13),14) we can obtain the intensity as a simple sum of the concerning polarizations of the scattered electro- intensities in a linearly polarized incident wave given magnetic wave. In the classical theory for Thomson by [KN-59], with replacing C in I0 by Ca and Cb. scattering, electron spins are not considered, and a Thus, we obtain the intensity for elliptically polarized linearly polarized incident wave remains to be incident wave as linearly polarized even after it is scattered by an e4 1 2 þ 2 þ 2 3) ¼ ðC2 þ C2Þ electron. The same condition holds, in Dirac and I 0 a b 4 m2c3r 2 ð1 þ Þ3 1 þ Gordon’s4) theories. On the other hand, in the Klein– 00 2 00 2 Nishina theory based on the Dirac equation, the 2½ðn CaÞ þðn CbÞ ; ½N-7 incident wave is linearly polarized, but the scattered wave is generally elliptically polarized, consisting of where O F (h8/mc2)(1 ! nnB), n denotes the incident two mutually incoherent elliptical polarizations, direction, and nB denotes the scattered direction. We produced by the transition of the electron to two then obtain the intensity of the doubly scattered mutually independent final states. This point is wave under the experimental conditions at that time; No. 6] How the Klein–Nishina formula was derived: Based on the Sangokan Nishina Source Materials 405 the linearly polarized incident plane wave along the (This might be a good problem for Mr. Nishina), x-axis is scattered to the y-axis direction, and is then and I, who had intended to attack that problem scattered again by the second electron on the y-axis. myself, agreed immediately. This secondly scattered wave is observed in the x–z plane. The electric fields of the two mutually As for Nishina, this problem was related to his incoherent elliptically polarized waves scattered by previous work at Copenhagen on an experimental the first electron are determined by [N-3]. Inserting study of the absorption spectra of X-rays. Since then the obtained electric fields into Ca and Cb in [N-7] and he had constantly paid attention to X-ray spectros- summing them, we obtain the intensity of the copy studies, although he became deeply interested in secondly scattered wave. Averaging over the direc- theory at the advent of quantum mechanics.16) tion of polarization of the linearly polarized incident Moreover, in the fall of 1927, he moved to Hamburg, wave, we obtain the averaged intensity, It, given by and then in collaboration with I.I. Rabi he theo- retically investigated the absorption coefficient of e8 I I ¼ 0 X-rays11),17) while learning quantum theory under t 2m4c8r2r02 ð þ Þ3 "#1 2 Pauli. He was thus deeply involved in X-ray research. 2ð2 þ 4 þ 3 2Þ Therefore, it was natural that the idea of treating sin2 þ ; ½N-14 2ð1 þ Þ2ð1 þ 2 Þ Compton scattering based on the Dirac equation occurred to Nishina in Hamburg where they encoun- where # denotes the angle between the direction of tered the appearance of the Dirac equation with great observation and the z-axis, r denotes the distance enthusiasm.11),18) Indeed he wrote to Dirac on 25th between the first and the second electron and rB is the Feb. 1928,19),20) “I hope to calculate Compton effect distance between the second electron and the point of according to your new theory”, while he was studying observation. The equation for It, based on Dirac’sor in Hamburg before he came back to Copenhagen in Gordon’s theories, contains only the first term, sin2 # March, 1928. in [N-14]. Note that Nishina’s result has an additional Here, in Sec. 5, we trace the approach Klein and constant term that is independent of #. Nishina took to obtain the Klein–Nishina formula using the “Sangokan Nishina Shiryo” (Building No. 3 – 5. The derivation process of the Klein Nishina Nishina source materials) retained in the Nishina “ formula revealed through the Sangokan Memorial Foundation, Tokyo. Their details have ” Nishina Source Materials been reported in References 21 and 22: how they Concerning the background of the start of Klein became noticed, how they were investigated, and a and Nishina’s work, Klein wrote in his private note15) summary of their contents. The main part of the as follows: Sangokan source materials consists of Nishina’s documents concerning his study abroad, especially Our nearer acquaintance, which then grew into in Copenhagen from 1923 to 1928, such as memo- friendship began in the spring 1928, I believe a randums concerning his investigations, colloquium little before Eastern[sic]. He had just returned notes, correspondences. from a visit to Hamburg then, where he had In the Nishina Memorial Foundation, a volun- met Pauli and Gordon, while I had been in tary investigation group, called “Tamaki Group”, was Cambridge to see Dirac, whose first paper on set up by Hidehiko Tamaki together with Fukutaro the electron had just appeared. At about the Shimamura, Hajime Takeuchi and Yuji Yazaki. same time also Gordon came on a little visit to Besides a historical investigation on Nishina’s study the Bohr institute. He and I had earlier worked abroad, they took up the task to list and assign on the Compton effect independently of each consecutive numbers to each of the Sangokan Nishina other — he with Schrödinger in Berlin, I in Shiryo, for which H. Takeuchi was responsible. In Copenhagen — his paper with the complete 1991, they prepared a pamphlet called “Sangokan solution appearing before mine was finished. Nishina Shiryo no Naiyou Ichiran” (the contents of And naturally we had both of us been thinking the Sangokan Nishina source materials). This pam- how this effect would come out according to phlet was edited by H. Takeuchi, T. Iwaki, H. Dirac’s new theory. I remember Gordon saying, Tamaki, F. Shimamura and Y. Yazaki. In 1994 this when we all three were standing together: “Das pamphlet was printed by the Nishina Memorial mag ein gutes Problem für Herrn Nishina sein” Foundation. 406 Y. YAZAKI [Vol. 93, The Sangokan source materials contain docu- somewhat different, and the obtained form of the ments on the Klein–Nishina formula, with assigned perturbed solution is also apparently different from numbers of [166] through [218]. These documents that in their joint paper. contain Klein’s memorandums, Nishina’s memo- Source [198]: Calculation memorandums (from the randums, and correspondences between Klein and 1st stage to the middle of the 3rd stage) Nishina, Waller and Nishina, Skobeltzyn and Nishina, The method taken to find the perturbed solu- Skobeltzyn and Jacobsen. tions is the same as that taken in their joint paper, 5.1 Source materials concerning to Klein– though here are contained in part errors in sign. Nishina’s joint paper. No draft or manuscript of Source [204]: Calculation memorandums (the 1st the Klein–Nishina paper was found in the sources. stage) This is probably because Klein undertook the writing The method taken to find the perturbed solution of the final manuscript, although it is clear from is the same as that in their joint paper with the Klein’s letters to Nishina (Ref. 23 and 24) that they correct answer. discussed closely until they reached the final stage of Source [212]: Calculation memorandums by Klein the manuscript, just before Nishina’s departure from (the 1st stage) Copenhagen. Nevertheless, we find a note in Nishina’s (i) the method in terms of the Fourier trans- handwriting (source No. [174]) corresponding to a formation semidraft with expository writing in German, in (ii) the same method as their joint paper are given. which the main points before the calculation of the (b) The second stage (Expression of the vector intensity of the scattered wave were elaborately potential of the scattered wave). stated. This indicates important suggestions con- Source [197]: Calculation memorandums (from the cerning the development of their ideas to treat the 1st stage to the middle of the 3rd stage) final states of the electron, discussed later in Sec. 6. The development process is discussed in detail, Now we divide the evolution of Klein–Nishina’s and the result is derived correctly, except for the theory into four stages, with sources at each stage: point discussed below. 1. Finding solutions of the Dirac equation per- Source [198]: Calculation memorandums (from the turbed by an external electromagnetic field 1st stage to the middle of the 3rd stage) ([KN-23]–[KN-27]). The calculation is advanced under an incorrect 2. Formulating an expression of the vector poten- expression for J F ce?(;2<)A. In the aforementioned tial of the scattered wave ([KN-36]). note [174], only the result obtained in [197] is written 3. Formulating the vector potential and magnetic down, and in both sourcespffiffiffiffiffiffiffiffiffiffi the Jacobian of the field of the scattered wave in the form of a linear transformation factor, 1= 0, is missing. This combination of d F u(0)v(pB), s F u(0) Source [201-(i)]: Calculation memorandums (from Source [185]: The angular distribution of the total the 2nd stage to the middle of the 4th stage). intensity of the scattered wave (Values due to Klein– Following the primitive method of calculation Nishina and Dirac or Gordon are given.) already mentioned, |the vector potential|2 was Source [187]: Scattering coefficient and transmission calculated. They probably intended to take the attenuation factor (Values due to Klein–Nishina and average over the spin directions in the final state, Dirac or Gordon are given.) 0 0 ð 6¼ Þ and hence the cross terms uiuk i k of the 4 The experimental data are given in sources [186] components of the wave function of the electron in and [188]. the final state were dropped out, which is not valid. Source [186]: Data of the intensity ratio of .-rays In Ref. 23, Klein wrote to Nishina that he made emitted from RaC. an important improvement on their theory concern- Source [188]: Data of the angular distribution of ing two points after Nishina left Copenhagen: One scattering and the transmission attenuation factor. was related to the handling of the final states of In addition sequential letters including exper- the electron, which will be discussed later; the other imental data and preprints were sent from was a technical improvement of the calculation of Skobeltzyn,25) who measured the angular distribution |magnetic field|2 using the matrix 7 given in their of the scattered electrons using the Wilson chamber. joint paper (p. 865), which is the projection operator 5.2 Source materials on the Nishina’s single- 13),14) to the state of Source [175-(ii)]: Handwritten manuscript of “Wie eine genauere Betrachtung zeigt, gibt der Nishina’s Nature note. Ausdruck (14) die Tertiärstrahlintensität für Though there is little difference between the two den Fall, dass eins — gleichgültig welches — der sources, the published note is slightly different from beiden streuenden Elektronen unmagnetisiert both. In the manuscript, there is a passage that the ist, während das andere Elektron nach einer spin direction of the electron is noted, but in the beliebigen Richtung magnetisiert oder auch published note, this passage is deleted. Regarding the unmagnetisiert sein kann.” (As a more detailed error that C. Møller discovered, Nishina’s Nature consideration shows, Equation (14) ([N-14]) note was published without the correction, since it gives the expression of the intensity of the third was submitted before Møller’s check. Nishina sub- radiation in the following case, that one of the mitted the correction notice26) to Nature, however, two scattering electrons (whether for the 1st or on this point later, and Klein and Bohr provided the 2nd scattering) is not magnetized, while the intermediation for early publication of it. This is other electron is magnetized for an arbitrary indicated in the letter11),27) from Klein to Nishina. direction or this also can be unmagnetized.) Source [177]: The note including explanations in English. The calculations in [181] give confirmation of this This note seems to be made for preparation of statement. completing the paper for Zs. f. Phys. (Ref. 13), in Source [182]: Calculation memorandums which the process of the calculations are recorded in The calculation of the intensity of the scattered more detail and explanations in English are attached wave for an elliptically polarized incident wave (not concerning the important points. The contents are for the double scattering but for the single scattering) almost the same as the source [167]. is described with the spin direction in the initial state We find two sets of calculation memorandums of the electron fixed along Dz direction. For a linearly ([172] and [181]), in which the calculation of the polarized incident wave, it is shown in the Klein– whole process are recorded. These two are different in Nishina paper,8) that the intensity of the scattered the treatment of the initial conditions. wave is independent on the spin direction in the Source [172]: Calculation memorandums (for the initial state of the electron. On the other hand, for an whole process) elliptically polarized incident wave treated in source Similar to the published paper, the calculations [182], the intensity of the scattered wave depends on proceed under the condition that the spin direction in the spin direction.12) The calculation here shows the the initial state of the electron for the first scattering actual matter. Regarding this point, the following is fixed at a certain direction, l, and then the footnote was added in the Klein–Nishina paper averaging process is taken over the spin direction (Ref. 8, p. 864): Nishina would discuss this point in in the initial state of the electron for the second detail in a subsequent paper by Nishina. However, in scattering. the published paper13) by Nishina only the result of Source [181]: Calculation memorandums (for the averaging over the spin directions was presented in whole process) [N-7]. Although in [182] the dependence of the result Two calculation procedures are considered: on l is not explicitly described, where l denotes the (1) When the spin direction in the initial state of spin direction in the initial state of the electron, [182] the electron for both the first and the second is a noteworthy document because it proves that in scattering are set along the Dz direction, the same the case of l k z, at least, the calculations regarding result is obtained, as in [172], after being averaged this matter were certainly performed. Moreover, it is over the polarization of the linearly polarized shown in [182] that the Equation [N-7] is obtained incident wave. after averaging for l k Dz and l k !z. (2) When the spin direction for the first scattering As discussed above, we see that in the course is fixed along the general direction l and the spin of writing the Nishina’s single-name paper, several direction for the second scattering is fixed along the attempts were made concerning the electron spin, Dz direction, the same result as in [172] is also though not included in the published paper. obtained after averaging over the direction l.In Section 6 discusses the problem of the final states of the published paper (Ref. 13, p. 876) we find the the electron, which was the most elaborating problem statement: for Klein and Nishina. 410 Y. YAZAKI [Vol. 93, 1, A F A F 0), as a solution of the Dirac equation. fi 3 4 6. The problem of nal states For the final state with p º 0, however, eigenstates of Notes on notations: In Section 6 through Sec- to read a few days before you left Copenhagen, presence of a static magnetic field. (see ①, ④) I had tried to prove; that these functions for 4. However, they felt a little uneasy because the arbitrary velocities were those where A1 F 0 and A1 F 0 type and the A2 F 0 type solutions could A2 F 0 respectively.” not lead to some definite meaning, and the ⑥ “That this is wrong would follow from our problem remained as a pending issue. (see ②, considerations in the country already, but in ⑤, ⑨) the meantime I had forgotten some of our 5. At last Klein devised a method to derive the conclusions” solutions for p º 0 by performing the Lorentz ⑦ “I have limited myself to try to give a proper transformation on the solutions for p F 0, and treatment of the degeneration in question. The solved the problem. (This is stated in their joint result is the new §2 of the paper. I found this paper,8) §2). (see ⑦, ⑨) way of treating the degeneration rather con- The solutions for p º 0 obtained through the Lorentz venient, since it gives a simple way of finding transformation on the solutions for p F 0, A3 F 0 type and the A4 F 0 type were taken instead of to be zero, where H indicated the function the A1 F 0 type and the A2 F 0 type, because of the conjugate to the wave function, A.IfH is to be a different choice of ;i.) Hermitian conjugate of A, the above boundary As pointed out in Sec. 5.2, comparing Nishina’s condition cannot be satisfied. But Klein and handwritten manuscripts of his single-name paper Nishina seemed to have thought as follows: (Source [167]) with his published paper,13) we noticed H was to be determined as the solution of the different manners of regulating the final states in conjugate equation to that of A, and the several passages: the direction of the magnetic coefficients of the 4 components of H were to moment of the electron is more clearly specified in be determined independently of the coefficients the manuscript. This also is consistent with citations of the 4 components of A; therefore, they chose ③, ④ in the letter of 27th Oct. 1928.24) those coefficients so as to satisfy the boundary Eigenstates in the presence of a static magnetic condition, and obtained discrete energy eigen- field. Though not mentioned in their joint paper, values. We now know that this is wrong, but Klein and Nishina made considerable efforts in trying here we can see the situation at that time when to solve the Dirac equation under a static magnetic the treating method of the 4 components of the field. As discussed above, it is obvious that their wave function of the Dirac equation was not yet efforts were related to the problem of the final states. established. Klein seemed to put a lot of time on trials because his 6. Superposing Bessel function-type eigensolutions memorandum contained itemization of the whole thus obtained, they constructed a plain-wave trial procedure (Source [212-(iii)]), and his calcula- solution of a free electron in the limit of a tion memorandums had indicated many trials magnetic field H ! 0 and R !1. As a result, (Source [194], [205]–[209]). Because these calculation they showed that the A3 F 0 type and the memorandums consist of fragmental assemblies, we A4 F 0 type solutions were obtained only in the can hardly read the line of his thoughts from these. case of px F 0 and py F 0. But in Nishina’s note (Source [195]) the whole Nishina’s note closed with the above statements. procedure of the trials is systematically explained in Note that regarding the problem of the final states, English, like a draft of a paper, from which we can unless px F py F 0, the A3 F 0 type and the A4 F 0 conjecture the course of their trials to solve this type solutions cannot be obtained in the limit H ! 0, problem. We give an outline below: R !1. This seems to be consistent with the citation 1. At first, they wrote down the Dirac equation in ⑥ in the Klein’s letter dated 2nd Dec. 1928: “That the presence of the vector potential, which gives this (the A1 F 0 type and the A2 F 0 type solutions the static magnetic field parallel to z. are eigenfunctions in the presence of a magnetic field) 2. They then showed that the z component of the is wrong would follow from our considerations in the angular momentum, Jz F lz D (!/2) 1 @F @F X 2 _ ¼ ! _ ¼ ½ 2 vfnvni br ;ibr 7-2 wi!f ¼ f ; ½7-8 i! @b @br ! r n Ei En ! v ; Since br and i br are regarded to be canonically where denotes the interaction term and f denotes conjugate variables with each other, according to the density of states. Taking ðe=cÞx_ A for v, we can the general operating procedure for quantization we discuss the scattering of light accompanied by the request as transition of an electron inside of an atom (the dispersion). ½b ;i!b ¼i! ; ½b ;b ¼½b ;b ¼0 ½7-3 r s rs r s r s This theory of dispersion by Dirac can be applied Here, the second quantization is finished. Re- directly to Compton scattering with modifications ikr writing the above into a relation between Nr and 3r, on two points. One is to attach the factor e , which we obtain expresses the space dependence to the vector potential, and the other is to rewrite the Hamiltonian ½N ; ¼ ; ½N ;N ¼½ ; ¼0 ½7-4 r s rs r s r s of the electron into that of the Dirac equation as The Hamiltonian F can be rewritten with N and 3. e H ¼ c < p þ A mc2: ½7-9 The state of the whole system can be expressed by 1 c 3 assigning the occupation number of each eigenstate Here, the interaction term becomes e 1< A and of the single system, which can be written as the state of the electron is expressed by the solution ð 0 0 Þ N1;N2; ... . The Schrödinger equation is given by of the Dirac equation for a free electron. Then, the transition probability on the Compton scattering can @ ðN0 ;N0 Þ ! 1 2 ¼ ð Þ ð 0 0 Þ ½ be obtained by using Dirac’s theory of dispersion. i @ F N; N1;N2; 7-5 t Naturally, Klein and Nishina should have known According to the relation [Nr, 3s] F /rs, well about these works of Dirac concerning the i 0 0 0 quantization of an electromagnetic field, because e ! r ðN ;N ; ;N ; Þ 1 2 r Dirac wrote a paper on field quantization31) during ¼ ð 0 0 0 Þ ½ N1;N2; ;Nr 1; 7-6 his stay in Copenhagen and in the Nishina’s ! is derived and we can see that e i r= are the colloquium notes11) dated 26 Jan. and 28 Jan. 1927, operators that express the creation and annihilation Dirac’s lectures on this work are recorded. Dirac’s of particles. The basic setting of the field quantiza- paper was received on 2nd Feb. 1927. Moreover, in tion is thus finished. autumn of 1927, Klein worked with Jordan34) to At last, in Ref. 33 (1927), he developed a expand Dirac’s method of field quantization to a dispersion theory using second-order perturbation mutually interacting Bose system, so that he should theory after formulating the interaction between the be considerably interested in the problem of field electron and the electromagnetic field. By applying quantization. In fact, on the introduction of the field quantization to the photon system, we can write Klein–Nishina’s joint paper8) they commented on down the vector potential A with N and 3. In the Dirac’s radiation theory as “Man wird erwarten, dass long-wavelength approximation, the factor e’ikr can die eine Berücksichitigung der strahlungsdämpfung be dropped out, and we obtain erlaubt, in diesem Falle ein übereinstimmendes X 2 1=2 Resultat gibt, wenn es sich um die erste Näherung !c i i A ¼ e 3=2 ð 1=2 ! r þ ! r 1=2Þ ½ in bezug auf die Intensität der Primärstrahlung rL Nr e e Nr ; 7-7 12),47) r r handelt.” (It will be expected that the radiation where exponential factor 3r appears from rewriting theory given by Dirac, which can afford to take ! ! cos(2:8rt)as1=2ðei r= þ e i r= Þ, which is the factor radiation damping into account gives, in this case, 1=2 fi of time dependence, and the factor containing Nr the same result if the rst-order approximation in is derived by considering the relation between the the intensity of incident radiation is applied). energy flux and the Poynting vector of the photon Nevertheless, as far as Nishina’s source materials with frequency 8. The term of interaction between an are concerned, we could not find any traces on their electron and an electromagnetic field can be written trials of adopting Dirac’s theory. as ðe=cÞx_ A from the term ðe=mcÞp A, which 7.2 Reasons for Klein and Nishina’s adoption appears in the development of (1/2m)(p D e/c A)2. of the semi-classical treatment. Here we consider In second-order perturbation theory, the transition why Klein and Nishina adopted the semi-classical probability is derived as way instead of the more consistent method based on No. 6] How the Klein–Nishina formula was derived: Based on the Sangokan Nishina Source Materials 415 ZZZ quantization of the radiation field. For this, we report three reasons, as described below: kðrÞ k0 ðrÞdr The first reason: Klein was one of the founders of ZZZ k r k0 r ¼ 3 i i r ¼ 0 ½ the semi-classical treatment. As noted in Sec. 1, Klein L e e d k;k 7-10 formulated in his paper5) (1927) the general method ðL3Þ to treat the interaction between charged particles In this case, when the periodic boundary and an electromagnetic field on the basis of the semi- condition is applied, k takes the discrete value classical treatment; hence he should be called the ((2:/L)n1,(2:/L)n 2,(2:/L)n3). The other is the /- founder of the semi-classical treatment (the Klein– function-type normalization, which claims for con- Gordon equation was also derived here). In this tinuous values of k in an infinite space that paper, he also derived the conservation laws of energy 3=2 ik r kðrÞ¼ð2 Þ e and momentum upon Compton scattering on the ZZZ ZZZ 3 ik r ik0 r basis of that method, while Gordon published his kðrÞ k0 ðrÞdr ¼ð2 Þ e e dr paper,4) in which the intensity of the Compton 0 scattering was treated on the basis of the Klein– ¼ ðk k Þ½7-11 Gordon equation about two months before Klein’s In Gordon’s paper on Compton scattering,4) as paper. Mentioning these former works, Klein wrote, stated in Sec. 2.2, the /-function type normalization in his private note (Ref. 15) “And naturally we had is adopted, while in Dirac’s quantization of electro- both of us been thinking how this effect would come magnetic field, it is necessary to make the wave out according to Dirac’s new theory.” number k of radiation field discrete: When both Thus it was not doubtful that Klein was electron and radiation fields are taken into consid- attached to the semi-classical treatment, and he eration, the consistency of the concept of space is would like to adopt this method also in the case of the spoiled unless the wave vectors of the wave function Dirac equation. Moreover, as noted in Sec. 1, Klein of electron are treated as being discrete under the and Nishina intended to verify the validity of Dirac’s same boundary condition as that for radiation. For relativistic theory of the electron in their paper. For that purpose, the periodic boundary condition for a this purpose, it was certainly convenient to treat the Dirac electron should be adopted, as is usually done problem on the basis of the semi-classical treatment today. However, the concept of the periodic boun- using the Klein–Gordon equation and the Dirac dary condition might not be well understood when equation, and to check the difference of the results. Klein and Nishina were working on the Compton The second reason: The problem of normalization. scattering, because it was an artificial device made up We discuss the normalization of the solution of the for mathematical convenience. The other method to Dirac equation for a free electron. Klein wrote in his make wave vectors discrete is a standing-wave-type private note (Ref. 15), “I shall mention here one boundary condition that claims the value of the wave difficulty which now would seem trivial, namely that function to be zero at a boundary surface. This is not possible to include a Dirac electron in a box method has a definite physical meaning. Note that (and which led me later to a well-known paradoxon the density of states assigned by the set of discrete [sic]). Finally we decided on a procedure earlier used wavenumbers is the same for the standing-wave-type by Gordon on the advise of Pauli.” The passage “that and periodic boundary conditions. This is probably is not possible to include a Dirac electron in a box” the reason to accept the periodic boundary condition means that if we set A F 0 on a certain boundary even though the physical meaning was not clear. surface, such a solution allowing A º 0 inside of the However, the standing-wave-type boundary condi- surface does not exist, because the Dirac equation is tion could not be adopted for a Dirac electron a first order differential equation. Because of this because of the reason stated above. Hence, the statement, Klein and Nishina seemed to think that grounds for accepting the periodic boundary con- for a Dirac electron such normalization could not be dition became infirm. Namely, the validity for taking allowed as to produce a discrete wave number k of discrete wave vectors was ambiguous. Klein might the eikr type solution expressing the eigenstate of the have meant this situation when he mentioned “one momentum. There are two kinds of procedures to difficulty”. Here, we add some remarks concerning normalize the eikr type solution: One is the box the acceptance of the periodic boundary condition. normalization, which claims for an eigenfunction Born and Dirac adopted this early in their studies !3/2 ikr 3 Ak(r) F L e in a finite volume, L on quantum mechanics. In fact, Born devised the 416 Y. YAZAKI [Vol. 93, periodic boundary condition before quantum me- ses differed in the following points: Tamm adopted chanics appeared. He introduced it to calculate the the quantization of an electron field as well as an density of states in Ref. 35 dealing with the classical electromagnetic field relying on Heisenberg–Pauli’s treatment of lattice vibration. This might be the paper,32) while Waller adopted a method in which origin of the periodic boundary condition. He used it the electron field was not quantized. Both Waller and to discuss the probability interpretation of quantum Tamm emphasized that it was essentially necessary mechanics (Ref. 36), particularly in the calculation of to take the negative energy states into account as the density of states. Dirac also claimed the validity intermediate states.12),47) They stated that especially of the periodic boundary condition in Ref. 30 to in the limit of h8 = mc2, the contributions to the express functions defined in the finite region in terms intensity of the scattered wave were given exclusively of a Fourier series. On the other hand, in the paper by these negative energy intermediate states, and as (Ref. 37) arguing the Klein’s paradox, Klein stated a result the classical Thomson-scattering formula was that in the case of the Dirac equation, the standing- obtained. This indicates that negative energy states wave-type boundary condition could not be applied must be considered as physically meaningful states. without mentioning the periodic boundary condition, The necessity of the negative energy states in this which probably was not in his mind. manner was regarded as a paradox at that time, called In the Tamm’s paper,38) which is discussed later, “Waller’s paradox”. Another paradox was “Klein’s the periodic boundary condition was adopted for the paradox” 37): when an electron with a positive energy Dirac electron. However, he added an annotation entered into a step-type potential barrier, the part of explaining Born’s idea of a periodic lattice. Since this the electron invading inside of the barrier tended to paper was published in 1930, the periodic boundary have a negative energy state if the potential barrier condition did not seem to be so broadly accepted, was sufficiently high. Klein mentioned this point in his even then. private note,15) as sited in Sec. 7.2, in regard to the The third reason: Question on negative energy second reason. There are two letters from Waller to states. As noted in Sec. 3, there are two solutions of Nishina in the Sangokan Nishina Source: One was the Dirac equation for a given momentumpffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi value: one dated 17 May 1929,42) in which Waller wrote “I have with the positive energy eigenvalue m2c4 þ c2p2 been working chiefly on the scattering theory”. The andpffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi the other with the negative energy eigenvalue other was dated 25 Jan. 1930,43) in which Waller m2c4 þ c2p2. When Dirac’s relativistic electron wrote on the negative energy states “I have recently theory6) appeared, this negative energy state was not been doing some things on scattering which tends to interpreted validly, and it was generally regarded as be my hobby. It will perhaps interest you to hear that being physically meaningless. In fact, we can find the the Dirac scattering theory gives for free electron just words “den physikalish nicht sinnvollen negativen your and Klein’s formula, if states of negative energy Energiewerten” (physically meaningless negative en- are properly taken into account”. And in the margin ergy values) in the Klein–Nishina’s joint paper of this letter Nishina wrote a draft44) of a reply in (Ref. 8, p. 856). It was surely inevitable that in English, and referred to Waller’s work as follows: 1928, the above view was generally accepted, because “I am very much interested to hear about your results in 1930 Dirac presented his hole theory,40) which was on the scattering, and anxious to see the details of verified in 1932 by discovery of the positron. If Klein your work. From Heisenberg’s lecture at Copenhagen and Nishina had taken up Dirac’s scattering theory I also thought that the states of negative energy must as the basis, they should have taken into account all be taken into account in order to get the correct result of the possible states as intermediate states. Then, but I did not expect that you would obtain the same was it possible for them to exclude the negative formula as ours. I am looking forward to your paper.” energy states at this stage? They seemed to avoid Waller made a note on Heisenbeg’s lecture in the Dirac’s scattering theory because of this question. footnote of his paper (Ref. 41, p. 844) “In einer The answer to this question was given by Waller41) nichit veröffentlichten Untersuchung hat Prof. W. and Tamm,38) as described below. Heisenberg schon gefunden, dass in dem Grenzfall, 2 7.3 Treatments based on field quantization. wo h8s/mc gegen eins vernachlässigt wird, die In 1930 Waller41) and Tamm38) independently carried klassische Streuformel wegen der Zwischenzustände out calculations for Compton scattering on the basis negativer Energie erhalten wird” (In an unreleased of field quantization and scattering theory, and work Prof. W. Heisenberg had already found that in 2 reproduced the Klein–Nishina formula. Their analy- the limiting case of h8s/mc negligibly smaller than No. 6] How the Klein–Nishina formula was derived: Based on the Sangokan Nishina Source Materials 417 unity, the classical scattering formula is obtained by When Waller’s theory and Tamm’s theory are the negative energy intermediate states). It must be reconsidered on the basis of hole theory, the same before October 1928 that Nishina listened to this result of the scattering cross section can be obtained Heisenberg’s lecture in Copenhagen, and therefore with only simple formal changes.12),39) Waller, him- this Heisenberg’s foreknowledge must have been self, stated on this matter in the footnote of Ref. 41 obtained at rather an earlier time point. and Dirac did also the same in Ref. 40. As for the problem of normalization, which was From the above arguments it became clear that a worrying matter for Klein and Nishina, Waller the Klein–Nishina formula can be reproduced by did not seem to worry and used the /-function using field quantization or hole theory. It was already type normalization for the electron, while he dealt stated in Sec. 1 that this formula played an with the wave vector of the electromagnetic field as important role to support of the Dirac equation being discrete. Tamm, on the other hand, used the and, moreover, Waller’s theory and Tamm’s theory periodic boundary condition for both the electron should be recognized as having reinforced that role. and the electromagnetic field, and thus treated space – ’ consistently. 8. The equivalence between the Klein Nishina s Here, we examine how Waller and Tamm dealt theory and the theory based fi with the problem of choosing the final state of an on the eld quantization electron that Klein and Nishina worried about most. In Klein–Nishina’s theory, the negative energy First, Waller adopted four independent final states of solutions of the Dirac equation were not taken into an electron with a given momentum as four solutions account. On the contrary, it is already noted that of the same type as those of Klein and Nishina, in the theory based on the field quantization (see 45) namely one of A1, A2, A3, A4 being equal to zero, and for instance, W. Heitler 3rd ed. Chap. V, §22, used these in actual calculations. He clearly stated pp. 211–224), the negative energy states play a very that the orthogonality condition was necessary for important role as intermediate states. Nevertheless, choosing four final states, and moreover, added at the both theories give the same result for the scattering end that the intensity of the scattered wave did not cross section of a photon as the Klein–Nishina depend on the concrete form of the four final states formula. To clarify this seemingly paradoxical as long as they satisfied the orthonormal condition. matter, we rewrite here Klein–Nishina’s theory into On the other hand, Tamm established a calculation the form in which the intermediate states appear, and method, which was capable of obtaining the intensity thus reveal the equivalence between both theories. of the scattered wave using orthonormality alone, At first, in Eq. [KN-36], which gives the vector and did not use any concrete forms of the four potential A(p, pB) of the scattered wave (where p independent states. However, he also mentioned an and pB are the momentum of the electron in the example of concrete forms, which was the same as initial and final states, respectively), we rewrite that used by Waller, namely four solutions, each one as follows: uðpÞ!uyðpÞ;fðpÞ!g yðpÞ;vðpÞ! of which had one of A1, A2, A3, A4 equal to zero uðpÞ; 1< ! ,; ! !. We then obtain – – (Klein Nishina type solutions). A0ðp ! p0Þ Waller and Tamm might have reached the e i!0ðt rÞ y 0 y 0 orthogonality condition because they used Dirac’s / e c u ðpÞ½,gðp Þþg ðpÞ, uðp Þ; ½8-1 scattering theory (time-dependent perturbation r theory). Nevertheless, the following general view- where g(p)andg ðpÞ are modified parts of the point Tamm stated in his paper (Ref. 38, p. 549) solutions for a free electron interacting with the would not have been established at the time of field of the incident wave, as given in [KN-23]. The Klein–Nishina’s calculations: “Die vier Komponenten scattered wave propagating in the direction nB has den Wellenfunktion A können als Komponenten eines two directions of polarization. Denoting one of them Vektors in einen vier dimensoionalen Hilbertraum as eB, we obtain the intensity of the scattered aufgefapt werden” (The four components of wave radiation of polarization, eB,as function A can be grasped as components of a vector !02 I ¼ jA0ðp ! p0Þ e0j2 in a four dimensional Hilbert space). During this 4 c period of a little longer than one year, the conceptual / e2juyðpÞ½ð, e0Þgðp0Þþg yðpÞð, e0Þ uðp0Þj2 understanding of Dirac’s four-component wave func- ¼ 2j yðp0Þ½ yðp0Þð, e0Þþð, e0Þ ðpÞ ðpÞj2 ½ tion progressed very much. e u g g u : 8-2 418 Y. YAZAKI [Vol. 93, Determining g(p) and g ðpÞ as below, we will show Using the properties of matrix , the denominator in that the quantity in | |2 in Eq. [8-2] is just equal Eq. [8-10] is reduced to the c-number [E(p) ! !B]2 ! to the transition matrix element in the Heitler–like [E(p ! (!B/c)n)]2, and thus we need not find the method. inverse matrix. This was the reason why Klein and We first rewrite the fundamental equations Nishina used the quadratic Eq. [8-9] instead of appearing in Sec. 2. The Dirac equation is given by Eq. [8-3]. However, to reveal the equivalence to the @ Heitler–like method, it is rather convenient to adopt i! ðH e, AÞ ½ ðpÞþ ðpÞ ¼ 0 Eq. [8-8] without change. Substituting Eq. [8-8] into @t 0 0 1 Eq. [8-2], we obtain @ e H ¼ c, i! A ; ½8-3 y 0 1 0 0 @r jj2 ¼ a2e4 u ðp Þ ð, e Þ À Áð, e Þ c 0 ð ðp0Þ ! Þ p0 !!n E ! H0 c where A denotes the vector potential of the incident 1 2 þð, e0Þ À Áð, e Þ uðpÞ wave ð ðpÞþ! Þ p þ !!n 0 E ! H0 c ð n r Þ ð n r Þ A ¼ e ½ i! t c þ i! t c ½ a 0 e e 8-4 ½8-11 and A0(p) denotes the solution that expresses the Generally, in the eigenstate |ni of H0(p) that satisfies initial state of the electron as H0(p)|ni F E(p)|ni, there are two kinds of states: i ðp r ðpÞ Þ ðpÞ¼ ! E t poneffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi belongs to the positive energy value E 0ðpÞ¼uðpÞe c2p2 þ m2c4 and thep otherffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi belongs to the negative H ðpÞuðpÞ ðc, p þ mc2ÞuðpÞ¼EðpÞuðpÞ 0 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi energy value EðpÞ¼ c2p2 þ m2c4. For each kind EðpÞ¼ c2p2 þ m2c4 > 0 ½8-5 of states, there are two mutually independent states Using [8-4] for A,inthefirst-order approximation in in terms of the spin direction. These four eigenstates parameter a, Eq. [8-3] can be reduced to span a complete system. Therefore, denoting ∑p as @ the summation over these four eigenstates for a given 45) i! H ðpÞ¼ e, A ðpÞ½8-6 p following Heitler, we can insert ∑pB! !B n|nihn| @t 0 1 0 ( /c) for the first term in Eq. [8-11] and ∑pD(!B/c)n|nihn| for 2 Setting A1(p) in the following form the second term after 1/(E ! H0). Thus | | in the i!ðt n r Þ i!ðt n r Þ right-hand side of Eq. [8-11] reduces to ðpÞ¼½ ðpÞ c þ ðpÞ c ðpÞ ½ 1 g e g e 0 ; 8-7 X ðuyðp0Þð, e Þu Þðuy ð, e0ÞuðpÞÞ then g(p) and g ðpÞ are determined according to [8-4], 0 n nÀ Á ð ðp0Þ ! Þ p0 !!n ð Þ E ! En [8-5], [8-6], and [8-7] as n c 2 3 2 X ð yðp0Þð, e0Þ Þð y ð, e Þ ðpÞÞ 1 u un un À 0 uÁ 6 gðpÞ¼ ð e, e0Þa 7 þ ½8-12 6 !! 7 ðEðpÞþ!!Þ E p þ !!n 6 ðEðpÞ !!Þ H p n 7 ðnÞ n c 6 0 7 6 c 7 By the conservation laws 6 7 6 1 7 0 ðpÞ¼ ð , e Þ 0 !! 0 !! 6 g e 0 a 7 p þ n ¼ p þ n; 4 !! 5 c c ðEðpÞþ!!Þ H0 p þ n c Eðp0Þþ!!0 ¼ EðpÞþ!! ½8-13 ½ 8-8 we obtain pB ! (!B/c)n F p ! (!BB/c)nB, E(pB) ! In the Klein–Nishina’s joint paper, they rewrote the !B F E(p) ! !BB. linear order Eq. [8-3] into the quadratic equation as Using above, we rewrite the denominator of @ @ the first term of Eq. [8-12] as E(p) D !B ! i! H i! H ¼ 0; ½8-9 [E (p ! (!BB/c)nB) D !B D !BB]. We then recognize @t @t n that the first term of Eq. [8-12] coincides with the and determined g, g using Eq. [8-9] instead of transition matrix element in the Heitler theory,45) Eq. [8-3]. They thus obtained g(p)andg ðpÞ instead where the intermediate state corresponds to the of Eq. [8-8] as emission of a photon at the final state at first 1 and the electron having momentum p ! (!BB/c)nB. gðp޼ À ÁàÀ Áà ð ðpÞ ! Þ p !!n ð ðpÞ ! Þþ p !!n Likewise, the second term of Eq. [8-12] coincides with E ! H0 c E ! H0 c  À Áà the transition matrix element in which the inter- ðEðpÞ !!ÞþH p !!n ð e, e Þa: ½8-10 0 c 0 mediate state corresponds to the absorption of a No. 6] How the Klein–Nishina formula was derived: Based on the Sangokan Nishina Source Materials 419 photon of the initial state at first and an electron 5) Klein, O. (1927) Elektrodynamik und Wellenmecha- having momentum p D (!B/c)n. We can thus show nik vom Standpunkt des Korrespondenzprinzips. Zs. f. Phys. 41, 407–442. that Eq. [8-12] just coincides with the transition 6) Dirac, P.A.M. (1928) The quantum theory of matrix element in the Heitler theory, namely the electron. Proc. R. Soc. Lond. A 117, 610–624. Eq. [H-18] (Equation (18) in Ref. 45, p. 215) given by 7) Dirac, P.A.M. (1928) The quantum theory of 0 0 electron. Part II. Proc. R. Soc. Lond. A 118, Ek2 X ðu u Þðu u Þ ¼ 4 0 0 351–361. d e d 0 k2 þ k E 8) Klein, O. and Nishina, Y. (1929) Über die Streuung 0 0 ðu u00Þðu00 u Þ 2 von Strahlung durch freie Elektronen nach der þ 0 0 ½ neuen relativistischen Quantendynamik von Dirac. 00 ; H-18 k E Zs. f. Phys. 52, 853–868. where d? denotes the differential cross section of a 9) Klein, O. and Nishina, Y. (1928) The scattering of light by free electrons according to Dirac’s new photon scattered by an electron initially at rest and – 7 F 2 F !B F !BB relativistic dynamics. Nature 122, 398 399. mc , k0 ,andk . The reason why no 10) Bohr, N. (1934) Letter to Nishina, Y. 26. Jan. negative energy states appeared explicitly in Klein– Copenhagen. In Bohr, N. (1984) Y. Nishina’s Nishina’s theory was that they adopted a quadratic Correspondence with N. Bohr and Copenhageners; Eq. [8-9] to find the wave function of an electron 1928–1949. Publication No. 20, the Nishina – modified by the incident wave, and thus reducing the Memorial Foundation, Tokyo. pp. 31 33. In Bohr, N. (2006) Collected Correspondence of denominator of Eq. [8-10] to a c-number. As a result, Yoshio Nishina, Vol. I. (eds. Nakane, R., Nishina, in the calculation of matrix elements, it became Y., Nishina, K., Yazaki, Y. and Ezawa, H.). possible to use only the initial and final states having Misuzu Shobo, Tokyo, pp. 330–332 (translation positive energys only for the bras | i and kets h |. in Japanese). 11) Ezawa, H. (1990) The apostle who witnessed the Acknowledgements genesis. Nihon Butsuri Gakkaishi 45, 744–751 (in Japanese). The main part of this article is based on the 12) Ekspong, G. (1991) Oskar Klein and Yoshio Nishina. former two papers46),47) written in Japanese by the In Evolutionary Trends in the Physical Sciences present author in 1992. Concerning these papers, the (eds. Suzuki, M. and Kubo, R.). Springer-Verlag, – author would like to thank the late Dr. Hidehiko Berlin Heidelberg, pp. 25 34. 13) Nishina, Y. (1929) Die Polarisation der Compton- Tamaki, the late Dr. Fukutaro Shimamura, the late streuung nach der Diracschen Theorie des Elek- Dr. Hajime Takeuchi for inviting him to being a trons. Zs. f. Phys. 52, 869–877. member of their investigation group and giving him 14) Nishina, Y. (1928) The polarisation of Compton opportunities to access the Nishina Source Materials. scattering according to Dirac’s new relativistic Since these three persons were former collaborators of dynamics. Nature 122, 843. 15) Klein, O. (1975) To the memory of Yoshio Nishina. Dr. Yoshio Nishina in RIKEN, the author owed them A private note sent from O. Klein to the Nishina valuable information about Dr. Nishina. He would Memorial Foundation in Feb. 1975 according to also like to thank Dr. Hiroshi Ezawa for valuable the request by S. Tomonaga. Translation by discussions on these papers. He is also grateful to Koizumi, K. In Klein, O. (1990) Nihon Butsuri – Dr. Yoshimasa A. Ono for carefully reading and Gakkaishi 45, 720 721 (in Japanese). Also In Klein, O. (1992) Nishina Yoshio — Nihon no correcting his English manuscripts of this article. genshikagaku no akebono (the dawn of atomic Thanks are due to The Nishina Memorial Foundation science in Japan) (eds. Tamaki, H. and Ezawa, H.). for permitting the author to utilize Nishina Source Misuzu Shobo, Tokyo, pp. 93–97 (in Japanese). Materials retained there. 16) Koizumi, K. (1976) Nishina Yoshio in era of study in Europe. Shizen 31 (November), 58–67 (in References Japanese). 17) Nishina, Y. and Rabi, I.I. (1928) Der wahre 1) Compton, A.H. (1923) A quantum theory of the Absorptionskoeffizient der Röntgen Strahlen scattering of X-rays by light elements. Phys. Rev. nach der Quantentheorie. Verhandl. der Deutsch. 21, 483–502. Physik. Gesell. (3) 9,6–9. 2) Breit, G. (1926) A correspondence principle in the 18) Nishina, Y. (1928) Letter to Bohr, N. 19. Feb. Compton effect. Phys. Rev. 27, 362–372. Hamburg. In Nishina, Y. (1985) Y. Nishina’s 3) Dirac, P.A.M. (1926) Relativity quantum mechanics Letters to N. Bohr, G. Hevesy, and others; 1923– with an application to Compton scattering. Proc. 1928. Publication No. 21. the Nishina Memorial R. Soc. Lond. A 111, 405–423. Foundation, Tokyo, p. 22. Also In Nishina, Y. 4) Gordon, W. (1927) Der Comptoneffekt nach der (2006) Collected Correspondence of Yoshio Schrödingerschen Theorie. Zs. f. Phys. 40, 117–133. Nishina. Vol. I. (eds. Ezawa, H. et al.). Misuzu 420 Y. YAZAKI [Vol. 93, Shobo, Tokyo, pp. 80–81 (translation in Japanese). Tokyo, pp. 114–115 (translation in Japanese). 19) Brown, L.M. and Rechenberg, H. (1987) Paul Dirac 28) (for instance) Landau, L. and Lifshitz, E. (1974) and Werner Heisenberg–a partnership in science. Quantum Mechanics (Non Relativistic Theory) In Paul Adrien Maurice Dirac (eds. Krusunoglu, (Theoretical Physics Course, Vol. III, revised B.N. and Wigner, E.P.). Cambridge Univ. Press, edition) Chap. 15, §112. Cambridge et al., pp. 117–162. 29) Ezawa, H. and Takeuchi, H. (1991) A chronological 20) Nishina, Y. (1928) Letter to Dirac, P.A.M. 25. Feb. personal history of Yoshio Nishina In Nishina Hamburg. In Nishina, Y. (2011) Collected Corre- Yoshio — Nihon no genshi kagaku no akebono (the spondence of Yoshio Nishina, Sup. Vol. (eds. dawn of atomic science in Japan) (eds. Tamaki, H. Ezawa, H. et al.). Misuzu Shobo, Tokyo, pp. 15– and Ezawa, H.). Misuzu Shobo, Tokyo, pp. 273– 16 (translation in Japanese). 300 (in Japanese). 21) Tamaki, H. (1987) Our study on Yoshio Nishina. 30) Dirac, P.A.M. (1926) On the theory of quantum KAGAKUSI KENKYU Ser. II. 26, 184–186 (in mechanics. Proc. R. Soc. Lond. A 112, 661–677. Japanese). 31) Dirac, P.A.M. (1927) The quantum theory of the 22) Takeuchi, H. and Yazaki, Y. (1990) Source materials emission and absorption of radiation. Proc. R. Soc. of Dr. Yoshio Nishina. Nihon Butsuri Gakkaishi Lond. A 114, 243–265. 45, 766–769 (in Japanese). 32) Heisenberg, W. and Pauli, W. (1929) Zur Quanten- 23) Klein, O. (1928) Letter to Nishina, Y. 2. Dec. dynamik der Wellenfelder. Zs. f. Phys. 56,1–61. Copenhagen. Sangokan Source [358]. In Klein, O. 33) Dirac, P.A.M. (1927) The quantum theory of (1986) Supplement to the Publications No. 17, dispersion. Proc. R. Soc. Lond. A 114, 710–728. No. 20, No. 21. Publication No. 27. the Nishina 34) Jordan, P. and Klein, O. (1927) Zum Mehrkörper- Memorial Foundation, Tokyo, pp. 2–4. In Klein, O. problem der Quantentheorie. Zs. f. Phys. 45, 751– (2006) Collected Correspondence of Yoshio Nishina, 765. Vol. I. (eds. Ezawa, H. et al.). Misuzu Shobo, 35) Born, M. (1922) Atomtheorie des festen Zustandes Tokyo, pp. 104–108 (translation in Japanese). (Dynamik d. Kristallgitter); §II Dynamik In 24) Klein, O. (1928) Letter to Nishina, Y. 27. Oct. Encyklopedie der Mathematischen Wissenschaften. Copenhagen. Sangokan Source [173–2]. In Klein, V. Physik, 3teil. B.G.Teubner, Lipzig, pp. 574– O. (1986) Publication No. 27. the Nishina Memo- 596. rial Foundation, Tokyo, p. 1. In Klein, O. (2006) 36) Born, M. (1926) Quantenmechanik der Stop- Collected Correspondence of Yoshio Nishina vorgänge. Zs. f. Phys. 38, 803–827. Vol. I. (eds. Ezawa, H. et al.). Misuzu Shobo, 37) Klein, O. (1929) Die Reflexion von Elektronen an Tokyo, pp. 99–101 (translation in Japanese). einem Potentialsprung nach der relativistischen 25) Skobeltzyn, D. (1928) Letter to Jacobsen, J.C. 6. Sep. Dynamik von Dirac. Zs. f. Phys. 53, 157–165. Leningrad. Sangokan Source [344]; Skobeltzyn, D. 38) Tamm, Ig. (1930) Über die Wechselwirkung der (1928) Letter to Jacobsen, J.C. 10. Sep. Leningrad. freien Elektronen mit der Strahlung nach der Sangokan Source [345], In Skobeltzyn, D. (2006) Diracschen Theorie des Elektrons und nach der Collected Correspondence of Yoshio Nishina, Quantenelektrodynamik. Zs. f. Phys. 62, 545–568. Vol. I. (eds. Ezawa, H. et al.). Misuzu Shobo, 39) Ezawa, H. (1978) The hole theory. In Quantum Tokyo, pp. 83–85; pp. 85–86 (translation in Mechanics I (Iwanami Fundaments of Modern Japanese). Physics Course), 2nd ed., Vol. 3. (eds. Yukawa, H. Skobeltzn, D. (1928) Letter to Nishina, Y. 23. Sep. and Toyoda, T.). Iwanami Shoten, Tokyo, §7.7, Leningrad. Sangokan Source [346]; Skobeltzyn, D. pp. 420–429 (in Japanese). (1928) Letter to Nishina, Y. 2. Oct. Leningrad. 40) Dirac, P.A.M. (1930) A theory of electrons and Sangokan Source [347]; Skobeltzyn, D. (1928) protons. Proc. R. Soc. Lond. A 126, 360–365. Letter to Nishina, Y. 16. Oct. Leningrad. Sangokan 41) Waller, I. (1930) Die Streuung durch gebundene und Source [348]; Skobeltzyn, D. (1929) Letter to freie Elektronen nach der Diracschen relativisti- Nishina, Y. 24. Feb. Leningrad. Sangokan Source schen Mechanik. Zs. f. Phys. 61, 837–851. [379]; Skobeltzyn, D. (1929) Letter to Nishina, 42) Waller, I. (1929) Letter to Nishina, Y. 17. May Y. 12. Jul. Paris. Sangokan Source [416]. In Upsala. Sangokan Source [400]. In Waller, I. (2006) Skobeltzyn, D. (2006) Collected Correspondence Collected Correspondence of Yoshio Nishina, of Yoshio Nishina, Vol. I. (eds. Ezawa, H. et al.). Vol. I. (eds. Ezawa, H. et al.). Misuzu Shobo, Misuzu Shobo, Tokyo, pp. 86–89; pp. 89–93; p. 93; Tokyo, p. 139 (translation in Japanese). pp. 118–120; p. 151 (translation in Japanese). 43) Waller, I. (1930) Letter to Nishina, Y. 25. Jan. 26) Nishina, Y. (1929) Polarisation of Compton scatter- Upsala. Sangokan Source [469] In Waller, I. (2006) ing according to Dirac’s new relativistic dynamics. Collected Correspondence of Yoshio Nishina, Nature 123, 349. Vol. I. (eds. Ezawa, H. et al.). Misuzu Shobo, 27) Klein, O. (1929) Letter to Nishina, Y. 23. Jan. Tokyo, p. 191 (translation in Japanese). Copenhagen. Sangokan Source [375]. In Klein, O. 44) Nishina, Y. (1930) Letter to Waller, I. (hand written (1986) Publication No. 27, the Nishina Memorial draft) Tokyo. Sangokan Source [469]. In Nishina, Foundation, Tokyo, pp. 4–5. In Klein, O. (2006) Y. (2006) Collected Correspondence of Yoshio Collected Correspondence of Yoshio Nishina, Nishina, Vol. I. (eds. Ezawa, H. et al.). Misuzu Vol. I. (eds. Ezawa, H. et al.). Misuzu Shobo, Shobo, Tokyo, p. 194 (translation in Japanese). No. 6] How the Klein–Nishina formula was derived: Based on the Sangokan Nishina Source Materials 421 45) Heitler, W. (1954) The Quantum Theory of Radia- 47) Yazaki, Y. (1992) How was the Klein–Nishina tion, 3rd ed. Clarendon Press, Oxford, pp. 211– formula derived? II. KAGAKUSI KENKYU Ser. 224. II. 31, 129–137 (in Japanese). 46) Yazaki, Y. (1992) How was the Klein–Nishina formula derived? I. KAGAKUSI KENKYU Ser. II. 31,81–91 (in Japanese). (Received Dec. 23, 2016; accepted Mar. 23, 2017)