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How the Klein–Nishina Formula Was Derived: Based on the Sangokan Nishina Source Materials

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How the Klein–Nishina Formula Was Derived: Based on the Sangokan Nishina Source Materials 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) e204 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): 8c 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 À 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 À Þ3 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 ’ À 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 À grad ’ À Á 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.
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