2017 Volume 93 Issue 6 Pages 399-421
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.
In Compton scattering, the scattering of an X-ray by an electron, the wavelength of the scattered X-ray varies with the scattering angle. In 1923, using the light quantum theory for X-rays, A.H. Compton1) explained the wavelength shift upon scattering by using the conservation principle of energy and momentum of the light quantum and electron system. This analysis showed that the Compton effect was one of the important experimental facts confirming the quantum theory of light.
In his analysis concerning the angular distribution of the intensity of the scattered wave, Compton used relativistic theory and the Doppler effect. In 1926, Breit2) discussed this problem using the correspondence principle in the old quantum theory, and obtained the same result as what Dirac3) and subsequently Gordon4) obtained independently, using quantum mechanics.
They deduced the energy and momentum conservation law on the light-quantum and electron system quantum mechanically, which Compton hypothesized, and derived a formula giving the angular distribution of the intensity of the scattered wave. In 1927, Klein5) discussed how the interaction between an electron and an electromagnetic field including Compton scattering should be treated quantum mechanically. As for Compton scattering, he obtained energy-momentum conservation for the Compton scattering, but did not show the intensity distribution of the scattered wave. In 1927, Gordon4) and Klein5) independently developed the so-called Klein–Gordon equation to describe the behavior of an electron to include relativistic effects in Compton scattering. The energy and momentum relation of an electron used by Dirac3) was equivalent to that of the Klein–Gordon equation.
Soon after Dirac presented his relativistic electron theory,6),7) O. Klein and Y. Nishina succeeded to derive the famous Klein–Nishina formula,8),9) calculating the intensity distribution of the scattered wave in the Compton scattering based on the Dirac equation. The Klein–Nishina formula has been firmly accepted and widely used, even now. When Klein and Nishina started to attack this problem, they intended in part to confirm the validity of the Dirac equation, as stated in Introduction of Ref. 8. Recalling that time, Bohr wrote to Nishina in 193410),11) that “the striking confirmation which this formula has obtained became soon the main support for the correctness of Dirac’s theory when it was apparently confronted with so many grave difficulties.” Ekspong discussed in Ref. 12, that the experiments conducted to confirm the Klein–Nishina formula were to suggest the existence of then unknown phenomena, namely the pair production and annihilation of positive and negative electrons.
The following arguments in this article are based on the Sangokan Nishina Source, preserved by the Nishina Memorial Foundation. In Section 2 through Section 5, we survey Klein–Nishina’s theory and consider the process they undertook when deriving their formula. In Section 6 and Section 7, we look back their efforts to solve the most difficult problem for them, namely how to set the final states of the electron after scattering, and consider why they adopted the semi-classical method in treating electromagnetic waves. Finally, in Section 8, we discuss the equivalence between Klein–Nishina’s semi-classical method and the quantum field theoretical method.
In this section we give the energy and momentum conservation law on the Compton scattering presented by Compton, himself, and the equation of the frequency shift accompanying the scattering. The energy and momentum conservation laws were regarded as being a basic hypothesis in Compton’s1) and Breit’s2) papers. In Section 2 through Section 5 we adopt the notations used by Klein and Nishina8): h denotes Planck’s constant divided by 2π and ν denotes frequency multiplied by 2π, while m and −e are the mass and charge of an electron and c is the velocity of light in a vacuum.
The energy and momentum of a light quantum having frequency ν and propagating direction n are given by hν and (hν/c)n. Suppose the frequency and propagating direction of a light quantum before and after the scattering are ν, n and ν′, n′, respectively, and the momenta of an electron before and after the scattering are 0 and p, respectively, then the energy and momentum conservation laws are given by
| \begin{align} &h\nu + mc^{2} = h\nu' + \sqrt{c^{2}p^{2} + m^{2}c^{4}},\quad \\ &(h\nu/c)\boldsymbol{{n}} = (h\nu'/c)\boldsymbol{{n}}' + \boldsymbol{{p}} \end{align} | [2-1] |
| \begin{equation} \nu' = \frac{\nu}{1 + \alpha(1 - \cos\theta)},\quad \left(\alpha = \frac{h\nu}{mc^{2}}\right) \end{equation} | [2-2] |
Dirac3) treated Compton scattering based on quantum mechanics for the first time, using his q-number algebraic method. As the first step, he developed a general theory of a quantum mechanical multi-periodic system.
For simplicity, we take here a system in which the degree of freedom is unity. The periodic system is defined by the following properties 1), 2) and 3) (we consider a system consisting only electrons without any electromagnetic field):
| \begin{equation} I = \frac{e^{2}\nu'^{4}}{8\pi c^{3}r^{2}}|C_{\alpha}(J')|^{2}, \end{equation} | [2-3] |
We apply the above-mentioned general theory to an electron in an incident electromagnetic wave. Taking into account only the oscillating term, we can apply the definition of the periodic system to the electron considered here. Using his proper mathematical insight, Dirac succeeded to construct a set of canonical variables, J and w, which describe scattering with the original variables xi, pi, t, and E. The transition from J′ to J′ − h corresponds to a scattering process; in addition he showed that ν′ = [H(J′) − H(J′ − h)]/h, ΔJ = −h and ΔJ2 = ΔJ3 = 0, where J2 and J3 were remaining variables commutable with J corresponding to the energy and momentum conservation laws. Then the intensity of the scattered wave can be obtained using Eq. [2-3] as
| \begin{equation} I = \frac{e^{4}}{r^{2}m^{2}c^{4}}I_{0}\frac{\sin^{2}\phi}{[1 + \alpha (1 - \cos\theta)]^{3}}, \end{equation} | [2-4] |
We now explain Gordon’s theory in some detail, since his work has a close relation to Klein–Nishina’s work. Gordon treated the scattering of an electromagnetic wave by an electron based on the correspondence principle, which means that at first writing down the classical equations of the interaction between an electron and an electromagnetic wave, and then replacing the quantities, such as the current density or the charge density, with the corresponding quantum mechanical expressions. Gordon’s theory can be characterized by compromising the use of classical and quantum mechanical theories, which came to be called a “semi-classical treatment” later.11),12) Gordon discussed the dynamics of an electron in an electromagnetic field using purely classical theory and quantum-mechanical theory in parallel; he confirmed that the results based on quantum-mechanical theory for an electromagnetic field produced by an electron coincided with those based on classical theory in the limit of Planck’s constant, where h → 0. Gordon called this treatment as the “correspondence principle”, and took it as a basis for the validity of the semi-classical treatment, while Klein5) called the semi-classical treatment, itself, as the “correspondence principle method”.
(a) Klein–Gordon equation.In classical theory, the basic equations of an electron in an electromagnetic potential A and V are given by the relativistic Hamilton–Jacobis’ equation,
| \begin{equation*} -\frac{1}{c^{2}}\left(\frac{\partial W}{\partial t} - eV\right)^{2} {}+{} \sum_{k = 1}^{3}\left(\frac{\partial W}{\partial t} + \frac{e}{c}A_{k}\right)^{2} {}+{} m^{2}c^{2} = 0 \end{equation*} |
| \begin{equation*} \frac{\partial W}{\partial x_{k}} \to \frac{h}{i} \cdot \frac{\partial}{\partial x_{k}},\quad -\frac{\partial W}{\partial t} \to ih\frac{\partial}{\partial t} \end{equation*} |
We then obtain the following Klein–Gordon equation:
| \begin{align} &{\biggl[-\frac{1}{c^{2}}\biggl(ih\frac{\partial}{\partial t} + eV\biggr)^{2} + \sum_{k = 1}^{3}\biggl(\frac{h}{i}\cdot\frac{\partial}{\partial x_{k}} + \frac{e}{c}A_{k}\biggr)^{2} + m^{2}c^{2}\biggr]\psi} \\ &{\quad{}= 0} \end{align} | [2-5] |
We define ψ as a solution of Eq. [2-5] and φ as its complex conjugate. We can then express the charge density, ρ, and the current density, J, as
| \begin{align} &\rho = \frac{ie}{c^{2}}\left(\varphi\frac{\partial \psi}{\partial t} - \psi\frac{\partial \varphi}{\partial t} + \frac{2ie}{h}V\varphi\psi\right),\quad\\&\boldsymbol{{J}} = \frac{e}{i}\left(\varphi\mathrm{grad}\psi - \psi\mathrm{grad} \varphi - \frac{2i}{h}\cdot \frac{e}{c}\boldsymbol{{A}}\varphi\psi\right) \end{align} | [2-6] |
We conjecture that the electromagnetic potential produced by an electron can be obtained by inserting ρ and J into the expression of the retarded potential in classical electromagnetic theory,
| \begin{equation} \boldsymbol{{A}}' = \frac{1}{c}\int\frac{[\boldsymbol{{J}}]_{t - \frac{R}{c}}}{R}d\boldsymbol{{r}},\quad V' = \int\frac{[\rho]_{t - \frac{R}{c}}}{R}d\boldsymbol{{r}}, \end{equation} | [2-7] |
We assume here that the incident wave is a linearly polarized plane wave, and denote its electromagnetic potential as Φi = ai cos ν(t − nr/c), where i = 0, 1, 2, 3; Φ0 = V, Φk = Ak.
The solution of the Klein–Gordon Equation [2-5] can be obtained in the perturbed form of the solution of a free electron with momentum p,
| \begin{align*} \psi(\boldsymbol{{p}}) &= C(\boldsymbol{{p}})e^{\frac{i}{h}(\boldsymbol{{p}}\cdot\boldsymbol{{r}} - Et)}\biggl(1 + [\text{quantity of $O(a_{i})$}] \\ &\quad\times\sin \nu \left(t - \frac{\boldsymbol{{n}}\cdot\boldsymbol{{r}}}{c}\right)\biggr), \end{align*} |
| \begin{equation*} \Psi = \int z(\boldsymbol{{p}})C(\boldsymbol{{p}})\psi(\boldsymbol{{p}})d\boldsymbol{{p}}. \end{equation*} |
| \begin{align*} \Phi_{i}' &= \frac{1}{r}\Re\int [\text{quantity of $O(a_{i})$}]\\ &\quad \times z(\boldsymbol{{p}})z(\boldsymbol{{p}}')e^{\frac{i}{h}(\boldsymbol{{P}} - \boldsymbol{{P}}') \cdot \boldsymbol{{r}}' - i\nu'(t - \frac{r}{c})}d\boldsymbol{{p}}d\boldsymbol{{p}}'d\boldsymbol{{r}}', \end{align*} |
| \begin{align*} &\nu' = \frac{E + h\nu - E'}{h},\quad \boldsymbol{{P}} = \boldsymbol{{p}} + \frac{h\nu}{c}\boldsymbol{{n}} - \frac{E + h\nu}{c}\boldsymbol{{n}}',\quad\\ &\boldsymbol{{P}}' = \boldsymbol{{p}}' - \frac{E'}{c}\boldsymbol{{n}}'. \end{align*} |
| \begin{align} &\Phi_{i}'(\boldsymbol{{p}} \to \boldsymbol{{p}}') \\ &\quad= \frac{ec}{2r} \cdot \frac{[\text{quantity of $O(a_{i})$}]}{\sqrt{E\Delta E'\Delta'}} \cdot \cos\nu'\left(t - \frac{r}{c}\right), \end{align} | [2-8] |
Klein and Nishina8),9) calculated the intensity of the scattered wave in the Compton scattering based on the same framework as Gordon and Klein, except for adopting the Dirac equation6) as the equation for describing the behavior of an electron. In the following we briefly discuss how they did this. Note that we use the same equation numbers as those used by Klein and Nishina in their paper,8) for example [KN-1].
(a) Dirac equation.Under the electromagnetic field (A, V) the Dirac equation is written as
| \begin{align} &\left[\frac{1}{c}\left(ih\frac{\partial}{\partial t} + eV \right) + \rho_{1}\left(\boldsymbol{{\sigma,}} - ih\frac{\partial}{\partial \boldsymbol{{r}}} + \frac{e}{c}\boldsymbol{{A}}\right) + \rho_{3}mc \right]\psi \\ &\quad = 0, \end{align} | [KN-1] |
| \begin{equation} \rho = -e\varphi\psi,\quad\boldsymbol{{J}} = ec\varphi \rho_{1}\boldsymbol{{\sigma}}\psi, \end{equation} | [KN-3] |
The Dirac equation for a free electron is obtained by setting A = V = 0 in [KN-1]; and the solution is written in the following form:
| \begin{equation} \psi_{0}(\boldsymbol{{p}}) = v(\boldsymbol{{p}})e^{-\frac{i}{h}(Et - (\boldsymbol{{p}}\boldsymbol{{r}}))}, \end{equation} | [KN-4] |
| \begin{equation} \left[\frac{E}{c} + \rho_{1}(\boldsymbol{{\sigma}}\boldsymbol{{p}}) + \rho_{3}mc\right]v(\boldsymbol{{p}})=0. \end{equation} | [KN-5] |
| \begin{equation*} v^{+}(0) = (2\pi h)^{-3/2} \begin{pmatrix} 1\\ 0\\ 0\\ 0 \end{pmatrix} ,\quad v^{-}(0) = (2\pi h)^{-3/2} \begin{pmatrix} 0\\ 1\\ 0\\0 \end{pmatrix} . \end{equation*} |
| \begin{equation} v^{\pm}(\boldsymbol{{p}}) = S^{-1}(\boldsymbol{{p}})v^{\pm}(0). \end{equation} | [KN-18] |
We write down the perturbed solution of the Dirac equation under an incident radiation field described by the vector potential,
| \begin{equation} \boldsymbol{{A}} = \boldsymbol{{a}}e^{i\nu (t - (\boldsymbol{{n}}\boldsymbol{{r}})/c)} + \text{c.c.} \end{equation} | [KN-22] |
| \begin{equation} \psi(\boldsymbol{{p}}) = [1 + g(\boldsymbol{{p}})e^{i\nu(t - (\boldsymbol{{n}}\boldsymbol{{r}})/c)} + \bar{g}(\boldsymbol{{p}})e^{-i\nu(t - (\boldsymbol{{n}}\boldsymbol{{r}})/c)}]\psi_{0}(\boldsymbol{{p}}), \end{equation} | [KN-23] |
| \begin{align} g(\boldsymbol{{p}}) &= [\text{sum of three terms including ($\boldsymbol{{a}}\boldsymbol{{p}}$), }\\ &\quad\text{$\boldsymbol{{\sigma}}$ and $\boldsymbol{{a}}$, $\rho_{1}\boldsymbol{{\sigma}}$ and $\boldsymbol{{a}}$}] \end{align} | [KN-25] |
Superposing the perturbed solutions described above, we obtain the general solution
| \begin{equation} \Psi = \int\psi(\boldsymbol{{p}})d\boldsymbol{{p}},\quad \Phi = \int\varphi(\boldsymbol{{p}})d\boldsymbol{{p}} \end{equation} | [KN-28] |
| \begin{equation} \boldsymbol{{J}} = ec\Phi\rho_{1}\boldsymbol{{\sigma}}\Psi = ec\iint\varphi(\boldsymbol{{p}})\rho_{1}\boldsymbol{{\sigma}}\psi (\boldsymbol{{p}}')d\boldsymbol{{p}}d\boldsymbol{{p}}' \end{equation} | [KN-29] |
| \begin{align} \boldsymbol{{A}}(\boldsymbol{{p}}, \boldsymbol{{p}}') &= \frac{e(2\pi h)^{3}}{r\sqrt{\Delta\Delta'}}(e^{i\nu'(t - \frac{r}{c})}u(\boldsymbol{{p}})[\rho_{1}{\boldsymbol{{\sigma}}}g(\boldsymbol{{p}}') \\ &\quad+ f(\boldsymbol{{p}})\rho_{1}\boldsymbol{{\sigma]v}}(\boldsymbol{{p}}') + \textit{c.c.}) \end{align} | [KN-36] |
| \begin{equation} \boldsymbol{{H}}_{0} = [\text{sum of linear terms in $d$, $\boldsymbol{{s}}$}]e^{i\nu'(t - \frac{r}{c})} + \textit{c.c.} \end{equation} | [KN-44] |
The intensity of the scattered wave, I, is given by the time average of the Poynting’s vector,
| \begin{equation*} I = \frac{c}{4\pi}[\boldsymbol{{E}}_{0}\times \boldsymbol{{H}}_{0}]_{\text{time average}} = \frac{c}{4\pi}[\boldsymbol{{H}}_{0}^{2}]_{\text{time average}} \end{equation*} |
| \begin{equation*} v(\boldsymbol{{p}}') = v^{+}(\boldsymbol{{p}}')e^{-i\delta_{1}} + v^{-}(\boldsymbol{{p}}')e^{-i\delta_{2}} \end{equation*} |
| \begin{equation} \left[ \begin{array}{l} \overline{(\boldsymbol{{B}}\boldsymbol{{s}})(\boldsymbol{{C}}\bar{\boldsymbol{{s}}})} = \gamma[(\boldsymbol{{B}}\boldsymbol{{C}}) + i\boldsymbol{{l}} \cdot (\boldsymbol{{B}} \times \boldsymbol{{C}})]\\ \overline{(\boldsymbol{{B}}\boldsymbol{{s}})\bar{d}} = \overline{d(\boldsymbol{{B}}\bar{\boldsymbol{{s}}})} = \gamma (\boldsymbol{{l}}\boldsymbol{{B}}),\quad \text{etc.} \end{array} \right] \end{equation} | [KN-56] |
| \begin{equation} \overline{\boldsymbol{{H}}_{0}^{2}} = \frac{e^{4}}{m^{2}c^{4}r^{2}}\left(\frac{\nu'}{\nu}\right)^{3}\left[\left(\frac{\nu}{\nu'} + \frac{\nu'}{\nu}\right)\boldsymbol{{\varepsilon}}^{2} - 2(\boldsymbol{{n}}'\boldsymbol{{\varepsilon}})^{2}\right], \end{equation} | [KN-58] |
| \begin{align} I &= I_{0}\frac{e^{4}}{m^{2}c^{4}r^{2}} \cdot \frac{\sin^{2}\phi}{(1 + \alpha (1 - \cos\theta))^{3}}\\ &\quad\cdot\left[1 + \alpha^{2}\frac{(1 - \cos\theta)^{2}}{2\sin^{2}\phi (1 + \alpha (1 - \cos\theta))}\right] \end{align} | [KN-59] |
| \begin{equation*} \frac{d\sigma}{d\Omega} = \frac{1}{2} \cdot \frac{e^{4}}{m^{2}c^{4}}\left(\frac{\nu'}{\nu}\right)^{2}\left[\left(\frac{\nu}{\nu'} + \frac{\nu'}{\nu}\right) - 2(\boldsymbol{{n}}'\boldsymbol{{e}}_{0})^{2}\right] \end{equation*} |
Immediately after joint papers with Klein, Nishina by himself wrote subsequent papers13),14) concerning polarizations of the scattered electromagnetic wave. In the classical theory for Thomson scattering, electron spins are not considered, and a linearly polarized incident wave remains to be linearly polarized even after it is scattered by an electron. The same condition holds, in Dirac3) and Gordon’s4) theories. On the other hand, in the Klein–Nishina theory based on the Dirac equation, the incident wave is linearly polarized, but the scattered wave is generally elliptically polarized, consisting of two mutually incoherent elliptical polarizations, produced by the transition of the electron to two mutually independent final states. This point is clearly different from the result of Dirac’s and Gordon’s theories. Therefore, experiments on this can confirm the validity of Klein–Nishina theory and, moreover, of the Dirac equation. For this purpose, using the experimental conditions at that time, Nishina calculated the intensity distribution of the doubly scattered wave with the scattering direction being normal to each other on the result of the Klein–Nishina work.
Here, we follow Nishina’s calculation. First we rewrite the expression for the magnetic field of the scattered wave [KN-44] in Klein–Nishina’s paper8); we separate the scattered wave into two mutually independent elliptically polarized components in the following way. In this section we use the same equation numbers as those used in Nishina’s paper13):
| \begin{align} \boldsymbol{{H}}_{0} &\propto (\boldsymbol{{A}} - i\boldsymbol{{B}})e^{i\nu'(t - \frac{r}{c} + \delta_{1})} - (\boldsymbol{{D}} + i\boldsymbol{{C}})e^{i\nu'(t - \frac{r}{c} + \delta_{2})} \\ & \quad+ \textit{c.c.}, \end{align} | [N-2] |
| \begin{align} \boldsymbol{{E}}_{0} &\propto (\boldsymbol{{A}}' - i\boldsymbol{{B}}')e^{i\nu'(t - \frac{r}{c} + \delta_{1})} - (\boldsymbol{{D}}' + i\boldsymbol{{C}}')e^{i\nu'(t - \frac{r}{c} + \delta_{2})} \\ &\quad+ \textit{c.c.} \end{align} | [N-3] |
| \begin{equation} \boldsymbol{{E}} = (\boldsymbol{{\varepsilon}}_{a} + i\boldsymbol{{\varepsilon}}_{b})e^{i\nu(t - \frac{(\boldsymbol{{nr}})}{c})} + \textit{c.c.} \end{equation} | [N-5] |
| \begin{align} I &= \frac{e^{4}}{4\pi m^{2}c^{3}r'^{2}} \cdot \frac{1}{(1 + \beta)^{3}}\biggl[\frac{2 + 2\beta + \beta^{2}}{1 + \beta}(\boldsymbol{{\varepsilon}}_{a}^{2} + \boldsymbol{{\varepsilon}}_{b}^{2}) \\ &\quad- 2[(\boldsymbol{{n}}''\boldsymbol{{\varepsilon}}_{a})^{2} + (\boldsymbol{{n}}''\boldsymbol{{\varepsilon}}_{b})^{2}]\biggr], \end{align} | [N-7] |
| \begin{align} I_{t} &= \frac{e^{8}}{2m^{4}c^{8}r^{2}r'^{2}}\frac{I_{0}}{(1 + 2\alpha)^{3}}\\ &\quad\times\left[\sin^{2}\Theta + \frac{\alpha^{2}(2 + 4\alpha + 3\alpha^{2})}{2(1 + \alpha)^{2}(1 + 2\alpha)}\right], \end{align} | [N-14] |
Concerning the background of the start of Klein and Nishina’s work, Klein wrote in his private note15) as follows:
Our nearer acquaintance, which then grew into friendship began in the spring 1928, I believe a little before Eastern[sic]. He had just returned from a visit to Hamburg then, where he had met Pauli and Gordon, while I had been in Cambridge to see Dirac, whose first paper on the electron had just appeared. At about the same time also Gordon came on a little visit to the Bohr institute. He and I had earlier worked on the Compton effect independently of each other — he with Schrödinger in Berlin, I in Copenhagen — his paper with the complete solution appearing before mine was finished. And naturally we had both of us been thinking how this effect would come out according to Dirac’s new theory. I remember Gordon saying, when we all three were standing together: “Das mag ein gutes Problem für Herrn Nishina sein” (This might be a good problem for Mr. Nishina), and I, who had intended to attack that problem myself, agreed immediately.
As for Nishina, this problem was related to his previous work at Copenhagen on an experimental study of the absorption spectra of X-rays. Since then he had constantly paid attention to X-ray spectroscopy studies, although he became deeply interested in theory at the advent of quantum mechanics.16) Moreover, in the fall of 1927, he moved to Hamburg, and then in collaboration with I.I. Rabi he theoretically investigated the absorption coefficient of X-rays11),17) while learning quantum theory under Pauli. He was thus deeply involved in X-ray research. Therefore, it was natural that the idea of treating Compton scattering based on the Dirac equation occurred to Nishina in Hamburg where they encountered the appearance of the Dirac equation with great enthusiasm.11),18) Indeed he wrote to Dirac on 25th Feb. 1928,19),20) “I hope to calculate Compton effect according to your new theory”, while he was studying in Hamburg before he came back to Copenhagen in March, 1928.
Here, in Sec. 5, we trace the approach Klein and Nishina took to obtain the Klein–Nishina formula using the “Sangokan Nishina Shiryo” (Building No. 3 Nishina source materials) retained in the Nishina Memorial Foundation, Tokyo. Their details have been reported in References 21 and 22: how they became noticed, how they were investigated, and a summary of their contents. The main part of the Sangokan source materials consists of Nishina’s documents concerning his study abroad, especially in Copenhagen from 1923 to 1928, such as memorandums concerning his investigations, colloquium notes, correspondences.
In the Nishina Memorial Foundation, a voluntary investigation group, called “Tamaki Group”, was set up by Hidehiko Tamaki together with Fukutaro Shimamura, Hajime Takeuchi and Yuji Yazaki. Besides a historical investigation on Nishina’s study abroad, they took up the task to list and assign consecutive numbers to each of the Sangokan Nishina Shiryo, for which H. Takeuchi was responsible. In 1991, they prepared a pamphlet called “Sangokan Nishina Shiryo no Naiyou Ichiran” (the contents of the Sangokan Nishina source materials). This pamphlet was edited by H. Takeuchi, T. Iwaki, H. Tamaki, F. Shimamura and Y. Yazaki. In 1994 this pamphlet was printed by the Nishina Memorial Foundation.
The Sangokan source materials contain documents on the Klein–Nishina formula, with assigned numbers of [166] through [218]. These documents contain Klein’s memorandums, Nishina’s memorandums, and correspondences between Klein and Nishina, Waller and Nishina, Skobeltzyn and Nishina, Skobeltzyn and Jacobsen.
5.1 Source materials concerning to Klein–Nishina’s joint paper.No draft or manuscript of the Klein–Nishina paper was found in the sources. This is probably because Klein undertook the writing of the final manuscript, although it is clear from Klein’s letters to Nishina (Ref. 23 and 24) that they discussed closely until they reached the final stage of the manuscript, just before Nishina’s departure from Copenhagen. Nevertheless, we find a note in Nishina’s handwriting (source No. [174]) corresponding to a semidraft with expository writing in German, in which the main points before the calculation of the intensity of the scattered wave were elaborately stated. This indicates important suggestions concerning the development of their ideas to treat the final states of the electron, discussed later in Sec. 6.
Now we divide the evolution of Klein–Nishina’s theory into four stages, with sources at each stage:
Source [197]: Calculation memorandums (from the 1st stage to the middle of the 3rd stage)
The method taken to find the perturbed solution is essentially equivalent to that in the Klein–Nishina’s paper,8) though the procedure to reform the Dirac equation into the 2nd-order equation is somewhat different, and the obtained form of the perturbed solution is also apparently different from that in their joint paper.
Source [198]: Calculation memorandums (from the 1st stage to the middle of the 3rd stage)
The method taken to find the perturbed solutions is the same as that taken in their joint paper, though here are contained in part errors in sign.
Source [204]: Calculation memorandums (the 1st stage)
The method taken to find the perturbed solution is the same as that in their joint paper with the correct answer.
Source [212]: Calculation memorandums by Klein (the 1st stage)
Source [197]: Calculation memorandums (from the 1st stage to the middle of the 3rd stage)
The development process is discussed in detail, and the result is derived correctly, except for the point discussed below.
Source [198]: Calculation memorandums (from the 1st stage to the middle of the 3rd stage)
The calculation is advanced under an incorrect expression for J = ceϕ(ρ2σ)ψ. In the aforementioned note [174], only the result obtained in [197] is written down, and in both sources the Jacobian of the transformation factor, $1/\sqrt{\Delta \Delta '} $, is missing. This factor, which appeared in the Gordon’s paper,4) is necessary to obtain the correct intensity. In the Klein–Nishinas’ paper8) they stated that the integration was performed using the variable transformation “nach dem Vorgang von Gordon” (according to the precedent by Gordon) (Ref. 8, p. 861). However, since in the source [196] they at last took the Jacobian transformation factor into account, they must have realized the necessity of the factor sometime later.
Source [200]: Calculation memorandums (from the 2nd stage to the middle of the 3rd stage)
They adopted the expression of J, not ceϕ(ρ1σ)ψ, but the same form written in terms of wave function and its derivatives as that used by Klein5) and Gordon,4) and they calculated the vector potential. Klein’s calculation memorandums for deriving such an expression of J are found in the source [203], and Nishina’s calculation memorandums for the same purpose are found in fragmented assembly sources [201] and [214]. Though this expression of J is not convenient for the calculation, they might have been interested in it because of former theories by Klein and Gordon, and tried to examine such a procedure. We note that minor errors exist in the expression of J in the above-mentioned sources: [200], [203], [201] and [214].
(c) The third stage (Concrete expression of the vector potential and the magnetic field of the scattered wave).While in the second stage the expression of the vector potential contains 2nd-order terms in ρ1 and σ, in this third stage a technique to transform the vector potential into the linear combination of d = u(0)v(p′), s = u(0)σv(p′) and $\bar{d}$, $\bar{\boldsymbol{{s}}}$ was employed based on the properties of the spin operator, σ, and the solution of the Dirac equation for free electrons.
Source [192]: Calculation memorandums (from the 3rd stage to the middle of the 4th stage)
Correct expressions are derived, except for the Jacobian factor. The result obtained here is taken into the note [174], which is mentioned in the beginning of this subsection 5.1.
Sources [197], [198], [200]: Calculation memorandums (from the 1st stage to the middle of the 3rd stage)
The vector potential is transformed into linear forms in σi, though there remains ρi. Therefore, these sources should be regarded as representing the middle stage of the formulation.
Source [201-(i)]
The source [201] is an assembly of the fragmented calculation memorandums. We classified and reorganized it into several parts, each of which has some meaningful contents.
[201-(i)] is the calculation memorandums for the stages from the 2nd stage to the middle of the 4th stage. In the 2nd stage the vector potentials are not expressed in terms of d and s, as above, but are expressed by substituting the direct matrix expressions into ρ1 and σ in the expression of the vector potential [KN-36]: the vector potential is expressed in terms of the components of the wave functions of the initial and final states: u3, u4; $u'_{1}$, $u'_{2}$, $u'_{3}$ and $u'_{4}$. Here, a primitive calculation method is adopted instead of using the properties of spin operators. The result gives the correct expression of the vector potential, except for the Jacobian factor mentioned above. This method, however, does not give a clear view, and has to be judged as being inferior to that of their joint paper8) from the view point of the beauty of the theory. Therefore, this calculation method should be judged as no more than a primitive attempt.
(d) The fourth stage (|Magnetic field of the scattered wave|2, the intensity of the scattered wave).Since the expression for the magnetic field of the scattered wave obtained in the 3rd stage consists of 8 terms, each of which contains d and s in different ways and takes complicated forms containing scalar products and vector products, the algebraic calculation of the square of the magnetic field became very tedious. Regarding this point, Klein wrote an episode in Ref. 15:
But we still had to make some lengthy algebraic calculations in order to get the practical result. This involved the addition of a great many terms, and to be certain not to make any mistakes in the summation we decided to work separately each in his place, and finally our independent results agreed.
Source [189]: Calculation memorandums (the 4th stage, interrupted in the middle)
This source contains the same sort of calculation as those in their joint paper, mistakes in the calculations interrupted their calculation.
Source [196]: Calculation memorandums (the 4th stage, completed)
This is the only one source showing the procedure to the final result of the intensity of the scattered wave. The Jacobian is taken into account. These memorandums indicate that to proceed calculations properly they needed to manage to reshape the terms that appeared in the development, and to accumulate them into compact forms, using the relations between ν, ν′ (the frequencies of the incident and scattered wave) and n, n′ (the propagating directions of the incident and scattered wave), instead of blindly calculating.
There are two sources that show different approaches from their joint paper.
Source [192]: Calculation memorandums (from the 3rd stage to the middle of the 4th stage)
Including the rotation operator in spin space, and averaging over the spin direction of the electron in the initial state led them to derive the calculation rules for the intensity, similar to [KN-56]. On the other hand, in their joint paper they fixed the spin direction, l, in the initial state and for a linearly polarized incident wave, they showed that the final intensity of the scattered wave was independent of the spin direction, l. The method in [192] could have led to the correct result, but the calculation of the intensity is interrupted in the middle.
Source [201-(i)]: Calculation memorandums (from the 2nd stage to the middle of the 4th stage).
Following the primitive method of calculation already mentioned, |the vector potential|2 was calculated. They probably intended to take the average over the spin directions in the final state, and hence the cross terms $u'_{i}\overline{u'_{k}}(i \ne k)$ of the 4 components of the wave function of the electron in the final state were dropped out, which is not valid.
In Ref. 23, Klein wrote to Nishina that he made an important improvement on their theory concerning two points after Nishina left Copenhagen: One was related to the handling of the final states of the electron, which will be discussed later; the other was a technical improvement of the calculation of |magnetic field|2 using the matrix μ given in their joint paper (p. 865), which is the projection operator to the state of σz = 1/2 (or σz = −1/2) with E > 0 in the static system of the electron*. Indeed, this method of using the matrix μ was not adopted in the source [196].
According to the matters discussed above, we managed to understand how the formulation methods and the calculation techniques in each stage were developed through various gropings, especially the calculation techniques using the properties of the spin operators and handling the spin states.
The solution of the Dirac equation under a static magnetic field.Although not mentioned in their joint paper, there are sources showing that Klein and Nishina attempted to find a solution of the Dirac equation under a static magnetic field. We will discuss later this point.
Comparison with experiment.There are sources that show numerical calculations based on the Klein–Nishina formula compared with experimental data. At that time (1928–29), γ-rays emitted from RaC were used in experiments concerning Compton scattering. They contained several wavelength components, but there were no reliable data on the ratio of each component. Accordingly, in a Klein–Nishina’s note to Nature,9) they stated that a comparison with the experiment was difficult. Nishina performed numerical calculations using the value of the wavelength and the intensity rate of each component obtained at that time.
Source [183]: The angular distribution of the intensity of the scattered wave for each component of wavelength.
Source [185]: The angular distribution of the total intensity of the scattered wave (Values due to Klein–Nishina and Dirac or Gordon are given.)
Source [187]: Scattering coefficient and transmission attenuation factor (Values due to Klein–Nishina and Dirac or Gordon are given.)
The experimental data are given in sources [186] and [188].
Source [186]: Data of the intensity ratio of γ-rays emitted from RaC.
Source [188]: Data of the angular distribution of scattering and the transmission attenuation factor.
In addition sequential letters including experimental data and preprints were sent from Skobeltzyn,25) who measured the angular distribution of the scattered electrons using the Wilson chamber.
5.2 Source materials on the Nishina’s single-name papers.13),14)In the source materials we find handwritten manuscripts of Nishina’s single-name papers and their preparation notes. These calculation memorandums are well regulated because the methods that Nishina used were already established through joint work with Klein. Therefore, these notes have different features from those of the joint papers.
Here we pick out some noteworthy sources:
Source [167]: Handwritten manuscript for Zs. f. Phys.13)
This manuscript is different from the published paper in the following two points: First, Nishina took the two final states for the electron, and stated clearly about those that one had the same spin direction as that of the initial state, and the other had the opposite direction; however, in the published paper such definite expression was deleted. This is probably because, as written in the letter from Klein to Nishina,11),24) Klein corrected this point in Nishina’s manuscript. Second, there is an error in the calculation of double scattering of the electromagnetic wave by electrons, where the frequency of the incident wave for the second scattering is different from that for the first scattering. In [167] this modification of the frequency is not included, while in the published paper this error is corrected. On this point, Klein11),23) wrote to Nishina that Bohr suggested to C. Møller that he should check Nishina’s calculation memorandums, and he discovered the error. Klein submitted the corrected version of Nishina’s paper to Zs. f. Phys. In the above-mentioned letter, Klein added the words for relief “a thing one would easily forget”.
Source [170]: Typewritten manuscript of Nishina’s Nature note.14)
Source [175-(ii)]: Handwritten manuscript of Nishina’s Nature note.
Though there is little difference between the two sources, the published note is slightly different from both. In the manuscript, there is a passage that the spin direction of the electron is noted, but in the published note, this passage is deleted. Regarding the error that C. Møller discovered, Nishina’s Nature note was published without the correction, since it was submitted before Møller’s check. Nishina submitted the correction notice26) to Nature, however, on this point later, and Klein and Bohr provided intermediation for early publication of it. This is indicated in the letter11),27) from Klein to Nishina.
Source [177]: The note including explanations in English.
This note seems to be made for preparation of completing the paper for Zs. f. Phys. (Ref. 13), in which the process of the calculations are recorded in more detail and explanations in English are attached concerning the important points. The contents are almost the same as the source [167].
We find two sets of calculation memorandums ([172] and [181]), in which the calculation of the whole process are recorded. These two are different in the treatment of the initial conditions.
Source [172]: Calculation memorandums (for the whole process)
Similar to the published paper, the calculations proceed under the condition that the spin direction in the initial state of the electron for the first scattering is fixed at a certain direction, l, and then the averaging process is taken over the spin direction in the initial state of the electron for the second scattering.
Source [181]: Calculation memorandums (for the whole process)
Two calculation procedures are considered:
(1) When the spin direction in the initial state of the electron for both the first and the second scattering are set along the +z direction, the same result is obtained, as in [172], after being averaged over the polarization of the linearly polarized incident wave.
(2) When the spin direction for the first scattering is fixed along the general direction l and the spin direction for the second scattering is fixed along the +z direction, the same result as in [172] is also obtained after averaging over the direction l. In the published paper (Ref. 13, p. 876) we find the statement:
“Wie eine genauere Betrachtung zeigt, gibt der Ausdruck (14) die Tertiärstrahlintensität für den Fall, dass eins — gleichgültig welches — der beiden streuenden Elektronen unmagnetisiert ist, während das andere Elektron nach einer beliebigen Richtung magnetisiert oder auch unmagnetisiert sein kann.” (As a more detailed consideration shows, Equation (14) ([N-14]) gives the expression of the intensity of the third radiation in the following case, that one of the two scattering electrons (whether for the 1st or the 2nd scattering) is not magnetized, while the other electron is magnetized for an arbitrary direction or this also can be unmagnetized.)
The calculations in [181] give confirmation of this statement.
Source [182]: Calculation memorandums
The calculation of the intensity of the scattered wave for an elliptically polarized incident wave (not for the double scattering but for the single scattering) is described with the spin direction in the initial state of the electron fixed along +z direction. For a linearly polarized incident wave, it is shown in the Klein–Nishina paper,8) that the intensity of the scattered wave is independent on the spin direction in the initial state of the electron. On the other hand, for an elliptically polarized incident wave treated in source [182], the intensity of the scattered wave depends on the spin direction.12) The calculation here shows the actual matter. Regarding this point, the following footnote was added in the Klein–Nishina paper (Ref. 8, p. 864): Nishina would discuss this point in detail in a subsequent paper by Nishina. However, in the published paper13) by Nishina only the result of averaging over the spin directions was presented in [N-7]. Although in [182] the dependence of the result on l is not explicitly described, where l denotes the spin direction in the initial state of the electron, [182] is a noteworthy document because it proves that in the case of l ∥ z, at least, the calculations regarding this matter were certainly performed. Moreover, it is shown in [182] that the Equation [N-7] is obtained after averaging for l ∥ +z and l ∥ −z.
As discussed above, we see that in the course of writing the Nishina’s single-name paper, several attempts were made concerning the electron spin, though not included in the published paper. Section 6 discusses the problem of the final states of the electron, which was the most elaborating problem for Klein and Nishina.
Notes on notations: In Section 6 through Section 8 we adopt usual notations: (Planck constant)/2π = ħ, frequency = ν, and 2πν = ω.
As noted in Sec. 3, due to spin degeneration, there are two independent states in the solution of the Dirac equation for a free electron, even though the values of the energy and momentum are assigned. When we consider scattering problems, we must assign a set of two mutually orthogonal states for final states of an electron as two independent states, and sum up the contribution to the intensity from each final state. Today we usually assign these two states as two eigen states of helicity pσ/p belonging to eigen values ±1, but we can assign two arbitrary, mutually orthogonal states instead. Therefore, there is a simple and clear judgment condition for the validity of taking two states: the orthogonality of these two states. However, in 1928 when Klein and Nishina finished their calculation, the concept of helicity had not yet been introduced, and furthermore the concept of orthogonality was not well-defined for the 4-component wave-function solutions of the Dirac equation. It was only six months after the appearance of the Dirac equation, and Dirac’s electron theory, itself, was considered skeptically because of the existence of negative energy states. The physical meaning of these 4-component wave functions, and treating methods of these were not well established. Therefore, instead of taking the orthogonality condition, Klein and Nishina used the following argument to treat two final states of an electron after scattering: they gave each of the two set states its own physical meaning properly, and judged that setting as being valid if transitions of the electron from the initial state to the two final states were mutually exclusive in probability theory. For instance, the following two transitions to the states were mutually exclusive, one with the same spin direction as that of the initial state, and the other with the opposite spin direction to that of the initial state. And if they had dealt with the case of the non-relativistic Schrödinger equation, they could take the eigen states of σz with the eigenvalues ±1 as its solution for arbitrary momentum, i.e., setting the eigenstate σz = 1 as the initial state and the eigenstates σz = ±1 as the two final states could be considered to be valid. However, the situation was different for the Dirac equation. For the initial state with p = 0, they could take eigenstates of σz = ±1, namely, (ψ1 = 1, ψ2 = ψ3 = ψ4 = 0) and (ψ1 = 0, ψ2 = 1, ψ3 = ψ4 = 0), as a solution of the Dirac equation. For the final state with p ≠ 0, however, eigenstates of σz were not solutions of the Dirac equation unless px = py = 0. Hence, it was impossible to assume a simple meaning of the final states: that one was in the same spin direction as the initial state, and the other was in the opposite direction. As a result, Klein and Nishina seemed to have felt some apprehension about the validity of their setting of the two final states until the final stage. At last Klein solved the problem at the very final stage. We argue about what happened based on two letters of Klein to Nishina in 1928, and testify it on the basis of other source materials. The first letter,11),24) dated 27. Oct. 1928, was found among calculation memorandums with no envelop and no creases, starting with the following text: “Just a few lines accompanying the manuscripts, which I hope reach you in time.” This sentence indicates that this letter was sent along with a copy of the manuscripts of Nishina’s single-name paper,13) which was corrected in part by Klein before submitting to Zs. f. Phys. Nishina left Copenhagen in early October of 1928 and set sail for America on 31st Oct. from Leavre29) via London and Paris. Klein and Nishina’s paper8) and Nishina’s single-name paper13) were both received by Zs. f. Phys. on 30th Oct. 1928. We pick out the points related to the problem of the final states from this letter. The word “paper” in the passages below probably means Nishina’s single-name paper. The numbers ①, ②, etc. assigned to each passage are attached by the present author.
We now mention other sources that describe the circumstances presented above. First, the method of choosing two final states of the electron as the ψ1 = 0 type and the ψ2 = 0 type remained unchanged in Nishina’s single-name paper13): Nishina stated clearly in p. 870 “Für den Endzustand wollen wir als unabhängige Eigenfunktionen diejenigen wählen, wo entweder u1, v1 oder u2, v2 gleich Null sind” (For the final states, we would like to choose as independent eigenfunctions those in which whether u1, v1 or u2, v2 are equal to zero). Moreover, in Source [174], it is stated that when taking these final states the intensity of the scattered wave comes to be independent of the spin direction of the electron in the initial state. This fact might be one support of the validity of their choice of the final states as the ψ1 = 0 type and the ψ2 = 0 type, although those states could not be interpreted as the eigenfunctions under a static magnetic field. (In fact, in Source [174], the ψ3 = 0 type and the ψ4 = 0 type were taken instead of the ψ1 = 0 type and the ψ2 = 0 type, because of the different choice of ρi.)
As pointed out in Sec. 5.2, comparing Nishina’s handwritten manuscripts of his single-name paper (Source [167]) with his published paper,13) we noticed different manners of regulating the final states in several passages: the direction of the magnetic moment of the electron is more clearly specified in the manuscript. This also is consistent with citations ③, ④ in the letter of 27th Oct. 1928.24)
Eigenstates in the presence of a static magnetic field.Though not mentioned in their joint paper, Klein and Nishina made considerable efforts in trying to solve the Dirac equation under a static magnetic field. As discussed above, it is obvious that their efforts were related to the problem of the final states. Klein seemed to put a lot of time on trials because his memorandum contained itemization of the whole trial procedure (Source [212-(iii)]), and his calculation memorandums had indicated many trials (Source [194], [205]–[209]). Because these calculation memorandums consist of fragmental assemblies, we can hardly read the line of his thoughts from these. But in Nishina’s note (Source [195]) the whole procedure of the trials is systematically explained in English, like a draft of a paper, from which we can conjecture the course of their trials to solve this problem. We give an outline below:
Although Klein and Nishina’s attempt ended unsuccessfully, after all, due to the first-order approximation in the magnetic field, the ordinary differential equation that they obtained for one component of the wave function at the above-mentioned step 3 was completely correct, and should be regarded as reaching an important step for solving the Dirac equation in the presence of a magnetic field. Thus, this work is worth being evaluated as an exertion at that time. Today we know the exact solution28) of the differential equation in the presence of a magnetic field, to which Klein and Nishina reached. Applying an appropriate variable transformation, it can be reduced to a confluent hypergeometric differential equation. Solving it, we obtain a discrete set of energy eigenvalues (relativistic Landau level). Moreover, we can construct the ψ3 = 0 type and the ψ4 = 0 type solutions from the obtained exact solutions. These two solutions give a pair of values directed to the ±z direction as expectation values of the spin. However, unless pz = 0, these states do not give the maximum value of |⟨σz⟩|; hence, these cannot be considered to have any specific character in the presence of a magnetic field. In that sense, Klein and Nishina’s considerations may be judged as being valid, so that the ψ1 = 0 type and the ψ2 = 0 type solutions are not eigenstates under a static magnetic field.
The greatest problem for Klein and Nishina.As a conclusion of the investigation of Nishina’s source materials discussed in Sec. 5 and Sec. 6, we can state that the hardest problem to which Klein and Nishina made the greatest effort was a conceptual problem concerning the 4-component wave function solution of the Dirac equation, i.e., the problem of the physical meaning of the spin states. Although a semi-classical treatment of the interaction of electron and electromagnetic field had already been established by Gordon and Klein, spin was an entirely new matter, which did not appear in the Klein–Gordon equation. Hence, in the sources we find various groping and progressing processes for the spin operators and a treating method of the spin states, on both the intensity calculations and the verbal expressions. Though the tediousness of the intensity calculations has so far been emphasized as the hardest problem for Klein and Nishina, the greatest problem concerned the physical interpretation and the treating method of the spin states.
As already noted in Sec. 3 Klein and Nishina adopted a semi-classical way to treat the interaction between electron and electromagnetic field, when they attacked the Compton-scattering problem. It is usually understood that they adopted the semi-classical way because no other way existed at that time. However, looking back at the situation concerning theories in the period of spring to summer of 1928 when their work was performed, we can see that another way had already appeared at that time.12),47) To treat the problem of the Compton scattering in the framework of quantum mechanics more consistently, two theories were necessary, the theory of transition probability for scattering and quantum field theory of radiation, both of which were developed and published by Dirac during the period 1926–1927.
7.1 Dirac’s work.First, we look back this series of work by Dirac. Regarding the transition probability, Dirac developed time-dependent perturbation theory to treat the transition problem, and gave a formulation using an iterative approximation. (the last section of Ref. 30 (1926))
Then, in Ref. 31 (1927), he developed a theory that applied the second quantization to a general Boson system at first, and then to a photon system. This important work was a starting point of quantum field theory. The formulation in this work, however, was different from those of the generally used theory developed by Heisenberg and Pauli,32) starting from the Lagrangian form. Hence, we survey here the main points of Dirac’s work.
First, we take a single-body system with a Hamiltonian given by H0 + V, where V denotes the interaction term. We then develop the wave function, ψ, of this system with the eigenfunction ψr of H0, such as ψ = ∑r brψr, where |br|2 corresponds to the probability of this system being in the state r. If we consider a many body system consisting of a large assembly of the same single body system with no interaction with each other, then |br|2 gives the number of particles in the state r, Nr. Hence, we can write br as
| \begin{equation} b_{r} = e^{-\frac{i}{\hbar}\theta_{r}}N_{r}^{1/2},\quad b_{r}^{*} = N_{r}^{1/2}e^{\frac{i}{\hbar}\theta_{r}} \end{equation} | [7-1] |
The expressions of $\dot{b}_{r}$ and $i\hbar \dot{b}_{r}^{*}$ can be written in the form of the canonical equations of motion of which the Hamiltonian is $F = \sum\nolimits_{sr}b_{r}^{*} H_{rs}b_{s}$, as
| \begin{equation} \dot{b}_{r} = \frac{1}{i\hbar}\frac{\partial F}{\partial b_{r}^{*}},\quad i\hbar\dot{b}_{r}^{*} = -\frac{\partial F}{\partial b_{r}} \end{equation} | [7-2] |
Since br and $i\hbar b_{r}^{*}$ are regarded to be canonically conjugate variables with each other, according to the general operating procedure for quantization we request as
| \begin{equation} [b_{r},i\hbar b_{s}^{*}] = i\hbar\delta_{rs},\quad [b_{r},b_{s}] = [b_{r}^{*},b_{s}^{*}] = 0 \end{equation} | [7-3] |
Here, the second quantization is finished. Rewriting the above into a relation between Nr and θr, we obtain
| \begin{equation} [N_{r},\theta_{s}] = \delta_{rs},\quad [N_{r},N_{s}] = [\theta_{r},\theta_{s}] = 0 \end{equation} | [7-4] |
| \begin{equation} i\hbar\frac{\partial\psi(N_{1}',N_{2}'\cdots\cdots)}{\partial t} = F(N,\theta)\psi(N_{1}',N_{2}',\cdots\cdots) \end{equation} | [7-5] |
According to the relation [Nr, θs] = δrs,
| \begin{align} &e^{\pm\frac{i}{\hbar}\theta_{r}}\psi(N_{1}',N_{2}',\cdots,N_{r}',\cdots) \\ &\quad= \psi (N_{1}',N_{2}',\cdots,N_{r}'\pm 1,\cdots) \end{align} | [7-6] |
At last, in Ref. 33 (1927), he developed a dispersion theory using second-order perturbation theory after formulating the interaction between the electron and the electromagnetic field. By applying field quantization to the photon system, we can write down the vector potential A with N and θ. In the long-wavelength approximation, the factor e±ikr can be dropped out, and we obtain
| \begin{equation} \boldsymbol{{A}} = \sum_{r}\boldsymbol{{e}}_{r}L^{-3/2}\left[\frac{\hbar c^{2}}{\nu_{r}}\right]^{1/2}(N_{r}^{1/2}e^{\frac{i}{\hbar}\theta_{r}} + e^{-\frac{i}{\hbar}\theta_{r}}N_{r}^{1/2}), \end{equation} | [7-7] |
| \begin{equation} w_{i \to f} = \frac{2\pi}{\hbar}\biggl|\sum_{n}\frac{v_{fn}v_{ni}}{E_{i} - E_{n}}\biggr|^{2}\rho_{f}, \end{equation} | [7-8] |
This theory of dispersion by Dirac can be applied directly to Compton scattering with modifications on two points. One is to attach the factor eikr, which expresses the space dependence to the vector potential, and the other is to rewrite the Hamiltonian of the electron into that of the Dirac equation as
| \begin{equation} H = - c\rho_{1}\boldsymbol{{\sigma}}\left(\boldsymbol{{p}} + \frac{e}{c}\boldsymbol{{A}}\right) - \rho_{3}mc^{2}. \end{equation} | [7-9] |
Naturally, Klein and Nishina should have known well about these works of Dirac concerning the quantization of an electromagnetic field, because Dirac wrote a paper on field quantization31) during his stay in Copenhagen and in the Nishina’s colloquium notes11) dated 26 Jan. and 28 Jan. 1927, Dirac’s lectures on this work are recorded. Dirac’s paper was received on 2nd Feb. 1927. Moreover, in autumn of 1927, Klein worked with Jordan34) to expand Dirac’s method of field quantization to a mutually interacting Bose system, so that he should be considerably interested in the problem of field quantization. In fact, on the introduction of the Klein–Nishina’s joint paper8) they commented on Dirac’s radiation theory as “Man wird erwarten, dass die eine Berücksichitigung der strahlungsdämpfung erlaubt, in diesem Falle ein übereinstimmendes Resultat gibt, wenn es sich um die erste Näherung in bezug auf die Intensität der Primärstrahlung handelt.”12),47) (It will be expected that the radiation theory given by Dirac, which can afford to take radiation damping into account gives, in this case, the same result if the first-order approximation in the intensity of incident radiation is applied). Nevertheless, as far as Nishina’s source materials are concerned, we could not find any traces on their trials of adopting Dirac’s theory.
7.2 Reasons for Klein and Nishina’s adoption of the semi-classical treatment.Here we consider why Klein and Nishina adopted the semi-classical way instead of the more consistent method based on quantization of the radiation field. For this, we report three reasons, as described below:
The first reason: Klein was one of the founders of the semi-classical treatment.As noted in Sec. 1, Klein formulated in his paper5) (1927) the general method to treat the interaction between charged particles and an electromagnetic field on the basis of the semi-classical treatment; hence he should be called the founder of the semi-classical treatment (the Klein–Gordon equation was also derived here). In this paper, he also derived the conservation laws of energy and momentum upon Compton scattering on the basis of that method, while Gordon published his paper,4) in which the intensity of the Compton scattering was treated on the basis of the Klein–Gordon equation about two months before Klein’s paper. Mentioning these former works, Klein wrote, in his private note (Ref. 15) “And naturally we had both of us been thinking how this effect would come out according to Dirac’s new theory.”
Thus it was not doubtful that Klein was attached to the semi-classical treatment, and he would like to adopt this method also in the case of the Dirac equation. Moreover, as noted in Sec. 1, Klein and Nishina intended to verify the validity of Dirac’s relativistic theory of the electron in their paper. For this purpose, it was certainly convenient to treat the problem on the basis of the semi-classical treatment using the Klein–Gordon equation and the Dirac equation, and to check the difference of the results.
The second reason: The problem of normalization.We discuss the normalization of the solution of the Dirac equation for a free electron. Klein wrote in his private note (Ref. 15), “I shall mention here one difficulty which now would seem trivial, namely that is not possible to include a Dirac electron in a box (and which led me later to a well-known paradoxon [sic]). Finally we decided on a procedure earlier used by Gordon on the advise of Pauli.” The passage “that is not possible to include a Dirac electron in a box” means that if we set ψ = 0 on a certain boundary surface, such a solution allowing ψ ≠ 0 inside of the surface does not exist, because the Dirac equation is a first order differential equation. Because of this statement, Klein and Nishina seemed to think that for a Dirac electron such normalization could not be allowed as to produce a discrete wave number k of the eikr type solution expressing the eigenstate of the momentum. There are two kinds of procedures to normalize the eikr type solution: One is the box normalization, which claims for an eigenfunction ψk(r) = L−3/2eikr in a finite volume, L3
| \begin{align} &\iiint\psi_{\boldsymbol{{k}}}^{*}(\boldsymbol{{r}})\psi_{\boldsymbol{{k}}'}(\boldsymbol{{r}})d\boldsymbol{{r}} \\ &\quad = L^{-3}\iiint\nolimits_{(L^{3})}e^{-i\boldsymbol{{k}}\cdot\boldsymbol{{r}}}e^{i\boldsymbol{{k}}'\cdot\boldsymbol{{r}}}d\boldsymbol{{r}} = \delta_{\boldsymbol{{k}},\boldsymbol{{k}}'} \end{align} | [7-10] |
In this case, when the periodic boundary condition is applied, k takes the discrete value ((2π/L)n1, (2π/L)n2, (2π/L)n3). The other is the δ-function-type normalization, which claims for continuous values of k in an infinite space that
| \begin{equation} \psi_{\boldsymbol{{k}}}(\boldsymbol{{r}}) = (2\pi)^{-3/2}e^{i\boldsymbol{{k}}\cdot\boldsymbol{{r}}} \end{equation} |
| \begin{align} \iiint\psi_{\boldsymbol{{k}}}^{*}(\boldsymbol{{r}})\psi_{\boldsymbol{{k}}'}(\boldsymbol{{r}})d\boldsymbol{{r}} &= (2\pi)^{-3}\iiint e^{-i\boldsymbol{{k}} \cdot \boldsymbol{{r}}}e^{i\boldsymbol{{k}}'\cdot \boldsymbol{{r}}}d\boldsymbol{{r}} \\ &= \delta (\boldsymbol{{k}} - \boldsymbol{{k}}') \end{align} | [7-11] |
In Gordon’s paper on Compton scattering,4) as stated in Sec. 2.2, the δ-function type normalization is adopted, while in Dirac’s quantization of electromagnetic field, it is necessary to make the wave number k of radiation field discrete: When both electron and radiation fields are taken into consideration, the consistency of the concept of space is spoiled unless the wave vectors of the wave function of electron are treated as being discrete under the same boundary condition as that for radiation. For that purpose, the periodic boundary condition for a Dirac electron should be adopted, as is usually done today. However, the concept of the periodic boundary condition might not be well understood when Klein and Nishina were working on the Compton scattering, because it was an artificial device made up for mathematical convenience. The other method to make wave vectors discrete is a standing-wave-type boundary condition that claims the value of the wave function to be zero at a boundary surface. This method has a definite physical meaning. Note that the density of states assigned by the set of discrete wavenumbers is the same for the standing-wave-type and periodic boundary conditions. This is probably the reason to accept the periodic boundary condition even though the physical meaning was not clear. However, the standing-wave-type boundary condition could not be adopted for a Dirac electron because of the reason stated above. Hence, the grounds for accepting the periodic boundary condition became infirm. Namely, the validity for taking discrete wave vectors was ambiguous. Klein might have meant this situation when he mentioned “one difficulty”. Here, we add some remarks concerning the acceptance of the periodic boundary condition. Born and Dirac adopted this early in their studies on quantum mechanics. In fact, Born devised the periodic boundary condition before quantum mechanics appeared. He introduced it to calculate the density of states in Ref. 35 dealing with the classical treatment of lattice vibration. This might be the origin of the periodic boundary condition. He used it to discuss the probability interpretation of quantum mechanics (Ref. 36), particularly in the calculation of the density of states. Dirac also claimed the validity of the periodic boundary condition in Ref. 30 to express functions defined in the finite region in terms of a Fourier series. On the other hand, in the paper (Ref. 37) arguing the Klein’s paradox, Klein stated that in the case of the Dirac equation, the standing-wave-type boundary condition could not be applied without mentioning the periodic boundary condition, which probably was not in his mind.
In the Tamm’s paper,38) which is discussed later, the periodic boundary condition was adopted for the Dirac electron. However, he added an annotation explaining Born’s idea of a periodic lattice. Since this paper was published in 1930, the periodic boundary condition did not seem to be so broadly accepted, even then.
The third reason: Question on negative energy states.As noted in Sec. 3, there are two solutions of the Dirac equation for a given momentum value: one with the positive energy eigenvalue $\sqrt{m^{2}c^{4} + c^{2}p^{2}} $ and the other with the negative energy eigenvalue $ - \sqrt{m^{2}c^{4} + c^{2}p^{2}} $. When Dirac’s relativistic electron theory6) appeared, this negative energy state was not interpreted validly, and it was generally regarded as being physically meaningless. In fact, we can find the words “den physikalish nicht sinnvollen negativen Energiewerten” (physically meaningless negative energy values) in the Klein–Nishina’s joint paper (Ref. 8, p. 856). It was surely inevitable that in 1928, the above view was generally accepted, because in 1930 Dirac presented his hole theory,40) which was verified in 1932 by discovery of the positron. If Klein and Nishina had taken up Dirac’s scattering theory as the basis, they should have taken into account all of the possible states as intermediate states. Then, was it possible for them to exclude the negative energy states at this stage? They seemed to avoid Dirac’s scattering theory because of this question. The answer to this question was given by Waller41) and Tamm,38) as described below.
7.3 Treatments based on field quantization.In 1930 Waller41) and Tamm38) independently carried out calculations for Compton scattering on the basis of field quantization and scattering theory, and reproduced the Klein–Nishina formula. Their analyses differed in the following points: Tamm adopted the quantization of an electron field as well as an electromagnetic field relying on Heisenberg–Pauli’s paper,32) while Waller adopted a method in which the electron field was not quantized. Both Waller and Tamm emphasized that it was essentially necessary to take the negative energy states into account as intermediate states.12),47) They stated that especially in the limit of hν ≪ mc2, the contributions to the intensity of the scattered wave were given exclusively by these negative energy intermediate states, and as a result the classical Thomson-scattering formula was obtained. This indicates that negative energy states must be considered as physically meaningful states. The necessity of the negative energy states in this manner was regarded as a paradox at that time, called “Waller’s paradox”. Another paradox was “Klein’s paradox”37): when an electron with a positive energy entered into a step-type potential barrier, the part of the electron invading inside of the barrier tended to have a negative energy state if the potential barrier was sufficiently high. Klein mentioned this point in his private note,15) as sited in Sec. 7.2, in regard to the second reason. There are two letters from Waller to Nishina in the Sangokan Nishina Source: One was dated 17 May 1929,42) in which Waller wrote “I have been working chiefly on the scattering theory”. The other was dated 25 Jan. 1930,43) in which Waller wrote on the negative energy states “I have recently been doing some things on scattering which tends to be my hobby. It will perhaps interest you to hear that the Dirac scattering theory gives for free electron just your and Klein’s formula, if states of negative energy are properly taken into account”. And in the margin of this letter Nishina wrote a draft44) of a reply in English, and referred to Waller’s work as follows: “I am very much interested to hear about your results on the scattering, and anxious to see the details of your work. From Heisenberg’s lecture at Copenhagen I also thought that the states of negative energy must be taken into account in order to get the correct result but I did not expect that you would obtain the same formula as ours. I am looking forward to your paper.” Waller made a note on Heisenbeg’s lecture in the footnote of his paper (Ref. 41, p. 844) “In einer nichit veröffentlichten Untersuchung hat Prof. W. Heisenberg schon gefunden, dass in dem Grenzfall, wo hνs/mc2 gegen eins vernachlässigt wird, die klassische Streuformel wegen der Zwischenzustände negativer Energie erhalten wird” (In an unreleased work Prof. W. Heisenberg had already found that in the limiting case of hνs/mc2 negligibly smaller than unity, the classical scattering formula is obtained by the negative energy intermediate states). It must be before October 1928 that Nishina listened to this Heisenberg’s lecture in Copenhagen, and therefore this Heisenberg’s foreknowledge must have been obtained at rather an earlier time point.
As for the problem of normalization, which was a worrying matter for Klein and Nishina, Waller did not seem to worry and used the δ-function type normalization for the electron, while he dealt with the wave vector of the electromagnetic field as being discrete. Tamm, on the other hand, used the periodic boundary condition for both the electron and the electromagnetic field, and thus treated space consistently.
Here, we examine how Waller and Tamm dealt with the problem of choosing the final state of an electron that Klein and Nishina worried about most. First, Waller adopted four independent final states of an electron with a given momentum as four solutions of the same type as those of Klein and Nishina, namely one of ψ1, ψ2, ψ3, ψ4 being equal to zero, and used these in actual calculations. He clearly stated that the orthogonality condition was necessary for choosing four final states, and moreover, added at the end that the intensity of the scattered wave did not depend on the concrete form of the four final states as long as they satisfied the orthonormal condition. On the other hand, Tamm established a calculation method, which was capable of obtaining the intensity of the scattered wave using orthonormality alone, and did not use any concrete forms of the four independent states. However, he also mentioned an example of concrete forms, which was the same as that used by Waller, namely four solutions, each one of which had one of ψ1, ψ2, ψ3, ψ4 equal to zero (Klein–Nishina–type solutions).
Waller and Tamm might have reached the orthogonality condition because they used Dirac’s scattering theory (time-dependent perturbation theory). Nevertheless, the following general viewpoint Tamm stated in his paper (Ref. 38, p. 549) would not have been established at the time of Klein–Nishina’s calculations: “Die vier Komponenten den Wellenfunktion ψ können als Komponenten eines Vektors in einen vier dimensoionalen Hilbertraum aufgefaßt werden” (The four components of wave function ψ can be grasped as components of a vector in a four dimensional Hilbert space). During this period of a little longer than one year, the conceptual understanding of Dirac’s four-component wave function progressed very much.
When Waller’s theory and Tamm’s theory are reconsidered on the basis of hole theory, the same result of the scattering cross section can be obtained with only simple formal changes.12),39) Waller, himself, stated on this matter in the footnote of Ref. 41 and Dirac did also the same in Ref. 40.
From the above arguments it became clear that the Klein–Nishina formula can be reproduced by using field quantization or hole theory. It was already stated in Sec. 1 that this formula played an important role to support of the Dirac equation and, moreover, Waller’s theory and Tamm’s theory should be recognized as having reinforced that role.
In Klein–Nishina’s theory, the negative energy solutions of the Dirac equation were not taken into account. On the contrary, it is already noted that in the theory based on the field quantization (see for instance, W. Heitler45) 3rd ed. Chap. V, §22, pp. 211–224), the negative energy states play a very important role as intermediate states. Nevertheless, both theories give the same result for the scattering cross section of a photon as the Klein–Nishina formula. To clarify this seemingly paradoxical matter, we rewrite here Klein–Nishina’s theory into the form in which the intermediate states appear, and thus reveal the equivalence between both theories.
At first, in Eq. [KN-36], which gives the vector potential A(p, p′) of the scattered wave (where p and p′ are the momentum of the electron in the initial and final states, respectively), we rewrite as follows: $u(\boldsymbol{{p}}) \to u^{\dagger }(\boldsymbol{{p}}),f(\boldsymbol{{p}}) \to \bar{g}^{\dagger }(\boldsymbol{{p}}),v(\boldsymbol{{p}}) \to u(\boldsymbol{{p}}),\rho _{1}{\boldsymbol{{\sigma}}} \to {\boldsymbol{{\alpha}}},\nu \to \omega $. We then obtain
| \begin{align} &\boldsymbol{{A}}'(\boldsymbol{{p}} \to \boldsymbol{{p}}') \\ &\quad\propto \frac{e}{r}e^{i\omega'(t - \frac{r}{c})}u^{\dagger}(\boldsymbol{{p}})[{\boldsymbol{{\alpha}}} g(\boldsymbol{{p}}') + \bar{g}^{\dagger}(\boldsymbol{{p}})\boldsymbol{{\alpha]u}}(\boldsymbol{{p}}'), \end{align} | [8-1] |
| \begin{align} I &= \frac{\omega'^{2}}{4\pi c}|\boldsymbol{{A}}'(\boldsymbol{{p}} \to \boldsymbol{{p}}') \cdot \boldsymbol{{e}}'|^{2} \\ &\propto e^{2}|u^{\dagger}(\boldsymbol{{p}})[(\boldsymbol{{\alpha}}\cdot \boldsymbol{{e}}')g(\boldsymbol{{p}}') + \bar{g}^{\dagger}(\boldsymbol{{p}})(\boldsymbol{{\alpha}}\cdot\boldsymbol{{e}}')]u(\boldsymbol{{p}}')|^{2}\\ &= e^{2}|u^{\dagger}(\boldsymbol{{p}}')[g^{\dagger}(\boldsymbol{{p}}')(\boldsymbol{{\alpha}}\cdot \boldsymbol{{e}}') + (\boldsymbol{{\alpha}}\cdot\boldsymbol{{e}}')\bar{g}(\boldsymbol{{p}})]u(\boldsymbol{{p}})|^{2}. \end{align} | [8-2] |
We first rewrite the fundamental equations appearing in Sec. 2. The Dirac equation is given by
| \begin{align} &\left[i\hbar \frac{\partial}{\partial t} - (H_{0} - e\boldsymbol{{\alpha}}\cdot\boldsymbol{{A}})\right][\psi_{0}(\boldsymbol{{p}}) + \psi_{1}(\boldsymbol{{p}})] = 0\\ &H_{0} = c\boldsymbol{{\alpha}}\cdot\left(-i\hbar \frac{\partial}{\partial\boldsymbol{{r}}} - \frac{e}{c}\boldsymbol{{A}}\right), \end{align} | [8-3] |
| \begin{equation} \boldsymbol{{A}} = a\boldsymbol{{e}}_{0}[e^{i\omega (t - \frac{\boldsymbol{{n}}\cdot \boldsymbol{{r}}}{c})} + e^{-i\omega (t - \frac{\boldsymbol{{n}}\cdot \boldsymbol{{r}}}{c})}] \end{equation} | [8-4] |
| \begin{align} \psi_{0}(\boldsymbol{{p}}) & = u(\boldsymbol{{p}})e^{\frac{i}{\hbar}(\boldsymbol{{p}} \cdot \boldsymbol{{r}} - E(\boldsymbol{{p}})t)}\\ H_{0}(\boldsymbol{{p}})u(\boldsymbol{{p}}) &\equiv (c\boldsymbol{{\alpha}}\cdot\boldsymbol{{p}} + \beta mc^{2})u(\boldsymbol{{p}}) = E(\boldsymbol{{p}})u(\boldsymbol{{p}})\\ E(\boldsymbol{{p}}) &= \sqrt{c^{2}p^{2} + m^{2}c^{4}} > 0 \end{align} | [8-5] |
| \begin{equation} \left[i\hbar\frac{\partial}{\partial t} - H_{0}\right]\psi_{1}(\boldsymbol{{p}}) = -e\boldsymbol{{\alpha}}\cdot\boldsymbol{{A}}\psi_{0}(\boldsymbol{{p}}) \end{equation} | [8-6] |
| \begin{equation} \psi_{1}(\boldsymbol{{p}}) = [g(\boldsymbol{{p}})e^{i\omega (t - \frac{\boldsymbol{{n}} \cdot \boldsymbol{{r}}}{c})} + \bar{g}(\boldsymbol{{p}})e^{-i\omega (t - \frac{\boldsymbol{{n}} \cdot \boldsymbol{{r}}}{c})}]\psi_{0}(\boldsymbol{{p}}), \end{equation} | [8-7] |
| \begin{equation} \begin{bmatrix} g(\boldsymbol{{p}}) = \cfrac{1}{(E(\boldsymbol{{p}}) - \hbar \omega) - H_{0}\biggl(\boldsymbol{{p}} - \cfrac{\hbar \omega}{c}\boldsymbol{{n}}\biggr)}(- e\boldsymbol{{\alpha}}\cdot\boldsymbol{{e}}_{0})a\\ \bar{g}(\boldsymbol{{p}}) = \cfrac{1}{(E(\boldsymbol{{p}}) + \hbar \omega) - H_{0}\biggl(\boldsymbol{{p}} + \cfrac{\hbar \omega}{c}\boldsymbol{{n}}\biggr)}(-e\boldsymbol{{\alpha}}\cdot \boldsymbol{{e}}_{0})a \end{bmatrix} \end{equation} | [8-8] |
| \begin{equation} \left(-i\hbar\frac{\partial}{\partial t} - H\right)\left(i\hbar \frac{\partial}{\partial t} - H\right)\psi = 0, \end{equation} | [8-9] |
| \begin{align} {g(\boldsymbol{{p}})} & {{}= \cfrac{1}{\bigl[(E(\boldsymbol{{p}}) - \hbar \omega) - H_{0}\bigl(\boldsymbol{{p}} - \tfrac{\hbar \omega}{c}\boldsymbol{{n}}\bigr)\bigr]\bigl[(E(\boldsymbol{{p}}) - \hbar\omega) + H_{0}\bigl(\boldsymbol{{p}} - \tfrac{\hbar \omega}{c}\boldsymbol{{n}}\bigr)\bigr]}}\\ &\quad {{}\times{} \bigl[(E(\boldsymbol{{p}}) - \hbar \omega) + H_{0}\bigl(\boldsymbol{{p}} - \tfrac{\hbar \omega}{c}\boldsymbol{{n}}\bigr)\bigr](-e\boldsymbol{{\alpha}}\cdot \boldsymbol{{e}}_{0})a.} \end{align} | [8-10] |
| \begin{align} {|\quad|^{2}} &{{}= a^{2}e^{4}\biggl|u^{\dagger}(\boldsymbol{{p}}')\biggl[(\boldsymbol{{\alpha}}\cdot\boldsymbol{{e}}_{0})\cfrac{1}{(E(\boldsymbol{{p}}') - \hbar\omega) - H_{0}\bigl(\boldsymbol{{p}}' - \tfrac{\hbar\omega}{c}\boldsymbol{{n}}\bigr)}(\boldsymbol{{\alpha}}\cdot\boldsymbol{{e}}')}\\ &\quad{{}+{} (\boldsymbol{{\alpha}}\cdot \boldsymbol{{e}}')\cfrac{1}{(E(\boldsymbol{{p}}) + \hbar\omega) - H_{0}\bigl(\boldsymbol{{p}} + \tfrac{\hbar\omega}{c}\boldsymbol{{n}}\bigr)}(\boldsymbol{{\alpha}}\cdot\boldsymbol{{e}}_{0})\biggr]u(\boldsymbol{{p}})\biggr|^{2}} \end{align} | [8-11] |
| \begin{align} &\Biggl|\sum_{(n)}\cfrac{(u^{\dagger}(\boldsymbol{{p}}')(\boldsymbol{{\alpha}}\cdot\boldsymbol{{e}}_{0})u_{n})(u_{n}^{\dagger}(\boldsymbol{{\alpha}}\cdot\boldsymbol{{e}}')u(\boldsymbol{{p}}))}{(E(\boldsymbol{{p}}') - \hbar\omega) - E_{n}\bigl(\boldsymbol{{p}}' - \tfrac{\hbar\omega}{c}\boldsymbol{{n}}\bigr)} \\ &\quad+ \sum_{(n)}\cfrac{(u^{\dagger}(\boldsymbol{{p}}')(\boldsymbol{{\alpha}} \cdot \boldsymbol{{e}}')u_{n})(u_{n}^{\dagger}(\boldsymbol{{\alpha}}\cdot \boldsymbol{{e}}_{0})u(\boldsymbol{{p}}))}{(E(\boldsymbol{{p}}) + \hbar\omega) - E_{n}\bigl(\boldsymbol{{p}} + \tfrac{\hbar\omega}{c}\boldsymbol{{n}}\bigr)}\Biggr|^{2} \end{align} | [8-12] |
| \begin{align} &\boldsymbol{{p}}' + \frac{\hbar\omega'}{c}\boldsymbol{{n}}' = \boldsymbol{{p}} + \frac{\hbar\omega}{c}\boldsymbol{{n}},\quad \\ &E(\boldsymbol{{p}}') + \hbar\omega' = E(\boldsymbol{{p}}) + \hbar \omega \end{align} | [8-13] |
Using above, we rewrite the denominator of the first term of Eq. [8-12] as E(p) + ħω − [En(p − (ħω′/c)n′) + ħω + ħω′]. We then recognize that the first term of Eq. [8-12] coincides with the transition matrix element in the Heitler theory,45) where the intermediate state corresponds to the emission of a photon at the final state at first and the electron having momentum p − (ħω′/c)n′. Likewise, the second term of Eq. [8-12] coincides with the transition matrix element in which the intermediate state corresponds to the absorption of a photon of the initial state at first and an electron having momentum p + (ħω/c)n. We can thus show that Eq. [8-12] just coincides with the transition matrix element in the Heitler theory, namely the Eq. [H-18] (Equation (18) in Ref. 45, p. 215) given by
| \begin{align} d\phi &= e^{4}\frac{Ek^{2}}{\mu k_{0}^{2}}d\Omega\biggl[\sum \biggl(\frac{(u^{*}\alpha u')(u'^{*}\alpha_{0}u_{0})}{\mu + k_{0} - E'} \\ &\quad+ \frac{(u^{*}\alpha_{0}u'')(u''^{*}\alpha u_{0})}{\mu - k - E''}\biggr)\biggr]^{2}, \end{align} | [H-18] |
The main part of this article is based on the former two papers46),47) written in Japanese by the present author in 1992. Concerning these papers, the author would like to thank the late Dr. Hidehiko Tamaki, the late Dr. Fukutaro Shimamura, the late Dr. Hajime Takeuchi for inviting him to being a member of their investigation group and giving him opportunities to access the Nishina Source Materials. Since these three persons were former collaborators of Dr. Yoshio Nishina in RIKEN, the author owed them valuable information about Dr. Nishina. He would also like to thank Dr. Hiroshi Ezawa for valuable discussions on these papers. He is also grateful to Dr. Yoshimasa A. Ono for carefully reading and correcting his English manuscripts of this article. Thanks are due to The Nishina Memorial Foundation for permitting the author to utilize Nishina Source Materials retained there.
