Abstract
In this review article, we consider the formal description of electron polarization, various methods of producing and detecting it, as well as experiments with polarized electrons. Issues of greatest interest to modern physics will be discussed in particular detail. An earlier review of this subject was written by Rosenfeld. Among other reviews, by Mott and Massey and by Sommerfeld, the former is concerned mainly with polarization due to Coulomb scattering by heavy nuclei, and we refer readers interested in a more detailed account of this subject than that given in §3 of the present article to it.
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ELECTRON POLARIZATION.
THEORY AND EXPERIMENT
H. A. Tolhoek *)
§ 1. INTRODUCTION
The term “electron polarization” may be applied to the properties of electron beams caused by the preferential orientation of the spins of the electrons. These properties are to some extent analogous to the properties of polarized light. The hypothesis of the electron spin was put forward, within the framework of the old quantum theory, by Uhlenbeck and Goudsmit in order to explain a number of spectral regularities. The principal milestones in the subsequent development of the theory of the rotating electron were Pauli’s nonrelativistic theory of spin, Dirac’s wave equation, and also the recent discovery of the anomalous magnetic moment of the electron and its explanation by quantum electrodynamics. The basic facts testifying in favor of the theory are, to a known extent, indirect, in particular the facts relating to the bound electron. The polarization properties of beams of free electrons, however, are an unquestionable consequence of the existence of the electron spin, in which it manifests itself in the most direct way.
In this review article we shall consider the formal description of electron polarization, the various methods of producing and detecting it, and also experiments with polarized electrons. Questions of greatest interest for contemporary physics will be discussed in especially great detail. An earlier review of this question was written by Rosenfeld[^1]. Of other reviews, by Mott and Massey[^2] and by Sommerfeld[^3], the first deals mainly with polarization due to Coulomb scattering by heavy nuclei; to it we refer the reader interested in a more detailed exposition of this question than that given in § 3 of the present article.
§ 2. FORMAL DESCRIPTION OF ELECTRON POLARIZATION.
COMPARISON WITH PHOTON POLARIZATION. INFLUENCE OF SLOWLY VARYING ELECTROMAGNETIC FIELDS ON ELECTRON POLARIZATION
In quantum mechanics the description of the polarization of beams of electrons and photons is to a considerable extent analogous[^4–^12]. Since we shall be concerned only with the question of wave polarization, we shall compare only plane waves corresponding to quanta with the same momentum $\mathbf{p}$. A definite state of polarization, specified by the wave function $\psi$, may be
) H. A. Tolhoek, Reviews of Modern Physics 28*, 277 (1956).
is written in the form
\[ \psi=c_1\psi_1+c_2\psi_2, \tag{2,1} \]
where \(\psi_1\) and \(\psi_2\) are two orthogonal wave functions which correspond, in the case of electrons, to two opposite spin orientations, either perpendicular, or parallel and antiparallel to the momentum \(\mathbf p\), and in the case of photons to two plane-polarized waves with mutually perpendicular directions of polarization, or else to two right- and left-circularly polarized waves.
The wave function \(\psi\) in the form (2,1) characterizes a completely polarized beam. If we have some ideal polarization detector (recording, for example, only photons linearly polarized in a definite plane), then it can be described by the wave function of the quanta recorded by it with one-hundred-percent efficiency:
\[ \psi^{\mathrm{det}}=c_1^{\mathrm{det}}\psi_1+c_2^{\mathrm{det}}\psi_2. \tag{2,2} \]
The probability that the detector will record a quantum with wave function (2,1) is equal to
\[ W=\left|\langle \psi^{\mathrm{det}}\mid \psi\rangle\right|^2 =\left|c_1^{\mathrm{det}*}c_1+c_2^{\mathrm{det}*}c_2\right|^2 . \tag{2,3} \]
Partially polarized light is represented not by a single wave function, but by an “ensemble” of pure states, each of which is characterized by one wave function. The quantum-mechanical description of such an “ensemble” is given by a density matrix (or statistical operator) \(\rho\) (see, for example, \(^{5,13}\)), which in our case of two basis states is two-rowed:
\[ \left\|\rho_{rs}\right\|= \left\| \begin{matrix} \rho_{11} & \rho_{12}\\ \rho_{21} & \rho_{22} \end{matrix} \right\|. \tag{2,4} \]
The matrix \(\rho\) is Hermitian, has positive or zero eigenvalues, and is normalized so that
\[ \rho_{11}+\rho_{22}=1. \tag{2,5} \]
To the special case of a completely polarized beam, characterized by the wave function (2,1), there corresponds the density matrix
\[ \rho= \left\| \begin{matrix} |c_1|^2 & c_1c_2^*\\ c_1^*c_2 & |c_2|^2 \end{matrix} \right\|. \tag{2,6} \]
Similarly, a detector described by the wave function \(\psi^{\mathrm{det}}\) (2,2) can be characterized by the density matrix \(\rho^{\mathrm{det}}\). The registration probability (2,3) can now be written as the trace of the product of both density matrices:
\[ W=\operatorname{Sp}[\rho\rho^{\mathrm{det}}]. \tag{2,7} \]
The identity of (2,3) and (2,7) is easily verified. In the general case, the expectation value of the operator \(a\) for the state \(\rho\) is given by the expression
\[ \langle a\rangle=\operatorname{Sp}[a\rho]. \tag{2,8} \]
The density matrix (2.6) of a pure state can always be brought by a unitary transformation to the simple form:
\[ \rho = \begin{Vmatrix} 1 & 0\\ 0 & 1 \end{Vmatrix}. \tag{2.9} \]
However, the use of the density matrix is essential only in the case when we are dealing with a quantum-mechanical “mixture” representing an “ensemble” of states. The matrix \(\rho\) can always be diagonalized, and in the general case instead of (2.9) we obtain
\[ \rho = \begin{Vmatrix} \rho' & 0\\ 0 & \rho'' \end{Vmatrix} \tag{2.10} \]
(\(\rho' > \rho'' > 0\) can always be chosen). At the same time a pure state \(\psi = c_1\psi_1 + c_2\psi_2\) may be regarded as a “coherent” superposition of \(\psi_1\) and \(\psi_2\), while the density matrix corresponding to a quantum-mechanical mixture
\[ \begin{aligned} \rho &= \begin{Vmatrix} \rho' & 0\\ 0 & \rho'' \end{Vmatrix} = \rho'' \begin{Vmatrix} 1 & 0\\ 0 & 1 \end{Vmatrix} + (\rho' - \rho'') \begin{Vmatrix} 1 & 0\\ 0 & 1 \end{Vmatrix} =\\ &= (1-P) \begin{Vmatrix} \frac12 & 0\\ 0 & \frac12 \end{Vmatrix} + P \begin{Vmatrix} 1 & 0\\ 0 & 1 \end{Vmatrix} \tag{2.11} \end{aligned} \]
(\(P=\rho' - \rho''\))—as an “incoherent” superposition of unpolarized,
\[ \begin{Vmatrix} \frac12 & 0\\ 0 & \frac12 \end{Vmatrix}, \]
and completely polarized,
\[ \begin{Vmatrix} 1 & 0\\ 0 & 1 \end{Vmatrix}, \]
beams with weights respectively \(1-P\) and \(P\). We may call \(P\) \((0 \leq P \leq 1)\) the degree of polarization. In the general case the density matrix is characterized by three independent parameters, for example \(P\) and the complex ratio \(c_2/c_1\), which corresponds to the polarization state of the completely polarized component mixed with the unpolarized one. Since in practice a beam contains a large number of quanta, for its description one must add a fourth parameter—the total intensity, since \(\psi\) and \(\rho\) refer to the state of only one quantum. We may say that we perform a measurement of the polarization of a physical beam with respect to some orthogonal basis \((\psi_1,\psi_2)\), if we determine the intensities \(I_1\) and \(I_2\) in the detector positions \(\psi^{\mathrm{det}}=\psi_1\) and \(\psi^{\mathrm{det}}=\psi_2\). The result of this measurement can be expressed in the form
\[ \frac{I_1}{I_2}=\frac{\rho_{11}}{\rho_{22}}, \tag{2.12} \]
so that
\[ P(\psi_1,\psi_2)\equiv \rho_{11}-\rho_{22}=\frac{I_1-I_2}{I_1+I_2}. \tag{2.13} \]
For a complete description of polarization one must determine the three parameters mentioned, for example by measuring the polarization with respect to three linearly independent bases. For photons one may take as the latter:
\[ \left. \begin{array}{l} \text{a) two states of linear polarization with mutually}\\ \text{perpendicular planes of polarization, b) two}\\ \text{other states of linear polarization, making}\\ \text{angles } \pi/4 \text{ with the planes of polarization in the preceding}\\ \text{case, and c) two states of right- and left-}\\ \text{circularly polarized light.} \end{array} \right\} \tag{A} \]
Another way of describing the state of polarization is given by the expectation values of the Pauli matrices \(\boldsymbol{\sigma}\). We shall denote them by \(\zeta_1,\zeta_2\), and \(\zeta_3\). For a pure state one may write
\[ \rho= \left\| \begin{array}{cc} c_1c_1^* & c_1c_2^*\\ c_2c_1^* & c_2c_2^* \end{array} \right\| =\frac{1}{2}(1+\boldsymbol{\zeta}\cdot\boldsymbol{\sigma}), \tag{2.14} \]
where
\[ \zeta_1=c_1c_2^*+c_2c_1^*,\quad \zeta_2=i(c_1c_2^*-c_2c_1^*),\quad \zeta_3=|c_1|^2-|c_2|^2 . \tag{2.15} \]
A two-dimensional unitary transformation of the pair of complex numbers \((c_1,c_2)\) corresponds to a real three-dimensional orthogonal transformation in the space of the vectors \(\boldsymbol{\zeta}\). For pure states \(\boldsymbol{\zeta}\) is a unit vector, while for mixtures \(|\boldsymbol{\zeta}|<1\), and it is easy to show that
\[ P=|\boldsymbol{\zeta}|. \tag{2.16} \]
If the density matrix characterizing the polarization detector is also written in the form (2.14):
\[ \rho^{\mathrm{det}}=\frac{1}{2}\left(1+\boldsymbol{\zeta}^{\mathrm{det}}\cdot\boldsymbol{\sigma}\right), \tag{2.17} \]
then the probability of registration by the detector takes the form
\[ W=\operatorname{Sp}[\rho\rho^{\mathrm{det}}] =\frac{1}{2}\left(1+\boldsymbol{\zeta}\cdot\boldsymbol{\zeta}^{\mathrm{det}}\right), \tag{2.18} \]
which is valid both for pure and for mixed states (we note that \(\operatorname{Sp}[\sigma_1]=\operatorname{Sp}[\sigma_2]=\operatorname{Sp}[\sigma_3]=0;\ \operatorname{Sp}[1]=2\)). Orthogonal (pure) states are characterized by unit vectors \(\boldsymbol{\zeta}\) directed in opposite directions.
Up to this point the formal description has been equally applicable to electrons and to photons. From now on we must consider these two cases separately.
A. Formulation for Electrons
In the case of electrons, it is, of course, well known that \(\boldsymbol{\zeta}\) represents the direction of the electron spin (i.e., of the spin angular momentum) in the nonrelativistic Pauli theory. Often \(\boldsymbol{\zeta}\) is specified by the angles \(\chi,\omega\)
in polar coordinates, namely
\[ \zeta_1=\sin\chi\cos\omega,\qquad \zeta_2=\sin\chi\sin\omega,\qquad \zeta_3=\cos\chi, \tag{2.19} \]
so that
\[ \frac{c_2}{c_1}=\operatorname{tg}\left(\frac{\chi}{2}\right)e^{i\omega}. \tag{2.20} \]
In the theory of the relativistic Dirac electron, instead of the two-component Pauli wave functions we have four-component ones. However, a definite state of polarization is still characterized by means of (2.1), where now \(\psi_1\) and \(\psi_2\) are two mutually orthogonal solutions corresponding to positive energies. Up to a constant factor, the spin angular momentum (its spatial components) is given by the expression \(\psi^{*}\boldsymbol{\sigma}\psi\), and the magnetic moment by the quantity \(\psi^{*}(-\beta\boldsymbol{\sigma})\psi\) (for measurements in the laboratory frame). The direction \(\boldsymbol{\zeta}\), according to (2.15), is determined by the expression \(\psi^{*}\left[\frac{1}{2}(1-\beta)\boldsymbol{\sigma}\right]\psi\), which is not fully relativistically covariant. Nevertheless, in the Dirac theory we shall continue to use the vector \(\boldsymbol{\zeta}\) obtained in this way. \(\boldsymbol{\zeta}\) may be regarded as the direction of the spin when the electron is transformed to rest. It is convenient to deal with \(\boldsymbol{\zeta}\) for the following reasons: a) \(\boldsymbol{\zeta}\) can be calculated only from the large components \(\psi_3\) and \(\psi_4\) of the Dirac wave function, b) the direction \(\boldsymbol{\zeta}\) remains unchanged when the electron is accelerated in an electric field (see below), c) an unpolarized beam may be regarded as an ensemble of electrons with spins \(\boldsymbol{\zeta}\) isotropically distributed in all directions.
If, however, one chooses the directions of the spin angular momentum or of the magnetic moment in the laboratory coordinate system, then statements a), b), c) no longer hold. We shall speak of transverse polarization of an electron beam if the spin direction is perpendicular to the direction of the momentum, and of longitudinal polarization in the case of their parallelism or antiparallelism. As the three independent bases relative to which \(P(\psi_1,\psi_2)\) is defined (cf. (2.13)), we may choose (cf. the analogous situation for photons, mentioned earlier as ((A))):
\[ \begin{array}{ll} \text{a)} & \text{two states of transverse polarization with opposite spin directions,}\\ \text{b)} & \text{two other states of transverse polarization, rotated relative to the preceding ones by an angle } \pi/2,\\ \text{c)} & \text{two opposite states of longitudinal polarization (parallel and antiparallel).} \end{array} \tag{B} \]
As we have just seen, the polarization of electron beams can be described in the Dirac theory by means of two-row matrices if one considers exclusively states with positive energy. However, in the calculation of certain polarization effects on the basis of the Dirac equation and perturbation theory, a four-row representation is also required. It is defined as follows:
If
\[ \psi=ue^{\frac{i}{\hbar}px} \tag{2.21} \]
represents a wave with polarization \(\boldsymbol{\zeta}\) (normalized to one particle
in a unit volume), then we set
\[ P_{\lambda\mu}^{(+)}(\zeta)=u_\lambda u_\mu^* \tag{2.22} \]
\(\lambda,\mu=1,\ldots,4\) denote the Dirac components, \((+)\) indicates the choice of solutions with positive energies only). An explicit expression for \(P_{\lambda\mu}^{(+)}(\zeta)\) can be written with the aid of the Dirac matrices \(\rho\) and \(\sigma\) in the form \(^{11,14,15}\)
\[ P^{(+)}(\zeta)=\frac{1}{4}\left[ 1-\frac{cp}{E}\zeta\rho_1-\frac{mc^2}{E}\rho_3+K\sigma -\frac{cp}{E}\rho_1\sigma+\frac{c}{E}(p\zeta)\rho_2\sigma+J\rho_3\sigma \right], \tag{2.23} \]
where
\[ \left. \begin{aligned} K&=\frac{mc^2}{E}\,\zeta+\frac{c^3}{E(E+mc^2)}\,\mathbf p(\mathbf p\zeta),\\ J&=-\zeta+\frac{c^2}{E(E+mc^2)}\,\mathbf p(\mathbf p\zeta). \end{aligned} \right\} \tag{2.24} \]
\(E=(p^2c^2+m^2c^4)^{1/2}\) is the energy of the electron, \(m\) its rest mass. \(P^{(+)}(\zeta)\) can also be written in an explicitly relativistically covariant form (see the Appendix).
B. Formulation for Photons
The derivation of equations (2.1)—(2.18) is equally valid both for photons and for electrons. In the case of photons, the wave function \(\psi\) may be regarded as a complex vector potential. To introduce a notational distinction from the electron case, one may replace \(c_1,c_2\) by \(a_1,a_2\), \(\zeta\) by \(\xi\), and \(\sigma\) by \(\omega\). Then, for photons, (2.1) is rewritten in the form
\[ \mathbf A e^{ikx}=a_1\mathbf A_1 e^{ikx}+a_2\mathbf A_2 e^{ikx}. \]
To establish the connection with the usual optical notation, set
\[ \left. \begin{aligned} \omega_1&= \begin{pmatrix} 1&0\\ 0&-1 \end{pmatrix}, & \omega_3&= \begin{pmatrix} 0&-i\\ i&0 \end{pmatrix}, \\ \omega_2&= \begin{pmatrix} 0&1\\ 1&0 \end{pmatrix}, \end{aligned} \right\} \tag{2.25} \]
so that, as for \(\sigma_1,\sigma_2,\sigma_3\),
\[ \omega_k^2=1,\qquad \omega_1\omega_2=i\omega_3\quad \text{etc.} \tag{2.26} \]
It follows that, for a pure state,
\[ \rho= \begin{Vmatrix} a_1a_1^*&a_1a_2^*\\ a_2a_1^*&a_2a_2^* \end{Vmatrix} =\frac{1}{2}(1+\xi\omega), \tag{2.27} \]
where
$$ \xi_1=|a_1|^2-|a_2|^2,\quad \xi_2=a_1a_2^*+a_2a_1^*,\quad \xi_3=i(a_1a_2^*-a_2a_1^*). \tag{2.28} $$
We can represent the real vector $\boldsymbol{\xi}$ by means of two angles $\alpha$ and $\beta$:
$$ \begin{aligned} \xi_1&=\cos 2\beta \cos 2\alpha,\\ \xi_2&=\cos 2\beta \sin 2\alpha,\\ \xi_3&=\sin 2\beta. \end{aligned} \tag{2.29} $$
These angles determine nothing other than the position of an ideal analyzer for light (Fig. 1), consisting of a quarter-wave plate and a Nicol prism. A similar description of photon polarization goes back to Stokes (see $^{6,16}$). Positive and negative unit vectors in the directions 1, 2 can be assigned to the polarization states corresponding to items a), b), c) in group (A).
Fig. 1. Diagram of an ideal analyzer of polarized light, consisting of a quarter-wave plate and a Nicol prism. Light propagates “slowly” through the quarter-wave plate when its electric vector lies in plane 1, and “rapidly” when this vector is in plane 2; light is transmitted by the Nicol prism when the electric vector is located in the plane indicated by cross-hatching. The state of polarization can be characterized by the angles $\alpha$ and $\beta$, corresponding to the position of the analyzer for maximum transmission of light, and by the intensities of the light that has passed through the analyzer at its maximum and minimum transmissions (taken from $^6$).
in directions 1, 2 may be assigned to the polarization states corresponding to items a), b), c) in group (A). It should be emphasized that the electron spin $\boldsymbol{\zeta}$ is a direction in physical space; the three-dimensional vector $\boldsymbol{\xi}$, however, defines the “space of polarizations,” which differs from real physical space. The meaning of the directions of the coordinate axes in $\xi$-space is not determined by the choice of a coordinate system in physical space, but requires additional fixing—for example, by choosing some plane (say, one playing the role of the scattering plane in some processes) so that $A_1$ defines linear polarization in this plane. What has been said follows from the fact that, in invariant expressions for physical processes, $\boldsymbol{\zeta}$ can enter only in invariant combina-
...tions with other vectors of physical space, such as the momentum \(\mathbf p\), which gives, for example, \(\mathbf p\cdot \boldsymbol{\zeta}\) or \(\mathbf p\times \boldsymbol{\zeta}\), whereas the components \(\xi\) may enter into them separately (see, for example, the expressions for Compton scattering in § 4).
B. Description of transition probabilities by means of density matrices\(^{10,11,14}\)
Several remarks can be made concerning the writing of transition probabilities by means of density matrices (here we again return to a joint formulation for electrons and photons) for calculations in which polarization effects must be taken into account. If the system is prepared in the state \(\rho_0\) at the time \(t=0\), then in the time \(t\) it will pass into the state \(\rho(t)\), where
\[ \rho(t)=e^{-\frac{iHt}{\hbar}} \rho_0 e^{\frac{iHt}{\hbar}} \tag{2,30} \]
and \(H\) is the Hamiltonian. The probability of obtaining a registration in the detector \(\rho^{\mathrm{det}}\) during the time \(t\) will then be given by the expression
\[ P(0\to \mathrm{det})=\operatorname{Sp}\,[\rho^{\mathrm{det}}\rho(t)] =\operatorname{Sp}\left[\rho^{\mathrm{det}} e^{-\frac{iHt}{\hbar}} \rho_0 e^{\frac{iHt}{\hbar}}\right]. \tag{2,31} \]
In scattering experiments we are interested in transition rates
\[ R(0\to \mathrm{det})=\frac{1}{t}P(0\to \mathrm{det}) \]
(for small times \(t\)), which determine effective cross sections. It is assumed here that \(\rho^{\mathrm{det}}\) is a partial form of the density matrix referring only to the registration of scattered states. The well-known formula of perturbation theory may be written with the aid of the density matrix in the form
\[ R(0\to \mathrm{det})=\frac{2\pi}{\hbar}\,d_f(E)\operatorname{Sp}\,[\rho^{\mathrm{det}}\Omega\rho_0\Omega^{+}]. \tag{2,32} \]
Here \(d_f(E)\) is the density of final states, and \(\Omega\) is the perturbing part of the Hamiltonian (in the first approximation of perturbation theory). Let us note the symmetric position of \(\rho_0\) and \(\rho^{\mathrm{det}}\) in the formulas.
When calculating polarization effects in those physical processes in which photons and electrons participate, according to (2,32) the dependence of \(\rho_0\) and \(\rho^{\mathrm{det}}\) on polarization is specified by operators of the form (2,33) and (2,27), and in order to obtain the final results one must take traces over the Dirac matrices \(\rho\) and \(\sigma\) for electrons and over the matrices \(\omega\) for photons. The convenience of expressing the polarization dependence of the transition probability through the vectors \(\boldsymbol{\zeta}\) and \(\boldsymbol{\xi}\) is that, if it is necessary to average the result over some unobserved polarizations, this is accomplished by simply omitting the term with \(\boldsymbol{\zeta}\) and \(\boldsymbol{\xi}\) for the corresponding quantum; the result proves to be linear in all \(\boldsymbol{\zeta}\) and \(\boldsymbol{\xi}\) (whereas, when represented by the ratios \(c_1/c_2\), it is quadratic), and orthogonal states are determined by opposite directions of \(\boldsymbol{\zeta}\) and \(\boldsymbol{\xi}\). If, however, for the polarization one uses the representation \((c_1,c_2)\), this is not so simple. Nevertheless, in obtaining the result for photons the representation \((a_1,a_2)\) has the advantage over the representation \(\boldsymbol{\xi}\) in that the former determines a direction in physical space, which can form simple invariant combinations with other physical vectors.
I. The Influence of Slowly Varying Electric and Magnetic Fields on Polarized Electron Beams
This influence consists, in the general case, in a change of the momentum vector \(\mathbf p\) and of the spin direction \(\boldsymbol\zeta\) \(^{2,17-23}\). A discussion of this effect, having a quantitative quantum-mechanical character, can be carried out by considering the influence of an external electromagnetic field on the solutions of the Dirac equation which, as the electric charge \(e\) tends to zero, pass into plane waves. In doing this it is sufficient to consider only small deviations from plane waves (of first order in \(e\)), for which we shall in this way obtain a differential law that completely determines the “constant refraction” and the change of polarization in an electromagnetic field. The results for the Dirac electron can be obtained in the following way. Let us consider separately transverse (i.e., perpendicular to \(\mathbf p\)) and longitudinal (i.e., parallel to \(\mathbf p\)) electric fields \(\mathfrak E\) and magnetic fields \(\mathfrak B\). Let the resulting change in the kinetic momentum be denoted by \(\Delta\boldsymbol\pi\) (it is necessary to distinguish between the kinetic momentum \(\boldsymbol\pi\) and the total momentum \(\mathbf p\), related to \(\boldsymbol\pi\) by the relation \(\boldsymbol\pi=\mathbf p-\dfrac{e}{c}\mathbf A\) in the case when a vector potential exists), by \(\Delta\gamma\) the angle through which the beam is deflected, and by \(\Delta\alpha\) the angle through which \(\boldsymbol\zeta\) is rotated, if it is followed over a distance \(x\) in the \(x\)-direction.
a) Transverse electric field \(\mathfrak E\) in the \(y\)-direction
\[ \Delta\pi_x=0,\quad \Delta\pi_z=0, \]
\[ \Delta\pi_y=(eE\mathfrak E/pc^2)x,\quad \Delta\gamma(\mathfrak E)=(eE\mathfrak E/p^2c^2)x, \tag{2,33} \]
\[ \left. \begin{aligned} \Delta\alpha(\mathfrak E_\perp)&=[e\mathfrak E/(E+mc^2)]x,\\ &\text{rotation about the } z\text{-axis.} \end{aligned} \right\} \tag{2,34} \]
Hence it follows that
\[ \Delta\alpha(\mathfrak E_\perp)/\Delta\gamma(\mathfrak E)=E_{\mathrm{kin}}/E, \]
where
\[ E_{\mathrm{kin}}=E-mc^2. \tag{2,35} \]
Consequently, we see that in the nonrelativistic approximation
\[ \left(\frac{\Delta\alpha(\mathfrak E)}{\Delta\gamma(\mathfrak E)}\to \frac12\left(\frac{v}{c}\right)^2 \right. \]
as \(v\to 0\)) the beam is deflected, while the spin direction remains unchanged. This is shown in Fig. 2. Thus, deflection of a longitudinally polarized beam of low-energy electrons through an angle \(\pi/2\) transforms its polarization into transverse polarization. This can be compared with the transformation of circular polarization of light into linear polarization, effected by a quarter-wave plate. The conversion of longitudinal polarization into transverse polarization may also be effected by means of a transverse electric field in the case of electrons possessing relativistic energies, but the angle of deflection of the beam must then be
\[ \frac{\pi}{2}\left(1-\frac{E_{\mathrm{kin}}}{E}\right). \]
Fig. 2. Deflection of an electron beam in an electric field (nonrelativistic approximation). The dotted lines correspond to the field lines; the short arrows indicate spin orientations. The polarization changes from longitudinal to transverse.
b) Longitudinal electric field \(\mathfrak{E}\)
\[ \Delta \pi_y=0,\qquad \Delta \pi_z=0,\qquad \Delta \pi_x=(eE\mathfrak{E}/pc^2)x, \tag{2.36} \]
\[ \Delta \alpha(\mathfrak{E}_{\parallel})=0. \tag{2.37} \]
Consequently, acceleration (or deceleration) by a longitudinal electric field leaves the polarization of the electron (the direction \(\xi\)) unchanged.
Fig. 3. Deflection of a beam of electrons in a magnetic field. The magnetic field is directed perpendicular to the plane of the drawing; the short arrows indicate the orientation of the spin. The polarization remains longitudinal.
c) Transverse magnetic field \(\mathfrak{B}\) in the direction \(z\)
\[ \Delta \pi_x=0,\qquad \Delta \pi_z=0, \]
\[ \left. \Delta \pi_y=-(e\mathfrak{B}/c)x,\qquad \Delta \gamma(\mathfrak{B})=-(e\mathfrak{B}/pc)x, \right\} \tag{2.38} \]
\[ \left. \begin{array}{l} \Delta \alpha(\mathfrak{B}_{\perp})=-(e\mathfrak{B}/pc)x,\\ \text{rotation about the axis }z, \end{array} \right\} \tag{2.39} \]
\[ \frac{\Delta \alpha(\mathfrak{B}_{\perp})}{\Delta \gamma(\mathfrak{B})}=1. \tag{2.40} \]
It follows from this that a transverse magnetic field rotates the direction of the beam with the same velocity as the direction of the electron spin (Fig. 3), so that it leaves unchanged, for example, the state of transverse polarization. This statement is valid both for relativistic and for nonrelativistic electrons.
d) Longitudinal magnetic field \(\mathfrak{B}\)
\[ \Delta \pi_x=0,\qquad \Delta \pi_y=0,\qquad \Delta \pi_z=0, \tag{2.41} \]
\[ \Delta \alpha(\mathfrak{B}_{\parallel})=-(e\mathfrak{B}/pc)x,\quad \text{rotation about the axis }x. \tag{2.42} \]
The direction and magnitude of the momentum do not change, but the electron spin precesses about the axis \(x\). This is analogous to the rotation of the plane of polarization of light by a quartz plate (or by a sugar solution).
The rotations of the electron spin indicated in a), b), c) were calculated directly from the Dirac equation. For a real electron the value of \(g\) is not exactly equal to 2, and we must multiply the velocity of spin precession (which is a consequence of its magnetic moment), according to quantum electrodynamics, by
\[ \frac{g}{2}=1+\frac{\alpha}{2\pi}+\cdots . \]
As a result, spin precession in a magnetic field takes place somewhat faster than the changes in the orbital motion, so that deviations from transverse polarization may occur, though not earlier than approximately after
\[ \frac{1}{4}\left[\frac{g}{2}-1\right]^{-1}\simeq 250 \]
“cyclotron” revolutions of the electron trajectory. A mathematical description of the motion of an electron possessing an anomalous magnetic moment was considered in detail by Mendelevich and Nizov \(^{23}\). The corresponding description can also be obtained by means of the Foldy–Wouthuysen transformation \(^{24}\).
The fact that the influence of electric and magnetic (nearly homogeneous) fields on polarized electron beams can be represented simply as a rotation of the spin vector \(\xi\) about some axis has an immediate consequence. If an unpolarized beam, which
can be regarded as an “ensemble” of electrons with \(\zeta\), distributed isotropically in all directions, passes through such an electric or magnetic field, then the distribution of the directions of \(\zeta\) remains isotropic, and therefore the beam remains unpolarized.
It might seem possible to try to obtain polarized electrons by a method similar to that used in the Stern–Gerlach experiment, in which, however, the electrons would pass through a strongly inhomogeneous magnetic field (varying on macroscopic scales). However, a well-known argument due to Bohr and Mott (see \(^{2,25}\), where a more precise quantitative treatment of the question is carried out) shows that splitting of an electron beam according to spin orientations cannot be obtained in this way: the inhomogeneity of the magnetic field causes such considerable spreading of the charged electron beam (the particles in the Stern–Gerlach experiment were electrically neutral) that the splitting due to different orientations of the magnetic moment in the inhomogeneous magnetic field becomes completely unobservable.
One may discuss the additional possibility of obtaining polarized electrons by reflection of an electron beam by means of an externally imposed change of potential, by analogy with the polarization of light upon reflection from a mirror (the Mott effect). Consideration of this question in a number of theoretical papers also led to a simple proof of the impossibility of this \(^{21,22,26–31}\) (this proof is reproduced in \(^{3}\)).
Thus, it appears, generally speaking, that all the above-mentioned proposals, in which the electromagnetic fields vary on macroscopic scales, are unsuitable for producing electron polarization. In the next section the possibility of obtaining polarized electrons with the aid of fields varying on microscopic scales will be discussed in detail.
There is no sharp distinction between variations of fields on macro- and microscopic scales. Electron polarization can be produced by a combination of very strong magnetic fields (of order \(10^{4}\) oersted) and very weak electric fields (of order \(10^{-4}\) e), necessary for removing the degeneracy from discrete quantum states. This proposal was made by Bloch and Dicke (for further discussion see \(^{7}\)).
§ 3. POLARIZATION OF ELECTRONS BY COULOMB SCATTERING ON NUCLEI. THEORY AND EXPERIMENT
At the end of the preceding paragraph the impossibility was considered of obtaining polarized electrons with the aid of various devices in which the electric and magnetic fields vary on macroscopic scales. Mott \(^{25,32}\) first showed that electron polarization can be produced by the Coulomb field of nuclei. Here we are dealing with a strongly inhomogeneous electric field, varying on microscopic scales. The physical causes underlying the polarization effect consist in the fact that scattering through a certain angle depends on the spin–orbit coupling caused by the interaction of the electron magnetic moment with the magnetic field arising when the electron moves in an electric field. This effect can be calculated quantitatively by considering the solutions of the Dirac equation with a spherically symmetric electric potential corresponding to the scattering of electron waves. The potential in this case is taken either in the form of the unperturbed Coulomb field of the nucleus, or
is corrected for the screening of the nucleus by the atomic electrons. The corresponding solutions of this equation may be written in asymptotic form (if the incident waves propagate in the direction of positive \(z\)):
\[ \psi_\lambda=b_\lambda e^{ikz}+\frac{e^{ikr}}{r}u_\lambda(\theta,\varphi) \tag{3,1} \]
(\(\lambda=1,2,3,4\) are Dirac indices). The asymptotic behavior in the case of a purely Coulomb field differs somewhat from that given by (3,1) in the radial dependence (see \(^{3}\)), but this is immaterial for what follows. If, for convenience, we put \(b_3=A\) and \(b_4=B\), then in the general case it can be shown that, for a spherically symmetric potential, \(u_3\) and \(u_4\) have the form
\[ \begin{aligned} u_3(\theta,\varphi)&=Af(\theta)-Bg(\theta)e^{-i\varphi},\\ u_4(\theta,\varphi)&=Bf(\theta)+Ag(\theta)e^{i\varphi}. \end{aligned} \tag{3,2} \]
The functions \(f(\theta)\) and \(g(\theta)\) must be determined from the detailed solution of the Dirac equation. It is important to note that (3,2) characterizes the scattering completely: the formula contains the dependence on direction as well as on polarization. One can, for example, calculate the total intensity \(I(\theta,\varphi)\) scattered in the direction \((\theta,\varphi)\), if the initial polarization is specified by the parameters \(A\) and \(B\):
\[ I(\theta,\varphi)=\frac{|u_3|^2+|u_4|^2}{|A|^2+|B|^2}. \tag{3,3} \]
However, the result obtained in this way does not fully reflect the dependence of the scattering on direction and polarization. We must therefore introduce into consideration the initial polarization of the beam, as well as the polarization observed after scattering. The differential effective cross section, including the dependence on both these polarizations, may be written as \(I(\mathbf p_1,\mathbf p_2,\boldsymbol\zeta_1,\boldsymbol\zeta_2)\), where \(\boldsymbol\zeta_1\) and \(\boldsymbol\zeta_2\) are the initial and final polarizations of the electron, and \(\mathbf p_1\) and \(\mathbf p_2\) are the corresponding momenta (below we introduce \(\mathbf n_1\) and \(\mathbf n_2\), unit vectors in the directions of \(\mathbf p_1\) and \(\mathbf p_2\), so that \(\mathbf n_1\mathbf n_2=\cos\theta\)). Accordingly, let us call \(\rho_1\) and \(\rho_2\) the density matrices for the initial state and for the detector sensitive to the polarization of the scattered electron:
\[ \rho_1=\frac{1}{2}(1+\boldsymbol\zeta_1\cdot\boldsymbol\sigma), \tag{3,4} \]
\[ \rho_2=\frac{1}{2}(1+\boldsymbol\zeta_2\cdot\boldsymbol\sigma). \tag{3,5} \]
The relation between \((A,B)\) and \((u_3,u_4)\) given by (3,2) can be expressed with the aid of the operator
\[ \Omega=f(\theta)+ig(\theta)[\sigma_x\sin\varphi-\sigma_y\cos\varphi], \tag{3,6} \]
with whose aid the effective cross section depending on polarization can be written in a form analogous to (2,32):
\[ I(\mathbf p_1,\mathbf p_2,\boldsymbol\zeta_1,\boldsymbol\zeta_2)=\operatorname{Sp}[\rho_2\Omega\rho_1\Omega^{+}]. \tag{3,7} \]
The result obtained by taking the traces of the matrices \(\sigma\) can be expressed in an invariant form, valid in any coordinate system:
\[ \begin{aligned} I(\mathbf p_1,\mathbf p_2,\boldsymbol\zeta_1,\boldsymbol\zeta_2)=\;& \frac{1}{2}\,\bar I(\theta)[1+\boldsymbol\zeta_1\cdot\boldsymbol\zeta_2] +\frac{1}{2}\,\frac{D(\theta)}{\sin\theta}\,[\boldsymbol\zeta_1(\mathbf n_1\times \mathbf n_2)+\boldsymbol\zeta_2(\mathbf n_1\times \mathbf n_2)]\\ &+\frac{1}{2}\,\frac{F(\theta)}{\sin\theta}\,[-(\boldsymbol\zeta_1\cdot\mathbf n_2)(\boldsymbol\zeta_2\cdot\mathbf n_1) +(\boldsymbol\zeta_1\cdot\mathbf n_1)(\boldsymbol\zeta_2\cdot\mathbf n_2)]\\ &+\frac{1}{2}\,\frac{G(\theta)}{\sin\theta}\times\\ &\quad\times\{(\mathbf n_1\cdot\mathbf n_2)[(\boldsymbol\zeta_1\cdot\mathbf n_2)(\boldsymbol\zeta_2\cdot\mathbf n_1) +(\boldsymbol\zeta_1\cdot\mathbf n_1)(\boldsymbol\zeta_2\cdot\mathbf n_2)]\\ &\qquad-[(\boldsymbol\zeta_1\cdot\mathbf n_2)(\boldsymbol\zeta_2\cdot\mathbf n_2) +(\boldsymbol\zeta_1\cdot\mathbf n_1)(\boldsymbol\zeta_2\cdot\mathbf n_1)]\}, \end{aligned} \tag{3,8} \]
where the abbreviations have been used
\[ \begin{gathered} \bar I(\theta)=|f|^2+|g|^2,\quad D(\theta)=i(fg^*-f^*g),\\ F(\theta)=fg^*+f^*g,\quad G(\theta)=2|g|^2. \end{gathered} \tag{3,8a} \]
This formula, with respect to its dependence on \(\boldsymbol\zeta_1\) or \(\boldsymbol\zeta_2\), has a form analogous to (2,18), which makes it possible easily to interpret any case of partial polarization. Consequently, the notation in the form (3,8) represents the scattering and its dependence on polarization in a completely explicit form, so that the further consideration of polarization in single or double scattering can be carried out using only (3,8) and without returning to the wave function (3,2). Examining (3,8), one can arrive at once at the following conclusions:
-
If a polarized electron is scattered whose spin direction was initially perpendicular to the scattering plane, then upon scattering it remains unchanged.
-
If an initially unpolarized electron is scattered, then after scattering it acquires a certain degree of polarization
\[ a(\theta)=D(\theta)/\bar I(\theta), \]
and the direction \(\boldsymbol\zeta\) after scattering becomes perpendicular to the scattering plane.
- The ratio of the intensities in the two directions at a certain scattering angle \(\theta\), for a given plane of polarization of the incident beam with degree of polarization \(P\) (and with \(\boldsymbol\zeta\) perpendicular to the scattering plane), is given by the expression
\[ \frac{1+Pa(\theta)}{1-Pa(\theta)}. \]
- The intensity after double scattering of an unpolarized beam through angles \(\theta_1\) and \(\theta_2\) is given by the formulas
\[ \bar I(\theta_1,\theta_2,\varphi)=1+\delta\cos\varphi,\quad \delta(\theta_1,\theta_2)=a(\theta_1)a(\theta_2), \tag{3,9} \]
where \(\varphi\) is the angle between the planes of the first and the second scattering.
- An experiment on depolarizing and ordinary double scattering makes it possible to determine only two of the four real functions (namely \(\bar I(\theta)\) and \(D(\theta)\)), which can be formed according to (3,8a) from the two complex functions \(f(\theta)\) and \(g(\theta)\). However, it follows from (3,8) that the other two functions, in principle, cannot be observed. A measurement could be arranged, for example, with the aid of an experiment on triple scattering, performed so that in the second scattering
one could observe both the initial \((\xi_1)\) and the final \((\xi_2)\) polarizations.
We have digressed somewhat from the consequences of (3.2), in order to clarify the state of affairs in the polarization of electrons by Coulomb scattering. (To obtain a more complete picture of the theory of the question, one must also make use of the works \(^{8,23}\).) For our purpose it will be useful to consider the situation arising in double scattering (see Fig. 4; for simplicity we take everywhere the scattering angle \(\pi/2\), \(a=a\left(\theta=\frac{\pi}{2}\right)\)). Let us choose points \(Q, R, S, T, U\), and \(V\) lying in one plane. The beams are scattered at the points \(R\) and \(T\), and the intensities are measured at \(U\) and \(V\); Fig. 4 also gives the intensities of the various beams; \(I\), \(I'\), and \(I''\) are constants.
Fig. 4. Intensities for the case of double scattering (at right angles) and for spin orientations “up” and “down.” The beams are scattered at the points \(R\) and \(T\). The intensities for double scattering of an unpolarized beam are obtained by summation, so that the resulting intensities at \(U\) and \(V\) are different.
The relative intensities measured at the points \(U\) and \(V\) when working with an initially unpolarized beam can be determined by means of the incoherent superposition of positions \(a\) and \(b\) in Fig. 4. Thus we have
\[ \left. \begin{aligned} I_U &= \frac{1}{2} I''\left[(1+a)^2+(1-a)^2\right]=I''(1+a^2),\\ I_V &= \frac{1}{2} I''\left[(1+a)(1-a)+(1+a)(1-a)\right]=I''(1-a^2), \end{aligned} \right\} \tag{3.10} \]
and consequently,
\[ \frac{I_U}{I_V}=\frac{1+a^2}{1-a^2}=\frac{1+\delta}{1-\delta}, \tag{3.11} \]
where
\[ \delta=a^2, \]
which is a special case of (3.9). It is seen that the intensity at \(U\) is greater than at \(V\); according to (3.9), double scattering has a minimum at \(\varphi=0\) and a maximum at \(\varphi=\pi\), in contrast to the experiment on double scattering of photons (Barkla’s experiment), in which maxima are observed at \(\varphi=0,\pi\), and minima at \(\varphi=\frac{1}{2}\pi\) and \(\varphi=\frac{3}{2}\pi\).
We shall confine ourselves to these results and refer to the theoretical calculations of the numerical values of \(\delta\) and \(a\). Mott \(^{25}\) obtained for \(\delta\), in double scattering under the conditions \(\theta_1=\theta_2=\pi/2\) by a nucleus with charge \(Ze\), the value
\[ \delta\simeq(\alpha Z)^2\beta^2(1-\beta^2)(2-\beta^2)^{-2} \qquad (\beta=v/c), \tag{3.12} \]
which is valid only for \(aZ \ll 1\). From this expression it is seen that \(\delta \to 0\) as \(\beta \to 0\) and as \(\beta \to 1\); this remains true also for large \(Z\). But only in the latter case do the values of \(\delta\) become sufficiently large for them to be measured. The calculations necessary for obtaining exact values of \(\delta\) were carried out by Mott \(^{32}\) in 1932, who found \(\delta\) as a function of \(v/c\) for \(\theta_1=\theta_2=\pi/2\) for gold \((Z=79)\); for mercury \((Z=80)\)—by Bartlett and Watson in 1940 \(^{33,34}\)—both for the unperturbed Coulomb field. Sauter \(^{35}\) obtained Mott’s result \(^{25}\) by a somewhat different method and estimated the effect of screening by atomic electrons in the first approximation. The latter was calculated numerically by Bartlett and Welton \(^{36}\), Tassi and Mott \(^{37}\) for the case \(\theta_1=\theta_2=\pi/2\), Mott \(^{38}\), and Mott and Tassi \(^{39}\). Bartlett and Welton, and also Mott, considered the dependence of \(\delta\) on the scattering angle*). Some theoretical results are given in Table I.
Table I
Theoretical values of the asymmetry (2008) in a double-scattering experiment for a number of scattering angles \(\vartheta_1=\vartheta_2=\vartheta\) at various electron energies \(E_{\mathrm{kin}}\).
Scattering by gold \((Z=79)\) and mercury \((Z=80)\)
| Author | \(E_{\mathrm{kin}}\) | \(45^\circ\) | \(60^\circ\) | \(75^\circ\) | \(90^\circ\) | \(105^\circ\) | \(120^\circ\) | \(135^\circ\) | \(150^\circ\) | \(165^\circ\) |
|---|---|---|---|---|---|---|---|---|---|---|
| Bartlett and Welton \(^{36}\) \((Z=80)\) | 100 kev | 4.8 | 9.6 | 19.8 | 22.6 | 24.0 | 17.2 | 5.2 | ||
| Mott and Tassi \(^{39}\) \((Z=79)\) | 121 kev | 0 | 1 | 6 | 16 | 25 | 33 | 32 | 25 | 14 |
| Mott \(^{38}\) \((Z=79)\) | 392 kev | 10 | 1 | 0 | 16 | 70 | 90 | 140 | 190 | 10 |
| Sherman \(^{40}\) \((Z=80)\) | 128 kev \((\beta=0.6)\) |
0.0 | 0.8 | 5.1 | 14.7 | 26.9 | 36.0 | 34.9 | 22.7 | 7.1 |
| Sherman \(^{40}\) \((Z=80)\) | 205 kev \((\beta=0.7)\) |
0.0 | 1.0 | 5.2 | 14.0 | 26.5 | 38.0 | 41.0 | 30.0 | 10.2 |
| Sherman \(^{40}\) \((Z=80)\) | 341 kev \((\beta=0.8)\) |
0.1 | 1.0 | 4.5 | 11.7 | 23.4 | 36.8 | 45.9 | 39.8 | 15.8 |
| Sherman \(^{40}\) \((Z=80)\) | 661 kev \((\beta=0.9)\) |
0.1 | 0.7 | 2.7 | 7.2 | 15.3 | 27.8 | 43.1 | 51.0 | 28.9 |
It can be seen that for the asymmetry in backward scattering and at energies of about 400 kev, very high values are predicted. The values of \(\delta\) for positrons were calculated by Massey \(^{41}\). The polarization effect for positrons is so much less significant than for electrons that it proves almost inaccessible to experimental observation. In Figs. 5 and 6 are shown the results of calculations by Bartlett and Watson \(^{33,34}\) and Massey \(^{41}\); in them are given the curves \(\delta\) and \(a=\sqrt{\delta}\) as functions of the electron energy. The accuracy of these old theoretical values, shown in the table and in the graphs, should not be overestimated.
For a long time (until 1942) the experimental results gave no evidence of a polarization effect in double scattering of electrons. The carefully performed experiments of Diamond \(^{42,43}\), J. P. Thomson \(^{44}\), and Richter \(^{45}\) led to negative results. The same may also be said of earlier experiments \(^{46-49}\); the asymmetry,
*) Similar calculations of the polarization asymmetry were performed on the Univac computer by Sherman \(^{40}\) for \(Z=13, 48\), and 80 and for the scattering of electrons by point nuclei.
observed in these experiments is now attributed to the influence of the apparatus itself. The difficulty of the experiments lies in the absence of certainty that truly single scattering is being observed. The experiments were mainly carried out with thin gold foils. A polarization magnitude smaller than for the case of single scattering may be caused by the following reasons: (a) inelastic scattering with ionization or excitation of the scattering atom; (b) exchange scattering—if the scattered electron exchanges positions with an atomic electron, then the polarization must be absent; (c) multiple scattering—if the final scattering angle is composed of many small scattering angles at each stage, then in all the latter the polarization will be negligibly small and no overall effect will exist; (d) plural scattering—we speak of it if the total scattering is the result of two processes occurring through rather large angles.
Fig. 5. Relative asymmetry \(200\delta\), in percent, in the double-scattering experiment as a function of energy for electrons and positrons \((Z=80)\) (on the basis of data calculated by Massey \(^{41}\), Bartlett and Watson \(^{34}\)). Scattering angles \(90^\circ\).
Examples of geometry occurring in plural scattering are shown in Fig. 7. The contribution of the latter to the total scattering is especially significant if the first scattering occurs in the direction of the foil and if both scattering angles are smaller than the total scattering angle (since the differential cross section increases rapidly as the angle decreases). The influence of causes a), b), c) and the depolarization caused by them were discussed by Rose and Bethe \(^{50}\). Wenzel \(^{51}\) had still earlier given a criterion for foil thickness in the case when multiple Coulomb scattering is negligible in comparison with the full scattering. From this criterion it may be concluded that foils of thickness of order \(10^{-5}\,\mathrm{cm}\), used in the above-mentioned experiments, were sufficiently thin for the first three causes of depolarization to be neglected. The negative results of the early experiments are now considered a consequence of the depolarizing influence of plural scattering for foils that were in the reflection position at an angle \(\pi/4\). (Kikuchi \(^{52}\) obtained positive results, using
Fig. 6. Relative asymmetry \(200(a)\), in percent, in the single-scattering experiment of an initially polarized beam as a function of energy for electrons and positrons \((Z=80)\) (on the basis of calculations cited under Fig. 5). Scattering angle \(90^\circ\).
using a polarizing foil of thickness \(10^{-3}\) cm; at present it is considered proven that the asymmetry he found was of instrumental origin.)
Even before the role of plural scattering was established, searches were made for a form of the nuclear potential that would make it possible to explain the negative results of the double-scattering experiment \(^{53,54}\). It was found that the form of the nuclear potential required for this explanation is rather improbable. Landau emphasized the importance of depolarizing multiple scattering \(^{55}\); now, however, it appears that plural scattering is of greater significance.
The importance of plural scattering and the noticeable deviations of the sum of the reflection and transmission coefficients from unity were first investigated and discovered by Chase, Cox, and Herzl \(^{56,57}\), and by Petukhov and
Fig. 7. Geometry of plural scattering for various arrangements. Solid lines are incident and singly scattered electrons; dashed lines are electrons scattered plurally at the same total scattering angle. The most important case is that in which the first scattering occurs in the plane of the scattering foil.
Vyshinskii \(^{58}\). Further investigations were carried out by Ryu \(^{59,60}\) (the treatment of the question in \(^{61}\) contains an error). The ratio of the actual scattering intensities (including plural scattering) to the intensity for the case of only single scattering may be taken to be \(1+r\) and \(1+t\), respectively, in the reflection and transmission arrangements. If scattering through an angle \(\pi/4\) exhibits a polarization effect that is negligible in comparison with scattering through an angle \(\pi/2\), then the quantity \(\delta\), characterizing the asymmetry in double scattering, is evidently reduced to the values \(\delta/(1+r)^2\), \(\delta/(1+t)^2\).
The ratio \(R=\dfrac{1+r}{1+t}\) was determined experimentally under certain conditions. Shull, Chase, and Myers \(^{62}\) obtained, for a foil of thickness \(4.1\cdot10^{-5}\) cm at an electron energy of 400 keV, the value \(R=1.55\); Ryu, Hashimoto, and Nonaka \(^{60}\), for a foil of thickness \(5\cdot10^{-6}\) cm at 100 keV, obtained \(R=1.4\) (for other results see the references). Theoretical calculations give a rough explanation of the values obtained. Taking into account the possibility of plural scattering, Shull, Chase, and Myers in 1942 \(^{62}\) made the first double-scattering experiment, which ended successfully. To avoid depolarization in plural scattering, in later experiments the foils
were set in positions \(b\), \(d\), or \(e\) (Fig. 7). It is clear from the drawing that an electron scattered repeatedly in these positions must undergo scattering through an angle of at least \(\pi/2\) or more, so that for it too a substantial polarization effect should be expected. Whatever the asymmetry introduced by the apparatus, it is not easy to exclude it in the experiments. In this respect the symmetric arrangement of the foil of the polarization detector and of the two counters has advantages (see Fig. 7, positions \(d, e\)).
The most important results on the asymmetry in double scattering are summarized in Table II. Comparing the theoretical and experimental
Table II
Experimental values of the asymmetry in experiments on double scattering through angles \(\vartheta_1\) and \(\vartheta_2\) as a function of the electron energy \(E\)
| Author | Scattering foils, element and position of foil | Foil thickness | Electron energy (keV) | \(\vartheta_1\) | \(\vartheta_2\) | Observed asymmetry \(200\delta\) |
|---|---|---|---|---|---|---|
| Shull, Chase, and Myers | Au—Au, transmission position at \(45^\circ\) | \(4.1\cdot10^{-5}\ \mathrm{cm}\) | 400 | \(90^\circ\) | \(90^\circ\) | \(12\pm2\) |
| Shinohara and Ri \(^{63}\); Ri*) \(^{59,60,64-66}\) | Au—Au \((a)\ (b)\) | \(5\cdot10^{-6}\ \mathrm{cm}\) | 90 | \(90^\circ\) | \(78^\circ\) | \(9.0\pm0.6\) |
| Shinohara and Ri \(^{63}\); Ri*) \(^{59,60,64-66}\) | Au—Au \((a)\ (b)\) | \(5\cdot10^{-6}\ \mathrm{cm}\) | 60 | \(105^\circ\) | \(105^\circ\) | \(7.9\pm2.2\) |
| Shinohara and Ri \(^{63}\); Ri*) \(^{59,60,64-66}\) | Au—Au \((a)\ (b)\) | \(5\cdot10^{-6}\ \mathrm{cm}\) | 80 | \(105^\circ\) | \(105^\circ\) | \(9.9\pm1.6\) |
| Shinohara and Ri \(^{63}\); Ri*) \(^{59,60,64-66}\) | Au—Au \((a)\ (b)\) | \(5\cdot10^{-6}\ \mathrm{cm}\) | 100 | \(105^\circ\) | \(105^\circ\) | \(13.2\pm2.2\) |
| Shinohara and Ri \(^{63}\); Ri*) \(^{59,60,64-66}\) | Au—Au \((a)\ (b)\) | \(5\cdot10^{-6}\ \mathrm{cm}\) | 120 | \(105^\circ\) | \(105^\circ\) | \(14.7\pm1.7\) |
| Shinohara and Ri \(^{63}\); Ri*) \(^{59,60,64-66}\) | Au—Au \((a)\ (b)\) | \(5\cdot10^{-6}\ \mathrm{cm}\) | 60 | \(120^\circ\) | \(120^\circ\) | \(9.1\pm2.2\) |
| Shinohara and Ri \(^{63}\); Ri*) \(^{59,60,64-66}\) | Au—Au \((a)\ (b)\) | \(5\cdot10^{-6}\ \mathrm{cm}\) | 80 | \(120^\circ\) | \(120^\circ\) | \(11.8\pm2.0\) |
| Shinohara and Ri \(^{63}\); Ri*) \(^{59,60,64-66}\) | Au—Au \((a)\ (b)\) | \(5\cdot10^{-6}\ \mathrm{cm}\) | 100 | \(120^\circ\) | \(120^\circ\) | \(14.0\pm1.3\) |
| Shinohara and Ri \(^{63}\); Ri*) \(^{59,60,64-66}\) | Au—Au \((a)\ (b)\) | \(5\cdot10^{-6}\ \mathrm{cm}\) | 120 | \(120^\circ\) | \(120^\circ\) | \(17.8\pm0.8\) |
| Shinohara and Ri \(^{63}\); Ri*) \(^{59,60,64-66}\) | Au—Au \((a)\ (b)\) | \(5\cdot10^{-6}\ \mathrm{cm}\) | 60 | \(135^\circ\) | \(135^\circ\) | \(7.3\pm1.0\) |
| Shinohara and Ri \(^{63}\); Ri*) \(^{59,60,64-66}\) | Au—Au \((a)\ (b)\) | \(5\cdot10^{-6}\ \mathrm{cm}\) | 80 | \(135^\circ\) | \(135^\circ\) | \(13.5\pm1.0\) |
| Shinohara and Ri \(^{63}\); Ri*) \(^{59,60,64-66}\) | Au—Au \((a)\ (b)\) | \(5\cdot10^{-6}\ \mathrm{cm}\) | 100 | \(135^\circ\) | \(135^\circ\) | \(14.3\pm1.1\) |
| Shinohara and Ri \(^{63}\); Ri*) \(^{59,60,64-66}\) | Au—Au \((a)\ (b)\) | \(5\cdot10^{-6}\ \mathrm{cm}\) | 120 | \(135^\circ\) | \(135^\circ\) | \(16.2\pm2.1\) |
| Louisell, Pidd, and Crane \(^{67,68}\) | Au—Au \((a)\ (b)\) | \(0.135\ \mathrm{mg/cm^2}\) | 420 | \(90^\circ\) | \(78^\circ\) | \(8.9\pm1\) |
| Louisell, Pidd, and Crane \(^{67,68}\) | Au—Au \((a)\ (b)\) | \(0.135\ \mathrm{mg/cm^2}\) | 420 | \(90^\circ\) | \(78^\circ\) | \(5.5\pm1\) |
| Louisell, Pidd, and Crane \(^{67,68}\) | Au—Au \((a)\ (b)\) | \(0.23\ \mathrm{mg/cm^2}\) | 420 | \(90^\circ\) | \(78^\circ\) | \(5.5\pm1\) |
\((a)\) The transmission position is at an angle \(90^\circ-\frac{1}{2}\vartheta\) to the beam.
\((b)\) Perpendicular to the beam.
*) Ri measured a considerably larger number of values of \(200\delta\); however, the values given here cover almost the entire range of parameter measurements used by him; gold \(Z=79\), silver \(Z=47\).
results presented in Tables I and II, one can see the following. The value of \(\delta\), obtained by Shull and co-workers at \(400\ \mathrm{keV}\) for the case \(\vartheta_1=\vartheta_2=90^\circ\), is well explained by the theory. The experiments of Ri and co-workers show qualitative agreement with the theory in the following respects: \(\delta\) in the measured range of \(\vartheta\) and \(E_{\mathrm{kin}}\) increases with energy and with increasing scattering angle. In calculating \(\delta\) from the values of \(\vartheta\) and \(E_{\mathrm{kin}}\), in all cases the results obtained were not corrected for the theoretically calculated shielding; it is clear, however, that quantitatively the growth of \(\delta\) with increasing \(\vartheta\),
the \(P_{\mathrm{I}}\) found was much less noticeable than theoretically predicted; moreover, the experimental values obtained by him are much lower than the theoretical ones, sometimes even by almost a factor of two. The influence of multiple scattering in these experiments was reduced by choosing the favorable geometrical arrangement mentioned above. It is not excluded that some part of the discrepancies may nevertheless be attributed to residual multiple scattering. However, it is evident that it is not capable of fully explaining the observed discrepancy.\(^{60}\) The results given in Table II at \(\theta_2 = 78^\circ\) appear too high in comparison with the theoretical data of Table I. The approximate proportionality of \(a\) and \(Z\), which follows from the theory, is confirmed by the results of Luesell, Pidd, and Crane within the limits of experimental errors.
A discussion of these results indicates the desirability of carrying out further theoretical and experimental investigations of the effect. In particular, the following would be especially important: an experimental determination of \(\delta\) for \(\theta = 90^\circ,\ldots,150^\circ\) and at energies of \(400\ \mathrm{keV}\), in order to check the high values of \(\delta\) obtained for these parameter values; a more accurate theoretical estimate of the values of \(\delta\) as a function of \(E\) and \(\theta\); and, finally, a careful theoretical investigation of the dependence of \(\delta\) on the screening field.
For detecting the polarization of electrons produced by a method other than scattering, scattering by gold foil may be used (see § 5). In other analogous cases the attainable intensity of polarized electron beams will often be considerably limited, as a result of which the problem of intensity in measurements of scattered electrons may arise. In the latter case the most preferable geometry may be \(d\) (Fig. 7), used in the experiments of Luesell, Pidd, and Crane. Single scattering in such an arrangement occurs partly in the forward direction (large effective cross sections), which may make it possible to use even comparatively thick foils, since the multiply scattered electrons in this case should also reveal the polarization effect quite distinctly. On the other hand, backscattering (position \(e\), Fig. 7) may have the advantage of a larger polarization effect, but leads to lower intensities (see Table I). Therefore the choice of position \(d\) or \(e\) should be made only after a very careful theoretical and experimental investigation. In this way the number of electrons may be reduced down to \(10^{-5}\)—\(10^{-4}\) of their total number incident on the detector foil, of course in the case of high quality of the latter.
Here one may point to the negative results of a large number of experiments carried out in 1929–1935 with the aim of detecting the effect of electron polarization in double reflection from mirrors or in double scattering by means of Debye–Scherrer diffraction.\(^{71–79}\)* These results were theoretically explained in investigations of the scattering of electrons by a periodic electric field (a crystal lattice) in a number of works.\(^{29,31,80–83}\) In this case the polarization effect arose only in the second approximation of perturbation theory, but at the same time depended on the size of the crystal and was vanishingly small, except for crystals consisting of so small a number of atoms that in fact one could be speaking only of Coulomb scattering by atoms.
* It is appropriate to note that we have not included among the cited works Rupp’s results, since subsequent verification revealed their invalidity. See \(^{69,70}\).
§ 4. POLARIZATION RELATIONS IN THE CASE OF COMPTON SCATTERING. THE CONNECTION BETWEEN THE POLARIZATION OF ELECTRONS AND THE CIRCULAR POLARIZATION OF PHOTONS \(^{6,10,11,84,95}\)
In phenomena in which both photons and electrons participate, one should in general expect a close connection between the polarization of the electrons and the circular polarization of the photons, since both are connected with the spin angular momentum. This statement is very well illustrated by the polarization effects in Compton scattering. The differential effective cross section of Compton scattering, including all polarization effects, can be expressed in the following form:
\[ \frac{d\sigma}{d\Omega} = \frac{k^3}{k_0^2}\, r_0^2\, \Phi(\mathbf{k}_0,\mathbf{k},\boldsymbol{\xi}^{0},\boldsymbol{\xi},\boldsymbol{\zeta}^{0},\boldsymbol{\zeta}), \tag{4,1} \]
where \(r_0=e^2/mc^2\) is the classical electron radius, \(\mathbf{k}_0\) is the initial momentum of the photon, \(\mathbf{n}_0=\mathbf{k}_0/|\mathbf{k}_0|\), \(\mathbf{k}\) is the final momentum of the photon, \(\mathbf{n}=\mathbf{k}/|\mathbf{k}|\), \(\boldsymbol{\xi}^{0}\) is the polarization vector of the initial photon, and \(\boldsymbol{\xi}\) that of the final photon, while \(\boldsymbol{\zeta}^{0}\) and \(\boldsymbol{\zeta}\) are respectively the polarization vectors of the initial and final electrons.
In order to fix the meaning of \(\boldsymbol{\xi}^{0}\) and \(\boldsymbol{\xi}\), let us define the states \(1\) and \(2\) for the initial and final photons. We choose
\[ \left. \begin{aligned} \mathbf{A}^{0}_{1}=\mathbf{A}_{1} &= \frac{\mathbf{k}_{0}\times \mathbf{k}}{|\mathbf{k}_{0}\times \mathbf{k}|}, \\[4pt] \mathbf{A}^{0}_{2} &= \frac{\mathbf{k}_{0}\times \mathbf{A}^{0}_{1}}{|\mathbf{k}_{0}|}, \qquad \mathbf{A}_{2} = \frac{\mathbf{k}\times \mathbf{A}_{1}}{|\mathbf{k}|}. \end{aligned} \right\} \tag{4,2} \]
In this way we define the directions \(1\) and \(2\) as corresponding to photons linearly polarized in the scattering plane and perpendicular to it.
The complete expression for the effective cross section is complicated, since it contains a very large number of independent vectors. \(\Phi(\mathbf{k}_0,\mathbf{k},\boldsymbol{\xi}^{0},\boldsymbol{\xi},\boldsymbol{\zeta}^{0},\boldsymbol{\zeta})\) is a linear function of the polarization vectors and may be divided into 16 terms, according to the 16 different possible choices of sets of polarization vectors. We shall write
\[ \Phi=\Phi_{0}+\Phi_{1}+\Phi_{2}+\Phi_{3}+\Phi_{4}, \tag{4,3} \]
where \(\Phi_{0}\) does not depend on the polarizations, while
\[ \left. \begin{aligned} \Phi_{1} &= \Phi_{1}(\boldsymbol{\zeta}) + \Phi_{1}(\boldsymbol{\zeta}^{0}) + \Phi_{1}(\boldsymbol{\xi}^{0}) + \Phi_{1}(\boldsymbol{\xi}), \\ \Phi_{2} &= \Phi_{2}(\boldsymbol{\xi}^{0},\boldsymbol{\xi}) + \Phi_{2}(\boldsymbol{\zeta}^{0},\boldsymbol{\zeta}) + \Phi_{2}(\boldsymbol{\zeta}^{0},\boldsymbol{\xi}^{0}) + \Phi_{2}(\boldsymbol{\xi},\boldsymbol{\zeta}) + \Phi_{2}(\boldsymbol{\xi}^{0},\boldsymbol{\zeta}) + \Phi_{2}(\boldsymbol{\xi},\boldsymbol{\zeta}^{0}). \end{aligned} \right\} \tag{4,4} \]
\(\Phi_{3}\) and \(\Phi_{4}\) depend respectively on three and four polarization vectors.
In a real experiment not all polarizations will be observed; at the present time experiments have been carried out in which no more than two polarizations were observed. To obtain effective cross sections in some specified experiment, one should average over the unobserved initial polarizations and sum over the unobserved final polarizations. Since \(-\boldsymbol{\xi}\) and \(-\boldsymbol{\zeta}\) correspond to states orthogonal to \(\boldsymbol{\xi}\) and \(\boldsymbol{\zeta}\), the averaging and summation simply annul the unobserved terms.
For example, if \(\boldsymbol{\xi}^{0}\) and \(\boldsymbol{\zeta}^{0}\) are observed, then the experimental effective cross section is equal to
\[ \frac{d\sigma_{\mathrm{expt}}}{d\Omega} = \frac{1}{2}\frac{k^{2}}{k_{0}^{2}} r_{0}^{2} \left[ \Phi(\boldsymbol{\xi}^{0},\boldsymbol{\xi},\boldsymbol{\zeta}^{0},\boldsymbol{\zeta}) + \Phi(\boldsymbol{\xi}^{0},-\boldsymbol{\xi},\boldsymbol{\zeta}^{0},\boldsymbol{\zeta}) +\right. \]
\[ \left. +\Phi(\boldsymbol{\xi}^{0},\boldsymbol{\xi},-\boldsymbol{\zeta}^{0},\boldsymbol{\zeta}) + \Phi(\boldsymbol{\xi}^{0},-\boldsymbol{\xi},-\boldsymbol{\zeta}^{0},\boldsymbol{\zeta}) \right], \tag{4,5} \]
or else
\[ \frac{d\sigma_{\mathrm{expt}}}{d\Omega} = 2\frac{k^{3}}{k_{0}^{2}} r_{0}^{2} \left[ \Phi_{0} + \Phi_{1}(\boldsymbol{\xi}^{0}) + \Phi_{1}(\boldsymbol{\zeta}) + \Phi_{2}(\boldsymbol{\xi}^{0},\boldsymbol{\zeta}) \right]. \tag{4,6} \]
The results obtained for \(\Phi_{0},\ldots,\Phi_{2}\) (in units in which \(mc^{2}=1,\ \hbar=1,\ c=1\); it is assumed that the electron is initially at rest) are as follows:
\[ \Phi_{0} = \frac{1}{8} \left[ (1+\cos^{2}\theta) + (k_{0}-k)(1-\cos\theta) \right], \tag{4,7} \]
\[ \Phi_{1}(\boldsymbol{\xi}^{0}) = \frac{1}{8}\xi_{1}^{0}\sin^{2}\theta, \tag{4,8} \]
\[ \Phi_{1}(\boldsymbol{\xi}) = \frac{1}{8}\xi_{1}\sin^{2}\theta, \tag{4,9} \]
\[ \Phi_{1}(\boldsymbol{\zeta}^{0}) = \Phi_{1}(\boldsymbol{\zeta}) = 0, \tag{4,10} \]
\[ \Phi_{2}(\boldsymbol{\xi}^{0},\boldsymbol{\xi}) = \frac{1}{8} \left[ (1+\cos^{2}\theta)\xi_{1}^{0}\xi_{1} + 2\cos\theta(\xi_{2}^{0}\xi_{2}+\xi_{3}^{0}\xi_{3}) +\right. \]
\[ \left. + (k_{0}-k)\cos\theta(1-\cos\theta)\xi_{3}^{0}\xi_{3} \right], \tag{4,11} \]
\[ \Phi_{2}(\boldsymbol{\xi}^{0},\boldsymbol{\zeta}^{0}) = -\frac{1}{8}\xi_{3}^{0}(1-\cos\theta)\, \boldsymbol{\zeta}^{0}\!\cdot(\mathbf{k}_{0}\cos\theta+\mathbf{k}), \tag{4,12} \]
\[ \Phi_{2}(\boldsymbol{\xi},\boldsymbol{\zeta}^{0}) = -\frac{1}{8}\xi_{3}(1-\cos\theta)\, \boldsymbol{\zeta}^{0}\!\cdot(\mathbf{k}_{0}+\mathbf{k}\cos\theta), \tag{4,13} \]
\[ \Phi_{2}(\boldsymbol{\xi}^{0},\boldsymbol{\zeta}) = -\frac{1}{8}\xi_{3}^{0}(1-\cos\theta) \left[ \boldsymbol{\zeta}\!\cdot(\mathbf{k}_{0}\cos\theta+\mathbf{k}) - \right. \]
\[ \left. - \frac{(1+\cos\theta)(k_{0}+k)}{k_{0}-k+2}\, \boldsymbol{\zeta}\!\cdot(\mathbf{k}_{0}-\mathbf{k}) \right], \tag{4,14} \]
\[ \Phi_{2}(\boldsymbol{\xi},\boldsymbol{\zeta}) = -\frac{1}{8}\xi_{3}(1-\cos\theta) \left[ \boldsymbol{\zeta}\!\cdot(\mathbf{k}_{0}+\mathbf{k}\cos\theta) - \right. \]
\[ \left. - \frac{(1+\cos\theta)(k_{0}+k)}{k_{0}-k+2}\, \boldsymbol{\zeta}\!\cdot(\mathbf{k}_{0}-\mathbf{k}) \right], \tag{4,15} \]
\[ \Phi_{2}(\boldsymbol{\zeta}^{0},\boldsymbol{\zeta}) = \frac{1}{8} \left\{ (\boldsymbol{\zeta}^{0}\!\cdot\boldsymbol{\zeta}) \left[ (1+\cos^{2}\theta) + \frac{1}{2}(k_{0}-k)\sin^{2}\theta \right] -\right. \]
\[ -\frac{1}{2}(k_{0}-k) \left[ \boldsymbol{\zeta}^{0}(\mathbf{n}_{0}+\mathbf{n})\, \boldsymbol{\zeta}(\mathbf{n}_{0}+\mathbf{n}) + \boldsymbol{\zeta}^{0}(\mathbf{n}_{0}\times\mathbf{n})\, \boldsymbol{\zeta}(\mathbf{n}_{0}\times\mathbf{n}) \right] + \]
\[ + \frac{1}{2}(k_{0}+k)(1+\cos\theta) (\boldsymbol{\zeta}^{0}\times\boldsymbol{\zeta})(\mathbf{n}_{0}\times\mathbf{n}) + \]
\[ + \frac{(1+\cos\theta)(k_{0}-k)}{k_{0}-k+2} \times \boldsymbol{\zeta}(\mathbf{k}_{0}-\mathbf{k})\, \boldsymbol{\zeta}^{0}(\mathbf{n}_{0}+\mathbf{n}) - \]
\[ \left. - \frac{1+2\cos\theta-\cos^{2}\theta}{k_{0}-k+2}\, \boldsymbol{\zeta}(\mathbf{k}_{0}-\mathbf{k})\, \boldsymbol{\zeta}^{0}(\mathbf{k}_{0}-\mathbf{k}) \right\}. \tag{4,16} \]
We shall discuss three polarization effects in Compton scattering in which polarized electrons take part; these effects appear to be the most accessible to experimental verification.
A. Production and detection of circular polarization of gamma quanta by means of Compton scattering of polarized electrons
We shall be interested in the effective cross sections \(d\sigma(\xi^0,\zeta^0)\) and \(d\sigma(\xi,\zeta^0)\). Let us set, for example,
\[ \frac{d\sigma(\xi^0,\zeta^0)}{d\Omega} = \frac{1}{2}\frac{k^2}{k_0^2}r_0^2 \left[ (1+\cos^2\theta) + (k_0-k)(1-\cos\theta) + \xi_1^0\sin^2\theta - \xi_3^0(1-\cos\theta)\zeta^0(k_0\cos\theta+k) \right]. \tag{4,17} \]
In the case when the effective cross section is determined for photons polarized circularly (without linear polarization), and the photon polarization vector is parallel (or antiparallel) to \(\mathbf{k}_0\), the expression for it is simplified:
\[ \frac{d\sigma(\xi^0,\zeta^0)}{d\Omega} = \frac{1}{2}\frac{k^2}{k_0^2} \left[ (1+\cos^2\theta) + (k_0-k)(1-\cos\theta) - P(1-\cos\theta)\cos\theta\,(k_0+k) \right]. \tag{4,18} \]
Fig. 8. Relative change in the differential effective cross section of circular scattering caused by completely spin-polarized initial electrons, with spin directed along the incident circularly polarized photon. This change is represented as a function of the photon scattering angle \(\theta\) (taken from \(^{86}\)).
Here \(P\) is the product of the degrees of polarization of the photon and the electron (\(P\) is positive for a right-polarized photon and for the case of electron spin parallel to \(\mathbf{k}_0\)). This effective cross section can be written in the form
\[ \frac{d\sigma}{d\Omega} = \frac{d\sigma_0}{d\Omega} + P\,\frac{d\sigma_1}{d\Omega}, \tag{4,19} \]
where \(d\sigma_0\) is given by the Klein–Nishina formula without polarization terms, while \(d\sigma_1\) is the term sensitive to spin and to circular polarization. Integration over \(d\Omega\) gives, for the polarization-dependent part of the total effective cross section:
\[ \frac{\sigma_1}{2\pi r_0^2} = \frac{1+4k_0+5k_0^4}{k_0(1+2k_0)^2} - \frac{1+k_0}{2k_0^2}\ln(1+2k_0). \tag{4,20} \]
The ratio \((d\sigma_1/d\Omega):(d\sigma_0/d\Omega)\) shown in Fig. 8 as a function of angle shows the change in sign of the polarization effect in the differential effective cross section for forward and backward scattering. Figure 9 shows the polarization-dependent part \(\sigma_1\) of the total effective cross section for Compton scattering. The change in sign of \(\sigma_1\), which occurs at \(1.25\,mc^2=0.65\) MeV, can be explained by the difference in signs of \(d\sigma_1/d\Omega\) for forward and backward scattering, and also by the fact that, with increasing
energy, forward Compton scattering becomes increasingly predominant.
The possibility of experimentally detecting effects depending on \(\xi^0\) in Compton scattering presupposes the presence, in the initial state, of polarized electrons at rest. Such electrons exist in magnetized ferromagnetic materials. The binding of these electrons does not play an essential role at the energies most often used in studies of the Compton effect. Unfortunately, the degree of polarization \(P\), averaged over all atomic electrons, is never very high. For iron at saturation magnetization,
\[ P \simeq \frac{2}{26} \simeq 8\%. \]
The effect of circular polarization in the Compton scattering of a polarized electron was first observed by Gunston and Page \(^{86}\) in 1953.\(^*\) These authors determined the difference in the transmissions \(T\) of an iron rod 30 cm long and 3.8 cm in diameter for gamma rays of energy \(2.62\,\text{MeV}\) (\(\mathrm{ThC''}\)) in the cases of magnetized and nonmagnetized iron. For a rod of length \(L\) a corresponding difference was found also for initially unpolarized photons, which can be explained as follows: taking the transmission of the nonmagnetic sample as unity, we have that the left- and right-polarized components of the unpolarized beam give exponential factors in the expression for the transmission of the magnetic rod, whose sum is equal to
Fig. 9. The polarization-sensitive part \(\sigma_1\) of the total effective cross section for Compton scattering for the same polarization values as in Fig. 8 (in units \(2\pi r_0^2\)). \(\sigma_0\) is the part of the cross section independent of polarization (taken from \(^{86}\)).
Fig. 10. Arrangement of the apparatus for the polarization experiment of Gunston and Page. The change in the transmission of gamma rays from a strong source through a magnetic iron rod is measured (change in the total effective cross section of Compton scattering) (taken from \(^{86}\)).
\[ \frac{1}{2} e^{-NL\nu\sigma_1} + \frac{1}{2} e^{+NL\nu\sigma_1} = \operatorname{ch} NL\nu\sigma_1 \simeq 1 + \frac{1}{2}(NL\nu\sigma_1)^2, \tag{4,21} \]
where \(N\) is the number of atoms in \(1\,\text{cm}^3\), and \(\nu\) is the number of polarized electrons per atom (Fig. 10). From (4,21) it follows that, for the relative change in the transmission \(T\) upon magnetization, one may write approximately
\[ \frac{T(\nu)-T(0)}{T(0)} \simeq \frac{1}{2}(NL\nu\sigma_1)^2. \tag{4,22} \]
Further, it follows from this that the ratio of the transmissions for left- and right-polarized photons is given by
\[ R = e^{2NL\nu\sigma_1}. \tag{4,23} \]
\(^*\) The positive result in the experiment on detecting circular polarization in Compton scattering, reported earlier by Clay and Hereford \(^{87}\), is now considered to be due to the apparatus itself (private communication).
The measured change in the pulse-counting rate reached \(0.59 \pm 0.09\%\), which, together with the value \(\nu = 2.06\), gives the experimental value \(\sigma_1/2\pi r_0^2 = 0.089 \pm 0.007\), in good agreement with the theoretical value \(0.093\). It follows from (4.23) that gamma rays that have passed through the rod must possess a degree of circular polarization \(P \simeq 10\%\) (at an intensity of about \(10\) pulses/sec).
Fig. 11. Schematic of the apparatus used in the Kamerlingh Onnes Laboratory in Leiden for producing and measuring circularly polarized gamma rays. The change in the differential effective cross section of Compton scattering is determined by varying the relative orientation of the direction of circular polarization of the gamma rays, emitted by a polarized nucleus in crystal \(L_1\), and the direction of magnetization of the scattering iron \(S\). Magnet \(M_p\) (with coil \(B\)) determines the direction of polarization of the nuclei; magnet \(M_s\) (with coil \(W\)) fixes the direction of magnetization of \(S\). The gamma rays are recorded by the NaJ(Tl) crystal \(C_1\), coupled to photomultiplier \(EM1\) (taken from \(^{88}\)).
This same effect was used in the Kamerlingh Onnes Laboratory in Leiden in 1955\(^{89}\) to detect circular polarization of \(1.17\) and \(1.33\) MeV gamma rays from polarized \(Co^{60}\) nuclei (Fig. 11). A \(Co^{60}\) source with an activity of \(110\) microcuries was introduced into crystals cooled by adiabatic demagnetization to a temperature below \(0.006^\circ\) K. In this way a degree of polarization reaching \(75\%\) was obtained. It was detected by means of the polarization effect in the differential effective cross section for Compton scattering forward. The experimentally determined changes in counting rate, associated with changing the relative orientation of the circular polarization of the photons and the polarization of the electrons, reached \(3\%\). The agreement between the calculated theoretical value (also taking into account the mechanism of nuclear polarization in the crystal) and the experimental value was satisfactory. In 1955 Trumpy also reported the detection of circular polarization of gamma rays from the capture of polarized neutrons, carried out by means of transmission measurements analogous to those performed in the experiments of Gunstead and Page\(^{88}\).
B. Production of polarized electrons by means of Compton scattering of circularly polarized photons
The effective cross section corresponding to this process is equal to
\[ \frac{d\sigma'(\xi^0,\zeta)}{d\Omega} = \frac{1}{4}\frac{k^2}{k_0^2} r_0^2 \left\{ (1+\cos^2\theta) + (k_0-k)(1-\cos\theta) + \xi_1^0\sin^2\theta - \xi_3^0(1-\cos\theta) \left[ \zeta(k_0\cos\theta+k) - \frac{(1+\cos\theta)(k_0+k)}{k_0-k+2}\, \zeta(k_0-k) \right] \right\}. \tag{4.24} \]
If it is written in the form
\[ \frac{d\sigma(\zeta^0,\zeta)}{d\Omega}=S[1+\zeta\cdot\zeta^{\mathrm{scatt}}], \tag{4.25} \]
then one may say that \(\zeta^{\mathrm{scatt}}\) represents the polarization state of the Compton-scattered electron. We write:
\[ \zeta^{\mathrm{scatt}}=P_p^{\mathrm{scatt}}\mathbf e_p+P_q^{\mathrm{scatt}}\mathbf e_q, \tag{4.26} \]
where \(\mathbf e_p\) and \(\mathbf e_q\) are unit vectors in the directions perpendicular to the vectors
\[ \begin{aligned} \mathbf p&=\mathbf k_0-\mathbf k,\\ \mathbf q&=\mathbf k-\frac{1}{p^2}\mathbf p(\mathbf k\cdot\mathbf p). \end{aligned} \tag{4.27} \]
For diagrams of \(P_p^{\mathrm{scatt}}\) and \(P_q^{\mathrm{scatt}}\) (which may be called the degrees of transverse and longitudinal polarization) we refer the reader to work \(^{85}\). The total degree of polarization \(P^{\mathrm{scatt}}=[(P_p^{\mathrm{scatt}})^2+(P_q^{\mathrm{scatt}})^2]^{1/2}\) is shown in Fig. 12. All data were calculated for \(\xi_1^0=0\) and \(\xi_3^0=1\) (complete circular polarization) for the initial photon.
B. Correlation of directions between the initial and final electrons
If the polarizations of the initial and final electrons are observed, then for the corresponding effective cross section one may write:
\[ \frac{d\sigma(\zeta^0,\zeta)}{d\Omega} = 2\frac{k^2}{k_0^2}r_0^2[\Phi_0+\Phi_2(\zeta^0,\zeta)] = F(1+\zeta\cdot\zeta^{\mathrm{scatt}}). \tag{4.28} \]
It determines the polarization vector \(\zeta^{\mathrm{scatt}}\) of the scattered electron and its degree of polarization \(P^{\mathrm{scatt}}=|\zeta^{\mathrm{scatt}}|\). Let us note consequences of the general formulas: a) in the classical limit \(k_0\simeq 0\), \(P^{\mathrm{scatt}}\simeq 1\) and \(\zeta^{\mathrm{scatt}}\simeq \zeta^0\); b) for forward scattering of the photon \(\cos\theta=1\), \(P^{\mathrm{scatt}}=1\) and \(\zeta^{\mathrm{scatt}}=\zeta^0\); c) if \(\zeta^0\) is perpendicular to the scattering plane, then
\[ \begin{aligned} P^{\mathrm{scatt}} &= \frac{1+\cos^3\theta}{(1+\cos^3\theta)+(k_0-k)(1-\cos\theta)}\, \zeta^0_{\perp},\\ \zeta^{\mathrm{scatt}}&=P^{\mathrm{scatt}}\zeta^0, \end{aligned} \tag{4.29} \]
where \(\zeta^0_{\perp}\) is the absolute value of \(\zeta^0\).
Fig. 12. Production of polarized electrons by Compton scattering of completely circularly polarized photons on unpolarized electrons. The total degree of polarization of the final electrons is given as a function of the photon scattering angle.
The curves for \(P^{\mathrm{scatt}}\), calculated according to formula (4.29), are shown in Fig. 13. By analogy with Compton scattering one should expect a correlation
between the polarization of the electron and the circular polarization of the photon of gamma radiation also in other quantum processes involving photons and electrons, such as bremsstrahlung, pair production, and positron annihilation.
Fig. 13. Obtaining polarized electrons by means of Compton scattering of unpolarized photons on initially polarized electrons perpendicular to the plane of scattering. The degree of polarization of the final electrons is given as a function of the photon scattering angle.
§ 5. METHODS OF OBTAINING FREE POLARIZED ELECTRONS
We shall now list and briefly discuss the principal possibilities for obtaining free polarized electrons:
a) The only experimentally confirmed method to date for obtaining polarized electrons is Coulomb scattering by heavy nuclei, considered in § 3.
b) Compton electrons knocked out by photons polarized in a circle will, in general, possess considerable polarization (see § 4). These photons can be produced by passing gamma quanta from a very strong source through a magnetized iron rod, or else with the aid of polarized radioactive nuclei emitting gamma rays (see also § 4). Numerical estimates of this effect can be obtained. The basic problem consists in attaining, in experiment, such intensities as would make measurements of electron polarization possible.
c) Compton electrons knocked out by photons that have passed through magnetized iron (or another ferromagnetic material; for the theory of the question see § 4). The difficulty in observing this effect is also the problem of intensity. Moreover, the degree of polarization should in this case be at least about 8% (the average degree of polarization of electrons in iron magnetized to saturation), so that the measurement will not be a very easy matter even at sufficient intensities.
d) The knocking out of bound polarized electrons from magnetized iron can be carried out by methods other than Compton scattering. For this purpose one may use the photoelectric effect (see ²¹), cold emission and secondary emission, or electron scattering in magnetized iron. These methods, as far as the author knows, have not been considered in detail theoretically. An attractive property of the photoelectric effect is apparently the possibility of obtaining a degree of polarization of the ejected photoelectrons exceeding the average degree of polarization over all electrons. With an appropriate choice of photon energy it is possible to make impossible the emission of unpolarized photoelectrons from the deeper shells of the atom, which considerably increases the averaged polarization of the photoelectrons.
In what has been said, it was implicitly assumed that the three methods indicated above should not alter too strongly the polarization of the bound electrons in the process of ejection. We can estimate the order of magnitude of the probability of a spin flip in the photoeffect according to Mott’s calculations².
Let us suppose that a bound electron is in the ground state in the Coulomb field of a nucleus with charge \(Ze\). It has a velocity of the order
\[ v \simeq a Z c . \tag{5,1} \]
The action exerted by the electric field \(\mathfrak{E}\) on the magnetic moment \(\mu = e\hbar/2mc\) of an electron moving with this velocity has an order of magnitude
\[ M = \frac{v}{c}\mathfrak{E}\mu = a Z \mathfrak{E}\frac{e\hbar}{2mc} = \frac{e^3}{2mc^2}\mathfrak{E} Z . \tag{5,2} \]
In a spin flip the spin angular momentum undergoes a change by the amount \(\hbar\), which occurs during a time \(T\) of the order \(\hbar/M\):
\[ T \simeq \frac{\hbar}{M} = \frac{2\hbar mc^2}{Z\mathfrak{E}e^3}. \tag{5,3} \]
Let us compare this time with the time \(t\) during which the action of the field leads to the liberation of the bound electron from the ground state in the Coulomb field with binding energy \(Z^2(me^4/\hbar^2)\), corresponding to a momentum \(p \simeq me^2 Z/\hbar\). The time \(t\), required in order that the force \(e\mathfrak{E}\) produce a change of momentum \(p\), is of the order
\[ t \simeq \frac{p}{e\mathfrak{E}} = \frac{Zem}{\mathfrak{E}\hbar}. \tag{5,4} \]
Comparing \(T\) and \(t\), we see that
\[ \frac{T}{t} \simeq \frac{1}{(aZ)^3}. \tag{5,5} \]
Consequently, in the case of small \(Z\) one may conclude that \(T \gg t\), so that during the emission of the photoelectron a spin flip has no time to occur. It is permissible to say that the photoeffect is associated with the electric vector and that the influence of the angular momentum of the photon (polarization) on the polarization of the electron is negligible. This condition is satisfied, for example, for visible light causing photoemission of outer electrons in an atom. Estimate (5,5) shows that \(T\) can become of the same order as \(t\) for high values of \(Z\) and for electrons in lower bound states. Consequently, it may turn out that some photoelectrons, liberated from atoms with high atomic numbers by circularly polarized gamma rays, are polarized.
d) Besides a magnetized iron rod, another source of bound polarized electrons may be the optical process indicated by Kastler⁹⁰ and carried out by him with collaborators⁹¹, and also by Dicke and Hawkins¹²,⁹². If, for example, sodium vapor is placed in a magnetic field, then the Zeeman splitting removes the degeneracy of levels having different magnetic quantum numbers. By exciting a considerable number of atoms by irradiating the vapor with light of the resonance frequency directed along the magnetic field, one can produce a noticeable “polarization” of the atoms. In this method, the imple-
...is the polarization of the bound electrons and nuclei. Polarized electrons that are in excited states can be knocked out upon absorption of a second photon by the atom. If the energy of the latter is chosen sufficiently low, so that only excited electrons can be knocked out, then it will even be possible to obtain completely polarized electrons (of low energy). However, since both photons must be absorbed by one and the same atom, a difficulty again arises in obtaining a sufficient electron intensity. This circumstance was pointed out by Dike (1950, private communication), who implemented the method and with its aid obtained currents of \(10^5\) electrons/sec, possessing a calculated degree of polarization of 20–30%. However, he did not succeed in detecting the polarization experimentally. Since Coulomb scattering by heavy nuclei as a polarization detector proves possible at electron energies somewhat above 50 keV, it becomes necessary either to accelerate the polarized low-energy electrons before they enter the detector, or else to use polarization detectors sensitive to electrons possessing low energies.
e) Beta radiation emitted by polarized \(\beta\)-radioactive nuclei must, in the general case, be polarized\(^{14}\). Some nuclei can be polarized in crystals cooled by adiabatic demagnetization to temperatures of a few hundredths of a degree (see, for example, \(^{93}\)). If a \(\beta\)-radioactive nucleus changes the value of its spin in beta decay (which had a definite direction before emission), then this angular momentum is transferred to the electron and neutrino emitted in the decay process. This causes a preferential orientation of the spins of the emitted electrons. Following the general principles indicated in § 2, and proceeding from the basic propositions of the theory of \(\beta\)-decay, one can calculate the degree of polarization using perturbation theory and Dirac wave functions. In particular, four-row matrices (2.23) are used. As an example we give an expression for the transition probability with emission of a \(\beta\)-electron having energy \(E\) (including the rest mass) and momentum \(\mathbf p\), when a detector is used to detect the spin direction \(\boldsymbol{\zeta}^{\mathrm{det}}\) (the formula given is valid for the pure Gamow–Teller interaction; units are used in which \(\hbar, c, m = 1\)):
\[ P(E,\mathbf p,\boldsymbol{\zeta}^{\mathrm{det}}) = \frac{G^2}{16\pi^4}\,pEq^2 \left\{ |\sigma|^2 \left[ 1+\frac{A}{E}\left(\boldsymbol{\eta}\cdot\boldsymbol{\zeta}^{\mathrm{det}}\right) \right] + \frac{A}{E(E+1)} \left(\boldsymbol{\eta}\cdot\mathbf p\right) \left(\boldsymbol{\zeta}^{\mathrm{det}}\cdot\mathbf p\right) \right\}. \tag{5,6} \]
Here \(G\) is the Fermi interaction constant, \(\int\sigma\) is the nuclear matrix element, \(q\) is the neutrino momentum; the unit vector \(\boldsymbol{\eta}\) is the axis of polarization of the nuclei (the axis of rotational symmetry of their orientation); \(A\) is proportional to the polarization of the nuclei. If \(I_i\) and \(I_f\) are the initial and final spins of the nucleus, then we have
\[ A= \begin{cases} f_1, & \text{if } I_i=I_f+1 \quad (I_i\geqslant 1),\\[4pt] f_1/(I_i+1), & \text{if } I_i=I_f \quad \left(I_i\geqslant \dfrac12\right),\\[4pt] -\,f_1 I_i/(I_i+1), & \text{if } I_i=I_f-1 \quad (I_i\geqslant 0), \end{cases} \tag{5,7} \]
where
\[ f_1=\sum_{M=-I_i}^{I_i} \frac{M}{I_i}\,a_M \qquad (-1\leq f_1\leq 1), \tag{5,8} \]
and where \(a_M\) is the probability that the initial nucleus has magnetic quantum number \(M\) relative to the axis \(\boldsymbol{\eta}\). (The normalization is chosen so that we can write the result (5,6), as was done in § 2, in the form
\(\frac12[1+\boldsymbol{\zeta}\cdot\boldsymbol{\zeta}^{\mathrm{det}}]\).) The value of \(\boldsymbol{\zeta}\) for the spin of electrons emitted with momentum \(\mathbf p\), in this notation, is equal to
\[ \boldsymbol{\zeta} = A\left[ \frac{1}{E}\,\boldsymbol{\eta} + \frac{1}{E(E+1)}(\boldsymbol{\eta}\cdot \mathbf p)\mathbf p \right]. \tag{5,9} \]
Therefore, in particular, we have:
1) \(\mathbf p \perp \boldsymbol{\eta}\) (emission of electrons perpendicular to the axis of polarization of the nuclei; \(E\simeq 1\) in our units for low-energy electrons)
\[ \boldsymbol{\zeta} = \frac{A}{E}\boldsymbol{\eta} \left( \begin{array}{l} \text{transverse polarization,}\\ \text{degree of polarization } P=A/E \end{array} \right), \tag{5,10} \]
2) \(\mathbf p \parallel \boldsymbol{\eta}\) (electrons are emitted in the direction of the axis of polarization of the nuclei)
\[ \boldsymbol{\zeta} = A\boldsymbol{\eta} \left( \begin{array}{l} \text{longitudinal polarization,}\\ \text{degree of polarization } P=A \end{array} \right). \tag{5,11} \]
The experimental situation is such that values of \(f_1\) and \(A\) of order unity can be achieved, for example for \( \mathrm{Co}^{60}\) nuclei. Therefore one may say that the electrons emitted in polarization experiments with these nuclei will be significantly polarized. So far no attempts have been made to detect the indicated polarization of the electrons; this is difficult for the following reasons: a) the \(\beta\)-particles emitted by the source emerge only from the surface of the crystal; b) since in these experiments the source is placed in a cryostat, detection of the polarization must also be carried out inside the cryostat, or else the \(\beta\)-particles must be brought out of it—and both are very difficult to accomplish experimentally; c) it seems very difficult to obtain electron intensities sufficient for detecting their polarization.
g) With the combined use of very strong magnetic fields (from \(10^3\) to \(10^4\) oersted) and very weak electric fields (from \(10^{-5}\) to \(10^{-4}\) volt), electrons with a definite polarization can be “captured.” The latter can then be freed from the potential well with preservation of their polarization. Although space charge severely limits the number of electrons that can be captured by the field, rapid alternation of their capture and release can make it possible to obtain currents of polarized electrons sufficient for many experiments: for example, 30 electrons can be captured in each cycle, and if this operation is repeated 3000 times per second, the result will be a current of about \(10^5\) electrons per second. This method is essentially limited to low-energy electrons, but, of course, they can subsequently be accelerated without loss of polarization. Details of the mechanism of electron capture by a combined field were described by Bloch and Dike\(^{94}\) (see § 7).
§ 6. METHODS FOR DETECTING ELECTRON POLARIZATION
Since in any experiment on electron polarization both a polarizing device and an analyzer are necessary, one should also enumerate the possible principles of operation of the latter.
a) As was already indicated in § 3, up to the present time there exists only one successfully applied method for detecting electron polarization—namely, Coulomb scattering by heavy nuclei.
b) Another phenomenon that depends to some extent on the polarization of the incident electrons is scattering by magnetized iron. One may try to detect this effect either in the total effective cross section (passing electrons through a foil of magnetized iron), or in the differential effective cross section at scattering angles that make it possible to obtain sufficient sensitivity of the polarization detector. One may expect that at low energies quantum-mechanical exchange scattering, which presupposes a dependence on the relative orientations of the spins of the incident and bound electrons, will prove more effective than method a). An attempt to use the latter effect was made by Dike in 1950 (private communication), who obtained polarized electrons by method e (§ 5) and tried to detect the polarization by passing the electrons (after acceleration) through a thin (\(\simeq 10^{-6}\) cm) magnetized iron foil. A theoretical estimate of the polarization effect made it possible to expect at least a small, but measurable, magnitude; however, the effect was not detected. An earlier attempt of this kind was made by Myers and Cox in 1929. They passed the \(\beta\)-rays of radium through two foils of magnetized iron with varying directions of magnetization \(^{95}\). They did not succeed in detecting any observable polarization effect in the transmission.
c) It may also prove possible to measure the spin angular momentum carried by electrons with longitudinal polarization by mechanical means (L. Marton—private communication). If these electrons are directed onto a freely suspended disk, one can measure its twisting as a function of the polarization. In this case, however, the possibility must be excluded that the twisting is due to the linear momentum of the electrons incident on the disk, or to asymmetry of the geometry. This method may prove accessible for electrons of low energy.
d) When atoms are excited by exchange scattering with polarized electrons of low energy, the excited state may turn out to be polarized (i.e., acquire an orientation of the total angular momentum). Such atoms can be introduced into a magnetic field, in which they will pass to lower states with the emission of circularly polarized light (E. C. Doyoff—private communication). If this circular polarization is recorded, then the polarization of the incident electrons will follow unambiguously from it.
It is interesting that it is easier to create electron polarization than to devise an experiment for detecting it. Apart from the four proposals mentioned, there are almost no others. In the resonance experiments proposed by Bloch and Dike (see § 7), the detection (as well as the creation) of polarization envisages the joint use of a strong magnetic and a weak electric field; however, this method of registration depends on the specific energies at which the polarized electrons are produced, and therefore it cannot be regarded as a general method for detecting electron polarization.
§ 7. EXPERIMENTS WITH POLARIZED ELECTRONS. DETERMINATION OF THE \(g\)-FACTOR OF THE FREE ELECTRON
In the preceding sections various possibilities for obtaining and registering polarized free electrons were discussed. In addition to them, one can also pose other experiments with polarized electrons. The various attainable goals of the corresponding experiments may be summarized as follows: 1) Verification of the theory of electron polarization (see, for example, § 3). 2) Determination of the \(g\)-factor of the free electron (see below). 3) Indirect determination of the degree of polarization of bound electrons—a quantity whose knowledge may be useful for solid-state physics. (Experiments on circular polarization in Compton scattering, described in § 4, make it possible in principle to determine the averaged degree of polarization of bound electrons. Similar information could be obtained if, for example, it were possible to measure the degree of polarization of electrons emitted as a result of the photoelectric effect from magnetized iron—see § 5.) 4) Detection of the polarization of \(\beta\)-particles emitted by \(\beta\)-radioactive nuclei can give useful information on the mechanism of nuclear polarization (see § 5).
The experiments carried out up to the present time have pursued only the first two of the named goals. In the remainder of this paragraph we shall consider in more detail the arrangement of experiments carried out, or so far only proposed, for determining the value of the \(g\)-factor of the free electron. This determination is of particular interest in connection with the experimental verification of the value
\[ g = 2\left(1+\frac{\alpha}{2\pi}+\ldots\right), \]
given by quantum electrodynamics and differing from \(g=2\), which follows from Dirac’s theory, by only one thousandth. This value was found to agree with the value of \(g\) for bound electrons in an atom, but its independent determination for the free electron is meaningful, being yet another and, possibly, more precise test.
The experiments proposed for measuring the value of \(g\) of the free electron may be divided into two groups:
A. Application of a constant magnetic field between the stages of creating and detecting electron polarization; in the experiment the angle of precession of the magnetic moment in the magnetic field is measured.
B. Determination of the frequency at which a spin flip takes place, by means of a resonance experiment in a constant magnetic field. For this purpose, after the creation of a definite state of motion and polarization of the electrons and before they reach the detector for detecting the spin flip, a radio-frequency field is applied.
A1. An experiment of this kind was carried out by Louisell, Pidd, and Crane\(^{96–98}\) in the following way. An experiment was arranged on double scattering of electrons with energy \(420\ \text{keV}\) by gold foil of thickness \(0.135\ \text{mg}/\text{cm}^2\) as polarizer and analyzer (Fig. 14). Polarization was obtained by scattering through an angle of \(90^\circ\); the polarizer was set in a position for transmission at an angle of \(45^\circ\). The scattered beam fell perpendicularly on the analyzer foil; two electron counters were placed so as to cover a scattering angle of \(78^\circ\). The detectors consisted of scintillation counters made up of an anthracene crystal, a lucite light guide, and a photomultiplier. The foils of the polarizer and analyzer were separated by a distance of \(725\ \text{cm}\); the electrons passed through a brass tube of diameter \(15\ \text{cm}\); a layer of copper covering the brass tube played the role of a solenoid through which a current was passed
at 60 amperes, producing a homogeneous magnetic field of strength about 120 oersteds. This magnetic field caused precession of the electron spin, which during the time the electron traversed the tube managed to make almost five complete revolutions (\(\simeq 1800^\circ\)). The precession was measured by rotating the analyzer counters about the axis of the brass tube. In this way the phase of the polarization asymmetry was determined for several
Fig. 14. Diagram of the experiment of Louisell, Pidd, and Crane on double scattering with a magnetic field applied between two scattering acts. The electron spin precesses in a plane perpendicular to the beam direction (taken from \(^{96}\)).
values of the current in the solenoid. Instrumental asymmetry was eliminated by comparing the counting rates that occurred for a polarizer foil made of aluminum (small \(Z\)), which produced negligible electron polarization. In addition to spin precession, the magnetic field also produced periodic focusing of the slightly divergent beam (aperture \(2.25^\circ\)). The focusing occurred at distances at which the electron, in cyclotron (orbital) motion perpendicular to the axis of the tube, completed an integer number of revolutions. Consequently, if the value of \(g\) were equal to 2, focusing should have occurred at the same positions that corresponded to an integer number of revolutions of the precessing spin. Hence \(g\) could be determined by comparing the spin precession with the focusing of the electrons, which could be established independently of one another. In carrying out such a relative measurement there is no need, for example, to perform an absolute calibration of the magnetic field. The experiment gave the value \(g = 2.00 \pm 0.01\), with an accuracy to within half a percent.
A2. Crane proposed an experimental arrangement that would make it possible to carry out measurements with an accuracy of about \(10^{-5}\) \(^{99}\). Its construction has already been begun. Instead of using a constant electron current, the experiment produces a pulsed beam captured by a constant magnetic field (of the betatron-field type),
in which it makes from 1000 to 10,000 revolutions before the second scattering. The number of revolutions of the pulsed beam can be found from the flight time \(\tau\). As was indicated in § 2, the anomalous magnetic moment of the electron can cause a change of the transverse polarization of the electrons into longitudinal polarization (and conversely) after approximately 250 revolutions. If the magnetic field is chosen appropriately, then one can hope to measure the resulting asymmetry as a function of the number of revolutions, as shown in Fig. 15. The quantity \(1/N_0\) will give the deviation of \(g/2\) from unity. Thus, if the quantity \(\tau\) can be measured with an accuracy of up to \(1\%\), then the \(g\)-factor can be determined with an accuracy of up to \(10^{-5}\).
The fundamental theoretical investigation of various aspects of the double-scattering experiment in a magnetic field was carried out by Case and Mendlowitz \(^{23,100,101}\). In particular, they considered the depolarization that may arise for polarized electrons after many cyclotron revolutions. They came to the conclusion that a carefully set up experiment could allow determination of the \(g\)-factor with an accuracy of at least \(1\cdot 10^{-5}\).
Fig. 15. Expected asymmetry in the second double-scattering experiment proposed by Crane for measuring the anomalous magnetic moment of the electron. The electron makes many revolutions in a magnetic field, which changes its polarization from transverse to longitudinal after approximately 250 revolutions. Therefore the observed asymmetry should vary with a period of approximately 1000 revolutions; the period \(N_0\) is a direct indication of the change in the ratio \(g\) from the Dirac value.
B1. If the electrons in the double-scattering experiment could be captured by a magnetic field so as to make in it a sufficient number of cyclotron revolutions before the second scattering, then an experiment of a resonant character could become possible (see \(^{1,17}\)). To obtain an accuracy in determining the value of \(g\) of \(1\cdot 10^{-5}\), it appears necessary to make at least \(10^5\) cyclotron revolutions. In this case there is no need to use a pulsed electron beam, since no measurement of the flight time is required. However, in connection with the problem of capture, the use of some periodic field appears more attractive, since it is clear from the above that the resonance frequencies for the cyclotron motion \(\omega_c\) and for the spin flip \(\omega_s\) must be very close to one another. A relative measurement of both frequencies directly gives the value of \(g\). To carry out an accurate measurement of \(\omega_s\), however, it is necessary that this measurement not be distorted by the nearby strong resonance at \(\omega_c\). Since this requires very little perturbed cyclotron motion, the realization of such an experiment may be associated with great difficulties.
B2. Another resonance experiment with low-energy electrons was proposed by Bloch \(^{94*}\). As in the preceding proposal, the value of \(g\) must be determined by measuring the ratio \(\omega_c/\omega_s\). However, this method is based on
* In the following exposition we use material from a private communication by Dr. O. Frisch, who also considered this experiment.
use of the ground states of an electron in a homogeneous magnetic field. If a field of magnitude \(\mathfrak{B}\) is applied in the \(z\) direction, then the proper energies entering into the Hamiltonian are characterized by two quantum numbers \(l\) and \(m_s\) (we give the result in the nonrelativistic approximation):
\[ E_{l,m_s}=\frac{p_z^2}{2m}+(2l+1+gm_s)\mu_0\mathfrak{B}, \tag{7,1} \]
where \(l=0,1,2,\ldots\) are the quantum numbers corresponding to the orbital motion of the electron in the \(x,y\) directions; \(m_s=\pm \frac12\) is the quantum number for the spin angular momentum in the \(z\) direction; \(\mu_0\) is the Bohr magneton.
The wave functions are bounded in the \(x\) and \(y\) directions, but not in \(z\). The motion in the latter direction, however, can be restricted to a certain region by introducing an electric trapping potential (see below), so that \(p\) proves negligible. The solutions obtained for large \(l\) correspond to circular motion of electrons in a magnetic field. For small \(l\), for example for \(l=0,1\), smeared wave packets are obtained.
For \(l=0\) we have a wave packet rigidly bounded to a region of radius \(r_0=0.8\cdot 10^{-5}\ \mathrm{cm}\) at \(\mathfrak{B}=1000\) oersted. Neglecting in (7,1) the term \(p_z^2/2m\), we obtain a system of energy levels which, for \(g=2\), is expressed as:
\[ E_{l,m_s}= \begin{cases} 0 & \text{for } l=0,\quad m_s=-\dfrac12,\\[4pt] 2\mu_0\mathfrak{B} & \text{for } l=0,\quad m_s=+\dfrac12 \text{ and } l=1,\quad m_s=-\dfrac12,\\[4pt] 4\mu_0\mathfrak{B} & \text{for } l=1,\quad m_s=+\dfrac12 \text{ and } l=2,\quad m_s=-\dfrac12. \end{cases} \tag{7,2} \]
For \(\mathfrak{B}=1000\) oersted, \(\omega_c\approx \omega_s \simeq 1.8\cdot 10^{10}\ \mathrm{sec}^{-1}\), \(\hbar\omega=2\mu_0\mathfrak{B}=10^{-5}\ \mathrm{eV}\). The degeneracy is removed by an anomaly in the magnetic moment. If the magnetic field varies slowly along \(z\), then the energy begins to play the role of a magnetic potential energy, displacing the electron toward the value of \(z\) corresponding to the least possible energy
\[ V_m^{l,m_s}=[2l+1+gm_s]\mu_0\mathfrak{B}(z). \tag{7,3} \]
Suppose that an additional electric trapping potential is imposed,
\[ V_e=-e\varphi(z), \tag{7,4} \]
where \(\varphi(z)\) may be written approximately near the origin in the form
\[ \varphi(z)\simeq az^2-\frac12\alpha(x^2+y^2), \tag{7,5} \]
or, for a more extended region near the \(z\) axis, in the form of the expression
\[ \varphi(z)\simeq az^2-\beta z^4-\frac12\alpha(x^2+y^2)+3\beta z^2(x^2+y^2)+\frac38\beta(x^2+y^2)^2, \tag{7,6} \]
which describes a potential that recurs after some distance from the origin, so that \(V_e\) has a known depth
(Fig. 16). If both potentials \(V_m\) and \(V_e\) exist simultaneously, and if a considerable gradient \(\partial \mathfrak{B}/\partial z\) is established, then a portion of the electrons will be freed from the electric field by the gradient of the magnetic field (this may occur even when all states are absent except \(l=0,\ m_s=1/2\); Fig. 16).
Fig. 16. Electric trapping potential \(V_e\) and “ejecting” magnetic potentials \(V'_m\) and \(V''_m\) in Bloch’s proposed experiment for measuring the magnetic moment of the electron. The magnetic ejecting potential \(V'_m\), which is a consequence of the imposed gradient of the magnetic field (and depends on the state of motion of the electron), will eject all electrons from the electric “trap” \(V_e\); the magnetic potential \(V''_m\) must retain a portion of the electrons in the “trap.”
The experiment consists in repetitions of the following cycle (throughout the entire experiment the homogeneity and constancy of the field are maintained):
-
Trapping stage: the electric trapping potential (which is retained also in stages 2–4) and the gradient \(\partial \mathfrak{B}/\partial z\) are set so that electrons in states up to some \(l'\), \(m_s=+1/2\), and \(l''=l'+1\), \(m_s=-1/2\), are captured.
-
Removal of the gradient \(\partial \mathfrak{B}/\partial z\); the electrons still remain in the trapped state.
-
Application of a radio-frequency field, which will cause transitions \(l'\to l'+1\), \(l''\to l''+1\) at the resonance frequency \(\omega_c\), and transitions \(m_s=-1/2\to m_s=+1/2\) at the resonance frequency \(\omega_s\).
-
Secondary switching-on of the gradient \(\partial \mathfrak{B}/\partial z\); in the case of resonance at \(\omega_c\), the states \(l'+1,\ m_s=+1/2\) and \(l''+1,\ m_s=-1/2\) are “ejected” from the field; for resonance at \(\omega_s\), the states \(l'',\ m_s=+1/2\) are “ejected.” By observing with the aid of a photomultiplier the liberation of electrons from the field after their corresponding acceleration, one can determine the values of the resonance frequencies \(\omega_c\) and \(\omega_s\).
The realization of such an experiment is associated with many difficulties: the depth of the trapping potential must be very small (of the order of \(10^{-5}\) V); such potentials can be obtained along the axis of a cylinder (of radius, for example, \(5\ \text{cm}\)) consisting of insulated rings, each of which is at a potential of \(0.02\) V relative to the neighboring—
than this. In each cycle only a very small number of electrons can be captured (say, from 5 to 10), and one of the limiting factors here is the space charge. However, if the repetition of the capture and ejection cycles is carried out at a high rate, for example 180 or more times per second, then a number of ejected electrons sufficient for the observation of resonance can be produced. For this purpose it is also necessary that the states of motion of the captured electrons have appreciable lifetimes; therefore a sufficiently high vacuum must be created (\(10^{-7}\) mm Hg), so that collisions are rare and the exchange of energy with the medium can be taken into account only through radiation. However, despite all these difficulties, the experiment is nevertheless quite feasible.
B3. A resonance experiment, very similar to that described in B2, was proposed in 1947–1949 by Dike, who also made an attempt at its practical realization (private communication). In this experiment there are likewise stages of electron capture, application of a radio-frequency field, and ejection of electrons from the field. However, the details of the first stage (for which, as in Bloch’s experiment, carefully adjusted electric and magnetic fields are used) are somewhat different. In Dike’s experiment sharp cyclotron resonances were obtained, but the attempt to obtain spin resonances ended in failure. This latter fact can be understood only by taking into consideration the fine details of the experiment.
§ 8. POLARIZATION OF POSITRONS
Positrons can be polarized analogously to electrons. Coulomb scattering by heavy nuclei is a means of producing and detecting polarization for electrons and positrons alike. The effect for positrons was calculated by Mott (\(^{41}\), see also § 3) and was found to be considerably smaller for them than for electrons. This can be qualitatively explained by recalling that the most effective region of polarization effects is located near nuclei, into which positrons, owing to their positive charge, penetrate to a much lesser degree than electrons. The other methods of obtaining and detecting electron polarization described in §§ 5 and 6 cannot be applied to positrons, with the exception of the method of investigating \(\beta\)-rays from polarized radioactive nuclei (for example, the methods of polarization of \(\beta^{+}\)-emitters \(Mn^{52}\), \(Co^{56}\), and \(Co^{58}\) are sufficiently well known).
One may ask whether the process of their annihilation might make it possible to create methods for detecting polarized positrons. If polarized positrons form positronium atoms, then the \(^{3}S\) state of the latter must possess a preferential spatial orientation (the \(^{1}S\) state is spherically symmetric). The annihilation of polarized positrons stopping in matter with polarized atomic electrons may reveal effects depending on the relative orientation of the spins. However, the practical use of any of these possibilities for detecting positron polarization appears to be very difficult.
§ 9. CONCLUDING REMARKS
The question of the polarization of free electrons is connected with a very fundamental property of matter, which is formulated as the existence of spin in the electron. Although the theory of a number of important consequences of this question was developed as early as about 1930, qualitative agreement between theo-
theory and experiment in this field was achieved only by 1942. This was due in large measure to the fact that, at last, it proved possible to give a satisfactory explanation of the causes of the negative results that had pursued a number of earlier experiments. Generally speaking, experiments with polarized electrons are by no means easy, and many theoretical predictions have not been tested experimentally up to the present day. Despite its considerable intrinsic interest, this field of research has always lain somewhat aside from the main directions of experimental work, perhaps because physicists already possessed evidence confirming the quantum phenomenon of the rotating electron, obtained from other experimental data. We hope that in the next few years it will be possible successfully to carry out experiments to determine the \(g\)-factor of the free electron with high accuracy, which will connect this field of research with the latest developments of quantum electrodynamics.
APPENDIX
RELATIVISTIC FORM OF THE ELECTRON-SPIN PROJECTION OPERATOR
The spin projection operator \(P^{(+)}(\zeta)\) can be written in an explicitly relativistically covariant form with the aid of the Dirac matrices \(\gamma\). Instead of \(u_\lambda\), take a positive-energy solution \(w_\lambda\) with the same spin direction, but normalized to one particle in unit volume in the rest system. If we write
\[ \overline{P}^{(+)}_{\lambda\mu}(\zeta)=-w_\lambda \overline{w}_\mu \qquad (\overline{w}_\mu=w_\mu^*\beta_3), \tag{1} \]
then we shall have the relation
\[ \overline{P}^{(+)}(\zeta)=-P^{(+)}(\zeta)\beta_3\frac{E}{mc^2}; \tag{2} \]
\(\overline{P}^{(+)}(\zeta)\) can be written in the form
\[ \overline{P}^{(+)}(\zeta)=\frac{1}{4} \left[ 1-\frac{i}{mc}(p_\mu\gamma^\mu)-s_\mu\gamma^{(\mu)}-m_{\mu\nu}\gamma^{(\mu\nu)} \right], \tag{3} \]
where
\[ p_\mu=\left[\mathbf{p},\,\frac{i}{c}E\right], \tag{4} \]
\[ \left. \begin{aligned} s_\mu&=\left(\mathbf{s}',\,\frac{i}{mc}(\mathbf{p}\cdot\boldsymbol{\zeta})\right),\\ \mathbf{s}'&=\boldsymbol{\zeta}+\frac{1}{m(E+mc^2)}(\mathbf{p}\cdot\boldsymbol{\zeta})\mathbf{p}, \end{aligned} \right\} \tag{5} \]
\[ \gamma^{(\mu)}= \left( i\gamma^2\gamma^3\gamma^4,\, i\gamma^3\gamma^1\gamma^4,\, i\gamma^1\gamma^2\gamma^4,\, -i\gamma^1\gamma^2\gamma^3 \right) \tag{6} \]
and where \(p_\mu\) is a 4-vector, and \(s_\mu\) is a 4-(pseudo)vector. The expressions for \(\overline{\psi}\gamma^{(\mu)}\psi\) and for \(\overline{\psi}\gamma^{(\mu\nu)}\psi\) are respectively a 4-(pseudo)vector and a 4-tensor. The product of two antisymmetric 4-tensors \(T_{\mu\nu}\) and \(S_{\mu\nu}\) can, with the aid of two pairs of 3-vectors \((\mathbf{F},\mathbf{G})\) and \((\mathbf{F}',\mathbf{G}')\), be written in the form
\[ T_{\mu\nu}S_{\mu\nu}=\mathbf{F}\cdot\mathbf{F}'-\mathbf{G}\cdot\mathbf{G}'. \tag{7} \]
For \(m_\mu\), these vectors are
\[ \begin{aligned} \mathbf F=\mathbf m'&=\frac{E}{mc^2}\,\boldsymbol\zeta-\frac{1}{m(E+mc^2)}(\mathbf p\cdot\boldsymbol\zeta)\mathbf p,\\ \mathbf G&=-\frac{1}{mc}(\mathbf p\times\boldsymbol\zeta). \end{aligned} \tag{8} \]
The corresponding terms in \(\gamma^{(\mu\nu)}\) are given by the expressions
\[ \begin{aligned} \mathbf F'&=(i\gamma^2\gamma^3,\quad i\gamma^3\gamma^1,\quad i\gamma^1\gamma^2),\\ \mathbf G'&=(-\gamma^1\gamma^4,\quad -\gamma^2\gamma^4,\quad -\gamma^3\gamma^4). \end{aligned} \tag{9} \]
The vectors \(\mathbf s'\) and \(\mathbf m'\) determine, respectively, the directions of the spin angular momentum and the magnetic moment of an electron with momentum \(\mathbf p\) (for \(\mathbf p=0\) we have \(\boldsymbol\zeta=\mathbf s=\mathbf m\)), for which \(\boldsymbol\zeta\) gives the direction of the spin in the coordinate system in which the electron is at rest. If the wave function is transformed according to
\[ \psi'=S\psi \tag{10} \]
under a Lorentz transformation, then the corresponding transformation for \(\overline P^{(+)}(\boldsymbol\zeta)\) will be given by the expression
\[ \overline P^{(+)\prime}(\boldsymbol\zeta)=S\overline P^{(+)}(\boldsymbol\zeta)S^{-1}. \tag{11} \]
\(\overline P^{(+)}(\boldsymbol\zeta)\) may be regarded as the spin projection operator \(\boldsymbol\zeta\) for positive values of energy and spin; it may be written as the product
\[ \overline P^{(+)}(\boldsymbol\zeta)=\overline P^{(+)}\overline P(\boldsymbol\zeta)=\overline P(\boldsymbol\zeta)\overline P^{(+)}, \tag{12} \]
where \(\overline P^{(+)}\) is the projection operator for positive energies, and \(\overline P(\boldsymbol\zeta)\) is the projection operator onto the spin direction \(\boldsymbol\zeta\). These operators can be expressed in the following form:
\[ \begin{aligned} \overline P^{(+)}&=\frac12\left[1-\frac{i}{mc}\,p_\mu\gamma^\mu\right],\\ \overline P(\boldsymbol\zeta)&=\frac12\left[1-s_\mu\gamma^{(\mu)}\right]. \end{aligned} \tag{13} \]
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