SPIN PRECESSION OF AN ELECTRON MOVING IN CROSSED FIELDS

V. G. Bagrov, V. A. Bordovitsyn

(Presented by Prof. V. A. Sokolov)

The paper studies the influence of the anomalous magnetic moment of the electron on the behavior of its spin during motion in constant and homogeneous orthogonal electric and magnetic fields.

Motion of an Electron in Crossed Fields

The motion of a relativistic electron in an electromagnetic field is described by the Dirac equation

\[ i\hbar \frac{\partial \Psi}{\partial t}=H\Psi,\quad H=c(\vec{\alpha}\vec{P})+\rho_3mc^2-e\varphi, \tag{1} \]

where

\[ \vec{P}=-i\hbar\nabla+\frac{e}{c}\vec{A}; \]

\(\vec{\alpha}, \rho\) are Dirac matrices, and \(\vec{A}, \varphi\) are the potentials of the external field.

Let us consider the motion of an electron in constant and homogeneous orthogonal electric and magnetic fields under the condition \(E<H\). We choose the coordinate system so that the \(z\)-axis is directed along the magnetic field and the \(y\)-axis along the electric field, and we shall take

\[ E=H\sin\eta\left(-\frac{\pi}{2}<\eta<\frac{\pi}{2}\right). \]

In this case the field potentials may be chosen in the form

\[ A_x=-yH,\quad A_y=A_z=0,\quad \varphi=-yH\sin\eta. \]

With this choice of potentials, the integrals of motion are the total energy of the system \(H\), the projection of the momentum on the magnetic field \(P_3\), and the operator

\[ p_1=-i\hbar\frac{\partial}{\partial x}. \]

Consequently, the wave function, in addition to (1), may be made to satisfy the equations

\[ H\Psi=c\hbar K\Psi,\quad P_3\Psi=\hbar K_3\Psi,\quad p_1\Psi=\hbar k_1\Psi. \tag{2} \]

Taking these requirements into account, the wave function has the form

\[ \Psi(t)=\frac{1}{\sqrt{L_xL_z}}e^{-icKt+ik_1x+iK_3z}\psi(\tau) \]

\[ \tau=\sqrt{\gamma\cos\eta}\left(y+\frac{K\sin\eta-k_1}{\gamma\cos^2\eta}\right),\quad \gamma=\frac{eH}{\hbar c}. \tag{3} \]

The spinor \(\psi(\tau)\) can be written in the form

\[ \psi_i(\tau)=\alpha_i U_2(\tau)+\beta_i U_{n-1}(\tau),\quad i=1,2,3,4,\quad n=0,1,\ldots . \tag{4} \]

Here \(U_n(\tau)\) are Hermite functions, related to the Hermite polynomials \(H_n(\tau)\) by the relation

\[ U_n(\tau)=\frac{1}{\sqrt{2^n n!}}\sqrt[4]{\frac{\gamma\cos\eta}{\pi}}\,e^{-\frac{\tau^2}{2}}H_n(\tau). \]

The coefficients \(\alpha_i\) and \(\beta_i\) are connected with two arbitrary constants \(A\) and \(B\) by the relations:

\[ \begin{aligned} \alpha_1&=-\frac{\sqrt{2\gamma n\cos\eta}\,A+k_3B}{\sqrt{2d(d+k_0)}}\sin\frac{\eta}{2}, & \beta_1&=A\sqrt{\frac{1}{2}\left(1+\frac{k_0}{d}\right)}\cos\frac{\eta}{2}, \\[6pt] \alpha_2&=B\sqrt{\frac{1}{2}\left(1+\frac{k_0}{d}\right)}\cos\frac{\eta}{2}, & \beta_2&=\frac{-\sqrt{2\gamma n\cos\eta}\,B+k_3A}{\sqrt{2d(d+k_0)}}\sin\frac{\eta}{2}, \\[6pt] \alpha_3&=B\sqrt{\frac{1}{2}\left(1+\frac{k_0}{d}\right)}\sin\frac{\eta}{2}, & \beta_3&=\frac{-\sqrt{2\gamma n\cos\eta}\,B+k_3A}{\sqrt{2d(d+k_0)}}\cos\frac{\eta}{2}, \\[6pt] \alpha_4&=-\frac{\sqrt{2\gamma n\cos\eta}\,A+k_3B}{\sqrt{2d(d+k_0)}}\cos\frac{\eta}{2}, & \beta_4&=A\sqrt{\frac{1}{2}\left(1+\frac{k_0}{d}\right)}\sin\frac{\eta}{2}. \end{aligned} \tag{5} \]

Here the notation introduced is

\[ d=\frac{K-k_1\sin\eta}{\cos\eta},\qquad K=k_1\sin\eta+\cos\eta\sqrt{k_0^2+k_3^2+2\gamma n\cos\eta}, \]

\[ k_0=\frac{mc}{\hbar}. \tag{6} \]

The function (3) is normalized to unity if \(A\) and \(B\) satisfy the condition

\[ AA^{+}+BB^{+}=1. \tag{7} \]

In addition to the three integrals of motion (2), there must exist a fourth, spin integral of motion. As established in [1], as such an integral of motion one may choose either the operator of transverse polarization of the electron spin

\[ L(\psi)=\Pi_2\sin\eta\sin\psi+\Pi_3\cos\psi+E_2\sin\eta\cos\psi-E_2\sin\psi, \tag{8} \]

or the operator of longitudinal polarization

\[ T(\psi)=(\vec{\sigma}\vec{P})\sin\psi+T_3\cos\psi-T_1\sin\eta\sin\psi. \tag{9} \]

In both formulas \(\psi\) is an arbitrary angle, and the matrix vectors introduced are

\[ \vec{\Pi}=mc\vec{\sigma}+\rho_2[\vec{\sigma}\vec{P}],\qquad \vec{E}=-\rho_2[\vec{\sigma}\vec{P}],\qquad \vec{T}=mc\rho_3\vec{\sigma}+\rho_1\vec{P}. \tag{10} \]

If, in addition to (2), one requires that \(\Psi\) satisfy the equation

\[ L(\psi)\Psi=\zeta\hbar\lambda_1\cos\eta\,\Psi,\qquad \zeta=\pm 1, \]

then, taking (7) into account, we obtain

\[ A=\zeta\sqrt{\frac{1}{2}(1+\zeta q_1)}\,e^{-i\Phi/2}, \qquad q_1=\frac{d(d+k_0)-k_3^2}{\lambda_1(d+k_0)}\cos\psi \tag{11} \]

\[ B=\sqrt{\frac{1}{2}(1-\zeta q_1)}\,e^{i\Phi/2}, \qquad \tg\Phi=\frac{d+k_0}{k_3}\tg\psi \]

\[ \lambda_1\cos\eta=\sqrt{(K-k_1\sin\eta)^2-\cos^2\eta\,(k_3^2+k_0^2\sin^2\psi)}. \]

For longitudinal polarization, subjecting the function \(\Psi\) to the equation

\[ T(\psi)\Psi=\zeta\hbar\lambda_2\Psi,\qquad \zeta=\pm1, \]

we obtain

\[ A=-\zeta\delta\sqrt{\frac{1}{2}(1+\zeta q_2)},\qquad \delta=\frac{(d+k_0)\cos\eta\sin\psi+k_3\cos\psi} {\left|(d+k_0)\cos\eta\sin\psi+k_3\cos\psi\right|}, \]

\[ B=\sqrt{\frac{1}{2}(1-\zeta q_2)},\qquad q_2=\frac{(d+k_0)(k_0\cos\psi+k_3\cos\eta\sin\psi)+k_3^2\cos\psi} {\lambda_2(d+k_0)}, \]

\[ \lambda_2=\sqrt{k_0^2(\cos^2\psi-\cos^2\eta\sin^2\psi)+\left[(K-k_1\sin\eta)\sin\psi+k_3\cos\psi\right]^2}. \]

The anomalous magnetic moment of the electron and spin precession

The presence of an anomalous magnetic moment in the electron leads to the fact that the set of spin integrals of motion is sharply reduced (see [1]). Indeed, as Pauli showed [2], the anomalous magnetic moment of the electron can be taken into account by modifying Hamiltonian (1)

\[ H=c(\vec{\alpha}\vec{P})+\rho_3mc^2-e\varphi-\hbar c\mu\left[\rho_3(\vec{\sigma}\vec{H})+\rho_2'(\vec{\sigma}\vec{E})\right], \tag{12} \]

where \(\hbar c\mu\) is the anomalous part of the electron’s moment. The operators (2) remain integrals of motion in this case as well. However, the operator \(T(\psi)\) (see (9)) does not commute with (12) for any value of \(\psi\). The operator (9) of transverse polarization \(L(\psi)\) commutes with (12) only for \(\psi=0\), if the vector \(\vec{\Pi}\) is defined, in contrast to (10), as follows:

\[ \vec{\Pi}=mc\vec{\sigma}+\rho_2[\vec{\sigma}\vec{P}]+\hbar\mu(\vec{H}-[\vec{\sigma}\vec{E}]). \]

Formulas (3–6) retain their form if, for the energy \(K\), one uses the expression (\(\zeta'=\pm1\) is the eigenvalue of \(L(0)\))

\[ K_{\zeta'}=k_1\sin\eta+\cos\eta \sqrt{k_3^2+\left(\sqrt{k_0^2+2\gamma h\cos\eta}+\zeta'\mu H\cos\eta\right)^2}. \]

The quantities \(A\) and \(B\) are determined by formulas (11) for \(\psi=0\). It is not obvious in advance how, with the passage of time, the spin described by the operators (8) and (9), which is conserved when the anomalous moment is not taken into account, will change.

To solve this question we proceed as follows. From the wave functions \(\Psi_{\zeta'}(t)\), which are eigenfunctions of the operator \(L(0)\) with the anomalous magnetic moment of the electron taken into account, we construct the nonstationary (\(K_{\zeta'}\ne K_{-\zeta'}\)) wave function

\[ \Psi(t)=C(\zeta,\zeta')\Psi_{\zeta'}(t)+C(\zeta,-\zeta')\Psi_{-\zeta'}(t) \]

so that at \(t=0\) it is an eigenfunction of the operator \(L(\psi)\) (or \(T(\psi)\)),

\[ L(\psi)\Psi(0)=\zeta\hbar\lambda_1\cos\eta\,\Psi(0). \tag{13} \]

If we require that \(\Psi(t)\) be normalized to unity, then from (13) the coefficients \(C(\zeta,\zeta')\) are determined uniquely. Having thus determined \(\Psi(t)\), we find the mean value \(\overline{L(\psi)}\) (or \(\overline{T(\psi)}\)) over these functions. This mean value will, in the general case, depend on time. Consequently, the problem of spin precession under the influ-

…of the anomalous moment of the electron will be solved. Straightforward, but cumbersome, calculations lead for the operator \(L(\psi)\) to the following expression:

\[ C(\zeta,\zeta')=\frac{1}{2}\left\{\zeta\zeta'\sqrt{(1+\zeta q_1)(1+\zeta' q')}e^{-i\frac{\Phi}{2}}+ \right. \]

\[ \left. +\sqrt{(1-\zeta q_1)(1-\zeta' q')}e^{i\frac{\Phi}{2}}\right\},\qquad q'=q_1(\psi=0), \]

and for longitudinal polarization (the operator \(T(\psi)\)) to the coefficients

\[ C(\zeta,\zeta')=\frac{1}{2}\left\{\zeta\zeta'\sqrt{(1+\zeta q_2)(1+\zeta' q')}+\sqrt{(1-\zeta q_2)(1-\zeta' q')}\right\},\qquad q'=q_2(\psi=0). \]

The mean value \(\overline{L(\psi)}\) with respect to the functions \(\Psi(t)\) has the form

\[ \overline{L(\psi)}=\xi\hbar\lambda_1\cos\eta\left(\cos^2 s+\sin^2 s\cos\Omega t\right),\qquad \Omega=c(K_1-K_{-1}), \tag{14} \]

where one must set

\[ \cos s=q_1q'-\sqrt{(1-q_1^2)(1-q'^2)}\cos\Phi. \]

For the quantity \(\overline{T(\psi)}\), formula (14) retains its form if in it one replaces \(\lambda_1\cos\eta\to\lambda_2\) and sets

\[ \cos s=q_2q'+\sqrt{(1-q_2^2)(1-q'^2)}. \tag{15} \]

From formulas (14) and (15), in the special case \(\psi=\frac{\pi}{2}\), \(\eta=0\), the result of works [3, 4] follows.

To accuracy up to terms linear in \(\mu\), we have

\[ \Omega=2c\mu H\cos^2\eta\sqrt{1-\beta_3^2} =\omega_0\frac{H}{H_0}\cos^2\eta\sqrt{1-\beta_3^2},\qquad \beta_3=\frac{k_3\cos\eta}{K-k_1\sin\eta}, \]

\[ \omega_0=2ck_0=2.472\cdot10^{20}\ \text{Hz},\qquad H_0=\frac{k_0}{\mu}=7.707\cdot10^{16}\ \text{Oe}. \]

Thus, for fields \(H\sim10^4\) Oe the frequency \(\Omega\sim10^8\) Hz, which may even exceed the orbital frequency of electrons in accelerators. Consequently, despite the small magnitude of the anomalous magnetic moment of the electron, its influence on the behavior of the spin during the motion of an electron in crossed fields may be quite substantial.

REFERENCES

1. I. M. Ternov, V. G. Bagrov, V. A. Bordovitsyn. Izvestiya vuzov, “Fizika,” 4, 41, 1967.
2. W. Pauli. Rev. of Mod. Phys. 13, 203, 1941.
3. I. M. Ternov, V. S. Tumanov. Izvestiya vuzov, “Fizika,” 1, 155, 1960.
4. I. M. Ternov, V. G. Bagrov, R. A. Rzaev, Yu. I. Klimenko. Izvestiya vuzov, “Fizika,” 6, 111, 1964.