POLARIZATION OF FAST PROTONS AND NEUTRONS

I. I. Levintov

POLARIZATION IN SCATTERING

The noncentral character of nuclear forces leads, as is well known, to a strong dependence of the interaction on the total angular momentum of the nucleon system. In particular, energy levels differing by different orientations of the spin relative to the orbital angular momentum (by the value of \((l, s)\)) are separated from one another by several MeV (spin–orbit interaction). Thus, the dependence of nuclear forces on the orientation of the spin relative to the orbit in a system of nucleons is close, in order of magnitude, to the dependence of the forces on the orientation of the spins (cf. the singlet and triplet states of the deuteron). One of the consequences of this in the theory of nuclear structure is the peculiar order of filling of nuclear shells1.

The spin–orbit interaction also manifests itself in processes of elastic scattering and nuclear reactions and, in the case of light elements, leads to the appearance of resonances associated with different orientations of \(l\) and \(s\) in the intermediate state. Important experimental consequences of the manifestation of \(l\)- and \(s\)-interaction in nuclear reactions are features in the character of the angular distributions and the polarization of the reaction products. In the case of particles with spin \(1/2\), polarization \(\mathbf{P}\) is understood to mean a vector equal to the mean value of the particle spin,

\[ \mathbf{P}=\frac{\psi^{*}\boldsymbol{\sigma}\psi}{\psi^{*}\psi}, \]

where \(\boldsymbol{\sigma}\) is the Pauli spin operator. Thus, if the direction of the predominant orientation of the spins coincides with the \(Z\) axis, then the magnitude of the polarization is

\[ |\mathbf{P}|=\frac{n^\uparrow-n^\downarrow}{n^\uparrow+n^\downarrow}, \]

where \(n^\uparrow\), \(n^\downarrow\) are the fractions of particles with spins oriented along and against \(Z\), respectively.

I. I. LEVINTOV

The possibility of polarization of fast nucleons arising in the scattering of unpolarized beams by unpolarized targets was pointed out by Schwinger\(^2\). The basic results of the quantum-mechanical theory of polarization of fast nucleons were obtained in the works of Wolfenstein\(^3{}^4\), Lepore\(^5\), and Dalitz\(^6\). In the present review the questions of polarization will be considered in the simplest example of the scattering of particles with spin \(1/2\) by particles with spin 0.

Elastic scattering and the polarization of nuclear particles with spin \(1/2\) in the presence of spin–orbit coupling admit a clear qualitative interpretation, correctly reflecting the main features of these processes.

These visual representations are no more than another way of writing complicated formulas, but they are useful, since they reflect the essence of the matter, which is obscured in a formal exposition.

Let us first consider spinless scattering, i.e., the case when the interaction of the particles does not depend on the orientation of their spin and orbital angular momenta. In the case of nuclear forces, for which spin interactions are very large, this is in fact true only for particles with zero spin.

1. Elastic scattering of uncharged particles with spin 0

Using the results of the exact theory (see, for example, Mott and Massey\(^7\)), we represent the wave function of the incident particles \(\psi_{\text{inc}}\) in the form of an infinite plane wave moving along the \(Z\) axis (the \(Z\) axis coincides with the direction of the incident beam):

\[ \psi_{\text{inc}} = e^{ikz}, \tag{1} \]

where

\[ k=\frac{2\pi}{\lambda}=\frac{p}{\hbar}=\frac{\sqrt{2\mu E}}{\hbar}, \]

and \(\mu\) and \(E\) are the reduced mass and the energy in the center-of-gravity system (\(\mu\)-system):

\[ E=\frac{M_1}{M_1+M_2}E_{\text{lab}},\qquad \mu=\frac{M_1M_2}{M_1+M_2}. \]

The wave function of the scattered particles, which is the result of solving the Schrödinger equation for a central force field, is represented at large distances from the center as a diverging wave of the form

\[ \psi_{\text{sc}}=\frac{1}{r}e^{ikr}f(\theta) =\frac{1}{r}e^{ikr}\sum_{0}^{l_{\max}}(2l+1)e^{i\delta_l}\sin\delta_l P_l^{0}(\cos\theta), \tag{2} \]

where \(f(\theta)\) is normalized in such a way that the scattering cross section

at angle \(\theta\): \(\sigma(\theta)=\psi_{\mathrm{scatt}}^{*}\psi_{\mathrm{scatt}}=|f(\theta)|^{2}\) and the total cross section

\[ \sigma_{\mathrm{tot}}=\int_{0}^{\pi}|f(\theta)|^{2}\cdot \sin\theta\cdot d\theta =\frac{4\pi}{k^{2}}\sum_{0}^{l_{\max}}(2l+1)\cdot \sin^{2}\delta_{l}. \tag{3} \]

The expressions written above are usually interpreted in the following way. Each individual particle, represented by an infinite plane wave \(e^{ikz}\), can be scattered while simultaneously having a set of different angular momenta \((l)\) relative to the \(z\)-system. The scattered \(l\)-waves have a phase shift \((\delta_l)\) relative to the unperturbed incident wave and an amplitude \(\sim \sin\delta_l\), which is determined by the interference of the incident and scattered \(l\)-wave. In the case of nuclear (short-range) forces, at a given particle energy only a limited number of \(l\)-waves participate in the scattering, i.e. only a few phases \(\delta_l\) differ from zero. This is due to the fact that the angular momentum of the particle is \(\hbar l \simeq |[\mathbf r,\mathbf p]|=r_0p\), where \(r_0\) is the impact parameter and \(p\) is the momentum. Thus, scattering with angular momentum \(l\) may be regarded as occurring from a scattering ring situated in a plane normal to the axis of the incident beam and having diameter \(2r_0\simeq 2\frac{\hbar l}{p}\simeq \frac{\lambda}{2}l\). If a particle with momentum \(p\) has so large an angular momentum that \(r_0>R\), where \(R\) is the range of action of the nuclear forces, then there is no interaction and the phase shift is \(\delta_l=0\).

Since \(R\simeq r_{\mathrm n}A^{1/3}\), where \(r_{\mathrm n}\) is the “radius” of a nucleon, equal to \(1.5\cdot10^{-13}\ \mathrm{cm}\), and \(A\) is the atomic number, then, for given energy and mass of the incident particle, the maximum angular momentum \(l_{\max}\), for which \(\delta_l\ne0\), in general increases with the mass of the nucleus, since a particle with a large impact parameter enters the region of action of the nuclear forces owing to the increase of the nuclear radius. The maximum value \(l_{\max}\), for which \(\delta_l\ne0\), may be roughly estimated by putting:

\[ l_{\max}\simeq \frac{R}{\lambda} =1.4\,\frac{r_{\mathrm n}m_{\mathrm n}^{1/2}}{\hbar}\, A^{2/3}\left(\frac{A_0}{A+A_0}\right)^{1/2}E^{1/2}, \]

where \(m_{\mathrm n}\) is the mass of a nucleon, \(A\) and \(A_0\) are the atomic numbers of the target nucleus and the incident nucleus, and \(E\) is the energy of the incident particle.

The phase shift depends on the energy of the incident particle and is determined by the interaction potential \(U(r)\):

\[ \sin\delta_l\simeq-\int_{0}^{\infty}\psi_{\mathrm{scatt}}^{*}\,U(r)\,\psi_{\mathrm{inc}}\,dr, \tag{4} \]

so that attractive forces \((U(r)<0)\) correspond to \(\delta>0\), and repulsive forces to \(\delta<0\). Attractive forces lead, at

at certain energies of the interacting particles, to resonance effects. In the presence of a resonance

\[ \tg \delta_l = - \frac{\Gamma_l}{E_{\mathrm{res}} - E}, \tag{5} \]

where \(\Gamma_l\) and \(E_{\mathrm{res}}\) are the width and energy of the resonance. Consequently, at the energy corresponding to the resonance, \(\delta_{\mathrm{res}} = \frac{\pi}{2}\).

Thus, phase shifts completely characterize the interaction, since their magnitudes, on the one hand, determine the dependence of the cross sections and angular distributions of the process on energy and, on the other hand, are uniquely connected with the character of the nuclear forces. Therefore, the ultimate task of any scattering experiment is (at least in principle) the determination of the set of phase shifts and of their dependence on energy.

The angular distribution of the amplitudes of waves scattered with angular momentum \(l\) is determined by spherical functions of the type \(P_l(\cos \theta)\) (Legendre polynomials).

For understanding polarization it is important to interpret graphically the angular dependence \(P_l(\cos \theta)\).

The fact that, in the expansion of the scattered wave in eigenfunctions of the angular momentum, only polynomials of the type \(P_l(\cos \theta)\) enter is connected with the axial symmetry of scattering and means that the scattered waves correspond to particles with an angular-momentum vector whose projection on the axis of incidence \(Z\) is equal to 0.

In the case of scattering of classical particles by central forces, the equality to zero of the projections of the angular momenta on the axis of the incident beam is obvious. Figure 1 shows classical particles with impact parameter \(r_0\), scattered in the \(XY\) plane. Their angular momenta, equal to the vector product \([\mathbf r,\mathbf p]\), are normal to the scattering plane and, consequently, have zero projection on the axis of the incident beam. In addition, we note that the projection of the angular momentum of the right-hand particle on the axis normal to the scattering plane is \(+r_0p\), while the projection of the angular momentum of the left-hand particle on the same axis is \(-r_0p\).

Fig. 1

This character of classical scattering suggests considering also the wave-mechanical scattering in a definite plane (for example \(X'\), \(Y'\)), where \(X'\) coincides with the axis of the incident

beam. In such a treatment the scattered wave corresponding to angular momentum \(l\), having zero projection on the axis of the incident beam (this wave has the angular dependence of the amplitude \(P_l(\cos\theta)\), see (2)), is represented as a superposition of waves having definite projections of the angular momentum on the axis \(Z'\), normal to the scattering plane. This expansion is equivalent to a rotation of the former coordinate system, in which the axis of the incident beam coincided with the axis \(Z\), about the axis \(Y\) through an angle \(\frac{\pi}{2}\), and it can easily be carried out using, for example, the addition theorem for Legendre polynomials.

As a result one obtains:

\[ (\psi_l)_{\mathrm{scatt}}(\theta) = \frac{1}{r}(2l+1)\sin\delta_l e^{i(kr+\delta_l)} P_l^0(\cos\theta) = \frac{1}{r}\sin\delta_l \sum_{m=-l}^{m=+l} a_l^m e^{i(kr+\delta_l+m\theta)} . \tag{6} \]

In equality (6) the left-hand side corresponds to a wave having zero projection of the angular momentum on the axis of the incident beam, while the right-hand side gives the required representation of this wave as a group of \((2l+1)\) waves belonging to definite values of the projection of the angular momentum on the axis normal to the scattering plane (the axis \(Z'\)). The projections of the angular momentum on the axis \(Z'\) take the values

\[ mh = lh,\ (l-1)h,\ldots,\ 0,\ldots,\ -(l-1)h,\ -lh. \]

Waves with projections of angular momentum \(>0\) correspond, in the classical picture, to the passage of particles to the right of the scattering center, and conversely.

The expansion coefficients \(a_l^m=[P_l^m(\cos 90^\circ)]^2\) represent the probability amplitudes of the various values of the projection of the angular momentum on the axis \(Z'\) (normal to the scattering plane) for a particle scattered in the plane \(X'\), \(Y'\) with angular momentum \(l\). The values of these coefficients for \(S\)-, \(P\)-, and \(D\)-waves are as follows:

\[ \begin{aligned} &S\text{-wave }(l=0)\qquad &&m=0, &&a_0^0=1,\\[4pt] &P\text{-wave }(l=1)\left\{ \begin{aligned} &m=+1, &&a_1^{+1}=1,\\ &m=0, &&a_1^{0}=0,\\ &m=-1, &&a_1^{-1}=1, \end{aligned}\right.\\[6pt] &D\text{-wave }(l=2)\left\{ \begin{aligned} &m=+2, &&a_2^{+2}=9,\\ &m=+1, &&a_2^{+1}=0,\\ &m=0, &&a_2^{0}=1/4,\\ &m=-1, &&a_2^{-1}=0,\\ &m=-2, &&a_2^{-2}=9. \end{aligned}\right. \end{aligned} \]

There exists an exact geometrical interpretation of the scattering process represented in the form of expansion (6). However, this interpretation is not simple and therefore poorly serves the main purpose of this work—to give a visual representation of the wave-mechanical essence of the processes of scattering and polarization. We shall therefore restrict ourselves to presenting a simplified, but very clear, qualitative picture that correctly reflects all the features of the processes.

We shall do this first for \(P\)-scattering. Since \(P_1(\cos\theta)=\cos\theta\), the scattered \(P\)-wave \((r\to\infty)\) is

\[ \psi_{P\,\mathrm{scatt}} = \frac{1}{r}\sin\delta_1\cdot e^{i(kr+\delta_1)}\cos\theta = \]

\[ = \frac{1}{2}\cdot\frac{1}{r}\sin\delta_1 \left\{ e^{i(kr+\delta_1+\theta)} + e^{i(kr+\delta_1-\theta)} \right\}. \tag{7} \]

Previously (the left-hand side of (7)) the scattered \(P\)-wave was described by a single radial wave \(\dfrac{1}{r}e^{i(kr+\delta)}\) with an amplitude depending on \(\theta\) as \(\cos\theta\). Now (the right-hand side of (7)) the same \(P\)-scattering is represented as a superposition of two interfering waves with amplitudes \(a_1^{\pm 1}=1\), independent of the scattering angle \(\theta\). But, in compensation, the path difference of these waves, i.e. the difference of the exponents in expression (7), depends on \(\theta\). When \(\theta=0\) (forward scattering), the path difference is zero and the waves reinforce one another, the amplitude being positive \((\cos 0=+1)\). When \(\theta=\dfrac{\pi}{2}\), the path difference is equal to \(\pi\) and the waves cancel one another; when \(\theta=\pi\) (backward scattering), the path difference is \(2\pi\) and the waves again reinforce one another, but the amplitude becomes negative \((\cos\pi=-1)\).

On the other hand, one of these waves, \(e^{i(kr+\delta_1+\theta)}\) (which we shall henceforth denote by \(\psi_1^{+1}\)), corresponds to an angular momentum whose projection on the \(Z'\) axis is \(+1h\), while the other, \(\psi_1^{-1}\sim e^{i(kr+\delta_1-\theta)}\), has angular momentum \(-1h\). Finally, since \(P\)-scattering is under consideration, the waves must depart from a scattering ring of diameter \(\sim \dfrac{\lambda}{2}\).

Figure 2a shows a scattering ring of diameter \(\dfrac{\lambda}{2}\). The waves \(\psi_{1\,\mathrm{inc}}^{+1}\) and \(\psi_{1\,\mathrm{inc}}^{-1}\), representing the incident particle, approach the scattering ring from left to right along the \(X'\) axis toward the points \(a\) and \(b\). The scattered waves \(\psi_{1\,\mathrm{scatt}}^{-1}\), \(\psi_{1\,\mathrm{scatt}}^{+1}\) leave the points \(a\) and \(b\) in the \(Z'Y'\) plane. The arrows correspond to the mean values of the angular-momentum vectors associated with the waves \(\psi_1^{\pm1}\). Their projections on the axis of the incident beam are zero, while on the \(Z'\) axis they are equal to \(\pm 1h\). The phase shift \(\delta_1\) in scattering is shown as a bending of the scattered \(\psi_1^{\pm1}\) waves in comparison with the corresponding incident wav-

by us at the points \((a, b)\)*) . In Fig. 2a the amplitude of the scattered waves must be \(\sim \sin\delta_1\) (in the drawing this condition is not satisfied).

Fig. 2a.

It should be emphasized that the incident \(\psi_1^{+1}\) and \(\psi_1^{-1}\) waves approach the ring with the same phase. This circumstance is connected with the fact that both waves belong to one and the same particle and, consequently, are coherent.

To obtain the resulting amplitude of the wave scattered through the angle \(\theta\), one must rotate both waves \(\psi_1^{\pm 1}\) in the plane \(X', Y'\) about the arrows of the moments through the required angle. As a result there arises a path difference, which leads to mutual cancellation for scattering at \(\theta = 90^\circ\) (Fig. 2b) and to reinforcement for \(\theta = 0, \pi\).

Fig. 2b.

In \(P\)-scattering at \(\theta > \dfrac{\pi}{2}\), the amplitude of the scattered waves changes sign, while remaining equal in absolute value to the amplitude of the waves scattered forward through the angle \(\pi-\theta\). This circumstance follows from the oddness of the \(P\)-function (change of sign upon reflection at the origin of coordinates) and does not appear upon rotation of \(P\)-waves through \(\theta > \dfrac{\pi}{2}\) in the present simplified graphical representation. Therefore, for a correct repre—

*) In fact the phase shift occurs gradually (continuity of \(\psi, \dot{\psi}\)) and is measured as the phase difference of the incident and scattered waves at infinity.

of the construction one should draw the back-scattered waves, in the case of \(P\)-scattering, with the opposite phase.

Arguments analogous to the preceding ones lead directly to representing \(S\)-scattering in the form of a single wave emerging from the scattering center (Fig. 3). In contrast to \(P\)-waves, the \(S\)-wave is even, and therefore its phase is not reversed upon scattering at \(\theta > \dfrac{\pi}{2}\). In Fig. 3 it is clear that the amplitude of the \(S\)-wave does not depend on \(\theta\).

Fig. 3.

Waves corresponding to higher moments should be represented as issuing from rings of diameter \(\dfrac{l\lambda}{2}\). Accordingly, when observed at different \(\theta\), the amplitude of the resulting scattered wave with moment \(l\) will pass through zero \(l\) times (zeros of Legendre polynomials). Waves corresponding to odd

Fig. 4.

\(l\) have, in particular, a zero at \(\theta = \dfrac{\pi}{2}\), whereas waves corresponding to even \(l\) have a maximum at \(\theta = \dfrac{\pi}{2}\). For example, \(D\)-scattering is shown in Fig. 4. In the present case, in addition to waves with projections of the moment on the \(Z'\) axis equal to \(\pm 2\hbar\), there also appears ...

wave with zero projection on this axis. Since this wave also has angular momentum equal to \(2h\), the zero projection on the \(X'\) and \(Y'\) axes means that it can be represented as a superposition of two waves with projections \(\pm 2h\) on the \(Y'\) axis, as is also shown in Fig. 4. The phase difference of these waves does not change when \(\theta\) is varied in the plane \(X',\,Y'\).

Since \(D\)-waves are even, the scattering is symmetric with respect to \(\theta=\dfrac{\pi}{2}\), and consequently, in the present case, as in the case of \(S\)-waves, it is not necessary to change the sign of the phase upon passing through \(\theta=\dfrac{\pi}{2}\).

If \(l\) angular momenta are involved in the scattering, then the amplitude of the resultant wave in the direction \(\theta\) is determined by the interference of all

\[ \sum_{0}^{l_{\max}}(2l+1) \]

partial waves corresponding to different \(l\) and \(m\), since they are all coherent. Therefore the scattering intensity observed at an angle \(\theta\) cannot be associated with a wave belonging to any one value of \(l\).

If among the scattered waves there are waves of different parity (for example, \(S\) and \(P\)), then the forward and backward scattering in the c.m. system is not symmetric.

Thus, the visual interpretation of scattering given here agrees qualitatively with the results of the exact theory.

Fig. 5.

Let us dwell on the question of determining the phases. If \(l\) angular momenta take part in the scattering, then the differential cross section may be represented in the form

\[ \sigma(\theta)\simeq \left| \sum_{l=0}^{l_{\max}}(2l+1)+e^{i\delta_l}\sin\delta_l P_l(\cos\theta) \right|^2 = \sum_{0}^{2l_{\max}} a_n\cos^n\theta, \tag{8} \]

where \(a_n\) depend quadratically on \(\delta_l\). Expanding the experimental results \(\sigma(\theta)\) in powers of \(\cos\theta\), one can find the coefficients \(a_n\) and thus obtain \(2l_{\max}+1\) quadratic equations for determining \(l_{\max}+1\) phases. The system of \(2l_{\max}+1\) quadratic

equations makes it possible to determine uniquely the absolute magnitudes and relative signs of the \(l_{\max}+1\) phases. Thus, from angular-distribution data alone, in the presence of only short-range forces it is impossible to establish whether they are attractive or repulsive. For this it is necessary to know the absolute sign of at least one phase. If, besides short-range forces, there are Coulomb forces, then the uncertainty in the sign of the phases of the short-range forces is removed, since the sign of the Coulomb phases is known if the charge of the particles is known.

2. Scattering of Unpolarized Uncharged Particles with Spin \(1/2\) by a Nucleus with Spin 0

(for example, \(n\) by \(\mathrm{He}^4\))

Before turning to questions of scattering of spin particles, let us consider the definition of a polarized and an unpolarized beam \(^{3,7}\).

For this purpose let us note that the polarization vector \(\mathbf P\), equal to the mean value of the spin of the particle \(\dfrac{\psi^{*}\boldsymbol{\sigma}\psi}{\psi^{*}\psi}\), may have any direction in space relative to the \(Z\) axis, whereas the “projection” of the spin on the \(Z\) axis takes only two values. Therefore a fully polarized beam of particles with spin \(1/2\) in some direction \(\mathbf P\) is defined as a pure state having a function which is a superposition of two partial states (spin functions) with eigenvalues of the spin directed along and opposite to the \(Z\) axis, i.e.

\[ \psi_{\text{polariz}}=A_{+\frac12}\psi^\uparrow + A_{-\frac12}\psi^\downarrow, \tag{9} \]

where \(A_{\pm\frac12}\) are amplitudes with definite phase factors of the type \(a_{\pm\frac12}e^{i\alpha}\), depending on the direction of the polarization vector \(\mathbf P\) relative to \(X, Y, Z\). The amplitudes \(A_{\pm\frac12}\) are normalized in such a way that

\[ \left|A_{+\frac12}\right|^{2}+\left|A_{-\frac12}\right|^{2}=1. \]

Thus, for example, if the direction of propagation of the polarized beam and the direction of polarization \(\mathbf P\) coincide with the \(X\) axis, then each polarized particle may be represented as a superposition of two coherent plane waves \(\psi^\uparrow\) and \(\psi^\downarrow\) with equal amplitudes \(a_{+\frac12}\) and \(a_{-\frac12}\), shifted in phase by \(\dfrac{\pi}{2}\), the wave having the spin direction along the \(Z\) axis \((\psi^\uparrow)\) being shifted in phase relative to the wave \((\psi^\downarrow)\) by \(\delta=+\dfrac{\pi}{2}\) (Fig. 5, a). If the polarization \(\mathbf P\) is directed opposite to the \(X\) axis, then the waves change places (Fig. 5, b).

In particular, for a beam polarized along the \(Z\) axis, one of the amplitudes in the expansion (9) vanishes, and we have simply \(\psi^\uparrow\) for a beam polarized along the \(Z\) axis, and \(\psi^\downarrow\) for a beam polarized opposite to the \(Z\) axis (Fig. 5, c, d).

A natural (unpolarized) beam may be regarded as a mixture of particles with opposite directions of polarization. Thus, a natural beam should be represented as a mixed state composed of mutually incoherent waves of type (9), possessing opposite directions of polarization*). The choice of the direction of the \(Z\) axis and of the direction of polarization for the incoherent components of the natural (unpolarized) mixture is completely arbitrary and is determined by the nature of the problem. For a graphical representation of scattering processes it is convenient to direct the \(Z\) axis normal to the scattering plane and to assume that the natural beam consists of a mixture of particles polarized along and opposite to the \(Z\) axis. Thus, to each particle polarized along \(Z\) there corresponds a plane wave \(\psi^\uparrow\), and to each particle polarized opposite to \(Z\), a wave \(\psi^\downarrow\). These waves are mutually incoherent, since in the present case they correspond to different particles.

Let us now consider the scattering of neutrons \(\left(s=\dfrac{1}{2}\right)\) by \(\mathrm{He}^4\) \((s=0)\). Analysis of the experimental data\(^{8,9}\) on the angular distribution of \(n-\mathrm{He}^4\) scattering at energies of several MeV permits one to conclude that \(S\)- and \(P\)-waves take part in the scattering. Since the \(S\)-level of \(\mathrm{He}^5\), if it exists, must lie very high, the convergence of the unexcited \(\mathrm{He}^4\) nucleus and the neutron (or proton) in an \(S\)-state is impossible. In other words, in the scattering of the neutron \(S\)-wave, repulsive forces act and, consequently, the phase shift \(\delta_S<0\). Conversely, \(P\)-wave scattering is associated with the presence of two broad levels (\(\Gamma\sim 1\ \mathrm{MeV}\)) at neutron energies \(\sim 1\) and \(\sim 4\ \mathrm{MeV}\). It is assumed that the \(n-\mathrm{He}^4\) system at these energies may be roughly regarded as an unperturbed nucleus with a neutron “rotating” about it, possessing relative orbital angular momentum \(l=1\), the levels at \(1\ \mathrm{MeV}\) and \(4\ \mathrm{MeV}\) differing in the relative orientation of the neutron spin and the orbital angular momentum. It is precisely to the level

*) Proceeding from this definition, an unpolarized beam may be described by a superposition of states

\[ c=\sum_\lambda \left\{ A_{1/2,\lambda}S^{1/2}+A_{-1/2,\lambda}S^{-1/2}\right\}\varepsilon_\lambda, \]

where \(\varepsilon_\lambda\) is the so-called statistical variable, defined in such a way that \(\varepsilon_i^*\varepsilon_l=\delta_{il}\). In any result of a physical measurement, the polarization \(\varepsilon_\lambda\) necessarily enters in the form \(\varepsilon_i^*\varepsilon_l\). This corresponds to the fact that it is impossible simultaneously to detect one and the same particle in states with opposite polarizations.

\(E_{\mathrm{res}}\sim 1\) MeV corresponds to the parallel orientation of spin and orbit (in the usual spectroscopic notation), \({}^{2}P_{3/2}\), while the level at \(E_{\mathrm{res}}\sim 4\) MeV corresponds to the antiparallel orientation of spin and orbit \(({}^{2}P_{1/2})\). Accordingly (see formula (5)), the phase shifts of the \(P\)-waves \(\delta_{p_{3/2}}\) and \(\delta_{p_{1/2}}\), for the states \({}^{2}P_{3/2}\) and \({}^{2}P_{1/2}\), also differ substantially. Thus, a strong spin–orbit splitting of the \(P\)-phase takes place.

Let us examine the behavior of the \(P\)-wave in the scattering of an unpolarized neutron beam. Let the energy of the incident neutrons correspond to the energy of the level of the intermediate nucleus \(\mathrm{He}^{5*}\), which has total angular momentum \(j=l+s=\dfrac{1}{2}\) and parity \((-1)\) (the \({}^{2}P_{1/2}\) level), i.e., let us put the phase \(\delta_{p_{1/2}}=\dfrac{\pi}{2}\). We shall, for the time being, set all the remaining phases equal to zero. Thus, only \(P\)-waves with spin oriented antiparallel to the orbital angular momentum are predominantly scattered, i.e., the waves \(\psi_{1}^{+-\,1\downarrow}\) and \(\psi_{1}^{-\,1\uparrow}\). This case is shown graphically in Fig. 6. In Fig. 6, to each incident wave \(\psi_{1}^{+-\,1\downarrow}\) or \(\psi_{1}^{-\,1\uparrow}\) one ought to add a second wave with the opposite spin direction, but since these waves would correspond to the parallel orientation of spin and orbital angular momentum and, by the condition, would not be scattered \((\delta_{p_{3/2}}=0)\), we have omitted them.

Fig. 6.

In Fig. 6, in contrast to Fig. 2, the incident waves \(\psi_{1}^{+-\,1\downarrow}\) and \(\psi_{1}^{-\,1\uparrow}\) correspond to different particles, since they have different directions of polarization. In other words, the moments of interaction time for the particles corresponding to the waves \(\psi_{1}^{+-\,1\downarrow}\) and \(\psi_{1}^{-\,1\uparrow}\) are not correlated, i.e., the incident waves \(\psi_{1}^{+-\,1\downarrow}\) and \(\psi_{1}^{-\,1\uparrow}\) are incoherent. The difference of their phases changes irregularly with time. In Fig. 6 this is reflected by the fact that the incident waves \(\psi_{1}^{+-\,1\downarrow}\) and \(\psi_{1}^{-\,1\uparrow}\) approach the scattering ring with different (arbitrary) phases.

As a result of scattering, the waves \(\psi_{1}^{+-\,1\downarrow}\) and \(\psi_{1}^{-\,1\uparrow}\) acquire identical phase shifts (in our example \(\delta_{p_{1/2}}=\dfrac{\pi}{2}\)), but remain

incoherent, since it is clear that incoherence is preserved under the action of any perturbations. But this means that when the scattering angle \(\theta\) is varied from 0 to \(\pi\), no definite path difference arises between the scattered waves \(\psi_1^{+1}\downarrow\) and \(\psi_1^{-1}\uparrow\). In other words, the time-averaged amplitude of the sum of the incoherent scattered waves, for any \(\theta\), remains equal to the sum of their amplitudes, since the interference terms in the time sum are equal to zero.

Thus, in contrast to the case of scattering of particles without spin (Fig. 26), when for \(P\)-waves at \(\theta=90^\circ\) there was complete disappearance of the amplitude, because the waves \(\psi_1^{+1}\) and \(\psi_1^{-1}\) were coherent and at \(\theta=90^\circ\) their path difference was equal to \(\pi\), the interaction in the presence of strong spin-orbit coupling (only one phase \(\delta_{P_{1/2}}\) different from zero) leads to spherically symmetric scattering.

From the present consideration it may seem that also in the case of scattering of a pure \(P_{3/2}\)-wave the angular distribution of the amplitude should be spherically symmetric. The exact theory shows that for waves with \(j>1/2\) this is not so.*) Thus, for example, at an angle of \(90^\circ\) the intensity of \(P_{3/2}\)-scattering falls to \(1/4\) of its value at \(\theta=0\). The anisotropy of scattering of pure waves with \(j>1/2\) (in particular, of \(P_{3/2}\)-scattering) can be understood qualitatively if one recalls that waves associated with a definite direction of polarization (spin) of the incident beam relative to the axis \(Z'\) are represented as a superposition of coherent waves corresponding to different values of the projection of the total angular momentum \(j\) on the axis \(Z'\), and, consequently, are scattered from several points on the scattering circle (cf. \(D\)-scattering, see Fig. 4).

*) It can be shown that for \(P_{1/2}\) waves the consideration presented here agrees with the results of the exact theory. Following Lepore\(^5\), the amplitude of the scattered wave in the scattering of an unpolarized beam of particles with spin \(1/2\) by particles with spin 0 can be written as an incoherent mixture:

\[ \frac{1}{r}e^{ikr} f(\theta)\chi_{1/2}^{+1/2}, \qquad \frac{1}{r}e^{ikr} f(\theta)\chi_{1/2}^{-1/2}, \]

where \(\chi_{1/2}^{\pm 1/2}\) are spin functions, and \(f(\theta)=A(\theta)+(\boldsymbol{\sigma},\mathbf{n})B(\theta)\). Here \(A(\theta)\) and \(B(\theta)\) are given by formulas (11), (12), \(\mathbf{n}\) is the unit vector normal to the scattering plane, and \(\boldsymbol{\sigma}\) is the vector spin operator. For \(P_{1/2}\) we have from (11) and (12):

\[ A(\theta)=\cos\theta\cdot e^{i\delta_{1/2}}\sin\delta_{1/2}; \qquad B(\theta)=i\sin(\theta)\cdot e^{i\delta_{1/2}}\sin\delta_{1/2}. \]

Further, choosing the spin quantization axis along \(\mathbf{n}\), we have \((\boldsymbol{\sigma},\mathbf{n})=\sigma_z\). After simple transformations we obtain for the scattered wave the incoherent mixture:

\[ \left. \begin{aligned} \frac{1}{r}e^{i(kr+\delta_{1/2})}\sin\delta_{1/2}(\cos\theta+i\sin\theta)\chi_{1/2}^{-1/2} &= \frac{1}{r}\sin\delta_{1/2}e^{i(kr+\delta_{1/2}+\theta)}\chi_{1/2}^{-1/2},\\ \frac{1}{r}e^{i(kr+\delta_{1/2})}\sin\delta_{1/2}(\cos\theta-i\sin\theta)\chi_{1/2}^{+1/2} &= \frac{1}{r}\sin\delta_{1/2}e^{i(kr+\delta_{1/2}-\theta)}\chi_{1/2}^{+1/2}, \end{aligned} \right\} \]

i.e. precisely the equivalent of our graphical representations.

In real scattering processes there usually participate simultaneously several waves belonging to different values of the orbital \(l\) and total \(j\) momenta. Therefore the scattering intensity observed at an angle \(\theta\) cannot be associated with a wave belonging to any single value \(l, j\).

Theory leads to the angular dependence of the cross section \(\sigma(\theta)\) for the scattering of unpolarized particles of spin \(1/2\) by particles of spin 0, having the following form:

\[ \sigma(\theta)=AA^{*}+BB^{*}, \tag{10} \]

where

\[ A=\frac{\lambda}{2\pi^{1/2}}\sum_{l=0}^{l_{\max}} \left[(l+1)e^{i\delta_l^{+}}\sin\delta_l^{+} +l\,e^{i\delta_l^{-}}\cdot\sin\delta_l^{-}\right]P_l(\cos\theta), \tag{11} \]

\[ B=-\frac{i\lambda}{2\pi^{1/2}}\sum_{l=0}^{l_{\max}} \left[e^{i\delta_l^{+}}\cdot\sin\delta_l^{+} -e^{i\delta_l^{-}}\cdot\sin\delta_l^{-}\right]P_l^{1}(\cos\theta). \tag{12} \]

\(\delta_l^{\pm}\) are defined as the phase shifts in states with spin parallel or antiparallel to the orbital momentum.

Let us determine how uniquely the set of phases \(\delta_l^{\pm}\) is determined from the quantities \(\sigma(\theta)\). For this purpose, note that \(P_l'(\cos\theta)\) in (12) contains \(\cos\theta\) to a power no higher than \(l_{\max}-1\). An immediate consequence of this is that \(\sigma(\theta)\) in spin scattering, just as in spinless scattering, includes terms with \(\cos\theta\) to powers no higher than \(2l_{\max}\), i.e.,

\[ \sigma(\theta)=\sum_{0}^{2l_{\max}} a_n' \cos^n\theta, \tag{13} \]

where each coefficient \(a_n'\) determined from experiment depends quadratically on the phase shifts, and consequently from the data on the angular distributions one can, also in the case of spin scattering, obtain only \(2l_{\max}+1\) quadratic equations for determining the phases. Meanwhile, the number of unknown phases in our case is also equal to \(2l_{\max}+1\), since to each value of the orbital momentum with \(l>0\) there correspond two phases. Thus, in the case of interaction of particles of spin \(1/2\) with particles of spin 0, the phases are determined from the data on angular distributions ambiguously both in sign and in absolute value. For an unambiguous determination it is necessary to have another \(2l_{\max}+1\) quadratic equations.

In conclusion, we note that expression (13) is valid for the scattering of particles with spin of any magnitude[^11]. On the other hand, the number of possible phase shifts, and consequently also the ambiguity of phase analysis from elastic-scattering data, increases very rapidly with increasing spin of the interacting particles.

3. Axial asymmetry in the scattering of polarized particles with spin \(1/2\) by particles with zero spin

Consider the scattering of a polarized beam of neutrons with spin directed, for example, opposite to \(Z'\), i.e. opposite to the normal to the scattering plane, by \(\mathrm{He}^4\) at an energy corresponding to the \(P_{1/2}\) resonance, and take into account the existence of potential \(S_{1/2}\)-scattering with phase \(\delta_0\). Since the incident beam is polarized (a pure state), the same particle takes part in both \(S_{1/2}\)- and \(P_{1/2}\)-scattering; that is, the scattered \(S_{1/2}\)- and \(P_{1/2}\)-waves are coherent. Thus, we shall consider the case in which waves with the same \(j=1/2\), but with different parity, interfere (the \(S_{1/2}\)-wave has parity \((+)\), and the \(P_{1/2}\)-wave has parity \((-)\)).

In Fig. 7 only those terms of the incident wave for which interaction actually takes place are shown. The polarized \(S'_{1/2}\)-wave falls on the scattering ring (i.e. the wave arriving at the center of the scattering ring and having spin directed opposite to \(Z'\)) and the polarized \(P'_{1/2}\)-wave (i.e. the \(P\)-wave with spin directed antiparallel to the orbital angular momentum). The waves are scattered at \(\theta=90^\circ\). It is clear from Fig. 7 that, for a certain relation between the phases \(\delta_{S'_{1/2}}\) and \(\delta_{P'_{1/2}}\), the waves scattered to the right (left) mutually cancel, while the waves scattered to the left (right) mutually reinforce one another; that is, there is an axial asymmetry of scattering in the plane normal to the direction of polarization.

Fig. 7.

For a given phase relation, the direction of the asymmetry \((\sigma_{\text{right}}-\sigma_{\text{left}}\gtrless 0)\), obviously, depends on the spin direction (along or opposite to \(Z'\)) in the incident wave.

Asymmetry can also arise in the interference of other waves, for example \(P_{1/2}\)- and \(P_{3/2}\)-waves, in the case where the scattering phases \(\delta_{p_{1/2}}\) and \(\delta_{p_{3/2}}\) are not equal. In this case scattering at \(90^\circ\) will not give asymmetry. Figure 8 shows \(P\)-waves scattered at \(90^\circ\), corresponding to total spin \(j=3/2\) and \(j=1/2\). The scattering ring is rotated so that the axis of the incident beam is directed

Fig. 8.

normal to the plane of the drawing. It is clear from Fig. 8 that, for any values of the phases \(\delta_{1/2}\) and \(\delta_{3/2}\), the path difference between the waves \(P_{1/2}\) and \(P_{3/2}\) scattered at \(90^\circ\) to the right remains equal in absolute value to the path difference of the waves \(P_{1/2}\) and \(P_{3/2}\) scattered at \(90^\circ\) to the left, and consequently, at \(\theta=90^\circ\) there is no asymmetry. In the interference of \(P_{1/2}-{}^{3/2}\)-waves, the maximum asymmetry occurs near \(45^\circ\) (in the \(\psi\)-system).

4. Polarization in the scattering of unpolarized beams (spin \(1/2\)) by an unpolarized target (spin 0)

In complete analogy with the preceding case of scattering of polarized waves \(\psi\downarrow\), we consider the scattering of an unpolarized neutron beam by He\(^4\), with simultaneous participation in the scattering of, for example, \(S\)- and \(P\)-waves. In the present case the initial state differs from the scattering of a polarized beam (see Fig. 7) by the presence of a second system of waves \(\psi\uparrow\) with spin directed along \(Z'\) (Figs. 9a and 9b), and the two systems \(\psi\downarrow\) and \(\psi\uparrow\) are incoherent. Owing to the incoherence of \(\psi\downarrow\) and \(\psi\uparrow\), the scattering of \(\psi\downarrow\) may be considered independently of \(\psi\uparrow\). Obviously, if the phase relation \(\delta_{s_{1/2}}\) and \(\delta_{p_{1/2}}\) is taken to be the same as for Fig. 7, then the scattering of \(\psi\uparrow\) will give an asymmetry of the same magnitude as that produced by \(\psi\downarrow\) (see Fig. 7), but in the opposite direction. The waves \(\psi\uparrow\) and \(\psi\downarrow\) scattered at \(\theta=90^\circ\) are shown in Figs. 9a and 9b. The complete scattering pattern

POLARIZATION OF FAST PROTONS AND NEUTRONS

of an unpolarized beam is obtained if the scattering rings of Fig. 9,a and 9,b are superposed. Thus, for certain phase relations the scattering of an unpolarized beam can lead, for certain \(\theta\), to a complete or partial spatial separation of \(\psi_{\downarrow\,\mathrm{scat}}\) and \(\psi_{\uparrow\,\mathrm{scat}}\), i.e., to polarization.

Fig. 9.

Let us note that the system consisting of an unpolarized beam of neutrons \(+\mathrm{He}^4\) can be interpreted as a mixed state consisting of two pure substates, shown in Fig. 9,a and 9,b.

The general picture of the distribution of polarization directions in the interaction of an unpolarized beam with an unpolarized target is shown in Fig. 10. In the exact theory it is shown that the polarization is always directed normally to the scattering plane. For the case of \(P\)-scattering this is obvious, since the spin direction of the scattered waves is parallel or antiparallel to the direction of the angular-momentum vector, whose mean value for \(P\)-waves is normal to the scattering plane (see § 1).

Fig. 10.

It can be shown\(^3\) that the scattering of a beam polarized along the incident axis does not differ from the scattering of an unpolarized beam. Finally, it is clear that the scattering intensity of an unpolarized beam on an unpolarized target is azimuthally symmetric.

Using Figs. 7 and 9 as examples, one can formulate the principal conditions for the occurrence of polarization in the scattering of unpolarized particles, or of axial asymmetry in the scattering of polarized particles: 1) at least one state with \(l \ne 0\) must participate in the scattering; 2) interference of at least

measure two states with different \(j\) or parity; 3) it is necessary that the interaction and, consequently, the phases of at least one of the interfering states with the given \(l\), depend on the orientation of the spin relative to the orbital angular momentum. Otherwise the scattering does not differ from the case with zero spin. In the rigorous theory it is shown that these conditions for the occurrence of asymmetry and polarization apply not only to the case of scattering of particles with spin \(1/2\) by a nucleus with zero spin, but also to the scattering of particles with arbitrary spin by any unpolarized targets.

The general expression for the polarization \(\mathbf P_{\rm scatt}(\theta)\) and the angular distribution \(\sigma(\theta,\varphi)\) in the scattering of a beam of particles with spin \(1/2\) and with initial polarization \(\mathbf P_{\rm inc}(\theta)\) by nuclei with zero spin was obtained by Lepore \(^5\). Namely:

\[ \sigma(\theta,\varphi)=(AA^*+BB^*)\left[1+\frac{A^*B+B^*A}{A^*A+B^*B}(\mathbf P_{\rm inc},\mathbf n)\right], \tag{14} \]

\[ \mathbf P_{\rm scatt}(\theta)= \frac{AA^*\mathbf P_{\rm inc}+(AB^*+BA^*)\mathbf n+i(A^*B-B^*A)[\mathbf P_{\rm inc}\mathbf n]} {AA^*+BB^*+(A^*B+B^*A)(\mathbf P_{\rm inc},\mathbf n)} + \]

\[ +\frac{BB^*[2(\mathbf P_{\rm inc}\mathbf n)\mathbf n-\mathbf P_{\rm inc}]} {AA^*+BB^*+(A^*B+B^*A)(\mathbf P_{\rm inc},\mathbf n)} . \tag{15} \]

In the case of an unpolarized incident beam

\[ \mathbf P_{\rm scatt}(\theta)=\frac{AB^*+BA^*}{AA^*+BB^*}\,\mathbf n . \tag{16} \]

In these expressions \(\mathbf n\) is the unit vector normal to the scattering plane,

\[ \mathbf n=\frac{[\mathbf k,\mathbf k_0]}{\sin\theta}, \]

where \(\mathbf k_0,\ \mathbf k\) are the directions of the incident and scattered beams, and \(A,\ B\) are quantities determining the amplitude of the scattered wave (see (11), (12)) and depending on the scattering phases \(\delta_l^\pm\), corresponding to states with parallel or antiparallel orientations of the spin and orbit. Each of the phases \(\delta_l^\pm\) is the sum

\[ \delta_l^\pm=\alpha_l+\beta_l^\pm+\gamma_l^\pm, \tag{17} \]

where \(\alpha_l\) is the phase shift due to the Coulomb field, which for small \(Z\) and large scattering angles, because of the weakness of the interaction of the magnetic moment of the proton with the magnetic orbital moment, does not depend on the orientation of the spin relative to the orbital angular momentum, \(\alpha_l=\arg\Gamma(l+ia)\), where \(\Gamma\) is the gamma function, \(a=Z_1Z_2e^2\mu/k\hbar^2\); \(\mu\) is the reduced mass of the system,

\[ \mu=\frac{m_1m_2}{m_1+m_2}, \]

\(Z_1,\ Z_2\) are the charges of the interacting particles, \(\beta_l^\pm\) is the phase shift due to potential scattering, and \(\gamma_l^\pm\) is the phase shift determined by resonant scat-

Near resonance

\[ \tan \gamma_l^{\pm}=\frac{\Gamma_l^{\pm}}{E_l^{\pm}-E}, \tag{18} \]

where \(E\) is the energy of the incident particle, \(E_l^{\pm}\) is the resonance energy, and \(\Gamma_l^{\pm}\) is the resonance width. For the scattering of nucleons by \(\mathrm{He}^4\) at energies of the order of several MeV, the phase shifts due to the nuclear interaction, i.e. \(\xi_l^{\pm}\) and \(\gamma_l^{\pm}\), differ from zero only for the \(S\)- and \(P\)-waves. In this case the formulas for \(\sigma(\theta)\) and \(P(\theta)\) can be reduced to the following form\(^3\):

\[ \begin{aligned} \sigma(\theta)=\lambda^2 \Bigl\{& \left|1-\frac{\eta}{2s^2}\exp[-i\eta\ln s^2]+\sin\delta_0 e^{i\delta_0}+\right.\\ &\left.+\cos\theta\,[2\sin\delta_{3/2}\exp[i(\delta_{3/2}+\sigma_1)]]+\right.\\ &\left.+\sin\delta_{1/2}\exp[i(\delta_{1/2}+\sigma_1)]\right|^2 +\sin^2\theta\,\sin^2(\delta_{3/2}-\delta_{1/2})\Bigr\}, \end{aligned} \tag{19} \]

\[ P(\theta)\cdot\sigma(\theta)= \]

\[ \begin{aligned} ={}&-2\lambda^2\sin\theta\cdot\sin(\delta_{3/2}-\delta_{1/2}) \Bigl\{\sin\delta_0\cdot\sin(\delta_{3/2}+\delta_{1/2}+\sigma_1-\sigma_0)\\ &-\left(\frac{\eta}{2s^2}\right)\cdot \sin(\delta_{3/2}+\delta_{1/2}+\sigma_1+\eta\ln s^2) +3\cos\theta\cdot\sin\delta_{3/2}\cdot\sin\delta_{1/2}\Bigr\}\mathbf n . \end{aligned} \tag{20} \]

Here \(\delta_0\), \(\delta_{1/2}\), and \(\delta_{3/2}\) are the phases for the \(S_{1/2}\)-, \(P_{1/2}\)-, and \(P_{3/2}\)-waves, due to the purely nuclear interaction,

\[ \eta=\lambda\,\frac{Z_1Z_2e^2}{\hbar^2}\,\mu, \qquad \sigma_1=2\arctan\eta, \qquad s=\sin\frac{\theta}{2}. \]

Putting \(\eta=0\) in (19) and (20), we directly obtain the formulas for the scattering and polarization of neutrons by nuclei with zero spin, taking account of \(S\)- and \(P\)-scattering.

The method of detecting and measuring the polarization of a beam of fast particles with spin \(1/2\) consists in measuring the axial asymmetry of scattering or of another polarizing process. Suppose, for example, that a beam of neutrons or protons with polarization \(\mathbf P_0\) (the direction of \(\mathbf P_0\) coincides with the \(X\)-axis) and energy \(E\) propagates along the \(Z\)-axis and is scattered by an “analyzer” nucleus with spin 0, placed at the origin of coordinates (Fig. 11). The scattering cross section of the polarized beam by the “analyzer” nucleus depends, in general, on the azimuthal angle \(\varphi\). From

Fig. 11.

(10), (14), (16) we find:

\[ \sigma(E,\theta,\psi)=\sigma(\theta)\left[1+P(\theta,E)_{\text{scat}}(\mathbf P_{0},\mathbf n)\right], \tag{21} \]

where \(\sigma(\theta)\) and \(P(\theta,E)\) are the cross section and polarization that would occur in the scattering of an unpolarized beam with the same energy by the “analyzer” nucleus, and \(\mathbf n\) is the normal to the scattering plane.

It follows from (21) that, in the case of scattering through an angle \(\theta\) in the plane normal to the direction of polarization in the incident beam (the \(Z,Y\) plane), the ratio \(R\) of the intensity of the wave scattered to the right to the intensity of the wave scattered to the left is

\[ R= \frac{\sigma\left(E,\theta,+\frac{\pi}{2}\right)} {\sigma\left(E,\theta,-\frac{\pi}{2}\right)} = \frac{1+P(\theta,E)_{\text{scat}}P_{0}} {1-P(\theta,E)_{\text{scat}}P_{0}}, \tag{22} \]

whence, knowing \(R\) and \(P(\theta,E)_{\text{scat}}\), one can determine \(P_{0}\). If \(P(\theta,E)_{\text{scat}}\) and \(P_{0}\sim 1\), then \(R\) may be very large.

Let us consider the phenomenon of polarization in the scattering of particles with spin \(1/2\) by a particle with spin 0 from the point of view of determining phase shifts. For this purpose, instead of the polarization \(P(\theta)\), it is convenient to introduce the quantity \(\sigma(\theta)P(\theta)\), which may be called the “effective cross section for spin orientation.” From (10) and (16) we find \(\sigma(\theta)P(\theta)=AB^{*}+BA^{*}\), where \(A,B\) are given in (11), (12). Noting that the maximum power of \(\cos\theta\) entering into \(A\) is equal to \(l_{\max}\), while the maximum power of \(\cos\theta\) entering into \(B\) is equal to \(l_{\max}-1\), we immediately obtain:

\[ \sigma(\theta)P|P(\theta)|=AB^{*}+BA^{*} = \sum_{0}^{l_{\max}-1} a''_{n}\cdot \cos^{n}\theta\cdot \sin\theta, \tag{23} \]

where the coefficients \(a''_{n}\), which depend quadratically on the phase shifts, differ from \(a'_{n}\) in (13). Thus, by expanding the experimental values of \(\sigma(\theta)|P(\theta)|\) in powers of \(\cos\theta\), one can obtain \(2l_{\max}\) quadratic equations for determining \(2l_{\max}+1\) phase shifts.

On the other hand, from data on the angular distribution of scattering \(\sigma(\theta)\), one can obtain \(2l_{\max}+1\) other quadratic equations for determining the phases.

From this there follows an important conclusion. Experimental data on the angular distribution of the polarization \(P(\theta)\), together with data on \(\sigma(\theta)\) in elastic scattering of particles with spin \(1/2\) by a particle with spin 0, determine a system of \(4l_{\max}+1\) quadratic equations, from which the magnitudes and relative signs of all \(2l_{\max}+1\) phases are uniquely determined. As in the case of scattering of spinless particles, in order to determine the true sign of the phases, additional ...

... additional data on the sign of at least one phase. Thus, in order to obtain maximal information on the interaction of spin particles, combined studies of scattering and polarization are necessary.

An interesting example of the application of polarization to phase analysis is the establishment of the order of succession of the levels of \(\mathrm{Li}^{5*}\). An analysis of the data on \(p\)—\(\mathrm{He}^{4}\) scattering leads to the conclusion that, in the energy region up to \(\sim 5\) MeV, nuclear scattering is determined by \(S\)- and \(P\)-waves, while the dependence \(\sigma(\theta)\) on energy indicates the potential character of \(S\)-scattering and the resonance character of \(P\)-scattering. The phase analysis of \(P\)-scattering admits a twofold interpretation. If it is assumed that the \(S\)-phase is negative in the energy region \(0\)—\(5\) MeV, then the ambiguity of the \(P\)-phases is manifested in the fact that the data on \(\sigma(\theta)\) are equally well compatible both with the assumption that the energy of the \(P_{3/2}\) level is greater than the energy of the \(P_{1/2}\) level (the so-called normal order of succession of the levels), with
\[ (E_{\mathrm{res}})_{P_{3/2}} \simeq 3.5\ \text{MeV}, \]
and
\[ (E_{\mathrm{res}})_{P_{1/2}} \simeq 2.3\ \text{MeV}, \]
and with the assumption that
\[ (E_{\mathrm{res}})_{P_{3/2}} < (E_{\mathrm{res}})_{P_{1/2}} \]
(the inverted order of succession of the levels), with
\[ (E_{\mathrm{res}})_{P_{3/2}} \simeq 3\ \text{MeV}, \]
and
\[ (E_{\mathrm{res}})_{P_{1/2}} \gg 3.5\ \text{MeV}. \]
The question of which order of succession of the levels exists is in fact of great importance.

The point is that the assumption of the existence of a strong dependence of the nuclear forces on the orientation of the spin and the orbit makes it possible naturally to explain the observed order of filling of nuclear shells; moreover, both theoretical considerations and experimental values of the “magic” numbers indicate that the splitting of nuclear levels due to spin-orbit coupling occurs in such a way that levels with angular momentum
\[ j = l + \frac{1}{2} \]
have a greater binding energy, i.e., lie lower than levels with angular momentum
\[ j = l - \frac{1}{2}. \]

Fig. 12.

In other words, in nuclear shells there is an inverted order of succession of the levels. Since the splitting of the \(P_{1/2}\) and \(P_{3/2}\) levels in \(\mathrm{Li}^{5*}\) is the simplest case of spin-orbit interaction, the experimental determination of the order of succession of these levels is of great interest.

The true order of succession of the \(P\)-levels was established in an experiment on double scattering \(p\)—\(\mathrm{He}^{4\,12}\) (Fig. 12). In this experiment

protons, once scattered by He\(^4\) and thus partially polarized, undergo secondary scattering by He\(^4\), which gives an axial asymmetry.

Fig. 13.

Figure 13 gives the values of \(P(90^\circ)\) for the scattering of protons by He\(^4\) at \(90^\circ\) (in the c.m. system) as a function of \(E_{p,\text{labor.}}\). Curve 1 was calculated from the phases with the reverse order of the levels, and curve 2 from the phases with the normal order of the Li\(^5\) levels\(^*\). It is easy to see that the experiment on double scattering of protons with energy \(\sim 3.5\) MeV should give completely different results depending on which order of the levels is correct.

Fig. 14.

Figure 14 shows the scheme of the experiment. A beam of protons with energy 3.1 MeV (current 3 µA) enters a chamber filled with helium at atmospheric pressure. Protons scattered once (\(\theta_1 = 90^\circ\) in the c.m. system) in direction 2 are selected by a system of slits. These protons turn out to be partially polarized. Their polarization \(P_1(90^\circ)\) depends essentially on the order of the levels and is equal to
\[ P_1(90^\circ)_{\text{norm}} \simeq -0.9 \]
for the normal order of levels.

levels and \(P_1(90^\circ)_{\mathrm{rev}} \simeq -0.35\) for the inverted order (see Fig. 13). The direction of the second scattering (\(\theta_2 = 90^\circ\) in the \(y\)-system) is selected by two thick-layer plates. As a result of the first scattering and the subsequent slowing down in \(\mathrm{He}^4\), the protons are slowed down, and the second scattering takes place at an energy of \(\sim 2\) MeV, which corresponds to a polarization \(P_2(90^\circ)_{\mathrm{norm}} \simeq +0.9\) for the normal order and \(P_2(90^\circ)_{\mathrm{rev}} \simeq -1\) for the inverted order of level sequence (see Fig. 13). Since the second scattering is undergone by protons already partially polarized in the first scattering, a certain axial asymmetry must occur for scattering forward and backward relative to the initial proton beam in the plane of the first scattering. In other words, the ratio of the number of protons striking emulsion 1 to the number of protons striking emulsion 2 must depend substantially on the true order of the levels.

Using formula (22), we immediately obtain that, in the case of the normal order of the levels,

\[ R_{\mathrm{norm}} = \frac{\sigma_{\mathrm{forward}}}{\sigma_{\mathrm{backward}}} = \frac{1 + P_1(90^\circ)P_2(90^\circ)} {1 - P_1(90^\circ)P_2(90^\circ)} = \]

\[ = \frac{1 + (-0.9)(+0.9)} {1 - (-0.9)(+0.9)} \simeq 0.1, \]

and for the inverted order of the levels

\[ R_{\mathrm{rev}} = \frac{1 + (-0.35)(-1)} {1 - (0.35)(-1)} \simeq 2. \]

The table gives the results obtained by the authors with a 24-hour exposure, from which the correctness of the inverted order of the levels follows.

Initial energy (MeV)	Number of tracks on plate No. 1	Number of tracks on plate No. 2	\(R\)	Theoretical values: inverted doublet	Theoretical values: normal doublet
3.25	364	191	1.9	2.6	1/20.2
3.5	61	33	1.85	1.9	1/6.6

In double-scattering experiments, a beam of particles polarized as a result of the first scattering passes, before the second scattering, through a considerable layer of material. In this connection the question of depolarization arises\(^3\).

If a completely polarized beam passes through a substance, then the percentage of depolarization \((1 - P_f)\) after passage

through a layer of thickness \(dt\) in a medium containing \(N\) scattering centers per unit volume, is determined by:

\[ 1-P_f=S_{\mathrm{depol}}\cdot N\cdot dt, \tag{24} \]

where \(P_f\) is the mean polarization of particles that have not undergone scattering plus the polarization of particles scattered through an angle \(\theta<\theta_0\), and \(S_{\mathrm{depol}}\) is the depolarization coefficient. The quantity \(\theta_0\) depends on the specifics of the experiment. In the examples given below, \(\theta_0\) is taken to be \(5.7^\circ\).

Depolarization in passing through matter may be determined by two principal processes:

1) by rotation of the magnetic moment of the polarized particle as a result of the action of the chaotic magnetic fields of the atoms of the substance and of the magnetic fields associated with the relative motion of the particle charge and the charges of the substance \(z\). The percentage decrease of the polarization of \(n\) and \(p\) due to these effects in traversing \(1\ \mathrm{cm}\) of gas under normal conditions is of the order

\[ S_{\mathrm{depol}}<3.6\cdot 10^{-9}(z^2+2z)\%, \]

2) by spin rotation in nuclear collisions. For example, in \((n-p)\)-interaction in the \(S\) state, collisions \(\uparrow\uparrow\) with cross section \(2\sigma_t\) are possible, which, owing to conservation of the projection of the total angular momentum of the system \(j_z\), do not lead to depolarization, and collisions \(\uparrow\downarrow\) with cross section \(\sigma_s+\sigma_t\), which lead to a change of the spin direction in \(1/2\) of the cases. Thus, the probability of depolarization in each \((n-p)\)-collision is

\[ \simeq \frac{\tfrac{1}{2}(\sigma_s+\sigma_t)}{\sigma_s+3\sigma_t}\simeq \frac{1}{2}, \]

where \(\sigma_s,\ \sigma_t\) are the triplet and singlet cross sections:

\[ \sigma_s=63\cdot 10^{-24}\ \mathrm{cm},\qquad \sigma_t=4.4\cdot 10^{-24}\ \mathrm{cm}. \]

For example, the value of \(1-P_f\) for a proton completely stopped in hydrogen with \(E_{\mathrm{initial}}=7\ \mathrm{MeV}\) turns out to be equal to \(\sim 10^{-5}\). Thus, if charged polarized particles have an energy for which the ionization range \(R_i\) is smaller than the nuclear range \(R_{\mathrm{n}}\) (for protons \(R_i\) approaches \(R_{\mathrm{n}}\) at \(E_p\sim 10^9\ \mathrm{eV}\)), then they can be slowed without depolarization.

Conversely, the energy of polarized neutrons cannot be reduced without substantial depolarization.

At present, phase values are known only for the scattering \(p-\mathrm{He}^4\), \(n-\mathrm{He}^4\), \(n-\mathrm{O}^{16}\), and \(n-\mathrm{C}^{12}\) in the energy range from \(1\) to \(5\ \mathrm{MeV}\); moreover, data sufficiently accurate for calculating polarization are apparently available only for \(p-\mathrm{He}^4\) and \(n-\mathrm{He}^4\) scattering.

In Fig. 15 are given the values of the nuclear-scattering phases for the system \(p-\mathrm{He}^4\), obtained by Critchfield and Dodder\({}^{8}\) from an analysis of the data of Freier et al.\({}^{13}\) on angular scattering of \(p-\mathrm{He}^4\). The phases are indicated in degrees, and the energy in the laboratory coordinate system. In Fig. 16 are given the phase values for \(n-\mathrm{He}^4\).\({}^{14}\)

The procedure of phase analysis of data on angular distributions is rather complicated and is carried out by expanding the $\sigma(\theta)$ obtained in the experiment in spherical functions of zero and first

$E_{\text{MeV}}$	$S$	$P_{3/2}$	$P_{1/2}$	$D_{5/2}$	$D_{3/2}$
5.81	$-48.2^\circ$	$112.3^\circ$	$39.3^\circ$	$-0.42^\circ$	$-1.7^\circ$
9.48	$-70.5^\circ$	$105.8^\circ$	$53.3^\circ$	$-3.3^\circ$	$-8.42^\circ$

Fig. 15.

orders. The errors in the absolute values of the phases are determined mainly by inaccuracies in the measurements of $\sigma(\theta)$, and also by the procedure of phase analysis[^15]. For the data on $\mathrm{p} — \mathrm{He}^4$ the errors of $\delta_{P_{1/2}}$ and $\delta_{P_{3/2}}$ amount to $\sim 2^\circ$, and for $\delta_{S_{1/2}}$, $\sim 5^\circ$. The errors of the phases for $\mathrm{n} — \mathrm{He}^4$ are not indicated by the authors, but are probably rather large.

Fig. 16.

It should be noted in general that phase analysis of data on $\sigma(\theta)$ is characterized by low stability, i.e., by large errors in the phase for comparatively small errors in the value of $\sigma(\theta)$. This is especially noticeable far from resonances, and also in the case of broad and overlapping resonances, as, for example, in $\mathrm{n} — \mathrm{He}^4$ scattering.

The inaccuracy is aggravated by the fact that data on $\sigma(\theta)$ can be obtained with sufficient accuracy only in a limited interval of $\theta$. Even in the favorable case of $\mathrm{n} — \mathrm{He}^4$, when the neutron scattering angle is determined from the recoil energy of $\mathrm{He}^4$ in an ioniza-

tion chamber or proportional counter, reliable data on $\sigma(\theta)$ can be obtained only in the interval $\theta = 70^\circ$—$180^\circ$ in the c-system[^14].

In this connection we point out that polarization measurements could serve as important supplementary material not only in the sense of removing ambiguity, but also—and chiefly—for a substantial refinement of the phase curves obtained from $\sigma(\theta)$. The point is that an analysis of expressions (19) and (20) shows that polarization effects and effects connected with the angular distribution of the cross section have different sensitivities to the magnitude $\sin(\delta_{3/2}-\delta_{1/2})$, i.e., to the magnitude of the spin-orbit splitting. It is precisely the polarization

Fig. 17.

effects that are proportional to $|\sin(\delta_{3/2}-\delta_{1/2})|$, whereas the cross-section effects are proportional to $|\sin(\delta_{1/2}-\delta_{1/4})|^2$. Thus polarization is a sensitive indicator of the spin-orbit splitting of levels, and comparatively crude polarization measurements could lead to a substantial refinement of the phases.

In the case of broad resonances, qualitative polarization measurements can sometimes decide the question of whether there is one resonance and potential scattering or two overlapping resonances. In the case of one resonance plus potential scattering, an approximate course of the phases near the resonance is shown in Fig. 17a. In the case of two overlapping resonances, it is shown in Fig. 17b. For the interference of two states the magnitude of the polarization (see formula (20)) is

$$ |\mathbf{P}| \sim \sin(\delta_1-\delta_2)\cdot \sin\delta_1\cdot \sin\delta_2. \tag{25} $$

From (25) and Fig. 17 it follows that, for overlapping levels, the polarization does not change sign when the energy is varied within the resonance “hump,” whereas in the case of interference of potential and resonant scattering it does change sign.

Figures 18 and 19 show graphs, calculated on the basis of the phase data of Figs. 15 and 16, of the polarization of protons and neutrons as a function of $\theta$ at different $E_{p,n}$. $E$ is given in the laboratory system, and $\theta$ in the c-system.

POLARIZATION OF FAST PROTONS AND NEUTRONS

The calculation of the polarization of neutrons and protons was carried out using formulas (19) and (20). Because of the inaccuracy of the \(n-\mathrm{He}^4\) phase shifts, the data of Fig. 19 are only of an indicative character.

It follows from Figs. 18 and 19 that, at certain energies and angles, the polarization may approach \(100\%\).

Thus, scattering by \(\mathrm{He}^4\) is one of the methods for obtaining and analyzing polarized beams. From the point of view of detecting the polarization of particles produced in nuclear reactions,

Fig. 18.

protons are more convenient than neutrons, since they can be slowed down (as a result of ionization braking) without depolarization to energies at which the polarization and, consequently, the azimuthal asymmetry in scattering by \(\mathrm{He}^4\) appear especially effectively.

Neutrons are slowed only in nuclear collisions, which lead to substantial depolarization, and therefore neutron scattering by helium can effectively detect polarization only for neutrons in a certain energy interval.

From the experimental point of view it is important to emphasize that the character of the interfering states necessary for the occurrence of polarization in scattering need not necessarily be connected with the nuclear interaction.

We have considered cases in which polarization arises through interference of the \(S\)-wave and one of the \(P\)-resonances \((P_{1/2})\) of the \(\mathrm{He}^{5*}\) nucleus, or as a result of interference of the levels \(P_{1/2}\) and \(P_{3/2}\). If, at a given energy, only one state with a definite \(j\) is realized,

I. I. Levintov

(sharp and high resonance), polarization is nevertheless possible in proton scattering as a result of the interference of this state with Coulomb scattering.

There are also some other interactions leading to polarization. When fast neutrons pass near a nucleus with large \(Z\), there is an interaction of the neutron’s intrinsic magnetic moment with the magnetic moment caused by the relative motion of the charge \(Z\). This purely electromagnetic interaction

Fig. 19.

depends substantially on the orientation of the neutron spin relative to the orbital angular momentum of the neutron’s motion with respect to the nucleus and, despite its smallness, leads to the appearance of waves scattered through small angles, which can interfere with nuclear diffraction scattering (the latter, for nuclei with large \(A\) and small angles, may be regarded as free of spin–orbit interaction). As a result, appreciable polarization effects occur within scattering angles of the order of \(5^\circ\) ^{16}.

The magnitude of the polarization \(\mathbf{P}_{\text{scatt}}\) is determined by the expression

\[ \mathbf{P}_{\text{scatt}} = \frac{ 2\left(\dfrac{k}{4\pi}\right)\sigma_{\text{tot}}\cdot \gamma \cdot \operatorname{ctg}\dfrac{\theta}{2} }{ \sigma^{(0)}+\gamma^2 \operatorname{ctg}^2\dfrac{\theta}{2} }\,\mathbf{n}. \tag{26} \]

Here \(\mathbf n\) is the normal to the scattering plane,

\[ \mathbf n=\frac{[\mathbf k,\mathbf k_0]}{k^2\sin\theta}, \]

where \(\mathbf k_0\) and \(\mathbf k\) are the directions of the incident and scattered neutrons,

\[ \gamma=\frac12\,\mu\left(\frac{\hbar}{mc}\right)\left(\frac{Ze^2}{\hbar c}\right), \]

where \(\mu\) is the magnetic moment of the neutron, \(m\) and \(Z\) are the mass and charge of the scattering nucleus, and \(\theta\) is the scattering angle in the \(c\)-system. \(\sigma_{\rm tot}\) is determined as the total cross section according to the black-nucleus model and is taken equal, for \(kR\ll1\), to \(\sigma_{\rm tot}=4\pi R^2\), and for \(kR\gg1\)

Fig. 20.

\[ \sigma_{\rm tot}=2\pi R^2, \]

where

\[ k=\frac{2\pi}{\lambda}; \]

\(R\) is the nuclear radius. \(\sigma(0)\) is the differential nuclear cross section for \(\theta=0\), taken equal, for \(kR\ll1\), to

\[ \sigma(0)=\frac{\sigma_{\rm tot}}{4\pi}, \]

and for \(kR\gg1\),

\[ \sigma(0)=\frac{\sigma_{\rm tot}}{2}\,\frac{\pi}{\lambda^2}R^2. \]

It follows from (26) that \(P_{\rm scatt}\) assumes its maximum value for angles \(\theta_0\) determined by the relations

\[ \tg\frac{\theta_0}{2}= \begin{cases} \dfrac{Ze^2}{2Mc^2}\,\dfrac{k_n}{R}, & \text{for } kR\ll1,\\[6pt] \dfrac{Ze^2}{Mc^2}\,\dfrac{k_n}{kR}, & \text{for } kR\gg1, \end{cases} \tag{27} \]

and the magnitude of the polarization at the optimum angle \(\theta_0\) is

\[ P(\theta_0)= \begin{cases} kR, & \text{for } kR\ll1,\\ \sim 1, & \text{for } kR\gg1. \end{cases} \]

By means of extrapolation Schwinger estimates \(P(\theta)\) for the scattering of neutrons with \(E\simeq1\) MeV by Pb (Fig. 20).

The author expresses his deep gratitude to Ya. B. Zel’dovich, A. S. Kompaneets, and Ya. A. Smorodinsky for discussion of questions of polarization.

References

1. M. G. Mayer, Phys. Rev. 75, 1969 (1949).
2. J. S. Scwinger, Phys. Rev. 69, 681 (1946).
3. L. Wolfenstein, Phys. Rev. 75, 1664 (1949).
4. L. Wolfenstein, Phys. Rev. 75, 342 (1949).
5. J. Lepore, Phys. Rev. 79, 137 (1950).
6. Dalitz, Proc. Phys. Soc. A 65, 175 (1952).
7. Mott and Massey, The Theory of Atomic Collisions. Moscow, IL, 1951.
8. C. Critchfield and D. Dodder, Phys. Rev. 76, 602 (1949).
9. Landau and Smorodinsky, Journ. of Phys. USSR 8, 154 (1944).
10. R. Adair, Phys. Rev. 86, 155 (1952).
11. C. N. Jang, Phys. Rev. 74, 764 (1948).
12. M. Heusingveld and G. Freier, Phys. Rev. 85, 81 (1951).
13. Freier, Lampi, Sleater, and Williams, Phys. Rev. 75, 1345 (1949).
14. P. Huber, E. Baldinger, Helv. Phys. Acta 25, 437 (1952).
15. D. Dodder, Gammel, Phys. Rev. 88, 520 (1952).
16. J. S. Schwinger, Phys. Rev. 73, 407 (1948).