POLARIZATION OF FAST NUCLEONS*

L. Wolfenstein

1. INTRODUCTION

Nuclear beams extracted from accelerators of various types are usually unpolarized, i.e., the directions of the spins of the particles in such beams are distributed at random. Recently a considerable number of experiments have been performed with polarized beams, in which the spins of the nucleons have a preferred orientation in a definite direction. The present article is devoted to the theoretical methods used in the analysis of such experiments, and to the corresponding results at high energies (nucleons with energies from 100 to 400 MeV).

Definition of polarization. The spin state of a particle with spin \(1/2\) is characterized by a Pauli spinor \(\binom{a_1}{a_2}\). For each such state there exists a direction \(\boldsymbol{\mu}\), along which the spin is directed, such that if we take the \(z\)-axis parallel to \(\boldsymbol{\mu}\), then the Pauli spinor has \(a_2 = 0\). If the unit vector \(\boldsymbol{\mu}\) is defined by the spherical coordinates \(\theta_\mu, \varphi_\mu\), then

\[ \begin{aligned} a_1 &= C \cos \frac{\theta_\mu}{2}\, e^{- i\varphi_\mu/2},\\ a_2 &= C \sin \frac{\theta_\mu}{2}\, e^{ i\varphi_\mu/2}, \end{aligned} \tag{1} \]

where \(2|C|^2\) is the magnitude of the spinor.** A beam of particles with spin \(1/2\) is a collection of many particles, each of which is in some spin state characterized by a Pauli spinor. If all particles are in one and the same state, the beam is called completely polarized, and its polarization can be characterized by a Pauli spinor; however, in most cases this does not occur, since usually the method of producing the beam does not ensure its complete polarization.

We must therefore determine what operation can define the polarization of a beam in the general case; for example, must we determine the spin states of each of the particles in the beam? We require of our definition that it completely determine the results of any experiment that can be carried out. (It is assumed that the intensity and energy of the beam are known.) As an example, let us consider the Stern–Gerlach experiment, where the inhomogeneity of the field is directed along the \(z\)-axis. The difference between the number of particles in two

* Annual Review of Nuclear Science 6 (1956). Translation by Yu. P. Kumekin.
** In what follows it will not always be normalized to unity.

of the beams formed is determined by the expression

\[ \frac{ \sum\limits_n \left\{ |a_1^{(n)}|^2 - |a_2^{(n)}|^2 \right\} }{ \sum\limits_n \left\{ |a_1^{(n)}|^2 + |a_2^{(n)}|^2 \right\} } =\overline{\langle \sigma_z\rangle}, \tag{2a} \]

where the sum over \(n\) is the sum over the states of all particles in the initial beam; \(\langle \sigma_z\rangle\) is the expectation value of the Pauli operators for any one of these states, and the bar above denotes averaging over these states. This experiment does not determine the results of Stern–Gerlach experiments with fields directed along the \(x\) and \(y\) axes (except for the special case \(\langle\sigma_z\rangle=1\)). The results of these additional experiments are determined by the expressions

\[ \frac{ 2\operatorname{Re}\sum\limits_n a_1^{(n)*} a_2^{(n)} }{ \sum\limits_n \left( |a_1^{(n)}|^2+|a_2^{(n)}|^2 \right) } =\overline{\langle \sigma_x\rangle}, \tag{2b} \]

\[ \frac{ 2\operatorname{Im}\sum\limits_n a_1^{(n)*} a_2^{(n)} }{ \sum\limits_n \left( |a_1^{(n)}|^2+|a_2^{(n)}|^2 \right) } =\overline{\langle \sigma_y\rangle}. \tag{2c} \]

(One might think that one of these experiments is impossible because one of the axes is directed along the direction of motion, but the spin can be rotated with respect to the direction of motion by means of a homogeneous magnetic field (see Sec. 4) placed in front of the Stern–Gerlach apparatus.) These three experiments determine the vector \(\overline{\langle\sigma\rangle}\), from which the result of a Stern–Gerlach experiment can be found in the general case. Thus, the polarization of the beam can be determined if the direction of \(\overline{\langle\sigma\rangle}\) and the magnitude of the polarization \(P\) are given, where

\[ P^2=\overline{\langle\sigma\rangle}\cdot\overline{\langle\sigma\rangle} =\overline{\langle\sigma_x\rangle}^{\,2} +\overline{\langle\sigma_y\rangle}^{\,2} +\overline{\langle\sigma_z\rangle}^{\,2}. \tag{2d} \]

Since an arbitrary spin operator corresponding to any possible spin state can be represented as a linear combination of the unit operator, \(\sigma_x\), \(\sigma_y\), and \(\sigma_z\), its mean value can be determined by equations (2). This proves that \(\overline{\langle\sigma\rangle}\) is indeed the complete definition of the polarization that we need. It is impossible to distinguish two beams with the same value of \(\overline{\langle\sigma\rangle}\), even if they have been obtained in entirely different ways. An unpolarized beam (\(P=0\)), therefore, may be represented as an equal mixture of two completely but oppositely polarized beams, and any partially polarized beam \((1>|P|>0)\) may be represented as a mixture containing a fraction \(|P|\) of a beam completely polarized in the direction \(\overline{\langle\sigma\rangle}\) and a fraction \((1-|P|)\) of an unpolarized beam.

Such a treatment is completely analogous to the treatment used for the polarization of light. The three Stern–Gerlach experiments are analogous to the three experiments: two for determining the plane of polarization and one for determining circular polarization, which are necessary for analyzing the polarization of light. The three components of \(\overline{\langle\sigma\rangle}\) are direct analogues of the Stokes parameters \(^{1,2}\).

POLARIZATION OF FAST NUCLEONS

It may be noted that the mean value of the spin is insufficient for determining the polarization of particles with spin 1. In this case, in the Stern–Gerlach experiment there are three beams, and therefore it is additionally necessary to determine the fraction of undeflected particles. This fraction is determined by the mean value of an operator that transforms as an irreducible tensor of rank two, and therefore, in order to determine this fraction in all possible Stern–Gerlach experiments, five parameters are needed, which in all gives eight parameters required for the complete determination of the polarization of particles with spin unity[^3].

Experiments on double scattering. In reality, Stern–Gerlach experiments are not a practical method for obtaining or analyzing the polarization of fast nucleons because of the smallness of their magnetic moment; moreover, it can be shown[^4] that, according to the uncertainty principle, the usual semiclassical treatment of Stern–Gerlach experiments is inapplicable to a beam of such charged particles as protons. Another method, first proposed by Mott for electrons, is the production of a polarized beam by scattering an unpolarized beam from a suitable target. If the interaction in scattering couples the spin and orbital angular momenta, then it is possible that particles of the beam with spin up will be scattered preferentially to the left (right), whereas particles with spin down will be scattered preferentially to the right (left). (We note that classically left and right scattering correspond to orbital angular momentum down and up, respectively.)

The performance of two such scatterings, constituting a double-scattering experiment, can be used first for polarizing the beam and then for analyzing the polarization. The first successful experiment of this kind was carried out by Shull[^6] with electrons, at Mott’s suggestion.

A simplified treatment of such an experiment is given in the table. An unpolarized beam of \(2N\) particles is scattered at the first target in such a way that a fraction \(f_1\) of all particles falls on target 2. It is assumed, however, that

Intensities in double scattering (target with spin zero)

Beam	Spin direction	Number of particles
Incident . . . . . . . . . . . . .	Up	\(N\)
Incident . . . . . . . . . . . . .	Down	\(N\)
Scattered once to the left . . .	Up	\(N f_1(1+P_1)\)
Scattered once to the left . . .	Down	\(N f_1(1-P_1)\)
Scattered once to the right . . .	Up	\(N f_1(1-P_1)\)
Scattered once to the right . . .	Down	\(N f_1(1+P_1)\)
Scattered twice to the left \((LL)\)	Up	\(N f_1 f_2(1+P_1)(1+P_2)\)
Scattered twice to the left \((LL)\)	Down	\(N f_1 f_2(1-P_1)(1-P_2)\)
Scattered first to the left, then to the right \((LR)\) . . . . . . .	Up	\(N f_1 f_2(1+P_1)(1-P_2)\)
Scattered first to the left, then to the right \((LR)\) . . . . . . .	Down	\(N f_1 f_2(1-P_1)(1+P_2)\)

of all particles scattered to the left, a fraction \(f_1(1+P_1)\) of the particles with spin up reaches target 2, and a fraction \(f_1(1-P_1)\) of the particles with spin down. Thus, the beam incident on target 2 has polarization \(P_1\) in the “up” direction. The second scatterer is characterized analogously by the quantities \(f_2\) and \(P_2\), where \(P_2\) expresses the property of target 2 to scatter “up” spins preferentially to the left. It follows that the beam scattered twice to the left \((LL)\) contains more particles than the beam that was first scattered to the left and then to the right \((LR)\): the left-right asymmetry in the second scattering is determined by the ratio

\[ e=\frac{(LL)-(LR)}{(LL)+(LR)} =\frac{1+P_1P_2-(1-P_1P_2)}{1+P_1P_2+(1-P_1P_2)} =P_1P_2 . \tag{3} \]

In particular, if the two scatterings are identical and we neglect the energy loss in the first scattering, then we have \(P_1=P_2=P\), and from equation (3) it follows that

\[ P=\pm\sqrt{e}. \tag{3a} \]

This gives us a method, commonly used at high energies, for determining the parameter \(P\) of the target; however, this method does not determine the sign of \(P\). We note that the symbol \(P\) is used to express both the magnitude of the polarization of the beam and the analyzing power of the given scattering.

Although our result, expressed by equation (3), is correct, the arguments underlying it are incomplete. It was tacitly assumed that the particles do not change the direction of spin upon scattering. In Section 3 it will be shown that this is true for targets with spin zero, but is, generally speaking, false for other targets. The general case will be considered in Section 4, where experiments on triple scattering are also discussed. It should be noted that attempts to relate the sign of the polarization to the sign of the spin-orbit coupling by means of semiclassical reasoning are not very reliable.

Notation

Below are the notations used in describing a single scattering.

\(\mathbf{k}\) — unit vector in the direction of the incident beam in the laboratory system,

\(\mathbf{k}'\) — unit vector in the direction of the scattered beam in the laboratory system,

\(\mathbf{p}\) — initial momentum in the center-of-mass system,

\(\mathbf{p}'\) — final momentum in the center-of-mass system,

\(\Theta,\ \varphi\) — spherical coordinates of \(\mathbf{k}'\) with respect to \(\mathbf{k}\) as the polar axis,

\(\theta,\ \varphi\) — spherical coordinates of \(\mathbf{p}'\) with respect to \(\mathbf{p}\) as the polar axis,

\(\mathbf{n}\) — unit vector perpendicular to the scattering plane,

\[ \mathbf{n}=\frac{\mathbf{k}\times\mathbf{k}'}{\sin\Theta} =\frac{\mathbf{p}\times\mathbf{p}'}{p^2\sin\theta}, \tag{4a} \]

\(\mathbf{s}\) — unit vector \(=\mathbf{n}\times\mathbf{k}'\),

\[ \tag{4b} \]

\(\hbar\boldsymbol{\varkappa}\) — change of momentum \(=\mathbf{p}'-\mathbf{p}\),

\(\mathbf{K}\) — unit vector in the direction of \(\boldsymbol{\varkappa}\),

\(\mathbf{P}\) — unit vector in the direction of \((\mathbf{p}'+\mathbf{p})\).

The indices 1, 2, 3 will be used to denote the first, second, and third scatterings, respectively, but the index 2 may be omitted if this does not lead to ambiguity. We define the azimuthal angle of the second scattering by the relations

\[ \cos\varphi_2=\mathbf{n}_1\cdot\mathbf{n}_2;\qquad \sin\varphi_2=\mathbf{n}_1\times\mathbf{n}_2\cdot\mathbf{k}_2 . \tag{4c} \]

2. REVIEW OF POLARIZATION EXPERIMENTS

Schwinger \(^{7,8}\) proposed polarizing beams of fast neutrons by means of scattering; this method is based on the use of the large spin-orbit coupling in the interaction of a neutron with a nucleus, which has been observed through the large splitting of the “fine structure” of nuclear levels. For example, the scattering of neutrons with energy \(\sim 1\) MeV by \(\mathrm{He}_4\) can be described by means of a broad \(J=3/2\) resonance corresponding to the ground state of \(\mathrm{He}_5\); the state \(J=1/2\) lies at least 2 MeV higher in energy. Calculations showed that neutrons can become strongly polarized as a result of the interference of this split resonant scattering and potential scattering of the \(S\)-wave \(^{9,10}\). While this experiment had not yet been performed with neutrons, a similar method was applied to protons by a group of authors in Minnesota \(^{11,12}\), who carried out the first successful experiments on the polarization of protons in 1951; the polarization was produced and analyzed in double scattering by helium. Because of the influence of Coulomb effects, the resonance on \(\mathrm{Li}_5\), corresponding to \(J=3/2\), which was studied in the experiment, required proton energies from 2 to 3 MeV.

The fact that spin-orbit coupling is significant in nuclear interactions suggests that the products of many nuclear reactions may be polarized and that many scattering experiments are quite feasible. This was clearly demonstrated by successful experiments, including the following:

1) It was shown that protons obtained in the \(d-d\) reaction at a deuteron energy of 300 keV are polarized \(^{13}\) (a left-right asymmetry was observed in their scattering by helium). Similarly, polarization of \(d-d\) neutrons was observed in resonant scattering by carbon and oxygen used as an analyzer \(^{14}\).

2) By scattering on oxygen used as an analyzer, it was shown \(^{15}\) that neutrons from the reaction \(\mathrm{Li}^7(p,n)\mathrm{Be}^7\) have a polarization of about 50%. Polarized neutrons with an energy of 400 keV, obtained by this method, were scattered by various targets from carbon to bismuth, and in many cases a left-right asymmetry was observed \(^{16}\).

A general formula for polarization effects in nuclear reactions was given by Blin-Stoyle \(^{17a}\), Satchler \(^{17b}\), and Simon and Welton \(^{18}\).

It was proposed to use fast polarized beams to study the spin dependence of the nucleon-nucleon interaction. If the interaction is central, then, irrespective of the presence of a spin-spin interaction, the differential scattering cross section of a polarized beam will be the same as for an unpolarized one, provided, of course, that the target is unpolarized. However, noncentral interactions, which strongly couple the spin and orbital angular momenta, can give rise to a left-right asymmetry in the scattering of a polarized beam \(^{19}\). This asymmetry cannot be large if the scattering is almost entirely due to the \(S\)-wave; consequently, polarized nucleon beams with energies considerably greater than 20 MeV are required.

The first successful attempt to polarize high-energy protons by means of scattering was carried out by Oxley et al. in Rochester in 1952 \(^{20}\). Since then, external proton beams with polarizations from 45 to 90% and with intensities from \(10^4\) to \(10^6\) particles/\(\mathrm{cm}^2\) sec have been obtained at various accelerators with energies from 130 MeV to 570 MeV \(^{21-25}\). In a typical experimental arrangement (Fig. 1) the internal unpolarized beam is elastically scattered by target 1 made of beryllium or carbon through an angle \(\Theta_1\), ranging from 10 to 20°, then analyzed by a magnet and collimated before the second scattering on target 2. In ordinary double-scattering experiments the beam scattered through the angle \(\Theta_2\)

to the right and to the left, is registered by a telescope containing such an absorber that only elastically or nearly elastically scattered protons remain. Since we are interested only in the left-right asymmetry, it is not necessary to determine the registration efficiency of the telescopes, but it is very important that there be no false asymmetry between left and right scattering. The results of scattering on complex nuclei and on hydrogen are discussed in Sections 3 and 5.

Beams of high-energy neutrons are usually obtained by charge-exchange scattering of protons on neutrons inside light nuclei. Considering such a collision inside the nucleus as quasielastic \(n-p\) scattering, one may hope that

Fig. 1. Layout of the cyclotron at Berkeley, showing the trajectory of the polarized beam. Scattering at target 1 takes place to the left, and the spin directed out of the plane of the drawing is the spin directed “upward.”

neutrons emitted at some angle in the direction of the incident protons are polarized because of the noncentral \(np\) interaction. The first experiment to detect such polarization was carried out by Bowers\(^{26}\) and showed that the magnitude of the polarization is small. Neutron beams with polarization from 10 to 20% were obtained in this way at energies from 100 to 400 MeV\(^{27–30}\) and were used mainly to study \(np\) scattering (Section 5). The small magnitude of the polarization has the consequence that the asymmetry between left and right scattering amounts to only a few percent, so that large errors may arise from small errors in setting up the apparatus. A group of authors at Harwell\(^{31}\) has recently overcome this difficulty by performing every second scattering on one side, first with neutrons with spin “up” and then with neutrons with spin “down.” The spin of the neutrons is rotated by a magnetic field around which the magnetic moment precesses.

The present article does not include the polarization of thermal neutrons and the use of polarized nuclei as radioactive sources or targets. These questions are considered in the review\(^{32}\).

3. SCATTERING ON A SPINLESS TARGET

We shall first consider the special case of scattering of nucleons by nuclei with zero spin, such as helium or carbon. Existing experiments show that the polarization of nucleons in scattering by complex nuclei varies slowly with atomic weight and, apparently, does not depend on the spin of the nuclei\(^{33}\). Experiments on triple scattering on aluminum\(^{34,35}\)

also show that \(D\) (see Section 4) is approximately equal to unity, as should be the case for nuclei with zero spin. Therefore, in the nonrelativistic approximation we shall carry out the usual treatment and apply it to scattering by any complex nuclei.

Use of the scattering-matrix amplitudes. In this case the wave function of the stationary state of elastic scattering may be written as a spinor with components

\[ \Psi_j^{(n)}=e^{\frac{i\mathbf p\cdot \mathbf r}{\hbar}}a_j^{(n)} +\frac{e^{\frac{ipr}{\hbar}}}{r}\sum_l {\cal M}_{jl}(\theta,\varphi)a_l^{(n)}, \tag{5} \]

where \(a_1^{(n)}, a_2^{(n)}\) determine the spin state of the incident wave, and the \(2\times 2\) matrix \(M(\theta,\varphi)\), which relates the spinor of the scattered beam to the spinor of the incident one, plays the role of the ordinary scattering amplitude \(f(\theta)\). At any given angle the matrix \(M\) may be represented in the form of an expansion in Pauli matrices:

\[ M=g\cdot 1+h_1\sigma_x+h_2\sigma_y+h_3\sigma_z=g+\mathbf h\boldsymbol\sigma, \tag{6} \]

where \(1\) is the unit matrix, which will not be written henceforth. The \(x, y,\) and \(z\) axes must be determined by physical vectors; in the scattering process there are only two such vectors: \(\mathbf p\) and \(\mathbf p'\). Since the only axial vector that can be constructed from them is \(\mathbf n\), the most general form of \(M\) is

\[ M=g(\theta)+\boldsymbol\sigma\cdot\mathbf n\,h(\theta), \tag{7} \]

where \(g\) and \(h\) are arbitrary complex functions of the energy and of the scattering angle \(\theta\). Terms of the type \(\boldsymbol\sigma\mathbf p\) do not enter here, since they change sign under spatial reflections.

If the initial direction of the spin \(\boldsymbol\mu\) is specified by the spherical coordinates \(\theta_u,\varphi_u\), and the \(z\)-axis is directed along \(\mathbf n\), then the spinor multiplying \(e^{ipr/\hbar}/r\) in equation (5), according to (1) and (7), is

\[ \chi= \begin{pmatrix} C(g+h)\cos\dfrac{\theta_u}{2}\,e^{-\frac{i\varphi_u}{2}}\\[6pt] C(g-h)\sin\dfrac{\theta_u}{2}\,e^{\frac{i\varphi_u}{2}} \end{pmatrix}. \tag{8} \]

From equation (8) we obtain:

a) The differential scattering cross section

\[ I=\frac{\chi^+\chi}{2|C|^2}=|g|^2+|h|^2+2\,\mathbf n\boldsymbol\mu\,\operatorname{Re}(g^*h), \tag{9} \]

where we have written \(\mathbf n\boldsymbol\mu\) instead of \(\cos\theta_u\). Averaging over two opposite directions of \(\boldsymbol\mu\), we obtain the cross section for an incident unpolarized beam

\[ I_0=|g|^2+|h|^2. \tag{10a} \]

The polarization of the incident beam adds to the cross section a term \(I_p\), which may be written as

\[ I_p=I_0P\,\mathbf n\boldsymbol\mu, \tag{11a} \]

\[ P=\frac{2\operatorname{Re}(g^*h)}{|g|^2+|h|^2}. \tag{10b} \]

It is seen that, for a given scattering angle \(\theta\), the polarization of the incident beam, directed along \(\mathbf n\), gives an azimuthal dependence of the cross section in the form \(\cos\varphi\), where

\(\varphi\) is the usual azimuthal angle (equation (4c)). Since \(I_0\) and \(P\) determine the quantities \(g+h\) and \(g-h\), we can write the equations

\[ \left. \begin{aligned} g+h &= \sqrt{I_0(1+P)}\, e^{\,i\left(\alpha-\frac{\beta}{2}\right)},\\ g-h &= \sqrt{I_0(1-P)}\, e^{\,i\left(\alpha+\frac{\beta}{2}\right)}, \end{aligned} \right\} \tag{12} \]

which determine the phase factors \(\alpha\) and \(\beta\). Namely,

\[ \cos\beta=\frac{|g|^2-|h|^2}{I_0\sqrt{1-P^2}};\qquad \sin\beta=\frac{2\operatorname{Im}h^*g}{I_0\sqrt{1-P^2}}. \tag{10c} \]

b) Direction of the spin \(\boldsymbol{\mu}_f\) of the scattered particles. The spherical coordinates \(\theta'_u,\ \varphi'_u\), which determine \(\boldsymbol{\mu}_f\), may be obtained by substituting expressions (12) into equation (8) and comparing the result obtained with relations (1):

\[ \left. \begin{aligned} \varphi'_u &= \varphi_u+\beta,\\ \cos\theta'_u &= \frac{\cos\theta_u+P}{1+P\cos\theta_u}. \end{aligned} \right\} \tag{13} \]

Hence it follows that, in scattering, the spin of the particles is inclined toward the \(z\)-axis (the normal \(\mathbf n\)), and this inclination is determined by the quantity \(P\) and the initial \(z\)-component of the spin, while the direction of the projection of the spin onto the scattering plane is rotated through the angle \(\beta\). Averaging over two opposite directions \(\mathbf u\), we obtain that for an unpolarized incident beam

\[ \overline{\langle\boldsymbol{\sigma}\rangle}_f=P\mathbf n. \tag{11b} \]

From these considerations the results of the table follow immediately.

In order to determine \(\beta\), we must measure the rotation of the projection of the spin vector onto the scattering plane. Such an experiment is shown in the second diagram of Fig. 2: the scattering plane is chosen so that the spin direction of the incident beam lies in it; and, since simple analyzers register in the final polarization only the component perpendicular to the direction of the scattered beam \(\mathbf k'\), the analyzer is chosen so as to register the polarization component along \(\mathbf S\). If the incident beam is completely polarized, then this component is called \(R\) (see Section 4), and in the present case

\[ R=\sin\theta'_u\cos(\varphi'_u-90^\circ-\Theta) =\sqrt{1-P^2}\cos(\beta-\Theta), \tag{14} \]

where we have used (13), with the \(x\)-axis directed along \(\mathbf k\), \(\theta_u=90^\circ\), \(\varphi_u=90^\circ\). The other two components are

\[ \overline{\langle\boldsymbol{\sigma}\rangle}\cdot\mathbf n=P \]

and

\[ \overline{\langle\boldsymbol{\sigma}\rangle}\cdot\mathbf k' \equiv R' =\sqrt{1-P^2}\sin(\Theta-\beta). \]

Thus, measurement of \(R\) requires three scatterings; the first scattering field—

Fig. 2. Three experiments on triple scattering. The direction of polarization is indicated by an arrow on the beam incident on the second scatterer (\(\odot\) is an arrow directed behind the plane of the drawing). The arrow on the scattered beam denotes the normal to the scattering plane and, consequently, the measured component of the polarization. The equation for this component is given for the case in which the incident beam is completely polarized. The diagrams refer to the laboratory coordinate system.

...polarizes the beam, the second is investigated, and the third analyzes the final polarization. The plane of the third scattering is determined by the fact that it is perpendicular to S.

Single, double, and triple scattering determine \(I_0\), \(P\), and \(\beta\), and thus, according to (12), determine for the given scattering angle \(\theta\) the complex amplitudes \(g\) and \(h\), apart from the common phase factor \(\alpha\). This phase factor can be determined only from the interference of nuclear and Coulomb scattering (except for the case \(\theta = 0\), when it can be determined from the optical theorem).

Scattering by a spin-orbit potential in the Born approximation. To calculate \(M\) we may assume that the incident nucleon interacts with the nucleus by means of a central potential supplemented by a spin-orbit term

\[ H' = V(r) + W(r)\,\frac{\boldsymbol{\sigma}}{2}\cdot \frac{\mathbf{L}}{\hbar}, \tag{15a} \]

where \(\mathbf{L}\) is the orbital angular-momentum operator. In most cases it is convenient to express \(W(r)\) in terms of a potential \(Y(r)\):

\[ W(r) = -\frac{1}{r}\left(\frac{\hbar}{mc}\right)^2 \frac{dY}{dr}, \tag{15b} \]

so that

\[ H' = V(r) - \left(\frac{\hbar}{mc}\right)^2 \frac{1}{r}\frac{dY}{dr}\, \frac{\boldsymbol{\sigma}}{2}\cdot \frac{\mathbf{L}}{\hbar}. \tag{15c} \]

Examples of such a potential:

1) The electromagnetic interaction of a moving nucleon with a nucleus of charge \(Ze\) in the nonrelativistic approximation is determined by an expression of the type (15), where \(V(r)= zZe^2/r\), and

\[ Y(r) = -\frac{1}{2}(2\mu - z)\frac{Ze^2}{r}, \tag{15d} \]

where \(\mu\) is the magnetic moment of the nucleon in units of the nuclear magneton, and \(z=1\) for protons, \(z=0\) for neutrons.

2) The interaction of a nucleon with a static potential \(V(r)\) is supplemented by the relativistic spin-orbit coupling due to Thomas precession*) according to the formula \(Y(r)=\frac{1}{2}V(r)^{36,37,38}\). If \(V(r)\) is the ordinary nucleon-nucleus interaction, then this magnitude of the spin-orbit coupling proves too small to explain the experimental results, although recent theories using an effective mass lead to a reduction of this discrepancy\(^{39}\).

3) The Mayer-Jensen shell model of nuclei\(^{40}\) assumes a large spin-orbit coupling in the interaction of low-energy nucleons with the nucleus.

An analogous spin-orbit-coupling term was proposed by Fermi\(^{41}\) to explain polarization experiments at high energy. Fermi\(^{42}\) pointed out that this spin-orbit coupling must be concentrated at the nuclear surface, since if the nucleon is surrounded on all sides by nuclear matter, then, to the extent that it can “sense” it, it does not “know” its position relative to the center of the nucleus and therefore does not “know” the sign of \(\mathbf{L}\). Such a concentration follows from equation (15b) if \(Y(r)\) is directly proportional to \(V(r)\). Various attempts have been made to relate this spin-orbit coupling to the noncentral part of the nucleon-nucleon interaction\(^{43-47}\).

*) It may be noted that the term \(\frac{1}{2}\frac{zZe^2}{r}\) may be attributed to Thomas precession.

Now we shall calculate, in the Born approximation, the contribution \(M_{LS}\) to \(M\) from the second term in equation (15c) for scattering with change of momentum from \(\mathbf p\) to \(\mathbf p'\):

\[ M_{LS}=\frac{m}{2\pi\hbar^{2}}\left(\frac{\hbar}{mc}\right)^{2} \int e^{-i\mathbf p'\mathbf r/\hbar}\frac{1}{r}\frac{dY}{dr}\frac{\boldsymbol\sigma}{2}\cdot\mathbf r\times \frac{\boldsymbol\nabla}{i}e^{i\mathbf p\mathbf r/\hbar}\,d\tau= \]

\[ =\frac{m}{2\pi\hbar^{2}}\left(\frac{\hbar}{mc}\right)^{2} \frac{\boldsymbol\sigma}{2}\cdot\frac{\mathbf p}{\hbar}\times \int e^{-i\boldsymbol\kappa\mathbf r}\boldsymbol\nabla Y(r)\,d\tau . \]

Integrating by parts, we have:

\[ M_{LS}=-\frac{m}{2\pi\hbar^{2}}\left(\frac{\hbar}{mc}\right)^{2} \frac{\boldsymbol\sigma}{2}\cdot\frac{\mathbf p}{\hbar}\times \int \boldsymbol\nabla\left(e^{-i\boldsymbol\kappa\mathbf r}\right)Y(r)\,d\tau= \]

\[ =-\frac{i}{2}\left(\frac{P}{mc}\right)^{2}\sin\theta\,Y(k)\,\boldsymbol\sigma\cdot\mathbf n, \tag{16} \]

where we used

\[ \mathbf p\times \hbar\boldsymbol\kappa=\mathbf p\times\mathbf p'=p^{2}\sin\theta\,\mathbf n \]

and introduced the notation of the Fourier transform

\[ x(k)=-\frac{m}{2\pi\hbar^{2}}\int e^{-i\mathbf k\cdot\mathbf r}X(r)\,d\tau . \]

Here \(p\) and \(x\) are the quantities \(\mathbf p\) and \(\boldsymbol\kappa\), respectively. The final result of the Born approximation has the form:

\[ g(\theta)=v(k), \tag{17a} \]

\[ h(\theta)=i\eta^{2}\sin\theta\,\frac{Y(k)}{2}, \tag{17b} \]

\[ \hbar\kappa=2p\sin\frac{\theta}{2};\qquad \eta=\frac{p}{mc}. \]

The physical picture of these calculations was given by Fermi\(^{42}\).

The interpretation of nucleon scattering by nuclei by means of the optical model\(^{48,49,50}\) uses a complex potential \(V(r)\) with an imaginary part corresponding to absorption,

\[ V(r)=V_{0}(r)(1+i\varepsilon). \tag{18} \]

We shall follow the usual assumption that the spin-dependent part of the potential is real and is given by equation (15b), where

\[ Y(r)=\gamma V_{0}(r), \tag{19} \]

although the use of a complex \(Y(r)\) has recently been proposed both on the basis of theoretical calculations\(^{47}\) and for bringing the theory into agreement with experimental results\(^{51}\). Using (18) and (19) for the potential, we obtain\(^{41,52,53}\) from (10) and (17) that

\[ \begin{aligned} I_{0}&=v_{0}^{2}(k)\left[1+\varepsilon^{2}+\frac{\gamma^{2}\eta^{4}}{4}\sin^{2}\theta\right],\\[6pt] P&=-\frac{\eta^{2}\gamma\varepsilon\sin\theta} {1+\varepsilon^{2}+\dfrac{\gamma^{2}\eta^{4}}{4}\sin^{2}\theta},\\[6pt] \sin\beta&=-\frac{P}{\varepsilon\sqrt{1-P^{2}}}. \end{aligned} \tag{20} \]

It should be noted that the polarization \(P\) vanishes if the potentials \(V(r)\) and \(Y(r)\) are both real (i.e., if \(\varepsilon = 0\)); in this case the amplitudes of scattering of spins “up” and “down” differ in phase, but not in magnitude, since \(h\) is \(90^\circ\) out of phase with \(g\). The vanishing of the polarization in the Born approximation for a Hermitian Hamiltonian can be proved in the general case (see reference 13 in paper \(^{18}\)). In this case the spin–orbit coupling still manifests itself in triple scattering, as can be shown by substituting the equation for \(\left(P/\varepsilon\right)\) into the equation for \(\sin \beta\).

More accurate calculations \(^{45,54–61}\), for the most part based on the WKB method, showed that the results of the Born approximation can be completely incorrect in the region of the diffraction minimum, but nevertheless can be used for a rough estimate everywhere.

Comparison of theory with experiment. We shall compare the results of the model with a complex potential and with spin–orbit coupling with the experimental results in the region of \(300\ \mathrm{MeV}^{34}\). In the Born approximation, as is evident from (20), the polarization as a function of angle does not depend on \(V_0(r)\) and attains its maximum value

\[ P_{\max}=\frac{\varepsilon}{\sqrt{1+\varepsilon^2}} \tag{21a} \]

at the angle \(\theta_{\max}\), which is determined from the relation

\[ \sin \theta_{\max}=\frac{2\sqrt{1+\varepsilon^2}}{\gamma \eta}. \tag{21b} \]

Application of (21a) and (21b) to the data on the scattering of protons with energy \(313\ \mathrm{MeV}\) by carbon gives for \(\varepsilon\) a value of about 1.0 and for \(\gamma\) about 16;

Fig. 3. Polarization \(P\) as a function of the laboratory scattering angle for scattering on carbon of protons with energy \(313\ \mathrm{MeV}\). The solid curve represents exact calculations \(^{54}\) (with finite angular resolution taken into account) obtained for a rectangular potential; the dashed curve represents calculations in the Born approximation \(^{41}\) for the same potential.

this value of \(\gamma\) is more than 30 times greater than the value \(1/2\) given by Thomas precession. The results of the Born approximation \(^{41}\) for \(\gamma=15\) and \(\varepsilon=0.6\) are shown in Fig. 3 for comparison with the experimental data.

In exact calculations the polarization should depend on the potential \(V_0(r)\). If we take the same quantities \(\gamma=15\) and \(\varepsilon=0.6\) and a rectangular well of radius \(3.2 \times 10^{-13}\ \mathrm{cm}\) and depth \(27\ \mathrm{MeV}\) for \(V_0(r)\), then the exact calcula-

...give the polarization shown by the solid curve in Fig. 3^54. For comparison with the experimental data, the experimental angular resolution was taken into account in constructing the theoretical curve. The exact results are characterized by large fluctuations of the polarization in the region of the expected diffraction minimum. It is easy to understand why such fluctuations should occur. Consider the scattering of a completely polarized beam: particles scattered to the left experience quite a different effective potential than those scattered to the right, and as a result the diffraction minimum to the left occurs at a somewhat smaller scattering angle than for scattering to the right. If the scattering angle corresponds to the diffraction minimum to the left, then almost all the particles are scattered to the right and we have a large negative value of \(P\). At a somewhat larger scattering angle the situation changes and we have a large positive value of \(P\). This is shown in Fig. 4,

Fig. 4. Differential scattering cross section of 76% polarized protons with an energy of 313 MeV on carbon to the left (triangles) and to the right (circles)^35. The solid curves correspond to exact calculations (with angular resolution taken into account) and were obtained with the same potential as in the case of Fig. 3.

where the scattering cross sections to the left and to the right are given for 76% polarized protons according to exact calculations. Only in the Born approximation, under assumption (19), do the calculations give identical diffraction curves for scattering to the left and to the right.

Contrary to these rigorous theoretical calculations, experiments on carbon give neither a diffraction minimum in the cross section nor a large decrease in polarization. Better agreement with the data was obtained^45,57,61 by replacing the rectangular well for \(V_0(r)\) by a smoother curve proposed by Saxon and Woods^49 to fit the cross-section data at 20 MeV. In these calculations the polarization fluctuation decreases in magnitude, and a careful experiment on aluminum at an energy of 313 MeV^34 showed that such a fluctuation does indeed exist. Sternheimer^56 obtained agreement with the aluminum data by using the Saxon–Woods potential with rounded edges for \(V_0(r)\) and expressions (18) and (19) with \(\varepsilon = 1\) and \(\gamma\) about .12 (Fig. 4). It should be noted that the theoretical differential cross section and the results for triple scattering obtained with such a potential do not satisfy the data equally well. In fact, the experimental and theoretical results in the region of the theoretical diffraction minimum and at large angles must be compared with great caution, even if

taken into account (which was not done in Fig. 5). The point is that in most experiments it is impossible to distinguish elastic scattering from inelastic scattering occurring from the first excited levels; this inelastic scattering may have a cross-section maximum in the region of the elastic-scattering minimum^{62,63}. Thus, the scattering data at these angles may in fact be mainly data on inelastic scattering. In the latest experiments on the scattering by carbon of polarized protons with an energy of 220 MeV^{64}, inelastic scattering from the first excited levels was separated out, and it was found to be considerably larger than elastic scattering for angles greater than \(25^\circ\). It turned out that the polarization \(P(\theta)\) for purely elastic scattering has large fluctuations, actually becoming slightly negative between \(25^\circ\) and \(30^\circ\), and then reaching a value close to 1 at \(35^\circ\); such behavior of the polarization is characteristic of the results of most theoretical calculations.

Fig. 5. Polarization \(P\) for protons with energy 300 MeV scattered by Al^{34}. The solid curve represents Sternheimer’s theoretical calculations^{57}. Angular resolution of the order of \(1^\circ\) is not taken into account in the theoretical curve.

At 300 MeV, experiments on triple scattering by carbon and aluminum were carried out to determine the parameter \(\beta^{34}\). Theoretical calculations^{57} give a value of \(\beta\) that depends strongly on angle. Contrary to this, the experimental results give an almost constant value of \(\beta\) between \(8^\circ\) and \(22^\circ\) for aluminum; however, significant fluctuations of \(\beta\) could have been missed because of poor angular resolution.

Dependence of polarization on energy. Analogous results for polarization in scattering by complex nuclei were obtained at energies from 130 MeV to 420 MeV. Figure 6 shows the maximum value of the polarization \(P_{\max}\) and the corresponding scattering angle \(\theta_{\max}\) as functions of the scattering energy on carbon. The following remarks should be made concerning this figure:

1) The black circles represent experiments in which the polarization was measured only at one or two angles, not necessarily at \(\theta_{\max}\), so that they give only a lower bound for \(P_{\max}\).

2) Determination of the absolute value of \(P\) requires knowledge of the beam polarization, and it cannot be determined accurately because of the difficulty of determining the first scattering angle, which occurs inside the cyclotron (Fig. 1). Allowance for this error is included in the error indicated in Fig. 6. Since the polarization is preserved when a high-energy beam is slowed down^{9}, then in each

laboratory can verify the determination of the beam polarization by comparison with data from other laboratories; such experiments \(^{65,66,67}\) have given good agreement, so that comparison of the results of different laboratories raises no doubts as to their correctness.

3) The value of \(P_{\max}\) may be too small because of the finite angular resolution of the experiment (usually \(\pm 1^\circ\)) or because of the presence of inelastic processes.

4) The error in \(\Theta_{\max}\) arises mainly from the smooth maximum of the curve \(P(\Theta)\), which makes the determination of \(\Theta_{\max}\) difficult.

In Fig. 6 the following data are of interest:

1) \(P_{\max}=0.8\) and is almost unchanged between \(130\ \text{Mev}\) and \(240\ \text{Mev}\), while \(\Theta_{\max}\) varies approximately inversely proportional to the energy. This

Fig. 6. Maximum polarization \(P_{\max}\) and scattering angle \(\theta_{\max}\) for maximum polarization in scattering on carbon. Black triangles — \(P_{\max}\); black circles — lower limit of \(P_{\max}\); white triangles — \(\theta_{\max}\). \(C\) — Carnegie \(^{16}\); \(B\) — Berkeley \(^{34}\); \(B'\) — Berkeley \(^{66}\); \(R\) — Rochester \(^{64}\); \(H\) — Harwell \(^{33}\). (The latest results at Harwell raise the value of \(P_{\max}\) at \(135\ \text{Mev}\) to 0.96 and at \(70\ \text{Mev}\) to 0.3.)

is in agreement with calculations in the Born approximation (see (21)) for the specified values of the optical parameters \(\varepsilon\) and \(\gamma\).

2) Beginning at about \(250\ \text{Mev}\), \(P_{\max}\) begins gradually to decrease, at least up to \(420\ \text{Mev}\), while \(\Theta_{\max}\) continues to decrease slowly. Such behavior can be explained in the Born approximation by a decrease of both parameters \(\varepsilon\) and \(\gamma\) by approximately a factor of 2. A more detailed study of this dependence on energy apparently has not been carried out. However, the work of Watson \(^{47}\) shows that the imaginary part \(W(r)\) at these energies may play an important role.

3) Below \(130\ \text{Mev}\) the polarization falls rapidly; thus an attempt to obtain a polarized beam in scattering of protons with energy \(80\ \text{Mev}\) \(^{68}\) ended in failure. It must be emphasized, however, that all the data below \(130\ \text{Mev}\) refer to \(30^\circ\), whereas, apparently, at these energies a large polarization should be observed at larger scattering angles. Indirect confirmations of this were obtained in the analysis \(^{3}\) of the polarization of deuterons with energy \(100\text{--}200\ \text{Mev}\) in scattering on carbon \(^{69}\). Sternheimer \(^{58}\) showed that this decrease of polarization at energies below \(130\ \text{Mev}\) can be quantitatively explained by a decrease of the optical parameter \(\varepsilon\) with decreasing energy;

such a reduction was also proposed$^{70,71}$ in order to satisfy the data on neutron scattering. However, even with this reduced value of $\varepsilon$, at some angles a large polarization (greater than 50%) is possible, as was shown in calculations by Gammel et al. for neutron scattering at an energy of $14\ \text{MeV}^{72}$. Moreover, a polarization of the order of 20%, observed in the scattering of neutrons with an energy of $0.5\ \text{MeV}$, can be explained by a very small value of $\varepsilon$ in the model of nuclear reactions of Feshbach et al.$^{50}$.

Dependence of the polarization on the target. Polarized protons were scattered on various targets from helium to tantalum at $300\ \text{MeV}^{34}$ and from beryllium to bismuth at $130\ \text{MeV}^{33}$. The general conclusion from the results obtained is that, with increasing atomic number from carbon to iron, a gradual decrease in the value of $P_{\max}$ is observed in the first polarization maximum, while at somewhat larger angles a second maximum appears. The decrease in the value of $P_{\max}$ was attributed to Coulomb effects$^{45}$, and the fact that this quantity is a slowly varying function of the nuclear radius was interpreted$^{60}$ as confirmation that the spin-orbit interaction is concentrated at the surface of the nucleus.

Sign of the polarization. If it is assumed that $\gamma$ is positive in accordance with the shell model, it follows from this that the polarization $P$ is positive, since the sign is determined by (11) and (4a); this means that nucleons with spin “up” are predominantly scattered to the left. It is easy to show that this result is obtained if one uses the Born approximation to compute $h(\theta)$ for the real potential of the spin-orbit coupling: then from (17b), (10b), and (19) it follows that

\[ I_0P = \eta^2 \gamma \sin\theta\, v_0(k)\,\operatorname{Im} g(\theta); \]

at small angles $\operatorname{Im} g(\theta)$ is positive, since $\operatorname{Im} g(0)$ is determined by the optical theorem, and thus the sign of $P$ at small angles is the same as the sign of $\gamma$. The double-scattering experiments considered up to now determine only the product $P_1P_2$ and cannot give the sign of $P$. In order to determine the sign of $P$, polarized protons were slowed down to energies below $10\ \text{MeV}$ and scattered on helium (see Section 1); in this case the sign of the parameter $P$ is known from a phase analysis of the data$^{11}$. The results of these experiments show that the polarization is indeed positive$^{73,74}$. It is interesting to note that the same sign of the polarization follows from data on p—p and n—p scattering at small angles (Section 5).

Electromagnetic scattering. In addition to the specific nuclear interaction, we must consider scattering in the Coulomb field of the nucleus. This should be especially important at small angles or near a diffraction minimum. The matrix $M_c$ for scattering in the Coulomb field includes a spin-dependent part due to the interaction of the magnetic moment of the nucleon (15d), and is given, for an infinitely heavy point nucleus with small atomic number $Z$, by expression (7), where

\[ g_c(\theta)= -\frac{1}{2}\frac{n}{\eta}\left(-\frac{\hbar}{mc}\right) \left\{ z\csc^2\left(\frac{\theta}{2}\right)-\frac{\eta^2}{E}\mu_1\right\} \times \]

\[ \times \exp\left[2iz\left\{n\ln\csc\left(\frac{\theta}{2}\right)+\eta_0\right\}\right], \tag{22a} \]

\[ h_c(\theta)= -\frac{i}{2}\left(\frac{Ze^2}{\hbar c}\right) \left(\frac{\hbar}{mc}\right)\mu_1\operatorname{ctg}\left(\frac{\theta}{2}\right) \exp\left[2iz\left\{n\ln\csc\left(\frac{\theta}{2}\right)+\eta_0\right\}\right], \tag{22b} \]

where

\[ n=\frac{Ze^2}{\hbar v_z} =\frac{Ze^2}{\hbar c}\cdot\frac{E}{\eta}, \qquad E=\eta^2+1=\frac{\text{total energy}}{mc^2}, \]

\[ \mu_1=\mu-\frac{zE}{E+1}, \qquad \eta_0=\arg\Gamma(1+in). \]

Equation (22) can be derived without the exponential phase factor by applying the Born approximation to the Dirac equation including the Pauli moment. It was obtained together with the phase factor by Garren[^75] as the first term of an expansion in powers of \(\eta\). An analogous result for small angles and low velocities was obtained by Heckrotte[^51] in the \(WKB\) approximation.

The simplest case is the polarization of neutrons \((Z=0)\) scattered through such a small angle that the scattering of the magnetic moment \(h_c(\theta)\) has the same order of magnitude as the nuclear scattering. In this case the results of the Born approximation should be valid even for large \(Z\)[^8]. From (22b) and (10b) we obtain

\[ I_0P=\left(\frac{Ze^2}{\hbar c}\right)\mu\eta\operatorname{ctg}\frac{\theta}{2}\cdot \operatorname{Im}\left(\frac{\hbar g_N(\theta)}{p}\right), \tag{23} \]

where the nuclear scattering \(g_N(\theta)\) is assumed to be spin-independent for such small angles. The factor \(\operatorname{Im}\left(\frac{\hbar g_N(\theta)}{p}\right)\) at these small angles may be replaced by its value at \(0^\circ\), which is equal to the forward scattering cross section divided by \(4\pi\). Thus all the factors in (23) can be determined, i.e., a theoretical calibration of the polarized-neutron analyzer can be carried out. This method was used by Bock and Wilson[^30], who determined the sign and magnitude of the polarization of a neutron beam of energy \(100\) MeV in scattering by uranium through angles from \(1/3^\circ\) to \(1^\circ\).

In the case of proton scattering, Coulomb effects were included in many calculations (including Sternheimer’s calculations, shown in Fig. 5); these effects strongly influence the fluctuation of \(P(\theta)\) in the region of the expected diffraction minimum. However, the influence of the magnetic moment was not taken into account in all calculations (which followed the method of Gattsek and Riddell[^76]), with the exception of Heckrotte’s work[^51] on the polarization of \(300\) MeV protons scattered by carbon through small angles[^34]. In this analysis \(P(\theta)\) is the sum of a purely nuclear term and a positive term (analogously to (23)) due to the interference of \(h_c(\theta)\) and \(g_n(\theta)\). The conclusion was drawn that, in order to satisfy the shape of the experimental curve for \(P(\theta)\) between \(2\) and \(6^\circ\), these two terms must have the same sign. This result once again confirms that the purely nuclear polarization has a positive sign.

4. General formalism. In Section 3 we dealt with a precisely defined initial spin state, which gave a precisely defined final spin state, after which averaging over the initial spins or summation over the final spins was performed where necessary. If the target also has spin, this operation becomes too inconvenient, and a more general formalism becomes desirable, one that deals more with operators than with wave functions.

The density-matrix formalism[^77,^78]. Consider a system of two particles with spins \(s\) and \(s_t\), and define the polarization by generalizing the treatment carried out in Section 1. An arbitrary spin state \((n)\) is a linear combination of \((2s+1)(2s_t+1)\) basis states of the system and can be represented by a vector \(a_i^{(n)}\), where \(i\) runs from \(1\) to \((2s+1)(2s_t+1)\). For example, if \(s=s_t=\frac12\), then the four basis states may be three triplet and one singlet state. An arbitrary operator may be written as a linear combination of Hermitian basis matrices \(S^\mu\), where \(\mu\) runs from \(1\) to \((2s+1)^2(2s_t+1)^2\). These matrices satisfy the orthogonality requirements:

\[ \operatorname{Sp}(S^\mu S^\nu)=(2s+1)(2s_t+1)\delta_{\mu\nu}. \tag{24} \]

For example, if particles with spin \(1/2\) are incident on a target consisting of particles with spin \(1/2\), then the sixteen basic matrices will be

\[ 1\,1_t;\qquad \sigma_\alpha\,1_t;\qquad 1\,\sigma_{t\alpha};\qquad \sigma_\alpha\,\sigma_{t\beta}, \]

where the first matrix acts on the spinor of the incident particle, the second (with the index \(t\)) acts on the spinor of the target particle, and \(\alpha\) and \(\beta\) take the values \(x,y,z\). The expectation value \(\langle S^\mu\rangle_n\) of one of these matrices in the state \((n)\) is determined by the relation

\[ \sum_j \left|a_j\right|^2 \langle S^\mu\rangle_n = \left(a_1^{(n)*}\ a_2^{(n)*}\ \ldots\right) \begin{pmatrix} S_{11}^{\mu} & S_{12}^{\mu} & \ldots \\ S_{21}^{\mu} & \ldots & \ldots \\ \ldots & \ldots & \ldots \end{pmatrix} \begin{pmatrix} a_1^{(n)} \\ a_2^{(n)} \\ \vdots \end{pmatrix} = \sum_k \sum_j a_j^{(n)} a_k^{(n)*} S_{kj}^{\mu}, \tag{25} \]

where \(\sum_j \left|a_j^{(n)}\right|^2\) characterizes the relative probability or intensity of the state \((n)\).

Any vector \(a_i^{(n)}\) defines a completely polarized state, i.e., one in which each particle is in a completely definite spin state. But, generally speaking, none of the particles is completely polarized, so that we must carry out an averaging over all states \((n)\) of the composite system that are incoherently mixed with one another, each entering with its relative weight \(\sum_i \left|a_i^{(n)}\right|^2\). In particular, from (25) we obtain the following average for \(\langle S^\mu\rangle\):

\[ \overline{\langle S^\mu\rangle} = \frac{ \displaystyle \sum_n \sum_k \sum_j a_j^{(n)} a_k^{(n)*} S_{kj}^{\mu} }{ \displaystyle \sum_n \sum_j \left|a_j^{(n)}\right|^2 }. \tag{26} \]

The mean values \(\overline{\langle S^\mu\rangle}\) for the complete set of basic matrices can be used to determine the polarization of the composite system (cf. definition (20)). Defining the density matrix*) \(^{79}\) by the relation

\[ \rho_{jk}=\sum_n a_j^{(n)} a_k^{(n)*}, \tag{27} \]

we obtain

\[ \overline{\langle S^\mu\rangle} = \frac{\operatorname{Sp}(\rho S^\mu)}{\operatorname{Sp}\rho}. \tag{28} \]

From (24) and (28) we obtain the expansion of \(\rho\) in the matrices \(S^\mu\):

\[ \rho = \frac{\operatorname{Sp}\rho}{(2s+1)(2s_t+1)} \sum_\mu \overline{\langle S^\mu\rangle}\,S^\mu . \tag{29} \]

The density matrix is therefore a convenient method for determining the state of polarization. One of the matrices \(S^\mu\) is always the identity,

*) It should be noted that the sum over \(n\) is a sum over all states that are incoherently mixed with one another in the process of obtaining the beam, so that this sum may include nonorthogonal states, and identical states may enter the sum several times. In the case of a beam of particles, this sum may be regarded as a sum over the states of the individual particles in the beam.

the mean value of which is equal to 1, so that the polarization is determined by \([(2s+1)^2(2s_t+1)^2-1]\) parameters. Let us note, for comparison, that in order to specify a completely polarized state (apart from its normalization and overall phase) \([2(2s+1)(2s_t+1)-2]\) parameters are required. The factor \(\operatorname{Sp}\rho\) determines the normalization of the density matrix and is usually taken to be proportional to the intensity. In the case of nucleons and a spinless target \(s=\frac12\), \(s_t=0\), and \(S^\mu=1;\ \sigma_x,\ \sigma_y,\ \sigma_z\), while the density matrix, according to equation (29), is

\[ \rho=\frac12\operatorname{Sp}(\rho) \begin{bmatrix} 1+\overline{\langle\sigma_z\rangle} & \overline{\langle\sigma_x\rangle}-i\overline{\langle\sigma_y\rangle}\\ \overline{\langle\sigma_x\rangle}+i\overline{\langle\sigma_y\rangle} & 1-\overline{\langle\sigma_z\rangle} \end{bmatrix}. \]

The density matrix of an unpolarized system is proportional to the unit matrix, and for a completely polarized system one can show that \(\operatorname{Sp}(\rho^2)=(\operatorname{Sp}\rho)^2\).

The wave function describing elastic scattering, corresponding to any initial spin state \(a_j^{(n)}\), can be expressed, as before, according to (5). If the incident beams are an incoherent mixture of states \((n)\), then the beam scattered in some direction \((\theta,\varphi)\) will also be a mixture of states, which can be described by the density matrix \(\rho_f\), obtained from (5) and (27):

\[ \rho_{f\,jk} = \sum_n \left(\sum_l M_{jl}a_l^{(n)}\right) \left(\sum_r M_{kr}a_r^{(n)}\right)^* = \sum_l\sum_r M_{jl}\rho_{i\,lr}M^*_{kr}, \]

or, in matrix notation,

\[ \rho_f=M\rho_i M^+, \tag{30} \]

where \(\rho_i\) is the density matrix of the initial beam. Substituting (30) into (28) and using (29) for \(\rho_i\), we obtain

\[ I\langle S^\mu\rangle_f = \frac{1}{(2s+1)(2s_t+1)} \sum_\nu \langle S^\nu\rangle_i\operatorname{Sp}(MS^\nu M^+S^\mu), \tag{31} \]

where

\[ I=\frac{\operatorname{Sp}\rho_f}{\operatorname{Sp}\rho_i} \tag{31a} \]

is the differential scattering cross section.

Relation (31) can also be extended to reaction processes, if \(M\) is made a matrix (in the general case rectangular) which transforms the spin space in the initial channel into the spin space of the final channel (see the special case in \({}^{80}\)).

Experiments on double scattering. These experiments can be described as two special cases of (31):

a) In the second scattering (analyzer) we are interested in the differential cross section \(I\) for the scattering of a polarized nucleon beam with \(\langle\sigma\rangle_i=P_i\mathbf{l}\) on an unpolarized target with spin \(s_t\):

\[ I=\frac{1}{2(2s_t+1)} \left\{\operatorname{Sp}(MM^+)+P_i\mu\cdot\operatorname{Sp}(M\sigma M^+)\right\} =I_0+P_iI_p. \tag{32} \]

The first term \(I_0\) is obtained by substituting into the right-hand side of (31) that matrix \(S^\nu\) which is equal to the unit matrix; this term is the differential cross section for an unpolarized beam. The second term \(P_iI_p\) is obtained by substituting those \(S^\nu\) which are equal to the three components \((\sigma\cdot\mathbf{l})\); this term is

contribution to the cross section from the initial polarization \(P_i\). All other \(\overline{\langle S^\nu\rangle}\) are equal to zero, since the target is unpolarized. Let us note that the formulas for the case of a spinless target (10a), (11a), and (10b) can be obtained by substituting (7) into (32).

b) In the first scattering (polarizer) an unpolarized beam is incident on an unpolarized target. In this case all \(\overline{\langle S^\nu\rangle_i}\) are equal to zero, except for the unit matrix, and the polarization of the scattered nucleon beam is

\[ I_0\,\overline{\langle \boldsymbol{\sigma}\rangle}_f = \frac{1}{2(2s_t+1)}\operatorname{Sp}(MM^+\boldsymbol{\sigma}). \tag{33} \]

We shall formulate three theorems concerning these quantities.

Theorem 1\(^{9}\). a) The contribution \(I_P\) from the initial polarization (in the direction \(\boldsymbol{\psi}\)) to the differential cross section always has an azimuthal dependence proportional to \(\cos\varphi(=\mathbf n\cdot\boldsymbol{\psi})\). It follows from this that polarization in the direction of motion cannot be recorded by a single scattering.

b) The polarization \(\overline{\langle\boldsymbol{\sigma}\rangle}_f\) (see (33)) produced by a polarizer is directed along the normal \(\mathbf n\) to the scattering plane.

Theorem 2\(^{77,78}\). The quantities \(I_P\) and \(\overline{\langle\boldsymbol{\sigma}\rangle}_f\), which characterize, respectively, the analyzing and polarizing powers, can be expressed through the single-scattering parameter \(P(\theta)\) by means of (11a) and (11b) (derived in Section 3 for the case of a spinless target). Comparing them with (32) and (33), one may write this theorem as follows:

\[ \operatorname{Sp}M\boldsymbol{\sigma}M^+ = \operatorname{Sp}MM^+\boldsymbol{\sigma} = 2(2s_t+1)I_0P\mathbf n. \tag{34} \]

The first equality is not obvious, since \(M\), \(M^+\), and \(\boldsymbol{\sigma}\) do not commute with one another. It follows from this that, in double-scattering experiments, the differential cross section in the second scattering is

\[ I_2=I_{02}(1+P_1P_2\cos\varphi_2), \tag{35} \]

where the indices 1 and 2 denote the scattering angle, energy, etc., for the first and second scatterings, respectively. From this the expression (3) for the asymmetry \(\varepsilon\) follows immediately.

Theorem 3\(^{9}\). If \(L_{\max}\) is the maximum orbital angular momentum that must be taken into account, then

\[ I_0P = \sum_{n=0}^{2L_{\max}-1} a_n\cos^n\theta\cdot\sin\theta, \tag{36} \]

where \(\theta\) is the scattering angle in the center-of-mass system.

The proof of these theorems is based on the invariance of \(M\) with respect to spatial rotations, reflections, and time reversal. We can write the matrix \(M\) in the form (6), where \(g\) and \(\mathbf h\) will now be operators (depending on \(\mathbf p\) and \(\mathbf p'\)) in the spin space of the target nuclei. The invariance conditions show that \(g\) transforms as a scalar and does not change sign under time reversal, whereas \(\mathbf h\) must transform as an axial vector and change sign under time reversal, so that the product \(\mathbf h\boldsymbol{\sigma}\) is invariant. Using (6), we obtain

\[ \left. \begin{aligned} \operatorname{Sp}M\boldsymbol{\sigma}M^+ &= 2\operatorname{Sp}'\left(\mathbf h g^+ + {}'g\,\mathbf h^+ - i\mathbf h\times\mathbf h^+\right),\\ \operatorname{Sp}MM^+\boldsymbol{\sigma} &= 2\operatorname{Sp}'\left(\mathbf h^+ g + g^+\mathbf h - i\mathbf h^+\times\mathbf h\right). \end{aligned} \right\} \tag{37} \]

Here \(\operatorname{Sp}'\) denotes the trace in the spin space of the target particles, since

we have already taken the trace in the inclined spin space. After we take identical traces in (37), we are left with functions of \(\mathbf p\) and \(\mathbf p'\). Let us note that the last terms in each of these traces must transform as axial vectors and must not change sign under time reversal (since \(\mathbf h\) enters them twice). But since the only axial vector that can be formed, \((\mathbf p \times \mathbf p')\), changes sign under time reversal, this means that the last term in each of the traces must be equal to zero. The remaining terms are linear in \(\mathbf h\) and must be proportional to \((\mathbf p \times \mathbf p') = \mathbf n p^2 \sin\theta\), after which theorem 1 follows from (32) and (33). The dependence of the type \(\sin\theta\) in (36) also follows from these arguments*). This means that if we decompose \(\mathbf h\) into the components (37), then only the component \(\mathbf h\cdot\mathbf n\) contributes, so that we replace \(\mathbf h\) by this component. Taking into account that the trace of a product of two operators does not depend on the order of this product, we immediately obtain (34), where

\[ I_0 P = \operatorname{Sp}'\left(g^{+}\mathbf h\cdot\mathbf n + gh^{+}\cdot\mathbf n\right)\cdot \frac{1}{2s_t+1}. \tag{38} \]

It should be noted that the validity of (34), or of theorem 2, depends on invariance with respect to time reversal, which allowed us to omit the term \(\mathbf h \times \mathbf h^{+}\) in (37). This means that theorem 2 may be valid even not for elastic scattering, provided only that the polarization and its analysis are inverse reactions. But we cannot use theorem 2 for polarization in inelastic scattering on nuclei, since in this case the inverse reaction must begin on an excited nucleus.

Triple scattering\(^{75,82,83}\). Further information on the matrix \(M\) can be obtained from experiments on triple scattering. Such experiments serve to determine how the second scattering changes the direction and (or) the magnitude of the polarization of protons; thus, the first scattering serves as a polarizer, and the third as an analyzer.

If we use (31) for the second scattering in order to relate the polarization of the scattered inclined beam \(\langle \boldsymbol{\sigma}\rangle_f\) to the polarization of the incident beam \(\langle \boldsymbol{\sigma}\rangle_i\), then the most general relation will be

\[ I_2\langle \boldsymbol{\sigma}\rangle_f = I_{02}\{[P_2 + D\langle \boldsymbol{\sigma}\rangle_i\cdot\mathbf n_2]\mathbf n_2 + [A\langle \boldsymbol{\sigma}\rangle_i\cdot\mathbf k_2 + R\langle \boldsymbol{\sigma}\rangle_i\cdot(\mathbf n_2\times\mathbf k_2)]\mathbf s_2 + \]

\[ + [A'\langle \boldsymbol{\sigma}\rangle_i\cdot\mathbf k_2 + R'\langle \boldsymbol{\sigma}\rangle_i\cdot(\mathbf n_2\times\mathbf k_2)]\mathbf k'_2, \tag{39} \]

where the coefficients \(P_2, D\), etc. are arbitrary functions of the scattering angle and the energy. The term \(P_2\) is obtained by substituting \(S'\), equal to the unit matrix, into the right-hand side of (31), and if the incident beam is unpolarized, this will be the only term different from zero, and \(I_2=I_{20}\), i.e. we obtain (11b). The other terms correspond to \(S'\) equal to one of the components of the spin operator of the incident beam. The proof that (39) is the most general relation is based on the requirement that the right-hand side transform as an axial vector, which forbids terms of the form \(\langle\boldsymbol{\sigma}\rangle\cdot\mathbf s_2\mathbf n_2\), etc. Thus, triple scattering uses five new parameters \(D, A, R, A'\), and \(R'\). From considerations of time reversal it can be shown that only four of them are independent: in the case of an infinitely heavy target nucleus they are connected by the relation \(R'=-A\), while in the case of two particles of equal mass (non-

*) We shall not prove here theorem 3, which is a direct generalization of Yang’s theorem and others concerning the angular distribution in unpolarized cases\(^{81}\).

POLARIZATION OF FAST NUCLEONS

(for relativistic ones) the relation holds

\[ \frac{A+R'}{A-R'}=\operatorname{tg}\frac{\theta}{2}. \tag{40} \]

Since the polarization along the direction of motion cannot be determined by a single scattering alone, the third scattering is carried out so as to register either \(\overline{\langle\sigma\rangle}_f\cdot\mathbf n_2\), or \(\overline{\langle\sigma\rangle}_f\cdot\mathbf s_2\). The corresponding asymmetries in the triple scattering are denoted by \(e_{3n}\) and \(e_{3s}\), defined as

\[ \frac{I_3(+)-I_3(-)}{I_3(+)+I_3(-)}, \]

where \(I_3(\pm)\) corresponds to \(\cos\varphi_3=\pm 1\) \((\cos\varphi_3=\mathbf n_2\cdot\mathbf n_3\) for \(e_{3n}\); \(\cos\varphi_3=\mathbf s_2\cdot\mathbf n_3\) for \(e_{3s}\)). If the beam incident on the second scatterer is obtained as the result of single scattering, then the initial polarization is directed along \(\mathbf n_1\) and, thus, is perpendicular to \(\mathbf k_2(=\mathbf k'_1)\). In this case \(I_2\) is given by relation (35). From (39) it is seen that only \(D\) and \(R\) can be determined in simple triple scattering:

1) \(D\)-experiment. If the third scattering measures \(e_{3n}\), then from (35) and (39) we have

\[ e_{3n}=P_3\overline{\langle\sigma\rangle}_f\cdot\mathbf n_2 =\frac{P_3(P_2+DP_1\cos\varphi_2)}{1+P_1P_2\cos\varphi_2}. \tag{41} \]

To measure \(D\), one chooses \(\cos\varphi_2=1\) (or \(-1\)), which means that all three scattering planes are parallel (Fig. 2a). In this case, if the incident beam is completely polarized, i.e. \(P_1=1\), then

\[ \overline{\langle\sigma\rangle}_f=\mathbf n_2\,\frac{P_2+D}{1+P_2}. \]

It is seen that the parameter \(D\) determines how much the second scattering depolarizes the beam, since \(D=1\) is necessary for there to be no depolarization. In fact, the direction of polarization may even change, since it is possible that \(D\) will be less than \((-P_2)\). It is easy to show that the value of \(D\) is bounded by

\[ -1+2|P_2|\leq D\leq 1. \]

In the case of a spinless target, from (8) at \(\theta_u=0\) it follows that \(D=1\).

2) \(R\)-experiment. If the third scattering measures \(e_{3s}\), then we have

\[ e_{3s}=P_3\overline{\langle\sigma\rangle}_f\cdot\mathbf s_2 =\frac{P_3RP_1\sin\varphi_2}{1+P_1P_2\cos\varphi_2}. \tag{42} \]

To measure \(R\), one chooses \(\sin\varphi_2=1\), which means that all three scattering planes are successively perpendicular to one another (Fig. 2b); thus, the second scattering must be upward or downward with respect to the plane of the first scattering. It is easy to show that the value of \(R\) is bounded by the relation

\[ |R|\leq\sqrt{1-P_2^2}. \]

If a magnetic field directed perpendicular to \(\mathbf n_1\) and \(\mathbf k'_1\) is placed between the first and second scatterers, then \(A\) can be determined. The magnetic field deflects the particles (upward or downward relative to the plane of the first scattering) by an angle \(\delta\) and simultaneously rotates the spin through an angle \(\Delta\), which is greater than \(\delta\) because of the anomalous magnetic moment of the nucleon. Thus the spin is rotated relative to the direction of motion by an angle \(^{75,84}\):

\[ \Delta-\delta=(\mu-z)\frac{eH}{mc}t=(\mu-1)E\delta\quad\text{for protons}, \tag{43} \]

where \(mc^{2}E\) is the total energy of the proton (including its rest energy). In particular, it is possible to rotate the spin so that it will be directed along the direction of motion before the second scattering (Fig. 2). Similarly one can measure the parameter \(R'\), by placing a magnetic field between the second and third scatterer. To determine \(A'\), two magnetic deflections are necessary. Thus, carrying out three scatterings in the presence of magnetic deflection can completely determine all the parameters in (39), and consequently nothing can be learned from a larger number of scatterings.

Correlation experiment. To obtain further information about the matrix \(M\), we must consider a case of the kind contained in (31), where either \(\overline{(S^{v})}_{i}\) or \(\overline{(S^{u})}_{f}\) depends on the spin of the target as well as on the spin of the incident particle. This means that either the target must be polarized, or it is necessary to measure the polarization of the recoil particles in correlation with the polarization of the scattered nucleons. We shall consider a possible correlation experiment in the case of nucleon-nucleon scattering on an unpolarized target without using a magnetic field\(^{83,85}\). For particles scattered through an angle \(\theta\), the component \(\overline{\langle \boldsymbol{\sigma}\rangle}\) along the directions \(\mathbf n\) or \(\mathbf s\), perpendicular to \(\mathbf k'\), is measured. The recoil particles (denoted by the subscript \(t\)) have direction \(\mathbf k'_t\), and for them the polarization \(\overline{\langle \boldsymbol{\sigma}_t\rangle}\) is measured along the directions \(\mathbf n\) or \(\mathbf s_t(=\mathbf n \times \mathbf k'_t)\). In nucleon-nucleon scattering in the nonrelativistic approximation \(\mathbf k'=\mathbf s_t=\mathbf P\) and \(\mathbf k'_t=-\mathbf s=-\mathbf K\). In a correlation experiment we are interested in the product \(\langle \boldsymbol{\sigma}\boldsymbol{\sigma}_t\rangle_f\), in which we can measure the \(\mathbf{nn}\), \(\mathbf{KP}\), \(\mathbf{Kn}\), and \(\mathbf{nP}\) components. Considering only these four components and restricting ourselves only to the components of the incident spin \(\langle\boldsymbol{\sigma}\rangle_i\) perpendicular to the direction of motion of the incident beam, one can obtain the most general expression

\[ I_2 \langle \overline{\boldsymbol{\sigma}\boldsymbol{\sigma}}_t\rangle_f = I_{02}\bigl(C_{nn}+C^p_{nn}\langle\boldsymbol{\sigma}\rangle_i\cdot\mathbf n_2\bigr)\mathbf n_2\mathbf n_2 + I_{02}\bigl(C_{KP}+C^p_{KP}\langle\boldsymbol{\sigma}\rangle_i\cdot\mathbf n_2\bigr)\mathbf K_2\mathbf P_2 + \]

\[ \quad + I_{02}\{C^p_{Kn}\langle\boldsymbol{\sigma}\rangle_i\cdot(\mathbf n_2\times\mathbf k_2)\}\mathbf K_2\mathbf n_2 + I_{02}\{C^p_{nP}\langle\boldsymbol{\sigma}\rangle_i\cdot(\mathbf n_2\times\mathbf k_2)\}\mathbf n_2\mathbf P_2, \]

where the coefficients \(C_{nn}\), etc., are arbitrary functions of \(\theta\) and of the energy. Terms of the type \(\langle\boldsymbol{\sigma}\rangle_i\cdot\mathbf n_2\mathbf K_2\mathbf n_2\) will not enter here, since \(\langle\overline{\boldsymbol{\sigma}\boldsymbol{\sigma}}_t\rangle\) is even with respect to inversion of coordinates. Thus there are two different experiments of this type which start with an unpolarized beam, and four using a polarized beam.

Relativistic effects. A consideration by means of Pauli spinors is nonrelativistic. However, the polarization of free Dirac particles can be determined by means of Pauli spinors in the system in which the particle is at rest\(^{4}\). The relativistic formalism, developed by Stapp\(^{85}\), reduces to equations very similar to those given in this article. Stapp showed that the nonrelativistic formalism gives correct results if the target may be regarded as infinitely heavy, or if all scatterings occur in one plane. Relativistic corrections (apart from the usual relativistic kinematics) are needed in the equations for \(R\), \(A\), \(C_{KP}\), etc., in nucleon-nucleon scattering.

5. NUCLEON-NUCLEON SCATTERING

It is well known that the nucleon-nucleon interaction has a complicated spin dependence, expressed in the \(n\)—\(p\) system at low energies by the difference between the singlet and triplet ground states and by the quadrupole moment of the deuteron. From measurement of the differential cross section of an unpolarized beam it is difficult to determine the nature of the spin-dependent interaction in scattering, since in this case all the initial ...

and the final spin states. The various polarization experiments considered in Section 4 prove very useful in obtaining further information.

Formalism of \(p\)—\(p\) scattering. In proton–proton scattering the amplitude \(M\) is a \(4 \times 4\) matrix in the composite spin space. Invariance considerations\(^{77,78}\) show that only five of the sixteen matrix elements are independent, and that the most general form of the matrix \(M\) is

\[ M = BS + C(\boldsymbol{\sigma}+\boldsymbol{\sigma}_{t})\cdot \mathbf{n} + \frac{1}{2}G(\boldsymbol{\sigma}\cdot \mathbf{K}\,\boldsymbol{\sigma}_{t}\cdot \mathbf{K} + \boldsymbol{\sigma}\cdot \mathbf{P}\,\boldsymbol{\sigma}_{t}\cdot \mathbf{P})T + \]

\[ + \frac{1}{3}H(\boldsymbol{\sigma}\cdot \mathbf{K}\,\boldsymbol{\sigma}_{t}\cdot \mathbf{K} - \boldsymbol{\sigma}\cdot \mathbf{P}\,\boldsymbol{\sigma}_{t}\cdot \mathbf{P}) + N\boldsymbol{\sigma}\cdot \mathbf{n}\boldsymbol{\sigma}_{t}\cdot \mathbf{n}T, \tag{44} \]

where \(S\) and \(T\) are projection operators which select the singlet or triplet state, respectively. The five amplitudes \(B, C, G, H\), and \(N\) depend only on the scattering angle \(\theta\) for any given energy; it can also be shown that, owing to the identity of the protons, \(B, H\), and \(C\) are even functions of \(\cos\theta\), while \(G\) and \(N\) are odd. For each scattering angle \(\theta \leqslant 90^\circ\) we may hope to determine experimentally nine real parameters—the moduli of the five amplitudes and their relative phases. Nine experiments at any given angle will give us nine quadratic equations for these nine unknowns. The remaining ambiguity is still very large; on the other hand, it may turn out that a smaller number of experiments already imposes strong restrictions on these amplitudes. These experiments are listed below in order of increasing difficulty:

1. Cross section of an unpolarized beam \(I_0(\theta)\).
2. Polarization \(P(\theta)\).
3. Triple scattering in one plane \((\theta \leqslant 90^\circ)\): \(D(\theta)\).
4. Triple scattering in successive perpendicular planes \((\theta \leqslant 90^\circ)\): \(R(\theta)\).
5. Triple scattering with a magnetic field before the second scattering \((\theta \leqslant 90^\circ)\): \(A(\theta)\).
   6–7. Correlation experiments with an unpolarized beam: \(C_{nn}; C_{KP}\).
   8–10. \(D, R, A\) for \(\theta > 90^\circ\).
   11–12. Triple scattering with a magnetic field after the second scattering \((\theta \leqslant 90^\circ\) and \(\theta \geqslant 90^\circ)\): \(R'(\theta)\).
   13–15. Correlation experiments with a polarized beam.

Since, for a proton scattered in the direction \((\theta,\varphi)\), there is a recoil proton in the direction \((\pi-\theta,\pi+\varphi)\), the differential cross section \(I(\theta,\varphi)\) must be equal to \(I(\pi-\theta,\pi+\varphi)\). This gives \(I_0(\theta)=I_0(\pi-\theta)\) and \(P(\theta)=-P(\pi-\theta)\). The minus sign arises because the contribution from the polarization to the cross section is proportional to \(P(\theta)\cos\varphi\) (35). However, in the case of triple-scattering experiments the results for \(\theta < 90^\circ\) and \(\theta > 90^\circ\) are independent, since the two emitted protons need not have the same polarization. If the final state were a pure triplet (or pure singlet) state, then there would be no difference between these two polarizations, and for triple scattering the relations \(D(\theta)=D(\pi-\theta)\), \(R(\theta)=R'(\pi-\theta)\), and \(A(\theta)=-A'(\pi-\theta)\) would hold. Violation of these equalities is a measure of singlet–triplet interference. Expressions for most of these experimental quantities in terms of amplitudes may be found in Refs. \(^{80,83,85}\).

Owing to the small range of nuclear forces, it is possible to describe scattering by means of a comparatively small number of phase shifts. Therefore it is not necessary to determine nine experimental quantities for

of one angle, since experiments at different angles are in fact not independent. Expressions for the amplitudes and experimental quantities in terms of phase shifts are given in \(^{75,85,86,87}\).

\(p\)—\(p\)-polarization experiments. Scattering of polarized protons by hydrogen for determining \(P(\theta)\) was carried out at various energies between 140 MeV and 430 MeV \(^{20-24,88-90}\); the results are summarized in Fig. 7. Polarization is of special interest if it is considered together with the differential cross section, which is approximately equal to \(3.7\) mb/ster independently of the magnitude of the angle and energy in this range. Thus, at 140 MeV the isotropic cross section can be explained by singlet scattering of the \(S\)-wave alone, but the observed substantial polarization requires that at least half of the scattering be due to triplet states. Moreover, even at 140 MeV the product of the polarization and the cross section \(I_0\) does not vary as \(\sin\theta \cos\theta\), so that, according to equation (36), triplet scattering is due not only to \(P\)-waves, but requires some contribution from \(F\)-waves. All the results can be satisfied by using the first two nonvanishing terms in (36):

Fig. 7. Polarization \(P\) as a function of the angle of \(p\)—\(p\)-scattering \(\theta\) in the center-of-mass system: crosses—140 MeV \(^{24,88}\); black circles—310 MeV \(^{90}\); squares—415 MeV \(^{23}\). (When using the data from works \(^{23}\) and \(^{24}\), the beam polarization was taken as 0.53 and 0.64, respectively, and not that given in those works. The new values were communicated by the authors.)

\[ I_0 P = (a_1 \cos\theta + a_3 \cos^3\theta)\sin\theta, \tag{45} \]

where the coefficients in mb/ster according to \(^{89,89,23}\) are\(^*\)

\[ \begin{array}{rclcl} 140\ \text{MeV} &\quad& a_1 = 1.11 \pm 0.12 && a_3 = 1.28 \pm 0.20,\\ 310\ \text{MeV} && a_1 = 2.0 \pm 0.3 && a_3 = 1.16 \pm 0.4,\\ 415\ \text{MeV} && a_1 = 1.6 \pm 0.3 && a_3 = 3.0 \pm 0.5. \end{array} \]

Such a large polarization is the first confirmation of the noncentral character of the proton-proton interaction in the triplet state. The smoothness of the differential cross section can also be partly explained by a noncentral triplet interaction; thus, for example, the presence of spin-orbit coupling in \({}^3P\)-scattering leads to a term \(\sin^2\theta\) in the cross section in addition to the usual term \(\cos^2\theta\).

In experiments on triple scattering it is necessary that the scattered protons have an energy greater than 100 MeV so that their polarization can be readily analyzed (Fig. 6). Therefore up to now these experiments have not been

\[ \overline{\phantom{xxxxxxxxxxxxxxxx}} \]

\(^*\) The value for the energy 140 MeV was obtained from all the data up to March 1956 \(^{88}\), and only some of them are shown in Fig. 7. At high energies there are indications of the presence of the term \(a_5\); thus, although the data can be satisfied without this term, a somewhat better agreement is obtained if \(a_3\) is smaller and \(a_5\) is almost the same as \(a_3\). The errors quoted reflect only the statistics of the experiment and do not include errors from the possible presence of the term \(a_5\).

POLARIZATION OF FAST NUCLEONS

were carried out at energies below 300 MeV. The most complete information on proton–proton scattering was obtained at Berkeley, where the values of $D$, $R$, and $A$ were measured at an energy of 310 MeV for angles between 20 and 80° in the center-of-mass system$^{90,91,92}$. $D$ was also measured at an energy of 420 MeV$^{93}$. Further data$^{94,95}$ on the proton–proton system at high energies are provided by measurements$^{96,97}$ of asymmetry in meson production in the reaction $\mathrm{p}+\mathrm{p}\to \pi^{+}+\mathrm{d}$ using a polarized beam.

Many attempts have been made to analyze the experiments on proton–proton scattering. The most ambitious attempt is the attempt to explain the scattering by a static potential model. It was shown that qualitative agreement with the observed polarization can be obtained by using either tensor forces or directly a spin–orbit coupling for the noncentral interaction$^{98}$, but no potential was found that would give quantitative agreement with the data. The latter fact may not have any substantial significance, since the ratio of the number of potentials for which calculations were performed to the number of all possible potentials is undoubtedly close to zero.

A considerably less broad, but nevertheless very difficult, problem is the attempt to find phase shifts from the experimental data$^{85,88,92,99-104}$. At energies of the order of 100 MeV one may hope that it is necessary to consider only four phase shifts: the singlet $S$ phase shift $\delta_{0}$ and three phase shifts $\delta_{1}^{0}$, $\delta_{1}^{1}$, $\delta_{1}^{2}$ for the $P$ waves, where the upper index denotes the total angular momentum $J$. There are four results that can be used to obtain these phase shifts: two coefficients in the nuclear differential cross section (which should have the form $a+b\cos^{2}\theta$), the polarization (i.e. $a_{1}$ (36)), and the effect of Coulomb–nuclear interference on the cross section. Unfortunately, the lowest energy at which polarization measurements were carried out is 140 MeV, where an $F$ wave is clearly present. Seeking agreement with experiment while neglecting $F$ waves, Taylor$^{88}$ obtained at 140 MeV:
$\delta_{0}=22^\circ$, $\delta_{1}^{0}=-40^\circ$, $\delta_{1}^{1}=10^\circ$, $\delta_{1}^{2}=7^\circ$. Lomon and Feshbach$^{103}$ obtained very similar results by a somewhat different approach, but they took $\delta_{1}^{1}=0^\circ$. These results may change substantially if $D$ and $F$ waves are taken into account$^{102}$. At an energy of 310 MeV the phase analysis was carried out including the $^{1}S_{0}$, $^{3}P_{0}$, $^{1}D$ and $^{3}F$ waves; this gives eight phase shifts and also a mixing parameter between the $^{3}P_{2}$ and $^{3}F_{2}$ states. The large amount of data made it possible to attempt to find all these nine parameters; four sets of phase shifts were obtained that give good agreement with the data$^{92}$. All these sets have small values for the $F$ phase shifts, which perhaps indicates that neglecting large partial waves at these energies is justified. An interesting result is the negative value of the singlet phase, which is in qualitative agreement with the hard-core model for the nuclear potential.

Equation (45) cannot be used at small angles, where the effect of Coulomb interference may be observed. This effect was calculated by Harren$^{105}$, who indicated that further information for determining the phase shifts can be obtained from accurate measurements of polarization at small angles. It is interesting to note that in this effect an important role is played by the interaction with the magnetic moment of the proton.

n—p-polarization experiments. In the case of neutron–proton scattering the most general form of $M(\theta,\varphi)$ is given in the form (44) supplemented by a term of the type $(\boldsymbol{\sigma}-\boldsymbol{\sigma}_{t})\mathbf{n}$, which is absent for identical particles. We shall agree that $\boldsymbol{\sigma}_{t}$ is the proton spin and $\boldsymbol{\sigma}$ the neutron spin. The number of different polarization experiments for n—p scattering is greater than for p—p scattering, since we must distinguish experiments with polarized-

neutrons and experiments with polarized protons. Thus, in this case there are two different double-scattering experiments:

1) scattering of polarized neutrons by hydrogen, and
2) scattering of polarized protons by neutrons. The cross section in the first case is given by (32), while in the second case in (32) one must interchange \(\boldsymbol{\sigma}\) and \(\boldsymbol{\sigma}_t\). Obviously, these two cases will differ only if \(M\) is not symmetric with respect to \(\boldsymbol{\sigma}\) and \(\boldsymbol{\sigma}_t\); thus, the difference is entirely explained by the term \((\boldsymbol{\sigma}-\boldsymbol{\sigma}_t)\cdot \mathbf n\) in \(M\). According to the hypothesis of charge symmetry, the scattering amplitude \(M\) should not change under interchange of the proton and neutron; thus this term must be equal to zero, and these two double-scattering experiments become identical. Both of these experiments were performed at approximately the same energy:

1) At Carnegie \({}^{106}\) the left-right asymmetry of recoil protons was measured using a \(16\%\) polarized neutron beam with an effective energy of \(350\) MeV.

2) At Berkeley \({}^{90,107}\) polarized protons with energy \(310\) MeV were scattered by deuterium, and the left-right asymmetry of recoil neutrons was measured (or of scattered protons counted in coincidence with recoil neutrons).

The identical results in these two experiments may be regarded as a confirmation of charge symmetry.

Using further charge independence, we may write that the matrix \(M\) in scattering is

\[ M_{\mathrm{np}}=\frac{1}{2}(M_0+M_T), \]

where \(M_T\) is the scattering of a two-nucleon system with total isotopic spin \(T\), and each \(M_T\) may be written separately in the form (44). For \(p-p\) scattering we simply have \(M_{\mathrm{pp}}=M_1\). If we take some result for \(n-p\) scattering, for example \((I_0P)_{\mathrm{np}}\), then it may be regarded as the sum of three terms:

1) a term due only to the \(T=1\) state;
2) a term due only to the \(T=0\) state; and
3) interference between the \(T=1\) and \(T=0\) states. The first of these terms can be determined directly from \(p-p\) scattering. The last two terms differ in that the second term includes interference between states with the same parity (for example \({}^3S\) and \({}^3D\)), while the third includes interference between states with opposite parity (for example \({}^3S\) and \({}^3P\)). As a result, the second term gives a contribution to \(I_0P\) that is even with respect to \(90^\circ\), and the third an odd one. Thus we can isolate \(I_0P\) for pure \(T=0\) scattering \({}^{108}\):

\[ (I_0P)_{00}(\theta)=2(I_0P)_{\mathrm{np}}(\theta)-2(I_0P)_{\mathrm{np}}(\pi-\theta)-(I_0P)_{\mathrm{pp}}(\theta). \tag{46} \]

The results for \(n-p\) scattering at an energy of \(310\) MeV were presented \({}^{107}\) in the form (36)

\[ (I_0P)_{\mathrm{np}}=\sin\theta\{-0.34+0.45\cos\theta+1.30\cos^2\theta+2.93\cos^3\theta\}\ \mathrm{mb/sr}, \]

where the coefficients have a large (unspecified) error. Comparing this result with (46) and (45) at an energy of \(310\) MeV, we obtain

\[ (I_0P)_{00}=10\sin\theta\cos^3\theta\ \mathrm{mb/sr}. \tag{47} \]

Although this result also contains a considerable error, the following statements apparently hold:

1) There is definitely a significant noncentral interaction acting in the \(T=0\) states at an energy of \(300\) MeV; one may

expect that this interaction is associated with tensor forces, which manifest themselves in \(T=0\) states at zero energies.

2) In n—p scattering at an energy of \(310\) MeV, \(D\)-states and, perhaps, higher states make a significant contribution; if we neglect the higher states, then the \({}^{3}D_{3}\)-state is very important. We note that (47) cannot be obtained from \({}^{3}D\)—\({}^{3}S\)-interference or from \(D_{1}\)—\(D_{2}\)-interference.

A very interesting preliminary result on n—p polarization was obtained at \(100\) MeV \(^{88,109}\), which can be written as

\[ (I_{0}P)_{\mathrm{np}}=\sin\theta\,[1.1\pm0.2+(3.5\pm0.7)\cos\theta]\ \mathrm{mb}/\mathrm{ster}. \tag{48} \]

If we assume that \((I_{0}P)_{\mathrm{pp}}\) at an energy of \(100\) MeV is smaller than at \(140\) MeV, then it gives a small contribution to (46), so that we shall be satisfied with a reasonable estimate of this quantity. We then obtain that, at an energy of \(100\) MeV,

\[ (I_{0}P)_{00}=-13\sin\theta\cos\theta\ \mathrm{mb}/\mathrm{ster}; \]

this result is apparently too large and, of course, very inaccurate, but it indicates a considerable noncentral interaction in \(T=0\) scattering at \(100\) MeV. Apparently it is most reasonable to attribute it chiefly to \({}^{3}S\)—\(D\)-interference \(^{103,104}\).

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