Spin Physics and Polarized Fusion: Where We Stand

Abstract

A summary of the present status of nuclear fusion is given with emphasis on utilizing spin-polarized particles as fuel. The reactions considered are those concerning the four- and five-nucleon systems and especially the D + D reactions for which the status of the theory and the experimental data are presented. Recent progress has been achieved by microscopic calculations of the D + D reactions. New aspects concern e.g. the increased cross-sections at very low energies by electron screening. The need to get more experimental data is pointed out.

Appendix

The cross-sections for the two cases spin-1 on spin-1/2 and spin-1 on spin-1 consist of the unpolarized cross-sections, a number of vector and tensor analyzing powers and spin-correlation coefficients. The coordinate system chosen is with the z axis in the incident momentum direction, the y axis in the direction of the normal to the scattering plane and the x axis forming a righthanded system with both. The Cartesian representation is chosen.

All “polarized” cross-sections are $\phi$ dependent and this dependence must be worked out for the contribution from each observable (for the determination of this “azimuthal complexity” see [4]) and is not given here.

All observables compatible with parity conservation and hermiticity are listed. For the special spin-1 on spin-1 case of deuterons on deuterons a number of terms are redundant due to the identity of the entrance-channel particles.

2.1.1 Spin-1 Beam on Spin-1/2 Target

$$\begin{aligned} \left( \frac{d\sigma (\varTheta ,\phi )}{d\varOmega }\right) _{\Phi =0}=\left( \frac{d\sigma (\varTheta )}{d\varOmega }\right) _0&\left\{ 1 +\frac{3}{2}A_y^{(b)}(\varTheta )p_y+A_y^{(t)}q_y+\frac{1}{2}A_{zz}^{(b)}(\varTheta )p_{zz}+\frac{2}{3} A_{xz}^{(b)}(\varTheta )p_{xz} \right. \nonumber \\&\;\;+ \frac{1}{6}A_{xx-yy}^{(b)}(\varTheta )p_{xx-yy}\nonumber \\&\;\;+ \frac{3}{2}\left[ C_{x,x}(\varTheta )p_xq_x + C_{y,y}(\varTheta )p_yq_y + C_{z,z}(\varTheta )p_zq_z \right. \nonumber \\&\qquad \quad + \left. C_{z,x}(\varTheta ) p_zq_x+C_{x,z}(\varTheta ) p_xq_z \right] \nonumber \\&\;\;+ \frac{2}{3}\left[ C_{xy,x}(\varTheta )p_{xy}q_{x}+C_{yz,x}(\varTheta )p_{yz}q_x \right. \nonumber \\&\qquad \quad + \left. C_{xz,y}(\varTheta )p_{xz}q_{y}+C_{xy,z}(\varTheta )p_{xy}q_z +C_{yz,z}(\varTheta )p_{yz}q_{z} \right] \nonumber \\&\;\;+ \left. \frac{1}{6} C_{xx-yy,y}(\varTheta )p_{xx-yy}q_y +\frac{1}{2} C_{zz,y}(\varTheta )p_{zz}q_y\right\} . \end{aligned}$$

(2.14)

The symbols b, t, p, and q designate the beam, the target, and the beam and target polarizations, respectively. Here 18 observables are measurable: one unpolarized cross-section, two vector and three tensor analyzing powers, and 12 correlation coefficients.

2.1.2 General Spin-1 on Spin-1 Case

$$\begin{aligned} \left( \frac{d\sigma (\varTheta ,\phi )}{d\varOmega }\right) _{\Phi =0} = \left( \frac{d\sigma (\varTheta )}{d\varOmega }\right) _0&\left\{ 1 + \frac{3}{2}\left[ A_y^{(b)}(\varTheta )p_y + A_y^{(t)}q_y\right] +\frac{1}{2}\left[ A_{zz}^{(b)}(\varTheta )p_{zz}+A_{zz}^{(t)}(\varTheta )q_{zz}\right] \right. \nonumber \\&\;\;+ \frac{1}{6}\left[ A_{xx-yy}^{(b)}(\varTheta )p_{xx-yy}+A_{xx-yy}^{(t)}(\varTheta )q_{xx-yy}\right] \nonumber \\&\;\;+ \frac{2}{3}\left[ A_{xz}^{(b)}(\varTheta )p_{xz}+A_{xz}^{(t)}(\varTheta )q_{xz}\right] \nonumber \\&\;\;+ \frac{9}{4}\left[ C_{y,y}(\varTheta )p_yq_y + C_{x,x}(\varTheta )p_xq_x + C_{x,z}(\varTheta )p_xq_z\right. \nonumber \\&\qquad \quad {\left. +C_{z,x}(\varTheta ) p_zq_x+C_{z,z}(\varTheta ) p_zq_z\right] }\nonumber \\&\;\;+\frac{3}{4}\left[ C_{y,zz}(\varTheta )p_yq_{zz}+C_{zz,y}(\varTheta )p_{zz}q_y\right] \nonumber \\&\;\;+ C_{y,xz}(\varTheta )p_yq_{xz}+C_{xz,y}(\varTheta )p_{xz}q_y +C_{x,yz}(\varTheta )p_{x}q_{yz}\nonumber \\&\;\;+ C_{yz,x}(\varTheta )p_{yz}q_{x}+C_{z,yz}(\varTheta )p_{z}q_{yz} +C_{yz,z}(\varTheta )p_{yz}q_{z}\nonumber \\&\;\;+\frac{1}{4}\left[ C_{y,xx-yy }(\varTheta )p_yq_{xx-yy} + C_{xx-yy,y}(\varTheta )p_{xx-yy}q_y \right. \nonumber \\&\qquad \quad {\left. +C_{zz,zz}(\varTheta )p_{zz}q_{zz}\right] }\nonumber \\&\;\;+\frac{1}{3}\left[ C_{zz,xz}(\varTheta )p_{zz}q_{xz}+C_{xz,zz}(\varTheta )p_{xz}q_{zz}\right] \nonumber \\&\;\;+\frac{1}{12}\left[ C_{zz,xx-yy}(\varTheta )p_{zz}q_{xx-yy}+C_{xx-yy,zz}(\varTheta )p_{xx-yy}q_{zz}\right] \nonumber \\&\;\;+\frac{4}{9}\left[ C_{xz,xz}(\varTheta )p_{xz}q_{xz}+C_{yz,yz}(\varTheta )p_{yz}q_{yz}\right] \nonumber \\&\;\;+\frac{8}{9}\left[ C_{xy,yz}(\varTheta )p_{xy}q_{yz}+C_{yz,xy}(\varTheta )p_{yz}q_{xy}\right] \nonumber \\&\;\;+\frac{16}{9} C_{xy,xy}(\varTheta )p_{xy}q_{xy} \nonumber \\&\;\;+\frac{1}{9}\left[ C_{xz,xx-yy}(\varTheta )p_{xz}q_{xx-yy}+C_{xx-yy,xz}(\varTheta )p_{xx-yy}q_{xz} \right] \nonumber \\&\;\;+\frac{1}{36} C_{xx-yy,xx-yy}(\varTheta )p_{xx-yy}q_{xx-yy} \nonumber \\&\;\;+ \frac{1}{2}\left[ C_{x,xy}(\varTheta )p_xq_{xy} + C_{xy,x}(\varTheta )p_{xy}q_x + C_{z,xy}(\varTheta )p_zq_{xy}\right. \nonumber \\&\qquad \quad {\left. +C_{xy,z}(\varTheta ) p_{xy}q_z\right] \left. \right\} }. \end{aligned}$$

(2.15)

Here the number of observables (in parentheses for D $+$ D) is 41 (24): one unpolarized cross-section, two (one) vector and six (three) tensor analyzing powers, and 32 (19) correlation coefficients.