Parity and Time-reversal Invariance Violation In Neutron-Nucleus Scattering

Let us start from a simple qualitative explanation of the origin of the enhanced parity violating effect. For simplicity, we will initially assume that nuclear spin is zero. Resonances which are excited by $\ell=0$ neutrons are called $s$-wave resonances and have positive parity, while resonances excited by $\ell=1$ neutrons are called $p$-wave resonances and have negative parity. In such a compound nucleus close to $p$-wave resonance, the wave function can be written as

$\displaystyle\psi_{p}^{\prime}=\psi_{p}+\sum_{s}\beta_{s}\psi_{s},$

(2)

with the mixing coefficient

$\displaystyle\beta_{s}=\frac{W_{sp}}{E_{p}-E_{s}},$

(3)

where $W_{sp}=\matrixelement{s}{W}{p}$ is the matrix element of the weak interaction mixing $s$-wave and $p$-wave resonances. We can present the parity violating effect in terms of the widths of the $s$-wave and $p$-wave resonances. Since the width $\Gamma^{n}$ is proportional to the amplitude squared, the amplitudes of $s$ and $p$-wave captures can be represented as $\sqrt{\Gamma_{s}^{n}}\eta_{s}$ and $i\sqrt{\Gamma_{p}^{n}}\eta_{p}$ respectively where $\eta_{s}$ and $\eta_{p}$ are the sign factors of each amplitude (equal to $\pm 1$). The cross section $\sigma$ is proportional to the square of the amplitude:

$\displaystyle\sigma_{\pm}\propto\left|\pm i\sqrt{\Gamma_{p}^{n}}\eta_{p}+\sum_{s}\frac{W_{sp}}{E_{p}-E_{s}}\sqrt{\Gamma_{s}^{n}}\eta_{s}\right|,$

(4)

where $\sigma_{\pm}$ are the cross sections for the positive and negative helicities. The $\pm$ sign in the $p$-wave amplitude corresponds to a positive or negative helicity, see Section II.2. It follows that the longitudinal asymmetry $P$ can be expressed as [1]

$\displaystyle P=\frac{\sigma_{+}-\sigma_{-}}{\sigma_{+}+\sigma_{-}}=2\sum_{s}\frac{iW_{sp}}{E_{s}-E_{p}}\sqrt{\frac{\Gamma_{s}^{n}}{\Gamma_{p}^{n}}},$

(5)

where the sign coefficients $\eta_{s}$ and $\eta_{p}$ have been moved inside the matrix element $W_{sp}$. Note, in the standard defintion of the angular wave functions, $W_{sp}$ is imaginary. A proper derivation of (5) including the case of a non-zero nuclear spin will be presented in Section II.2. Here we assume that the non-resonant part of $(\sigma_{+}+\sigma_{-})$ has been subtracted, as it has been done in the PV neutron transmission experiments.

Upon analysis of (5), two reasons for the enhancement of the PV effect can be identified. Firstly, in a nucleus excited by neutron capture, the interval $E_{s}-E_{p}$ between the chaotic compound states (resonances) of opposite parity is very small. This contribution is labelled dynamical enhancement, and it enhances the mixing of these states (by the weak PV interaction between nucleons) by three orders of magnitude. Secondly, the admixture between the large $s$-wave amplitudes $\sqrt{\Gamma_{s}^{n}}$ and the small $p$-wave amplitudes $\sqrt{\Gamma_{p}^{n}}$ allows neutron capture in the $s$-wave channel to contribute to the $p$-wave resonance. At small neutron energies the $s$-wave amplitude is three orders of magnitude larger than the p-wave amplitude ($\sqrt{\Gamma_{s}^{n}/\Gamma_{p}^{n}}\sim 10^{3}$). This contribution is referred to as kinematic enhancement. The ratio of the strength of the weak interaction to that of the strong interaction is $\sim 10^{-7}$. As a result of these two $10^{3}$ factors acting together, the factor of enhancement is as large as $\sim 10^{6}$, see [1, 2, 3, 4, 6].

The relative difference of the neutron cross sections for positive and negative helicities has been measured in a range of nuclei. These measurements have been done in various polarised neutron transmission experiments, by flipping the neutron spin orientation $\mathbf{s}$ from parallel to anti-parallel to neutron momentum $\mathbf{p}$ by a magnetic field pulse. Once the spin has been flipped, the number of neutrons passing through a material can be counted, i.e. measuring the correlation $\mathbf{s}\cdot\mathbf{p}$, see e.g. [9]. Further, there has also been measurements of PV correlations in neutron radiative capture, $(n,\gamma)$ reactions. The theory of these correlations is presented in [3, 4].

In this section, we follow the derivation in Ref. [3], in which the authors use a resonance diagrammatic technique to calculate the forward elastic scattering amplitude with parity violation $f_{\text{p.v}}$. Let us consider the parity violating effects in neutron optics. The angle of neutron spin rotation around the direction of motion in matter is

$\displaystyle\varphi=\frac{2\pi N_{0}l}{k}2\ \text{Re}f_{\text{p.v}},$

(6)

where $N_{0}$ is density of the target atoms, $l$ is the neutron path length, $k$ is the neutron momentum. The difference in total cross sections $\Delta\sigma$ for right and left handed polarised neutrons can be expressed in terms of $f_{\text{p.v}}$

$\displaystyle\Delta\sigma=\sigma_{+}-\sigma_{-}=\frac{4\pi}{k}2\ \text{Im}f_{\text{p.v}},$

(7)

The main contribution to the forward elastic scattering amplitudes comes from mixing by the PV weak interaction of the wave functions of compound resonances. Other contributions, such as neutron scattering on the parity violating potential of the nucleus are not enhanced by small energy denominators, and may be neglected.

We will now present a calculation of this resonance amplitude. In this derivation, we will skip the summation over all $s$-wave and $p$-wave resonances for brevity. Let $\mathbf{n}_{k}$ denote the neutron’s initial momentum direction and $I$ be the angular momentum of the initial nucleus. $\mathbf{J}=\mathbf{I}+\mathbf{j}$ is the spin of the compound resonance and $\mathbf{j}=\mathbf{l}+\mathbf{s}$ denotes the momentum of the $p$-wave neutron at which the capture occurs. Let us begin by writing down the wave-function of the incident neutron

$\displaystyle e^{ikr}|\chi_{\alpha}>$

$\displaystyle=\sum_{l,m}4\pi i^{l}j_{l}(kr)Y^{*}_{lm}(\mathbf{n}_{k})Y_{lm}(\mathbf{n}_{k})|\chi_{\alpha}>,$

(8)

where $|\chi_{\alpha}>$ is the neutron spinor with spin projection $\alpha$ and $j_{l}(kr)$ is the spherical Bessel function. In neutron-nucleus scattering, the capture amplitude of the neutron into the $s$-resonance is

$\displaystyle C^{JJ_{z}}_{II_{z}\frac{1}{2}\alpha}\eta_{s}\sqrt{\Gamma_{s}^{n}(E)},$

(9)

while the capture amplitude of the neutron into the $p$-resonance is

$\displaystyle\sum_{jj_{z}m}C^{JJ_{z}}_{II_{z}jj_{z}}C^{jj_{z}}_{1m\frac{1}{2}\alpha}\sqrt{4\pi}\ Y_{1m}^{*}(\mathbf{n}_{k})i\eta_{j}\sqrt{\Gamma_{p_{j}}^{n}(E)},$

(10)

where $C^{JJ_{z}}_{II_{z}\frac{1}{2}\alpha}$ is the Clebsch-Gordon coefficient; $\Gamma_{pj}^{n}$ is the neutron width corresponding to the emission of a neutron with momentum $j$ and $\eta_{s,j}=\pm 1$ is, as defined above, the sign of the amplitude [3]. We further define the Green function of a compound nucleus

$\displaystyle\frac{1}{E-E_{c}+\frac{1}{2}i\Gamma_{c}}.$

(11)

Now, in conjunction with the above rules, we may write the forward elastic scattering amplitude near the $s$-resonance [3];

$\displaystyle f(0)=-\frac{1}{2k}C_{II_{z}\frac{1}{2}\alpha}^{JJ_{z}}\sqrt{\Gamma_{s}^{n}(E)}\frac{1}{E-E_{s}+\frac{1}{2}i\Gamma_{s}}C_{II_{z}\frac{1}{2}\alpha}^{JJ_{z}}\sqrt{\Gamma_{s}^{n}(E)},$

(12)

where $-1/2k$ is the common factor for the scattering amplitudes, given the neutron momentum $k$. Next, summing over $J_{z}$ (using the standard relations for the coupling of angular momenta) and averaging over $I_{z}$, we obtain the standard Breit-Wigner Formula [3]

$\displaystyle f(0)=-\frac{1}{2k}\frac{g\Gamma_{s}^{n}(E)}{E-E_{s}+\frac{1}{2}i\Gamma_{s}},$

(13)

where the factor

$\displaystyle g=\frac{(2J+1)}{2(2I+1)},$

(14)

appears after averaging over the initial nucleus’ spin projections. Performing a similar calculation for the $p$-wave resonance yields

$\displaystyle\begin{split}f(0)&=-\frac{1}{2k}\sum_{jj_{z}m\atop\tilde{j}\tilde{j_{z}}\tilde{\dot{m}}}C_{II_{z}jj_{z}}^{JJ_{z}}C_{1m_{2}^{1}\alpha}^{jj_{z}}\sqrt{4\pi}Y_{1m}^{*}\left(\mathbf{n}_{k}\right)\sqrt{\Gamma_{p_{j}}^{n}(E)}\\ &\times\frac{1}{E-E_{p}+\frac{1}{2}\Gamma_{p}}C_{II_{z}\tilde{j}\tilde{j_{z}}}^{JJ_{z}}C_{1m_{2}^{1}\alpha}^{\tilde{j}\tilde{j_{z}}}\sqrt{4\pi}Y_{1\tilde{m}}\left(\mathbf{n}_{k}\right)\sqrt{\Gamma_{p\tilde{j}}^{n}(E)},\end{split}$

(15)

which, after summation over $J_{z}$ and averaging over $I_{z}$ is again equal to

$\displaystyle f(0)=-\frac{1}{2k}\frac{g\Gamma_{p}^{n}(E)}{E-E_{p}+\frac{1}{2}i\Gamma_{p}},$

(16)

where $\Gamma_{p}^{n}=\Gamma_{p_{1/2}}^{n}+\Gamma_{p_{3/2}}^{n}$. Using the above calculations as guides, we may now write down the forward elastic scattering amplitude with parity violation

$\displaystyle\begin{split}f_{\mathrm{p.v.}}(0)&=-\frac{2}{2k}C_{II_{z}\frac{1}{2}\alpha}^{JJ_{z}}\eta_{s}\sqrt{\Gamma_{s}^{n}(E)}\frac{1}{E-E_{s}+\frac{1}{2}i\Gamma_{s}}\times W_{sp}\frac{1}{E-E_{p}+\frac{1}{2}\Gamma_{p}}\\ \times&(-i)\sum_{jj_{z}m}C_{II_{z}jj_{z}}^{JJ_{z}}C_{1m\frac{1}{2}\alpha}^{ij_{z}}\sqrt{4\pi}Y_{1m}\left(\mathbf{n}_{k}\right)\eta_{j}\sqrt{\Gamma_{pj}^{n}(E)},\end{split}$

(17)

where $W_{sp}$ is the weak interaction matrix element between compounds states. Once again, we may sum over $J_{z}$ and average over $I_{z}$ to obtain [3]

$\displaystyle f_{\mathrm{p.v.}}(0)=\pm\frac{1}{2k}\frac{2g\sqrt{\Gamma_{s}^{n}(E)}iW_{sp}\sqrt{\Gamma_{p\frac{1}{2}}^{n}(E)}\eta_{s}\eta_{\frac{1}{2}}}{\left(E-E_{s}+\frac{1}{2}i\Gamma_{s}\right)\left(E-E_{p}+\frac{1}{2}i\Gamma_{p}\right)}.$

(18)

The sign of this expression corresponds to the positive or negative helicity neutron (respectively). We also note that the term with $j=3/2$ vanishes after summation over $J_{z}$. The sign factor $\eta_{s}\eta_{1/2}$ can be excluded by means of the redefinition of the states $s$ and $p$ (i.e. we introduce it into the matrix element $W_{sp}$).

Note that all neutron widths are energy dependent, and should be taken at neutron energy $E$. If the energy $E$ is close to $p$-wave resonance, then they must take the form $\Gamma_{s}^{n}(E_{p})$ and $\Gamma_{p}^{n}(E_{p})$.

Equation (18) can now be related to the total cross section via the optical theorem

$\displaystyle\sigma=\frac{4\pi}{k}\text{Im}f(0).$

(19)

Upon application of the optical theorem to the forward scattering amplitude with parity violation, we yield the predicted longitudinal asymmetry (5).

In this section, we will present our calculation of the forward scattering amplitude of neutron-nucleus scattering with time and parity violation. In the case of PTRIV effects, the spin of the target nucleus and neutron momentum are perpendicular. As such, it is convenient to set the nuclear target spin $I$ along the $z$-axis, thus $I=I_{z}$. Neutron spin is along $x$-axis, so the spinor has equal amplitudes 1/2 and -1/2 along $z$ -axis, $\frac{1}{\sqrt{2}}[\ket{s_{z}=1/2}+\ket{s_{z}=-1/2}]$, and neutron momentum is along $y$ axis, therefore we substitute $\theta=\pi/2$ and $\phi=\pi/2$ as the arguments of the $\ Y_{1m}^{*}(\theta,\phi)$ in Eq. (10). In this case, we have contributions to the capture amplitude from both the $s$-wave and $p$-wave resonances, with a total spin of $J=I+1/2$ or $J=I-1/2$. We will make the substitutions $M_{s}=\sqrt{\Gamma_{s}^{n}}\eta_{s},M_{p,j}=\sqrt{\Gamma_{p,j}^{n}}\eta_{j}$ for brevity. Following the method described above, we can write the time and parity violating amplitude $f_{\text{t.p.v}}$

$\displaystyle f_{\text{t.p.v}}$

$\displaystyle=\pm\frac{1}{k}\frac{M_{s}W_{sp}^{T,P}(\alpha_{1/2,J}M_{p,1/2}+\alpha_{3/2,J}M_{p,3/2})}{\left(E-E_{s}+\frac{1}{2}i\Gamma_{s}\right)\left(E-E_{p}+\frac{1}{2}i\Gamma_{p}\right)},$

(20)

where the angular coefficients $\alpha_{i.J}$ are

$\displaystyle\begin{split}\alpha_{1/2,J=I+1/2}&=\frac{I}{2I+1},\\ \alpha_{3/2,J=I+1/2}&=-\frac{\sqrt{I(2I+3)}}{2(2I+1)},\\ \alpha_{1/2,J=I-1/2}&=\frac{I}{2I+1},\\ \alpha_{3/2,J=I-1/2}&=-\frac{I}{2(2I+1)}\sqrt{\frac{2I-1}{I+1}}.\end{split}$

(21)

The calculation of this amplitude is presented in Appendix A.1. In this amplitude we should sum over s-wave resonances (if the energy is close to a p-wave resonance). However, for comparison with Ref. [79] it is instructive to present the ratio of the two scattering amplitudes $f_{\text{t.p.v}}$ (20) and $f_{\text{p.v}}$ (18) taking into account only one s-wave resonance:

$\displaystyle\frac{f_{\text{t.p.v}}}{f_{\text{p.v}}}$

$\displaystyle=\kappa\frac{W_{sp}^{T,P}}{W_{sp}},$

(22)

where the angular coefficient of the ratio $\kappa$ includes amplitudes of the partial neutron widths which depend on spin channels $J=I\pm 1/2$, and can be determined via our calculations above;

$\displaystyle\kappa(I\pm 1/2)=\frac{\left(\alpha_{1/2,J}M_{p,1/2}+\alpha_{3/2,J}M_{p,3/2}\right)}{gM_{p,1/2}},$

(23)

where $g$ is as defined by Equation (14),

$\displaystyle g=\frac{2J+1}{2(2I+1)}=\begin{cases}\frac{I+1}{2I+1},&J=I+\frac{1}{2},\\ \frac{I}{2I+1},&J=I-\frac{1}{2}.\end{cases}$

(24)

Hence, calculations yield

	$\displaystyle\begin{split}\kappa(I+1/2)&=\frac{I}{I+1}-\frac{M_{p,3/2}}{M_{p,1/2}}\frac{\sqrt{I(2I+3)}}{2(I+1)},\end{split}$			(25)
	$\displaystyle\begin{split}\kappa(I-1/2)&=1-\frac{M_{p,3/2}}{M_{p,1/2}}\frac{\sqrt{2I-1}}{2\sqrt{I+1}}.\end{split}$			(26)

Our results for the factors $\kappa(I\pm 1/2)$ seem to be in agreement with the values calculated by in Ref.  [79], however their calculations do not include the sign factors $\eta_{s},\eta_{j}$ from the amplitudes $M_{s}=\sqrt{\Gamma_{s}^{n}}\eta_{s}$ and $M_{p,j}=\sqrt{\Gamma_{p,j}^{n}}\eta_{j}$ (their results are expressed in terms of $\sqrt{\Gamma_{s}^{n}}$ and $\sqrt{\Gamma_{p,j}^{n}}$ assuming that both $M_{s}$ and $M_{p,j}$ are positive; this assumption is not justified). Ref.  [79] limits their calculation to one s-wave resonance. This approximation is not justified in the case of the statistical theory. Finally, the experiments measure the relative difference $P$ of the number of passing neutrons for opposite orientation of neutron spin. The relative difference contains the $p$-wave amplitude in the denominator, which also should be calculated.

Now, in a similar way, we may also calculate the parity violating forward scattering amplitude for a polarised target. In this case, we require neutron momentum and spin to be parallel, and both perpendicular to the nuclear target spin. Setting once again nuclear target spin along the $z$-axis, we have $I=I_{z}$. We once again have neutron spin along the $x$-axis, meaning the spinor is $\frac{1}{\sqrt{2}}[\ket{s_{z}=1/2}+\ket{s_{z}=-1/2}]$. However now, we require neutron momentum to also be along the $x$-axis, and thus we substitute $\theta=\pi/2$ and $\phi=0$ as the arguments of the $Y_{1m}^{*}(\theta,\phi)$. Using a similar method to Section II.3, we yield the following for the forward scattering amplitude with parity violation with a polarised target (see Appendix A.2)

$\displaystyle f_{\text{p.v}}^{\text{P}}$

$\displaystyle=\pm\frac{1}{k}\frac{M_{s}iW_{sp}(\delta_{1/2,J}M_{p,1/2}+\delta_{3/2,J}M_{p,3/2})}{\left(E-E_{s}+\frac{1}{2}i\Gamma_{s}\right)\left(E-E_{p}+\frac{1}{2}i\Gamma_{p}\right)},$

(27)

where the angular coefficients $\delta_{i.J}$ are

$\displaystyle\begin{split}\delta_{1/2,J=I+1/2}&=-\frac{I+1}{2I+1},\\ \delta_{3/2,J=I+1/2}&=-\frac{2I-1}{2(2I+1)}\sqrt{\frac{I}{2I+3}},\\ \delta_{1/2,J=I-1/2}&=-\frac{I}{2I+1},\\ \delta_{3/2,J=I-1/2}&=\frac{I}{2(2I+1)}\sqrt{\frac{2I-1}{I+1}}.\end{split}$

(28)

Now we can present the ratio of the PTRIV and PV forward scattering amplitudes for a polarised target. We start from the results in the two-resonance approximation:

$\displaystyle\frac{f_{\mathrm{t.p.v}}}{f_{\mathrm{p.v}}^{P}}=\frac{W_{sp}^{T,P}}{iW_{sp}}\left[\frac{\alpha_{1/2,J}M_{p,1/2}+\alpha_{3/2,J}M_{p,3/2}}{\delta_{1/2,J}M_{p,1/2}+\delta_{3/2,J}M_{p,3/2}}\right].$

(29)

Specifically, for $J=I+1/2$, we have

$\displaystyle\begin{split}\frac{f_{\mathrm{t.p.v}}}{f_{\mathrm{p.v}}^{P}}=\\ \frac{W_{sp}^{T,P}}{iW_{sp}}\frac{-2I\sqrt{2I+3}M_{p,1/2}+(2I+3)\sqrt{I}M_{p,3/2}}{2(I+1)\sqrt{2I+3}M_{p,1/2}+(2I-1)\sqrt{I}M_{p,3/2}},\end{split}$

(30)

while for $J=I-1/2$ we have a very simple result,

$\displaystyle\frac{f_{\mathrm{t.p.v}}}{f_{\mathrm{p.v}}^{P}}=-\frac{W_{sp}^{T,P}}{iW_{sp}}$

(31)

A more accurate treatment requires summation over $s$-wave resonances in the numerator and denominator since the PTRIV and PV matrix elements are not proportional to each other (according to our statistical theory calculation [44], the relative correlator between $W_{sp}^{T,P}$ and $W_{sp}$ is 0.1). Therefore, in the above expressions, we should make the substitution

$\displaystyle\frac{W_{sp}^{T,P}}{iW_{sp}}\,\,\rightarrow\,\,\frac{\sum_{s}A_{sp}W_{sp}^{T,P}}{\sum_{s}A_{sp}iW_{sp}}\,,$

(32)

where

$\displaystyle A_{sp}=2\frac{1}{E_{s}-E_{p}}\sqrt{\frac{\Gamma_{s}^{n}}{\Gamma_{p}^{n}}}.$

(33)

The width $\Gamma_{p}^{n}$ is a common factor and may be cancelled. However, after statistical averaging the ratio of PTRIV and PV effects for $J=I-1/2$ will again be given by a simple expression containing only root-mean-squared values of the PTRIV and PV matrix elements (see below). Eq. (31) is also valid for the octupole doublet and doorway state mechanisms (the permanent sign contribution) where the two-level approximation is justified (see below).

We will now present our results for the $p$-wave amplitude in the configurations above which use a polarised target. Firstly, let us consider the case when neutron momentum, neutron spin and target spin are all perpendicular to each other, i.e. for the configuration presented in section II.3. This calculation was performed using the method presented in Section II.2, from which we yield (see Appendix A.3)

$\displaystyle f_{p}=-\frac{1}{2k}\frac{\beta_{1,J}M_{p,1/2}^{2}+\beta_{13,J}M_{p,1/2}M_{p,3/2}+\beta_{3,J}M_{p,3/2}^{2}}{E-E_{p}+\frac{1}{2}i\Gamma_{p}},$

(34)

where, in the case when $J=I+1/2$

$\displaystyle\begin{split}\beta_{1,J=I+1/2}&=\frac{I+1}{2I+1}=g,\\ \beta_{13,J=I+1/2}&=\frac{\sqrt{I}}{\sqrt{2I+3}}\frac{2I-1}{2I+1},\\ \beta_{3,J=I+1/2}&=\frac{2I^{2}+5I+9}{2(2I+3)(2I+1)},\end{split}$

(35)

and in the case when $J=I-1/2$

$\displaystyle\begin{split}\beta_{1,J=I-1/2}&=\frac{I}{2I+1}=g,\\ \beta_{13,J=I-1/2}&=-\frac{I}{2I+1}\sqrt{\frac{2I-1}{I+1}},\\ \beta_{3,J=I-1/2}&=\frac{I(I+4)}{(2I+1)(2I+3)}.\end{split}$

(36)

Next, we consider the case when neutron momentum and neutron spin are parallel to each other, and both perpendicular to the nuclear target spin (the configuration presented in section II.4). A similar calculation verifies that the $p$-wave amplitude in this configuration coincides with Eq. (34) above.

The expressions presented above contain the unknown ratio of the $p_{1/2}$ and $p_{3/2}$ capture amplitudes, $T=M_{p,3/2}/M_{p,1/2}$. This ratio may be extracted via measurement of the ratio of the total $p$-wave resonance cross sections (transmission probabilities) for the polarised and unpolarised target, i.e. from the ratio of the forward scattering amplitudes in Eqs. (34) and (16):

$$\frac{\sigma(polarised)}{\sigma(unpolarised)}=\frac{1}{g}(\beta_{1,J}R_{1/2}^{2}+\beta_{13,J}R_{1/2}R_{3/2}+\beta_{3,J}R_{3/2}^{2})\,.$$

(37)

where

$\displaystyle R_{j}=\frac{M_{p,j}}{\sqrt{M^{2}_{p_{1/2}}+M^{2}_{p_{3/2}}}}.$

(38)

Here $M_{p,1/2}^{2}+M_{p,3/2}^{2}=M_{p}^{2}=\Gamma_{p}^{n}$. Note that $R_{1/2}^{2}+R_{3/2}^{2}=1$. This ratio $T=M_{p,3/2}/M_{p,1/2}$ may also be extracted from the ratio of the PV forward scattering amplitudes for a polarised and unpolarised target, which is equal to

$$\frac{\Delta\sigma(polarised)}{\Delta\sigma(unpolarised)}=\frac{1}{g}\left[(\delta_{1/2,J}+\delta_{3/2,J}T)\right].$$

(39)

In this section, we follow Bowman et al. [81] in their analysis using the statistical mechanism of parity violation in nuclear resonances, the effects of which are produced by the mixing of opposite parity states. As such, individual weak matrix elements between $s$- and $p$-wave resonances can be considered to be mean-zero Gaussian random variables, with variance $W^{2}=\expectationvalue{|W_{sp}|^{2}}$.

Let us first consider the case of targets with spin $I=0$. In such cases, we have for $s$-wave resonances, $J=1/2^{+}$, while for $p$-wave resonances, $J=1/2^{-},3/2^{-}$. As $p$-wave resonances with $3/2^{-}$ cannot mix with the $1/2^{+}$ $s$-wave levels via a parity violating interaction, these levels will not show parity violation.

For a given $p$-wave level, the observed asymmetry has contributions from many $s$-wave levels, as per Eq. (5). We may rewrite this equation as

$\displaystyle P=\sum_{s}iA_{sp}W_{sp},$

(40)

where

$\displaystyle A_{sp}=2\frac{1}{E_{s}-E_{p}}\sqrt{\frac{\Gamma_{s}^{n}}{\Gamma_{p}^{n}}},$

(41)

in order to separate factors which are taken from experimental data and not affected by averaging. Squaring Equation (40) yields

$\displaystyle\begin{split}P^{2}&=\left|\sum_{s}A_{sp}W_{sp}\right|^{2}\\ &=\sum_{s}A_{sp}^{2}|W_{sp}|^{2}+\sum_{s\neq i}A_{ip}A_{sp}W_{ip}W_{sp},\\ \frac{P}{A_{J}^{2}}&=\frac{1}{A_{J}^{2}}\sum_{s}A_{sp}^{2}|W_{sp}|^{2}+\frac{1}{A_{J}^{2}}\sum_{s\neq i}A_{ip}A_{sp}W_{ip}W_{sp},\end{split}$

(42)

where we define $A_{J}^{2}\equiv\sum_{s}A_{sp}^{2}$. Now, given that each $W_{sp}$ is statistically independent, upon averaging over matrix elements the cross terms vanish. We can rewrite this expression in terms of $W$ [81]

$\displaystyle W^{2}$

$\displaystyle=\expectationvalue{\frac{P^{2}}{A_{J}^{2}}}.$

(43)

Thus, we conclude that each $P/A_{J}$ is also a Gaussian random variable, with mean zero and variance $|W|^{2}$. This means that given several experimental measurements for the quantity $P$, $\expectationvalue{|W_{sp}|^{2}}\equiv W^{2}$ can be extracted.

Let us now consider neutron scattering from an unpolarised target with nonzero spin, $I\neq 0$. Given parity $\pi$, these targets have spin and parity $(I\pm 1/2)^{\pi}$ in $s$-wave levels and $(I\pm 1/2)^{-\pi},(I\pm 3/2)^{-\pi}$ in $p$-wave levels. As the weak interaction is a scalar, angular momentum conservation implies only $I\pm 1/2$ has a nonzero P-odd effect. Only the $j=1/2$ $p$-wave capture amplitude contributes to the effects of parity violation, see Section II.2. The projectile $\mathbf{j}$ is further coupled to the spin of the target nucleus $\mathbf{I}$ to form the total spin $\mathbf{J}=\mathbf{j}+\mathbf{I}$. Once again defining $M_{s}\equiv\eta_{s}\sqrt{\Gamma_{s}^{n}}$ and $M_{p}\equiv\eta_{p}\sqrt{\Gamma_{p}^{n}}$ as the neutron decay amplitudes of the levels $s$ and $p$ respectively, we can rewrite Eq. (5) as (see Section II.2)

$\displaystyle P=R_{1/2}\sum_{s}\frac{2iW_{sp}}{E_{s}-E_{p}}\sqrt{\frac{\Gamma_{s}^{n}}{\Gamma_{p}^{n}}},$

(44)

where $R_{1/2}$ is given by Eq. (38). Let us rewrite Equation (44) in the form

$\displaystyle P$

$\displaystyle=\sum_{s}i\tilde{W}_{sp}A_{sp},$

(45)

where $\tilde{W}_{sp}\equiv W_{sp}R_{1/2}$ and coefficients $A_{sp}$ are given by Eq. (33). Thus, in a similar method to the case when $I=0$, we obtain

$\displaystyle\expectationvalue{|\tilde{W}_{sp}|^{2}}$

$\displaystyle=\expectationvalue{\frac{P^{2}}{A_{J}^{2}}},$

(46)

where $A_{J}^{2}\equiv\sum_{s}A_{sp}^{2}$. Assuming that $W_{sp}$ and $R_{1/2}$ are statistically independent, their product can be averaged separately. Then, the average squared matrix elements between compound states are

$\displaystyle W^{2}$

$\displaystyle=\frac{1}{\expectationvalue{R_{1/2}^{2}}}\expectationvalue{\frac{P^{2}}{A_{J}^{2}}}.$

(47)

$\expectationvalue{R_{1/2}^{2}}$ can be extracted experimentally, and for masses near $A=110$, it typically lies between 0.6 and 0.8 [9].

In this section, we consider the PTRIV configuration, and perform a similar calculation to that above. Firstly, we note that the $p$-wave forward scattering amplitude for neutron momentum, neutron spin and target spin perpendicular to each other, does not coincide with Eq. (16). As per the result presented in section II.6, the $p$-wave forward scattering amplitude in this case is given by Equation (34). Using the calculation of the PTRIV amplitude in section II.3, we can now write PTRIV effect in a similar form to that of the unpolarised P-odd, T-even effect (5)

$\displaystyle\begin{split}P_{T,P}&=\left[\frac{(\alpha_{1/2,J}M_{p,1/2}+\alpha_{3/2,J}M_{p,3/2})M_{p}}{\beta_{1,J}M_{p,1/2}^{2}+\beta_{13,J}M_{p,1/2}M_{p,3/2}+\beta_{3,J}M_{p,3/2}^{2}}\right]\sum_{s}\frac{W^{T,P}_{sp}}{E_{s}-E_{p}}\sqrt{\frac{\Gamma_{s}^{n}}{\Gamma_{p}^{n}}},\\ \end{split}$

(48)

where $\alpha_{p,J}$ are the angular factors for the PTRIV amplitude in Eq. (21) and $W_{sp}^{T,P}$ is the matrix element of the PTRIV interaction. We can further rewrite this equation in the familiar form

$\displaystyle P_{T,P}=\sum_{s}\tilde{W}_{sp}^{T,P}A_{sp},$

(49)

where $A_{sp}$ is as defined in (33), and

$\displaystyle\tilde{W}_{sp}^{T,P}=\left[\frac{\alpha_{1/2,J}R_{1/2}+\alpha_{3/2,J}R_{3/2}}{\beta_{1,J}R_{1/2}^{2}+\beta_{13,J}R_{1/2}R_{3/2}+\beta_{3,J}R_{3/2}^{2}}\right]W_{sp}^{T,P},$

where $R_{j}$ is as defined in Eq. (38). Once again, we aim to determine the average of this relation. In the same way as above, we yield

$\displaystyle\expectationvalue{(\tilde{W}_{sp}^{T,P})^{2}}=\expectationvalue{\frac{P_{T,P}^{2}}{A_{J}^{2}}}.$

(50)

Assuming that $R_{1/2},R_{3/2}$ and $W_{sp}^{T,P}$ are statistically independent, the products $R_{1/2}^{2}W_{sp}^{T,P}$ and $R_{3/2}^{2}W_{sp}^{T,P}$ can be averaged separately, and using the same method as Section III.2, we may determine the average squared matrix elements of the T,P-odd interaction. Neglecting the terms linear in $R_{1/2}$ or $R_{3/2}$ (as they have a random sign), assuming that $\expectationvalue{R_{j}^{4}}\approx\expectationvalue{R_{j}^{2}}^{2}$ and averaging the numerator and denominator separately gives

$\displaystyle\begin{split}\expectationvalue{(\tilde{W}_{sp}^{T,P})^{2}}&\approx\left[\frac{\alpha_{1/2,J}^{2}\expectationvalue{R_{1/2}^{2}}+\alpha_{3/2,J}^{2}\expectationvalue{R_{3/2}^{2}}}{\beta_{1,J}^{2}\expectationvalue{R_{1/2}^{2}}^{2}+\beta_{13,J}^{2}\expectationvalue{R_{1/2}^{2}}\expectationvalue{R_{3/2}^{2}}+\beta_{3,J}^{2}\expectationvalue{R_{3/2}^{2}}^{2}}\right]\expectationvalue{(W_{sp}^{T,P})^{2}}.\end{split}$

(51)

Thus, we conclude that the average squared matrix elements of time and parity violating effects are