Polarization Observables in R-Matrix Theory =========================================== This chapter develops the theory behind the vector analyzing power as AZURE2 computes it. It assumes the R-matrix formalism of Lane and Thomas but nothing about spin algebra beyond angular-momentum coupling. Why an analyzing power is worth measuring ----------------------------------------- A differential cross section is a sum of squared amplitudes. Squaring destroys phase, and phase is where much of the physics lives: whether two levels interfere constructively or destructively, whether a resonance is a :math:`3/2^-` or a :math:`5/2^+`, whether a broad background amplitude has the sign that a direct-capture calculation predicts. A cross section constrains these things only weakly, because a fit can usually trade a phase against a width and land on the same curve. A polarization observable is different. It is built from an *interference* between amplitudes that differ in the spin projection of the projectile, and it appears in the numerator of a ratio whose denominator is the cross section. The normalization cancels, the overall phase cancels, and what survives is precisely the relative phase. Where a cross section has a smooth maximum, an analyzing power can swing from :math:`+1` to :math:`-1` over a few tens of keV. This is why the elastic scattering of polarized protons from :sup:`12`\ C is used as a beam polarimeter, and why the analyzing power of :sup:`9`\ Be(p,d) or :sup:`9`\ Be(p,\ :math:`\alpha`) adds information that no amount of cross-section data supplies. For astrophysics the payoff is indirect but real: a spin and parity assignment that a polarization measurement settles is one fewer degree of freedom in the extrapolation of an S-factor to stellar energies. The spin structure of the scattering amplitude ---------------------------------------------- For spinless particles a single complex function :math:`f(\theta)` carries all the information, and :math:`d\sigma/d\Omega = |f|^2`. With spin, the amplitude becomes a matrix in spin space. Write the entrance channel spin as :math:`s` with projection :math:`\nu`, and the exit channel spin as :math:`s'` with projection :math:`\nu'`. The object we need is .. math:: M_{s'\nu' s\nu}(\theta), the amplitude for the system to enter with spin projection :math:`\nu` and leave with :math:`\nu'`. The channel spin is the vector sum of the two intrinsic spins, :math:`\mathbf{s} = \mathbf{j}_1 + \mathbf{j}_2`, so for a spin-1/2 projectile on a spin-0 target :math:`s = 1/2` and :math:`\nu = \pm 1/2` is just the projectile's own spin projection. Every observable follows from :math:`M` by a density-matrix construction. If the beam is described by a spin density matrix :math:`\rho_{\text{in}}`, the outgoing state is .. math:: \rho_{\text{out}} = M \rho_{\text{in}} M^{\dagger}, and any observable is a trace of :math:`\rho_{\text{out}}` against the operator that the detector measures. The unpolarized cross section is the trace with the identity, .. math:: \frac{d\sigma}{d\Omega} = \frac{1}{2s+1}\sum_{s'\nu'}\sum_{\nu} \left| M_{s'\nu' s\nu} \right|^2, averaging over entrance projections and summing over exit ones. The vector analyzing power is the trace with a Pauli matrix. Once :math:`M` exists there is no further angular-momentum algebra to do; this is the reason for organizing the calculation around it. Seyler's expression for the transition matrix --------------------------------------------- The step that makes this practical for an R-matrix code is due to R. G. Seyler [Seyler1969]_, whose Eq. (4) writes :math:`M` directly in terms of the Lane–Thomas collision matrix :math:`U` — the very object an R-matrix code already builds: .. math:: :label: seyler M_{s'\nu' s\nu}(\theta) = \frac{\sqrt{\pi}}{k}\Bigg[ -\,C(\theta)\,\delta_{ss'}\delta_{\nu\nu'} + i \sum_{J l l'} \sqrt{2l+1}\; (s\,l\,\nu\,0 | J\,\nu)\; (s'\,l'\,\nu'\,\nu-\nu' | J\,\nu)\\ \times\; e^{i(\omega_l + \omega_{l'})} \left( \delta_{ss'}\delta_{ll'} - U^J_{s'l' sl} \right) Y_{l'}^{\nu-\nu'}(\theta,0) \Bigg]. Each piece earns its place: *The Coulomb term* :math:`C(\theta)` is the Rutherford amplitude. It is diagonal in channel spin and its projection, since a pure Coulomb field does not touch spin, and it exists only when the entrance and exit pairs are the same. *The first Clebsch–Gordan coefficient* :math:`(s\,l\,\nu\,0|J\nu)` has the entrance orbital projection fixed at zero. This is the choice of quantization axis: the beam travels along :math:`\hat z`, so :math:`\mathbf{l}\cdot\hat z = 0` for the incoming wave, and the total projection is therefore :math:`\nu`, the channel-spin projection alone. *The second Clebsch–Gordan coefficient* :math:`(s'\,l'\,\nu'\,\nu-\nu'|J\nu)` enforces conservation of the total projection: whatever the exit channel spin takes as :math:`\nu'`, the exit orbital motion must carry :math:`\mu = \nu - \nu'`. *The spherical harmonic* :math:`Y_{l'}^{\nu-\nu'}` is the angular function of that outgoing orbital motion. This is the crucial structural point. For an unpolarized cross section only :math:`\mu = 0` ever appears and Legendre polynomials suffice — which is why AZURE2, like most R-matrix codes, had :math:`P_L(\cos\theta)` and nothing else. **Spin flip means** :math:`\nu \neq \nu'`, **which means** :math:`\mu \neq 0`, **which requires the associated Legendre functions.** No amount of rearrangement avoids this: a code that has only :math:`P_L` cannot produce a non-zero analyzing power, because the amplitudes that would interfere are not representable in it. *The bracket* :math:`\delta_{ss'}\delta_{ll'} - U^J_{s'l'sl}`, dressed with the Coulomb phases :math:`e^{i(\omega_l+\omega_{l'})}`, is the transition matrix. Section :doc:`polarization_implementation` shows that AZURE2 already forms exactly this quantity, phases included, so nothing has to be reconstructed. The vector analyzing power -------------------------- Specialize to a spin-1/2 projectile on a spin-0 target, which covers :math:`{}^{12}\mathrm{C}(\vec p, p)`. The channel spin is :math:`1/2`, and :math:`M` is a :math:`2\times 2` matrix in the projectile's spin projection for each exit configuration. Take the quantization axis for the polarization along .. math:: \hat{\mathbf{n}} = \frac{\mathbf{k}_{\text{in}} \times \mathbf{k}_{\text{out}}} {|\mathbf{k}_{\text{in}} \times \mathbf{k}_{\text{out}}|}, the normal to the scattering plane. This is the Madison convention, and it is the only sensible choice: parity conservation forbids a vector polarization along any other direction, so :math:`A_x = A_z = 0` identically and :math:`A_y` is the whole of the vector analyzing power. The observable is the trace of the outgoing density matrix against :math:`\sigma_y` acting in the *entrance* spin space, .. math:: A_y = \frac{\mathrm{Tr}\left( M \sigma_y M^{\dagger} \right)} {\mathrm{Tr}\left( M M^{\dagger} \right)}. Writing this out with :math:`(\sigma_y)_{+-} = -i` and :math:`(\sigma_y)_{-+} = +i` gives the form the code evaluates: .. math:: :label: ay A_y = \frac{2 \displaystyle\sum_{s'\nu'} \mathrm{Im}\!\left[\, M_{s'\nu',\,s\,+\frac{1}{2}}\; M^{*}_{s'\nu',\,s\,-\frac{1}{2}} \right]} {\displaystyle\sum_{s'\nu'} \left( \left|M_{s'\nu',\,s\,+\frac{1}{2}}\right|^2 + \left|M_{s'\nu',\,s\,-\frac{1}{2}}\right|^2 \right)}. The structure is worth pausing on. The numerator is an interference between the two entrance projections; the denominator is the cross section. If the two projections scatter identically the numerator vanishes. If they scatter differently but *in phase*, the imaginary part vanishes and :math:`A_y` is still zero. **A non-zero analyzing power requires both spin dependence and a relative phase**, which is exactly why it is a sharp probe of interference and why it peaks near resonances, where phases move fast. Four places where the analyzing power must vanish -------------------------------------------------- These are the checks that tell you an implementation is right, and each one fails differently when it is wrong. **At** :math:`\theta = 0` **and** :math:`\theta = 180^\circ`. The normal :math:`\hat{\mathbf{n}}` is undefined when :math:`\mathbf{k}_{\text{out}}` is parallel or antiparallel to :math:`\mathbf{k}_{\text{in}}`, so there is no direction for the polarization to point along. Formally, the spherical harmonic :math:`Y_{l'}^{\mu}` vanishes at :math:`\theta = 0` for every :math:`\mu \neq 0`, so no spin-flip amplitude survives. Every curve in Baumann's fig. 1 goes to zero at both ends. **In the pure Coulomb limit.** Rutherford scattering is spin-independent: the Coulomb term in :eq:`seyler` is diagonal in :math:`\nu`, so both projections acquire the same amplitude and the same phase. At energies well below any resonance :math:`A_y \to 0`. This is a useful trap — checking an implementation only at low energy will show a perfectly correct zero and prove nothing. **Without spin-orbit splitting.** If the nuclear interaction did not distinguish :math:`j = l + 1/2` from :math:`j = l - 1/2`, the sum over :math:`J` in :eq:`seyler` would collapse and the spin-flip amplitude would cancel. The analyzing power in :sup:`12`\ C(:math:`\vec p`,p) is large precisely because the :math:`p_{3/2}` and :math:`d_{5/2}` resonances are spin-orbit partners of states that lie elsewhere. **When only one partial wave contributes.** A single resonance with no background produces a common phase, which cancels in the ratio. The structure in :math:`A_y` between 1.6 and 1.8 MeV comes from the :math:`3/2^-` and :math:`5/2^+` resonances interfering *with each other* and with the non-resonant :math:`s_{1/2}` and :math:`p_{1/2}` phases. The bound :math:`|A_y| \le 1` follows from the Cauchy–Schwarz inequality applied to :eq:`ay` and is the cheapest sanity check available; a value outside :math:`[-1,1]` means the amplitude matrix is not a consistent set of amplitudes. The sign, and why it cannot be checked internally -------------------------------------------------- Every check listed above is invariant under :math:`A_y \to -A_y`. The bound holds, the zeros stay zeros, the Coulomb limit is still zero. The overall sign depends on a chain of conventions — the direction assigned to :math:`\hat{\mathbf{n}}`, the Condon–Shortley phase in the spherical harmonics, the ordering of the coupling in the Clebsch–Gordan coefficients — and getting any one of them backwards flips it without disturbing anything else. The sign therefore has to be fixed against measurement. The comparison used here is described in the next chapter: Baumann *et al.* mark four points where :math:`|A_y|` reaches unity, three positive and one negative, and reproducing that pattern — not merely the positions — is what pins the convention down. Analyzing power through a target of finite thickness ----------------------------------------------------- One subtlety has no analogue in cross-section work and is easy to get wrong. A cross section measured on a target of finite thickness is an integral: the beam loses energy as it goes, and the yield is .. math:: Y = \int_{E_{\text{back}}}^{E_{\text{surface}}} \frac{\sigma(E)}{\varepsilon(E)}\, dE with :math:`\varepsilon` the stopping power. An analyzing power is not an integral of this kind, because it is a *ratio*. Averaging :math:`A_y(E)` along the target with equal weight has no physical meaning. What the experiment returns is the ratio of the polarized and unpolarized yields, so the correct average is weighted by the cross section: .. math:: :label: aytarget \langle A_y \rangle = \frac{\displaystyle\int A_y(E)\,\sigma(E)\,dE} {\displaystyle\int \sigma(E)\,dE}. The depths where the reaction is likely contribute more, as they must. The consequence is quantitative and severe. In the :sup:`12`\ C + p system below 2 MeV the cross section is dominated by Rutherford scattering, which diverges as :math:`E^{-2}` at low energy, while :math:`A_y` there is essentially zero. A thick target therefore dilutes the analyzing power towards zero: for the gas target in the ``tests/13N`` evaluation, whose integration grid runs from 0.42 to 1.54 MeV, the weighted average of a resonant :math:`A_y \approx 0.8` comes out around :math:`10^{-6}`. This is physics, not a numerical artefact, and it is why analyzing-power measurements use thin targets — Baumann's were 85 nm of :sup:`12`\ C, about 3.1 keV of energy loss at 1.7 MeV. Scope ----- What is implemented covers particle channels with a spin-1/2 projectile and a target of **any** spin, which is what the vector analyzing power :math:`A_y` requires. Two extensions are not implemented and should not be assumed to work: *Tensor observables* (:math:`T_{20}`, :math:`T_{22}`, and the rest) need a spin-1 projectile and the corresponding rank-2 operators. The amplitude matrix built here is general enough in principle — it carries all channel spins and projections — but the observable side would have to be written. *Capture channels* need the photon analogue of :eq:`seyler`, for which the reference is Seyler and Weller [SeylerWeller1979]_. Their Eq. (12) carries a :math:`(-1)^M` phase that several earlier treatments drop, which is a warning about how delicate this bookkeeping is. .. [Seyler1969] R. G. Seyler, *Nuclear Physics* **A124** (1969) 253. .. [SeylerWeller1979] R. G. Seyler and H. R. Weller, *Physical Review C* **20** (1979) 453. .. [Baumann1992] R. Baumann *et al.*, *Nuclear Physics* **A542** (1992) 53.