Intensities, Moments and SDMEs — Script and Notebook

This repository contains a Wolfram Language script (SymbolicTest.wl) and a companion notebook (SymbolicTest.nb) for analytically generating spin–density matrix elements $\rho^{\alpha,ll'}_{mm'}$ (SDMEs), moments $H(LM)$, and intensities $I(\theta,\phi)$. Both produce LaTeX outputs collected under a local results/ directory. Can be edited to produce raw terminal/notebook outputs, but LaTeX outputs were chosen for the ease of directly exporting the results. Of note, our partial-wave expansions are always given in terms of the reflectivity basis and our target/nucleon helicities are given in terms of the index $k$. Here, the $k$-basis just refers to either the non-spin-flip $k=0$ state ($++$ or $--$) or the spin-flip $k=1$ state ($+-$ or $-+$).

Files

• SymbolicTest.wl: Runnable Wolfram Language script (CLI) with interactive prompts.
• SymbolicTest.nb: Interactive Mathematica notebook for exploration and generation.
• results/: Created on first run; contains LaTeX outputs:
  • MomentsAnalytic.tex: Contains the moments in terms of SDMEs
  • sdmes_expanded.tex: Contains the SDMEs, expanded in terms of the partial-waves
  • MomentsWaves.tex: Contains the moments, expanded in terms of partial-waves
  • IntensitiesAnalytic.tex: Contains the intensities in terms of the SDMEs and the corresponding angular functions

Requirements

• Wolfram Language/Mathematica or Wolfram Engine
• wolframscript available on PATH (for CLI script runs)

Run — Script (SymbolicTest.wl)

You can run the script either interactively (it prompts for lmax and lprimemax) or non‑interactively by piping values.

Interactive (prompts):

# Option A: run directly if executable
chmod +x SymbolicTest.wl
./SymbolicTest.wl

# Option B: via wolframscript
wolframscript -file SymbolicTest.wl

Non‑interactive (example with lmax = 0, lprimemax = 0):

printf "0\n0\n" | wolframscript -file SymbolicTest.wl

Outputs are written to results/ next to the script. The script prints progress messages and the export locations when done.

Note: The script currently computes and writes:

• Moments H in terms of SDMEs (MomentsAnalytic.tex)
• Expanded SDMEs in terms of partial waves (sdmes_expanded.tex)
• Moments H explicitly expanded in partial waves (MomentsWaves.tex)
• Analytic intensities $I(\theta,\phi)$ in terms of SDMEs + angular terms (IntensitiesAnalytic.tex)

Results directory resolution (CLI vs notebook)

The file export section creates results/ using NotebookDirectory[]. When running as a pure script (no notebook front end), NotebookDirectory[] may be unavailable. If you encounter an error creating the results folder, consider changing the results directory line in SymbolicTest.wl to this more robust variant:

resultsDir = FileNameJoin[{
  If[StringQ[$InputFileName] && $InputFileName != "",
     DirectoryName[$InputFileName],
     Directory[]
  ],
  "results"
}];
If[!DirectoryQ[resultsDir], CreateDirectory[resultsDir]];

If you want, I can apply this patch for you.

Run — Notebook (SymbolicTest.nb)

1. Open SymbolicTest.nb in Mathematica.
2. Set the values of lmax and lprimemax in the parameter cell (or at the prompts, if present).
3. Evaluate all cells (Kernel → Evaluation → Evaluate Notebook).
4. The outputs will be exported to the results/ directory next to the notebook.

Example: lmax = lprimemax = 0

This is the smallest configuration (S‑wave only, m = 0). You can generate it with either the script or the notebook.

Script (non‑interactive):

printf "0\n0\n" | wolframscript -file SymbolicTest.wl
ls -1 results/

Expected files:

IntensitiesAnalytic.tex
MomentsAnalytic.tex
MomentsWaves.tex
sdmes_expanded.tex

You can open these .tex files directly, copy the contents over or include them via \input{results/<file>.tex} in a LaTeX document.

Sample Output for lmax = lprimemax = 0

MomentsAnalytic.tex (excerpt):

% Auto-generated H^{\alpha}(L,M)
 \begin{align*} 
  H^{0}(00)  &= \rho _{00}^{0,00}\\ 
  H^{1}(00)  &= -\rho _{00}^{1,00}\\ 
  H^{2}(00)  &= -\rho _{00}^{2,00}\\ 
  H^{3}(00)  &= -\rho _{00}^{3,00}\\ 
  H^{4}(00)  &= -\rho _{00}^{4,00}\\ 
  H^{5}(00)  &= -\rho _{00}^{5,00}\\ 
  H^{6}(00)  &= -\rho _{00}^{6,00}\\ 
  H^{7}(00)  &= -\rho _{00}^{7,00}\\ 
  H^{8}(00)  &= -\rho _{00}^{8,00}\\ 
\end{align*}

$$% Auto-generated H^{\alpha}(L,M) \begin{align*} H^{0}(00) &= \rho _{00}^{0,00}\\ H^{1}(00) &= -\rho _{00}^{1,00}\\ H^{2}(00) &= -\rho _{00}^{2,00}\\ H^{3}(00) &= -\rho _{00}^{3,00}\\ H^{4}(00) &= -\rho _{00}^{4,00}\\ H^{5}(00) &= -\rho _{00}^{5,00}\\ H^{6}(00) &= -\rho _{00}^{6,00}\\ H^{7}(00) &= -\rho _{00}^{7,00}\\ H^{8}(00) &= -\rho _{00}^{8,00}\\ \end{align*}$$

sdmes_expanded.tex (excerpt):

    % Expanded SDMEs in reflectivity basis 
    \begin{align*}
    \rho _{00}^{0,00} &= 2 \left(\left| S_{T,0}^-\right| {}^2+\left| S_{T,0}^+\right| {}^2\right) \\ 
    \rho _{00}^{1,00} &= 2 \left(\left| S_{T,0}^-\right| {}^2-\left| S_{T,0}^+\right| {}^2\right) \\ 
    \rho _{00}^{2,00} &= 0 \\ 
    \rho _{00}^{3,00} &= 0 \\ 
    \rho _{00}^{4,00} &= 2 \left(\left| S_{L,0}^-\right| {}^2+\left| S_{L,0}^+\right| {}^2\right) \\ 
    \rho _{00}^{5,00} &= -2 i \sqrt{2} \Im\left(S_{L,0}^- S_{T,0}^-{}^*+S_{L,0}^+ S_{T,0}^+{}^*\right) \\ 
    \rho _{00}^{6,00} &= -2 \sqrt{2} \Im\left(S_{L,0}^- S_{T,0}^-{}^*\right) \\ 
    \rho _{00}^{7,00} &= -2 \sqrt{2} \Re\left(S_{L,0}^+ S_{T,0}^+{}^*\right) \\ 
    \rho _{00}^{8,00} &= -2 \sqrt{2} \Im\left(S_{L,0}^- S_{T,0}^-{}^*\right) \\ 
    \end{align*}

$$% Expanded SDMEs in reflectivity basis \begin{align*} \rho _{00}^{0,00} &= 2 \left(\left| S_{T,0}^-\right| {}^2+\left| S_{T,0}^+\right| {}^2\right) \\ \rho _{00}^{1,00} &= 2 \left(\left| S_{T,0}^-\right| {}^2-\left| S_{T,0}^+\right| {}^2\right) \\ \rho _{00}^{2,00} &= 0 \\ \rho _{00}^{3,00} &= 0 \\ \rho _{00}^{4,00} &= 2 \left(\left| S_{L,0}^-\right| {}^2+\left| S_{L,0}^+\right| {}^2\right) \\ \rho _{00}^{5,00} &= -2 i \sqrt{2} \Im\left(S_{L,0}^- S_{T,0}^-{}^*+S_{L,0}^+ S_{T,0}^+{}^*\right) \\ \rho _{00}^{6,00} &= -2 \sqrt{2} \Im\left(S_{L,0}^- S_{T,0}^-{}^*\right) \\ \rho _{00}^{7,00} &= -2 \sqrt{2} \Re\left(S_{L,0}^+ S_{T,0}^+{}^*\right) \\ \rho _{00}^{8,00} &= -2 \sqrt{2} \Im\left(S_{L,0}^- S_{T,0}^-{}^*\right) \\ \end{align*}$$

MomentsWaves.tex (excerpt):

% Auto-generated H^{\alpha}(L,M)
 \begin{align*} 
  H^{0}(00)  &= -2 \left(\left| S_{T,0}^-\right| {}^2+\left| S_{T,0}^+\right| {}^2\right) \\ 
  H^{1}(00)  &= 2 \left(\left| S_{T,0}^-\right| {}^2-\left| S_{T,0}^+\right| {}^2\right) \\ 
  H^{2}(00)  &= 0 \\ 
  H^{3}(00)  &= 0 \\ 
  H^{4}(00)  &= 2 \left(\left| S_{L,0}^-\right| {}^2+\left| S_{L,0}^+\right| {}^2\right) \\ 
  H^{5}(00)  &= -2 i \sqrt{2} \Im\left(S_{L,0}^- S_{T,0}^-{}^*+S_{L,0}^+ S_{T,0}^+{}^*\right) \\ 
  H^{6}(00)  &= -2 \sqrt{2} \Im\left(S_{L,0}^- S_{T,0}^-{}^*\right) \\ 
  H^{7}(00)  &= -2 \sqrt{2} \Re\left(S_{L,0}^+ S_{T,0}^+{}^*\right) \\ 
  H^{8}(00)  &= -2 \sqrt{2} \Im\left(S_{L,0}^- S_{T,0}^-{}^*\right) \\ 
\end{align*}

$$% Auto-generated H^{\alpha}(L,M) \begin{align*} H^{0}(00) &= -2 \left(\left| S_{T,0}^-\right| {}^2+\left| S_{T,0}^+\right| {}^2\right) \\ H^{1}(00) &= 2 \left(\left| S_{T,0}^-\right| {}^2-\left| S_{T,0}^+\right| {}^2\right) \\ H^{2}(00) &= 0 \\ H^{3}(00) &= 0 \\ H^{4}(00) &= 2 \left(\left| S_{L,0}^-\right| {}^2+\left| S_{L,0}^+\right| {}^2\right) \\ H^{5}(00) &= -2 i \sqrt{2} \Im\left(S_{L,0}^- S_{T,0}^-{}^*+S_{L,0}^+ S_{T,0}^+{}^*\right) \\ H^{6}(00) &= -2 \sqrt{2} \Im\left(S_{L,0}^- S_{T,0}^-{}^*\right) \\ H^{7}(00) &= -2 \sqrt{2} \Re\left(S_{L,0}^+ S_{T,0}^+{}^*\right) \\ H^{8}(00) &= -2 \sqrt{2} \Im\left(S_{L,0}^- S_{T,0}^-{}^*\right) \\ \end{align*}$$

IntensitiesAnalytic.tex (excerpt):

% Auto-generated H^{\alpha}(L,M)
 \begin{align*} 
  I^{0} &= \frac{1}{4 \pi }\left[\rho _{00}^{0,00}\right] \\ 
  I^{1} &= \frac{1}{4 \pi }\left[\rho _{00}^{1,00}\right] \\ 
  I^{2} &= \frac{1}{4 \pi }\left[\rho _{00}^{2,00}\right] \\ 
  I^{3} &= \frac{1}{4 \pi }\left[\rho _{00}^{3,00}\right] \\ 
  I^{4} &= \frac{1}{4 \pi }\left[\rho _{00}^{4,00}\right] \\ 
  I^{5} &= \frac{1}{4 \pi }\left[\rho _{00}^{5,00}\right] \\ 
  I^{6} &= \frac{1}{4 \pi }\left[\rho _{00}^{6,00}\right] \\ 
  I^{7} &= \frac{1}{4 \pi }\left[\rho _{00}^{7,00}\right] \\ 
  I^{8} &= \frac{1}{4 \pi }\left[\rho _{00}^{8,00}\right] \\ 
\end{align*}

$$% Auto-generated H^{\alpha}(L,M) \begin{align*} I^{0} &= \frac{1}{4 \pi }\left[\rho _{00}^{0,00}\right] \\ I^{1} &= \frac{1}{4 \pi }\left[\rho _{00}^{1,00}\right] \\ I^{2} &= \frac{1}{4 \pi }\left[\rho _{00}^{2,00}\right] \\ I^{3} &= \frac{1}{4 \pi }\left[\rho _{00}^{3,00}\right] \\ I^{4} &= \frac{1}{4 \pi }\left[\rho _{00}^{4,00}\right] \\ I^{5} &= \frac{1}{4 \pi }\left[\rho _{00}^{5,00}\right] \\ I^{6} &= \frac{1}{4 \pi }\left[\rho _{00}^{6,00}\right] \\ I^{7} &= \frac{1}{4 \pi }\left[\rho _{00}^{7,00}\right] \\ I^{8} &= \frac{1}{4 \pi }\left[\rho _{00}^{8,00}\right] \\ \end{align*}$$

Troubleshooting

• "Cannot create results directory": See the note above about replacing NotebookDirectory[] with a $InputFileName/Directory[] fallback when running from CLI.
• "wolframscript: command not found": Ensure Wolfram Engine or Mathematica is installed and wolframscript is on your PATH.
• Long runtimes for larger lmax/lprimemax: Start with small values (e.g., 0 or 1) to validate setup before scaling up.

Repro tips

• To rerun with different lmax, lprimemax, simply run the script again and enter new values when prompted.
• The script overwrites the .tex files in results/; delete the folder to start fresh if needed.