epsproc.geomFunc.afblmGeom module

epsproc.geomFunc.afblmGeom.AFwfExp(matE, AKQS=None, thres=0.01, thresDims='Eke', selDims={'Type': 'L', 'it': 1})

Implement (approximate) LF/AF wavefunction expansion,

\[\begin{split}\begin{equation} ^{AF}T_{\mu_{0}}^{p_{i}\mu_{i},p_{f}\mu_{f}}(\hat{k}_{L})=8\pi^{2}\sum_{K,Q,S}\sum_{l,m,\mu,\Lambda}A_{Q,S}^{K}I_{l,m,\mu}^{p_{i}\mu_{i},p_{f}\mu_{f}}(E)(-1)^{m-\Lambda}(-1)^{\mu-\mu_{0}}(-1)^{Q-S}\left(\begin{array}{ccc} l & 1 & K\\ -m & -\mu & -Q \end{array}\right)\left(\begin{array}{ccc} l & 1 & K\\ -\Lambda & -\mu_{0} & -S \end{array}\right)Y_{l\Lambda}^{*}(\hat{k}_{M}) \end{equation}\end{split}\]

Where each component is defined by fns. in :py:module:`epsproc.geomFunc.geomCalc` module.

01/02/21 version 1, this is pretty rough and ready, and possibly not correct.

See http://localhost:8888/lab/tree/SLAC/angular_streaking/AF_wavefns_method_dev_050121.ipynb for dev notes.

epsproc.geomFunc.afblmGeom.afblmXprod(matEin, QNs=None, AKQS=None, EPRX=None, EPRXresort=None, p=None, ep=None, BLMtable=None, BLMtableResort=None, lambdaTerm=None, polProd=None, AFterm=None, EfieldPol=None, EfieldBasis='rot', EfieldRotSel='z', thres=0.01, thresDims='Eke', selDims={'Type': 'L'}, sqThres=True, dropThres=True, sumDims=['mu', 'mup', 'l', 'lp', 'm', 'mp', 'S-Rp'], sumDimsPol=['P', 'R', 'Rp', 'p', 'R-p'], symSum=True, outputDims={'LM': ['L', 'M']}, degenDrop=True, SFflag=False, SFflagRenorm=False, BLMRenorm=1, squeeze=False, phaseConvention='E', basisReturn='BLM', verbose=0, **kwargs)

Implement \(\beta_{LM}^{AF}\) calculation as product of tensors.

\[\begin{eqnarray} \beta_{L,-M}^{\mu_{i},\mu_{f}} & =(-1)^{M} & \sum_{P,R',R}{[P]^{\frac{1}{2}}}{E_{P-R}(\hat{e};\mu_{0})}\sum_{l,m,\mu}\sum_{l',m',\mu'}(-1)^{(\mu'-\mu_{0})}{\Lambda_{R'}(\mu,P,R')B_{L,-M}(l,l',m,m')}I_{l,m,\mu}^{p_{i}\mu_{i},p_{f}\mu_{f}}(E)I_{l',m',\mu'}^{p_{i}\mu_{i},p_{f}\mu_{f}*}(E)\sum_{K,Q,S}\Delta_{L,M}(K,Q,S)A_{Q,S}^{K}(t) \end{eqnarray}\]

Where each component is defined by fns. in :py:module:`epsproc.geomFunc.geomCalc` module.

23/03/24 Updating for arb pol case (p!=0), changes as per working mfblmGeom.py 05/03/24

• Allow passing of ep.

• Allow direct passing for EPRXresort (skips default).

• Updated defaults for p, ep.

• Default case still p=0, now set via ep=[0,1,0] selection.

• Other defaults based on inputs, see p, ep help below.

• Added ‘R-p’ to sumDimsPol defaults.

• Added ‘polProd’ tp full basis return dict.

30/06/23 Debugging for 3D aligment case. Currently issues with handling of M and S-Rp dims here?

• Tested:

  • Removed sum over S-Rp (MF/AF)

  • Set sum over M (LF)

  • Various phase conventions.

• 3D alignment case (for OCS HOMO-1 test values) OK after fixing phaseCons[‘afblmCons’][‘negM’] = True (now corrected in source function).

  • Gives correct basis functions, but only L,M=even terms in test case (with Q!=0 and S!=0 terms in ADMs).

  • May still be phase issues for odd cases - TBC with further testing but probably not.

• Added outDims to allow for changes in output dim mapping (TODO: should use sph dim checks here too?)

• Added matEmult to output dataset for ‘Full’ basis return case.

• Fixed issue with inplace multiplication for Xarray coords (’*=’), not valid for XR 2022.3 (coordinates immutable).

24/11/22 Added sqThres = True, dropThres = True for use with INITIAL matEleSelector() call only - in some cases may get issues with dim drops here otherwise.

04/05/22 Added **kwargs, unused but allows for arb basis dict unpack and passing from other functions. May want to pipe back to Full basis return however.

06/08/21 Added basic handling for degenerate states, including degenDrop option.

Updated docs, but still rather messy!

27/07/21 Removed eulerAngs & RX input options, since these are redundant (and lead to confusion here!).

For cases where E-field and alignment distribution are rotated, set AKQS in rotated frame, see https://epsproc.readthedocs.io/en/latest/tests/ePSproc_frame_rotation_tests_Dec2019.html Also removed selDims={‘it’:1}, which can result in issues! In current code, adding ‘it’ to sumDims doesn’t work (dim issues somewhere), so may be best to treat independently…?

03/05/21 Tidying up a bit & improving/wrapping for fitting use (inc. basis function reuse).

10/09/20 Verified (but messy) version, with updated defaults.

01/09/20 Verified (but messy) version, including correct renormalisation routines.

15/06/20 In progress! Using mfblmXprod() as template, with just modified lambda term, and new alignment term, to change.

For basic use see the docs: https://epsproc.readthedocs.io/en/dev/demos/ePSproc_class_demo_161020.html#Compute-LF/AF-beta_{LM}-and-PADs

Dev code:

• afblmGeom_v1-ref_2020.py - Messy working v1 for reference, archived 04/05/21. Now working on tidier version.

• geometric_method_dev_pt3_AFBLM_090620.ipynb

• http://localhost:8888/lab/tree/dev/ePSproc/geometric_method_dev_Betas_090320.ipynb

• D:codeePSprocpython_devePSproc_MFBLM_Numba_dev_tests_120220.PY

Parameters:

• matE (Xarray) – Xarray containing matrix elements, with QNs (l,m), as created by readMatEle()

• settings (*** Optional calculation) –

• selDims (dict, default = {'Type':'L'}) – Selection parameters for calculations, may be be checked and appened herein.

• sumDims (list, default = ['mu', 'mup', 'l','lp','m','mp','S-Rp']) – Main summation dims, will be checked herein.

• sumDimsPol (list, default = ['P','R','Rp','p']) – Additional polarization summation dims.

• symSum (bool, default = True) – Sum over symmetries sets (=={Cont, Targ, Total}) if true.

• degenDrop (bool) – Flag to set dropping of degenerate components.

• thres (float, default = 1e-2) – Set threshold value, used to set input matrix elements and again for outputs.

• thresDims (str, default = 'Eke') – Set threshold dimension (set to be contiguous).

• verbose (bool or int) – Print output?

*** Optional renormalisation settings (mainly for testing only)

SFflagbool, default = False

Multiply input matrix elements by complex scale-factor if true.

SFflagRenormbool, default = False

Renorm output BLMs by complex scale-factor if true.

BLMRenormint, default = 1

Set different BLM renorm conventions. If 1 renorm by B00. See code for further details.

sqThresbool, default = True

Used for initial matrix element threholding call only. Default = True as previous code, but can cause issues in custom cases (usually accidentally dropping singleton mu coord). Added 24/11/22

dropThresbool, default = True

Used for initial matrix element threholding call only. Default = True as previous code, but can cause issues in custom cases (usually accidentally dropping singleton mu coord). Added 24/11/22

squeezebool, default = False

Squeeze output array after thresholding? Note: this may cause dim issues if True.

*** Optional input data/basis functions (mainly for fitting routine use)

QNsnp.array, optional, default = None

List of QNs as generated by genllpMatE(). Will be generated if not passed.

AKQSXarray, optional, default = None

Alignment parameters, as set by setADMs(). Defaults to isotropic case if not passed.

EPRX, EPRXresortXarray, optional, default = None

E-field parameters, as generated by EPR(). Defaults to normalised/unity field, pol = p (below).

plist or array, optional, default = None

Specify polarization terms p. Possibly currently only valid for p=0, TBC See https://epsproc.readthedocs.io/en/latest/methods/geometric_method_dev_260220_090420_tidy.html#E_{P,R}-tensor UPDATE 23/03/24: now supports all p terms via ep values, default p=[-1,0,1]

eplist or array, optional, default = None

Set E-field components. If p and ep are None, will default to ep=[0,1,0] for p=0 term only. If p is set, but ep=None, will default to ep=np.ones(len(p)).

BLMtable, BLMtableResortXarrays, optional, default = None

Beta calculation parameters, as defined by geomCalc.betaTerm(). BLMtableResort includes phase settings & param renaming as herein.

lambdaTermXarray, optional, default = None

Lambda term parameters, as defined by geomCalc.MFproj()

AFtermXarray, optional, default = None

Alignment term as defined by geomCalc.deltaLMKQS()

polProdXarray, optional, default = None

Polarization tensor as defined by EPRXresort * lambdaTermResort * AFterm

EfieldPolobject, default = None

Pass field settings as epsproc.efield.epol.EfieldPol object. Note this defaults to the ‘rot’ basis case, and z-pol geometry. Multiple polarization states are supported, but only a single pol geometry currently. Set EfieldBasis and EfieldRotSel arguments to override. See docs at https://epsproc.readthedocs.io/en/3d-afpad-dev/demos/Epol_class_demo_docs_030324-tidy.html

EfieldBasisstr, optional, default = ‘rot’

Only used for EfieldPol objects. Sets basis for E-field, see epsproc.classes.epol.setep().

EfieldRotSelstr, optional, default = ‘z’

Only used for EfieldPol objects. Sets pol geom selection for E-field, see epsproc.classes.epol.setep(). Note this is also used as a pol geom label in the output.

phaseConventionoptional, str, default = ‘E’

Set phase conventions with epsproc.geomCalc.setPhaseConventions(). To use preset phase conventions, pass existing dictionary.

basisReturnoptional, str, default = “BLM”

• ‘BLM’ return Xarray of results only.

• ‘Full’ return Xarray of results + basis set dictionary as set during the run.

• ‘Product’, as full, but minimal basis set with products only.

• ‘Results’ or ‘Legacy’ direct return of various calc. results Xarrays.

**kwargs, unused but allows for arb basis dict unpack and passing from other functions.

Returns:

• Xarray – Set of AFBLM calculation results

• dict – Dictionary of basis functions, only if basisReturn != ‘BLM’ (see basisReturn paramter notes).

Notes

Cross-section outputs now set as:

• XSraw = direct AF calculation output.

• XSrescaled = XSraw * sqrt(4pi)

• XSiso = direct sum over matrix elements

Where XSrescaled == XSiso == ePS GetCro output for isotropic distribution. Optionally set SFflag = True to multiply by (complex) scale-factor.