epsproc.geomFunc package

Module contents

ePSproc geomFunc

Codes for geometric functions:

  • Wigner 3j
  • Wigner D
  • Spherical harmonics
  • EPR tensors
  • BLM tensors

etc.

Status

In progress Feb. 2020 for MFBLM development. Will eventually supercede/subsume existing functions in sphCalc.py.

epsproc.geomFunc.EPR(QNs=None, p=None, ep=None, nonzeroFlag=True, form='2d', dlist=['l', 'lp', 'P', 'p', 'R-p', 'R'], phaseConvention='S')[source]

Define polarization tensor (LF) for 1-photon case.

Define field terms (from QM book, corrected vs. original S&U version - see beta-general-forms\_rewrite\_290917.lyx):

\[\begin{split}\begin{equation} E_{PR}(\hat{e})=[e\otimes e^{*}]_{R}^{P}=[P]^{\frac{1}{2}}\sum_{p}(-1)^{R}\left(\begin{array}{ccc} 1 & 1 & P\\ p & R-p & -R \end{array}\right)e_{p}e_{R-p}^{*} \end{equation}\end{split}\]
Parameters:
  • QNs (np.array, optional, default = None) – List of QNs [l, lp, P, m, mp, R] to compute 3j terms for. If not supplied all allowed terms for the one-photon case, l=lp=1, will be set.
  • p (list or array, optional, default = None) – Specify polarization terms p. If set to None, all allowed values for the one-photon case will be set, p=[-1,0,1]
  • ep (list or array, optional, default = None) – Relative strengths for the fields ep. If set to None, all terms will be set to unity, ep = 1
  • nonzeroFlag (bool, optional, default = True) – Drop null terms before returning values if true.
  • form (string, optional, default = '2d') – For options see ep.w3jTable()
  • phaseConvention (optional, str, default = 'S') –

    Set phase conventions:

    • ’S’ : Standard derivation.
    • ’R’ : Reduced form geometric tensor derivation.
    • ’E’ : ePolyScat, may have additional changes in numerics, e.g. conjugate Wigner D.

    See setPhaseConventions() for more details.

Examples

# Generate full EPR list with defaults >>> EPRtable = EPR()

# Return as Xarray >>> EPRtable = EPR(form = ‘xarray’)

Note

Currently not handling ep correctly! Should implement as passed Xarray for correct (p,p’) assignment.

epsproc.geomFunc.MFproj(QNs=None, RX=None, nonzeroFlag=True, form='2d', dlist=['l', 'lp', 'P', 'mu', 'mup', 'Rp', 'R'], phaseConvention='S')[source]

Define MF projection term, \(\Lambda_{R',R}(R_{\hat{n}})\):

\[\begin{split}\begin{equation} \Lambda_{R',R}(R_{\hat{n}})=(-1)^{(R')}\left(\begin{array}{ccc} 1 & 1 & P\\ \mu & -\mu' & R' \end{array}\right)D_{-R',-R}^{P}(R_{\hat{n}}) \end{equation}\end{split}\]

Then…

\[\begin{eqnarray} \beta_{L,-M}^{\mu_{i},\mu_{f}} & = & \sum_{P,R',R}{\color{red}E_{P-R}(\hat{e};\mu_{0})}\sum_{l,m,\mu}\sum_{l',m',\mu'}(-1)^{(\mu'-\mu_{0})}{\color{red}\Lambda_{R',R}(R_{\hat{n}};\mu,P,R,R')B_{L,-M}(l,l',m,m')}I_{l,m,\mu}^{p_{i}\mu_{i},p_{f}\mu_{f}}(E)I_{l',m',\mu'}^{p_{i}\mu_{i},p_{f}\mu_{f}*}(E) \end{eqnarray}\]
Parameters:phaseConvention (optional, str, default = 'S') – Set phase conventions: - ‘S’ : Standard derivation. - ‘R’ : Reduced form geometric tensor derivation. - ‘E’ : ePolyScat, conjugate Wigner D. See setPhaseConventions() for more details.

Notes

This is very similar to \(E_{PR}\) term.

Examples

>>> lTerm, lambdaTable, lambdaD, QNs = MFproj(form = 'xarray')
epsproc.geomFunc.Wigner3jQNs(QNs)[source]
epsproc.geomFunc.afblmXprod(matEin, QNs=None, AKQS=None, EPRX=None, p=[0], BLMtable=None, lambdaTerm=None, RX=None, eulerAngs=None, thres=0.01, thresDims='Eke', selDims={'Type': 'L', 'it': 1}, sumDims=['mu', 'mup', 'l', 'lp', 'm', 'mp', 'S-Rp'], sumDimsPol=['P', 'R', 'Rp', 'p'], symSum=True, SFflag=False, SFflagRenorm=False, BLMRenorm=1, squeeze=False, phaseConvention='S')[source]

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.

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.

Dev code:
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

TOTAL MESS AT THE MOMENT>>?>>>>?DFdas<>r ty

Parameters:phaseConvention (optional, str, default = 'S') – Set phase conventions with epsproc.geomCalc.setPhaseConventions(). To use preset phase conventions, pass existing dictionary.

Notes

Cross-section outputs now set as:

  • XSraw = direct AF calculation output.
  • XSrescaled = XSraw * SF * sqrt(4pi)
  • XSiso = direct sum over matrix elements

Where XSrescaled == XSiso == ePS GetCro output for isotropic distribution.

epsproc.geomFunc.betaTerm(QNs=None, Lmin=0, Lmax=10, nonzeroFlag=True, form='2d', dlist=['l', 'lp', 'L', 'm', 'mp', 'M'], phaseConvention='S')[source]

Define BLM coupling tensor

Define field terms (from QM book, corrected vs. original S&U version - see beta-general-forms\_rewrite\_290917.lyx):

\[\begin{split}\begin{equation} B_{L,M}=(-1)^{m}\left(\frac{(2l+1)(2l'+1)(2L+1)}{4\pi}\right)^{1/2}\left(\begin{array}{ccc} l & l' & L\\ 0 & 0 & 0 \end{array}\right)\left(\begin{array}{ccc} l & l' & L\\ -m & m' & M \end{array}\right) \end{equation}\end{split}\]
Parameters:
  • QNs (np.array, optional, default = None) – List of QNs [l, lp, L, m, mp, M] to compute 3j terms for. If not supplied all allowed terms for {Lmin, Lmax} will be set. (If supplied, values for Lmin, Lmax and mFlag are not used.)
  • Lmax (Lmin,) – Integer values for Lmin and Lmax respectively.
  • mFlag (bool, optional, default = True) – m, mp take all values -l…+l if mFlag=True, or =0 only if mFlag=False
  • nonzeroFlag (bool, optional, default = True) – Drop null terms before returning values if true.
  • form (string, optional, default = '2d') – For options see ep.w3jTable()
  • dlist (list of labels, optional, default ['l','lp','L','m','mp','M']) – Used to label array for Xarray output case.
  • phaseConvention (optional, str, default = 'S') – Set phase conventions: - ‘S’ : Standard derivation. - ‘R’ : Reduced form geometric tensor derivation. - ‘E’ : ePolyScat, may have additional changes in numerics, e.g. conjugate Wigner D. See setPhaseConventions() for more details.

Examples

>>> Lmax = 2
>>> BLMtable = betaTerm(Lmax = Lmax, form = 'xds')
>>> BLMtable = betaTerm(Lmax = Lmax, form = 'xdaLM')
epsproc.geomFunc.deltaLMKQS(EPRX, AKQS, phaseConvention='S')[source]

Calculate aligned-frame “alignment” term:

\[\begin{equation} \sum_{K,Q,S}\Delta_{L,M}(K,Q,S)A_{Q,S}^{K}(t) \end{equation}\]
\[\begin{split}\begin{equation} \Delta_{L,M}(K,Q,S)=(2K+1)^{1/2}(-1)^{K+Q}\left(\begin{array}{ccc} P & K & L\\ R & -Q & -M \end{array}\right)\left(\begin{array}{ccc} P & K & L\\ R' & -S & S-R' \end{array}\right) \end{equation}\end{split}\]

15/06/20 IN PROGRESS

Parameters:
  • EPRX (Xarray) – Polarization terms in an Xarray, as set by epsproc.geomCalc.EPR()
  • AKQS (Xarray) – Alignement terms in an Xarray, as set by epsproc.setADMs()
Returns:

  • AFterm (Xarray) – Full term, including multiplication and sum over (K,Q,S) (note S-Rp term is retained).
  • DeltaKQS (Xarray) – Alignment term \(\Delta_{L,M}(K,Q,S)\).
  • To do
  • —–
  • - Add optional inputs.
  • - Add error checks.
  • See other similar functions for schemes.

epsproc.geomFunc.genKQSterms(Kmin=0, Kmax=2, mFlag=True)[source]
epsproc.geomFunc.genKQStermsFromTensors(EPR, AKQS, uniqueFlag=True, phaseConvention='S')[source]

Generate all QNs for \(\Delta_{L,M}(K,Q,S)\) from existing tensors (Xarrays) \(E_{P,R}\) and \(A^K_{Q,S}\).

Cf. epsproc.geomFunc.genllpMatE(), code adapted from there.

Parameters:
  • matE (Xarray) – Xarray containing matrix elements, with QNs (l,m), as created by readMatEle()
  • uniqueFlag (bool, default = True) – Check for duplicates and remove (can occur with some forms of matrix elements).
  • mFlag (bool, optional, default = True) – m, mp take all passed values if mFlag=True, or =0 only if mFlag=False
  • phaseConvention (optional, str, default = 'S') – Set phase conventions with epsproc.geomCalc.setPhaseConventions(). To use preset phase conventions, pass existing dictionary. If matE.attrs[‘phaseCons’] is already set, this will be used instead of passed args.
Returns:

  • QNs1, QNs2 (two 2D np.arrays) – Values take all allowed combinations [‘P’,’K’,’L’,’R’,’Q’,’M’] and [‘P’,’K’,’L’,’Rp’,’S’,’S-Rp’] from supplied matE. Note phase conventions not applied to QN lists as yet.
  • To do
  • —–
  • - Implement output options (see dev. function w3jTable).

epsproc.geomFunc.genllL(Lmin=0, Lmax=10, mFlag=True)[source]

Generate quantum numbers for angular momentum contractions (l, lp, L)

Parameters:
  • Lmax (Lmin,) – Integer values for Lmin and Lmax respectively.
  • mFlag (bool, optional, default = True) – m, mp take all values -l…+l if mFlag=True, or =0 only if mFlag=False
Returns:

QNs – Values take all allowed combinations [‘l’,’lp’,’L’,’m’,’mp’,’M’] up to l=lp=Lmax, one set per row.

Return type:

2D np.array

Examples

>>> # Calculate up to Lmax = 2
>>> QNs = genllL(Lmax=2)
>>> # Use with w3jTable function to calculate Wigner 3j terms
>>> w3j = w3jTable(QNs = QNs)
  • Implement output options (see dev. function w3jTable).
epsproc.geomFunc.genllLList(Llist, uniqueFlag=True, mFlag=True)[source]

Generate quantum numbers for angular momentum contractions (l, lp, L) from a passed list, (m, mp, M)=0 or all allowed terms.

Parameters:
  • Llist (list) – Values [l, lp, L] to use for calculations.
  • uniqueFlag (bool, optional, default = True) – Drop duplicate [l,lp,L] sets from list.
  • mFlag (bool, optional, default = True) – m, mp take all values -l…+l if mFlag=True, or =0 only if mFlag=False
Returns:

QNs – Values take all allowed combinations [‘l’,’lp’,’L’,’m’,’mp’,’M’] up to l=lp=Lmax, one set per row.

Return type:

2D np.array

Examples

>>> # Set from an array
>>> QNs = genllLList(np.array([[1,1,2],[1,3,2],[1,1,2]]), mFlag = True)
>>> # Use with w3jTable function to calculate Wigner 3j terms
>>> w3j = w3jTable(QNs = QNs)
  • Implement output options (see dev. function w3jTable).
epsproc.geomFunc.genllpMatE(matE, uniqueFlag=True, mFlag=True, phaseConvention='S')[source]

Generate quantum numbers for angular momentum contractions (l, lp, L, m, mp, M) from sets of matrix elements.

Parameters:
  • matE (Xarray) – Xarray containing matrix elements, with QNs (l,m), as created by readMatEle()
  • uniqueFlag (bool, default = True) – Check for duplicates and remove (can occur with some forms of matrix elements).
  • mFlag (bool, optional, default = True) – m, mp take all passed values if mFlag=True, or =0 only if mFlag=False
  • phaseConvention (optional, str, default = 'S') – Set phase conventions with epsproc.geomCalc.setPhaseConventions(). To use preset phase conventions, pass existing dictionary. If matE.attrs[‘phaseCons’] is already set, this will be used instead of passed args.
Returns:

QNs – Values take all allowed combinations [‘l’,’lp’,’L’,’m’,’mp’,’M’] from supplied matE

Return type:

2D np.array

Examples

>>> # From demo matrix elements
>>> dataFile = os.path.join(dataPath, 'n2_3sg_0.1-50.1eV_A2.inp.out')  # Set for sample N2 data for testing
>>> # Scan data file
>>> dataSet = ep.readMatEle(fileIn = dataFile)
>>> QNs = genllpMatE(dataSet[0])
>>> # Use with w3jTable function to calculate Wigner 3j terms
>>> w3j = w3jTable(QNs = QNs)
  • Implement output options (see dev. function w3jTable).
epsproc.geomFunc.mfblmXprod(matEin, QNs=None, EPRX=None, p=[0], lambdaTerm=None, BLMtable=None, thres=0.01, thresDims='Eke', selDims={'Type': 'L', 'it': 1}, sumDims=['mu', 'mup', 'l', 'lp', 'm', 'mp'], sumDimsPol=['P', 'R', 'Rp', 'p'], symSum=True, SFflag=True)[source]
epsproc.geomFunc.mfblmXprod(matEin, QNs=None, EPRX=None, p=[0], lambdaTerm=None, BLMtable=None, thres=0.01, thresDims='Eke', selDims={'Type': 'L', 'it': 1}, sumDims=['mu', 'mup', 'l', 'lp', 'm', 'mp'], sumDimsPol=['P', 'R', 'Rp', 'p'], symSum=True, SFflag=True)[source]
epsproc.geomFunc.remapllpL(dataIn, QNs, form='dict', method='sel', dlist=['l', 'lp', 'L', 'm', 'mp', 'M'], verbose=0)[source]

Remap Wigner 3j table, with QNs (l,lp,L,m,mp,M) > tensor forms.

Tensors are stored by (l,lp,L) triples, with corresponding arrays [m,mp,M], as a dictionary, or Xarray dataarray or dataset.

Parameters:
  • dataIn (np.array) – Array of data values corresponding to rows (coords) in QNs.
  • QNs (np.array) – List of QNs [l, lp, L, m, mp, M] to compute 3j terms for. If not supplied all allowed terms for {Lmin, Lmax} will be set. (If supplied, values for Lmin, Lmax and mFlag are not used.)
  • form (str, optional, default = 'dict') – Type of output structure. - ‘dict’ : dictionary with keys (l,lp,L), coordinate tables - ‘3d’ : dictionary with keys (l,lp,L), 3D arrays indexed by [l+m, lp+mp, L+M]; this case also sets (0,0,0) term as ‘w3j0’. - ‘xdaLM’ : Xarray dataarray, with stacked dims [‘lSet’,’mSet’] - ‘xds’ : Xarray dataset, with one array per (l,lp,L) - ‘xdalist’ : List of Xarray dataarrays, one per (l,lp,L)
  • method (str, optional, default = 'sel') – Method for selection from input array. - ‘sel’ : use full selection routine, epsproc.selQNsRow(), should be most robust. - ‘n’ : sort based on np.unique reindexing. Should be faster, but possibly not robust…?
  • dlist (list, optional, default = ['l','lp','L','m','mp','M']) – Full list of dimension names to use/set.
Returns:

w3j – Wigner3j(l,lp,L,m,mp,M) values corresponding to rows (coords) in QNs, type according to input form. Data structure sorted by (l,lp,L) triples.

Return type:

Xarray, dictionary

epsproc.geomFunc.selQNsRow(QNs, QNmask, fields=None, verbose=1)[source]

Basic routine for selecting/filtering valid rows from an input array of QNs.

This is similar to methods used in Matlab version, but likely rather slow.

Parameters:
  • QNs (np.array) – Values to be filtered.
  • QNmask (list or np.array) – Values to be matched. Must be same dims as QNs, unless fields is also set. If set to None field will be skipped (i.e. match all values)
  • fields (list or np.array, optional, default = None) – Which fields to match in QNs.
  • verbose (int, optional, default = 1) – Print info to terminal.
Returns:

  • QNs[boolMask] (np.array of selected values.)
  • boolMask (np.array corresponding to selected values.)

Examples

>>> QNsMask, mask = selQNsRow(QNs,[None, None, None, 0, 0, 0], verbose = False)
>>> # With fields
>>> QNsMask, mask = selQNsRow(QNs,[0, 0, 0], fields = [3,4,5], verbose = False)
epsproc.geomFunc.setPhaseConventions(phaseConvention='S', typeList=False)[source]

Set phase convention/choices for geometric functions.

20/03/20 - first attempt. Aim to centralise all phase choices here to keep things clean and easy to debug/change.

Set as dictionary for each term, to be appended to results Xarray.

Parameters:
  • phaseConvention (optional, str, default = 'S') – Set phase conventions: - ‘S’ : Standard derivation. - ‘R’ : Reduced form geometric tensor derivation. - ‘E’ : ePolyScat, may have additional changes in numerics, e.g. conjugate Wigner D. If a dict of phaseConventions is passed they will simply be returned - this is for transparency/consistency over multiple fns which call setPhaseConventions()… although may be an issue in some cases.
  • typeList (optional, bool, default = False) – If true, return list of supported options instead of list of phase choices.

Note

If a dict of phaseConventions is passed they will simply be returned - this is for transparency/consistency over multiple fns which call setPhaseConventions()… although may be an issue in some cases.

epsproc.geomFunc.setPhaseConventions(phaseConvention='S', typeList=False)[source]

Set phase convention/choices for geometric functions.

20/03/20 - first attempt. Aim to centralise all phase choices here to keep things clean and easy to debug/change.

Set as dictionary for each term, to be appended to results Xarray.

Parameters:
  • phaseConvention (optional, str, default = 'S') – Set phase conventions: - ‘S’ : Standard derivation. - ‘R’ : Reduced form geometric tensor derivation. - ‘E’ : ePolyScat, may have additional changes in numerics, e.g. conjugate Wigner D. If a dict of phaseConventions is passed they will simply be returned - this is for transparency/consistency over multiple fns which call setPhaseConventions()… although may be an issue in some cases.
  • typeList (optional, bool, default = False) – If true, return list of supported options instead of list of phase choices.

Note

If a dict of phaseConventions is passed they will simply be returned - this is for transparency/consistency over multiple fns which call setPhaseConventions()… although may be an issue in some cases.

epsproc.geomFunc.w3jTable(Lmin=0, Lmax=10, QNs=None, mFlag=True, nonzeroFlag=False, form='2d', dlist=['l', 'lp', 'L', 'm', 'mp', 'M'], backend='par', verbose=0)[source]

Calculate/tabulate all wigner 3j terms for a given problem/set of QNs.

\[\begin{split}\begin{equation} \begin{array}{ccc} l & l' & L\\ m & m' & M \end{array} \end{equation}\end{split}\]

Where l, l’ take values Lmin…Lmax (default 0…10). \(\l-lp\<=L<=l+lp\) m, mp take values -l…+l if mFlag=True, or =0 only if mFlag=False

Parameters:
  • Lmax (Lmin,) – Integer values for Lmin and Lmax respectively.
  • QNs (np.array, optional, default = None) – List of QNs [l, lp, L, m, mp, M] to compute 3j terms for. If not supplied all allowed terms for {Lmin, Lmax} will be set. (If supplied, values for Lmin, Lmax and mFlag are not used.)
  • mFlag (bool, optional, default = True) – m, mp take all values -l…+l if mFlag=True, or =0 only if mFlag=False
  • nonzeroFlag (bool, optional, default = False) – Drop null terms before returning values if true.
  • form (string, optional, default = '2d') –
    Defines return format. Options are:
    • 2d, return 2D np.array, rows [l, lp, L, m, mp, M, 3j]
    • xarray, return xarray
      This is nice for easy selection/indexing, but may be problematic for large Lmax if unstacked (essentailly similar to nd case).
    • nd, return ND np.array, dims indexed as [l, lp, L, l+m, lp+mp, L+M], with values 3j.
      This is suitable for direct indexing, but will have a lot of zero entries and may be large.
    • ndsparse, return ND sparse array, dims indexed as [l, lp, L, l+m, lp+mp, L+M], with values 3j.
    Additional options are set via remapllpL(). This additionally sorts values by (l,lp,L) triples, which is useful in some cases.
    • ’dict’ : dictionary with keys (l,lp,L), coordinate tables
    • ’3d’ : dictionary with keys (l,lp,L), 3D arrays indexed by [l+m, lp+mp, L+M]; this case also sets (0,0,0) term as ‘w3j0’.
    • ’xdaLM’ : Xarray dataarray, with stacked dims [‘lSet’,’mSet’]
    • ’xds’ : Xarray dataset, with one array per (l,lp,L)
    • ’xdalist’ : List of Xarray dataarrays, one per (l,lp,L)
  • dlist (list of labels, optional, default ['l','lp','L','m','mp','M']) – Used to label array for Xarray output case.
  • backend (str, optional, default = 'par') – See Implementation note below.
Returns:

w3j – Wigner3j(l,lp,L,m,mp,M) values corresponding to rows (coords) in QNs, type according to input form.

Return type:

np.array, Xarray, dictionary

Currently set to run:
  • ‘vec’: w3jguVecCPU(), which uses sf.Wigner3j on the back-end, with additional vectorisation over supplied QNs via Numba’s @guvectorize.
  • ‘par’: w3jprange(), which uses sf.Wigner3j on the back-end, with parallelization over QNs via Numba’s @njit with a prange loop.
epsproc.geomFunc.w3jguVecCPU(QNs, w3j_QNs)[source]

Wrapper for 3j with vectorization via @numba.guvectorize([“void(int32[:,:], float64[:])”], ‘(n,m)->(n)’, target = ‘parallel’).

Parameters:
  • QNs (np.array) – Array of QNs to calculated Wigner 3j terms for, columns [l,lp,L,m,mp,M].
  • w3j_QNs (np.array) – Empty array to hold results (no return from @guvectorize). Create as w3j_QNs = np.zeros(QNs.shape[0])
epsproc.geomFunc.w3jprange(QNs)[source]

Wrapper for 3j with @numba.njit(parallel=True), using prange parallel loop.

In testing (Feb 2020) on an AMD Threadripper 1950X (16 core) this was (usually) fastest case, and beat vectorised version.

Parameters:QNs (np.array) – Array of QNs to calculated Wigner 3j terms for, columns [l,lp,L,m,mp,M].
Returns:w3j_QNs – Array of Wigner 3j results, one per row of input QNs.
Return type:np.array