# epsproc.geomFunc.geomCalc module¶

ePSproc geometric terms/functions

Collection of codes for geometric functions and tensors.

26/02/20 v1 Initial implementation.

epsproc.geomFunc.geomCalc.CG(QNs, dlist=['l', 'lp', 'L', 'm', 'mp', 'M'], form='xarray')[source]

Basic Clebsch-Gordan from 3j calculation, from table of input QNs (corresponding to CG term defn.).

This implements numerical defn. from Moble’s Spherical Functions, https://github.com/moble/spherical_functions/blob/master/spherical_functions/recursions/wigner3j.py

def clebsch_gordan(j_1, m_1, j_2, m_2, j_3, m_3)

(-1.)**(j_1-j_2+m_3) * math.sqrt(2*j_3+1) * Wigner3j(j_1, j_2, j_3, m_1, m_2, -m_3)

22/06/20 - barebones version for quick testing, should upgrade as per w3jTable (which is the back-end here in any case).

Parameters: QNs (np.array) – List of QNs [l, lp, L, m, mp, M] to compute 3j terms for. form (string, optional, default = 'xarray') – 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.
epsproc.geomFunc.geomCalc.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.geomCalc.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.geomCalc.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.geomCalc.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() 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.geomCalc.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. 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. Xarray, dictionary
epsproc.geomFunc.geomCalc.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.geomCalc.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. w3j – Wigner3j(l,lp,L,m,mp,M) values corresponding to rows (coords) in QNs, type according to input form. 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.