Angular::CkTable

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Lookup table for Ck_ab reduced matrix elements and the special 3j symbol (j_a, j_b, k; -1/2, 1/2, 0).

Pre-computes and caches the angular symbols needed for radial matrix-element evaluation. Three quantities are stored for each triple \((k, \kappa_a, \kappa_b)\):

• The reduced matrix element of the normalised spherical harmonic:

  \[ C^k_{ab} \equiv \redmatel{\kappa_a}{C^k}{\kappa_b} = (-1)^{j_a+\frac{1}{2}}\sqrt{[j_a][j_b]} \threej{j_a}{j_b}{k}{-\tfrac{1}{2}}{\tfrac{1}{2}}{0} \,\pi(l_a+l_b+k), \]

  where \([x]\equiv 2x+1\) and \(\pi\) encodes the parity selection rule. \(C^k_{ab}\) is antisymmetric under interchange up to a phase.
• The symmetrised (tilde) variant:

  \[ \widetilde C^k_{ab} = (-1)^{j_a+\frac{1}{2}} C^k_{ab}, \]

  which is fully symmetric: \(\widetilde C^k_{ab} = \widetilde C^k_{ba}\).
• The underlying special 3j symbol:

  \[ \threej{j_a}{j_b}{k}{-\tfrac{1}{2}}{\tfrac{1}{2}}{0}. \]

All table lookups take \(k\) and the Dirac quantum numbers \(\kappa_a, \kappa_b\) as input.

Note

• Table is filled up to a maximum \(2j\) at construction; call fill() to extend it afterwards.
• All symbols with \(2j \le \) max_tj() and \(k \le \) max_k() are guaranteed to be present; there are no gaps.
• Non-mutable accessors are thread-safe. The _mutable variants extend the table on demand but are not thread-safe.

Warning

Calling a non-mutable accessor with out-of-range indices is undefined behaviour; check against max_tj() and max_k() first.

#include <CkTable.hpp>

Public Member Functions
	CkTable (const int in_max_twoj=0)
	Constructs the table, pre-filling all symbols up to in_max_twoj.
void	fill (const int in_max_twoj)
	Extends the lookup table to cover all symbols up to in_max_twoj.
double	get_Ckab_mutable (int k, int ka, int kb)
	Returns Ck_ab; extends table if needed. Not thread-safe.
double	get_tildeCkab_mutable (int k, int ka, int kb)
	Returns tilde-Ck_ab; extends table if needed. Not thread-safe.
double	get_3jkab_mutable (int k, int ka, int kb)
	Returns 3j(j_a,j_b,k; -1/2,1/2,0); extends table if needed. Not thread-safe.
double	get_Ckab (int k, int ka, int kb) const
	Returns Ck_ab. Undefined behaviour if indices exceed max_tj() or max_k().
double	get_tildeCkab (int k, int ka, int kb) const
	Returns tilde-Ck_ab. Undefined behaviour if indices exceed max_tj() or max_k().
double	get_3jkab (int k, int ka, int kb) const
	Returns 3j(j_a,j_b,k; -1/2,1/2,0). Undefined behaviour if indices exceed max_tj() or max_k().
double	operator() (int k, int ka, int kb) const
	Equivalent to get_Ckab(k, ka, kb).
double	get_Lambdakab (int k, int ka, int kb) const
	Returns Lambda^k_ab := 3j(j_a,j_b,k; -1/2,1/2,0)^2 * parity(l_a+l_b+k).
int	max_tj () const
	Maximum value of 2j currently stored in the table.
int	max_k () const
	Maximum value of k currently stored in the table.

Constructor & Destructor Documentation

CkTable()

Angular::CkTable::CkTable ( const int  in_max_twoj = 0 )

inline

Constructs the table, pre-filling all symbols up to in_max_twoj.

Calls fill() internally; passing zero (the default) creates an empty table that can be extended later with fill() or the mutable accessors.

Parameters

in_max_twoj

Maximum value of \(2j\) to pre-compute.

Member Function Documentation

fill()

void Angular::CkTable::fill

(

const int

in_max_twoj

)

Extends the lookup table to cover all symbols up to in_max_twoj.

Called automatically on construction. Only needed explicitly when the required \(2j\) grows beyond the original maximum.

Parameters

in_max_twoj

New maximum value of \(2j\); no-op if already met.

get_Ckab_mutable()

double Angular::CkTable::get_Ckab_mutable	(	int	k,
		int	ka,
		int	kb
	)

Returns Ck_ab; extends table if needed. Not thread-safe.

get_tildeCkab_mutable()

double Angular::CkTable::get_tildeCkab_mutable	(	int	k,
		int	ka,
		int	kb
	)

Returns tilde-Ck_ab; extends table if needed. Not thread-safe.

get_3jkab_mutable()

double Angular::CkTable::get_3jkab_mutable	(	int	k,
		int	ka,
		int	kb
	)

Returns 3j(j_a,j_b,k; -1/2,1/2,0); extends table if needed. Not thread-safe.

get_Ckab()

double Angular::CkTable::get_Ckab	(	int	k,
		int	ka,
		int	kb
	)		const

Returns Ck_ab. Undefined behaviour if indices exceed max_tj() or max_k().

get_tildeCkab()

double Angular::CkTable::get_tildeCkab	(	int	k,
		int	ka,
		int	kb
	)		const

Returns tilde-Ck_ab. Undefined behaviour if indices exceed max_tj() or max_k().

get_3jkab()

double Angular::CkTable::get_3jkab	(	int	k,
		int	ka,
		int	kb
	)		const

Returns 3j(j_a,j_b,k; -1/2,1/2,0). Undefined behaviour if indices exceed max_tj() or max_k().

operator()()

double Angular::CkTable::operator() ( int  k, int  ka, int  kb  ) const

inline

Equivalent to get_Ckab(k, ka, kb).

get_Lambdakab()

double Angular::CkTable::get_Lambdakab	(	int	k,
		int	ka,
		int	kb
	)		const

Returns Lambda^k_ab := 3j(j_a,j_b,k; -1/2,1/2,0)^2 * parity(l_a+l_b+k).

This is the angular factor that appears in the Coulomb interaction after angular reduction, and equals \(\left(C^k_{ab}\right)^2 / ([j_a][j_b])\) up to a phase.

max_tj()

int Angular::CkTable::max_tj ( ) const

inline

Maximum value of 2j currently stored in the table.

max_k()

int Angular::CkTable::max_k ( ) const

inline

Maximum value of k currently stored in the table.

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The documentation for this class was generated from the following files:

• Angular/CkTable.hpp
• Angular/CkTable.cpp