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Intermediate abstract base class for all EM relativistic multipole operators.

These are the \( t^K_Q \tilde\gamma \), \( T^{(\sigma)}_{KQ} \tilde\gamma \) operators from the vector expansion:

\[ \begin{align} e^{i\vec{q}\cdot\vec{r}} &= \sqrt{4\pi}\sum_{KQ}\sqrt{[K]} \, i^K \, {Y^*_{KQ}}{(\hat q)} \, t^K_Q(q,r),\\ \vec{\alpha} \, e^{i\vec{q}\cdot\vec{r}} & = \sqrt{4\pi} \sum_{KQ\sigma} \sqrt{[K]} \, i^{K-\sigma} \, \vec{Y}_{KQ}^{(\sigma)*}(\hat{{q}}) \, T^{(\sigma)}_{KQ}, \end{align} \]

with \( \tilde\gamma = 1,\) \( \gamma^5, \) \( \gamma^0, \) \( i\gamma^0\gamma^5 \) for vector (V), axial(pseudo)-vector (A), scalar (S), or pseudoscalar (P).

Stores the construction parameters needed to reconstruct the concrete operator via the MultipoleOperator factory, enabling type-erased cloning without the caller knowing the derived type.

Specifically, the stored parameters are: a pointer to the radial grid, the current frequency omega (updated by each derived class's updateFrequency()), the Lorentz type character m_type, the component character m_comp, the low-q flag m_low_q, an optional Bessel table pointer m_jl (shallow-owned; shared between original and clone), and a form character m_form (used to distinguish the length-form electric operator VEk_Len from the velocity-form VEk).

All concrete EM multipole operators VEk, VMk, VLk, Phik, Sk, AEk, ALk, AMk, Phi5k, S5k, and their low-q counterparts derive from this class.

Note: This class does not implement angularF() and so remains abstract. Operators must be constructed via the concrete derived types or the MultipoleOperator factory.

#include <EM_multipole_base.hpp>

Inheritance diagram for DiracOperator::EM_multipole:

Public Member Functions
const SphericalBessel::JL_table *	jl () const
	Returns the precomputed Bessel table pointer (may be nullptr).

std::string	name () const override
	Returns a human-readable label, e.g. "T^E_1", "T^M5_2", "t_1", "P_1".

double	angularF (const int ka, const int kb) const override
	Angular factor linking the radial integral to the reduced matrix element: \( \langle a \\| h \\| b \rangle = F(\kappa_a,\kappa_b) \cdot \int \! dr \).

void	updateRank (int new_K) override
	Updates the tensor rank and adjusts parity accordingly.

std::unique_ptr< TensorOperator >	clone () const override
	Creates a fully independent copy of this operator at its current (rank, frequency) state via the MultipoleOperator factory.

	EM_multipole (const EM_multipole &)=default

EM_multipole &	operator= (const EM_multipole &)=default

	EM_multipole (EM_multipole &&)=default

EM_multipole &	operator= (EM_multipole &&)=default

Public Member Functions inherited from DiracOperator::TensorOperator
	TensorOperator (const TensorOperator &)=default

TensorOperator &	operator= (const TensorOperator &)=default

	TensorOperator (TensorOperator &&)=default

TensorOperator &	operator= (TensorOperator &&)=default

bool	freqDependantQ () const
	Returns true if the operator is frequency-dependent (requires updateFrequency() calls).

bool	isZero (int ka, int kb) const
	Returns true if <a\|h\|b> = 0 by rank/parity selection rules.

bool	isZero (const DiracSpinor &Fa, const DiracSpinor &Fb) const
	Overload taking DiracSpinors; forwards to isZero(ka, kb).

bool	selectrion_rule (int twoJA, int piA, int twoJB, int piB) const
	Returns true if the matrix element is non-zero by angular momentum and parity selection rules (arguments are 2j and pi as integers).

virtual void	updateFrequency (const double)
	Updates the operator for a new frequency omega.

const std::vector< double > &	getv () const
	Returns a const ref to the stored vector v.

double	getc () const
	Returns the "overall" constant c.

bool	imaginaryQ () const
	returns true if operator is imaginary (has imag MEs)

int	rank () const
	Rank k of operator.

int	parity () const
	returns parity, as integer (+1 or -1)

int	symm_sign (const DiracSpinor &Fa, const DiracSpinor &Fb) const
	returns relative sign between <a\|\|x\|\|b> and <b\|\|x\|\|a>

virtual std::string	units () const
	Returns units of operator as a string (usually au, may be MHz, etc.)

virtual double	angularCff (int kappa_a, int kappa_b) const
	Angular coefficient C_ff for the f_a*f_b term of the radial integral.

virtual double	angularCgg (int, int) const
	Angular coefficient C_gg for the g_a*g_b term of the radial integral.

virtual double	angularCfg (int, int) const
	Angular coefficient C_fg for the f_a*g_b term of the radial integral.

virtual double	angularCgf (int, int) const
	Angular coefficient C_gf for the g_a*f_b term of the radial integral.

double	angularCxy (uint8_t x, uint8_t y, int kappa_a, int kappa_b) const
	Dispatches to angularCff/fg/gf/gg based on component indices x, y.

virtual DiracSpinor	radial_rhs (const int kappa_a, const DiracSpinor &Fb) const
	Computes the right-hand spinor dF_b for the radial integral.

virtual double	radialIntegral (const DiracSpinor &Fa, const DiracSpinor &Fb) const
	Radial integral R_ab, defined by RME = angularF(a,b) * radialIntegral(a,b).

double	rme3js (int twoja, int twojb, int two_mb=1, int two_q=0) const
	3j-symbol factor linking the full ME to the RME.

double	rme3js (const DiracSpinor &Fa, const DiracSpinor &Fb, int two_mb=1, int two_q=0) const
	Overload of rme3js taking DiracSpinors.

DiracSpinor	reduced_rhs (const int ka, const DiracSpinor &Fb) const
	Returns angularF(ka,kb) * radial_rhs(ka,Fb); spinor-valued RME action on Fb, used in perturbation theory/TDHF.

DiracSpinor	reduced_lhs (const int ka, const DiracSpinor &Fb) const
	As reduced_rhs but for the conjugate direction; Fb * reduced_lhs(ka, Fb) = <b\|\|h\|\|a>.

double	reducedME (const DiracSpinor &Fa, const DiracSpinor &Fb) const
	Returns the reduced matrix element <a\|\|h\|\|b> = A_ab * R_ab.

double	fullME (const DiracSpinor &Fa, const DiracSpinor &Fb, std::optional< int > two_ma=std::nullopt, std::optional< int > two_mb=std::nullopt, std::optional< int > two_q=std::nullopt) const
	Returns "full" matrix element, for optional (ma, mb, q) [taken as int 2*]. If not specified, returns z-component (q=0), with ma=mb=min(ja,jb)

double	matel_factor (MatrixElementType type, int twoJa, int twoJb) const
	Returns the factor to convert a reduced ME to a different form (Reduced, Stretched, or HFConstant); see MatrixElementType.

double	matel_factor (MatrixElementType type, const DiracSpinor &Fa, const DiracSpinor &Fb) const
	Overload of matel_factor taking DiracSpinors.

Protected Member Functions
	EM_multipole (int rank_k, Parity pi, double constant, const std::vector< double > &vec, Realness RorI, bool freq_dep, const Grid grid, char type, char comp, bool low_q, const SphericalBessel::JL_table jl=nullptr, char form='V')
	Initialise the EM_multipole base layer.

Protected Member Functions inherited from DiracOperator::TensorOperator
	TensorOperator (int rank_k, Parity pi, double constant=1.0, const std::vector< double > &vec={}, Realness RorI=Realness::real, bool freq_dep=false)
	Constructs a specific tensor operator. Called by derived classes.

Protected Attributes
const Grid *	m_grid {nullptr}

double	m_omega
	Current frequency; cached by each derived updateFrequency().

char	m_type {}

char	m_comp {}

bool	m_low_q {}

char	m_form {}

const SphericalBessel::JL_table *	m_jl {nullptr}

Protected Attributes inherited from DiracOperator::TensorOperator
int	m_rank

Parity	m_parity

Realness	m_Realness

bool	m_freqDependantQ {false}

double	m_constant

std::vector< double >	m_vec

Constructor & Destructor Documentation

◆ EM_multipole()

DiracOperator::EM_multipole::EM_multipole	(	int	rank_k,
		Parity	pi,
		double	constant,
		const std::vector< double > &	vec,
		Realness	RorI,
		bool	freq_dep,
		const Grid *	grid,
		char	type,
		char	comp,
		bool	low_q,
		const SphericalBessel::JL_table *	jl = `nullptr`,
		char	form = `'V'`
	)

inlineprotected

Initialise the EM_multipole base layer.

Called exclusively from derived-class constructors.

Parameters

rank_k	Tensor rank of the operator.
pi	Parity.
constant	Overall multiplicative constant.
vec	Radial vector (usually gr.r(), sometimes empty).
RorI	Real or imaginary matrix elements.
freq_dep	True if the operator is frequency-dependent.
grid	Pointer to the radial Grid (stored for clone()).
type	Lorentz type: 'V', 'A', 'S', or 'P'.
comp	Component: 'E', 'M', 'L', or 'T'.
low_q	True for the low-momentum (long-wavelength) approximation.
jl	Optional pointer to a precomputed Bessel table.
form	'V' (velocity/V-form) or 'L' (length-form, VEk_Len only).

Member Function Documentation

◆ jl()

const SphericalBessel::JL_table * DiracOperator::EM_multipole::jl ( ) const

inline

Returns the precomputed Bessel table pointer (may be nullptr).

◆ name()

std::string DiracOperator::EM_multipole::name ( ) const

inlineoverridevirtual

Returns a human-readable label, e.g. "T^E_1", "T^M5_2", "t_1", "P_1".

Reimplemented from DiracOperator::TensorOperator.

◆ angularF()

double DiracOperator::EM_multipole::angularF	(	const int	ka,
		const int	kb
	)		const

inlineoverridevirtual

Angular factor linking the radial integral to the reduced matrix element: \( \langle a \| h \| b \rangle = F(\kappa_a,\kappa_b) \cdot \int \! dr \).

Returns \( C^K(\kappa_a, \kappa_b) \) or \( C^K(\kappa_a, -\kappa_b) \) depending on the Lorentz structure. The flipped form applies to: vector-magnetic (VM), all axial components except axial-magnetic (AE, AL, AT), and pseudoscalar (P).

Note: \( C^K(\kappa_a,-\kappa_b) = C^K(-\kappa_a,\kappa_b) \).; VMk_lowq overrides this with an extra \((_a+_b)\) prefactor.

Implements DiracOperator::TensorOperator.

Reimplemented in DiracOperator::VMk_lowq.

◆ updateRank()

void DiracOperator::EM_multipole::updateRank ( int new_K )

inlineoverridevirtual

Updates the tensor rank and adjusts parity accordingly.

Parity follows the tensor rank: even K gives even parity for most operators, but is flipped (even K odd parity) for vector-magnetic, all axial components except axial-magnetic, and pseudoscalar operators. The same flip rule as angularF() applies.

Note: Must be followed by a call to updateFrequency() to recompute the radial Bessel vectors for the new rank.

Reimplemented from DiracOperator::TensorOperator.

◆ clone()

std::unique_ptr< TensorOperator > DiracOperator::EM_multipole::clone ( ) const

overridevirtual

Creates a fully independent copy of this operator at its current (rank, frequency) state via the MultipoleOperator factory.

The Bessel table pointer m_jl is shallow-copied (the clone shares the same external table), but all computed radial vectors are regenerated independently on the first updateFrequency() call.

Returns: A std::unique_ptr<TensorOperator> to the cloned operator.

Reimplemented from DiracOperator::TensorOperator.

Member Data Documentation

◆ m_omega

double DiracOperator::EM_multipole::m_omega

protected

Initial value:

{

0.0}

Current frequency; cached by each derived updateFrequency().

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

DiracOperator/Operators/EM_multipole_base.hpp
DiracOperator/Operators/EM_multipole.cpp

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Public Member Functions

Protected Member Functions

Protected Attributes

Constructor & Destructor Documentation

◆ EM_multipole()

Member Function Documentation

◆ jl()

◆ name()

◆ angularF()

◆ updateRank()

◆ clone()

Member Data Documentation

◆ m_omega