Generalised hyperfine-structure operator, including relevant nuclear moment. More...

#include <hfs.hpp>

Inheritance diagram for DiracOperator::hfs:

Public Member Functions
	hfs (int in_k, double GQ, double rN_au, const Grid &rgrid, const RadialFunction &hfs_F=Hyperfine::pointlike_F(), bool MHzQ=true)
	Constructs hyperfine operator of multipolarity k.

std::string	name () const override final
	Returns "name" of operator (e.g., 'E1')

std::string	units () const override final
	Returns units of operator (usually au, may be MHz, etc.)

double	angularF (const int ka, const int kb) const override final
	angularF: links radiation integral to RME. RME = <a\|\|h\|\|b> = angularF(a,b) * radial_int(a,b)

double	angularCff (int, int) const override final
	Angular factor for f_a*f_b part of radial integral.

double	angularCgg (int, int) const override final
	Angular factor for g_a*g_b part of radial integral.

double	angularCfg (int, int) const override final
	Angular factor for f_a*g_b part of radial integral.

double	angularCgf (int, int) const override final
	Angular factor for g_a*f_b part of radial integral.

Public Member Functions inherited from DiracOperator::TensorOperator
bool	freqDependantQ () const

bool	isZero (const int ka, int kb) const
	If matrix element <a\|h\|b> is zero, returns true.

bool	isZero (const DiracSpinor &Fa, const DiracSpinor &Fb) const

bool	selectrion_rule (int twoJA, int piA, int twoJB, int piB) const

virtual void	updateFrequency (const double)
	Update frequency for frequency-dependant operators.

void	scale (double lambda)
	Permanently re-scales the operator by constant, lambda.

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

double	getc () const
	Returns a const ref to constant c.

int	get_d_order () const

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 DiracSpinor	radial_rhs (const int kappa_a, const DiracSpinor &Fb) const
	radial_int = Fa * radial_rhs(a, Fb) (a needed for angular factor)

virtual double	radialIntegral (const DiracSpinor &Fa, const DiracSpinor &Fb) const
	Radial part of integral R_ab = (Fa\|t\|Fb).

double	rme3js (const int twoja, const int twojb, int two_mb=1, int two_q=0) const
	ME = rme3js * RME.

DiracSpinor	reduced_rhs (const int ka, const DiracSpinor &Fb) const
	<a\|\|h\|\|b> = Fa * reduced_rhs(a, Fb) (a needed for angular factor)

DiracSpinor	reduced_lhs (const int ka, const DiracSpinor &Fb) const
	<b\|\|h\|\|a> = Fa * reduced_lhs(a, Fb) (a needed for angular factor)

double	reducedME (const DiracSpinor &Fa, const DiracSpinor &Fb) const
	The reduced matrix element.

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
	Converts reduced matrix element to different "type" (MatrixElementType)

double	matel_factor (MatrixElementType type, const DiracSpinor &Fa, const DiracSpinor &Fb) const

Additional Inherited Members
Protected Member Functions inherited from DiracOperator::TensorOperator
	TensorOperator (int rank_k, Parity pi, double constant=1.0, const std::vector< double > &vec={}, int diff_order=0, Realness RorI=Realness::real, bool freq_dep=false)
	Constructs a tensor operator description.

Protected Attributes inherited from DiracOperator::TensorOperator
int	m_rank

Parity	m_parity

int	m_diff_order

Realness	opC

bool	m_freqDependantQ {false}

double	m_constant

std::vector< double >	m_vec

Detailed Description

Generalised hyperfine-structure operator, including relevant nuclear moment.

Implements the nuclear multipole hyperfine operator of rank k using a specified finite-nucleus radial model.

By default, includes the nuclear moment, and relevant factors. Input parameter GQ is the g-factor (k=1) or nuclear moment (k>1), in units of nuclear magnetons * barns^power.

That is, it calculates:

\[ GQ * t^k \]

with:

\[ t^k = \frac{-1}{r^{k+1}}\,F(r) \begin{cases} C^k & \text{electric (even k)} \\ \sqrt{\frac{k+1}{k}} \mathbf{\alpha}\!\cdot\!\mathbf{C}^{(0)}_k & \text{magnetic (odd k)} \end{cases} \]

where F(r) is nuclear k-pole distribution (=1 for pointlike).

Convert to hyperfine constant by taking stretched state, and multiply by:

\[ M = \begin{cases} 1/J & k = 1 \\ 2 & k = 2 \\ -1 & k = 3 \\ 1 & k \geq 4 \\ \end{cases} \]

This factor (including the steched 3j symbol) is returned by Hyperfine::convert_RME_to_HFSconstant()

Units:

Assumes nuclear moments in units of:
magnetic moments: \(\mu_N\, b^{(k-1)/2}\)
electric moments: \(b^{k/2}\)
Matrix element are in MHz by default, otherwise in atomic units.

Here \( \mu_N \) is the nuclear magneton and \( b \) is the barn.

See, e.g., Xiao et al., Phys. Rev. A 102, 022810 (2020).

The radial part is constructed from Hyperfine::tk_radial().

Constructor & Destructor Documentation

◆ hfs()

DiracOperator::hfs::hfs	(	int	in_k,
		double	GQ,
		double	rN_au,
		const Grid &	rgrid,
		const RadialFunction &	hfs_F = `Hyperfine::pointlike_F()`,
		bool	MHzQ = `true`
	)

inline

Constructs hyperfine operator of multipolarity k.

Initialises the hyperfine interaction operator.

Parameters

in_k	Tensor rank k (multipolarity) of the hyperfine interaction (e.g., 1,2,3,... for M1, E2, M3,... etc.)
GQ	Nuclear g-factor for k=1, general moment for k>1
rN_au	Nuclear radius (a.u.), used for finite-size magnetisation models.
rgrid	Radial grid used to define the radial functions.
hfs_F	Radial magnetisation distribution function (see DiracOperator::Hyperfine).
MHzQ	If true outputs in MHz units; otherwise in atomic units.

See also: See also: DiracOperator::Hyperfine

Member Function Documentation

◆ name()

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

inlinefinaloverridevirtual

Returns "name" of operator (e.g., 'E1')

Reimplemented from DiracOperator::TensorOperator.

◆ units()

std::string DiracOperator::hfs::units ( ) const

inlinefinaloverridevirtual

Returns units of operator (usually au, may be MHz, etc.)

Reimplemented from DiracOperator::TensorOperator.

◆ angularF()

double DiracOperator::hfs::angularF	(	const int	,
		const int
	)		const

inlinefinaloverridevirtual

angularF: links radiation integral to RME. RME = <a||h||b> = angularF(a,b) * radial_int(a,b)

Implements DiracOperator::TensorOperator.

◆ angularCff()

double DiracOperator::hfs::angularCff	(	int	,
		int
	)		const

inlinefinaloverridevirtual

Angular factor for f_a*f_b part of radial integral.

Often constant, though sometimes depends on \(\kappa_a,\kappa_b\)
\[ \int_0^\infty v(r)\left( C_{ff}f_a(r)f_b(r) + C_{fg}f_a(r)g_b(r) + C_{gf}g_a(r)f_b(r) + C_{gg}g_a(r)g_b(r) \right)\,{\rm d}r \]

Reimplemented from DiracOperator::TensorOperator.

◆ angularCgg()

double DiracOperator::hfs::angularCgg	(	int	,
		int
	)		const

inlinefinaloverridevirtual

Angular factor for g_a*g_b part of radial integral.

Reimplemented from DiracOperator::TensorOperator.

◆ angularCfg()

double DiracOperator::hfs::angularCfg	(	int	,
		int
	)		const

inlinefinaloverridevirtual

Angular factor for f_a*g_b part of radial integral.

Reimplemented from DiracOperator::TensorOperator.

◆ angularCgf()

double DiracOperator::hfs::angularCgf	(	int	,
		int
	)		const

inlinefinaloverridevirtual

Angular factor for g_a*f_b part of radial integral.

Reimplemented from DiracOperator::TensorOperator.

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

DiracOperator/Operators/hfs.hpp

Public Member Functions

Additional Inherited Members

Detailed Description

Constructor & Destructor Documentation

◆ hfs()

Member Function Documentation

◆ name()

◆ units()

◆ angularF()

◆ angularCff()

◆ angularCgg()

◆ angularCfg()

◆ angularCgf()