DiracOperator::Hyperfine Namespace Reference

Detailed Description

Auxiliary functions for hyperfine operators; F(r) [nuclear distribution] and similar.

See DiracOperator::hfs for operator

Typedefs
using	RadialFunction = std::function< double(double r, double r_Nuc)>
	Type for radial function F(r, r_Nuc) – nuclear magnetisation/charge distribution.

Functions
std::vector< double >	tk_radial (int k, double rN, const std::vector< double > &r, const RadialFunction &hfs_F)
	Forms the hyperfine radial operator: F(r, r_N) / r^{k+1}.
RadialFunction	sphericalBall_F (int k)
	Spherical ball model for F(r, r_N) [default]. Uniformly distributed point k-poles.
RadialFunction	sphericalShell_F ()
	Spherical shell F(r, r_N): 0 for r < r_N, 1 for r > r_N.
RadialFunction	pointlike_F ()
	Pointlike F(r): F = 1 everywhere.
RadialFunction	VolotkaSP_F (double mu, double I_nuc, double l_pn, int gl, bool print=true)
	Volotka single-particle nuclear model: F(r, rN)
RadialFunction	uSP (double mu, double I_nuc, double l_pn, int gl, double n, double R, bool u_option, bool print=true)
	Elizarov single-particle magnetisation model [extended Volotka].
RadialFunction	doublyOddSP_F (double mut, double It, double mu1, double I1, double l1, int gl1, double I2, double l2, bool print=true)
	Volotka single-particle model for doubly-odd nuclei.
RadialFunction	generic_F (const Grid &grid, const std::vector< double > &rho, int k)
	Constructs F(r) from a numerical density rho(r) given on the radial grid.
double	convert_RME_to_HFSconstant_2J (int k, int tja, int tjb)
	Converts reduced matrix element to A/B coeficients (takes k, 2J, 2J)
double	convert_RME_to_HFSconstant (int k, int ka, int kb)
	Converts reduced matrix element to A/B coeficients (takes k, kappa, kappa)

Typedef Documentation

RadialFunction

using DiracOperator::Hyperfine::RadialFunction = typedef std::function<double(double r, double r_Nuc)>

Type for radial function F(r, r_Nuc) – nuclear magnetisation/charge distribution.

• F(r,rN) contains only finite nuclear size correction, not HFS radial function.
• F(r,rN) -> 1 for r > rN
• r and rN always in atomic units

The full HFS radial operator is t^k(r) = F(r, r_N) / r^{k+1}, formed by tk_radial. F(r) itself is defined by the chosen nuclear model; see sphericalBall_F, VolotkaSP_F, uSP, doublyOddSP_F, generic_F.

Function Documentation

tk_radial()

std::vector< double > DiracOperator::Hyperfine::tk_radial	(	int	k,
		double	rN,
		const std::vector< double > &	r,
		const RadialFunction &	hfs_F
	)

Forms the hyperfine radial operator: F(r, r_N) / r^{k+1}.

\[ t^k_{\rm radial}[r_i] = \frac{F(r_i, r_N)}{r_i^{k+1}} \]

See RadialFunction for the contract on F(r, r_N). Angular factors and signs are handled elsewhere.

Parameters

k	Multipole rank (1 = M1, 2 = E2, ...)
rN	Nuclear radius (a.u.); passed to hfs_F as r_N
r	Radial grid points (a.u.)
hfs_F	Nuclear magnetisation/charge distribution F(r, r_N); see RadialFunction and sphericalBall_F

Returns

Radial values of t^k(r)

sphericalBall_F()

RadialFunction DiracOperator::Hyperfine::sphericalBall_F

(

int

k

)

Spherical ball model for F(r, r_N) [default]. Uniformly distributed point k-poles.

Based on a simple multipole expansion, assuming the nucleus is composed of uniformly distributed point k-poles. Note: not everyone defines this the same way.

\[ F(r, r_N) = \begin{cases} (r/r_N)^{2k+1} & r < r_N \\ 1 & r > r_N \end{cases} \]

sphericalShell_F()

RadialFunction DiracOperator::Hyperfine::sphericalShell_F

(

)

Spherical shell F(r, r_N): 0 for r < r_N, 1 for r > r_N.

pointlike_F()

RadialFunction DiracOperator::Hyperfine::pointlike_F

(

)

Pointlike F(r): F = 1 everywhere.

VolotkaSP_F()

RadialFunction DiracOperator::Hyperfine::VolotkaSP_F	(	double	mu,
		double	I_nuc,
		double	l_pn,
		int	gl,
		bool	print = true
	)

Volotka single-particle nuclear model: F(r, rN)

Calculates the Bohr-Weisskopf magnetisation distribution using the Volotka single-particle model of Phys. Rev. Lett. 125, 063002 (2020). Assumes radial nuclear density is a step function. F goes to 1 as r_N -> 0. See RadialFunction for F(r) contract.

Parameters

mu	Nuclear magnetic moment (in nuclear magnetons)
I_nuc	Nuclear spin
l_pn	Orbital angular momentum of valence nucleon
gl	Orbital g-factor (1 = proton, 0 = neutron)
print	Print model details

Returns

Radial function \( F_{\mathrm{BW}}(r, r_N) \)

uSP()

RadialFunction DiracOperator::Hyperfine::uSP	(	double	mu,
		double	I_nuc,
		double	l_pn,
		int	gl,
		double	n,
		double	R,
		bool	u_option,
		bool	print = true
	)

Elizarov single-particle magnetisation model [extended Volotka].

Returns the radial magnetisation function F(r, r_N) for the model of Elizarov, A. A. et al., Opt. Spectrosc. 100, 361 (2006). Uses nuclear radial density u(r), with:

\[ u(r) = \begin{cases} u_0 (R-r)^n & \text{u1(r)} \\ u_0 r^n & \text{u2(r)} \end{cases} \]

n=0 should correspond to the Volotka model (see VolotkaSP_F). See RadialFunction for F(r) contract.

Parameters

mu	Nuclear magnetic moment (in nuclear magnetons)
I_nuc	Nuclear spin
l_pn	Orbital angular momentum of valence nucleon
gl	Orbital g-factor (1 = proton, 0 = neutron)
n	Power; usually 0, 1, or 2. 0 => Volotka
R	Nuclear radius
u_option	Selects radial form: true for u1(r), false for u2(r)
print	Print model details

Returns

Radial magnetisation function F(r, r_N)

Warning

Normalisation is done by numerical integration; may be unstable. Check that result returns to VolotkaSP_F as n -> 0.

doublyOddSP_F()

RadialFunction DiracOperator::Hyperfine::doublyOddSP_F	(	double	mut,
		double	It,
		double	mu1,
		double	I1,
		double	l1,
		int	gl1,
		double	I2,
		double	l2,
		bool	print = true
	)

Volotka single-particle model for doubly-odd nuclei.

From: Phys. Rev. Lett. 125, 063002 (2020).

Total magnetisation:

\[ g F(r) = 0.5 \left[ g_1 F_1(r) + g_2 F_2(r) + (g_1 F_1(r) - g_2 F_2(r)) K \right] \]

with

\[ K = \frac{[ I_1(I_1+1) - I_2(I_2+1) ]}{[ I(I+1) ]} \]

Returns F(r) (i.e., divided by total g).

g2 is obtained from g = 0.5 [ g1 + g2 + (g1 - g2) K ].

Note

F1, F2 are the Volotka single-particle functions (see VolotkaSP_F).

Parameters

mut	Total nuclear magnetic moment (in nuclear magnetons)
It	Total nuclear spin
mu1	Magnetic moment of nucleon 1
I1	Spin of nucleon 1
l1	Orbital angular momentum of nucleon 1
gl1	Orbital g-factor of nucleon 1 (1 = proton, 0 = neutron)
I2	Spin of nucleon 2
l2	Orbital angular momentum of nucleon 2
print	Print model details

Returns

Radial function F(r)

generic_F()

RadialFunction DiracOperator::Hyperfine::generic_F	(	const Grid &	grid,
		const std::vector< double > &	rho,
		int	k
	)

Constructs F(r) from a numerical density rho(r) given on the radial grid.

Computes the cumulative radial integral with rank-k weighting:

\[ F(r) = N \int_0^r x^{2k} \rho(x)\, \d x, \qquad N = \frac{1}{\int_0^\infty x^{2k} \rho(x)\, \d x} \]

so that \( F(r) \to 1 \) as \( r \to \infty \).

The x^{2k} weighting matches the radial scaling of the multipole operator of rank k (consistent with sphericalBall_F, where F ~ r^{2k+1} inside the nucleus).

The density rho need not be pre-normalised. The returned function ignores its second argument (r_Nuc), since the distribution is already encoded in rho. Assumes r is always a grid point (no interpolation).

Parameters

grid	Radial grid on which rho is defined.
rho	Nuclear density values on the grid points.
k	Multipole rank (1 = M1, 2 = E2, ...)

Returns

Radial distribution function F(r, r_Nuc).

convert_RME_to_HFSconstant_2J()

double DiracOperator::Hyperfine::convert_RME_to_HFSconstant_2J	(	int	k,
		int	tja,
		int	tjb
	)

Converts reduced matrix element to A/B coeficients (takes k, 2J, 2J)

convert_RME_to_HFSconstant()

double DiracOperator::Hyperfine::convert_RME_to_HFSconstant	(	int	k,
		int	ka,
		int	kb
	)

Converts reduced matrix element to A/B coeficients (takes k, kappa, kappa)