Nuclear Namespace Reference

Detailed Description

Data and useful functions for nuclear properties and potentials. Radii all in Fermi (fm, e-15m) from Nuclear Data Service: https://www-nds.iaea.org/.

Classes
struct	Isotope
	Isotope data: Z, A, r_rms/fm, I, pi, mu, Q. More...
class	Nucleus
	Stores set of nuclear parameters (all radii in fm) More...

Enumerations
enum class	ChargeDistro {   Fermi , spherical , point , Gaussian ,   custom , Error }
	Nuclear charge distribution options. More...

Functions
Isotope	findIsotopeData (int z, int a)
	Looks up + returns an isotope from the list. If not in list, partially blank.
std::vector< Isotope >	findIsotopeList (int z)
	Returns all known isotopes of given atom.
double	find_rrms (int z, int a)
	Looks up default value of r_rms for given isotope. Returns 0 if not found.
double	find_mu (int z, int a)
	As above, for dipole moment.
int	find_parity (int z, int a)
	As above, for parity.
double	find_spin (int z, int a)
	As above, for nuclear spin. Returns -1 if not found.
double	approximate_r_rms (int a, int z)
	Approximate rms radius from a for to Angeli data.
double	c_hdr_formula_rrms_t (double rrms, double t=default_t)
	Calculates c from rrms and t.
double	rrms_formula_c_t (double c, double t=default_t)
	Calculates rrms from c and t.
double	deformation_effective_t (double c, double t, double beta)
	Calculates effective skin thickness due to quadrupole deformation [See Eq. 8 of https://doi.org/10.1103/PhysRevA.100.032511.].
double	approximate_t_skin (int a)
	just returns 2.3
ChargeDistro	parseType (const std::string &str_type)
	Converts string to ChargeDistro (case-insensitive).
std::string	parseType (ChargeDistro type)
	Converts ChargeDistro to string.
std::ostream &	operator<< (std::ostream &ostr, const Nucleus &n)
Nucleus	form_nucleus (int Z, std::optional< int > A=std::nullopt, IO::InputBlock input={})
	Forms nucleus from Z, A, and the nuclear options in input.
std::vector< double >	sphericalNuclearPotential (double Z, double rnuc, const std::vector< double > &rgrid)
	Nuclear potential: uniformly charged sphere (ball) of radius rnuc.
std::vector< double >	GaussianNuclearPotential (double Z, double r_rms, const std::vector< double > &rgrid)
	Nuclear potential: Gaussian charge distribution.
std::vector< double >	fermiNuclearPotential (double Z, double t, double c, const std::vector< double > &rgrid)
	Nuclear potential: Fermi charge distribution.
std::vector< double >	fermiNuclearDensity_tcN (double t, double c, double Z_norm, const Grid &grid)
	Fermi charge distribution, rho(r), normalised to Z_norm.
std::vector< double >	formPotential (const Nucleus &nucleus, const std::vector< double > &r)
	Nuclear potential V(r) for given nucleus, at radial points r.

Variables
constexpr double	default_t = 2.30
constexpr auto	FourLn3 = 4.0 * 1.098612289
constexpr auto	Pi2 = M_PI * M_PI

---

Class Documentation

Nuclear::Isotope

struct Nuclear::Isotope

Class Members
int	Z	Atomic charge.
int	A	Atomic mass number (A = Z + N)
optional< double >	r_rms {}	root-mean-square charge radius, in Fermi (fm, e-15m)
optional< double >	I_N {}	Nuclear spin (in hbar)
optional< int >	parity {}
optional< double >	mu {}	Magnetic dipole moment, in nuclear magnetons.
optional< double >	q {}	Magnetic quadrupole moment, in barns.

Enumeration Type Documentation

ChargeDistro

enum class Nuclear::ChargeDistro

strong

Nuclear charge distribution options.

• Fermi : two-parameter Fermi distribution (c and t)
• spherical : uniformly charged sphere (ball)
• point : point-like nucleus
• Gaussian : Gaussian distribution
• custom : potential read from file (see formPotential)
• Error : unrecognised type (see parseType)

Function Documentation

findIsotopeData()

Isotope Nuclear::findIsotopeData	(	int	z,
		int	a
	)

Looks up + returns an isotope from the list. If not in list, partially blank.

findIsotopeList()

std::vector< Isotope > Nuclear::findIsotopeList

(

int

z

)

Returns all known isotopes of given atom.

find_rrms()

double Nuclear::find_rrms	(	int	z,
		int	a
	)

Looks up default value of r_rms for given isotope. Returns 0 if not found.

find_mu()

double Nuclear::find_mu	(	int	z,
		int	a
	)

As above, for dipole moment.

find_parity()

int Nuclear::find_parity	(	int	z,
		int	a
	)

As above, for parity.

find_spin()

double Nuclear::find_spin	(	int	z,
		int	a
	)

As above, for nuclear spin. Returns -1 if not found.

approximate_r_rms()

double Nuclear::approximate_r_rms	(	int	A,
		int	Z
	)

Approximate rms radius from a for to Angeli data.

c_hdr_formula_rrms_t()

double Nuclear::c_hdr_formula_rrms_t	(	double	rrms,
		double	t
	)

Calculates c from rrms and t.

rrms_formula_c_t()

double Nuclear::rrms_formula_c_t	(	double	c,
		double	t
	)

Calculates rrms from c and t.

deformation_effective_t()

double Nuclear::deformation_effective_t	(	double	c,
		double	t,
		double	beta
	)

Calculates effective skin thickness due to quadrupole deformation [See Eq. 8 of https://doi.org/10.1103/PhysRevA.100.032511.].

approximate_t_skin()

double Nuclear::approximate_t_skin

(

int

)

just returns 2.3

parseType() [1/2]

ChargeDistro Nuclear::parseType

(

const std::string &

str_type

)

Converts string to ChargeDistro (case-insensitive).

Also accepts: "spher*", "ball", "uniform" (spherical); "point*", "zero*" (point); "gaus*" (Gaussian). If unknown, prints a warning with the closest match.

Returns

ChargeDistro; ChargeDistro::Error if unknown.

parseType() [2/2]

std::string Nuclear::parseType

(

ChargeDistro

type

)

Converts ChargeDistro to string.

form_nucleus()

Nucleus Nuclear::form_nucleus	(	int	Z,
		std::optional< int >	A = std::nullopt,
		IO::InputBlock	input = {}
	)

Forms nucleus from Z, A, and the nuclear options in input.

Starts from the default nucleus for given Z and A (see Nucleus), then over-rides with any options given in input: type, rrms, c, t, beta, input_file, parameters. If c is given, it over-rides rrms. If A or r_rms is zero, the nucleus is set to point-like (except for custom).

Parameters

Z	Nuclear charge
A	Mass number; default A for Z if not given
input	Input block with nuclear options (see input.check() in .cpp)

Note

input_file only sets the file name: type must also be custom.

sphericalNuclearPotential()

std::vector< double > Nuclear::sphericalNuclearPotential	(	double	Z,
		double	rnuc,
		const std::vector< double > &	rgrid
	)

Nuclear potential: uniformly charged sphere (ball) of radius rnuc.

\[ V(r) = \begin{cases} -\frac{Z}{2 r_N}\left(3 - r^2/r_N^2\right) & r < r_N, \\ -Z/r & r \geq r_N. \end{cases} \]

rnuc = 0 gives a point-like nucleus.

Parameters

Z	Nuclear charge
rnuc	Nuclear charge radius, in fm (not r_rms: for a uniform sphere, r_N = sqrt(5/3) r_rms)
rgrid	Radial points (atomic units)

Returns

V(r) at each point of rgrid (atomic units).

GaussianNuclearPotential()

std::vector< double > Nuclear::GaussianNuclearPotential	(	double	Z,
		double	r_rms,
		const std::vector< double > &	rgrid
	)

Nuclear potential: Gaussian charge distribution.

\[ V(r) = -\frac{Z}{r}\,{\rm erf}\left(\sqrt{3/2}\;r/r_{\rm rms}\right). \]

Parameters

Z	Nuclear charge
r_rms	Root-mean-square charge radius, in fm
rgrid	Radial points (atomic units)

Returns

V(r) at each point of rgrid (atomic units).

fermiNuclearPotential()

std::vector< double > Nuclear::fermiNuclearPotential	(	double	Z,
		double	t,
		double	c,
		const std::vector< double > &	rgrid
	)

Nuclear potential: Fermi charge distribution.

For charge density

\[ \rho(r) = \rho_0 \left[ 1 + e^{(r - c)/a} \right]^{-1}, \qquad a = t/(4\ln 3), \]

normalised to Z. Expressed in terms of complete Fermi-Dirac integrals (GSL). V = -Z/r beyond r = 10c.

Parameters

Z	Nuclear charge
t	Skin thickness (90% to 10% fall-off), in fm
c	Half-density radius, in fm (not r_rms)
rgrid	Radial points (atomic units)

Returns

V(r) at each point of rgrid (atomic units).

fermiNuclearDensity_tcN()

std::vector< double > Nuclear::fermiNuclearDensity_tcN	(	double	t,
		double	c,
		double	Z_norm,
		const Grid &	grid
	)

Fermi charge distribution, rho(r), normalised to Z_norm.

\( \int \rho\, d^3r = 4\pi \int \rho(r)\, r^2 dr = Z_{\rm norm} \): Z_norm = Z for nuclear charge density; Z_norm = 1 for nuclear density. rho = 0 beyond r = 10c.

Parameters

t	Skin thickness, in fm
c	Half-density radius, in fm
Z_norm	Normalisation
grid	Radial grid

Returns

rho(r) on grid (atomic units).

formPotential()

std::vector< double > Nuclear::formPotential	(	const Nucleus &	nucleus,
		const std::vector< double > &	r
	)

Nuclear potential V(r) for given nucleus, at radial points r.

Calls one of the above, depending on nucleus type. If r_rms = 0, point-like (-Z/r) for all types except custom.

• Fermi: c from r_rms and t; if beta is non-zero, uses the effective skin thickness for the deformed nucleus (deformation_effective_t)
• spherical: radius r_N = sqrt(5/3) r_rms
• custom: reads {r, V(r)} (atomic units) from custom_pot_file(), and interpolates onto r. Beyond the last point, V = -Z_eff/r, with Z_eff from the last point. Not extrapolated to smaller r.

Returns

V(r) at each point of r (atomic units).

Warning

Aborts if the custom file is invalid (fewer than 3 points), or the type is invalid.