DiracODE Namespace Reference

Functions and classes used to solve the Dirac equation. More...

Classes
class	AsymptoticSpinor
	Performs asymptotic expansion for f and g at large r, up to order Nx in (1/r) More...
struct	DiracContinuumDerivative
	Derivative function for H-like; valid for continuum states at large r. More...

Functions
void	boundState (DiracSpinor &Fa, const double en0, const std::vector< double > &v, const std::vector< double > &H_off_diag={}, const double alpha=PhysConst::alpha, double eps=1.0e-14, const DiracSpinor *const VxFa=nullptr, const DiracSpinor *const Fa0=nullptr, double zion=1, double mass=1.0)
	Solves bound-state problem for local potential (en < 0) More...
void	regularAtOrigin (DiracSpinor &Fa, const double en, const std::vector< double > &v, const std::vector< double > &H_off_diag, const double alpha, const DiracSpinor *const VxFa=nullptr, const DiracSpinor *const Fa0=nullptr, double zion=1, double mass=1.0)
	For given energy en, solves DE with correct boundary conditions at the origin.
void	regularAtInfinity (DiracSpinor &Fa, const double en, const std::vector< double > &v, const std::vector< double > &H_off_diag, const double alpha, const DiracSpinor *const VxFa=nullptr, const DiracSpinor *const Fa0=nullptr, double zion=1, double mass=1.0)
	For given energy en, solves (local) DE with correct boundary conditions at infinity.
DiracSpinor	boundState (int n, int kappa, const double en0, const std::shared_ptr< const Grid > &gr, const std::vector< double > &v, const std::vector< double > &H_off_diag={}, const double alpha=PhysConst::alpha, double eps=1.0e-14, const DiracSpinor *const VxFa=nullptr, const DiracSpinor *const Fa0=nullptr, double zion=1, double mass=1.0)
void	regularAtOrigin_C (DiracSpinor &FaR, DiracSpinor &FaI, const std::complex< double > en, const std::vector< double > &v, const std::vector< double > &H_off_diag, const double alpha)
	For given complex energy en, solves DE with correct boundary conditions at the origin.
void	regularAtInfinity_C (DiracSpinor &FaR, DiracSpinor &FaI, const std::complex< double > en, const std::vector< double > &v, const std::vector< double > &H_off_diag, const double alpha)
	For given complex energy en, solves (local) DE with correct boundary conditions at infinity.
void	solveContinuum (DiracSpinor &Fa, double en, const std::vector< double > &v, double alpha, const DiracSpinor *const VxFa=nullptr, const DiracSpinor *const Fa0=nullptr)
	For given energy en (en > 0), solves Dirac eq. for continuum state (with energy normalisation). More...
std::pair< double, double >	numerical_f_amplitude (double en, int kappa, double alpha, double Zeff, double f_final, double g_final, double r_final, double dr)
	Finds the (numerical) amplitude of f(r) continuum Dirac solution at large r. More...
double	analytic_f_amplitude (double en, double alpha)
	Analytic amplitude of f(r) at very large r, for H-like Dirac continuum.
double	fitQuadratic (double x1, double x2, double x3, double y1, double y2, double y3)
	Fits a quadratic to three points {x,y}, assuming |y2| = max(|y1|,|y2|,|y3|). More...
DiracSpinor	solve_inhomog (const int kappa, const double en, const std::vector< double > &v, const std::vector< double > &H_mag, const double alpha, const DiracSpinor &source, const DiracSpinor *const VxFa=nullptr, const DiracSpinor *const Fa0=nullptr, double zion=1, double mass=1.0)
	Solves inhomogeneous Dirac equation. More...
void	solve_inhomog (DiracSpinor &Fa, const double en, const std::vector< double > &v, const std::vector< double > &H_mag, const double alpha, const DiracSpinor &source, const DiracSpinor *const VxFa=nullptr, const DiracSpinor *const Fa0=nullptr, double zion=1, double mass=1.0)
	Solves inhomogeneous Dirac equation. More...
void	solve_inhomog (DiracSpinor &Fa, DiracSpinor &Fzero, DiracSpinor &Finf, const double en, const std::vector< double > &v, const std::vector< double > &H_mag, const double alpha, const DiracSpinor &source, const DiracSpinor *const VxFa=nullptr, const DiracSpinor *const Fa0=nullptr, double zion=1, double mass=1.0)
	Solves inhomogeneous Dirac equation. More...

Detailed Description

Functions and classes used to solve the Dirac equation.

Function Documentation

boundState()

void DiracODE::boundState	(	DiracSpinor &	Fa,
		const double	en0,
		const std::vector< double > &	v,
		const std::vector< double > &	H_off_diag = {},
		const double	alpha = PhysConst::alpha,
		double	eps = 1.0e-14,
		const DiracSpinor *const	VxFa = nullptr,
		const DiracSpinor *const	Fa0 = nullptr,
		double	zion = 1,
		double	mass = 1.0
	)

Solves bound-state problem for local potential (en < 0)

\[ (H_0 + v - \epsilon_a)F_a = 0\]

en0 is initial energy guess (must be reasonably good). log_eps: log10(eps); eps is convergence target for energy.

• v is local potential (e.g., v = v_dir + v_nuc)
• H_off_diag is optional off-diagonal potential.
• alpha: \(\alpha = \lambda\alpha_0\) is the effective value of fine-structure constant

fitQuadratic()

double DiracODE::fitQuadratic	(	double	x1,
		double	x2,
		double	x3,
		double	y1,
		double	y2,
		double	y3
	)

Fits a quadratic to three points {x,y}, assuming |y2| = max(|y1|,|y2|,|y3|).

Used to find amplitude of sin(x); points must be close to max

numerical_f_amplitude()

std::pair< double, double > DiracODE::numerical_f_amplitude	(	double	en,
		int	kappa,
		double	alpha,
		double	Zeff,
		double	f_final,
		double	g_final,
		double	r_final,
		double	dr
	)

Finds the (numerical) amplitude of f(r) continuum Dirac solution at large r.

It does this by continuing ODE integration to large r until:
(a) wavelength and,
(b) amplitude
become constant. It starts from a solution hat has been already integrated out to rmax of regular radial grid. It uses linearly-spaced grid (dr), and assumes H-like potential (-Z/r).

solve_inhomog() [1/3]

DiracSpinor DiracODE::solve_inhomog	(	const int	kappa,
		const double	en,
		const std::vector< double > &	v,
		const std::vector< double > &	H_mag,
		const double	alpha,
		const DiracSpinor &	source,
		const DiracSpinor *const	VxFa = nullptr,
		const DiracSpinor *const	Fa0 = nullptr,
		double	zion = 1,
		double	mass = 1.0
	)

Solves inhomogeneous Dirac equation.

\[ (H_0 + v -\epsilon_a)F_a = S \]

with ‘source’ term, S. Solves for \(\psi_\kappa\) with angular momentum kappa. en = \(\epsilon\) is given. Note sign of S. Uses Green's method (see Method documentation).

solve_inhomog() [2/3]

void DiracODE::solve_inhomog	(	DiracSpinor &	Fa,
		const double	en,
		const std::vector< double > &	v,
		const std::vector< double > &	H_mag,
		const double	alpha,
		const DiracSpinor &	source,
		const DiracSpinor *const	VxFa = nullptr,
		const DiracSpinor *const	Fa0 = nullptr,
		double	zion = 1,
		double	mass = 1.0
	)

Solves inhomogeneous Dirac equation.

As above. Overload to accept/overwrite solution to Fa. kappa is taken from Fa.

solve_inhomog() [3/3]

void DiracODE::solve_inhomog	(	DiracSpinor &	Fa,
		DiracSpinor &	Fzero,
		DiracSpinor &	Finf,
		const double	en,
		const std::vector< double > &	v,
		const std::vector< double > &	H_mag,
		const double	alpha,
		const DiracSpinor &	source,
		const DiracSpinor *const	VxFa = nullptr,
		const DiracSpinor *const	Fa0 = nullptr,
		double	zion = 1,
		double	mass = 1.0
	)

Solves inhomogeneous Dirac equation.

As above. Overload to accept/overwrite solution to Fa. All these routines solve also for Fzero, Finf, which are solutions to homogeneous equation (H-en)Fa = 0 [reg @ origin, and infinity, respectively].

• The first two throw these solutions away, the third keeps them (in some cases they can be re-used)
• These Spinors are solved internally and over-written, they don't need to be solved first (i.e., they are out parameters, not in/out parameters)

solveContinuum()

void DiracODE::solveContinuum	(	DiracSpinor &	Fa,
		double	en,
		const std::vector< double > &	v,
		double	alpha,
		const DiracSpinor *const	VxFa = nullptr,
		const DiracSpinor *const	Fa0 = nullptr
	)

For given energy en (en > 0), solves Dirac eq. for continuum state (with energy normalisation).

Normalisation is achieved by continuuing solving ODE to very large r, and comparing asymptotic amplitude to that of analytic solution. We only keep solution on regular grid; extended part is not kept.