DiracODE Namespace Reference

Detailed Description

Functions and classes used to solve the Dirac equation.

Classes
class	AsymptoticSpinor
	Performs asymptotic expansion for f and g at large r, up to order Nx in (1/r). More...
struct	DiracContinuumDerivative
	H-like Dirac derivative matrix 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).
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)
	Factory overload: constructs a DiracSpinor(n, kappa, gr) and calls boundState().
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 Dirac equation 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 Dirac equation 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)
	Solves Dirac equation for a continuum state (en > 0) with energy normalisation.
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 and phase of f(r) for a continuum Dirac solution at large r.
double	analytic_f_amplitude (double en, double alpha)
	Analytic amplitude of f(r) at very large r for an H-like Dirac continuum state.
double	fitQuadratic (double x1, double x2, double x3, double y1, double y2, double y3)
	Fits a quadratic to three points and returns the interpolated maximum.
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 the inhomogeneous Dirac equation, returning the solution spinor.
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 the inhomogeneous Dirac equation, overwriting Fa.
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 the inhomogeneous Dirac equation, overwriting Fa and exposing the homogeneous solutions.

Function Documentation

boundState() [1/2]

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).

Solves \( (H_0 + v - \epsilon_a)F_a = 0 \) for the bound state. en0 is the initial energy guess (must be reasonably good). eps is the convergence target for the energy.

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

regularAtOrigin()

void DiracODE::regularAtOrigin	(	DiracSpinor &	Fa,
		const double	en,
		const std::vector< double > &	v,
		const std::vector< double > &	H_mag,
		const double	alpha,
		const DiracSpinor *const	VxFa,
		const DiracSpinor *const	Fa0,
		double	zion,
		double	mass
	)

For given energy en, solves DE with correct boundary conditions at the origin.

regularAtInfinity()

void DiracODE::regularAtInfinity	(	DiracSpinor &	Fa,
		const double	en,
		const std::vector< double > &	v,
		const std::vector< double > &	H_mag,
		const double	alpha,
		const DiracSpinor *const	VxFa,
		const DiracSpinor *const	Fa0,
		double	zion,
		double	mass
	)

For given energy en, solves (local) DE with correct boundary conditions at infinity.

boundState() [2/2]

DiracSpinor DiracODE::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  )

inline

Factory overload: constructs a DiracSpinor(n, kappa, gr) and calls boundState().

regularAtOrigin_C()

void DiracODE::regularAtOrigin_C	(	DiracSpinor &	FaR,
		DiracSpinor &	FaI,
		const std::complex< double >	en,
		const std::vector< double > &	v,
		const std::vector< double > &	H_mag,
		const double	alpha
	)

For given complex energy en, solves Dirac equation with correct boundary conditions at the origin.

regularAtInfinity_C()

void DiracODE::regularAtInfinity_C	(	DiracSpinor &	FaR,
		DiracSpinor &	FaI,
		const std::complex< double >	en,
		const std::vector< double > &	v,
		const std::vector< double > &	H_mag,
		const double	alpha
	)

For given complex energy en, solves Dirac equation with correct boundary conditions at infinity.

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
	)

Solves Dirac equation for a continuum state (en > 0) with energy normalisation.

Normalisation is achieved by continuing the ODE integration to very large r and comparing the asymptotic amplitude to that of the analytic solution. Only the solution on the regular grid is kept; the extended part is discarded.

Parameters

Fa	Output spinor (result stored here).
en	Continuum energy (must be > 0).
v	Local potential v(r).
alpha	Fine-structure constant.
VxFa	Optional exchange potential. If nullptr, ignored.
Fa0	Optional inhomogeneous source spinor. If nullptr, ignored.

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 and phase of f(r) for a continuum Dirac solution at large r.

Continues ODE integration beyond the regular grid until both the wavelength and amplitude become constant. Assumes an H-like potential (-Zeff/r) and a linearly-spaced extension grid with step dr.

Parameters

en	Continuum energy.
kappa	Orbital kappa quantum number.
alpha	Fine-structure constant.
Zeff	Effective nuclear charge.
f_final	Value of f at the end of the regular grid.
g_final	Value of g at the end of the regular grid.
r_final	Radial position at the end of the regular grid.
dr	Step size for the extended linear grid.

Returns

{amplitude, phase} of the asymptotic f(r) oscillation.

analytic_f_amplitude()

double DiracODE::analytic_f_amplitude	(	double	en,
		double	alpha
	)

Analytic amplitude of f(r) at very large r for an H-like Dirac continuum state.

Parameters

en	Continuum energy.
alpha	Fine-structure constant.

Returns

Analytic asymptotic amplitude.

fitQuadratic()

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

Fits a quadratic to three points and returns the interpolated maximum.

Assumes |y2| = max(|y1|, |y2|, |y3|); used to find the amplitude of a sinusoidal oscillation. The three points must be close to the maximum.

Parameters

x1,x2,x3	x-coordinates of the three points.
y1,y2,y3	y-coordinates of the three points.

Returns

Interpolated maximum value.

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 the inhomogeneous Dirac equation, returning the solution spinor.

Solves \( (H_0 + v - \epsilon_a) F_a = S \) for \( \psi_\kappa \) using Green's method (see Methods documentation). Note the sign convention for S.

Parameters

kappa	Angular momentum kappa of the solution.
en	Orbital energy \( \epsilon \).
v	Local potential v(r).
H_mag	Off-diagonal (magnetic) potential.
alpha	Fine-structure constant.
source	Inhomogeneous source term S.
VxFa	Optional exchange potential. If nullptr, ignored.
Fa0	Optional inhomogeneous source spinor. If nullptr, ignored.
zion	Effective ionic charge (default 1).
mass	Effective particle mass in atomic units (default 1 = m_e).

Returns

Solution spinor Fa.

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 the inhomogeneous Dirac equation, overwriting Fa.

As above; 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 the inhomogeneous Dirac equation, overwriting Fa and exposing the homogeneous solutions.

As above, but also returns the homogeneous solutions Fzero (regular at origin) and Finf (regular at infinity), which satisfy \( (H_0 + v - \epsilon)F = 0 \). The first two overloads discard these; this one keeps them for potential reuse. Fzero and Finf are out parameters – they are overwritten internally and do not need to be initialised before calling.