DiracOperator Namespace Reference

Detailed Description

Dirac operators: TensorOperator base class and derived implementations for single-particle (one-body) spherical tensor operators.

This namespace contains the TensorOperator base class and all derived operator classes. Operators act on DiracSpinor objects and compute reduced matrix elements (RMEs) of the form:

\[ \langle a \| \hat{h} \| b \rangle = A_{ab} \cdot R_{ab} \]

where \( A_{ab} \) is a purely angular factor and \( R_{ab} \) is a radial integral. New operators are implemented by deriving from TensorOperator (or ScalarOperator for rank-0 operators) and overriding the relevant virtual functions; see TensorOperator for details.

Operators are typically constructed via generate(), which takes a name string and an InputBlock; see GenerateOperator.hpp for the full list of available operators.

See TensorOperator class documentation for main descriptions. All usable tensor operators dervive from that virtual class.

From the command line, use

ampsci -o

to get a list of available operators. For a specific operator 'OperatorName', use:

ampsci -o OperatorName

to see available run-time options for that operator. These options are passed to generate().

Note

New operator classes must be registered in the operator_list in GenerateOperator.hpp to be accessible at runtime.

Namespaces
namespace	Hyperfine
	Auxillary Functions for hyperfine operatrs; F(r) [nuclear distribution] and similar.
namespace	multipole
	Helper functions for the multipole operators.

Classes
class	AEk
	Axial electric multipole operator: \( A^E_K = T^{(+1)}_K(q)\gamma^5 \). More...
class	AEk_lowq
	Low qr form of Axial electric multipole operator: \( A^E_K = T^{(+1)}_K(q)\gamma^5\). More...
class	ALk
	Axial longitudinal multipole operator: \( A^L_K = T^{(-1)}_K(q)\gamma^5 \). More...
class	ALk_lowq
	Low qr form of Axial longitudinal multipole operator: \( A^L_K = T^{(-1)}_K(q)\gamma^5\). More...
class	AMk
	Axial magnetic multipole operator: \( A^M_K = T^{(0)}_K(q)\gamma^5 \). More...
class	AMk_lowq
	Low qr form of Axial magnetic multipole operator: \( A^M_K = T^{(0)}_K(q)\gamma^5\). More...
class	E1
	Electric dipole operator: -|e|r = -er. More...
class	E1v
	Electric dipole operator, v-form: \( \frac{ie}{\omega \alpha} \vec{\alpha}\). More...
class	Ek
	E^k (electric multipole) operator, length form, with (qr<<1) approximation. More...
class	EM_multipole
	Intermediate abstract base class for all EM relativistic multipole operators. More...
class	fieldshift
	Field shift operator: dV = V(r, nuc2) - V(r, nuc1) More...
class	g0jL
	Matrix element of tensor operator: gamma^0 J_L(qr) C^L. More...
class	hfs
	Generalised hyperfine-structure operator, including relevant nuclear moment. More...
class	ig0g5jL
	Matrix element of tensor operator: i gamma^0gamma^5 J_L(qr) C^L. nb: i makes ME real. More...
class	ig5jL
	Matrix element of tensor operator: i gamma^5 J_L(qr) C^L. nb: i makes ME real. More...
class	j
	j (total angular momentum) operator More...
class	jL
	Matrix element of tensor operator: J_L(qr)*C^L. More...
class	l
	l (orbital angular momentum) operator More...
class	M1
	Magnetic dipole operator: <a||M1||b> More...
class	M1nr
	Magnetic dipole operator, in non-relativistic form: M1 = L + 2S. More...
class	MLVP
	Magnetic loop vacuum polarisation (Uehling vertex) More...
class	NullOperator
	Speacial operator: 0. More...
class	p
	Momentum operator p = -i grad. More...
class	Phi5k
	Temporal component of the axial vector multipole operator. More...
class	Phi5k_lowq
	Low qr form of Temporal component of the axial vector multipole operator: \( \Theta_K = \Phi^5_K = t^K(q)\gamma^5\) . NOTE: If K=0, omega should be (ea-eb); for K=1 should be q = alpha*omega! More...
class	Phik
	Temporal component of the vector multipole operator: \( \Phi_K = t^K(q) \). More...
class	Phik_lowq
	Low qr form of Temporal component of the vector multipole operator: \( \Phi_K = t^K(q)\). More...
class	PNCnsi
	Nuclear-spin independent PNC operator (Qw) More...
class	PNCnsi_const
	PNC nuclear-spin-independent operator with uniform (constant) nuclear density. More...
class	RadialF
	General function of r, even scalar operator. More...
class	s
	s (spin) operator More...
class	S5k
	Pseudoscalar multipole operator: t^k (i g^0 g^5) More...
class	S5k_lowq
	Low qr form of Pseudoscalar multipole operator: \( P_K = S^5_K = t^K(q)(i\gamma^0\gamma^5)\) NOTE: If K=0, omega should be (ea-eb); for K=1 should be q = alpha*omega! More...
class	ScalarOperator
	Rank-0 (scalar) tensor operator; derives from TensorOperator with k=0. More...
class	sigma_r
	Odd-parity rank-0 operator with radial function r (sigma_r) More...
class	Sk
	Scalar multipole operator: \( S_K = t^K(q)\gamma^0 \). More...
class	Sk_lowq
	Low qr form of Scalar multipole operator: \( S_K = t^K(q)\gamma^0\). More...
class	TensorOperator
	General tensor operator (virtual base class); all single-particle (one-body) tenosor operators derive from this. More...
class	VEk
	Vector electric multipole (V-form) operator: \( V^E_K = T^{(+1)}_K(q) \). More...
class	VEk_Len
	Vector electric multipole (transition) operator, Length-form: ( \( T^{(+1),{\rm Len}}_K \) ). More...
class	VEk_lowq
	Low qr form of Vector electric. Only for K=1 (zero otherwise) More...
class	VertexQED
	Effective VertexQED operator. More...
class	VLk
	Vector longitudinal multipole operator (V-form): \( V^L_K = T^{(-1)}_K(q) \). More...
class	VLk_lowq
	Low qr form of Vector longitudinal. More...
class	VMk
	Vector magnetic multipole operator: \( V^M_K = T^{(0)}_K(q) \). More...
class	VMk_lowq
	Low qr form of Vector magnetic. Only for K=1 (zero otherwise) More...
class	Vrad
	Flambaum-Ginges radiative potential operator. More...

Enumerations
enum class	Parity { even , odd , Error }
	Parity of operator. More...
enum class	Realness { real , imaginary , Error }
	Specifies whether an operator's matrix elements are real or imaginary. More...
enum class	MatrixElementType { Reduced , Stretched , HFConstant , Error }
	Type of matrix element returned. More...

Functions
std::unique_ptr< DiracOperator::TensorOperator >	generate (std::string_view operator_name, const IO::InputBlock &input, const Wavefunction &wf)
	Returns a unique_ptr (polymorphic) to the requested operator, with given properties.
void	list_operators ()
	List available operators.
std::unique_ptr< DiracOperator::TensorOperator >	generate_sigma_r (const IO::InputBlock &input, const Wavefunction &wf)
std::unique_ptr< DiracOperator::TensorOperator >	generate_E1 (const IO::InputBlock &input, const Wavefunction &wf)
std::unique_ptr< DiracOperator::TensorOperator >	generate_E1v (const IO::InputBlock &input, const Wavefunction &wf)
std::unique_ptr< DiracOperator::TensorOperator >	generate_E2 (const IO::InputBlock &input, const Wavefunction &wf)
std::unique_ptr< DiracOperator::TensorOperator >	generate_ialpha (const IO::InputBlock &input, const Wavefunction &)
std::unique_ptr< DiracOperator::TensorOperator >	generate_Ek (const IO::InputBlock &input, const Wavefunction &wf)
std::unique_ptr< DiracOperator::TensorOperator >	MultipoleOperator (const Grid &grid, int k, double omega, char type, char comp, bool low_q, const SphericalBessel::JL_table *jl=nullptr)
	Factory for relativistic multipole operators.
std::unique_ptr< DiracOperator::TensorOperator >	generate_Multipole (const IO::InputBlock &input, const Wavefunction &wf)
std::unique_ptr< DiracOperator::TensorOperator >	generate_fieldshift (const IO::InputBlock &input, const Wavefunction &wf)
std::unique_ptr< DiracOperator::TensorOperator >	generate_hfs (const IO::InputBlock &input, const Wavefunction &wf)
std::unique_ptr< DiracOperator::TensorOperator >	generate_l (const IO::InputBlock &input, const Wavefunction &)
std::unique_ptr< DiracOperator::TensorOperator >	generate_s (const IO::InputBlock &input, const Wavefunction &)
std::unique_ptr< DiracOperator::TensorOperator >	generate_M1 (const IO::InputBlock &input, const Wavefunction &wf)
std::unique_ptr< DiracOperator::TensorOperator >	generate_M1nr (const IO::InputBlock &input, const Wavefunction &)
std::unique_ptr< DiracOperator::TensorOperator >	generate_p (const IO::InputBlock &input, const Wavefunction &)
std::unique_ptr< DiracOperator::TensorOperator >	generate_pnc (const IO::InputBlock &input, const Wavefunction &wf)
std::unique_ptr< DiracOperator::TensorOperator >	generate_Vrad (const IO::InputBlock &input, const Wavefunction &wf)
std::unique_ptr< DiracOperator::TensorOperator >	generate_MLVP (const IO::InputBlock &input, const Wavefunction &wf)
std::unique_ptr< DiracOperator::TensorOperator >	generate_r (const IO::InputBlock &input, const Wavefunction &wf)
double	Pab (double pm, const std::vector< double > &t, const DiracSpinor &Fa, const DiracSpinor &Fb)
	Pab function: Int[ (fa*gb + pm*ga*fb) * t(r) , dr]. pm = +/-1 (usually)
double	Rab (double pm, const std::vector< double > &t, const DiracSpinor &Fa, const DiracSpinor &Fb)
	Rab function: Int[ (fa*fb + pm*ga*gb) * t(r) , dr]. pm = +/-1 (usually)
void	Pab_rhs (double pm, const std::vector< double > &t, DiracSpinor *dF, const DiracSpinor &Fb, double A=1.0)
	Pab_rhs function: dF_ab += A * t(r) * (g, pm*f) , pm=+/-1 (usually). NOTE: uses +=, so can combine. Ensure empty to begin.
void	Rab_rhs (double pm, const std::vector< double > &t, DiracSpinor *dF, const DiracSpinor &Fb, double A=1.0)
	Rab_rhs function: dF_ab += A * t(r) * (f, pm*g) , pm=+/-1 (usually). NOTE: uses +=, so can combine. Ensure empty to begin.
double	Vab (const std::vector< double > &t, const DiracSpinor &Fa, const DiracSpinor &Fb)
	Vab function: Int[ (fa*gb ) * t(r) , dr].
double	Wab (const std::vector< double > &t, const DiracSpinor &Fa, const DiracSpinor &Fb)
	Wab function: Int[ (ga*fb ) * t(r) , dr].
double	Gab (const std::vector< double > &t, const DiracSpinor &Fa, const DiracSpinor &Fb)
	Gab function: Int[ (ga*gb ) * t(r) , dr].
void	Gab_rhs (const std::vector< double > &t, DiracSpinor *dF, const DiracSpinor &Fb, double a)
	Gab_rhs function: dF += a * t(r) * (0, g_b). NOTE: uses +=, so can combine.
double	Gab (const DiracSpinor &Fa, const DiracSpinor &Fb)
	Gab = Int[ ga*gb , dr] - (just relativistic correction part of integral)
void	Gab_rhs (DiracSpinor *dF, const DiracSpinor &Fb, double a)
	Gab_rhs(r) += a*g_b(r). Note: uses += so may be sumulative.
double	Pab (double pm, const DiracSpinor &Fa, const DiracSpinor &Fb)
	Pab[1] function: Int[ (fa*gb + pm*ga*fb) , dr]. pm = +/-1 (usually)
double	Rab (double pm, const DiracSpinor &Fa, const DiracSpinor &Fb)
	Rab[1] function: Int[ (fa*fb + pm*ga*gb) , dr] = Int[ (pm-1)*ga*gb) , dr]. NOTE: assumes NOT diagonal, using orthogonality condition.
void	Pab_rhs (double pm, DiracSpinor *dF, const DiracSpinor &Fb, double A=1.0)
	Pab_rhs[1] function: dF_ab += A * (g, pm*f) , pm=+/-1 (usually).
void	Rab_rhs (double pm, DiracSpinor *dF, const DiracSpinor &Fb, double A=1.0)
	Rab_rhs[1] function: dF_ab += A * (f, pm*g) = dF_ab += A * (0, (pm-1)*g). NOTE: assumes NOT diagonal, using orthogonality condition.
Parity	parse_Parity (const std::string &s)
	Convert string to Parity.
std::string	parse_Parity (Parity p)
	Convert Parity to string.
Realness	parse_Realness (const std::string &s)
	Convert string to Realness.
std::string	parse_Realness (Realness r)
	Convert Realness to string.
MatrixElementType	parse_MatrixElementType (const std::string &s)
	Convert string to MatrixElementType.
std::string	parse_MatrixElementType (MatrixElementType t)
	Convert MatrixElementType to string.

Variables
brief i	vec

Enumeration Type Documentation

Parity

enum class DiracOperator::Parity

strong

Parity of operator.

Realness

enum class DiracOperator::Realness

strong

Specifies whether an operator's matrix elements are real or imaginary.

Operators must have purely real or purely imaginary reduced matrix elements. For imaginary operators, the imaginary part is computed and stored; the real part is identically zero. This distinction affects the Hermitian symmetry relation \( \langle b \| \hat{h} \| a \rangle = \pm \langle a \| \hat{h} \| b \rangle^* \) and the sign of \( \langle b \| \hat{h} \| a \rangle \) relative to \( \langle a \| \hat{h} \| b \rangle \) .

MatrixElementType

enum class DiracOperator::MatrixElementType

strong

Type of matrix element returned.

Specifies which form of matrix element is used or returned.

@value Reduced : Reduced matrix element @value Stretched : Matrix element for stretched states (m = j) @value HFConstant : Hyperfine (A,B,C...) constant

For off-diagonal, j:=min(ja,jb)

Function Documentation

generate()

std::unique_ptr< DiracOperator::TensorOperator > DiracOperator::generate	(	std::string_view	operator_name,
		const IO::InputBlock &	input,
		const Wavefunction &	wf
	)

Returns a unique_ptr (polymorphic) to the requested operator, with given properties.

From the command line, use

ampsci -o

to get a list of available operators. For a specific operator 'OperatorName', use:

ampsci -o OperatorName

to see available run-time options for that operator.

list_operators()

void DiracOperator::list_operators

(

)

List available operators.

MultipoleOperator()

std::unique_ptr< DiracOperator::TensorOperator > DiracOperator::MultipoleOperator	(	const Grid &	grid,
		int	k,
		double	omega,
		char	type,
		char	comp,
		bool	low_q,
		const SphericalBessel::JL_table *	jl = nullptr
	)

Factory for relativistic multipole operators.

Constructs and returns a specific multipole operator derived from DiracOperator::TensorOperator, based on the requested Lorentz structure, multipole component, and momentum-transfer regime.

These are the \( t^K_Q \tilde\gamma \), \( T^{(\sigma)}_{KQ} \tilde\gamma \) operators from the vector expansion:

\[ \begin{align} e^{i\vec{q}\cdot\vec{r}} &= \sqrt{4\pi}\sum_{KQ}\sqrt{[K]} \, i^K \, {Y^*_{KQ}}{(\hat q)} \, t^K_Q(q,r),\\ \vec{\alpha} \, e^{i\vec{q}\cdot\vec{r}} & = \sqrt{4\pi} \sum_{KQ\sigma} \sqrt{[K]} \, i^{K-\sigma} \, \vec{Y}_{KQ}^{(\sigma)*}(\hat{{q}}) \, T^{(\sigma)}_{KQ}. \end{align} \]

The operator corresponds to a spherical multipole of rank k with frequency/energy transfer omega. The type and component determine the Lorentz structure and spatial character of the interaction.

These are "frequency"-dependent operators, via momentum transfer, q. For electromagnetic interactions, \( \omega = q c\). The "updateFrequency()" function expects these units, so even when momentum transfer is not equal to energy exchange, we should pass \( qc \) to this function.

Parameters

grid	Radial grid on which the operator acts.
k	Multipole rank (total angular momentum of the operator).
omega	Energy (frequency) transfer.
type	Interaction type: 'V' : Vector 'A' : Axial-vector 'S' : Scalar 'P' : Pseudoscalar
comp	Multipole component (for V/A types): 'E' : Electric 'M' : Magnetic 'L' : Longitudinal 'T' : Temporal [caution - NOT transverse!] (Ignored for scalar and pseudoscalar operators.)
low_q	If true, construct the low-momentum (long-wavelength) approximation of the operator.
jl	Optional pointer to a precomputed spherical Bessel table. If provided, radial Bessel functions are taken from this table to avoid recomputation. If nullptr, they are generated internally as needed.

Returns

A std::unique_ptr to the requested TensorOperator.

Note

Length-form electric operators (if implemented separately) are only valid for vector-electric ('V','E') combinations.

Warning

Invalid combinations of type and comp may result in a nullptr being returned.

Pab() [1/2]

double DiracOperator::Pab	(	double	pm,
		const std::vector< double > &	t,
		const DiracSpinor &	Fa,
		const DiracSpinor &	Fb
	)

Pab function: Int[ (fa*gb + pm*ga*fb) * t(r) , dr]. pm = +/-1 (usually)

Note: does not know selection rules, so only evaluate if non-zero

Rab() [1/2]

double DiracOperator::Rab	(	double	pm,
		const std::vector< double > &	t,
		const DiracSpinor &	Fa,
		const DiracSpinor &	Fb
	)

Rab function: Int[ (fa*fb + pm*ga*gb) * t(r) , dr]. pm = +/-1 (usually)

Note: does not know selection rules, so only evaluate if non-zero

Pab_rhs() [1/2]

void DiracOperator::Pab_rhs	(	double	pm,
		const std::vector< double > &	t,
		DiracSpinor *	dF,
		const DiracSpinor &	Fb,
		double	A = 1.0
	)

Pab_rhs function: dF_ab += A * t(r) * (g, pm*f) , pm=+/-1 (usually). NOTE: uses +=, so can combine. Ensure empty to begin.

Note: does not know selection rules, so only evaluate if non-zero. Should have Fa*Pab_rhs = A * Pab.

Rab_rhs() [1/2]

void DiracOperator::Rab_rhs	(	double	pm,
		const std::vector< double > &	t,
		DiracSpinor *	dF,
		const DiracSpinor &	Fb,
		double	A = 1.0
	)

Rab_rhs function: dF_ab += A * t(r) * (f, pm*g) , pm=+/-1 (usually). NOTE: uses +=, so can combine. Ensure empty to begin.

Note: does not know selection rules, so only evaluate if non-zero. Should have Fa*Rab_rhs = A * Rab.

Vab()

double DiracOperator::Vab	(	const std::vector< double > &	t,
		const DiracSpinor &	Fa,
		const DiracSpinor &	Fb
	)

Vab function: Int[ (fa*gb ) * t(r) , dr].

Wab()

double DiracOperator::Wab	(	const std::vector< double > &	t,
		const DiracSpinor &	Fa,
		const DiracSpinor &	Fb
	)

Wab function: Int[ (ga*fb ) * t(r) , dr].

Gab() [1/2]

double DiracOperator::Gab	(	const std::vector< double > &	t,
		const DiracSpinor &	Fa,
		const DiracSpinor &	Fb
	)

Gab function: Int[ (ga*gb ) * t(r) , dr].

Gab_rhs() [1/2]

void DiracOperator::Gab_rhs	(	const std::vector< double > &	t,
		DiracSpinor *	dF,
		const DiracSpinor &	Fb,
		double	a
	)

Gab_rhs function: dF += a * t(r) * (0, g_b). NOTE: uses +=, so can combine.

Gab() [2/2]

double DiracOperator::Gab	(	const DiracSpinor &	Fa,
		const DiracSpinor &	Fb
	)

Gab = Int[ ga*gb , dr] - (just relativistic correction part of integral)

Gab_rhs() [2/2]

void DiracOperator::Gab_rhs	(	DiracSpinor *	dF,
		const DiracSpinor &	Fb,
		double	a
	)

Gab_rhs(r) += a*g_b(r). Note: uses += so may be sumulative.

Pab() [2/2]

double DiracOperator::Pab	(	double	pm,
		const DiracSpinor &	Fa,
		const DiracSpinor &	Fb
	)

Pab[1] function: Int[ (fa*gb + pm*ga*fb) , dr]. pm = +/-1 (usually)

Rab() [2/2]

double DiracOperator::Rab	(	double	pm,
		const DiracSpinor &	Fa,
		const DiracSpinor &	Fb
	)

Rab[1] function: Int[ (fa*fb + pm*ga*gb) , dr] = Int[ (pm-1)*ga*gb) , dr]. NOTE: assumes NOT diagonal, using orthogonality condition.

Pab_rhs() [2/2]

void DiracOperator::Pab_rhs	(	double	pm,
		DiracSpinor *	dF,
		const DiracSpinor &	Fb,
		double	a
	)

Pab_rhs[1] function: dF_ab += A * (g, pm*f) , pm=+/-1 (usually).

Rab_rhs() [2/2]

void DiracOperator::Rab_rhs	(	double	pm,
		DiracSpinor *	dF,
		const DiracSpinor &	Fb,
		double	a
	)

Rab_rhs[1] function: dF_ab += A * (f, pm*g) = dF_ab += A * (0, (pm-1)*g). NOTE: assumes NOT diagonal, using orthogonality condition.

parse_Parity() [1/2]

Parity DiracOperator::parse_Parity

(

const std::string &

s

)

Convert string to Parity.

parse_Parity() [2/2]

std::string DiracOperator::parse_Parity

(

Parity

p

)

Convert Parity to string.

parse_Realness() [1/2]

Realness DiracOperator::parse_Realness

(

const std::string &

s

)

Convert string to Realness.

parse_Realness() [2/2]

std::string DiracOperator::parse_Realness

(

Realness

r

)

Convert Realness to string.

parse_MatrixElementType() [1/2]

MatrixElementType DiracOperator::parse_MatrixElementType

(

const std::string &

s

)

Convert string to MatrixElementType.

parse_MatrixElementType() [2/2]

std::string DiracOperator::parse_MatrixElementType

(

MatrixElementType

t

)

Convert MatrixElementType to string.

Variable Documentation

vec

brief i DiracOperator::vec

Initial value:

{\alpha} matrix element: propto Electric dipole operator.
    i factored so that ME is real
class ialpha final : public TensorOperator {
public:
  ialpha() : TensorOperator(1, Parity::odd, 1.0, {}, Realness::real, true) {}
 
  std::string name() const override final { return "i*alpha"; }
  std::string units() const override final { return "au"; }
 
  double angularF(const int ka, const int kb) const override final {
    return Angular::Ck_kk(1, ka, kb);
  }
 
  double angularCff(int, int) const override final { return 0; }
  double angularCgg(int, int) const override final { return 0; }
  double angularCfg(int ka, int kb) const override final { return ka - kb - 1; }
  double angularCgf(int ka, int kb) const override final { return ka - kb + 1; }
}