#include <PrimGrid.hh>

Detailed Description

Definition at line 30 of file PrimGrid.hh.

Public Member Functions
	PrimGrid (const Lattice &p_lat, const Lattice &s_lat, Index NB=1)

	PrimGrid (const Lattice &p_lat, const Lattice &s_lat, const Eigen::Ref< const PrimGrid::matrix_type > &U, const Eigen::Ref< const PrimGrid::matrix_type > &Smat, Index NB)

Index	size () const

const matrix_type &	matrixU () const

const matrix_type &	invU () const

const Eigen::DiagonalWrapper < const PrimGrid::vector_type >	matrixS () const

int	S (Index i) const

Index	find (const Coordinate &_coord) const

Index	find (const UnitCell &_unitcell) const

Index	find (const UnitCellCoord &_coord) const

Index	find_cart (const Eigen::Ref< const Eigen::Vector3d > &_cart_coord) const

UnitCellCoord	get_within (const UnitCellCoord &_uccoord) const

Coordinate	coord (Index l, CELL_TYPE lat_mode) const

Coordinate	coord (const UnitCellCoord &bijk, CELL_TYPE lat_mode) const

UnitCell	unitcell (Index i) const

UnitCellCoord	uccoord (Index i) const

SymGroupRepID	make_permutation_representation (const SymGroup &group, SymGroupRepID basis_permute_rep) const

ReturnArray< Permutation >	make_translation_permutations (Index NB) const

const Array< Permutation > &	translation_permutations () const
	const access to m_trans_permutations. Generates permutations if they don't already exist. More...

const Permutation &	translation_permutation (Index i) const

SymOp	sym_op (Index l) const

Private Types
typedef Eigen::Matrix< long, 3, 3 >	matrix_type

typedef Eigen::Matrix< long, 3, 1 >	vector_type

Private Member Functions
UnitCellCoord	to_canonical (const UnitCellCoord &bijk) const

UnitCellCoord	from_canonical (const UnitCellCoord &bmnp) const

Private Attributes
Lattice const *	m_lat [2]
	m_lat[PRIM] is primitive lattice, lat[SCEL] is super lattice More...

long int	m_N_vol
	Number of primgrid lattice points in the supercell. More...

long int	m_NB
	Number of basis atoms in the primitive cell. More...

matrix_type	m_plane_mat

matrix_type	m_trans_mat

matrix_type	m_U
	The Smith Normal Form decomposition is: trans_mat = USV, with det(U)=det(V)=1; S is diagonal. More...

matrix_type	m_invU

Array< Permutation >	m_trans_permutations
	Permutations that describe how translation permutes sites of the supercell. More...

int	m_stride [2]

Eigen::Matrix< long, 3, 1 >	m_S

Member Typedef Documentation

typedef Eigen::Matrix<long, 3, 3> CASM::PrimGrid::matrix_type

private

Definition at line 32 of file PrimGrid.hh.

typedef Eigen::Matrix<long, 3, 1> CASM::PrimGrid::vector_type

private

Definition at line 33 of file PrimGrid.hh.

Constructor & Destructor Documentation

CASM::PrimGrid::PrimGrid	(	const Lattice &	p_lat,
		const Lattice &	s_lat,
		Index	NB = `1`
	)

Definition at line 16 of file PrimGrid.cc.

CASM::PrimGrid::PrimGrid	(	const Lattice &	p_lat,
		const Lattice &	s_lat,
		const Eigen::Ref< const PrimGrid::matrix_type > &	U,
		const Eigen::Ref< const PrimGrid::matrix_type > &	Smat,
		Index	NB
	)

Definition at line 84 of file PrimGrid.cc.

Member Function Documentation

Coordinate CASM::PrimGrid::coord	(	Index	l,
		CELL_TYPE	lat_mode
	)		const

Definition at line 223 of file PrimGrid.cc.

Coordinate CASM::PrimGrid::coord	(	const UnitCellCoord &	bijk,
		CELL_TYPE	lat_mode
	)		const

Definition at line 213 of file PrimGrid.cc.

Index CASM::PrimGrid::find ( const Coordinate & _coord ) const

Definition at line 144 of file PrimGrid.cc.

Index CASM::PrimGrid::find ( const UnitCell & _unitcell ) const

Definition at line 156 of file PrimGrid.cc.

Index CASM::PrimGrid::find ( const UnitCellCoord & _coord ) const

Definition at line 162 of file PrimGrid.cc.

Index CASM::PrimGrid::find_cart ( const Eigen::Ref< const Eigen::Vector3d > & _cart_coord ) const

Definition at line 171 of file PrimGrid.cc.

UnitCellCoord CASM::PrimGrid::from_canonical ( const UnitCellCoord & bmnp ) const

private

Convert canonical UnitCellCoord (bmnp) to UnitCellCoord (bijk) U*mnp = ijk

Definition at line 364 of file PrimGrid.cc.

UnitCellCoord CASM::PrimGrid::get_within ( const UnitCellCoord & bijk ) const

The m_plane_mat works because of the following:

prim_frac_coord = m_trans_mat * scel_frac_coord

m_trans_mat is integer, and (if scel_frac_coord is a lattice translation) then prim_frac_coord is also integer.

transf_mat.inverse() does the inverse mapping, but it is NOT integer. However,

 m_plane_mat = transf_mat.determinant()*transf_mat.inverse()

is integer. This is because it is simply the adjugate matrix (i.e., the transpose of the cofactor matrix, which can be obtained without using division).

This means that

  m_plane_mat*prim_frac_coord = m_N_vol*scel_frac_coord

again, for a lattice translation, the left hand side is integer, so the right hand side must be also. Moreover, the elements of the RHS should be on the interval [0,m_N_vol-1] if it is within the supercell.

Definition at line 200 of file PrimGrid.cc.

const matrix_type& CASM::PrimGrid::invU ( ) const

inline

Definition at line 115 of file PrimGrid.hh.

SymGroupRepID CASM::PrimGrid::make_permutation_representation	(	const SymGroup &	group,
		SymGroupRepID	basis_permute_rep
	)		const

Definition at line 247 of file PrimGrid.cc.

ReturnArray< Permutation > CASM::PrimGrid::make_translation_permutations ( Index NB ) const

Definition at line 308 of file PrimGrid.cc.

const Eigen::DiagonalWrapper< const PrimGrid::vector_type > CASM::PrimGrid::matrixS ( ) const

Definition at line 140 of file PrimGrid.cc.

const matrix_type& CASM::PrimGrid::matrixU ( ) const

inline

Definition at line 112 of file PrimGrid.hh.

int CASM::PrimGrid::S ( Index i ) const

inline

Definition at line 121 of file PrimGrid.hh.

Index CASM::PrimGrid::size ( ) const

inline

Definition at line 108 of file PrimGrid.hh.

SymOp CASM::PrimGrid::sym_op ( Index l ) const

Definition at line 375 of file PrimGrid.cc.

UnitCellCoord CASM::PrimGrid::to_canonical ( const UnitCellCoord & bijk ) const

private

Convert UnitCellCoord (bijk) to canonical UnitCellCoord (bmnp) mnp = invU * ijk

Convert UnitCellCoord (bijk) to canonical UnitCellCoord (bmnp) mnp = m_invU * ijk

Definition at line 348 of file PrimGrid.cc.

const Permutation& CASM::PrimGrid::translation_permutation ( Index i ) const

inline

Definition at line 159 of file PrimGrid.hh.

const Array<Permutation>& CASM::PrimGrid::translation_permutations ( ) const

inline

const access to m_trans_permutations. Generates permutations if they don't already exist.

Definition at line 147 of file PrimGrid.hh.

UnitCellCoord CASM::PrimGrid::uccoord ( Index i ) const

Definition at line 235 of file PrimGrid.cc.

UnitCell CASM::PrimGrid::unitcell ( Index i ) const

Definition at line 229 of file PrimGrid.cc.

Member Data Documentation

matrix_type CASM::PrimGrid::m_invU

private

Definition at line 49 of file PrimGrid.hh.

Lattice const* CASM::PrimGrid::m_lat[2]

private

m_lat[PRIM] is primitive lattice, lat[SCEL] is super lattice

Definition at line 35 of file PrimGrid.hh.

long int CASM::PrimGrid::m_N_vol

private

Number of primgrid lattice points in the supercell.

Definition at line 38 of file PrimGrid.hh.

long int CASM::PrimGrid::m_NB

private

Number of basis atoms in the primitive cell.

Definition at line 41 of file PrimGrid.hh.

matrix_type CASM::PrimGrid::m_plane_mat

private

The transformation matrix, 'trans_mat', is: lat[SCEL]->lat_column_mat() = (lat[PRIM]->lat_column_mat())*trans_mat; plane_mat = trans_mat.determinant()*trans_mat.inverse();

Definition at line 46 of file PrimGrid.hh.

Eigen::Matrix<long, 3, 1> CASM::PrimGrid::m_S

private

Definition at line 88 of file PrimGrid.hh.

int CASM::PrimGrid::m_stride[2]

private

============================================================================================== Because m_lat[SCEL]->lat_column_mat() = (m_lat[PRIM]->lat_column_mat())*trans_mat; and trans_mat=U*S*V we know that (m_lat[SCEL]->lat_column_mat())*V.inverse() = (m_lat[PRIM]->lat_column_mat())*U*S;

In other words, [(m_lat[PRIM]->lat_column_mat())*U] is a primitive lattice that perfectly tiles the equivalent super lattice [(m_lat[SCEL]->lat_column_mat())*V.inverse()] – (because S is diagonal)

We thus use (m,n,p) on the grid specified by [(m_lat[PRIM]->lat_column_mat())*U] as a canonical indexing

We can do this by manipulating fractional coordinates: trans_mat*super_frac_coord = prim_frac_coord

so U*S*V*super_frac_coord = prim_frac_coord

and finally S*V*super_frac_coord = U.inverse()*prim_frac_coord

meaning that multiplication of prim_frac_coord by invU gives the canonical index

This means that:

(m,n,p) = invU * (i,j,k) and (i,j,k) = U * (m,n,p)

where (i,j,k) are the UnitCellCoords relative to m_lat[PRIM], and (m,n,p) are canonical UnitCellCoords, relative to (m_lat[PRIM]->lat_column_mat())*U

Definition at line 87 of file PrimGrid.hh.

matrix_type CASM::PrimGrid::m_trans_mat

private

Definition at line 46 of file PrimGrid.hh.

Array<Permutation> CASM::PrimGrid::m_trans_permutations

mutableprivate

Permutations that describe how translation permutes sites of the supercell.

Definition at line 52 of file PrimGrid.hh.

matrix_type CASM::PrimGrid::m_U

private

The Smith Normal Form decomposition is: trans_mat = U*S*V, with det(U)=det(V)=1; S is diagonal.

Definition at line 49 of file PrimGrid.hh.

The documentation for this class was generated from the following files:

/Users/bpuchala/Work/codes/CASMcode_v0.2.X_reference/include/casm/crystallography/PrimGrid.hh
/Users/bpuchala/Work/codes/CASMcode_v0.2.X_reference/src/casm/crystallography/PrimGrid.cc

Detailed Description

Public Member Functions

Private Types

Private Member Functions

Private Attributes

Member Typedef Documentation

Constructor & Destructor Documentation

Member Function Documentation

Member Data Documentation