Particle Bunch. More...

#include <PartBunch.h>

Inheritance diagram for PartBunch:

Collaboration diagram for PartBunch:

Public Types
enum	{ Dim = Dimension }

typedef IpplParticleBase < Layout_t >	pbase_t

Public Types inherited from PartBunchBase< double, 3 >
enum	UnitState_t

typedef AbstractParticle < double, Dim >::ParticlePos_t	ParticlePos_t

typedef AbstractParticle < double, Dim > ::ParticleIndex_t	ParticleIndex_t

typedef AbstractParticle < double, Dim >::UpdateFlags	UpdateFlags_t

typedef AbstractParticle < double, Dim >::Position_t	Position_t

typedef std::pair< Vector_t, Vector_t >	VectorPair_t

Public Member Functions
	PartBunch (const PartData *ref)
	Default constructor. More...

	PartBunch ()=delete

	PartBunch (const PartBunch &)=delete

PartBunch &	operator= (const PartBunch &)=delete

	~PartBunch ()

void	runTests ()

void	initialize (FieldLayout_t *fLayout)

void	do_binaryRepart ()

double	getRho (int x, int y, int z)

const Mesh_t &	getMesh () const

Mesh_t &	getMesh ()

FieldLayout_t &	getFieldLayout ()

void	setBCAllPeriodic ()

void	setBCAllOpen ()

void	setBCForDCBeam ()

VectorPair_t	getEExtrema ()

void	computeSelfFields ()

void	computeSelfFields (int b)

void	computeSelfFields_cycl (double gamma)
	Calculates the self electric field from the charge density distribution for use in cyclotrons. More...

void	computeSelfFields_cycl (int b)
	Calculates the self electric field from the charge density distribution for use in cyclotrons. More...

void	resetInterpolationCache (bool clearCache=false)

void	swap (unsigned int i, unsigned int j)

Inform &	print (Inform &os)

Public Member Functions inherited from PartBunchBase< double, 3 >
	PartBunchBase (AbstractParticle< double, Dim > *pb)

	PartBunchBase (AbstractParticle< double, Dim > pb, const PartData ref)

	PartBunchBase (const PartBunchBase &rhs)=delete

virtual	~PartBunchBase ()

bool	getIfBeamEmitting ()

int	getLastEmittedEnergyBin ()

size_t	getNumberOfEmissionSteps ()

int	getNumberOfEnergyBins ()

void	Rebin ()

void	setEnergyBins (int numberOfEnergyBins)

bool	weHaveEnergyBins ()

void	switchToUnitlessPositions (bool use_dt_per_particle=false)

void	switchOffUnitlessPositions (bool use_dt_per_particle=false)

void	setDistribution (Distribution d, std::vector< Distribution > addedDistributions, size_t &np)

bool	isGridFixed ()

bool	hasBinning ()

void	setTEmission (double t)

double	getTEmission ()

bool	doEmission ()

bool	weHaveBins () const

void	setPBins (PartBins *pbin)

void	setPBins (PartBinsCyc *pbin)

size_t	emitParticles (double eZ)
	Emit particles in the given bin i.e. copy the particles from the bin structure into the particle container. More...

void	updateNumTotal ()

void	rebin ()

int	getNumBins ()

int	getLastemittedBin ()

void	setLocalBinCount (size_t num, int bin)

void	calcGammas ()
	Compute the gammas of all bins. More...

void	calcGammas_cycl ()

double	getBinGamma (int bin)
	Get gamma of one bin. More...

virtual void	setBinCharge (int bin, double q)
	Set the charge of one bin to the value of q and all other to zero. More...

virtual void	setBinCharge (int bin)
	Set the charge of all other the ones in bin to zero. More...

size_t	calcNumPartsOutside (Vector_t x)
	returns the number of particles outside of a box defined by x More...

void	calcLineDensity (unsigned int nBins, std::vector< double > &lineDensity, std::pair< double, double > &meshInfo)

void	setBeamFrequency (double v)

virtual void	boundp ()

void	boundp_destroy ()

size_t	boundp_destroyT ()

size_t	destroyT ()

virtual double	getPx (int i)

virtual double	getPy (int i)

virtual double	getPz (int i)

virtual double	getPx0 (int i)

virtual double	getPy0 (int i)

virtual double	getX (int i)

virtual double	getY (int i)

virtual double	getZ (int i)

virtual double	getX0 (int i)

virtual double	getY0 (int i)

virtual void	setZ (int i, double zcoo)

void	get_bounds (Vector_t &rmin, Vector_t &rmax)

void	getLocalBounds (Vector_t &rmin, Vector_t &rmax)

std::pair< Vector_t, double >	getBoundingSphere ()

std::pair< Vector_t, double >	getLocalBoundingSphere ()

void	push_back (OpalParticle p)

void	set_part (FVector< double, 6 > z, int ii)

void	set_part (OpalParticle p, int ii)

OpalParticle	get_part (int ii)

void	maximumAmplitudes (const FMatrix< double, 6, 6 > &D, double &axmax, double &aymax)
	Return maximum amplitudes. More...

void	setdT (double dt)

double	getdT () const

void	setT (double t)

void	incrementT ()

double	getT () const

double	get_sPos ()

void	set_sPos (double s)

double	get_gamma () const

double	get_meanKineticEnergy () const

Vector_t	get_origin () const

Vector_t	get_maxExtent () const

Vector_t	get_centroid () const

Vector_t	get_rrms () const

Vector_t	get_rprms () const

Vector_t	get_rmean () const

Vector_t	get_prms () const

Vector_t	get_pmean () const

Vector_t	get_pmean_Distribution () const

Vector_t	get_emit () const

Vector_t	get_norm_emit () const

Vector_t	get_halo () const

virtual Vector_t	get_hr () const

double	get_Dx () const

double	get_Dy () const

double	get_DDx () const

double	get_DDy () const

virtual void	set_meshEnlargement (double dh)

void	gatherLoadBalanceStatistics ()

size_t	getLoadBalance (int p) const

void	get_PBounds (Vector_t &min, Vector_t &max) const

void	calcBeamParameters ()

void	calcBeamParametersInitial ()

double	getCouplingConstant () const

void	setCouplingConstant (double c)

void	setCharge (double q)

void	setChargeZeroPart (double q)

void	setMass (double mass)

double	getEkin () const
	Need Ek for the Schottky effect calculation (eV) More...

double	getWorkFunctionRf () const
	Need the work function for the Schottky effect calculation (eV) More...

double	getLaserEnergy () const
	Need the laser energy for the Schottky effect calculation (eV) More...

double	getCharge () const
	get the total charge per simulation particle More...

double	getChargePerParticle () const
	get the macro particle charge More...

virtual void	setSolver (FieldSolver *fs)

bool	hasFieldSolver ()

std::string	getFieldSolverType () const

void	setStepsPerTurn (int n)

int	getStepsPerTurn () const

void	setGlobalTrackStep (long long n)
	step in multiple TRACK commands More...

long long	getGlobalTrackStep () const

void	setLocalTrackStep (long long n)
	step in a TRACK command More...

void	incTrackSteps ()

long long	getLocalTrackStep () const

void	setNumBunch (short n)

short	getNumBunch () const

void	setTotalNumPerBunch (size_t numpart, short n)

size_t	getTotalNumPerBunch (short n) const

void	setLocalNumPerBunch (size_t numpart, short n)

size_t	getLocalNumPerBunch (short n) const

void	countTotalNumPerBunch ()

void	setGlobalMeanR (Vector_t globalMeanR)

Vector_t	getGlobalMeanR ()

void	setGlobalToLocalQuaternion (Quaternion_t globalToLocalQuaternion)

Quaternion_t	getGlobalToLocalQuaternion ()

void	setSteptoLastInj (int n)

int	getSteptoLastInj ()

double	calcMeanPhi ()
	calculate average angle of longitudinal direction of bins More...

bool	resetPartBinID2 (const double eta)
	reset Bin[] for each particle according to the method given in paper PAST-AB(064402) by G. Fubiani et al. More...

bool	resetPartBinBunch ()

double	getdE ()

virtual double	getGamma (int i)

virtual double	getBeta (int i)

virtual void	actT ()

const PartData *	getReference () const

double	getEmissionDeltaT ()

Quaternion_t	getQKs3D ()

void	setQKs3D (Quaternion_t q)

Vector_t	getKs3DRefr ()

void	setKs3DRefr (Vector_t r)

Vector_t	getKs3DRefp ()

void	setKs3DRefp (Vector_t p)

void	iterateEmittedBin (int binNumber)

void	calcEMean ()

void	correctEnergy (double avrgp)

Inform &	print (Inform &os)

size_t	getTotalNum () const

void	setTotalNum (size_t n)

void	setLocalNum (size_t n)

size_t	getLocalNum () const

size_t	getDestroyNum () const

size_t	getGhostNum () const

unsigned int	getMinimumNumberOfParticlesPerCore () const

void	setMinimumNumberOfParticlesPerCore (unsigned int n)

ParticleLayout< double, Dim > &	getLayout ()

const ParticleLayout< double, Dim > &	getLayout () const

bool	getUpdateFlag (UpdateFlags_t f) const

void	setUpdateFlag (UpdateFlags_t f, bool val)

ParticleBConds< Position_t, Dimension > &	getBConds ()

void	setBConds (const ParticleBConds< Position_t, Dimension > &bc)

bool	singleInitNode () const

void	resetID ()

void	update ()

void	update (const ParticleAttrib< char > &canSwap)

void	createWithID (unsigned id)

void	create (size_t M)

void	globalCreate (size_t np)

void	destroy (size_t M, size_t I, bool doNow=false)

void	performDestroy (bool updateLocalNum=false)

void	ghostDestroy (size_t M, size_t I)

FMatrix< double, 2 Dim, 2 Dim >	getSigmaMatrix ()

double	getQ () const
	Access to reference data. More...

double	getM () const

double	getP () const

double	getE () const

ParticleType::type	getPType () const

double	getInitialBeta () const

double	getInitialGamma () const

void	resetQ (double q)

void	resetM (double m)

void	setPType (ParticleType::type)

Public Attributes
Field_t	rho_m
	scalar potential More...

VField_t	eg_m
	vector field on the grid More...

Public Attributes inherited from PartBunchBase< double, 3 >
ParticlePos_t &	R

ParticleIndex_t &	ID

ParticleAttrib< Vector_t >	P

ParticleAttrib< double >	Q

ParticleAttrib< double >	M

ParticleAttrib< double >	Phi

ParticleAttrib< Vector_t >	Ef

ParticleAttrib< Vector_t >	Eftmp

ParticleAttrib< Vector_t >	Bf

ParticleAttrib< int >	Bin

ParticleAttrib< double >	dt

ParticleAttrib< short >	PType

ParticleAttrib< int >	TriID

ParticleAttrib< short >	cavityGapCrossed
	particle just crossed cavity gap (for ParallelCyclotronTracker) More...

ParticleAttrib< short >	bunchNum

Vector_t	RefPartR_m

Vector_t	RefPartP_m

ParticleType::type	refPType_m

CoordinateSystemTrafo	toLabTrafo_m

int	myNode_m
	avoid calls to Ippl::myNode() More...

int	nodes_m
	avoid calls to Ippl::getNodes() More...

bool	fixed_grid
	if the grid does not have to adapt More...

PartBins *	pbin_m

std::unique_ptr< LossDataSink >	lossDs_m

std::unique_ptr< Inform >	pmsg_m

std::unique_ptr< std::ofstream >	f_stream

IpplTimings::TimerRef	distrReload_m
	timer for IC, can not be in Distribution.h More...

IpplTimings::TimerRef	distrCreate_m

double	dtScInit_m

double	deltaTau_m

bool	lowParticleCount_m
	if a local node has less than 2 particles lowParticleCount_m == true More...

IpplTimings::TimerRef	selfFieldTimer_m
	timer for selfField calculation More...

Private Member Functions
void	updateDomainLength (Vektor< int, 3 > &grid)

void	updateFields (const Vector_t &hr, const Vector_t &origin)

void	resizeMesh ()
	resize mesh to geometry specified More...

ParticleLayout< double, 3 > &	getLayout ()

const ParticleLayout< double, 3 > &	getLayout () const

Private Attributes
BConds< double, 3, Mesh_t, Center_t >	bc_m
	for defining the boundary conditions More...

BConds< Vector_t, 3, Mesh_t, Center_t >	vbc_m

bool	interpolationCacheSet_m

ParticleAttrib< CacheDataCIC < double, 3U > >	interpolationCache_m

Additional Inherited Members
Static Public Attributes inherited from PartBunchBase< double, 3 >
static const unsigned	Dimension

Protected Member Functions inherited from PartBunchBase< double, 3 >
size_t	calcMoments ()

void	calcMomentsInitial ()

double	calculateAngle (double x, double y)
	angle range [0~2PI) degree More...

Protected Attributes inherited from PartBunchBase< double, 3 >
IpplTimings::TimerRef	boundpTimer_m

IpplTimings::TimerRef	boundpBoundsTimer_m

IpplTimings::TimerRef	boundpUpdateTimer_m

IpplTimings::TimerRef	statParamTimer_m

IpplTimings::TimerRef	histoTimer_m

const PartData *	reference

UnitState_t	unit_state_

UnitState_t	stateOfLastBoundP_

double	centroid_m [2 *Dim]
	holds the centroid of the beam More...

FMatrix< double, 2 Dim, 2 Dim >	moments_m
	6x6 matrix of the moments of the beam More...

double	dt_m
	holds the timestep in seconds More...

double	t_m
	holds the actual time of the integration More...

double	eKin_m
	mean energy of the bunch (MeV) More...

double	dE_m
	energy spread of the beam in MeV More...

double	spos_m
	the position along design trajectory More...

Vector_t	globalMeanR_m

Quaternion_t	globalToLocalQuaternion_m

Vector_t	rmax_m
	maximal extend of particles More...

Vector_t	rmin_m
	minimal extend of particles More...

Vector_t	rrms_m
	rms beam size (m) More...

Vector_t	prms_m
	rms momenta More...

Vector_t	rmean_m
	mean position (m) More...

Vector_t	pmean_m
	mean momenta More...

Vector_t	eps_m
	rms emittance (not normalized) More...

Vector_t	eps_norm_m
	rms normalized emittance More...

Vector_t	halo_m

Vector_t	rprms_m
	rms correlation More...

double	Dx_m
	dispersion x & y More...

double	Dy_m

double	DDx_m
	derivative of the dispersion More...

double	DDy_m

Vector_t	hr_m
	meshspacing of cartesian mesh More...

Vektor< int, 3 >	nr_m
	meshsize of cartesian mesh More...

FieldSolver *	fs_m
	stores the used field solver More...

double	couplingConstant_m

double	qi_m

int	distDump_m
	counter to store the distribution dump More...

int	fieldDBGStep_m

double	dh_m
	Mesh enlargement. More...

double	tEmission_m
	in % how much the mesh is enlarged More...

std::unique_ptr< double[]>	bingamma_m
	holds the gamma of the bin More...

std::unique_ptr< size_t[]>	binemitted_m

int	stepsPerTurn_m
	steps per turn for OPAL-cycl More...

long long	localTrackStep_m
	step in a TRACK command More...

long long	globalTrackStep_m
	if multiple TRACK commands More...

short	numBunch_m
	current bunch number More...

std::vector< size_t >	bunchTotalNum_m
	number of particles per bunch More...

std::vector< size_t >	bunchLocalNum_m

int	SteptoLastInj_m

std::unique_ptr< size_t[]>	globalPartPerNode_m

Distribution *	dist_m

bool	dcBeam_m

double	periodLength_m

std::shared_ptr < AbstractParticle< double, Dim > >	pbase

Detailed Description

Particle Bunch.

Definition at line 30 of file PartBunch.h.

Member Typedef Documentation

typedef IpplParticleBase<Layout_t> PartBunch::pbase_t

Definition at line 33 of file PartBunch.h.

Member Enumeration Documentation

anonymous enum

Enumerator
Dim

Definition at line 34 of file PartBunch.h.

Constructor & Destructor Documentation

PartBunch::PartBunch ( const PartData * ref )

explicit

Default constructor.

Definition at line 54 of file PartBunch.cpp.

PartBunch::PartBunch ( )

delete

PartBunch::PartBunch ( const PartBunch & )

delete

PartBunch::~PartBunch ( )

Definition at line 62 of file PartBunch.cpp.

Member Function Documentation

void PartBunch::computeSelfFields ( )

virtual

Magnetic field in x and y direction induced by the eletric field

\[ B_x = \gamma(B_x^{'} - \frac{beta}{c}E_y^{'}) = -\gamma \frac{beta}{c}E_y^{'} = -\frac{beta}{c}E_y \]

\[ B_y = \gamma(B_y^{'} - \frac{beta}{c}E_x^{'}) = +\gamma \frac{beta}{c}E_x^{'} = +\frac{beta}{c}E_x \]

\[ B_z = B_z^{'} = 0 \]

Implements PartBunchBase< double, 3 >.

Definition at line 463 of file PartBunch.cpp.

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void PartBunch::computeSelfFields ( int b )

virtual

/brief used for self fields with binned distribution

Set initial charge density to zero. Create image charge potential field.

Set initial E field to zero.

Mesh the whole domain

Scatter charge onto space charge grid.

Calculate mesh-scale factor and get gamma for this specific slice (bin).

Scale charge density to get charge density in real units. Account for Lorentz transformation in longitudinal direction.

Scale mesh spacing to real units (meters). Lorentz transform the longitudinal direction.

Find potential from charge in this bin (no image yet) using Poisson's equation (without coefficient: -1/(eps)).

Scale mesh back (to same units as particle locations.)

The scalar potential is given back in rho_m and must be converted to the right units.

IPPL Grad numerical computes gradient to find the electric field (in bin rest frame).

Scale field. Combine Lorentz transform with conversion to proper field units.

Interpolate electric field at particle positions. We reuse the cached information about where the particles are relative to the field, since the particles have not moved since this the most recent scatter operation.

Magnetic field in x and y direction induced by the electric field.

\[ B_x = \gamma(B_x^{'} - \frac{beta}{c}E_y^{'}) = -\gamma \frac{beta}{c}E_y^{'} = -\frac{beta}{c}E_y \]

\[ B_y = \gamma(B_y^{'} - \frac{beta}{c}E_x^{'}) = +\gamma \frac{beta}{c}E_x^{'} = +\frac{beta}{c}E_x \]

\[ B_z = B_z^{'} = 0 \]

Now compute field due to image charge. This is done separately as the image charge is moving to -infinity (instead of +infinity) so the Lorentz transform is different.

Find z shift for shifted Green's function.

Find potential from image charge in this bin using Poisson's equation (without coefficient: -1/(eps)).

Scale mesh back (to same units as particle locations.)

The scalar potential is given back in rho_m and must be converted to the right units.

IPPL Grad numerical computes gradient to find the electric field (in rest frame of this bin's image charge).

Scale field. Combine Lorentz transform with conversion to proper field units.

Interpolate electric field at particle positions. We reuse the cached information about where the particles are relative to the field, since the particles have not moved since this the most recent scatter operation.

Magnetic field in x and y direction induced by the image charge electric field. Note that beta will have the opposite sign from the bunch charge field, as the image charge is moving in the opposite direction.

\[ B_x = \gamma(B_x^{'} - \frac{beta}{c}E_y^{'}) = -\gamma \frac{beta}{c}E_y^{'} = -\frac{beta}{c}E_y \]

\[ B_y = \gamma(B_y^{'} - \frac{beta}{c}E_x^{'}) = +\gamma \frac{beta}{c}E_x^{'} = +\frac{beta}{c}E_x \]

\[ B_z = B_z^{'} = 0 \]

Implements PartBunchBase< double, 3 >.

Definition at line 119 of file PartBunch.cpp.

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void PartBunch::computeSelfFields_cycl ( double gamma )

virtual

Calculates the self electric field from the charge density distribution for use in cyclotrons.

computeSelfFields_cycl()

See Also: ParallelCyclotronTracker

Warning: none yet

Comments -DW: I have made some changes in here: -) Some refacturing to make more similar to computeSelfFields() -) Added meanR and quaternion to be handed to the function so that SAAMG solver knows how to rotate the boundary geometry correctly. -) Fixed an error where gamma was not taken into account correctly in direction of movement (y in cyclotron) -) Comment: There is no account for image charges in the cyclotron tracker (yet?)!

set initial charge density to zero.

set initial E field to zero

mesh the whole domain

scatter particles charge onto grid.

Lorentz transformation In particle rest frame, the longitudinal length (y for cyclotron) enlarged

from charge (C) to charge density (C/m^3).

now charge density is in rho_m calculate Possion equation (without coefficient: -1/(eps))

retrive coefficient: -1/(eps)

calculate electric field vectors from field potential

Back Lorentz transformation CAVE: y coordinate needs 1/gamma factor because IPPL function Grad() divides by hr_m which is not scaled appropriately with Lorentz contraction in y direction only hr_scaled is! -DW

interpolate electric field at particle positions.

calculate coefficient

calculate B field from E field

Implements PartBunchBase< double, 3 >.

Definition at line 643 of file PartBunch.cpp.

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void PartBunch::computeSelfFields_cycl ( int bin )

virtual

Calculates the self electric field from the charge density distribution for use in cyclotrons.

computeSelfFields_cycl()

See Also: ParallelCyclotronTracker

Warning: none yet

Overloaded version for having multiple bins with separate gamma for each bin. This is necessary For multi-bunch mode.

Comments -DW: I have made some changes in here: -) Some refacturing to make more similar to computeSelfFields() -) Added meanR and quaternion to be handed to the function (TODO: fall back to meanR = 0 and unit quaternion if not specified) so that SAAMG solver knows how to rotate the boundary geometry correctly. -) Fixed an error where gamma was not taken into account correctly in direction of movement (y in cyclotron) -) Comment: There is no account for image charges in the cyclotron tracker (yet?)!

set initial charge dentsity to zero.

set initial E field to zero

get gamma of this bin

mesh the whole domain

scatter particles charge onto grid.

Lorentz transformation In particle rest frame, the longitudinal length (y for cyclotron) enlarged

from charge (C) to charge density (C/m^3).

now charge density is in rho_m calculate Possion equation (without coefficient: -1/(eps))

retrive coefficient: -1/(eps)

calculate electric field vectors from field potential

Back Lorentz transformation CAVE: y coordinate needs 1/gamma factor because IPPL function Grad() divides by hr_m which is not scaled appropriately with Lorentz contraction in y direction only hr_scaled is! -DW

Interpolate electric field at particle positions.

Calculate coefficient

Calculate B_bin field from E_bin field accumulate B and E field

Implements PartBunchBase< double, 3 >.

Definition at line 833 of file PartBunch.cpp.

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void PartBunch::do_binaryRepart ( )

virtual

Reimplemented from PartBunchBase< double, 3 >.

Definition at line 106 of file PartBunch.cpp.

References BinaryRepartition(), PartBunchBase< double, 3 >::boundp(), PartBunchBase< double, 3 >::get_bounds(), PartBunchBase< double, 3 >::pbase, PartBunchBase< double, 3 >::rmax_m, PartBunchBase< double, 3 >::rmin_m, and PartBunchBase< double, 3 >::update().

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PartBunch::VectorPair_t PartBunch::getEExtrema ( )

inlinevirtual

Here we emit particles from the cathode. All particles in a new simulation (not a restart) initially reside in the bin container "pbin_m" and are not part of the beam bunch (so they cannot "see" fields, space charge etc.). In pbin_m, particles are sorted into the bins of a time histogram that describes the longitudinal time distribution of the beam, where the number of bins is given by \(NBIN \times SBIN\). \(NBIN\) and \(SBIN\) are parameters given when defining the initial beam distribution. During emission, the time step of the simulation is set so that an integral number of these bins are emitted each step. Once all of the particles have been emitted, the simulation time step is reset to the value defined in the input file.

A typical integration time step, \(\Delta t\), is broken down into 3 sub-steps:

1) Drift particles for \(\frac{\Delta t}{2}\).

2) Calculate fields and advance momentum.

3) Drift particles for \(\frac{\Delta t}{2}\) at the new momentum to complete the full time step.

The difficulty for emission is that at the cathode position there is a step function discontinuity in the fields. If we apply the typical integration time step across this boundary, we get an artificial numerical bunching of the beam, especially at very high accelerating fields. This function takes the cathode position boundary into account in order to achieve smoother particle emission.

During an emission step, an integral number of time bins from the distribution histogram are emitted. However, each particle contained in those time bins will actually be emitted from the cathode at a different time, so will only spend some fraction of the total time step, \(\Delta t_{full-timestep}\), in the simulation. The trick to emission is to give each particle a unique time step, \(Delta t_{temp}\), that is equal to the actual time during the emission step that the particle exists in the simulation. For the next integration time step, the particle's time step is set back to the global time step, \(\Delta t_{full-timestep}\).

Implements PartBunchBase< double, 3 >.

Definition at line 1139 of file PartBunch.cpp.

References eg_m, max(), and min().

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