TabulatedMaterial

A metatype exposing the tabulations of a material.

Attributes

A TabulatedMaterial is a special instance of a Material for which physical properties relevant for the transport have been pre-computed (tabulated), e.g. the energy loss of the projectile. In addition to the base attributes of a Material it has the following extra attributes:

Name Type Description
name string Name of the material used e.g. when referring to from a Medium.
physics Readonly The Physics instance this TabulatedMaterial belongs to.
table Readonly Material tabulations (see below).
composite boolean Flag indicating if the instance is a composite material (true) or a base one (false).
materials CompositeMaterials or nil For composite materials the CompositeMaterials table contains the mass fractions of the base materials indexed by name.

Note

All attributes are Readonly, the base Material ones as well as the extra tabulated ones. There is one exception though. For composite materials the mass fractions of the base materials can be modified (see CompositeMaterials).

table

The table attribute contains the physics tabulations organized in three sub-tables corresponding to the three energy loss modes: 'csda', 'detailed' and 'hybrid'. The available tabulations are summarized below, where it must be understood that mode stands for one of csda, detailed or hybrid. E.g. table.csda.energy_loss returns the tabulation of the continuous energy loss for the CSDA mode. More detailed explanations of the various physical quantities are also provided hereafter.

Name Type Description
(mode).energy_loss Readonly Continuous Energy Loss (CEL) per unit grammage, \frac{dE}{dX}, in \text{GeV} \cdot \text{m}^2 \cdot \text{kg}^{-1} (see below).
(mode).kinetic_energy Readonly Tabulated projectile kinetic energies, E, in \text{GeV}.
(mode).grammage Readonly Total grammage for CEL, X, in \text{kg} \cdot \text{m}^{-2} (see below).
(mode).proper_time Readonly Total proper time for CEL, \tau, assuming a uniform density, in \text{kg} \cdot \text{m}^{-2} (see below).
(mode).cross_section Readonly Macroscopic cross-section for DELs, \Sigma, in m2/kg. Available only for detailed and hybrid modes (see below).

Note

The detailed and hybrid mode use the same tabulations. For convenience both notations can be used. Yet, they actualy refer to the same tables.

energy_loss

The simulation of physics interactions is split into two components: Discrete Energy Losses (DELs) and Continuous Energy Losses (CELs), see Niess et al., 2017 for more details. The splitting is done using a relative cutoff \alpha = 5\%, following Sokalski et al., 2001. The continuous energy loss per unit of grammage, \frac{dE}{dX}, is given as:

\frac{dE}{dX} = \frac{\mathcal{N}_A}{A} \int_0^{\alpha E}{q \frac{\partial \sigma(E, q)}{\partial q} dq}

with E the projectile kinetic energy, \frac{\partial \sigma}{\partial q} the total Differential Cross Section (DCS), \mathcal{N}_A Avogadro number and A the molar mass of the target material.

cross_section

The macroscopic cross-section for DELs, \Sigma, is defined per unit mass as:

\Sigma(E) = \frac{\mathcal{N}_A}{A} \int_{\alpha E}^E{\frac{\partial \sigma(E, q)}{\partial q} dq}

The macroscopic cross-section is related to the interaction length, \lambda, as:

\lambda = \frac{\rho}{\Sigma}

with \rho the medium density.

grammage

In the absence of DEL, the total grammage path length of the projectile is obtained by integrating its energy loss up to its initial energy, as:

X(E) = \int_0^E{\frac{1}{\frac{dE}{dX}} dE'}

The previous expression can also been used in order to compute the grammage variation, \Delta X, between two DELs as:

\Delta X = X(E_1) - X(E_0)

where E_0 if the energy after the first DEL and E_1 the energy before the second DEL occurs.

proper_time

For a uniform density and in the absence of DEL, the total proper time of the projectile is obtained by integration similarly to the grammage path length. Per unit mass, it writes:

\tau(E) = \int_0^E{\frac{1}{\beta \gamma \frac{dE}{dX}} dE'}

In order to get the proper time in unit of time one needs to divide the previous expression by \rho c, i.e. the constant medium density, \rho, times the vacuum speed of light, c.

See also

build, CompositeMaterials, Element, Material, Physics.

TabulatedMaterial.cross_section

Interpolation of the DELs macroscopic cross section, \Sigma, as function of the projectile kinetic energy, E. See above for more details.

Synopsis

TabulatedMaterial:cross_section(energy)

Arguments

Name Type Description
energy number Kinetic energy of the projectile, in GeV.

Returns

Type Description
number The macroscopic cross-section per unit mass, in \text{m}^2 \cdot \text{kg}^{-1} (see above).

See also

energy_loss, grammage, kinetic_energy, magnetic_rotation, proper_time, scattering_length.

TabulatedMaterial.energy_loss

Interpolation of the Continuous Energy Loss (CEL), \frac{dE}{dX}, as function of the projectile kinetic energy, E. See above for more details.

Synopsis

TabulatedMaterial:energy_loss(energy, (mode))

Arguments

Name Type Description
energy number Kinetic energy of the projectile, in GeV.
mode string Energy loss mode, one of 'csda', 'detailed' or 'hybrid'. Defaults to 'csda'.

Returns

Type Description
number The continuous energy loss per unit grammage, in \text{GeV} \cdot \text{m}^2 \cdot \text{kg}^{-1} (see above).

See also

cross_section, grammage, kinetic_energy, magnetic_rotation, proper_time, scattering_length.

TabulatedMaterial.grammage

Interpolation of the total grammage path length, X, as function of the projectile initial kinetic energy, E. See above for more details.

Synopsis

TabulatedMaterial:grammage(energy, (mode))

Arguments

Name Type Description
energy number Initial kinetic energy of the projectile, in GeV.
mode string Energy loss mode, one of 'csda', 'detailed' or 'hybrid'. Defaults to 'csda'.

Returns

Type Description
number The total grammage path length, in \text{kg} \cdot \text{m}^{-2} (see above).

See also

cross_section, energy_loss, kinetic_energy, magnetic_rotation, proper_time, scattering_length.

TabulatedMaterial.kinetic_energy

Interpolation of the projectile initial kinetic energy, E, as function of its total grammage path length, X. See above for more details.

Synopsis

TabulatedMaterial:kinetic_energy(grammage, (mode))

Arguments

Name Type Description
grammage number Total grammage path length, in \text{kg} \cdot \text{m}^{-2}.
mode string Energy loss mode, one of 'csda', 'detailed' or 'hybrid'. Defaults to 'csda'.

Returns

Type Description
number The projectile initial kinetic energy, in GeV (see above).

See also

cross_section, energy_loss, grammage, magnetic_rotation, proper_time, scattering_length.

TabulatedMaterial.magnetic_rotation

Interpolation of the total rotation angle of the projectile direction, \Theta, due to a constant magnetic field, B, as function of its initial kinetic energy, E. For a uniform magnetic field and medium density, \rho, if no DELs occur the rotation angle is related to the total proper time, \tau, as:

\Theta(E) = \frac{q B}{m} \tau(E)

with q (m) the projectile electric charge (rest mass). See above for more details on how the proper time is computed.

Synopsis

TabulatedMaterial:magnetic_rotation(energy)

Arguments

Name Type Description
energy number Initial kinetic energy of the projectile, in GeV.

Note

The magnetic rotation is only available for the CSDA mode.

Returns

Type Description
number Total rotation angle, in \text{rad} \cdot \text{kg} \cdot \text{m}^{-3} \cdot \text{T}^{-1}. Multiply by the constant magnetic field, B, and divide by the uniform medium density, \rho, in order to get the angle in radians.

See also

cross_section, energy_loss, grammage, kinetic_energy, proper_time, scattering_length.

TabulatedMaterial.proper_time

Interpolation of the total proper time, \tau, as function of the projectile initial kinetic energy, E. See above for more details.

Synopsis

TabulatedMaterial:proper_time(energy, (mode))

Arguments

Name Type Description
energy number Initial kinetic energy of the projectile, in GeV.
mode string Energy loss mode, one of 'csda', 'detailed' or 'hybrid'. Defaults to 'csda'.

Returns

Type Description
number The total proper time, in \text{kg} \cdot \text{m}^{-2} (see above).

See also

cross_section, energy_loss, grammage, kinetic_energy, magnetic_rotation, scattering_length.

TabulatedMaterial.scattering_length

Compute the first transport path length for Coulomb Multiple Scattering (MS), \lambda, as function of the projectile kinetic energy, E. The first path length is related to the standard deviation of the MS angle as:

\theta^2 = \frac{X}{2 \lambda}

with X the grammage path length. See Fernández-Varea et al., 1993, for more details.

Synopsis

TabulatedMaterial:scattering_length(energy)

Arguments

Name Type Description
energy number Kinetic energy of the projectile, in GeV.

Returns

Type Description
number The Coulomb MS first path length, in \text{kg} \cdot \text{m}^{-2}.

See also

cross_section, energy_loss, grammage, kinetic_energy, magnetic_rotation, proper_time.