Nuclear models

abstract type AbstractNuclearModel

Supertype for all (radially symmetric) nuclear models. The underlying charge distribution of the nuclear model is assumed to be normalized to unity (i.e. corresponding to $Z = 1$).

Each concrete nuclear model of type T <: AbstractNuclearModel should implement the following method

• potential(::T, r::Real) -> Float64: return the value of the potential at the radius r.

Additional, each nuclear model may also implement

• density(::T, r::Real) -> Float64: return the value of the (normalized) charge density at the radius r.

• rms(::T): return the root-mean-square radius $\sqrt{\langle r^2 \rangle}$ of the underlying charge distribution.

• from_rms(::Type{T}, rms): construct an instance of the nuclear model with the specified root-mean-square radius.

struct PointNucleus <: AbstractNuclearModel

Represent a point nucleus with the potential

\[V(r) = - Z / r, \quad \rho(r) = \frac{Z}{4\pi r^2} \delta(r)\]

This nuclear model has no parameters. Also, this model does not define a method for density due to the delta-function nature of the charge distribution.

Constructors

• PointNucleus(): construct an instance of PointNucleus

Examples

julia> using AtomicMiscellany: PointNucleus, rms, potential

julia> rms(PointNucleus())
0.0

julia> potential(PointNucleus(), 2)
-0.5

struct UniformShellNucleus <: AbstractNuclearModel

Represents a nuclear model where all the charge is unformly distributed over an infinitely thin shell at radius R. The nuclear potential and charge distribution are given by

\[V(r) = \begin{cases} -Z/R, & 0 \leq r \leq R \\ -Z/r, & r > R \end{cases}, \quad \rho(r) = \frac{Z}{4\pi r^2} \delta(r - R)\]

Note that this model does not define a method for density due to the delta-function nature of the charge distribution.

Constructors

• UniformShellNucleus(R::Real): construct a shell nucleus with radius R.

• from_rms(UniformShellNucleus, rms): constructs a shell nucleus with the radius determined from the root-mean-square radius $\sqrt{\langle r^2 \rangle}$ of the charge distribution

  \[R = \sqrt{\langle r^2 \rangle}\]

struct UniformSphericalNucleus <: AbstractNuclearModel

Represents a nuclear charge distribution where the charge is homogeneously distributed in a sphere of radius $R$. The potential and charge distributions are given by

\[V(r) = \begin{cases} -\frac{Z}{2R}\left[3 - \left(\frac{r}{R}\right)^2\right], & 0 \leq r \leq R \\ -\frac{Z}{r}, & r > R \end{cases}, \quad \rho(r) = \begin{cases} \frac{3Z}{4\pi R^3}, & 0 \leq r \leq R \\ 0, & r > R \end{cases} \]

Constructors

• UniformSphericalNucleus(R::Real): construct a homogeneous spherical nucleus with radius R.

• from_rms(UniformSphericalNucleus, rms): constructs a homogeneous spherical nucleus with the radius determined from the root-mean-square radius $\sqrt{\langle r^2 \rangle}$ of the charge distribution

  \[R = \sqrt{\frac{5}{3}} \sqrt{\langle r^2 \rangle}\]

struct GaussianNucleus <: AbstractNuclearModel

Represents a Gaussian-shaped nuclear charge distribution, size of which is determined by the parameter $R$`. The potential and charge distributions are given by

\[V(r) = - \frac{Z}{r} \mathrm{erf}\left(\frac{r}{R}\right), \quad \rho(r) = \frac{Z}{\pi^{3/2} R^3}\exp\left[-\left(\frac{r}{R}\right)^2\right] \]

Constructors

• GaussianNucleus(R::Real): constructs a Gaussian nuclear model of size R.

• from_rms(UniformSphericalNucleus, rms): constructs a Gaussian nucleus with the $R$ parameter determined from the root-mean-square radius $\sqrt{\langle r^2 \rangle}$ of the charge distribution

  \[R = \sqrt{\frac{2}{3}} \sqrt{\langle r^2 \rangle}\]

rms(JohnsonSoff1985, A::Real) -> Float64

Returns the root-mean-square radius (in atomic units) of a nucleus based on the fit from the JohnsonSoff1985 paper

\[\rm{rms}(A) = (0.836 A^{1/3} + 0.570)~\rm{fm}\]

where A is the atomic mass number of the isotope.

julia> using AtomicMiscellany: rms, JohnsonSoff1985

julia> rms(JohnsonSoff1985, 238)
0.00010867476884785438

potential(::AbstractNuclearModel, r::Real) -> Float64

Return the value of the (normalized) charge density at the radius r.

density(::AbstractNuclearModel, r::Real) -> Float64

Return the value of the (normalized) charge density at the radius r.

rms(::AbstractNuclearModel)

Return the root-mean-square radius $\sqrt{\langle r^2 \rangle}$ of the underlying charge distribution.

from_rms(::Type{<:AbstractNuclearModel}, rms)

Construct an instance of the nuclear model with the specified root-mean-square radius.

Can be used to bring various bindings related to nuclear models into the namespace.

julia> using AtomicMiscellany.NuclearModels

julia> parentmodule(rms), parentmodule(AbstractNuclearModel)
(AtomicMiscellany, AtomicMiscellany)

struct Andrae2000

Type to dispatch on data from the following paper:

• Andrae, D. "Finite Nuclear Charge Density Distributions in Electronic Structure Calculations for Atoms and Molecules." Physics Reports 336, no. 6 (October 2000): 413–525. https://doi.org/10.1016/S0370-1573(00)00007-7.

struct JohnsonSoff1985

Type to dispatch on data from the following paper:

• W.R. Johnson, Gerhard Soff, "The lamb shift in hydrogen-like atoms, 1 ⩽ Z ⩽ 110", Atomic Data and Nuclear Data Tables, Volume 33, Issue 3, 1985, Pages 405-446, https://doi.org/10.1016/0092-640X(85)90010-5

See also: rms(::Type{JohnsonSoff1985}, ::Real).