3.53 Finite Nuclear Models

This functionality was provided by Thanush Patlolla, high school researcher at Duke AIMS. FHI-AIMS runs standard calculations by assuming a point nucleus, or that the entire nuclear charge is concentrated at a point in the center of the atom. This, of course, is not entirely accurate, and FHI-AIMS offers the ability to specify a nuclear charge distribution for each species. AIMS calculates the potential associated with this nucleus and uses it instead of the standard $-\frac{Z}{r}$ potential throughout the code. (Of course, outside the nucleus, the two potentials are identical.) The potential is calculated by solving Poisson’s equation as detailed in Reference [10]:

-rV(r)=4\pi\left(\int_{0}^{r}\mathrm{d}ss^{2}\rho(s)+r\int_{0}^{\infty}\mathrm% {d}ss\rho(s)\right)

(3.283)

Species defaults are also available for each element and model type. These defaults are based on the following sources: [72], [97], [71]. The data from these sources is derived from elastic electron scattering experiments.
This functionality is available for all exchange-correlation functionals and all levels of relativity. We recommend using the Q4C level of relativity to best see the effects of the finite nuclear model.

A few model types, detailed below, are provided for the desired charge distribution to be fit to. (These are the model types used in the above reference and the defaults.) All model types are normalized such that the total charge is equal to the nuclear charge $Z$ .

3.53.1 Model Types

3.53.2 Harmonic Oscillator/Modified Harmonic Oscillator

\rho(r)=\rho_{0}\left(1+\alpha\left(\frac{r}{a}\right)^{2}\right)\exp\left(-% \left(\frac{r}{a}\right)^{2}\right)

(3.284)

Where $a$ is a free parameter. $\alpha$ is determined according to a formula (see above references) in the Harmonic Oscillator model, while it is a free parameter in the Modified Harmonic Oscillator. In either case, AIMS requires the input of both $a$ and $\alpha$ . Uses keyword HO or MHO.

3.53.3 Two-parameter Fermi

\rho(r)=\frac{\rho_{0}}{1+\exp\left(\frac{r-c}{z}\right)}

(3.285)

With $c$ and $z$ free parameters. Uses keyword 2pF.

3.53.4 Three-parameter Fermi

\rho(r)=\rho_{0}\frac{1+\frac{wr^{2}}{c^{2}}}{1+\exp\left(\frac{r-c}{z}\right)}

(3.286)

With $c$ $z$ , and $w$ free parameters. Uses keyword 3pF.

3.53.5 Three-parameter Gaussian

\rho(r)=\rho_{0}\frac{1+\frac{wr^{2}}{c^{2}}}{1+\exp\left(\frac{r^{2}-c^{2}}{z% ^{2}}\right)}

(3.287)

With $c$ , $z$ , and $w$ free parameters. Uses keyword 3pG.

3.53.6 Fourier-Bessel Expansion

\rho(r)=\begin{cases}\sum_{v}a_{v}j_{0}(v\pi r/R)&r\leq R\\ 0&r\geq R\end{cases}

(3.288)

With $v$ , $a_{v}$ , and $R$ free parameters. Uses keyword FB. This input is a little more complicated, see below.

3.53.7 Sum-of-Gaussian Expansion

A_{i}=\frac{ZeQ_{i}}{2\pi^{3/2}\gamma^{3}\left(1+\frac{2R_{i}^{2}}{\gamma^{2}}% \right)}

(3.289)

\rho(r)=\sum_{i}A_{i}\left[\exp\left(-\left(\frac{r-R_{i}}{\gamma}\right)^{2}% \right)+\exp\left(-\left(\frac{r+R_{i}}{\gamma}\right)^{2}\right)\right]

(3.290)

With $i$ , $Q_{i}$ , $R_{i}$ , and $\gamma$ free parameters. Uses keyword SOG. This input is a little more complicated, see below.

Subtags for species tag in control.in:

species sub-tag: isotope(control.in)

Usage: isotope A
Purpose: Nucleon number
A is a real number that specifies the nucleon number (mass number), or the total number of protons and neutrons (not to be confused with atomic mass).

species sub-tag: finite_nucleus(control.in)

Usage: finite_nucleus model_type parameters
Purpose: Specify the usage and model type of the finite nuclear model for a given species, and any associated parameters of the distribution. Specifics are detailed above in the description of each model type, but in general models have either 2 or 3 space separated parameters, or in the case of the Fourier-Bessel or Sum of Gaussians models, use a multi-line input with the keywords detailed below. The various keywords and number of parameters for each model type are listed above.

Example: finite_nucleus 3pF $4.920$ $5.6135$ $0.586824$ $0.09614$

finite_nucleus sub-tag: HO(control.in)

Usage: HO a alpha
Purpose: Harmonic Oscillator model inputs
a (fm) and alpha are parameters of the model.

finite_nucleus sub-tag: MHO(control.in)

Usage: MHO a alpha
Purpose: Modified Harmonic Oscillator model inputs
a (fm) and alpha are parameters of the model.

finite_nucleus sub-tag: 2pF(control.in)

Usage: 2pF c z
Purpose: Two-parameter Fermi model inputs
c (fm) and z (fm) are parameters of the model.

finite_nucleus sub-tag: 3pF(control.in)

Usage: 3pF c z w
Purpose: Three-parameter Fermi model inputs.
c (fm), z (fm), and w are parameters of the model.

finite_nucleus sub-tag: 3pG(control.in)

Usage: 3pG a alpha
Purpose: Three-parameter Gaussian Model inputs.
c (fm), z (fm), and w are parameters of the model.

species sub-tag: fb_series_terms(control.in)

Usage: fb_series_terms number_of_terms
Purpose: Specifies number of terms in the Fourier-Bessel expansion of the finite nuclear model.
number_of_terms is an integer (max. 2 characters) that specifies the number of terms in the expansion

species sub-tag: fb_coefficient(control.in)

Usage: fb_coefficient coefficient_value
Purpose: Nth usage of the keyword specifies the value of the nth coefficient in the Fourier-Bessel expansion. (Specified in inverse femtometers cubed).

species sub-tag: fb_radius(control.in)

Usage: fb_radius radius_value
Purpose: Specifies the cutoff radius of the finite nuclear model in a Fourier-Bessel expansion in femtometers.

Example:
finite_nucleus FB
fb_series_terms $3$
fb_coefficient $4.4846e-02$
fb_coefficient $6.1326e-02$
fb_coefficient $-1.6818e-03$
fb_radius $8.0$

species sub-tag: sog_series_terms(control.in)

Usage: sog_series_terms number_of_terms
Purpose: Specifies number of terms in the Sum-of-Gaussians expansion of the finite nuclear model.
number_of_terms is an integer (max. 2 characters) that specifies the number of terms in the expansion

species sub-tag: sog_coefficients(control.in)

Usage: sog_coefficients coefficient_value_1 coefficient_value_2
Purpose: ith usage of the keyword specifies the values of the ith coefficients $R_{i}$ and $Q_{i}$ in the Sum-of-Gaussians expansion.
coefficient_value_1 $R_{i}$ in femtometers
coefficient_value_1 $Q_{i}$

species sub-tag: sog_radii(control.in)

Usage: sog_radii coefficient_value_1 coefficient_value_2
Purpose: Specifies the rms radius and RP value for the distribution
coefficient_value_1 rms radius in femtometers
coefficient_value_1 $RP=\gamma\sqrt{\frac{3}{2}}$ in femtometers
Example:
sog_series_terms $3$
sog_coefficients $0.4$ $0.042870$
sog_coefficients $1.2$ $0.056020$
sog_coefficients $1.8$ $0.167853$
sog_radii $3.48$ $1.45$

species sub-tag: nuc_rho_step_size(control.in)

Usage: nuc_rho_step_size value specifies minimum log grid size for use with the finite nucleus.
value minimum log grid size in Bohr radii

species sub-tag: min_nuc_rho_cutoff(control.in)

Usage: min_nuc_rho_cutoff value specifies magnitude of nuclear charge (as a percentage of charge at $r=0$ at which the nuclear radius is set.
value cutoff value[1.0ex]