Target g3 Construction

A target g3 distribution represents the desired structural correlations for a supercell. It is built from a crystalline reference measurement by blurring and blending toward the random limit. Target construction is independent of supercell generation and is used for comparison.

Construction pipeline¶

Starting from a measured crystalline g3:

Reduce. Divide out the ideal density factor to obtain the reduced distribution $\tilde{g}_3$ , which approaches 1.0 in the random limit.
Blur in $\phi$ . Gaussian convolution along the angular axis with reflected boundaries at $\phi = 0$ and $\phi = \pi$ . The blur $\sigma$ grows linearly with radius:

\sigma_\phi(r) = \frac{\phi_{\sigma,\text{deg}}}{r_{\sigma\text{\_at}}} \cdot r

(1)

If r_sigma_at is omitted, phi_sigma_deg is interpreted as the slope directly ( $\sigma_\phi(r) = \phi_{\sigma,\text{deg}} \, r$ ).

Blur in $r$ . 1D Gaussian kernel applied sequentially along $r_{01}$ then $r_{02}$ (a separable 2D blur). The radial $\sigma$ scales with $r_{01}$ and $r_{02}$ respectively using the same r_sigma / r_sigma_at relation.
Blend toward random. Smooth Hermite cubic interpolation between the blurred crystalline distribution and the random limit:

\tilde{g}_3^{\text{target}} = (1 - m) \cdot \tilde{g}_3^{\text{blurred}} + m \cdot 1.0

(2)

where the mixing factor transitions from 0 to 1 between target_r_min and target_r_max:

m(r_\text{eff}) = s^2 (3 - 2s), \quad s = \text{clamp}\left(\frac{r_\text{eff} - r_\text{min}}{r_\text{max} - r_\text{min}},\; 0,\; 1\right)

(3)

with $r_\text{eff} = \max(r_{01}, r_{02})$ .

Un-reduce. Multiply back by the ideal density factor to recover the raw histogram form.

Parameters¶

Parameter	Description
`target_r_min`	Radius where the transition from crystalline to random begins
`target_r_max`	Radius where the distribution is fully random
`r_sigma`	Radial blur width in Å (at `r_sigma_at`)
`r_sigma_at`	Reference radius where the radial blur equals `r_sigma`
`phi_sigma_deg`	Angular blur width in degrees (at `r_sigma_at`)

Usage¶

Target construction is a pure G3Distribution operation with no dependence on Supercell:

dist = am.G3Distribution(atoms)
dist.measure_g3(
    r_max=10,
    r_step=0.1,
    phi_num_bins=90,
)
target = dist.target_g3(
    target_r_min=5.0,
    target_r_max=8.0,
    r_sigma=0.05,
    r_sigma_at=2.34,
    phi_sigma_deg=3.0,
)

# View the target
target.plot_g3()

# Use for comparison after generating a supercell
cell = am.Supercell(
    target,
    cell_dim_angstroms=(40, 40, 40),
)
cell.generate(shell_target, grain_size=15.0)
cell.measure_g3()
cell.plot_g3_compare()

The g2 pair distribution¶

The same blurring and blending pipeline is also applied to the pair distribution $g_2$ , using a 1D version of each operation. The target g2 is stored alongside the target g3 in the returned G3Distribution object.