Electron-capture decay rate of ${}^7\mathrm{Be}$ in cluster and crystal forms of beryllium: A first-principles study

To discuss the half-life of the electron-capture decay of ${}^7\mathrm{Be}$ in cluster and crystal forms of beryllium, we performed density functional calculations of the electron density $\rho \left(0\right)$ at the Be nucleus by assuming an inversely proportional relationship between them. We found that the electron density $\rho \left(0\right)$ decreases with cluster size in Be clusters and increases at an interstitial site in crystalline Be. The electron density $\rho \left(0\right)$ at the hexagonal close packed (HCP) lattice point, i.e., the substitutional site, is about 1.7% smaller than that at the center of ${\mathrm{C}}_{60}$, which perfectly coincides with the recent experimental evidence [Phys. Rev. C 108, L011301 (2023)] that the half-life of ${}^7\mathrm{Be}$ in metallic Be at $T=5\mathrm{K}$ is 1.7% longer than that of ${}^7\mathrm{Be}$@${\mathrm{C}}_{60}$ at $T=5\mathrm{K}$ (i.e., at the center of ${\mathrm{C}}_{60}$). This strongly suggests that at $T=5\mathrm{K}$ all ${}^7\mathrm{Be}$ stay at substitutional sites of the Be metal. For an interstitial Be atom, we found that the basal octahedral (BO) site is energetically most stable, and the basal split (BS) dumbbell structure is second most stable. Performing first-principles MD simulations of a system having an interstitial Be atom at room temperature, we found that (1) in a system having ${}^9\mathrm{Be}$ atoms only, BO can quite rapidly migrate through BS; (2) BS made of ${}^7\mathrm{Be}$ and ${}^9\mathrm{Be}$ atoms very rapidly changes into BO of ${}^7\mathrm{Be}$; and (3) ${}^7\mathrm{Be}$ stays very stably at a BO site and seldom changes its position. Therefore, ${}^7\mathrm{Be}$ stays more likely at BO than ${}^9\mathrm{Be}$ at room temperature. The electron density $\rho \left(0\right)$ at BO is 0.54% higher than $\rho \left(0\right)$ at a substitutional site, which is about double the experimental difference of 0.26% in the half-life of ${}^7\mathrm{Be}$ in Be metal between $T=293\mathrm{K}$ and $T=5\mathrm{K}$. This means that half of ${}^7\mathrm{Be}$ atoms are at BO sites, but the other half still remain at substitutional sites at $T=293\mathrm{K}$. So, we expect that the half-life of ${}^7\mathrm{Be}$ can be further shortened at higher temperatures. Performing unit cell relaxation of a Be crystal, we found that the difference in the total energy between BO and BS is only 0.03 eV. Thus, if there is no ${}^7\mathrm{Be}$, BO can very easily migrate throughout the crystal through the BO $\to$ BS $\to$ BO pathway at room temperature.

Figure 1

Calculated total energy per atom $E/N$ and average electron density $\rho \left(0\right)$ at the Be nucleus position versus the cluster size $N(=1–25)$. (BLYP result).

Figure 2

(a) Relative total energy (eV) and (b) electron density $\rho \left(0\right)$ at the Be nucleus of Be crystal versus the lattice constant $a$ (with $c/a$ fixed at 1.569) calculated by using BLYP, SCAN, and revTPSS.

Figure 3

Temporal change in electron densities $\rho \left(0\right)$ at the ${}^7\mathrm{Be}$ nucleus (red line) and ${}^9\mathrm{Be}$ nuclei (black line) during MD simulation at $T=293\mathrm{K}$. Horizontal dashed lines represent their average values. (BLYP result).

Figure 4

Unit cells where (a) one triangle or (b) all triangles are replaced with that of a smaller isolated ${\mathrm{Be}}_3$ in a perfect Be crystal with the original lattice constant fixed.

Figure 5

Eight conventional interstitial sites in HCP crystal. Tetrahedral (T), octahedral (O), crowdion (C), basal tetrahedral (BT), basal octahedral (BO), and basal crowdion (BC) sites in the upper panel, and split (S) and basal split (BS) sites in the lower panel.

Figure 6

Change in the relative (total) energy and the electron density $\rho \left(0\right)$ at the Be nucleus along the migration path coordinates (a) from BO (path length 0) to BS (path length 0.4 $\mathring{\mathrm{A}}$), (b) from BO (path length 0) to the crowdion site C (path length 1.45 $\mathring{\mathrm{A}}$), and (c) from BO (path length 0) to the octahedral interstitial site O (path length 0.9 $\mathring{\mathrm{A}}$), which is a saddle point. (BLYP result).

Figure 7

Migration path of the large blue interstitial atom from (a) BO, through (b) and (c), toward (d) the octahedral interstitial site O, which is the saddle point, determined by the transition state search using FlexTS. Six middle-sized moss green atoms represent an octahedron.

Figure 8

Difference in the electron density when the interstitial atom is at (a) BO and (b) O. The blue $\left(e^{-}\right)$ and red $\left(e^{+}\right)$ isosurface values are $\pm 0.02$ ${\mathring{\mathrm{A}}}^{-3}$.

Figure 9

Snapshots of MD of a system consisting of all ${}^9\mathrm{Be}$ showing a rapid migration of BO through BS: (a) initial (optimized) BO at 0 fs, (b) BS by including upper Be in the dumbbell at 1.8 ps, and (c) BO of the upper Be, leaving the lower Be to the lattice point, at 2.0 ps.

Figure 10

Snapshots of MD of a system consisting of ${}^9\mathrm{Be}$ (small and medium balls) except one ${}^7\mathrm{Be}$ (large ball): (a) initial (optimized) BS made up of ${}^7\mathrm{Be}$ and ${}^9\mathrm{Be}$ at 0 fs, (b) ${}^9\mathrm{Be}$ moving from BS to a lattice point at 0.3 ps, and (c) ${}^7\mathrm{Be}$ remaining at BO at 0.5 ps.

Figure 11

Half-life of ${}^7\mathrm{Be}$ experimentally measured by Ohtsuki et al. (after Ref. [29]).

Figure 12

Frenkel defect made of one BO (large blue ball) and one atomic vacancy (red circle) (a) on the same basal plane and (b) on different, adjacent basal planes. In (b), small light green and moss green balls are atoms on the different basal planes.