Effective radius of ground- and excited-state positronium in collisions with hard walls

We determine effective collisional radii of positronium (Ps) by considering Ps states in hard-wall spherical cavities. $B$-spline basis sets of electron and positron states inside the cavity are used to construct the states of Ps. Accurate Ps energy eigenvalues are obtained by extrapolation with respect to the numbers of partial waves and radial states included in the bases. Comparison of the extrapolated energies with those of a point-like particle provides values of the effective radius ${\rho }_{nl}$ of Ps($nl$) in collisions with a hard wall. We show that, for $1s, 2s$, and $2p$ states of Ps, the effective radius decreases with the increasing Ps center-of-mass momentum and find ${\rho }_{1s}=1.65$ a.u., ${\rho }_{2s}=7.00$ a.u., and ${\rho }_{2p}=5.35$ a.u. in the zero-momentum limit.

Figure 1

Extrapolation in $l_{\text{max}}$ for the lowest six energy eigenvalues for $J^{\mathrm{\Pi }}=0^{+}, R_c=10$ a.u., and $n_{\text{max}}=15$. Solid lines connect the data points to guide the eye, while the dashed lines show extrapolation to $l_{\max }\to \infty$ by using Eq. (20) or (21).

Figure 2

Extrapolation in $l_{\text{max}}$ for the lowest energy eigenvalue for $J^{\mathrm{\Pi }}=0^{+}, R_c=10$ a.u., and $n_{\text{max}}=15$. The solid line connects the data points to guide the eye, while the dashed line shows extrapolation to $l_{\max }\to \infty$ by using Eq. (20).

Figure 3

Extrapolation of the energy in $n_{\text{max}}$ for the lowest six energy eigenvalues for $J^{\mathrm{\Pi }}=0^{+}$ and $R_c=10$ a.u. Solid lines connect the data points to guide the eye, while the dashed lines show extrapolation to $n_{\max }\to \infty$ by using Eq. (22).

Figure 4

Extrapolation of the expected electron-positron separation in $l_{\text{max}}$ for the six lowest-energy eigenstates for $J^{\mathrm{\Pi }}=0^{+}, R_c=10$ a.u., and $n_{\text{max}}=15$. Solid lines connect the data points to guide the eye, while the dashed lines show extrapolation to $l_{\max }\to \infty$ by using Eq. (23) or (24).

Figure 5

Extrapolation of the expected contact density in $l_{\text{max}}$ for the six lowest-energy eigenstates for $J^{\mathrm{\Pi }}=0^{+}, R_c=10$ a.u., and $n_{\text{max}}=15$. Solid lines connect the data points to guide the eye, while the dashed lines show extrapolation to $l_{\max }\to \infty$ by using Eq. (25).

Figure 6

Dependence of the effective Ps($1s$) radius ${\rho }_{1s}\left(K\right)$ on the Ps center-of-mass momentum $K$. The dashed line is the linear fit, Eq. (26).

Figure 7

Dependence of the effective Ps($2s$) radius ${\rho }_{2s}\left(K\right)$ on the Ps center-of-mass momentum $K$. The dashed line is the linear fit, Eq. (27).

Figure 8

Dependence of the effective Ps($2p$) radius ${\rho }_{2p}\left(K\right)$ on the Ps center-of-mass momentum $K$.

Figure 9

Dependence of the effective Ps($2p$) radius ${\rho }_{2p}\left(K\right)$ on the Ps center-of-mass momentum $K$, including data for $J^{\mathrm{\Pi }}=1^{+}$ and $2^{+}$ states. The dashed line is the linear fit, Eq. (28).

Figure 10

Values of $\mathrm{\Delta }{E}_J$ and $\mathrm{\Delta }{E}_J^{\text{(pert)}}$ for the $2p[1,1]$ and $2p[1,2]$ states in a cavity of radius $R_c=10$ a.u.

Figure 11

Values of $\mathrm{\Delta }{E}_J$ and $\mathrm{\Delta }{E}_J^{\text{(pert)}}$ for the $2p[1,1]$ and $2p[1,2]$ states in a cavity of radius $R_c=12$ a.u.