- qspec.delta_r6 ( r , r_d , r_ref , r_ref_d , dr = None , dr_d = None , v6 = 1.0 , v6_ref = 1.0 )[source]
The difference of the mean square nuclear charge radius between two isotopes and its uncertainty calculated from the Barrett radii and elastic electron scattering form factors $$\begin{aligned} \delta\!\langle r^6\rangle &= \left(\frac{R}{V_6}\right)^6 - \left(\frac{R_\mathrm{ref}}{V_{6,\mathrm{ref}}}\right)^6\\[2ex] \Delta\delta\!\langle r^6\rangle &= \sqrt{\left(\frac{6 R^5\Delta R}{V_6^6}\right)^2 + \left(\frac{6 R_\mathrm{ref}^5\Delta R_\mathrm{ref}}{V_{6,\mathrm{ref}}^6}\right)^2}. \end{aligned}$$ If the uncertainty
dr_d
is specified, which is often much lower thanr_d
andr_ref_d
, an improved formula is used that gives smaller uncertainties $$\begin{aligned} \delta\!\langle r^6\rangle &= s(d + t), \quad s \coloneqq \frac{V_6}{R}\left[\left(\frac{R}{V_6}\right)^3 + \left(\frac{R_\mathrm{ref}}{V_{6,\mathrm{ref}}}\right)^3\right],\\ \quad d &\coloneqq \left(\frac{R}{V_6}\right)^4 - \left(\frac{R_\mathrm{ref}}{V_{6,\mathrm{ref}}}\right)^4 \quad t \coloneqq \left(\frac{R_\mathrm{ref}}{V_{6,\mathrm{ref}}}\right)^4 \left(1 - \frac{R}{R_\mathrm{ref}}\frac{V_{6,\mathrm{ref}}}{V_6}\right)\\[2ex] \Delta\delta\!\langle r^6\rangle &= \sqrt{\splitfrac{\left(s\Delta d\right)^2 + \left(\left[\left(-\frac{R_\mathrm{ref}}{V_{6,\mathrm{ref}}}\right)^3\frac{s}{V_6} + (d + t)\left(-\frac{s}{R} + 3\frac{R}{V_6^2}\right)\right]\Delta R\right)^2} {+ \left(\left[s\left(\frac{4t}{R_\mathrm{ref}} + \frac{R}{V_6}\frac{R_\mathrm{ref}^2}{V_{6,\mathrm{ref}}^3}\right) + 3(d + t)\frac{V_6}{R}\frac{R_\mathrm{ref}^2}{V_{6,\mathrm{ref}}^3}\right]\Delta R_\mathrm{ref}\right)^2}}. \end{aligned}$$ Here $d$ and $\Delta d$ are calculated using the small-uncertainty version of thedelta_r4
function for the mean-quartic nuclear charge radius.- Parameters:
-
- rarray_like
The Barrett radius $R$ of the first isotope (arb. units).
- r_darray_like
The uncertainty of the Barrett radius $\Delta R$ of the first isotope ([
r
]).- r_refarray_like
The Barrett radius $R_\mathrm{ref}$ of the second isotope ([
r
]).- r_ref_darray_like
The uncertainty of the Barrett radius $\Delta R_\mathrm{ref}$ of the second isotope ([
r
]).- drarray_like
The difference between the Barrett radii $\delta R = R - R_\mathrm{ref}$ of the first and second isotope ([
r
]).- dr_darray_like
The uncertainty of the difference between the Barrett radii $\Delta\delta R$ of the first and second isotope ([
r
]).- v6array_like
The shape factor $V_6$ of the first isotope.
- v6_refarray_like
The shape factor $V_{6,\mathrm{ref}}$ of the second isotope.
- Returns:
-
- (dr6, dr6_d)(ndarray, ndarray)
The difference of the mean-sextic nuclear charge radius between two isotopes $\delta\!\langle r^6\rangle$ and its uncertainty $\Delta\delta\!\langle r^6\rangle$ ([
r**6
]).