qspec.delta_r4  (  r ,  r_d ,  r_ref ,  r_ref_d ,  dr = None ,  dr_d = None ,  v4 = 1.0 ,  v4_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 $$\begin{aligned} \delta\!\langle r^4\rangle &= \left(\frac{R}{V_4}\right)^4 - \left(\frac{R_\mathrm{ref}}{V_{4,\mathrm{ref}}}\right)^4\\[2ex] \Delta\delta\!\langle r^4\rangle &= \sqrt{\left(\frac{4 R^3\Delta R}{V_4^4}\right)^2 + \left(\frac{4 R_\mathrm{ref}^3\Delta R_\mathrm{ref}}{V_{4,\mathrm{ref}}^4}\right)^2}. \end{aligned}$$ If the uncertainty dr_d is specified, which is often much lower than r_d and r_ref_d, an improved formula is used that gives smaller uncertainties $$\begin{aligned} \delta\!\langle r^4\rangle &= sd, \quad s \coloneqq \left(\frac{R}{V_4}\right)^2 + \left(\frac{R_\mathrm{ref}}{V_{4,\mathrm{ref}}}\right)^2, \quad d \coloneqq \left(\frac{R}{V_4}\right)^2 - \left(\frac{R_\mathrm{ref}}{V_{4,\mathrm{ref}}}\right)^2\\[2ex] \Delta\delta\!\langle r^4\rangle &= \sqrt{\left(s\Delta d\right)^2 + \left(\frac{2dR}{V_4^2}\Delta R\right)^2 + \left(\frac{2dR_\mathrm{ref}}{V_{4,\mathrm{ref}}^2}\Delta R_\mathrm{ref}\right)^2}. \end{aligned}$$ Here $d$ and $\Delta d$ are calculated using the small-uncertainty version of the delta_r2 function for the mean-square 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]).

v4array_like

The shape factor $V_4$ of the first isotope.

v4_refarray_like

The shape factor $V_{4,\mathrm{ref}}$ of the second isotope.

Returns:

(dr4, dr4_d)(ndarray, ndarray)

The difference of the mean-quartic nuclear charge radius between two isotopes $\delta\!\langle r^4\rangle$ and its uncertainty $\Delta\delta\!\langle r^4\rangle$ ([r ** 4]).