Power laws for nuclear observables

Often mass-dependent nuclear observables are summarized by fits to a power law in the nuclear mass $A$, as proportional to $A^{\alpha }$. In this work a simple justification of this usage is presented, also providing a simple expression for the exponent $\alpha$, with one known parameter being the beam-nucleon total cross section and another parameter being a nuclear size $r_0{A}^{1/3}$. Measured power-law exponents $\alpha$ for total and reaction cross sections are near this simple formulation using $r_0=1.6\mathrm{fm}$, whereas quasifree observables are near the simple expression using $r_0=1.2\mathrm{fm}$.

Figure 1

The fitted exponents for an $A^{\alpha }$ power law from the data in Tables 1, 2, 3 are plotted against $70\%$ of the total in-medium beam-nucleon cross sections SGT, as computed in the text. The results of Eq. (6) are shown as lines, with use of $r_0=1.6\mathrm{fm}$ compared to the total and reaction cross section exponents, and $r_0=1.2\mathrm{fm}$ compared to the quasifree cross sections.

Figure 2

The coefficients (in mb) of the $A^{\alpha }$ power law fits to the reaction and total cross section data of Tables 1, 2 are plotted against the beam-nucleon total cross sections, computed as described in the text. The line shows the result for a coefficient of 1. Total cross sections are shown as solid points and reaction cross sections as open points.

Figure 3

The coefficients $A_0$, $N_0$, or $Z_0$ of fits with a power law to measured effective numbers of nucleons for integrations of the quasifree (beam-nucleon) peaks in inclusive intermediate energy spectra from nuclei are plotted against the average beam-nucleon total in-medium cross sections. Solid points are for NCX and open points for SCX. The high point is for a ${\mathrm{K}}^{+}$ beam [27]. Data sources are found in Table 3.

Figure 4

As Fig. 1, but for fits to the results of Glauber calculations for $A=12$, 17, 56, 91, 119, 207 across a range of in-medium $(70\%)$ SGT. The black disk limit would be $\alpha =2/3$. Quasifree results are shown for NCX reactions.