 Solution

Use the supplied polynomial as an effective-atomic-number approximation for the alloys. Interpreting each quoted phosphorus number as an atomic fraction $x_P=N_P/(N_P+N_{\mathrm{Ni}})$,The gold reference yield is $\eta (79)=0.486996$. Substitution gives:

The ordering is gold brightest, then copper, NiP${}_z$, NiP${}_x$ and NiP${}_y$. The phosphorus-rich layer has the lowest effective atomic number.

There is a wording ambiguity in “atomic fraction of P relative to Ni.” If those numbers instead mean $r=N_P/N_{\mathrm{Ni}}$, first convert $x_P=r/(1+r)$, giving $\bar{Z}=(28+15r)/(1+r)$. The three alloy contrasts then become $0.4243$, $0.4361$, $0.4129$, respectively. The qualitative ordering is unchanged. Both calculations explicitly use $\eta (\bar{Z})$; this simple alloy approximation is not a full electron-transport simulation.