Maximum J Pairing and Asymptotic Behavior of the 3j and 9j Coefficients

We investigate the large $j$ behavior of certain $3j$ and $9j$ symbols, where $j$ is the total angular momentum of one particle in a given shell. Our motivation is the problem of maximum $J$ pairing in nuclei, along with the more familiar $J=0$ pairing. Maximum $J$ pairing leads to an increase in $J=2$ coupling of two protons and two neutrons relative to $J=0$. We find that a coupling unitary $9j$ symbol ($U9j$) is very weak as $j$ increases, leading to wavefunctions which are to an excellent approximation single $U9j$ coefficients. Our study of the large $j$ behavior of coupling unitary $9j$ symbols is through the consideration of the case when the total angular momentum $I$ is equal to $I_{\mathrm{max}}-2n$ and $I_{\mathrm{max}}\equiv4j-2$, where $n=0,1,2,...$. We here derive asymptotic approximations of coupling $3j$ symbols and find that the $3j\propto j^{-3/4}$ in the high $j$ limit. One major analytical tool we used is the Stirling Approximation. Through analytical, numerical, and graphical methods, we show the power law behavior of the coupling unitary $9j$ symbols in the $n/j\ll1$ limit, i.e. $U9j\propto j^{-n}$. Power-law behavior is evident if there is a linear dependence of $\ln{|U9j|}$ vs. $\ln{j}$. We also present some examples of percent errors in our approximations.