$D_5^{(1)}$- Geometric Crystal corresponding to the Dynkin spin node $i=5$ and its ultra-discretization

Let $g$ be an affine Lie algebra with index set $I=\{0,1,2,\cdots ,n\}$ and $g^L$ be its Langlands dual. It is conjectured that for each Dynkin node $i \in I \setminus {0}$theaffineLiealgebra$g$ has a positive geometric crystal whose ultra-discretization is isomorphic to the limit of certain coherent family of perfect crystals for $g^L$. In this paper we construct a positive geometric crystal $V(D_5^{(1)})$ in the level zero fundamental spin $D_5^{(1)}$- module $W({\varpi }_5)$. Then we define explicit $0$-action on the level $l$ known $D_5^{(1)}$- perfect crystal $B^{5,l}$ and show that ${B^{5, l}}_{l \geq 1}$isacoherentfamilyofperfectcrystalswithlimit$B^{5, \infty}$\mathrm{.}Finallyweshowthattheultra-discretizationof$V(D_5^{(1)})$ is isomorphic to $B^{5,\mathrm{\infty }}$ as crystals which proves the conjecture in this case.