$($a$)$ A small amount of SiO$2$ powder $(5$ wt$\%)$ is added to a fine YAG powder, in hopes of densifying the powder compact into a transparent polycrystaline YAG $($suitable for IR window material$)$ by forming a minor amount of liquid phase during firing. After mixing uniformly, the powder mixture is pressed into a pellet and rapidly heated in a furnace. What is the minimum temperature $($approximately$)$ at which a liquid phase might form? What is the corresponding liquid composition?

$($b$)$ After firing the pellet at $1800\backslash$deg C for many hours, the sample is cooled to room temperature at a rate slow enough to maintain equilibrium. Upon reheating, at what temperature would a liquid first form in this sample? What would be the liquid composition? $($Explain how you arrived at your answer.$)$

$($c$)$ In a separate study, fine Y$2$O$3$ and SiO$2$ powder are added to Al$2$O$3$ powder as liquid$-$phase sintering aids. The overall composition is $80$ wt $\%$ Al$2$O$3,15\%$ SiO$2$ and $5\%$ Y$2$O$3.$ If the sample is equilibrated at $1700\backslash$deg C$,$ what phases are present, and in what amounts?

$($d$)$ Toward the SiO$2-$rich end of the SiO$-$Y$2$O$3$ join, there exists a region of liquid immiscibility where two liquid phases are present at equilibrium. Can you determine from this phase diagram the exact compositions of the two liquids? The approximate composition? Explain.

$($e$)$ What is the most refractory $($highest melting$)$ three$-$phase composite one can prepare in this ternary system?