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Calculation of Ga Gas Particles Transmission Probability for Vertical Standing Crucible
The evaporated particles inside the crucible either pass through without striking the crucible walls or collide with the walls. The molecules that once collided with the walls go through one of the following processes; passing through the exit, returning to the entrance and condensing, or striking with the wall again.
Clausing derived the transmission probability for emitted gas particles from the crucible for the case of particles directly passing through the crucible without collision, and for the case of passing through the crucible with collisions .
The motion of gas particles that would evaporate inside the crucible can be divided into the following ways as depicted in Fig. 3: (a) movement from the liquid surface to the crucible opening end, (b) movement from the liquid surface to the wall surface, (c) movement from the wall surface to the crucible opening end, (d) movement from the wall surface to the upper wall surface, and (e) movement from the wall surface to the liquid surface. At low angles, u, the gas particle evaporates in a straight line without collision. At larger angles, depending on the location of evaporation from the liquid surface, the particle will strike the wall. From the wall there are three possible further processes that may consist of returning back to liquid surface (e), colliding again with the wall surface (d), or direct evaporation to outside of crucible (c).
The transmission probability is defined by Eq. (3).
where Wsr is the probability that gas particles move from the liquid surface to the wall surface and Wss is the probability that gas particles move from liquid surface to the crucible opening end. Here, w(x) is the probability of collision at x inside the crucible and is defined by Eq. (4)
where П is arbitrary point that collision occurs inside the crucible after the collision at x (Fig. 3), Wrr is the probability that gas particles move from the wall surface to the upper wall surface, and Wrs is the probability that gas particles move from the
Fig. 3 Schematic diagram of the movement of gas particles inside the crucible in four different cases a liquid surface —crucible opening end, b liquid surface—wall surface, c wall surface— crucible opening end, d wall surface—wall surface and e wall surface—liquid surface
wall surface to the crucible opening end. The possibility of particles returning to the liquid surface from the wall (case (e) in Fig. 3) is neglected in the Clausing calculation. The probability of collision at x, w(x) can be represented by Eq. (5)
When the distance between melt surface and crucible opening end, L is greater than the radius of melt surface, a is given by Eq. (6)
with v = 3Tw7and u v
Then the final transmission probability is represented by Eq. (7)
The transmission probability of Ga particles, considering the probability of collision for the MBE crucible is the function of the distance between melts surface and crucible opening end in Fig. 4 and the radius of crucible in Fig. 5 As L is deeper, more collisions with the crucible walls occur. Hence, the transmission probability decreases as L increases in Fig. 4 On the other hand, the dependence of radius is smaller in transmission probability as suggested in Fig. 5.
The evaporation rate at the presence of collision is defined by multiplying transmission probability in Eq. (6) to free evaporation rate in Eq. (1) and given by Eq. (8)
The evaporation rate is the evaporated weight per unit area of melt surface and unit time. The weight of evaporated amount can be converted to volume change during evaporation. The change of volume in the cylindrical crucible is surface area of melts times change of length of melts. If the surface area of melts is assumed as a constant in cylindrical crucible, the evaporation rate can be represented as a function of receding length in melts during evaporation as indicated in Eq. (9).
Fig. 4 The variation of transmission probability at the presence of collision in the crucible depending on length between melt surface and crucible end at r = 0.95 cm
Fig. 5 The variation of transmission probability at the presence of collision in the crucible depending on length between melt surface and crucible end at L = 5 cm
Then the proportion of evaporation amount can be derived by integrating the infinitesimal change of length of melt over time.
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