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3 Reaction between fermions and bosons Warmup (a) [2pts] Consider a 2D gas of non-interacting spin-1/2 fermions (i.e., with internal degeneracy g = 2) in an area A. Let their total density be ny. Suppose the fermions are non-relativistic, with mass my and dispersion h2 K2 Ef ( k ) 2mf (3.1) Calculate the relationship between B = 1/kBT, ny, and the fugacity zy = eBus (the two spin species have identical chemical potential). You should simplify your answer by introducing the thermal de Broglie wavelength f = V2Th2B/mf. (b) [2pts] Similarly, consider a 2D gas of non-interacting bosons with no internal degeneracy (i.e., spin-0) in an area A and with density no. Suppose their dispersion is given by Eb (k ) 12k2 A, (3.2) 2mb where A is some constant energy shift. Likewise, calculate the relationship between the quantities A, B, no, and zo = eBub. You should introduce do = V2Th28/mb. (c) [2pts] In the bosonic case, use part (b) to determine the maximum and minimum values of zb. At fixed temperature, what happens qualitatively to no as zo approaches these two limits? Fermion pairing Consider a closed system with temperature T and area A. Initialize a density no of spin-1/2 fermions f1/1 of mass my = m, with dispersion given by Eq. (3.1). However, unlike the warmup, imagine that the fermions can now undergo a mass-conserving reaction process (3.3) where b is a spin-0 boson with mass my = 2m. In other words, two fermions can pair up to form a boson, and each boson can likewise split into two fermions. In this context, the quantity A in Eq. (3.2) represents the binding energy won by pairing two fermions (i.e., A is a positive constant). Initially, there are no bosons. Let n, and nb now denote the densities after the system has reached equilibrium. (d [1pt] Due to the reaction process, the density of fermions and bosons are not separately conserved. However, given the mass conservation constraint, what is the relation between no, ny, and nb? (e) [2pts] What is the relation between zy and zo such that equilibrium is achieved? Hint: what is the relation between their chemical potentials? Provide a detailed explanation. (f ) [2pts] Given the difference in the fermion and boson masses, what is the relation between f and do? Use the previous parts to express Agno in terms of zf, B, and A (g) [2pts] At some fixed temperature T, assume that the initial density of fermions is large, such that x}no 1. (3.4) To leading order, what are no and ny? Your answer for ny should depend explicitly on A