Im stuck in this question. Please help.

In this question we consider the degeneracy of electrons and nucleons in three different types of star: Main Sequence, White Dwarfs and Neutron Stars. (Take the mass of hydrogen to be mH = 1.67 1027kg) and the mass of an electron to be me = 9.11 1031kg. (a) The density of states in k is given by, g(k)= V 4k2G 1 2 (2)3 particles. The number of states for a particle of spin s is given Electrons are spin- by the spin factor, G = 2s + 1. Sketch a graph of g(k) as a function of k. 1 (b) Use the result = (k)2/2m and the expression for g(k) to show that g(), the density of states in energy, is given by, 2m 3/2 g()=4 h2 V Sketch a graph of g() as a function of . (c) The Fermi-Dirac distribution function is given by, fFD() = 1 , e()/kBT +1 where is the Fermi Energy. Showthatatabsolutezero,fFD =1if(0). Hence show that the number of particles, N is given by, (0) 4 g()d. By substituting the expression from part (b), integrate this to show that, (0) = (2/2m)(32N/V )^2/3

d) The Fermi Temperature, TF is defined by, (0) = kBTF . We can use this definition to quantify the need for Fermi-Dirac statistics in a given system. The degeneracy parameter was given in the lectures as, 2 3/2 A=(N/V)(h/2mkBT) . Substitute the expression for (0) in part (c) to show the degeneracy parameter can be expressed by, We need to use Fermi-Dirac statistics when a gas of fermions has a degeneracy parameter A > 1. This is effectively equivalent to T

(i) In a white dwarf star of one solar mass the atoms are all ionised and contained in a sphere of radius 2 107 m. What is the Fermi energy of the electrons in eV? Are they relativistic? If the temperature of the white dwarf star is 107 K, are the electrons and nucleons in the star degenerate? (Hint: Particles with rest mass, m are non-relativistic if their Fermi momentum, pF = (32N/V )1/3 is less than mc, where c is the speed of light).

6. In this question we consider the degeneracy of electrons and nucleons in three different types of star: Main Sequence, White Dwarfs and Neutron Stars. (Take the mass of hydrogen to be mu = 1.67 x 10-27kg) and the mass of an electron to be me = 9.11 x 10-31 kg. (a) The density of states in k is given by, 9(k) = 0 347k2. G Electrons are spin-, particles. The number of states for a particle of spin s is given by the spin factor, G = 2s + 1. Sketch a graph of g(k) as a function of k. (b) Use the result e = (hk)2/2m and the expression for g(k) to show that g(e), the density of states in energy, is given by, g(e) = 44 (202) VVE Sketch a graph of g(e) as a function of . (c) The Fermi-Dirac distribution function is given by, ffp(e) = e(6-1)/k37 + 1' where u is the Fermi Energy. Show that at absolute zero, fed = 1 if M(0). Hence show that the number of particles, N is given by, (0) N = g()de. Jo By substituting the expression from part (b), integrate this to show that, M(O) = (h/2m)(312N/V)2/3 (d) The Fermi Temperature, TF is defined by, u(0) = kbTp. We can use this definition to quantify the need for Fermi-Dirac statistics in a given system. The degeneracy parameter was given in the lectures as, A= (N/V)(h2/24mkpT)3/2, Substitute the expression for u(0) in part (c) to show the degeneracy parameter can be expressed by, A= (8/3V+)(TF/T)3/2 ~ 1.50(TF/T)3/2. We need to use Fermi-Dirac statistics when a gas of fermions has a degeneracy parameter A > 1. This is effectively equivalent to T M(0). Hence show that the number of particles, N is given by, (0) N = g()de. Jo By substituting the expression from part (b), integrate this to show that, M(O) = (h/2m)(312N/V)2/3 (d) The Fermi Temperature, TF is defined by, u(0) = kbTp. We can use this definition to quantify the need for Fermi-Dirac statistics in a given system. The degeneracy parameter was given in the lectures as, A= (N/V)(h2/24mkpT)3/2, Substitute the expression for u(0) in part (c) to show the degeneracy parameter can be expressed by, A= (8/3V+)(TF/T)3/2 ~ 1.50(TF/T)3/2. We need to use Fermi-Dirac statistics when a gas of fermions has a degeneracy parameter A > 1. This is effectively equivalent to T