[SOLVED] Phys 362 statistical and thermal physics: homework 16

Consider a system of N spin-1/2 particles, each with magnetic dipole moment and in a
magnetic field of magnitude B. Let n+ the number of particles with spin up.a) Show that the temperature of the system satisfies
1
T = k
2B ln !
N E/B
N + E/B
and use the result to determine an expression for the energy equation of state E =
E(T,N).b) List the range of possible values of E and plot the temperature as a function of E over
the entire range. Describe the range of energies for which the temperature is positive
and that for which it is negative.c) Using the probabilities with which a single particle is in the spin up state or the spin
down state, determine the mean energy E for a single particle. How does this compare
to the expression for the energy, E, of the entire system that you obtained earlier?Consider a system of N spin-1/2 particles, each with magnetic dipole moment and in a
magnetic field of magnitude B. Suppose that the system is in equilibrium with an environment
at temperature T > 0.a) Explain whether the number of particles in the spin up state is larger than, smaller than
or the same as the number in the spin down state.b) Consider various spin systems in contact with environments at various temperatures.
As the temperature of the environment increases does the fraction of particles in the
spin up state increase, decrease or remain constant? Explain your answer.Consider a system of N spin-1/2 particles, each with magnetic dipole moment and in a
magnetic field of magnitude B. Suppose that the system is in equilibrium with an environment
at temperature T > 0.a) Consider a single particle in the ensemble. What temperature would guarantee that the
particle is in the spin-up state with certainty? Explain your answer.b) What temperature is such that the particle is equally likely to be in the spin up state
versus in the spin down state? Explain your answer.c) Nuclear magnetic resonance (NMR) uses such systems of spin-1/2 particles. In this case
the signal provided by any particle with spin-up cancels that provided by any particle
with spin down. In order to increase the signal strength would it be better to increase
or decrease the temperature of the system? Explain your answer.For an Einstein solid where q N, we found that
(N,q)
#eq
N
$N 1

2N .Recall that the energy of the solid is
E = !
%
q +
N
2
&
.a) Determine an expression for the temperature of the Einstein solid in terms of the total
energy of the solid.b) Use this to determine an expression for the energy equation of state of the solid E =
E(N,T). Does the energy have the expected behavior in terms of the number of particles?c) Using q N show that the result of the previous part implies that kT ! (this is
why it is called the high temperature limit). Determine the approximate energy of one
mole of such oscillators at a temperature of 300 K.For any Einstein solid
(N,q)
%
1 + q
N 1
&N1 %
1 +
N 1
q
&q

N + q 1
2(N 1)q

#
1 + q
N
$N %
1 +
N
q
&q

N + q
2Nq
whenever q 1 and N 1.In the low temperature limit q % N.
a) Show that if 1 % q % N then
(N,q)
%eN
q
&q 1
2q
and
S = kq !
ln %N
q
&
+ 1
.
2b) Use this to determine an expression for the temperature of the solid in terms of energy.
Use this to get an expression for the energy in terms of temperature.6 Gould and Tobochnik, Statistical and Thermal Physics, 4.24, page 210.Two collections of toy model molecules have the following energy distributions.
Prob A
State
01234
Prob B
State
01234
Which of these has the larger temperature? Explain your answer.Consider a particle which could be in one of five possible states; the energies of these states
are 0, #, 2#, 3# and 4#.
a) Suppose that # = 0.050 eV and T = 100 K. Determine the probabilities with which the
particle could be in each state and plot a bar graph of these.b) Suppose that # = 0.050 eV and T = 500 K. Determine the probabilities with which the
particle could be in each state and plot a bar graph of these.c) Suppose that # = 0.20 eV and T = 500 K. Determine the probabilities with which the
particle could be in each state and plot a bar graph of these.d) Use your results to describe qualitatively the appearance of the entire bar graph as the
temperature increases.e) Use your results to describe qualitatively the appearance of the entire bar graph as the
energy gap, #, increases.