(a) Consider an assembly of n weakly interacting magnetic atoms per unit volume at a temperature T and describe the situation classically. Then each magnetic moment u can make an arbitrary angle 0 with respect to a given direction (call it the z direction). In the absence of a magnetic field, the probability that this angle lies between 0 and 0+d0 is simply proportional to the solid angle 27 sin Od0 enclosed in this range. In the presence of a magnetic field H in the z-direction, this probability must further be proportional to the Boltzmann factor e-BE, where E is the magnetic energy of the moment µ making this angle 0 with the z-axis. Use this result to calculate the classical expression for the mean magnetic moment M, of the n atoms (per unit volume). [Remember that the energy of a classical magnetic moment µ in a magnetic field H is given by E = –µ·H = -µH cos 0.]
(a) Consider an assembly of n weakly interacting magnetic atoms per unit volume at a temperature T and describe the situation classically. Then each magnetic moment u can make an arbitrary angle 0 with respect to a given direction (call it the z direction). In the absence of a magnetic field, the probability that this angle lies between 0 and 0+d0 is simply proportional to the solid angle 27 sin Od0 enclosed in this range. In the presence of a magnetic field H in the z-direction, this probability must further be proportional to the Boltzmann factor e-BE, where E is the magnetic energy of the moment µ making this angle 0 with the z-axis. Use this result to calculate the classical expression for the mean magnetic moment M, of the n atoms (per unit volume). [Remember that the energy of a classical magnetic moment µ in a magnetic field H is given by E = –µ·H = -µH cos 0.]
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Need help with the following problem.
![(a) Consider an assembly of n weakly interacting magnetic atoms per unit volume at a
temperature T and describe the situation classically. Then each magnetic moment u
can make an arbitrary angle 0 with respect to a given direction (call it the z direction).
In the absence of a magnetic field, the probability that this angle lies between 0 and 0+d0
is simply proportional to the solid angle 27 sin Ode enclosed in this range. In the presence
of a magnetic field H in the z-direction, this probability must further be proportional to
the Boltzmann factor e-BE where E is the magnetic energy of the moment u making this
angle 0 with the z-axis. Use this result to calculate the classical expression for the mean
magnetic moment M, of the n atoms (per unit volume). [Remember that the
classical magnetic moment u in a magnetic field H is given by E = -µ·H = -µH cos 0.]
energy of
a
(b) Show that in the high temperature regime (kT > µH), the magnetization M; in part
(a) has the form M, = xH, where the susceptibility x is given by
nu?
X =
3kT
Thus a classical magnetic moment obeys Curie's Law, just like a quantum magnetic
moment, but x is a factor of 3 smaller classically than for a quantum spin-1/2 particle.
(For those who are interested, one can show that a quantum spin with spin J approaches
the classical result for J → .)](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F591ac009-156d-459c-82fc-cac6b9112e8c%2F6c422f4e-2864-4fce-819b-d18f6515b796%2Fuwneo89_processed.png&w=3840&q=75)
Transcribed Image Text:(a) Consider an assembly of n weakly interacting magnetic atoms per unit volume at a
temperature T and describe the situation classically. Then each magnetic moment u
can make an arbitrary angle 0 with respect to a given direction (call it the z direction).
In the absence of a magnetic field, the probability that this angle lies between 0 and 0+d0
is simply proportional to the solid angle 27 sin Ode enclosed in this range. In the presence
of a magnetic field H in the z-direction, this probability must further be proportional to
the Boltzmann factor e-BE where E is the magnetic energy of the moment u making this
angle 0 with the z-axis. Use this result to calculate the classical expression for the mean
magnetic moment M, of the n atoms (per unit volume). [Remember that the
classical magnetic moment u in a magnetic field H is given by E = -µ·H = -µH cos 0.]
energy of
a
(b) Show that in the high temperature regime (kT > µH), the magnetization M; in part
(a) has the form M, = xH, where the susceptibility x is given by
nu?
X =
3kT
Thus a classical magnetic moment obeys Curie's Law, just like a quantum magnetic
moment, but x is a factor of 3 smaller classically than for a quantum spin-1/2 particle.
(For those who are interested, one can show that a quantum spin with spin J approaches
the classical result for J → .)
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