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3.1) Use the Lorentz force to describe how the angular frequency w changes for a
circular electron orbit (e.g. a classical Bohr orbit) in the xy plane when a magnetic field
is applied along the z axis.
3.2) Derive the expressions for total energy En and radius of circular electron orbits of
radius rn in Bohr's model. Evaluate the constants E1 and r1 in SI units.
3.3) Calculate the classical angular momentum and the magnetic moment for a uniform
shell of charge -|e| and radius rotating with angular velocity w. What
is the gyromagnetic ratio for this particle? What is the surface velocity? (This
problem was considered by Max Abraham in 1903, more than 20 years before
Uhlenbeck and Goudsmit hypothesized that electrons spin with a gyromagnetic
ratio twice that for orbital motion).
3.4) Compare essential characteristics of Lenz's Law (macroscopic) and diamagnetism.
How are they similar and how different?
3.5) Calculate the dimension at which diamagnetism crosses over to 2 paramagnetism
for a metal with relaxation time,
mean free path, = mean drift velocity of change carriers.
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3.6) a) Evaluate the ratio of integrals in the classical expression in the notes for
b) Show that L(s) = coth (x)-1/s goes to x/3 in the limit that x approaches zero.
Discuss.
c) Show ; discuss.
3.7) Show that the units of
so that if N is number of atoms per unit volume, X is dimensionless. However, if N
is v Avogadro's number. (in which case x is the molar susceptibility
it must be multiplied by to compare with
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3.8) Calculate the paramagnetic susceptibility of diatomic oxygen at room
temperature and compare it with the experimental, room temperature (cgs) value
3.4 x 10-3.
3.9) Calculate the diamagnetic susceptibility of atomic He assuming
and compare with the room temperature cgs value
3.10) Derive a classical expression for the diamagnetic susceptibility of an electron in
circular orbit by considering the change in its angular momentum due to the electric field
induced as a B field is slowly turned on normal to the orbit plane in time dt.
3.11) Analyze the condition for which the paramagnetic and diamagnetic
susceptibilities are equal and opposite for a classical Bohr atom with orbital but not
spin magnetic moment. Discuss.
3.12) Calculate the orbital magnetic moment of an electron in a circular Bohr
orbit.
and compare the results. Which one is correct? Why?
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3.13) A very thin film (t < mean free path) may exhibit diamagnetism in an external field
perpendicular to the film plane. Explain how application of the field in the plane of the
film could change the sign of this effect.
3.14) Write the electronic configuration the spectroscopic notation
and effective magneton number
[J(J+1) + S(S+1) - L(L+1)]/2J(J+1) for Cr3+, Fe3+ and Co2+.
3.15) a) Show that Bj(x) reduces to the Langevin function L(x) , with x= mmB/kT, in
the limit J approaches infinity.
b) Show that B1/2(x) = tanh (x)
and thus in this limit
d) Show that as x approaches infinity, Bj(x) approaches 1, i.e. M = Nvg mB mJ .
Describe the physical significance of each case.
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3.16) The Landé g factor is used to account for the fact that is not collinear
with J = L + S because mL= mB ml = (eh /2m)ml whereas ms =2 mBms. Derive the
expression for g in terms of the quantum numbers l, s and j. Make use of the facts that
because L and S precess around J, then also precess around J and it is the
projection of on J that is measured.
3.17) What type(s) of magnetism would you expect to find and why in a) NaCl b)
MnSO4 . 4H2O c) Fe3O4 d) H2O (or Ne) e) metallic Cu?
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SOLUTIONS
3.1) I have two solutions to this problem. Which do you prefer?
FIRST: 3.1) The Lorentz force, F = q(v x B), is everywhere directed along the radius
vector. The torque on the orbital motion, T = dL/dt, is given by r x F, which is zero.
Hence, the Lorentz force cannot change the angular momentum or the angular
frequency . Alternatively, the circular orbit generates an angular momentum
and orbital magnetic moment that are directed parallel or antiparallel to the applied
field. The torque due to the B field acting on this moment
However, the presence of the magnetic field does add a new potential energy to the
problem that should change something other than
classically (which is the limit of our treatment so far), it can assume an orbit of a
different radius as B is applied. (If it is treated quantum mechanically, it may not
incrementally change its radius, but can only do so in quantized steps that
correspond to discrete energy differences.) So the energy of the system may be
written:
If the electron is described
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s -1) and radius (ro = 0.52 A) respectively. Thus,
How significant is the magnetic field term in the denominator?
Even classically, the change in orbit radius is insignificant. These contradicting results
underscore the limitations of the classical Bohr model.
SECOND: The Lorentz force, F = q(v x B), is everywhere directed along the radius
vector. A circular orbit would result from a central force
its total energy, E=T+V=
If the electron is described classically (which is the limit of our treatment so far), it can
assume an orbit of a different radius as B is applied. (If it is treated quantum
mechanically, it may not incrementally change its radius, but can only do so in quantized
steps that correspond to discrete energy differences.)
is conserved with application of B
The force balance describing the classical, field-induced change in the electron
orbit is
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To see how the angular frequency w varies with B we write the force balance as
where wL = eB/m which is of order , this equation suggests
the definition:
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If we assume the radius is unchanged with application of B, then this equation is
quadratic in w, and has the solution
or, considering the smallness of
At B = 0 this reduced to w = ±w . For o increasing B,
w increases or decreases as shown at right and
asymptotically approaches the straight line w = wL = e
B/m for very large B. For experimentally achievable
values of B,
If we do not assume r = ro, then we must solve the equation of motion for r. We assume
giving:
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which, if we take w = w0, gives:
Thus, application of a field increases or decreases the radius of the orbit depending on
the sign of the B field.
b) Use coth(s ) = 1/x + s/3 +…. The result is the classical Curie law for magnetization at
weak fields/high temperatures
c) Both sinh and cosh appraoch infinity for infinite argument, so coth(s Æ ∞) = 1. This
describes saturation, at which point the thermally averaged moment in the field direction
is equal to the magnitude of the moment:
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3.10) Using Stokes' theorem gives
3.11) 0 implies
The larger the paramagnetic moment squared, the larger the area of the diamagnetic orbit
must be to cancel it out at a given T.
Going further, for a classical orbit, µm = gL = ( e/2m) mvr = evr /2, so the condition
becomes
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This says that the higher the kinetic energy of the classical orbit, the higher the T at which
This is because higher EK requires a smaller orbit which has smaller area
and smaller diamagnetism; the small diamagnetism is only canceled out by the
paramagnetism when the temperature is large enough to suppress the Curie
paramagnetism. At lower temperatures, the paramagnetism dominates.
However, using w =
The latter is correct, classically. The former is wrong because it makes use of the
Einstein relation for the energy of a photon. This then gives w = 1/2 w which is
wrong as is the first estimate.
They differ by a factor of 2.
3.13) The diamagnetic contribution of free electrons in metals (between scattering
events) is small. But for thin films or wires, the presence of narrow boundaries changes
diamagnetism to paramagnetism for B perpendicular to the thin dimension. This effect
is illustrated below.
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3.14 Cr3+ = 3d3. l = 3, s = 3/2, j = l - s = 3/2, thus 4F3/2 g = 0.4, neff = 3.87 Fe3+ =
3d5. l = 0, s = 5/2, j = 5/2, thus: 6S5/2 g = 2 (l = 0) and neff = 5.92 Co2+ = 3d7. (3
holes whereas Cr3+ has 3e's)l = 3, s = 3/2, j = 9/2, thus: 4F9/2 g = 1.33, neff = 6.63
3.16) Experiment measures the projection of mJ on the direction J about which it
precesses:
cosC applied to the vector triangle L + S = J gives (writing √l(l+1) as L for convenience):
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cos(LJ) = [L2 + J2 - S2]/[2L J]
cos (SJ) = [S2 + J2 - L2]/[2S J]
uj = mB [(L2 + J2 - S2) + 2(S2 + J2 - 2L)]/2J
uJ = mB (3J2 - L2 + S2)/2J = mB J(2J2 + J 2+ S2 - L2)/2J2
uJ = mB J[1 + (J2 + S2 - L2)/2J2]
Restoring the proper quantum mechanical lengths for the vectors, leads to
3.17) a) NaCl: diamagnetic (temperature independent) because this ionic crystal
satisfies complete octets in each component. No paramagnetism.
b) MnSO4 . 4H2O: Mn is in state Mn ++ 3d5. It will not order ferromagnetically.
Superexchange giving antiferromagnetism is a possibility, but it is not observed, probably
because oxygen bonds primarily with the sulfur and so it is not available for mediating
exchange between Mn ions. So this is a paramagnetic salt with some diamagnetism supplied
by H20. x~1/T and M vs. H/T follows B5/2 (x).
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c) Fe 304: this is a classic antiferromagnetically coupled two sublattice system: Fe3+
is on one site (A) and Fe3+ and Fe2+ on the other (B). mA = -5µB and mB = +9µB so
net moment is 4 mB.
d) Ne is diamagnetic as is H 20. Former has complete octet l = s = 0, latter has covalent
bonds with paired spins. In both cases c < 0 and temperature independent
e) Cu is a Pauli paramagnet which is temperature independent, x > 0
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