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In-Class Exercises

Pb. 8.25 Find the Larmor frequency for an electron in a magnetic flux den-
–2
sity of 100 Gauss (10 Tesla).
Pb. 8.26 Similar to the electron, the proton and the neutron also have spin
as one of their internal degrees of freedom, and similarly attached to this
spin, both the proton and the neutron each have a magnetic moment. The
magnetic moment attached to the proton and neutron have, respectively, the
values µ = –1.91 µ and µ = 2.79 µ , where µ is called the nuclear magneton
N
N
n
p
N
–26
and is equal to µ = 0.505 × 10 Joule/Tesla.
N
Find the precession frequency of the proton spin if the proton is in the pres-
ence of a magnetic flux density of strength 1 Tesla.
Homework Problem

Pb. 8.27 Magnetic resonance imaging (MRI) is one of the most accurate
techniques in biomedical imaging. Its principle of operation is as follows. A
strong dc magnetic flux density aligns in one of two possible orientations the
spins of the protons of the hydrogen nuclei in the water of the tissues (we say
that it polarizes them). The other molecules in the system have zero magnetic
moments and are therefore not affected. In thermal equilibrium and at room
temperature, there are slightly more protons aligned parallel to the magnetic
flux density because this is the lowest energy level in this case. A weaker
rotating ac transverse flux density attempts to flip these aligned spins. The
energy of the transverse field absorbed by the biological system, which is pro-
portional to the number of spin flips, is the quantity measured in an MRI scan.
It is a function of the density of the polarized particles present in that specific
region of the image, and of the frequency of the ac transverse flux density.
In this problem, we want to find the frequency of the transverse field that
will induce the maximum number of spin flips.
The ODE describing the spin system dynamics in this case is given by:

d ψ = j[Ω ω t)σ + Ω ω t)σ + Ωσ ψ
dt ⊥ cos( 1 ⊥ sin( 2 3 ]

µ B µ B ⊥
where Ω = p 0 , Ω = p , µ is given in Pb. 8.26, and the magnetic flux
h ⊥ h p
density is given by

r
B = B cos(ω t ê + B sin(ω t ê + B ê
)
)
⊥
⊥
0 3
2
1
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