Biomedical Engineering Reference
In-Depth Information
Consider then a sample of
N
such two-state particles. (It is hard to do experiments on
free electrons, so we had better imagine a sample of hydrogen atoms; the only fly in the
ointment is that the nuclei also have a spin of
1
/
2
which I am going to ignore.) We keep the
temperature constant by surrounding the sample with a heat bath and keep the magnetic
field fixed. Figure 7.10 shows a few of the hydrogen atoms in the heat bath.
Heat Bath
B
A
Figure 7.10
The heat bath
The diagram should be pretty obvious: atom A has its electron in the lower spin state,
atomB has its electron in the higher spin state and so on. The heat bath keeps the temperature
of our system constant.
Do not get confused with electronic energy levels, though. If
N
u
and
N
l
are the number
of electrons in the upper and lower spin energy levels and
N
the total then
N
=
N
u
+
N
l
(7.23)
and the magnetic internal energy of the system is
U
=
N
u
ε
u
+
N
l
ε
l
(7.24)
We also know from the Boltzmann distribution law
N
exp
ε
i
k
B
T
−
N
i
=
(7.25)
Q
In order to simplify the algebra, I will take the magnetic energy zero to be ε
l
and the energy
difference ε
u
−
ε
l
to be
D
. With this notation, explicit expressions for
N
u
and
N
l
are
N
exp
D
k
B
T
−
N
N
l
=
exp
;
N
u
=
exp
(7.26)
D
k
B
T
D
k
B
T
1
+
−
1
+
−
Figure 7.11 shows how the populations vary with temperature.
