Chapter 5 – Detection and Imaging Tools that Use Nonoptical Waves 195
The magnitude of αm is small at ~(1/180) × ½ ≈ 0.01, so use a Taylor expansion for
the exponential (and using hint for sum of natural squares):
∴
≈
+
+
(
)
+
+
(
)
=
+
+
(
)
=−
=
=−
=
=
∑
∑
µ
γ
π
α
α
γ
π
α
h
m
m
m
m
h
m
m
m
m
m
m
m
2
1
1
2
1
1
1
1
1
2
−
=
=−
=
=−
=
=−
=
∑
∑
∑
∑
+
+
(
)
=
+
+
1
1
1
1
2
1
1
1
1
1
2
1
m
m
m
m
m
m
m
m
m
h
m
m
α
γ
π
α
(
)
(
)
( )
(
)
(
) /
+
+
=
+
+
(
)
+
+
(
)+
=
=−
=
=−
=
∑
∑
α
αγ
π
γ
m
h
I I
I
I
m
m
m
m
1
1
1
1
2
2
0
1 2
1
3
2
1
0
h
k T I I
B
B
2
6
1
π
+
(
)
Thus, if the number of magnetic moments per unit volume (i.e., per m3 in SI
units) is N
M
N
h
k T I I
B
B
0
2
2
6
1
=
+
(
)
γ
π
For each ethanol molecule, there are five 1H atoms, and a 1 mM concentration (i.e.,
1 mmol L−1) will give
N =
×
×
(
)×
×
×
≈
×
−
−
5
1 10
6 02
10
10
3 10
3
23
3
22
3
.
atoms m
Using the value for γ for 1H indicated by Table 5.1 is ~42.6 × 2π ≈ 268 MHz T−1.
Therefore,
M0
22
6 2
34
2
3 10
268
10
6 6
10
10
0 5
0 5
1
≈×
×
×
(
) ×
×
×
(
) ×
×
+
(
)
−
.
.
.
/
(6
10 23
7
1
π ×
×
×
≈
×
−
−
−
1.38
4)
7
10 Am
The magnetic susceptibility χ of this sample is thus ~(7 × 10−7)/10.2 ≈ 7 × 10−8.
c
Nuclear shielding of the 1H atoms in ethanol results in a chemical shift of the res
onance frequency, which is an average per 1H atom of ~2.2 ppm in TMS solvent.
Therefore, the B-field required to excite these atoms will be greater by this factor,
as will the resonance frequency, which scales with B. Therefore,
∆ν ≈
×
×
(
) =
−
434
10
2 2
10
955
6
6
.
Hz
5.5 TOOLS THAT USE GAMMA RAYS, RADIOISOTOPE
DECAYS, AND NEUTRONS
Several other high-energy particles can be used as biophysical probes. Alpha and beta
particles and gamma rays are relevant to the radioactive decay of isotopes, which can be
used as reporter tags in biomolecules, especially useful in investigating the kinetics of bio
chemical processes. Gamma rays are also relevant to Mössbauer spectroscopy. Also, struc
tural details of biomolecules can be investigated using the scattering/diffraction of thermal
neutrons.