Hyperpolarized Nuclei for NMR Imaging and Spectroscopy - Lunds ...

of interest, the applied magnetic field strength B 0, and the volumes of the
RF coil and the subject, but the results will differ depending on whether hyperpolarized
or non-hyperpolarized substances are used.
By using the principle of reciprocity (Hoult and Richards 1976, Insko et
al. 1998), the NMR signal voltage, S, across the terminals of the RF coil can
be calculated as:
16
S ∝ γ B PBVS 2
0 1
ˆ . [18]
Here, ˆ B 1 is the B 1 field at unit current through the RF coil and V S is the
sample volume. The noise voltage, N, across the terminals of the RF coil
originates both from the coil itself and from inductive losses in the sample
(Hoult and Lauterbur 1979), and can be expressed as
( )
N = 4 k Tb R + R
[19]
B C S
where b is the receiver bandwidth, R C the coil resistance, and R S is an
“equivalent sample resistance” (Hoult and Lauterbur 1979, Macovski 1996),
modeling the inductive sample losses. For a spherical body and a solenoidal
coil, R S is given by
R
S
2 2 2 5
πω µ 0nrS =
. [20]
2 2
30ρ r + l 4
( C C )
Here, r C, l C, and n are the radius, the length, and the number of turns of
the coil, respectively, r S the sample radius, and ρ the resistivity of the sphere.
For the solenoidal coil, the resistance (taking into account the frequencydependent
skin depth δ) and ˆ B 1 are given by (Hoult and Lauterbur 1979)
2
3σρCnrC RC
= ,
δlC
δ =
2ρC
µµ 0ω
Bˆ
1 =
2
nµ
0
2 2
r + l 4
C C
[21]
where σ is a proximity effect factor (typically 3–6 (Austin 1934)), and ρ C and
µ are the resistivity and relative permeability of the coil material. The calculation
of R C is valid as long as δ is smaller than the radius of the coil wire,