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(A) Magnetic Resonance Phenomena
A_I
Nuclei, Electrons, Spin, and Angular Momentum
By what properties can nuclei be distinguished? (Spin Dynamics p.5 (1st & 2nd edn.))
What is nuclear or electron spin? (Spin Dynamics p.6-9 (1st edn.) p. 5-8 (2nd edn.))
How is spin angular momentum quantised? (Spin Dynamics p.6-9 (1st edn.) p. 5-8 (2nd edn.))
PX388 Magnetic Resonance: Section A: Magnetic Resonance Phenomena
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What is necessary to see an EPR signal?
What are the nuclear spin species of the most commonly encountered chemical elements?
(Spin Dynamics p.10-15 (1st edn.) p. 9-15 (2nd edn.))
PX388 Magnetic Resonance: Section A: Magnetic Resonance Phenomena
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The NMR Properties of the nuclei of most relevance for organic and biological systems
Nucleus
I
Natural
Larmor Frequency
Abundance/ %
at 14.1 T/ MHz
1H
1/2
99.98
600.0
2H
1
0.02
92.1
12C
0
98.9

13C
1/2
1.1
150.9
14N
1
99.6
43.3
15N
1/2
0.4
60.8
16O
0
99.96

17O
5/2
0.04
81.3
The NMR Properties of nuclei of relevance for inorganic systems
Nucleus
I
Natural
Larmor Frequency
Abundance/ %
at 14.1 T/ MHz
6Li
1
7.4
88.3
7Li
3/2
92.6
233.2
10B
3
19.6
64.5
11B
3/2
80.4
192.5
16O
0
99.96

17O
5/2
0.04
81.3
23Na
3/2
100.0
158.7
27Al
5/2
100.0
156.3
28Si
0
95.3

29Si
1/2
4.7
119.2
31P
1/2
100.0
242.9
PX388 Magnetic Resonance: Section A: Magnetic Resonance Phenomena
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What is an isotopomer? (Spin Dynamics p.16-17 (1st and 2nd edn.))
How can isotopomers be separated? (Spin Dynamics p.17 (1st and 2nd edn.))
PX388 Magnetic Resonance: Section A: Magnetic Resonance Phenomena
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A_II
Nuclear and Electronic Paramagnetism, Magnetic Fields,
and the Larmor Frequency
Are nuclei and electrons inherently magnetic? (Spin Dynamics p.25-27 (1st edn.) p.25-26
(2nd edn.))
How does an individual spin behave in a magnetic field (microscopic behaviour)?
(Spin Dynamics p.27-30 (1st edn.), p.26-28 (2nd edn.))
PX388 Magnetic Resonance: Section A: Magnetic Resonance Phenomena
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What determines the frequency at which a nuclear or electron spin precesses? (Spin Dynamics
p.30 (1st edn.) p. 29 (2nd edn.))
From classical physics, the potential energy, U, (in units of Joules) of a magnetic
moment in a magnetic field of strength, (B = (0,0,Bz), is given by:
Using

U =   B = z Bz
[A4]
z =  Sz
[A3]
Sz = m ħ
[A2]
U =
[A5]
For a spin I = 1/2 nucleus, m = +1/2 or 1/2, i.e.,
Um = 1/2 =
[A6]
MR is a form of spectroscopy with the selection rule, m = 1:

U = Um = +1/2  Um = 1/2 =
[A7]
The energy of a photon, U, is related to its angular frequency  by:
U =
[A8]
It, thus, follows that the Larmor frequency, 0, (in rads1) is given by

0 =
PX388 Magnetic Resonance: Section A: Magnetic Resonance Phenomena
[A9]
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Larmor frequency: ESR notation
z = (ge B / ħ) Sz
[A10]
ge(free electron) = 2.002 (dimensionless)
[A11]
B (Bohr magneton) = |e| ħ / (2 me) = 9.274  1024 J T1
[A12]

where

Using Eqs. [A2] and [A4],
 U = Um = +1/2  Um = 1/2 =
[A13]
Using Eq. [A8], the Larmor frequency, 0, (in rads1) is given by

0 =
[A14]
NB:

 = (ge B / ħ)
(free electron) / (1H) = 658
0(free electron)/2 = 9.5 GHz at B0 = 0.34 T
PX388 Magnetic Resonance: Section A: Magnetic Resonance Phenomena
[A15]
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A_III
Magnetic Resonance Experiments: Thermal Equilibrium & Bulk Magnetisation
How does an ensemble of spins behave in a magnetic field (macroscopic behaviour)? (Spin
Dynamics p.34 (1st edn.), p. 32 (2nd edn.))
PX388 Magnetic Resonance: Section A: Magnetic Resonance Phenomena
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How big is the population difference?
For a proton spin in a 14.1 T magnet (referred to as B0) (from Eq. [A7]),
U| =  ℏ B0| = 2.675  108  1.055  1034  14.1 = 4.0  1025 J
Thermal energy at room temperature = kb T = 1.381  1023  298 = 4.1  1021 J
The probability that a nucleus or electron has a particular spin, pi = ni / N, where ni is the
number of nuclei/electrons with this spin, and N is the total number of nuclei.
By the Boltzmann distribution,
p
i

exp( U i )
k bT
[A16]
U
 exp( T )
k
j
j
p
p
i
j

b
exp( U i )
k bT
U )
exp(
kT
j
 exp(
U
)
k bT
b
For 1H in a 14.1 T magnet,
pi/pj =
For a free electron in a 14.1 T magnet,
U| =  ℏ B0| = 1.761  1011  1.055  1034  14.1 = 2.6  1022 J
pi/pj =
PX388 Magnetic Resonance: Section A: Magnetic Resonance Phenomena
[A17]
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How big is the bulk magnetisation? (Spin Dynamics p.23-25 & 34 (1st edn.), p. 23-24 & 32
(2nd edn.))
(For derivations, see further reading on course website: "Magnetisation and Susceptibility"
and "Curie's Law for a Spin I = 1/2 Nucleus or Electron")
PX388 Magnetic Resonance: Section A: Magnetic Resonance Phenomena
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What hinders the direct measurement of nuclear or electronic paramagnetism? (Spin
Dynamics p.36 (1st edn.) & p. 33 (2nd edn.))
PX388 Magnetic Resonance: Section A: Magnetic Resonance Phenomena
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Does a bulk magnetisation establish instantaneously? (Spin Dynamics p.32-35 (1st edn.)
p. 30-33 (2nd edn.))
PX388 Magnetic Resonance: Section A: Magnetic Resonance Phenomena
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A_IV Magnetic Resonance Experiments: Electromagnetic Irradiation and Induction
How can MR phenomena be experimentally observed?
PX388 Magnetic Resonance: Section A: Magnetic Resonance Phenomena
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What is the effect of an on-resonance electromagnetic pulse on the bulk magnetisation? (Spin
Dynamics p.36-37 (1st edn.) p.33-34 (2nd edn.))
PX388 Magnetic Resonance: Section A: Magnetic Resonance Phenomena
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PX388 Magnetic Resonance: Section A: Magnetic Resonance Phenomena
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What does the magnetisation do after the electromagnetic pulse is turned off? (Spin Dynamics
p.37 (1st edn.) p.34 (2nd edn.))
PX388 Magnetic Resonance: Section A: Magnetic Resonance Phenomena
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How is the NMR signal detected? (Spin Dynamics p.39 (1st edn.) p.36 (2nd edn.))
PX388 Magnetic Resonance: Section A: Magnetic Resonance Phenomena
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(A) Magnetic Resonance Phenomena: Key Facts

The quantisation of spin angular momentum:
Quantum mechanics states that the magnitude, |S|, and one component, Sz, can be specified:
|S| = ℏ √[I(I + 1)]
[A1]
Sz = m ℏ
[A2]
I is the spin quantum number and takes values 0, 1/2, 1, 3/2, …
m is the azimuthal quantum number and takes values +I, +I  1, … I.

Rules for predicting the spin quantum number of an isotope:
Isotopes with even mass number have integer spin (I = 0, 1, 2, …).
Isotopes with odd mass number have half-integer spin (I = 1/2, 3/2, 5/2…).
For isotopes with even mass number:
If there is an even number of protons and an even number of neutrons, I = 0.
If there is an odd number of protons and an odd number of neutrons, I is non-zero (1, 2, …).

Isotopomers:
Relative abundance equals the product of the relative abundance of each constituent isotope.
Can be separated due to their different mass, e.g., different diffusive mobility in the gas phase
(gas chromatography).


Magnetic moment of a nucleus
 = S
[A3]
 is called the magnetogyric or gyromagnetic ratio and is a constant for a specific isotope.


Larmor frequency (spin precession frequency):
0 =  Bz
0 refers to the angular frequency (in units of rad s1):
NB: frequency in units of Hz = (frequency in units of rad s1) / ().
PX388 Magnetic Resonance: Section A: Magnetic Resonance Phenomena
[A9]
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
Larmor frequency (spin precession frequency, EPR notation):

0 = ge B Bz / ℏ
[A14]

 = (ge B / ħ)
[A15]

Population difference between spin states with different m
depends on the energy difference and temperature according to the Boltzmann distribution:
Small in NMR spectroscopy (1 in 10,000 for 1H at 14.1 T).
Bigger population differences in EPR (since (free electron) / (1H) = 658); exploited in the
DNP (dynamic nuclear polarisation) technique to improve NMR sensitivity.


Bulk magnetisation, M (total induced magnetic moment per unit volume),
M = induced / V
[A18]
 = 0 M / Bapplied
[A19]
Magnetic susceptibility, 
0 is the magnetic constant (4  107 Hm1).
A paramagnetic material (M adds to Bapplied):
 is positive
A diamagnetic material (M reduces Btotal):
 is negative

Curie's Law for a spin I = 1/2 nucleus or electron:
M = Nv 2 ℏ2 B0/ 4kb T
[A20]
where Nv is the number of spins per unit volume.

The nuclear paramagnetism ( = 4  109 for 1H in H2O at 300 K) is dwarfed by the
electron diamagnetism ( = 9  106 for H2O).
PX388 Magnetic Resonance: Section A: Magnetic Resonance Phenomena
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
Relaxation (i.e., establishing of thermal equilibrium) occurs due to:
Small molecular magnetic fields that fluctuate rapidly due to molecular motion induce very
slight changes in the precession behaviour of the spins: relaxation is often on the order of
seconds, cf 2 ns for 1 period of free precession by a 1H spin at 14.1 Tesla – Larmor frequency
= 600 MHz).

Two magnetic fields are needed for a Magnetic Resonance Experiment:
B0: strong, static magnetic field (generates bulk magnetisation at thermal equilibrium)
B1: applied perpendicular to B0 as an oscillating (with frequency rf) electromagnetic wave
(radio-frequency for NMR, microwave for EPR).

In a pulsed magnetic resonance experiment:
An on-resonant (rf = 0) electromagnetic pulse causes the magnetisation to precess around
the direction of the pulse.
If the pulse is turned off after a 90 degree rotation, the net magnetic moment is perpendicular
to B0 – this is referred to as transverse magnetisation.
This transverse magnetisation precesses around B0 at the Larmor frequency.
The precessing transverse magnetisation induces an oscillating (at the Larmor frequency)
electric current to flow in the coil – this is the NMR signal.
(A) Magnetic Resonance Phenomena: Questions
1. The isotopes 50V, 113Cd, and 36Ar have (in no particular order) the spin quantum numbers
0, 1/2, and 6. The atomic numbers of Ar, V and Cd are 18, 23 and 48, respectively. What
are the spin quantum numbers of the three isotopes?
2.
(a) What are the isotopomers of H2O, and what are their natural abundances?
(b) If NMR is performed on H2O with a 4.7 T magnet, at what frequencies should the
spectroscopist look for NMR signals?
PX388 Magnetic Resonance: Section A: Magnetic Resonance Phenomena