Download Spin Angular Momentum Magnetic Moments

Nuclear Magnetism and NMR Spectroscopy - Ralph W. Adams
Lecture 1: Nuclear Magnetic Resonance – Theory and Techniques
Spin Angular Momentum
Magnetic nuclei possess an intrinsic angular momentum known as spin. Spin is quantized and has units ħ (where ħ
= h / 2π and h is the Planck constant, 6.63 x 10-34 m2 kg s-1).
I
Nuclide
12
The magnitude of spin angular momentum is �I (I + 1) ħ.
0
C, 16O
1
1/2
H, 13C, 15N, 29Si, 31P
The spin quantum number I can take a value I = 0, 1/2 , 1, 3/2, 2, 5/2, …
2
1
H, 14N
11
When discussing NMR we often use a special subset of language:
3/2
B, 23Na, 35Cl, 37Cl
17
2
O, 27Al
spin means magnetic nucleus
10
5/2
B
protons is often used in place of 1H nuclei
The spin quantum number, and proof of the quantum mechanical nature of spin, can be measured and shown by
performing a Stern-Gerlach experiment. Here, a beam of spins is passed through a magnetic field and onto a
detector. The number of positions of signal detected indicated the spin quantum number while the lack of a
continuum between points at which spins land shows that the system is quantized.
Spin angular momentum can be described using a vector I whose direction and magnitude are quantized. The
length of I is �I (I +1) ħ with 2I + 1 projections along the axis of the applied magnetic field. By convention the axis
along which the magnetic field is applied is labelled z.
The projection of I onto the z axis is labelled Iz.
Iz = m ħ where m = I, I - 1, I - 2, …, - I.
Magnetic Moments
The magnetic moment µ is related to the spin angular
momentum I by µ = γ I where γ is the magnetogyric ratio of
the nuclide.
Nuclide γ ( 106 rad s−1 T−1) ν (MHz T−1)
H
1
267.513
42.58
2
H
41.065
6.54
13
C
67.262
10.71
15
−27.116
−4.32
19
251.662
40.05
29
Si
−53.190
−8.47
P
108.291
17.24
N
F
31
Figure 1. Space quantization of the angular momentum
of spin-½ and spin-1.
Magnetic moment is parallel to spin angular momentum (if γ is positive, antiparallel if γ is negative).
The magnitude and orientation of γ and I are quantized.
Nuclear Magnetism and NMR Spectroscopy - Ralph W. Adams
Lecture 1: Nuclear Magnetic Resonance – Theory and Techniques
Energy levels
If we introduce a magnetic field B – a vector because it has direction and magnitude – then we can calculate
classically the energy E of a magnetic moment µ by taking the scalar product of the vectors µ and B.
E=-µ∙B
If the magnetic field is strong then the spin quantization axis coincides with the field direction.
E = - µz B where µz is the z component of µ. As µz = γ Iz = γ m ħ, E = - γ m ħ B.
This means that the energy of the nucleus is changed in proportion with the magnetic field strength B, the
magnetogyric ratio γ, and the z-component of the spin angular momentum.
The energy gap ΔE = γ ħ B with a corresponding frequency νNMR = (γ B)/(2 π).
Resonance frequencies and chemical shift
The magnetic field within an atom differs slightly from the external field, depending on the chemical environment.
This is the origin of chemical shift (δ).
At 9.4 T (T = Tesla, the SI unit of magnetic field strength), c. 100,000 stronger than the earth’s field,
2.675 x 108 rad s-1 T-1 x 9.4 T
2π
6
= 400.2 x 10 Hz
Larmor frequency, νNMR = (γ B)/(2 π) =
νNMR is in the radiofrequency part of the electromagnetic spectrum so we use the term R.F. field when discussing the
radiation required to irradiate NMR transitions.
We usually refer to the 1H NMR frequency rather than the magnetic field as 1H is the most commonly studied nucleus.
For uncoupled spins, the description of the NMR experiment can be easily described by rotating magnetisations.
Individual spins are not observed in a normal NMR experiment, rather it is a bulk magnetization vector – the sum of
all of the spin angular moments – which is manipulated and observed. The bulk magnetization vector cannot be
observed when it is directed along the z axis. For a basic NMR experiment a pulse of electromagnetic radiation at
an appropriate radiofrequency is applied to rotate the bulk magnetisation vector into the transverse plane –
conventionally labelled x,y. This allows Larmor precession – rotation around the z axis at the frequency νNMR – to be
observed, and it is this precession which is used to generate an NMR signal. The behaviour of an uncoupled spin
Nuclear Magnetism and NMR Spectroscopy - Ralph W. Adams
Lecture 1: Nuclear Magnetic Resonance – Theory and Techniques
can be approximated to the behaviour of a classical rotor such as a gyroscope or bicycle wheel and the
mathematics associated with these types of systems are applicable to many NMR systems.
The expanded region of the NMR spectrum (at 9.4 T) shows how chemical
shifts and NMR spectroscopy fit into the full electromagnetic spectrum.
Populations and polarizations
We can calculate a Boltzmann distribution across the lower (m = + ½) and upper (m = - ½) energy levels for 1H spins.
There is a convention of naming the lower and upper energy levels α and β respectively.
For nuclei with spin I = 1/2, the ratio of the number of nuclei in the upper energy level β to the number of nuclei in
the lower energy level α is determined by Boltzmann’s equation:
nβ / nα = exp(-ΔE / kBT)
ΔE = γ ħ B = 2.65 x 10-25 J and kBT = 4.14 x 10-21 J, so ΔE/kBT = 6.4 x 10-5
This means that the available thermal energy, kBT, is large compared to the energy required to reorient the spins
and that the population difference will be small. For this example it can be shown that the ratio nβ / nα corresponds
to nα = 15625 and nβ = 15624, so the population difference is only 1 in 31250 molecules. The result here shows that
the equivalent of only 1 in 104 – 106 nuclei (depending on γ, B, and T) will contribute to the observed NMR signal –
NMR is a very weak phenomenon.