Extra Notes to Lecture 1 – Matt Crump.
‘Modern Approaches to DNA/Protein Structure Determination’.
Different ways of describing NMR experiments:
 Quantum mechanical description - evolution of density operator (t) in time.
- product spin operators
 Bloch equations
 Energy level diagrams
 Vector models
Only the last two will be discussed here.
An Introduction to NMR. Dr Matthew Crump.
In the central square restaurant, Cambridge, Massachusetts in late 1945, Torrey, Pound and
Purcell discussed the possibility of detecting absorption of radio frequency energy by
protons polarised in a strong magnetic field. A sample would be placed in a strong static
magnetic field and a second magnetic field of quantum energy corresponding to the energy
splitting of the proton spins would be applied transverse to the static field.
Work was hampered by the fact all three men were ‘moonlighting’ and were officially
employed by the M.I.T. to write several books. Their equipment, such as low noise
amplifiers, was ‘borrowed’ from M.I.T. and moved to their laboratory. A large resonant
cavity was constructed and paraffin wax (a proton rich source) was chosen as a sample. A
large permanent magnet was borrowed from a colleague’s cosmic ray shed which they
calculated would give proton resonance at 30 MHz. On Thursday evening 13th December
they began their experiment but failed. Persevering, magnetic resonance was first detected
on Saturday 15th December (Purcell, E. M., Torrey, H. C. and Pound, R. V. (1946).
Resonance absorption by nuclear magnetic moments in solid. Phys Rev. 69, 37-38)
Almost simultaneous discovery in Stanford by Felix Bloch who allegedly had the idea
while attending a concert in the Symphony hall, Boston
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In 1950, during studies on 14N samples (ammonium and nitrate ions), Proctor made ‘the
surprising observation that the frequency of resonance in liquid samples depended strongly
on the chemical compound in which it was contained’ (Proctor, W. G. & Yu, F. C. (1950)
The dependence of nuclear magnetic resonance frequency upon chemical compound. Phys
Rev. 77, 717). An NMR spectrum of ethanol followed in 1951. (Arnold, J. T., Dharmatti,
S. S. & Packard, M. E. (1951). Chemical effects on nuclear induction signals from organic
compounds. J. Chem. Phys 19,507.).
First ‘high resolution’ NMR
oscillograph trace of ethanol.
Peaks from left to right represent
proton resonances from OH, CH2
and CH3 groups. The areas of the
peaks roughly matched the ratios of
the number of protons associated
with each peak.
The implications of this single trace
were incredible and following this
discovery, the physicists handed the
technique to chemists and the
phenomenon promptly changed
their world. Bloch and Purcell
shared nobel prize in 1952.
Amazingly Varian launched the
first spectrometer in 1953.
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Energy level diagrams for a two spin system, IS ( II = Is = 1/2).
Nuclear magnetism is a manifestation of nuclear spin angular momentum, an intrinsically
quantum mechanical property that has no classical analog.
Spin is characterised by the nuclear spin quantum number I.
nuclei with odd mass numbers have half integral spin numbers (1H, 13C, 15N, 31P, 19F).
nuclei with even mass numbers and even charge numbers have spin zero. (32S)
nuclei with even mass numbers and odd charge numbers have integral spin numbers (2H)
Nuclear spin angular momentum, I, is a vector quantity with magnitude given by
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The SI unit for magnetic field is B. The derived unit for describing magnetic field strength is the Tesla (T),
defined as 1T = 1 kg s-1 A-1.
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 2       2 I ( I  1)
I is the angular momentum quantum number and  is Planck’s constant divided by 2.
If we specify a I2 value, quantum mechanics restricts us as well to specifying the projection
of this vector along only one of three cartesian components of I. By convention the z-axis is
chosen and Iz is given by
Iz =  m
where m = (-I, -I+1,…..,I-1,I).
Therefore Iz has 2I+1 values. So a spin half has two energy levels, spin ‘up’ and spin
‘down’.
So far we talked about spin. Now the critical equation,
  
The magnetic moment  is colinear with the vector for the spin angular momentum vector.
 is the gyromagnetic ratio (T.s)-1. It tells us how much magnetism we get from so much
spin.
When placed in a magnetic field, the spin states of the nucleus have energies given by,
    
So placing the spins in the field, the magnetic moment cannot just align with the field (as a
compass would in a magnetic field) as it can only have certain discrete values along z
(taking our static field vertically along z in the laboratory frame).
To minimize the energy we maximize projection of  along the z-axis. Maximum is
 z   z  m   2 
When a nucleus is placed in an external magnetic field, Bo, 2I+1 energy levels (two for
spin- 1/2 ) are created with energies defined as:
E    z     2 
The allowed magnetic dipole transition fulfil a condition m= ± 1, therefore the energy
needed for the transition from one state into another is
E   [Joules]
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This is often expressed in frequency units (E = h) and called Larmor frequency:
w = -  Bo
[rad.s-1]
If a radio frequency field is applied at this frequency (MHz region) it can induce transitions
between the energy levels and NMR signal can be detected. This concept is similar to other
spectroscopies where different parts of the rf spectrum are used to excite different types of
transitions e.g. electronic states in UV or vibrational modes in IR spectroscopy.
Populations – Boltzmann distribution of states
Pi / Po  exp(  / kT )  exp(
 B
)
kT
k is Boltzmann’s constant (1.38 * 10–23 J K-1)
T is temperature in Kelvins.
So feed some numbers in,
Pi / Po  exp(
 (2.68  10 8  1 s 1 )(6.63  10 34 J  s)( 4.69 / kT )
2 (1.38  10  23 JK 1 )( 293K )
 exp( 3.28  10 5  0.999967
Therefore, for exactly 999967 in the upper state, there will be 1000000 in the lower state.
Only a 33 ppm excess. Expand exponential in the Boltzmann equation as a Maclaurin series
and get
B
Pi / Po  1 
kT
i.e. sensitivity is linearly proportional to B.
After the introduction of chemical shift, a Larmor frequency for isolated spins I and S,
respectively can be represented respectively:
I =  Bo(1 - I)/2
S =  Bo(1 - S)/2
The following is the spectrum of this system:
In the presence of a coupling constant, JIS, the resonance frequencies are shifted and the
spectrum looks like this:
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and the frequencies of individual lines are:
13 = I – 1/2 JIS
24 = I + 1/2 JIS
12 = S – 1/2 JIS
34 = S + 1/2 JIS
The two lines of the S spin correspond to the two possible states of the I spin and vice
versa. It is a convention to represents the spin with mI = ½ aligned parallel with Bo as  and
spin with mI = -½ aligned antiparallel as .
For a coupled two spin system, IS, there are four allowed transitions which frequencies are
derived from the following energy levels:
IS




Energy
E4 = + 0.5I + 0.5S + 0.25 JIS
E3 = + 0.5I - 0.5S - 0.25 JIS
E2 = - 0.5I + 0.5S- 0.25 JIS
E1 = - 0.5I - 0.5S+ 0.25 JIS
I - S >> | JIS | and |I + S| >> |I - S| we can draw the energy level diagram
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
S
I

2
m
1

3
S
0
I
1

-1
IS
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where m is the total magnetic quantum number. There are two allowed transitions (m= ±
1, single quantum) for the S spin (12, 34) and two for the I spin (13, 24) Other transitions
in the energy diagram:
Vector model - Another way to look at NMR experiments is in terms of precessing
magnetisation vectors.
Charged spinning nucleus behaves like a little electromagnet with the magnetic dipole
proportioned to the spin angular momentum, I.
=I
When placed into external magnetic field, Bo, the spinning nucleus it is not simply aligned
in the direction of this field, but start precessing around it. This motion is called Larmor
precession and is governed by the equation:
d/dt = - Bo x 
where the angular velocity is given by :
o = -  Bo
and because
o = 2L
L =  Bo/ 2
which is identical (except for the sign) to the frequency of the allowed transitions given at
the beginning of the lecture.
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