LECTURE-10
1.3 Nuclear Magnetic Resonance (NMR) Spectroscopy
So far, we have seen that mass spectrometry gives a molecule’s size and formula,
while infrared spectroscopy identifies a molecule’s functional groups. Nuclear
magnetic resonance spectroscopy complements these other techniques by mapping a
molecule’s carbon-hydrogen framework. Taken together, mass spectrometry, IR, and
NMR make it possible to determine the structures of even very complex molecules.
As is the case with other forms of optical spectroscopy, the signal in NMR
spectroscopy arises from a difference in the energy levels occupied by the nuclei in
the analyte. Here, we develop a general theory of nuclear magnetic resonance
spectroscopy that draws on quantum mechanics to explain these energy levels.
1.3.1 Principles of NMR
The quantum mechanical description of an electron is given by four quantum
numbers: the principal quantum number, n, the angular momentum quantum number,
l, the magnetic quantum number, ml, and the spin quantum number, ms. The first three
of these quantum numbers tell us something about where the electron is relative to the
nucleus and something about the electron's energy. The last of these four quantum
numbers, the spin quantum number ms, tells us something about the ability of an
electron to interact with an applied magnetic field. An electron has possible spins of
+1/2 or -1/2, which we often refer to as spin up, using an upwards arrow, ↑, to
represent it, or as spin down, using a downwards arrow, ↓to represent it.
Like an electron, a nucleus carries a charge and has a spin quantum number. The
overall spin, I, of a nucleus, is a function of the number of protons and neutrons that
make up the nucleus. The three simple rules for nuclear spin states are:
(i) If the number of neutrons and the number of protons are both even numbers, then the
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nucleus does not have a spin; thus, C, with six protons and six neutrons, has no
overall spin and I=0
(ii) If the number of neutrons plus the number of protons is an odd number, then the
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nucleus has a half-integer spin, such as 1/2 or 3/2; thus, C, with six protons and
seven neutrons, has an overall spin of I=1/2; this also is true for 1H.
(iii) If the number of neutrons and the number of protons are both odd numbers, then the
nucleus has an integer spin, such as 1 or 2; thus, 2H, with one proton and one neutron,
has an overall spin of I=1
The total number of spin states i.e., the total number of possible orientations of the
spin is equal to (2×I)+1. To be NMR active, a nucleus must have at least two spin
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states so that a change in spin states, and, therefore, a change in energy, is possible;
for this reason, 12C is NMR inactive because it has (2×0) +1 = 1 spin states, while 13C
is NMR active because it has (2×1/2) +1 = 2 spin states with values of m = +1/2 and
of m = −1/2. 2H for which there are (2×1) +1 = 3 spin states with values of m = +1/2,
m = 0, and m = −1/2 is also NMR active. In this section, our interest is in the NMR
spectra for 1H and for 13C, and we will limit ourselves to considering I = 1/2 and spin
states of m = +1/2 and of m = −1/2.
Spinning charged nuclei generate a magnetic field, like the field of a small bar
magnet. In the absence of an applied magnetic field, the nuclear spins are randomly
oriented and equally divided between their possible spin states (50% with a spin of
+1/2 and 50% with a spin of -1/2, both spins have the same energy and neither
absorption nor emission occurs). However, when a sample is placed in an applied
magnetic field, the nuclei twist and turn to align themselves either with the applied
magnetic field with spins of m = +1/2 or against the applied magnetic field with spins
of m = -1/2 of the larger magnet. More energy is needed for a proton to align against
the field than with it. Protons that align with the field are in the lower energy α-spin
state while those that align against the field are in the higher-energy β-spin state.
More nuclei are in the α-spin state than in the β-spin state. Although the difference in
the populations is very small (about 20 out of a million protons), it is sufficient to
form the basis of NMR spectroscopy.
When the sample is subjected to a pulse of radiofrequency (rf) radiation whose energy
corresponds to the difference in energy (ΔE) between the α- and β-spin states, energy
absorption occurs and the nuclei in the α-spin state are promoted to the β-spin state.
This transition is called “flipping” the spin.
When the nuclei undergo relaxation (i.e., return to their original state), they emit
electromagnetic signals whose frequency depends on the difference in energy (ΔE)
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between the α- and β-spin states. The NMR spectrometer detects these signals and
displays them as a plot of signal frequency versus intensity-an NMR spectrum. It is
because the nuclei are in resonance with the rf radiation that the term “nuclear
magnetic resonance” came into being. In this context, “resonance” refers to the
flipping back and forth of nuclei between the α- and β-spin states in response to the rf
radiation.
The energy difference (ΔE) between the α- and β-spin states depends on the strength
of the applied magnetic field (B0). The greater the strength of the magnetic to which
we expose the nucleus, the greater is the difference in energy between the α- and β-
spin states.
Elower = -(γhB0)/4π
Eupper = +(γhB0)/4π
where γ is the gyromagnetic ratio for the nucleus (a constant specific to each specific
nucleus), h is Planck’s constant and B0 is the strength of the applied magnetic field.
The difference in energy (ΔE) between the two states is
ΔE = Eupper - Elower
= {+(γhB0)/4π} - {-(γhB0)/4π}
= γhB0)/2π
Recall ΔE = hν (ν is the frequency of electromagnetic radiation needed to effect a
change in spin state)
ν = {γhB0)/2π}/h
= γB0/2π.
The equation above shows that the magnetic field is proportional to the operating
frequency (MHz). Therefore, if the spectrometer has a more powerful magnet, it will
have a higher operating frequency.
Today’s NMR spectrometers operate at frequencies between 60 and 900 MHz. The
operating frequency of a particular spectrometer depends on the strength of the built-
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in magnet. The greater the operating frequency of the instrument-and the stronger the
magnet-the better the resolution (separation of the signals) of the NMR spectrum.
Because each kind of nucleus has its own gyromagnetic ratio, different frequencies
are required to bring different kinds of nuclei into resonance. For example, an NMR
spectrometer with a magnet requiring a frequency of 300 MHz to flip the spin of a 1H
nucleus requires a frequency of 75 MHz to flip the spin of a 13C nucleus.
γ = 2.68*108 T-1s-1 for 1H and 6.69*107 T-1s-1 for 13C
νH = γHB0/2π
νC = γCB0/2π
νC/ νH = {γCB0/2π}/ γHB0/2π =γC/γH
νC = {γC/γH}* νH
= (6.69*107 T-1s-1)/(2.68*108 T-1s-1)* νH
= 0.25* νH
NMR spectrometers are equipped with radiation sources that can be tuned to different
frequencies so that they can be used to obtain NMR spectra of different kinds of
nuclei (1H, 13C, 15N, 19F, and 31P etc.).
Example
For a magnet of field strength 11.74 Tesla, the frequency needed to effect a change in spin
state for 1H, for which γ is 2.68×108 T-1s-1 is
ν = γB0/2π
= (2.68×108 T-1s-1)(11.74 T)2π = 5.01×108 s-1 = 500 MHz, which is in the radio frequency
range of the electromagnetic spectrum.
Exercise
(i) Determine the frequency required to cause a 1H nucleus to flip its spin when it is
exposed to a magnetic field of 1 T?
(ii) Calculate the magnetic field (in tesla) required to flip a 1H nucleus in an NMR
spectrometer that operates at 360 MHz. What magnetic field strength is required when
a 500 MHz instrument is used?
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