# Nuclear Magnetic Resonance

### NMR

NMR refers to Nuclear Magnetic Resonance

* Nuclear Magnetic Resonance (NMR) spectroscopy is the most important and widely used technique by chemists for identifying the structures of organic and inorganic compounds.

* It is the only spectroscopic technique for which we do a complete analysis and interpretation of the entire spectrum for a compound.

* Although hydrogen (1H) and carbon (13C) nuclei are most frequently studied, there are many elements which possess isotopes which have suitable nuclear spins to allow them to be studied by NMR methods, especially for inorganic compounds.

### Theory

NMR spectroscopy takes advantage od the magnetic properties of certain nuclei and records the absorption of energy between quantized nuclear energy levels

In addition to charge and mass, which all nuclei have, various nuclei also possess a property called nuclear spin, which means that they behave as if they were spinning. Since nuclei have a charge, they generate a magnetic field with an associated magnetic moment(µ). Nuclei with a non-zero magnetic moment can be studied by NMR spectroscopy.

The angular momentum of the spinning nuclear charge can be described by a nuclear spin quantum number(I). There are useful empirical rules relating mass number, atomic number(Z) and spin quantum number(I).

Since NMR depend on the existence of a nuclear spin, nuclei with I=0 have no NMR spectrum. From standpoint of generating NMR spectra, the most important class of nuclei are those with I=1/2, because these nuclei have a spherical charge distribution. Nuclei with I>1/2 have a non-spherical charge distribution by an electric quadrupole moment.

For nuclear spin I=0

* These nuclei do not interact with an applied magnetic field and are NOT NMR active.

* The nuclei with I = 0 will have an even number of protons and neutrons and have no net spin.

* The most commonly encountered nuclei: 12C and 16O, the most abundant isotopes of carbon and oxygen, respectively.

· Fortunately, these element have other isotopes, 13C (I = ½) and 17O (I = 5/2), respectively, which can be observed by NMR spectroscopy.

For nuclear spin I=1/2

* These nuclei do interact with an applied magnetic field and are NMR active.

* The nuclei with I = ½ will have an even number of protons and an odd number of neutrons, or vice-versa and have a non-zero magnetic moment.

* The most commonly encountered nuclei: 1H (ordinary hydrogen) and 13C (1% abundance). Over 90% of all NMR spectra reported in the chemical literature are 1H or 13C NMR and we will predominantly use these spectra in determining the structures of unknown organic and inorganic compounds.

· Other commonly observed nuclei: 19F and 31P. The analysis and interpretation of NMR spectra for I = ½ nuclei are based on a common set of rules, independent of the nature of the element.

For nuclear spin I>1/2

* These nuclei have both a non-zero magnetic moment, and are thus NMR active, and an electric quadrupole moment (Q).

* The nuclei with integer values of I (1, 2, 3…) will have both an odd number of protons and neutrons.

* The nuclei with fractional values of I (3/2, 5/2, 7/2…) will have an odd number of protons and an even number of neutrons, or vice-versa.

* The most commonly encountered nuclei: 2H (Deuterium, < 1% abundant) and 15N (the most abundant isotope of nitrogen).

· Unfortunately, the electric quadrupole moment results in very broad spectral peaks, making these nuclei more difficult to observe and interpret.

### Nuclear Spin Quantum Number

Rules for predicting the nuclear spin quantum number, I, from the numbers of protons and neutrons in a nucleus.

### Nuclear spin and magnets

The elementary particles, neutrons and protons, composing an atomic nucleus, have the intrinsic quantum mechanical property of spin. The overall spin of the nucleus is determined by the spin quantum number I. If the number of both the protons and neutrons in a given isotope are even then I = 0, i.e. there is no overall spin; just as electrons pair up in atomic orbitals, so do even numbers of protons and neutrons (which are also spin ½ particles and hence fermions) pair up giving zero overall spin. In other cases, however, the overall spin is non-zero. For example 27Al has an overall spin I = 5/2.

A non-zero spin is associated with a non-zero magnetic moment, μ, via

where the proportionality constant, γ, is the gyromagnetic ratio. It is this magnetic moment that is exploited in NMR.

Electron spin resonance is a related technique which exploits the spin of electrons instead of nuclei. The basic principles are otherwise similar.

### Values of spin angular momentum

The angular momentum associated with nuclear spin is quantized. This means both that the magnitude of angular momentum is quantized (i.e. I can only take on a restricted range of values), and also that the 'orientation' of the associated angular momentum is quantized. The associated quantum number is known as the magnetic quantum number, m, and can take values from +I to –I in integral steps. Hence for any given nucleus, there is a total of 2I+1 angular momentum states.

The z component of the angular momentum vector, Iz, is therefore:

where is Planck's reduced constant.

The z component of the magnetic moment is simply

### Spin behavior in a magnetic field

Consider nuclei which have a spin of one-half, like 1H, 13C or 19F. The nucleus has two possible spin states: m = ½ or m = -½ (also referred to as up and down or α and β, respectively). The energies of these states are degenerate—that is to say that they are the same. Hence the populations of the two states (i.e. number of atoms in the two states) will be exactly equal at thermal equilibrium.

If a nucleus is placed in a magnetic field, however, the interaction between the nuclear magnetic moment and the external magnetic field mean the two states no longer have the same energy. The energy of a magnetic moment μ when in a magnetic field B0 (the zero subscript is used to distinguish this magnetic field from any other applied field) is given by the negative scalar product of the vectors:

where the magnetic field has been oriented along the z axis.

Hence

As a result the different nuclear spin states have different energies in a non-zero magnetic field. In hand-waving terms, we can talk about the two spin states of a spin ½ as being aligned either with or against the magnetic field. If γ is positive (true for most isotopes) then m = ½ is the lower energy state.

The energy difference between the two states is

and this difference results in a small population bias toward the lower energy state.

### Resonance

Resonant absorption will occur when electromagnetic radiation of the correct frequency to match this energy difference is applied. The energy of a photon is E= hν, where ν is its frequency. Hence absorption will occur when

These frequencies typically correspond to the radio frequency range of the electromagnetic spectrum.

It is this resonant absorption that is detected in NMR.

### Nuclear shielding

It might appear from the above that all nuclei of the same nuclide (and hence the same γ) would resonate at the same frequency. This is not the case. The most important perturbation of the NMR frequency for applications of NMR is the 'shielding' effect of the surrounding electrons. In general, this electronic shielding reduces the magnetic field at the nucleus (which is what determines the NMR frequency). As a result the energy gap is reduced, and the frequency required to achieve resonance is also reduced. This shift of the NMR frequency due to the chemical environment is called the chemical shift, and it explains why NMR is a direct probe of chemical structure.

Unless the local symmetry is particularly high, the shielding effect depends on the orientation of the molecule with respect to the external field. In solid-state NMR, magic angle spinning is required to average out this orientation dependence. This is unnecessary in conventional NMR of molecules in solution since rapid molecular tumbling averages out the anisotropic component of the chemical shift.

### Relaxation

The process called population relaxation refers to nuclei that return to the thermodynamic state in the magnet. This process is also called T1 relaxation, where T1 refers to the mean time for an individual nucleus to return to its equilibrium state. Once the population is relaxed, it can be probed again, since it is in the initial state.

The precessing nuclei can also fall out of alignment with each other (returning the net magnetization vector to a nonprecessing field) and stop producing a signal. This is called T2 relaxation. It is possible to be in this state and not have the population difference required to give a net magnetization vector at its thermodynamic state. Because of this, T1 is always larger (slower) than T2. This happens because some of the spins were flipped by the pulse and will remain so until they have undergone population relaxation. In practice, the T2 time is the life time of the observed NMR signal, the free induction decay. In the NMR spectrum, meaning the Fourier transform of the free induction decay, the T2 time defines the width of the NMR signal. Thus, a nucleus having a large T2 time gives rise to a sharp signal, whereas nuclei with shorter T2 times give rise to more broad signals. The length of T1 and T2 is closely related to molecular motion.