Nuclear magnetic resonance

Nuclear magnetic resonance (NMR) is a physical phenomenon based upon the magnetic property of an atom's nucleus. NMR studies a magnetic nucleus, like that of a hydrogen atom, by aligning it with an external magnetic field and perturbing this alignment using an electromagnetic field. The response to the field (the perturbing), is what is exploited in NMR spectroscopy and magnetic resonance imaging.

Theory of nuclear magnetic resonance

Nuclear spin and magnets

Electrons, neutrons and protons, the three particles which constitute an atom, have an intrinsic property called spin. This spin is defined by the fourth quantum number for any given wave function obtained by solving relativistic form of the Schrödinger equation (SE). It represents a general property of particles which we can describe using the properties of electrons. Electrons flowing around a coil generate a magnetic field in a given direction; this property is what makes electric motors work. In much the same way electrons in atoms circulate around the nucleus, generating a magnetic field. This generated field has an angular momentum associated with it. It so turns out that there is also an angular momentum with the electron particle itself, denoted the spin, and this gives rise to the spin quantum number, ms.

Related Topics:
Electrons - Neutrons - Protons - Quantum number - Wave function - Schrödinger equation - Spin quantum number

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Spin angular momentum is quantized and can take different integer or half-integer values depending on what system is under study. If we solve the relativistic SE for the electron we get the values +½ and -½. Since the Pauli principle states that no two species can have the same quantum number, it is why only two electrons, paired antiparallel (one positive one negative), can appear in a single atomic orbital.

Related Topics:
Quantized - Integer - Pauli principle - Atomic orbital

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Like the electron, protons and neutrons also have a spin angular momentum which can take values of + ½ and ?½. In the atomic nucleus, protons can pair with other antiparallel protons, much in the same way electrons pair in a chemical bond. Neutrons do the same. Paired particles, with one positive and one negative spin, thus have a net spin of zero "0". We can see that a nucleus with unpaired protons and neutrons will have an overall spin, with the number unpaired contributing ½ to the overall nuclear spin quantum number, I. When this is larger than zero, a nucleus will have a spin angular momentum and an associated magnetic moment, μ, dependent on the direction of the spin. It is this magnetic moment that we manipulate in modern NMR experiments.

Related Topics:
Chemical bond - Spin quantum number

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It is worth noting here that nuclei can have more than one unpaired proton and one unpaired neutron, much in the same way electronic structure in transition metals can have many unpaired spins. For example 27Al has an overall spin I=5/2.

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NB: A technique related to NMR is electron spin resonance that exploits the spin of electrons instead of nuclei. The principles are otherwise similar.

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Values of spin angular momentum

The spin angular momentum of a nucleus can take ranges from +I to ?I in integral steps. This value is known as the magnetic quantum number, m. For any given nucleus, there is a total (2I+1) angular momentum states. Spin angular momentum is a vector quantity. The z component of which, denoted Iz, is quantised:

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:Iz = mh/2π

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where h is Planck's constant.

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The resultant magnetic moment of this nucleus is intrinsically connected with its spin angular momentum. In the absence of any external effects the magnetic moment of a spin ½ nuclei lies approximately 52.3° from the angular momentum axis or 127.7° for the opposing spin. This magnetic moment is intrinsically related to I with a proportionality constant γ, called the gyromagnetic ratio:

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:μ=γI

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Spin behaviour in a magnetic field

If we take the case of nuclei which have a spin of a half like 1H, 13C or 19F. The nucleus thus has two possible magnetic moments it could take, often referred to as up or down, +0.5 -0.5, or to be more in tune with physicists... α and β. The energies of each state are degenerate - that is to say that they are the same. The effect is that the number of atoms (population)in the up or α state is the same as the number of atoms in the β state.

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If we place a nucleus in a magnetic field the angular momentum axis coincides with the field direction. The resultant magnetic momenta, space quantised from the angular momentum axis, no longer have the same energy since one state has a z-component aligned with an external field and are lower in energy (positive I values) and the other opposes the external field and is higher in energy. This causes a population bias toward the lower energy states.

Related Topics:
Magnetic field - Angular momentum

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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 the negative scalar product of the vectors:

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:E = -μzB0

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We've already defined μz=γIz. So placing this in the above equation we get:

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:E = -mhγB0 / 2π

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Resonance

The energy gap between our α and β states is (hγB0)/2π. We get resonance between the states, therefore equalising populations, if we apply a radiofrequency with the same energy as the energy difference ΔE between the spin states. The energy of a photon is E = hν, where ν is its frequency.

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:ΔE = hγB0/2π

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I.e. the frequency of electromagnetic radiation required to produce resonance of an specific nucleus in a field B is:

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:ν = γB0/2π

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It is this frequency that we are concerned with, and detect in NMR. And it is this frequency which describes the sample we are observing. But importantly: it is this resonance that gives rise to the NMR spectrum

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Nuclear shielding

It would appear from the above equation that all nuclei of the same isotope, which take the same the gyromagnetic ratio (μ), resonate at the same frequency. This of course is not the case. Since the gyromagnetic ratio of a given isotope does not change we can only rationalise this by stating that the effect of the external magnetic field is different for different nuclei. Local effects of other nuclei, especially spin-active nuclei, and local electron effects shield each nucleus differently from the main external field.

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It was stated that the energy of a spin state is defined by E= -μzB0. We can see that by shielding the strength of the magnetic field, the experienced effect, or effective magnetic field at the nucleus is lower: Beffective < B0. Thus the energy gap is different, and hence the frequency required to achieve resonance deviates from the expected value.

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These differences due to nuclear shielding give rise to many peaks (frequencies) in an NMR spectrum. We can now see why NMR is a direct probe of chemical structure.

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An astute reader will notice that differences in shielding would occur between two identical molecules oriented differently! However, these differences are averaged out due to molecular motion.

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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 returns to its equilibrium state. Once the population is relaxed, it can be probed again, since it is in the initial state.

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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.

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This happens because some of the spins were flipped by the pulse and will remain so until they have undergone population relaxation.

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