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# NMR: Theory

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Nuclear magnetic resonance has been play an important role in the fields of physical techniques available to the chemist for more than 25 years. It is becoming a more and more useful method to probe the structure of molecules. The primary object of this module is to help readers understand some knowledge about NMR Theory.

### Introduction

Nuclear magnetic resonance, NMR, is a physical phenomenon of resonance transition between magnetic energy levels, happening when atomic nuclei are immersed in an external magnetic field and applied an electromagnetic radiation with specific frequency. By detecting the absorption signals, one can acquire nuclear magnetic resonance spectra. According to the positions, intensities and fine structure of resonance peaks, people can study the structures of molecules quantitatively. The size of molecules of interest varies from small organic molecules, to biological molecules of middle size, and even to some macromolecules such as nucleic acids and proteins. Apart from these commonly utilized applications in organic compound, NMR also play an important role in analyzing inorganic molecules, which makes NMR spectroscopy a powerful technique.

Here are some examples showing what type of information NMR can provide;

1H NMR spectrum will provide information about;

• The number of different types of hydrogen placed in the molecule,
• The integration of NMR spectra provides information about relative numbers of different types of hydrogen,
• The electronic environment of the different types of hydrogens,
• The number of hydrogen present in neighborhoods .

### Nuclear Spin Angular Momentum

Nuclear spin angular momentum was first reported by Pauli in 1924. The spin of a nucleus is an intrinsic property and represented with the quantum number I. Analogous to the angular momentum commonly encountered in electron, the angular momentum is a vector which can be described by a magnitude L and a direction, m. The magnitude is given by

$L=\bar{h}\sqrt{I(I+1)}$

The projection of the vector on the z axis (arbitrarily chosen), takes on descritized values according to m, where

m=-I, -I+1, -I+2,...+I

The angular momentum along the z-axis is now

$I_z=m\bar{h}$

Pictorially, this is represented in the figure below for certain values of I.

The spin of the nucleus comes from the protons and neutrons which have their own spin properties. The details of finding I are beyond the scope of this wiki, however several rules relating the number of protons and neutrons in the nucleus can give insights into the nature of I. For a nucleus to be NMR active, I>0. These rules are summarized in the table below.

Table 1. General rules for determination of nuclear spin quantum numbers

 Atomic Number Proton Number Neutron Number I examples even even even 0 $^{16}O$ even odd odd integral $^{2}H$ odd even odd Half-integral $^{13}C$ odd even $^{15}N$

### Magnetic Moment of a Nucleus

The nucleus is a charged particle which is moving in a loop thereby creating a magnetic moment due to Lorentz forces. The magnetic moment is related to the angular momentum

$\mu=\gamma I$

where $$\gamma$$ is the gyromagnetic ratio, a proportionality constant unique to each nucleus. The table below shows some of the gyromagnetic ratios for some commonly studies nuclei.

### Application of a Magnetic Field

The NMR experiment places a sample in a magnetic field. The application of a magnetic field splits the degenerate 2I+1 nuclear energy levels. The energy of a particular level is

$E=-\mu\cdot B_0$

where $$B_0$$ is the applied magnetic field. Along the z-direction, which we assume the magnetic field is applied,

$E=-\mu B_0$

by substitution,

$E=-m\bar{h}\gamma B_0$

The magnitude of the splitting therefore depends on the size of the magnetic field. In most labs this magnetic field is somewhere between 1 and 21T.

### Distribution of Particles Between Magnetic Quantum States

In the absence of a magnetic field the magnetic dipoles are oriented randomly and there is no net magnetization (vector sum of µ is zero). There is alignment on application of a magnetic field, more spins parallel to the field (lower energy) than antiparallel (higher energy), controlled by Boltzmann Distribution. The signal in NMR spectroscopy results from the difference between the energy absorbed by the spins that make a transition from the lower energy state to the higher energy state. The energy emitted by the spins which simultaneously makes a transition from the higher energy state to the lower energy state. The signal strength is thus proportional to the population difference between the states. NMR is a very sensitive form of spectroscopy since it is capable of detecting these very small population differences.

Figure 10. Spins configurations according to applied magnetic field

When a group of spins is placed in a magnetic field, each spin aligns in one of the two possible orientations either positive or negative. In sample, which contains a specific NMR-active nucleus, the nuclei will be distributed throughout the various spin states. The energy separation between these states is relatively small and  the energy from thermal collisions is sufficient to place many nuclei into higher energy spin states. The number of nuclei in each spin state can be described by the Boltzmann distribution. The Boltzmann equation expresses the relationship between temperature and the related energy as shown below.

$\large \frac{N_{upper}}{N_{lower}}=e^{\frac{-\Delta{E}}{kT}} = e^{\frac{-h\nu}{kT}}$

Where Nupper and Nlower represent the population of nuclei in upper and lowe energy states, E is the energy difference between the spin states, k is the Boltzmann constant (1.3805x10-23 J/Kelvin ) and T is the temperature in K. At room temperature, the number of spins in the lower energy level, N lower, slightly outnumbers the number in the upper level, N upper.

### The NMR Experiment

#### Selection Rules

The selection rule in NMR is

$\Delta m=\pm1$

For a nucleus with I=1/2 there is only one allowed transition.

#### Energy Transitions

Lets assume we have a compound in and we are interested in in a nucleus with I=1/2. From what we have learned, if we place that sample in a magnetic field, the sample with split into 2 energy levels, with a Boltzmann distribution of spins in the low energy and high energy states. We now need to apply electromagnetic radiation to promote the spins in low energy level to the high energy level. We can calculate the energy that is required to promote this transition, using the above equation for energy and the selection rule. The energy range is on the order of MHz. As the spins relax they go back to the ground state emitting radiation characteristic of energy gap.

### References

1. Claude H. Yoder, Charles D. Schaeffer,Introduction to multinuclear NMR : theory and application, Benjamin/Cummings Pub. Co., Menlo Park, Calif., 1987.
2. Robin K. Harris, Nuclear magnetic resonance spectroscopy : a physicochemical view, Pitman, Marshfield, Mass., 1983.
3. Jeremy K.M. Sanders and Brian K. Hunter, Modern NMR spectroscopy : a guide for chemists, Oxford University Press, New York, 1993.
4. David A.R. Williams ; editor, David J. Mowthorpe, Nuclear magnetic resonance spectroscopy, Published on behalf of ACOL, London, by J. Wiley, New York, 1986.
5. Frank A. Bovey, Lynn Jelinski, Peter A. Mirau, Nuclear magnetic resonance spectroscopy, Academic Press, San Diego, 1988.
6. R.J. Abraham, J. Fisher and P. Loftus, Introduction to NMR spectroscopy, Wiley, New York, 1988.
7. J.W. Akitt, NMR and chemistry : an introduction to modern NMR spectroscopy, Chapman & Hall, London; New York, 1992.

### Contributors

• Derrick Kaseman (UC Davis), Sureyya OZCAN, Siyi Du

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