PRETENSIONES DE LA DEFENSA

2.1.1. Principles of NMR

The nuclei for which NMR spectra can be measured have an intrinsic angular momentum called spin [13]. This property, combined with the nuclei’s positive charge creates a small magnetic field. When this small nuclear magnetic field is exposed to a strong external magnetic field (B0) each spinning nucleus attempts

to align its magnetic moment with the external field. This results in a torque being exerted on the nuclear spins causing a circular motion called precession at a frequency νo, similar to a traditional spinning top rotating due to the earth’s gravity [14, 15]. The precession frequency is proportional to the strength of the external magnetic field strength (B0) and the gyromagnetic ratio (γ) of the

nucleus, which is a constant defining the ratio of its spin angular momentum to its magnetic dipole moment.

vo =γB0/2π (2.1)

Magnetic field strengths of typical commercial NMR spectrometers cause the rate of precession to be in the radio frequency range (megahertz), and can be measured by applying a short radio signal pulse before monitoring the oscillating

2. Theory

current that is induced by the oscillating magnetization in the sample. The detected current is known as the Free Induction Decay (FID), and contains the precession frequencies of all nuclei in the sample. This measurement can allow nuclei in different magnetic environments to be detected and characterized.

2.1.1.2. A Quantum Model

The advantage of the classical model is its simplicity. However, it is unable to account for many important NMR phenomena, for which the more thorough, but more complex quantum model is required.

The intrinsic nuclear spin is described by a quantum number value (I) which can take values of 0, 1

2, 1. . . . For a nucleus to be NMR active I must be >0.

Such nuclei have an odd number of protons and/or neutrons. The higher the spin quantum number the more possible orientations (spin states) the nuclear spin can adopt in the external magnetic field. The most useful nuclei for NMR have I = 1₂ e.g. 1_H, 13_C, 15_N, 19_F, 31_{P which have two possible spin states (as}

determined through the simple formula (2(I)+1)). These spins can point ‘up’ (α) or ‘down’ (β) in the external magnetic field and are characterized by a magnetic quantum number, m(which can hold values of m=−I,−I+ 1, ..., 0, ..., I−1, I

). When these spin 1

2 nuclei are at thermal equilibrium in the absence of an

external magnetic field exactly one half of the nuclei will be in each state at any given time.

When an external magnetic field is introduced the distribution becomes unequal with slightly above 50% of the nuclei being in theαstate (aligned with B0). This

distribution occurs due to the αstate having a lower energy than the β state (for a spin with a positive γ). The energy of each spin is found to be related to the specific nuclei and the magnetic field strength and can be found through:

E =−hγB0m

Figure 2.1.:Quantum energy levels of a spin 1₂ nucleus in a magnetic field: The two quantum energy levels formed when a spin1

2 nucleus

is placed into a strong external magnetic field

As with other types of spectroscopy, only specific transitions are allowed and can take place. With NMR the allowed transitions are those which change the magnetic quantum number by a value of ±1.

When energy is applied which matches the energy gap (∆E) between theαand

β levels one of the lower energy α spins can be promoted to the higher energy state.

∆E =hvo =hγB0/2π (2.3)

This is similar to the equation seen in the classical model situation, with vo now being the resonant frequency instead of the precession frequency along with

h being Planck’s constant.

2.1.1.3. Experimental NMR spectroscopy

The ratio of populations of each spin state is found via a Boltzmann distribution:

2. Theory

where Pα and Pβ are the proportion of the total population of ‘up’ or ‘down’ spins respectively, ∆E is the energy gap between the ‘up’ and ‘down’ spins, k is the Boltzmann constant and T is the temperature in Kelvin (K).

As the ‘up’ spin will always have a slight population excess, there will be a net magnetic field along the direction of the external magnetic field, B0, which

defines the z-axis of the laboratory frame. This net magnetic field is represented by the net magnetization vector M.

In order to measure the specific resonant frequencies of the nuclei modern NMR spectrometers emit a high power radio frequency (RF) pulse, which if applied for an appropriate duration (typically a few µs) will rotate the net magnetization vector from the z-axis into the x-y plane. Once the magnetization has been rotated, M will precess in a cone as it returns towards the equilibrium position parallel with the z-axis. The precession occurs at the resonant frequency of the given nuclei, generating an oscillating voltage in the receiver coil of the NMR spectrometer’s probehead. This occurs for all nuclei of a given type (e.g. all 1_H

nuclei or all13_{C nuclei) simultaneously, subject to the bandwidth of the RF pulse}

being sufficiently wide. Since, in a typical sample the (e.g. 1_{H) nuclei rarely}

experience exactly the same external magnetic field (due to phenomena such as the chemical shift – see below) the resulting signal is therefore usually a sum of sine waves, known as a free induction decay (FID). The NMR signal captured as the FID is one of signal intensity vs. time. This can be converted to the standard NMR spectrum of intensity vs. frequency via a Fourier transform.