Nuclear magnetic resonance is an experimental technique of importance in molecular structure elucidation.

The nucleus of an atom might have a spin, or angular moment. Since a spinning charge generates a magnetic field, there is a magnetic moment associated with its moment (Also, the neutron has a magnetic moment). The magnitude of the nuclear magnetic moment is specified in terms of the magnetogyric ratio g = 2pm/Ih, where h is Planck’s constant, m is the magnetic dipole moment and I the spin quantum number. The permitted values of the angular moment are described by the magnetic quantum number m (m = I, (I-1), (I-2),….-I). The ^{1}H and ^{13}C nuclei, which are the two most important nuclei in PE, have spin I=˝, and therefore only these nuclei will be considered. (In deuteurated polymers ^{2}H, a spin n = 1 particle may give additional and relevant structure and dynamic information).

When a spin I=˝ nucleus is placed in a magnetic field, the spin will precess with or against the field direction (B_{0}). The energy of the nucleus depends on the spin direction as shown in Figure 1.

**Figure 1** *Energy levels of spin half nuclei.*

The following equation (Larmor equation) relates the precession frequency (w _{0}) to the magnetic field strength (B_{0});

(1)

The macroscopic equilibrium magnetization (M_{0}) along the external field, depend on strength of this field (B_{0}) and on the temperature (T) according to;

(2)

where N_{0 }is the total number of spins and k is Boltzmann constant. We define the direction of B_{0 }the z-axsis.

The nucleus is surrounded by electrons, which shields it from the influence of the external field. The field (B) seen by the nucleus will therefore be different from the external field (B = B_{0} (1-s )), where s defines the shielding tensor. Since s is sensitive to the electron environment of the nucleus observed, it will contain information about the molecular structure. The analysis of chemical shifts has proven to be very useful in solution NMR for identification purposes.

The precession of the magnetization around B_{0 }can not be observed directly. All NMR experiments consist of exciting the spins by a second field perpendicular to B_{0} and then observe the response to this excitation. This second field is located in a rotating frame, normal to B_{0 }and rotates with frequency w _{e}. The rotating field B_{1} gives the precessing spins coherence, and induces a detectable magnetic moment. In pulsed NMR the magnetization is tipped an angle a, which is proportional to the applied radio frequency pulse;

(3)

where t_{p} is the duration of the B_{1 }-field. The time behavior of the magnetization in the xy-plan is called the free induction decay (FID). If one applies a p/2 radio-frequency pulse the magnetization will rotate from the z-axis and into the y-axis as shown in Figure 2 a-c.

a) b) c)

**Figure 2** *The macroscopic magnetization a) before a rf-pulse, b) during the rf-pulse B _{1} in the rotating frame and c) after a p/2 pulse.*

After a p/2 rf-pulse the equilibrium magnetization M_{0} is located in the xy-plane (Figure 2 c) and starts to relax towards equilibrium along the y-axis (spin-spin relaxation or transverse relaxation) or along the B_{0} -field (Spin-lattice relaxation) according to equations 3 and 4;

(3)

where T_{2} is the spin-spin relaxation time. The growth along the z-axis is described by a time constant T_{1};

(4)

To obtain a quantitative NMR spectrum a time delay between succeessive pulses of the order of 5 times T_{1} must be applied. The signal M_{y }defines the Free- Induction Decay (FID).

Eq. 3 describes a situation where all spins are on resonance. If the sample contains groups of non-identical spins, the overall signal (FID) will be a super position of the contribution from the different groups. Since dipole – dipole interactions are the main reason for relaxation in PE it can be shown that T_{1} and T_{2} depend on the motion of the nuclei, as characterized by a correlation time (t _{c});^{ }

(5)

(6)

The correlation time is roughly defined as the time taken for a molecule to rotate one radian or move a distance of the order of its own dimension. Usually it is assumed that the correlation time depend on temperature according to an Arrhenius type of function;

(7)

where E is the energy barrier and n _{0 }is the attempt frequency. An increase in temperature means an increased mobility and hence a decrease in correlation time. It should be pointed out that Eq 6 breaks down for "rigid" lattice motion. The relaxation time dependence on correlation time is illustrated in Figure 3.

**Figure 3** *T _{2} and T_{1 }as a function of correlation time (t _{c}).*

Molecules characterized by long correlation times (crystalline phase of a polymer) will have a short T_{2} and long T_{1} (Eq. 5 and 6).

The dipole-dipole coupling between spins can induce flip-flop transition, and thus transfer magnetization from one part of the system to the other, and thereby affects the measured spin-lattice relaxation. This "transport" of magnetization is referred to as spin-diffusion.

Fourier transformation is a mathematical operation, which for instance transforms a time signal (FID) (f(t)) into a frequency spectrum (F(w )) according to;

(8)

If the rf-pulses used in NMR is perfectly rectangular and infinitely short, the response signal could be acquired immediately after the pulse. However, due to a finite duration of the rf-pulse, and pulse breakthrough, the receiver must be turned off during the pulse and a short time after the rf-pulse. The first part of the FID is therefore destroyed. The frequency spectrum obtained from Fourier transforming a FID which does not contain the initial data points will lead to a truncated frequency spectrum.

a) b)

**Figure 3.4** *a) The Free Induction Decay (FID) of PE and b) its corresponding frequency spectrum (after removing the initial data points of the FID). *

By Fourier transforming Eq 3. it can be shown that the line width (Dn ; full width at half-maximum) of the resonance line can be is expressed by T_{2} according to;

(9)