by Nikolai V. Shokhirev

Prof. F. Ann Walker Research Group, Department of Chemistry, University of Arizona, Tucson, Arizona 85721, USA

## 1D NMR Basics

## Contents

- Introduction
- Bloch Equation
- Free Precession
- Numeric Experiments
- Fourier spectroscopy
- Continuous Wave Spectroscopy
- "Rotating" Frame
- Stationary Solution
- Saturation Effect
- Adiabatic Passage
- Multi-nuclei NMR spectra
- Pulse vs. CW Spectroscopy
## 2D NMR Basics

## Chemical Exchange

## 1D NMR with Chemical Exchange

## Programs

Nuclear Magnetic Resonance phenomenon is determined by the interaction of magnetic nuclei with the external magnetic field and surrounding media. The nuclei are treated as small magnets with magnetic moments . The energy of the interaction (also called the Hamiltonian) is

The interaction between nuclei is also taken into account where it is necessary.

In general, the interaction of magnetic moment with external magnetic field is called Zeeman interaction (named after Dutch physicist Pieter Zeeman (1865-1943))

In the above formula
is a nucleus spin vector,
is a conversion factor (the gyromagnetic or magnetogyric ratio).
The small (usually of the order of 10^{-6})
correction reflects
the interaction of the nucleus with the rest of a molecule. It is called a chemical shift.
The chemical shift is different for different molecules and for different position of the atom in a molecule. The information about chemical shifts allows specification of a molecule (chemical composition, structure, interactions and processes). Namely this stimulated initial and still the main interest to the NMR spectroscopy.

The above Hamiltonian and the interaction with bulk media determines the spin dynamics, which can be detected in NMR experiments.

Felix Bloch (Swiss-born American physicist (1905-1983)) obtained his famous equation combining magnetic moment dynamics with the simplest variant of interaction with media:

The energy exchange with the media (relaxation) is described in terms of the following relaxation matrix:

The equilibrium magnetization is

Resolving the he vector product we get:

The Bloch equation does not reflect all peculiarities of NMR spectroscopy but is very useful both in continuous wave and pulse experiments.

In this case the external magnetic field has only a static component (along the axis z), the Bloch equation reduces to the following

Here is the Larmor frequency. This frequency is named after the British scientist Sir Joseph Larmor (1857-1942).

The solution of this equation is

It describes the rotation of the transversal magnetization in the xy-plane. Its absolute value decays as . The longitudinal (z-component) magnetization exponentially approaches the equilibrium value with the decay time .

Equating the real and imaginary parts we get for the x and y components

Note, that only the longitudinal relaxation determines energy dissipation processes

At this point you can download the program Bloch and experiment with magnetization precession.

Each 1D NMR experiment consists of two sections: preparation and detection. During preparation the spin system is set to a defined state. It is usually done by applying short resonance RF pulses. During detection the resulting signal is recorded.

The signal decays due to the T_{2} relaxation and is therefore called free induction decay (FID).
The equilibrium magnetization is directed along the external magnetic field (z-axis).
Usually the preparation is a short RF-pulse,
which rotates the initial magnetization M_{z} onto the xy-plane. This pulse is called 90° or
pulse. In
addition the rotation axis can be indicated (e.g. x-pulse).

The time-dependent signal can be converted into the frequency domain by the Fourier transformation

The initial signal can be restored from the amplitudes at specific frequencies by the invert Fourier transformation:

The intensity of the detected signal is

It has the highest value at (resonance) :

NMR Spectrum |

This technique is also called **Pulse NMR Spectroscopy**

In this case the external magnetic field has the static component and the rotating field :

The field
is **continuously** applied. The Bloch equation reduces to the following

where is the Larmor frequency, the rotation frequency of free (without relaxation, and rotating field) magnetic nuclei. Here is also introduced . This is the rotating field amplitude converted in a frequency scale by the factor .

The first two equations can be combined into one for a complex magnetization

where .

It is convenient to present the transversal (x and y) magnetization in the following form

This transformation can be interpreted in two ways:

- Mathematical. This is the separation of rotating (usually relatively fast) and slow-changing dependencies.
- Physical. The are the coordinates of magnetization in the rotating (with magnetic field) coordinate frame. It is also assumed that (rotation around the z-axis).

Rotation frame approach seems to be more pictorial and is often used in the description of magnetic spectroscopy. However it can be confusing in the case of multi-frequency experiments (different frames rotate with different rates) or when other processes are involved (e.g. chemical reactions) because there is no material rotation. Actually both approaches are equivalent.

The advantage of the equations for the primed variables (or for magnetization in the rotating coordinate system) is that they do not contain the oscillation dependencies:

After a long (compared with relaxation) time, the x and y components of magnetization will rotate with external field frequency (or and the z component will reach it stationary value.

In the other words, and their time derivatives are equal to zero.

This is equivalent to the system of three real equations:

The solution of the above equations is

Returning to the original variables (or back to the laboratory system) we get

The z component remains the same.

The negative
means delay in rotation of the transversal magnetization from the rotation of the magnetic field. The magnetization rotates behind the magnetic field due to "friction" or dissipation of energy. In NMR spectroscopy it means the absorption of irradiated electromagnetic field. Hence, the dependence of
on frequency represents an NMR spectrum.

The highest value of magnetization is at
:

The maximal value is

At small irradiation amplitude , the spectrum intensity is proportional to the amplitude
. It reaches its maximum value at and decreases with further increase of the amplitude.

This phenomenon is known as a saturation effect (energy pumping rate exceeds system ability of absorbing and dissipating the energy). It also causes the broadening of a spectrum. The width at half-height is expressed as

For example, at line width is 41% larger its initial value. The saturation factor should be made as small as possible in order to achieve better resolution (narrow sharp lines).

The above expressions are derived under the assumption that all relaxation already happened. It means that the recording of spectra (changing the frequency) should be slow enough. In spectroscopy it is called slow adiabatic passage.

In this case the Hamiltonian consists of the Zeeman interactions of nuclei and interactions between nuclear spins.

We will not discuss such spectra in our Basic Tutorials.

In the case of two nuclei without interaction the spectrum consists of two lines:

1D two-line spectrum |

The two 1D techniques are essentially the same because they probe the same Hamiltonian. The differences are in the method of excitation of a spin system, in detection its response and the processing of the signal. However the advantages and disadvantages of each method depends on technical implementation. The pulse approach allows easier development of multi dimension spectroscopy and NMR imaging.

©Nikolai V. Shokhirev, F. Ann Walker, 2001-2007