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# Electron spin resonance

 Contents

## Overview

Electron Paramagnetic Resonance (EPR) or Electron Spin Resonance (ESR) is a spectroscopic technique which detects species that have unpaired electrons, generally meaning that it must be a free radical, if it is an organic molecule, or that it has transition metal ions if it is a inorganic complex. Because most stable molecules have a closed-shell configuration without a suitable unpaired spin, the technique is less widely used than nuclear magnetic resonance (NMR).

The basic physical concepts of the technique are analogous to those of NMR, but instead of the spins of the atom's nuclei, electron spins are excited. Because of the difference in mass between nuclei and electrons, weaker magnetic fields and higher frequencies are used, compared to NMR. For electrons in a magnetic field of 0.3 tesla, spin resonance occurs at around 10 GHz.

EPR is used in solid-state physics, for the identification and quantification of radicals (i.e., molecules with unpaired electrons), in chemistry, to identify reaction pathways, as well as in biology and medicine for tagging biological spin probes.

Since radicals are very reactive, they do not normally occur in high concentrations in biological environments. With the help of specially designed nonreactive radical molecules that attach to specific sites in a biological cell, it is possible to obtain information on the environment of these so-called spin-label or spin-probe molecules.

To detect some subtle details of some systems, high-field-high-frequency electron spin resonance spectroscopy is required. While ESR is affordable for a medium-sized academic laboratory, there are few scientific centers in the world offering high-field-high-frequency electron spin resonance spectroscopy, among them ILL in Grenoble, France and one in Tallahassee, USA.

## EPR theory

### Units and constants

Magnetic Field is described by some constants and units:

• Magnetic induction $\bar{B}$ in teslas (T)
• Magnetic flux density $\bar{H}$ in amperes per metre (A/m)
• $\bar{B}$ and $\bar{H}$ relationship:

$\bar{B}=\mu_0\bar{H}$

Furthermore, in describing EPR, following units are very important:

• Planck's constant $h=6.63\times 10^{-34} \mbox{ J} \mbox{ s}$
• Boltzmann constant $k=1.38\times 10^{-23} \mbox{ J} \mbox{ K}^{-1}$
• Bohr magneton $\mu_B=9.27\times 10^{-24} \mbox{ J} \mbox{ T}^{-1}$

### Basics

EPR is based on Zeeman effect, which depends on fision of energetic levels in paramagnetic molecules lies in variable magnetic field. After induction a molecule with magnetic moment μ in magnetic field $\bar{B}$ it gains energy E.

$E=-\mu\times\bar{B}$

For the atom with the completely impact moment $\bar{J}$ the completely dipole magnetic moment $\bar{\mu}=-g\beta\bar{J}$, where g - spectroscopic fision factor, β - Bohr magneton. After placed into magnetic field the atom gains energy:

$E_J=g\beta\bar{J}\times\bar{B}$

The $\bar{J}$ vector of any atom can possess only some allowed orientations determined by MJ quantum number values collection. It is the reason why energy linked with orientation of atom is limited to MJ dependent collection.

$E_J=g\times M_J\beta B$

where: MJ=J, J-1, J-2,... -J.
In the external magnetic field the fision of single J level for 2J+1 sublevels is declined, and this phenomenon is called Zeemanic fission.
When constant magnetic field $\bar{B}$is added to variable magnetic field $\bar{B_1}$ with ν frequency, in the probe moves between zeemanic levels can be reached. This movement is depending on:

• $\bar{B_1}\perp\bar{B}$
• ΔMJ = 1 only neighbouring levels movement are avalible
• hν = gβB condition of energetic fitting

In practice single paramagnetic probe never occures but only population of probes with many paramagnetic centers. If this configuration of probes is in thermic equlibrium, statistical placeing is described by Boltzmann distribution.

### Boltzman distribution

$\frac{n_{M_J+1}}{n_{M_J}}$= exp$\left ( -\frac{E_{M_J+1}-{E_{M_J}}}{kT} \right )$=exp $\left (-\frac{\Delta E}{kT}\right )$

where $n_{M_J+1}$ - number of probes on $n_{M_J}$level
k - Boltzmann constant
T - temperature in kelvins

For X-band (ν = 10GHz) and room temerature, $\frac{n_{M_J+1}}{n_{M_J}}$= 0.998, because of the lower level has more electrons than higher one and transitions from lower to higher level are more probable.
The fundamental equation in EPR theory is:

$h\nu = g\times\beta\times B$

Observations of EPR signal can be leaded by resonance energy absorption measurements depending on electromagnetic radiation frequency ν in constant external magnetic field induction. Otherwise measurements can be provided inversely by changing of magnetic field B in constant ν frequency . Due to technical considerations, second way is more comfortable.
In the case of clear spin magnetism (g = 2.0) and ν frequency, energy absorption occurs in magnetic field induction:

$B = \frac{\nu [Hz]}{2.8\times 10^{10}} [T]$

For the common used frequency ν = 9.5GHz (X band of microwaves), resonance can be occurred for B= 0.34 T (3400 Gs).

## EPR spectrum parameters

### Spectral fission factor g

Knowledge of g parameter gives us information about paramagnetic center electron structure. For free radicals and solid or liquid state ions, orbital electron moment is hard-linked with environment and cannot occures in whichever arrangement in magnetic field. This is due to orbital magnetic moment freezing phenomena in cristal network. Due to this phenomena g parameter of EPR signal for many paramagnetic atoms in condensate state has different value (g = 2.0023). In the case of free ion, the g parameter has isotropic properties. In crystal g value is depended on external magnetic field vector.

### Resonance linewidth definition

Resonance linewidths are defined in magnetic induction units B and are measured along the x axis, from line center to y value crossing chosen point of spectrum. These defined widths are called halfwidths and possess some advantages: for asymmetric lines values of left and right halfwidth can be given. Halfwidth ΔBh is distance measured from center of line to the point in which absorption value has half of maximal absorption value in the center of resonance line. First inclination width $\Delta B_{\frac{1}{2}}$ is a distance from center of the line to the point of maximal absorption curve inclination. In a practical approach, full definition of linewidth is used. In the case of symmetric lines, halfwidth $\Delta B_{\frac{1}{2}} = 2\Delta B_h$, and full inclination width ΔBmax = 2ΔB1s