## Walker group units of measures

### Magnetic resonance in SI units

#### Definitions

The energy of interaction of a nucleus in a diamagnetic molecule with the external magnetic field is called the Nuclear Zeeman Hamiltonian

The Hamiltonian is also often expressed in frequency units ( -units):

or in the units of angular frequency :

here

 is the nuclear magnetic moment, is the nuclear magneton, without c in denominator! is the dimensionless nuclear spin so that the angular moment is, is the gyromagnetic ratio g N is the nuclear g-factor (dimensionless),

Remark. Traditionally, in NMR spectroscopy g N  and  are considered as constants and all influence of a molecular surrounding is assigned to the shielding tensor  (often called the chemical shift tensor).

Pure electronic Zeeman Hamiltonian can be written as

Here

 is the electron magnetic moment, is the Bohr magneton, without c in denominator! me is the electron rest mass, is the dimensionless electron spin (so that the electron angular moment is ), is the electron g-tensor.

If the g-tensor is proportional to the identity matrix

then it is called isotropic (or scalar) and g is called the g-factor.

He Hamiltonian can be expressed in the frequency scale

here

is the free electron gyromagnetic ratio.

Remark. Traditionally, in EPR spectroscopy all dependence on molecular structure and its electronic state is assigned to the g-tensor.

For S > 1/2, so-called electron zero-field splitting can be important. It is written in the following conventional form

Total electronic and nuclear Hamiltonian is

is called the hyperfine interaction between a nucleus and unpaired electron spin. Here  is the tensor of the hyperfine interaction.

The HFI interaction has two contributions:

• Isotropic contact Fermi term

where   is the electronic density at the nucleus.

• Anisotropic dipole-dipole interaction

here   is the vector from a nucleus to electron in the molecular coordinate frame. In the above expressions the HFI tensor has dimension of energy. It can be also expressed in other related units:

 here is the hyperfine tensor in Hz and is the hyperfine tensor in cm -1.

#### The use of field units

For simplicity, assume that both g and A are scalar and the magnetic field is directed along the z-axis

The EPR transition energy for    is

There are 2 I + 1 lines in the spectrum corresponding to different projections of the nuclear spin. The distance between lines in the energy scale is A. The distance between lines in the frequency scale is

.

However, in EPR experiments the frequency is kept constant and the field is changed. The resonance occurs at the following field strength values:

The distance between lines in the magnetic field scale is

.

Combining with the above expression for the frequency scale, we get

or

#### Conversion of Units

The above relation means that Hertzs can be recalculated to Teslas and back. Similar relations can be established between various other scales. Note, that Hertz and Tesla, have different dimensions, so we cannot equate them. Below the sign <=> means "corresponds to".

 A [MHz] <=> 2.8024944 (g / ge) A [G] A [MHz] <=> 28.024944 (g / ge) A [mT] A [MHz] <=> 13.996241 g A [mT] A [MHz] <=> 2.99792458 e+4 A [1/cm] A [1/cm] <=> 0.3335641 e-4 A [MHz] A [1/cm] <=> 4.668643 e-4 g A [mT]

Some other useful relations

1 G  =  0.1 mT
1 T  =  10 kG
1 mT  =  10 G

ABC Tutorials | Data Processing | Indirect Measurements | NMR Tutorials