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Dipolar Interactions: Hyperfine Structure Interaction and Fine Structure Interactions

By Betül Çalişkan and Ali Cengiz Çalişkan

Submitted: October 10th 2019Reviewed: February 23rd 2020Published: March 25th 2020

DOI: 10.5772/intechopen.91791

Downloaded: 15

Abstract

The interaction between the nuclear spin and the electron spin creates a hyperfine structure. Hyperfine structure interaction occurs in paramagnetic structures with unpaired electrons. Therefore, hyperfine structure interaction is the most important of the fundamental parameters investigated by electron paramagnetic resonance (EPR) spectroscopy. For EPR spectroscopy the two effective Hamiltonian terms are the hyperfine structure interaction and the electronic Zeeman interaction. The hyperfine structure interaction has two types as isotropic and anisotropic hyperfine structure interactions. The zero-field splitting term (electronic quadrupole fine structure), the nuclear Zeeman term, and the nuclear quadrupole interaction term are among the Hamiltonian terms used in EPR. However, their effects are not as much as the term of the hyperfine structure interaction. The zero-field splitting term and the nuclear quadrupole interaction term are the fine structure terms. The interaction of two electron spins create a zero-field splitting, the interaction between the two nucleus spins form the nuclear quadrupole interaction. Hyperfine structure interaction, zero-field interaction, and nuclear quadrupole interaction are subclasses of dipolar interaction. Interaction tensors are available for all three interactions.

Keywords

  • dipolar interaction hyperfine structure
  • isotropic hyperfine structure
  • anisotropic hyperfine structure
  • the zero-field splitting
  • the nuclear quadrupole interaction
  • the electronic Zeeman interaction
  • the nuclear Zeeman term
  • EPR

1. Dipolar interactions

Dipolar interaction occurs due to the interaction between the two spins. If one spin becomes an electron spin and the other spin becomes a nucleus spin, this interaction is called a hyperfine structure interaction. If two of the spins are electron spin or both are nucleus spin, this interaction is called fine structure interaction. The dipolar interaction Hamiltonian is expressed as

ϰ=μ1.μ2r33μ1.rμ2.rr5E1

where μ1and μ2are the magnetic dipole moments for each spin (electron spin or nucleus spin).

1.1 Hyperfine structure interaction

The interaction between the magnetic dipole moment of the nucleus and the magnetic dipole moment of the electron gives the hyperfine structure interaction. There are two types of hyperfine structure interaction. These are isotropic hyperfine interaction and anisotropic hyperfine interaction.

1.1.1 Isotropic hyperfine structure

Isotropic superfine interaction is also known as Fermi contact interaction. The Hamiltonian term of isotropic hyperfine structure interaction is expressed as

ϰ=gegNβeβN8π3S.I.δrE2

where ge= g-value of the electron, gN= g-value of the nucleus, βe= Bohr magneton, βN= nuclear magneton, S=electron spin operator, I= nuclear spin operator, and δr= Dirac delta function for the distance between the electron and the nucleus.

In a shorter way, it is expressed as

ϰ=aS.IE3

The isotropic hyperfine constant is written as

a=8π3gegNβeβNδrE4

Here ais called the isotropic hyperfine constant, Sis the spin angular momentum of the electron, and Iis the spin angular momentum of the nucleus.

1.1.2 Anisotropic hyperfine structure

Anisotropic hyperfine interaction is also called dipolar interaction or dipole–dipole interaction. The Hamiltonian term of anisotropic hyperfine structure interaction is expressed as

ϰ=gegNβeβN3S.rI.rr5S.Ir3E5

More specifically, the expression of the anisotropic hyperfine interaction in the Cartesian coordinate is written as

ϰ=gegNβeβN3x2r2r5IxSx+3y2r2r5IySy+3z2r2r5IzSz+3xyr5IxSy+IySx+3yzr5IySz+IzSy+3xzr5IxSz+IzSxE6

In a shorter way, it is expressed as

ϰ=S.A0.IE7

where A0is called the anisotropic hyperfine coupling tensor. The tensor is expressed in two ways as diagonal elements and non-diagonal elements. The diagonal elements of the tensor is expressed as

Aii0=gegNβeβN3i2r2r5,i=x,y,zE8

The non-diagonal elements of the tensor is expressed as

Aij0=gegNβeβN3ijr5,i,j=x,y,zE9

The sum of the isotropic and anisotropic terms fully expresses the hyperfine structure interaction Hamiltonian and is expressed as

ϰ=aS.I+S.A0.I=S.A.IE10

where Ais the general hyperfine structure tensor.

Figure 1 shows the formation of the hyperfine structure splittings. Figure 2 shows the formation of an EPR spectrum due to the hyperfine structure splittings.

Figure 1.

The formation of the hyperfine structure splittings.

Figure 2.

The formation of an EPR spectrum due to the hyperfine structure splittings.

1.2 Fine structure interaction

The fine structure is seen in two ways. The first is the fine structure interaction between two electron spins. The second is the fine structure interaction between the two nucleus spins. The fine structure interaction between two electron spin is also referred to as zero-field interaction or zero-field splitting. The interaction between two nuclear spin is called nuclear quadrupole interaction.

1.2.1 Zero-field splitting (interaction)

This interaction between two electron spins is the dipolar interaction. When writing Hamiltonian for zero-field interaction, the magnetic dipole moments in Eq. (1) are arranged for two electron spins. In this case, the Hamiltonian of the zero-field splitting is written as

ϰ=ge2βe2S1.S2r33S1.rS2.rr5E11

More specifically, the expression of the anisotropic hyperfine interaction in the Cartesian coordinate is written as

x=ge2βe2r23x2r5S1xS2x+r23x2r5S1yS2y+r23z2r5S1ZS2Z3xyr5S1xS2y+S1yS2x3yzr5S1yS2z+S1zS2y3xzr5S1zS2x+S1xS2zE12

In a shorter way, it is expressed as

ϰ=S1.D.S2E13

In general, the Hamiltonian of the zero-field splitting is written as

ϰ=S.D.SE14

where Dis called the zero-field splitting tensor or the spin–spin coupling tensor. The tensor is expressed in two ways as diagonal elements and non-diagonal elements. The diagonal elements of the tensor is expressed as

Dii=ge2βe2r23i2r5,i=x,y,zE15

The non-diagonal elements of the tensor is expressed as.

Dij=ge2βe23ijr5,i,j=x,y,zE16

The zero-field splittings for s = 1/2, s = 1, and s = 3/2 are shown in Figure 3 .

Figure 3.

The zero-field splittings for (a) s = 1/2, (b) s = 1, and (c) s = 3/2.

1.2.2 Nuclear quadrupole interaction

The interaction between the nucleus spins is known as the nuclear quadrupole interaction. The effects of nuclear quadrupole interaction can be observed on the energy levels of the hyperfine structure for a nucleus with I1. The Hamiltonian of the nuclear quadrupole interaction is expressed as

ϰ=eQ6I2I1α,β=x,y,zVαβ32IαIβ+IβIαδαβI2E17

where Vαβis the component of the field gradient tensor and eQis the nuclear quadrupole moment, and it is a measure of the deviation of charge distribution from spherical symmetry. The nuclear quadrupole moment is expressed as

eQ=ρN3z2r2dVE18

where eis the proton charge,ρNis the distribution function of the nuclear charge, zis the z-coordinate of the charge element a distance rfrom the origin. The integral was taken over the volume of the nucleus.

In general, the nuclear quadrupole interaction Hamiltonian is written as

ϰ=I.P.IE19

where Pis called the nuclear quadrupole coupling tensor.

The nuclear quadrupole splittings are shown in Figure 4 .

Figure 4.

The nuclear quadrupole splittings for (a) Hquadrupole ≠ 0, H = 0 and (b) for HZeeman ≫ Hquadrupole.

2. Effective Hamiltonian terms in electron paramagnetic resonance spectroscopy

The hyperfine structure Hamiltonian term, electron Zeeman Hamiltonian term, nuclear Zeeman Hamiltonian term, the term of the zero-field splitting, and the term of the nuclear quadrupole interaction are Hamiltonian terms in EPR Spectroscopy. However, in EPR spectroscopy, the electron Zeeman term and the hyperfine structure term are effective Hamiltonian terms. Therefore, the effect of the terms other than the electron Zeeman term and the hyperfine structure term is not taken into account, since the effect is minimal compared to these two terms. The electron Zeeman term and the nuclear Zeeman term have not been mentioned before. Therefore, it will be explained briefly below.

The electron Zeeman interaction occurs as a result of the interaction of the magnetic dipole moment caused by the spin of the electron with the applied magnetic field:

ϰ=μs.HE20
ϰ=γsS.HE21

where γsis the gyromagnetic ratio of electron spin and is written as.

γs=gsβe=gsβein the atomic unit system=1E22

where gsis the spectroscopic splitting factor of the electron spin and is written as

gs=2E23
ϰ=gsβeS.HE24
ϰ=gsβeS.HE25

The nuclear Zeeman interaction occurs as a result of the interaction of the magnetic dipole moment caused by the spin of the nucleus with the applied magnetic field:

ϰ=μI.HE26
ϰ=γII.HE27

where γIis the nuclear gyromagnetic ratio and is written as.

γI=gIβN=gIβNin the atomic unit system=1E28

where gIis the spectroscopic splitting factor of the nucleus spin and is written as

gI=1E29
ϰ=gIβNI.HE30
ϰ=gIβNI.HE31

The general spin Hamiltonian for EPR spectroscopy can be written as

ϰ=gsβeS.H+S.A.IgIβNI.H+I.P.I+S.D.SE32

The effective spin Hamiltonian for EPR spectroscopy can be written as [1, 2, 3, 4, 5, 6, 7, 8, 9].

ϰ=gsβeS.H+S.A.IE33

3. Conclusion

Dipolar interaction can be seen in three ways. These are the hyperfine structure interaction, the zero-field splitting interaction, and the nuclear quadrupole interaction. Each interaction involves the interaction of two spins. The interaction between a nucleus spin and an electron spin is mentioned in the hyperfine structure interaction. The interaction of two electron spins is mentioned in the zero-field splitting interaction. The interaction of two nuclear spins is mentioned in the nuclear quadrupole interaction. The last two interactions are also known as fine structure interactions.

The hyperfine structure interaction is an important interaction for EPR spectroscopy. In EPR spectroscopy, the effect of the hyperfine structure interaction is taken into account together with the electron Zeeman interaction [10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24]. In addition, nuclear Zeeman interaction, the zero-field interaction, and the nuclear quadrupole interaction have an effect on EPR spectroscopy. However, their effects are negligible.

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Betül Çalişkan and Ali Cengiz Çalişkan (March 25th 2020). Dipolar Interactions: Hyperfine Structure Interaction and Fine Structure Interactions [Online First], IntechOpen, DOI: 10.5772/intechopen.91791. Available from:

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