PH5462 – Magnetism in Solids
Lecture – 7
Magnetic Resonance
Compared to the nucleus the electronic properties are strongly dependent on their surroundings as they have much larger
spatial extent than the nucleus and they require much smaller energy to get excited.
Atoms and and molecules which have one or more unpaired electrons gives rise to Electron Spin Resonance (ESR). As the
unpaired electron gives rise to parmagnetism the technique is also called Electron Paramagnetic Resonance (EPR)
classes of substances or circumstances in which one may expect to find resonances
1. Materials containing atoms of the transition elements with incomplete inner shells; as, for example, the iron group or rare
earths.
2. Ordinary metals, the conduction electrons.
3. Ferro-- and ferrimagnets.
4. Imperfections in insulators, which may trap electrons or holes. For example, the F-center (electron trapped at the site of a
missing halogen ion in an alkali halide) or donor and acceptor sites of semiconductors.
The principal terms in the electron Hamiltonian will consist of:
1. The electron kinetic energy.
2. The electron potential energy. Often it is convenient to divide this into a "free ion" potential energy plus one due to the
crystalline surroundings, the so-called crystalline potential. Such a decomposition makes sense provided there is such a thing
as a "free ion", but, as remarked above, it would not have meaning for an F-center.
3. The spin-orbit coupling. An electron moving in an electric field E experiences a coupling of the spin to the orbital motion
4. The coupling of the electron spin and orbital magnetic moments to an externally applied magnetic field.
5. The magnetic coupling of the nuclear spin to the electronic spin and orbital moments.
6. The coupling of the nuclear electrical quadrupole moment to the electronic charge.
Spin-Orbit Coupling and Crystalline Fields
The role played by some of the phenomena mentioned above can be understood using an example.
Consider an atom at the origin of a set of coordinates, possessing a single p-electron. It is surrounded by two
positive and two negative charges of equal magnitude and at equidistant from the origin as shown in the figure.
𝑧
𝑦
−𝑞
+𝑞 +𝑞 𝑥
−𝑞
Neglecting nuclear coupling, we have for the Hamiltonian of the electron of charge 𝑞 (𝑞 negative),
1 𝑞 2
ℋ= 𝑝 − 𝐴 + 𝑉0 + 𝑉1 + 𝜆𝐿 ∙ 𝑆 + 2𝛽 𝐻 ∙ 𝑆
2𝑚 𝑐
Where, 𝐴 is the vector potential associated with the applied static magnetic field 𝐻, 𝑉0 is the potential of the "free
atom", 𝑉1 is the potential due to the four charges, 𝜆𝐿 ∙ 𝑆 represents the spin-orbit coupling and 2𝛽 𝐻 ∙ 𝑆 is the
contribution from the coupling of the electron spin moment to the external field. The symbol 𝛽 is used for Bhor
magneton to express the electron magnetic moment. It is related to 𝛾𝑒 the electron gyromagnetic ratio, and 𝜇𝑒 the
spin magnetic moment, by the equation.
𝜇𝑒 = −𝛾𝑒 ħ𝑆 = −2𝛽𝑆 𝑜𝑟 𝛾𝑒 ħ = −2𝛽
The negative sign indicates that the magnetic moment of electron is directed opposite to spin angular momentum.
Expanding the first term in the Hamiltonian and neglecting the diamagnetic contribution, we can rewrite the
Hamiltonian as,
𝑝2
ℋ= + 𝛽𝐻 ∙ 𝐿 + 𝑉0 + 𝑉1 + 𝜆𝐿 ∙ 𝑆 + 2𝛽 𝐻 ∙ 𝑆
2𝑚
𝑝2
The Kinetic energy term ( ) and the free atom potential term 𝑉0 gives the main contribution to the Hamiltonian
2𝑚
and the remaining terms can be treated by perturbation theory. Among the remaining terms the 𝐻 ∙ 𝐿 and 𝐻 ∙ 𝑆
terms are negligible for laboratory fields when compared to the other two terms.
Among the remaining two perturbation terms, the term 𝑉1 dominates in certain cases (transition metals) and Spin-
Orbit interaction dominates in some other materials (rare earths).
First let us consider the situation when 𝑉1 dominates. It will lift the orbital degeneracy. The resultant energy levels ar
shown in figure are two fold degenerate due to electron spin. We denote the wave functions as 𝑥𝑓(𝑟)𝑢𝑚 and so on.
where the function 𝑢𝑚 is a spin function.
𝑦𝑓(𝑟)𝑢𝑚 If there were no spin-orbit coupling. The spin would be quantized
independently of the orbital state so that the 𝑢𝑚 ’s would be the
𝑧𝑓(𝑟)𝑢𝑚
usual eigen functions of 𝑆𝑧 ′ , where z' is the magnetic field direction.
𝑥𝑓(𝑟)𝑢𝑚
We examine the sorts of matrix elements these terms possess. "There are two sorts: those that connect the same orbital state,
and those that connect different orbital states. The former are clearly the more important, if they exist, since the orbital splittings
are so large. We have, then. matrix elements such as
∗
𝑥𝑓(𝑟)𝑢𝑚 𝛽𝐻𝑧 𝐿𝑧 𝑥𝑓(𝑟)𝑢𝑚′ 𝑑𝜏𝑑𝜏𝑆
∗ 𝜆𝐿 𝑆 𝑥𝑓(𝑟)𝑢 ′ 𝑑𝜏𝑑𝜏
𝑥𝑓(𝑟)𝑢𝑚 𝑧 𝑧 𝑚 𝑆
where d𝜏 stands for an integral over spatial coordinates. and 𝑑𝜏𝑆 over spin variables. Both the integrals involve
𝑥𝑓(𝑟)𝐿𝑧 𝑥𝑓 𝑟 dτ
This integral vanishes. The only non vanishing matrix elements of terms 𝛽𝐻 ∙ 𝐿 𝑎𝑛𝑑 𝜆𝐿 ∙ 𝑆 connect states differing in the orbital
energy. They have therefore, no effect in first order.
The spin is coupled to states in which the electron has no preferential circulation. The average magnetic field due to orbital
motion seen by the spin vanishes.
But the 𝛽 𝐻 ∙ 𝐿 term will induce some orbital circulation. The spin will not therefore, experience a strictly zero field due to orbital
motion.
Now let us consider both the terms containing orbital angular momentum, 𝛽𝐻 ∙ 𝐿 + 𝜆𝐿 ∙ 𝑆. Neither of the two
terms give rise to first order contribution.
In second and higher orders they perturb the wave function. Though the spin-orbit term is stronger than the 𝛽 𝐻 ∙ 𝐿
term, a detailed analysis shows that the net effect depends on the interplay between the two energies.
The effective Hamiltonian is given by,
𝜆 𝜆
ℋ𝑒𝑓𝑓 = 2𝛽 𝑆 𝐻 + 𝑆 𝐻
𝐸𝑥 − 𝐸𝑧 𝑦 𝑦 𝐸𝑥 − 𝐸𝑦 𝑧 𝑧
Combining this with the Zeeman term 2𝛽 𝐻 ∙ 𝑆, we obtain a spin Hamiltonian for the ground orbital state,
ℋ = 𝛽 𝑔𝑥𝑥 𝐻𝑥 𝑆𝑥 + 𝑔𝑦𝑦 𝐻𝑦 𝑆𝑦 + 𝑔𝑧𝑧 𝐻𝑧 𝑆𝑧
𝜆 𝜆
Where, 𝑔𝑥𝑥 = 2 ; 𝑔𝑦𝑦 = 2 1 − ; 𝑔𝑧𝑧 = 2 1 −
𝐸𝑧 −𝐸𝑥 𝐸𝑦 −𝐸𝑥
The combined effect of the spin orbit coupling and orbital Zeeman energy is as though the real field 𝐻 were replaced by an
effective field 𝐻𝑒𝑓𝑓, given by
𝑔𝑥𝑥 𝑔𝑦𝑦 𝑔𝑧𝑧
ℋ𝑒𝑓𝑓 = 𝑖𝐻𝑥 + 𝑗𝐻𝑦 + 𝑘𝐻𝑧
2 2 2
With the resonance given by,
ℋ = 2𝛽 ℋ𝑒𝑓𝑓 ∙ 𝑆
Since 𝑔𝑥𝑥 , 𝑔𝑦𝑦 , and 𝑔𝑧𝑧 are in general different, the effective field differs from the actual field in both magnitude and direction. If
we denote by 𝑧" the direction of the effective field, it is clear that a coordinate transformation will change the above equation
into the form,
ℋ = 2𝛽ℋ𝑒𝑓𝑓 𝑆𝑧 ′′
where ℋ𝑒𝑓𝑓 is the magnitude of ℋ𝑒𝑓𝑓 . The resonant frequency 𝜔0 therefore satisfies the condition
ħ𝜔0 = 2𝛽ℋ𝑒𝑓𝑓
= 𝛽 𝐻𝑥2 𝑔𝑥𝑥
2
+ 𝐻𝑦2 𝑔𝑦𝑦
2
+ 𝐻𝑧2 𝑔𝑧𝑧
2
= 𝛽𝐻 𝛼12 𝑔𝑥𝑥
2
+ 𝛼22 𝑔𝑦𝑦
2
+ 𝛼32 𝑔𝑧𝑧
2
where 𝛼1 , 𝛼2 𝑎𝑛𝑑 𝛼3 are the cosines of the angle between 𝐻 and the 𝑥, 𝑦, and z axes.
Usually it is written as,
ħ𝜔0 = 𝑔𝛽𝐻
where the "𝑔-factor" is defined by the equation
2 2 2
𝑔= 𝑔𝑥𝑥 𝛼1 + 𝑔𝑦𝑦 𝛼22 + 𝑔𝑧𝑧
2 𝛼2
3
For a given orientation of 𝐻, the splitting of the spin states is directly proportional to the magnitude of 𝐻.
Frequently one talks about the 𝑔-shift, a term that refers to the difference between 𝑔 and the free spin value of 2.
Positive values of 𝜆 make 𝑔 less than or equal to 2, whereas negative 𝜆 values make 𝑔 greater than or equal to 2.
The positive 𝜆 values are associated with atomic shells which are less than half-full and negative 𝜆 values with those
more than half-full.
The 𝑔 shift increases with nuclear charge 𝑍.
Its magnitude also depends on the magnitude of the splitting to the excited states to which the orbital angular
momentum couples.
The g-shift arises because of the interplay between the spin-orbit and orbital Zeeman interactions. It is analogous to
the chemical shift that arises from the interplay between nuclear spin-electron orbit coupling and the electron
orbital Zeeman interaction.
In both cases the spin (electron or nuclear) experiences both the applied magnetic field and a sort of induced
magnetic field.
All such phenomena involving the interplay of two interactions can be viewed also as an application of a generalized
form of second-order perturbation theory.
Defining the perturbation term by 𝐻𝑝𝑒𝑟𝑡 by
𝐻𝑝𝑒𝑟𝑡 = 𝜆𝐿 ∙ 𝑆 + 𝛽𝐻 ∙ 𝐿
The perturbation effectively adds a term ℋ𝑛𝑒𝑤 to the Hamiltonian, which has matrix elements between states 0
and 0′
For the example system which we have considered these elements have the same orbital part 𝑥𝑓(𝑟) but may differ
in spin function.
0 ℋ𝑝𝑒𝑟𝑡 𝑛 𝑛 ℋ𝑛𝑒𝑤 0′
0 ℋ𝑛𝑒𝑤 0′ =
𝐸0 − 𝐸𝑛
𝑛
0 𝜆𝐿 ∙ 𝑆 𝑛 𝑛 𝛽𝐻 ∙ 𝐿 0′ 0 𝛽𝐻 ∙ 𝐿 𝑛 𝑛 𝜆𝐿 ∙ 𝑆 0′ 0 𝛽𝐻 ∙ 𝐿 𝑛 𝑛 𝛽 𝐻 ∙ 𝐿 0′
= + +
𝐸0 − 𝐸𝑛 𝐸0 − 𝐸𝑛 𝐸0 − 𝐸𝑛
𝑛
The first two terms on the right give the 𝑔-shift we have calculated.
The last two terms shift the two spin states equally. They do not, therefore, either produce a splitting of the doubly degenerate
ground state or contribute to the 𝑔-shift. (If the spin were greater than 1/2, however, such a term could give a splitting of the
ground spin state even when 𝐻 = 0.)
The last two terms are just what we should have had if either perturbation were present by itself. Our previous calculation of the
𝑔-shift did not give them because it treated the effect of one term of the perturbation 𝛽𝐻 ∙ 𝐿 on the other 𝜆𝐿 ∙ 𝑆.
In the example we have discussed so far, we have considered the crystalline potential 𝑉1 to be much larger than the spin-orbit
coupling constant 𝜆. As a result, the orbital angular momentum is largely quenched, and the g-value is very close to the spin-only
value of 2. This situation corresponds to the iron group atoms as well as to many electron and hole centers.
Now we will consider the case were the spin-orbit coupling is strong and crystal field is relatively much weaker as in rare earth
atoms. If the spin-orbit coupling is dominant, the situation is in first approximation similar to that of a free atom. The
Hamiltonian will be identical to that of free atom except for the additional term 𝑉1 .
𝑝2
ℋ= + 𝑉0 + 𝜆𝐿 ∙ 𝑆 + 𝛽𝐻 ∙ 𝐿 + 2𝛽𝐻 ∙ 𝑆 + 𝑉1
2𝑚
The total angular momentum, 𝐽 is given by ,𝐽 = 𝐿 + 𝑆
𝜆 2
𝜆𝐿 ∙ 𝑆 = 𝐽 − 𝐿2 − 𝑆 2
2
With eigen value,
𝜆
𝐸𝑆𝑂 = 𝐽 𝐽 + 1 − 𝐿 𝐿 + 1 − 𝑆(𝑆 + 1)
2
The spacing between the levels,
3
∆𝐸 = 𝐽𝜆 𝐽=
3 2
The energy levels are shown in figure for 𝐽 = 3
2 ∆𝐸 = 𝜆
2
1
𝐽=
2
We consider next the effect of 𝑉1 . Here it becomes convenient to assume a specific function. Assuming that the potential arises
from charges external to the atom, the potential in the region of the atom can be expressed as,
𝑉1 = 𝐶𝑙𝑚 𝑟 𝑖 𝑌𝑙𝑚
𝑙,𝑚
where the 𝑌𝑙𝑚 ’s are spherical harmonics and the 𝐶𝑙𝑚 ’s are constants.
For the four charge configuration we considered (see the figure in the 3rd page), it vanishes on the 𝑧-axis and changes sign if we
replace 𝑥 by 𝑦 and 𝑦 by −𝑥 (a coordinate rotation). It is a maximum on the 𝑥-axis and a minimum on the 𝑦-axis for a given
distance from the origin. The lowest 𝑙 in the series of (11.39) is clearly 𝑙 = 2. Of the five 𝑙 = 2 functions,
𝑥𝑦, 𝑥𝑧, 𝑦𝑧, 3𝑧 2 − 𝑟 2 , 𝑎𝑛𝑑 𝑥 2 − 𝑦 2 only the last is needed. We have, therefore, as an approximation, insofar as terms for 𝑙 > 2
are not required,
𝑉1 = 𝐴(𝑥 2 − 𝑦 2 )
Where, A is constant.
We have, then, to consider the effect of 𝑉1 on the states shown in the energy level diagram. Two sorts of matrix elements will be
important: those entirely within a given 𝐽, such as 𝐽𝑀𝐽 𝑉1 𝐽𝑀𝐽′ and those connecting the different 𝐽 states. The former will be
the more important because they connect degenerate states. We can compute the matrix elements internal to a given 𝐽 by
means of the Wigner-Eckart theorem.
𝐽𝑀𝐽 𝑉1 𝐽𝑀𝐽′ = 𝐴 𝐽𝑀𝐽 𝑥 2 − 𝑦 2 𝐽𝑀𝐽′ = 𝐶𝐽 𝐽𝑀𝐽 𝐽𝑥2 − 𝐽𝑦2 𝐽𝑀𝐽′
This is equivalent to our replacing 𝑉1 by the operator ℋ1 .
ℋ1 = 𝐶𝐽 𝐽𝑥2 − 𝐽𝑦2
Taking into account the effect of field and using Wigner-Eckart theorem, we can write,
𝐽𝑀𝐽 𝐿 + 2𝑆 𝐽𝑀𝐽′ = 𝑔𝐽 𝐽𝑀𝐽 𝐽 𝐽𝑀𝐽′
Where 𝑔𝐽 is the Lande 𝑔 factor
𝐽 𝐽 + 1 + 𝑆 𝑆 + 1 − 𝐿(𝐿 + 1)
𝑔𝐽 = 1 +
2𝐽(𝐽 + 1)
The effective Hamiltonian,
ℋ𝑒𝑓𝑓 = 𝐶𝐽 𝐽𝑥2 − 𝐽𝑦2 + 𝑔𝐽 𝛽𝐻 ∙ 𝐽
If the field is axial, then,
ℋ𝑒𝑓𝑓 = 𝐶𝐽′ 3𝐽𝑧2 − 𝐽2 + 𝑔𝐽 𝛽𝐻 ∙ 𝐽
The two terms on the right hand side lifts the 2𝐽 + 1 fold degeneracy of each 𝐽 state.
It is equivalent to the solution of the problem of a nucleus possessing a quadrupole moment acted on by an electric field
gradient and a static magnetic field
Hyperfine Structure
Hyperfine structure arises due to the interaction of electron with nearby nuclei. It is similar to the case considered in
the context of NMR. We have to distinguish between 𝑆 states and non 𝑆 states.
For 𝑆 states,
8𝜋
ℋ𝐼𝑆𝑟 = 𝛾𝑒 𝛾𝑛 ħ2 𝐼 ∙ 𝑆𝛿 𝑟
3
Non 𝑆 states, 3
𝛾𝑒 𝛾𝑛 ħ2 3 𝐼∙𝑟 𝑠∙𝑟
ℋ𝐼𝑆𝑟 = −𝐼∙𝑆
𝑟3 𝑟2
The possible transitions produced by an alternating field are found by considering the matrix elements of the
magnetic operator ℋ𝑚 (𝑡)
ℋ𝑚 𝑡 = 𝛾𝑒 ħ𝑆𝑥 − 𝛾𝑒 ħ𝐼𝑥 𝐻𝑥 𝑐𝑜𝑠 𝜔𝑡
between states. We find in this way that the 𝑆𝑧 part of ℋ𝑚 𝑡 connects states with ∆𝑚𝑠 = ±1, ∆𝑚𝐼 = 0, whereas
the 𝐼𝑧 part connects ∆𝑚𝐼 = ±1, ∆𝑚𝑠 = 0. We can consider these respectively to represent electron resonance and
nuclear resonance. The transitions are allowed only if 𝜔 satisfies the conservation of energy.
𝑔𝑧𝑧 𝛽𝐻0 + 𝐴𝑧 𝑚𝐼
𝜔𝑒 =
ħ
for electron resonance and for nuclear resonance,
𝐴 𝑧 𝑚𝑠
𝜔𝑛 = 𝛾𝑛 𝐻0 +
ħ
The effect of the hyperfine coupling on the electron resonance is seen to be equivalent to the addition of an extra magnetic field
proportional to the 𝑧-component of the nuclear spin.
Since the nucleus can take up only quantized orientation, the electron resonance is split into 2𝐼 + 𝐼 (equally spaced) lines. If the
nuclei have no preferential orientation, the lines corresponding to various values 𝑚𝐼 occur with equal probability.
Electron Nuclear Double Resonance(ENDOR)
After saturating the ESR
levels, the NMR transition is
induced to alter the 4
population in levels 2 and 3 NMR
to induce ESR transitions. 3
This will give sharp ESR lines.
The figure shows a system
with electron spin ½ and
nuclear spin 1/2 ESR Transitions
NMR
