MODULE 159-1(ESR)

I. Given:

(a) System == Unpaired Electrons

(b)Spin Angular Momentum S = ½ in units of :

(c) The objective of Lecture is to determine the magnitude of the splitting of energy levels in magnetic field by the interaction between magnetic dipoles and an externally applied magnetic field Bz.

II. What the students must know:

The background knowledge for the lecture is freshman physics and some physical chemistry if you did not get it there. The volume magnetic susceptibility X is a dimensionless tensor which gives magnetization M induced by a magnetic field of strength H:

M = X. H, (P.1)

where Xm = Vm X is called the molar magnetic susceptibility. When the field changes by a finite amount in say the z direction z, an energy level, identified by the quantum numbers represented by (i) changes as:

W(i) = W0(i) + W1(i) z+ W2(i) z2 +.., (P.2)

the value of the magnetic moment in the direction of the chage is:

Mz (i)= - = - W1(i) -2 W2(i) z +.., (P.3)

the molal susceptibility is then:

Xm = N0 (P.4).

The magnetic flux density B is related to the applied magnetic field by:

B = (H + M)= H.(1 + X), (P.5)

where = 4 E-7 J s2/C2/m is called the vacuum permeability. The plots of 1/X versus T identify several cases:

  1. M increases the magnitude of H (i.e., X>0) and the material is said to be paramagnetic. The paramgnetic susceptibility is given by the Curie-Weiss Law: Xm = Constant /(T- ) where vanishes for isotropic substances.
  2. M decreases the magnitude of H (i.e., X<0) and the material is said to be diamagnetic.
  3. Ferromagnetic materials (e.g., iron) undergo spontaneous macroscopic domain magnetization by the alignment of spins below a critical temperature =Tc called the Curie temperature, e.g., Xm = Constant /(T- ), T> Tc>0 with =1043 K for iron, 627 for Ni, 292 K for Gd, and 69 K for EuO.
  4. Ferrimagnetism and antiferromagnetism are cases of macroscopic spontaneous magnetizations. In antiferomagnets the spins are aligned antiparallel in domains and ferrimagnetism is a mixture of parrallel and antiparallel alignments that do not cancel completely. The sign of determines the property, Xm = Constant /(T+TN ), T> TN for antiferromagnetic materials.

The interaction energy between a magnetic moment (do not confuse with the permeability of vacuum) and an external flux field Bz is:

Wmagnetic =-M.H =

-e. g. Bz = -g e Bz cos (^ Bz), (P.6)

where for a single free electron the moment is:

e=- ge S B,

B == = |e|/2me is called the Bohr magneton, me is the electron mass, ge is the g-factor, (^ Bz) is the angle between the molecule's moment and the external field. The g factor was first introduced semi-empirically, then it was justified theoretically. Any particle possessing spin the obeys relation (P.6), i.e., an electron, a proton, etc.. with:
, (P.7)

where,


is called the magnetogyric ratio, mF is the projection of the total the spin quantum number Fi along the direction of the magnetic field, mi is the mass of the particle and qi its charge.

III.

For a free electron, F = S = ½, the energy levels in a magnetic field are obtained by applying relation (P.6):

E+(Sz=1/2) = ge Sz Hz , (1.1+)
E-(Sz=-1/2) = - ge Sz Hz, (1.1-)

The energy difference between the above levels is said to be the splitting in a magnetic field. Now the field Hz seen by the spins may be different from that applied in the laboratory. This is because there are other molecules surrounding the spin and they produce an inductive field, i.e.,

H.S = B.(1 + ).S, (1.2)

where is called a shielding tensor. It is more important in nuclear magnetic interactions than for free electrons. It becomes important again in solids which are antiferromagnetic or ferromagnetic. In electron spin resonance then the shielding is incorporated into the g-tensor, i.e.,

E+(Sz=1/2) = S .ge. Bz , (1.3+)
E-(Sz=-1/2) = S .ge. Bz, (1.3-)

and the energy separation between the corresponding levels becomes:

, (1.4)

The Bohr condition for inducing transitions between the states requires that an energy equal to the separation be applied, i.e., if w is the frequency of an applied external oscillation field:

Bx == 2 B1 cos (wt), (1.5)

in a direction normal to the field Bz. Transitions can be induced when conservation of energy and momentum is achieved by the radiation photons and the system, i.e.,

, (1.6)

IV.

There is a lot more to actually understanding the measurement of an absorption spectrum. This involves quantum mechanics at a level already learned in a course in physical chemistry. But at first we must notice that the ratio of Bohr magneton to the nuclear magneton absolute values varies inversely proportional to their masses, i.e., the splitting of free electron spin states in a magnetic field is greater than that for protons because the g-factors are of the same order of magnitude but:

|e|/|proton| == mp/me = 1838.282000(37)

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