Tabish September 2003 Aim: To determine the Land´ g-factor using Electron Spin Resonance. e Apparatus: ESR setup which includes Helmholtz coils, R.F. oscillator and the test sample, and in addition, a cathode ray oscilloscope (CRO).

Theory

Background

Suppose a particle having a magnetic moment µ is placed in a uniform magnetic ﬁeld of intensity B, then the Hamiltonian can be written as ˆ H=g e ˆ J · B, 2mc

where g is the Land´ g-factor, which is 1 for orbital angular momentum, and 2 for spin angular e e¯ h momentum. The factor 2mc , sometimes written as µB , is called Bohr magneton, if the particle in question is an electron. If the particle is a nucleon, then the factor is called the nuclear magneton. If the angular momentum J results from a combination of an orbital angular momentum and a spin, then g would be given by the Land´ formula: e g =1+ j(j + 1) + s(s + 1) − l(l + 1) , 2j(j + 1)

where l, s and j represent the magnitude of the orbital, the spin and the total angular momenta, respectively. Remember that j can go from l − s to l + s. Conventionally, the static magnetic ﬁeld is assumed to be pointing along the z−axis, which modiﬁes the above equation to e ˆ ˆ Jz B. H=g 2mc Let us now consider an atom which has an electronic ground state with total angular momentum j = 1/2 and an excited state with j = 3/2 (see ﬁgure 2).There is only a single transition which can be induced by the absorption of radiation of frequency ω12 = (E2 − E1 )/¯ . As the energy does not depend h on the angular momentum states, the ground state is doubly degenerate corresponding to eigenvalues ±1/2 ˆ of Jz and the excited state is quadruply degenerate corresponding to eigenvalues +3/2, 1/2, −1/2, −3/2 of ˆ Jz . 1

Electronic excited state Electronic transition

j=3/2 ESR

Electronic ground state

j=1/2

ESR

Zeeman effect

If one now applies a magnetic ﬁeld B along the z-axis, each of the angular momentum states acquires a diﬀerent energy. The ground state energy level thus splits into two sublevels and the excited state level into four sublevels. This is called Zeeman splitting. Now instead of a single transition of frequency ω12 = (E2 − E1 )/¯ , many transitions of frequencies close to ω12 h are possible. Experimentally this is seen as a splitting a single absorption or emission line into several closely spaced lines. This is called Zeeman eﬀect. As one would have noticed, transition should also be possible between the sublevels of the same energy level. It is indeed possible and this phenomenon is known as electron spin resonance (ESR).

Electron Spin Resonance

Let us try to understand the phenomenon of ESR in somewhat more detail. As ESR invloves transitions only between the sublevels of one energy level, we will not bother about the Hamiltonian of the atom/molecule which gives us the energy levels. We will only worry about the part of the Hamiltonian which is the result of the applied magnetic ﬁeld B, which gives us the sublevels. For simplicity, we will consider one electron with angular momentum j, in a magnetic ﬁeld B. In addition we have an electromagnetic ﬁeld of frequency ω in the direction perpendicular to B. The time-dependent Hamiltonian can thus be written as ˆ H=g eB ˆ ˆ ˆ Jz + V0 eiωt + V0† e−iωt , 2mc

ˆ where V0 represents the interaction of the electromagnetic ﬁeld with the electron. The electromagnetic ﬁeld is supposed to be very weak compared to the applied static ﬁeld B, and so one can use time-dependent perturbation theory to study this problem. The states ˆ that we will use are the eigenstates of Jz : ˆ Jz |m = hm|m , ¯ where m will take 2j + 1 values, from −j to +j. The energy of these levels is given by g where n

eB ˆ Jz |n = 2mc

n |n

,

=

geB¯ n h 2mc

= gBµB n.

In time-dependent perturbation theory, we know that the time-dependent interaction can cause transition between various |m states. The transition rate per unit time, from i th level to...