Enzo Martinelli

Rabi resonance method is a technique developed by Isidor Isaac Rabi for measuring the nuclear spin. The atom is placed in a static magnetic field and a perpendicular rotating magnetic field.

We present a classical treatment in here.

Theory

when only the static magnetic field (B₀) is turned on. the spin will be precess around it with Larmor frequency ν₀ and the corresponding angular frequency is ω₀.

According to mechanics, the equation of motion of the spin J is:

$\frac{d\overrightarrow{J}}{dt}=\overrightarrow{\mu }\times \overrightarrow{B}$

$\overrightarrow{\mu }=g\frac{{\mu }_B}{\hslash }\overrightarrow{J}=\gamma \overrightarrow{J}$

where μ is the magnetic moment.

g is g-factor, which is dimensionless and reflecting the environmental effect on the spin.

Solving gives the angular frequency (Larmor frequency) with the magnetic field pointing on z-axis:

${\omega }_0=-\frac{\gamma }{\hslash }{B}_0$

The minus sign is necessary. it reflects that the J is rotating in left-hand when the thumb is pointing as the H field.

when turned on the rotating magnetic field (B_R), with angular frequency ω. In the rotating frame of the rotating field, the equation of motion is:

$\frac{d\overrightarrow{J_R}}{dt}=\frac{d\overrightarrow{J}}{dt}-\overrightarrow{J}\times \omega$

or

$\frac{d\overrightarrow{J_R}}{dt}=\frac{\gamma }{\hslash }\overrightarrow{J}\times (\overrightarrow{B_0}+\overrightarrow{B_R})-\overrightarrow{J}\times \omega$

if $\frac{\gamma }{\hslash }{B}_0=\omega$, the static field was cancelled, and only the "sleeping" rotation frame. and the spin is now precess around H_R with angular frequency Rabi frequency

${\omega }_R=\frac{\gamma }{\hslash }{H}_R$

Since the rotating field is perpendicular to the static field. the spin in rotating fame is now able to flip between up and down.

by sweeping ω_R, one can obtain a maximum flipping and determine the magnetic moment.

Experiment

the experiment setup contains 3 parts, an inhomogeneous magnetic field in front, the rotating field at the middle, and another inhomogeneous magnetic field at the end.

atoms after passed the first inhomogeneous field will split into 2 beams corresponding the spin up and spin down state. selecting 1 beams (spin up state, for example) and let it pass the rotating field. If the rotating field has frequency (ω) equal to the Larmor frequency, it will produced a high intensity of the other beam (spin down state). by sweeping the frequency to obtain a maximum intensity, one can find out the Larmor frequency and the magnetic moment of the atom.

References and notes

http://www.colorado.edu/physics/phys7550/phys7550_sp07/extras/Ramsey90_RMP.pdf

See also

• Rabi frequency
• Rabi cycle
• Rabi problem