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:

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

${\vec {\mu }}=g{\frac {\mu _{B}}{\hbar }}{\vec {J}}=\gamma {\vec {J}}$

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:

$\omega _{0}=-{\frac {\gamma }{\hbar }}B_{0}$

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:

${\frac {d{\vec {J_{R}}}}{dt}}={\frac {d{\vec {J}}}{dt}}-{\vec {J}}\times \omega$

or

${\frac {d{\vec {J_{R}}}}{dt}}={\frac {\gamma }{\hbar }}{\vec {J}}\times ({\vec {B_{0}}}+{\vec {B_{R}}})-{\vec {J}}\times \omega$

if ${\frac {\gamma }{\hbar }}B_{0}=\omega$, 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

$\omega _{R}={\frac {\gamma }{\hbar }}H_{R}$

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