Rabi method

The Rabi method is one of Isidor Isaac Rabi developed technique for the measurement of the nuclear spins . The atom is in a static magnetic field , perpendicular to it a rotating magnetic field is applied.

theory

According to classical mechanics, the equation of motion of the spin in a magnetic field is : ${\displaystyle {\vec {J}}}$ ${\displaystyle {\vec {B}}}$

${\displaystyle {\frac {\mathrm {d} {\vec {J}}}{\mathrm {d} t}}={\vec {\mu }}\times {\vec {B}}}$

With

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

in which

• ${\displaystyle {\vec {\mu }}}$: the magnetic moment
• ${\displaystyle \mu _{\mathrm {B} }}$: Bohr's magneton
• ${\displaystyle g}$: the dimensionless Landé factor , which reflects environmental effects on the spin
• ${\displaystyle \hbar }$: the reduced Planck quantum of action
• ${\displaystyle \gamma }$: the gyromagnetic ratio .

If only the static magnetic field is switched on, the spin precesses with the Larmor frequency or the corresponding angular velocity : ${\displaystyle B_{0}}$ ${\displaystyle f_{0}}$ ${\displaystyle \omega _{0}}$

${\displaystyle \omega _{0}=-\,\gamma B_{0}.}$

The negative sign is necessary. It means that the spin rotates left-handed when the thumb points in the direction of the magnetic field. ${\displaystyle J}$

If one in which the angular velocity of rotating reference frame of the rotating magnetic field changes, one obtains the transformed equation of motion: ${\displaystyle \omega }$ ${\displaystyle B_{\mathrm {R} }}$

${\displaystyle {\frac {\mathrm {d} {\vec {J}}_{\mathrm {R} }}{\mathrm {d} t}}={\frac {\mathrm {d} {\vec {J}}}{\mathrm {d} t}}-{\vec {J}}\times {\vec {\omega }}}$

or

${\displaystyle {\frac {\mathrm {d} {\vec {J}}_{\mathrm {R} }}{\mathrm {d} t}}=\gamma \,{\vec {J}}\times ({\vec {B}}_{0}+{\vec {B}}_{\mathrm {R} })-{\vec {J}}\times {\vec {\omega }}.}$

If so , then the precession of the static field is canceled, the spin precesses in the rotating reference system only around the axis of the magnetic field : ${\displaystyle \omega =\gamma B_{0}}$ ${\displaystyle B_{\mathrm {R} }}$

${\displaystyle {\frac {\mathrm {d} {\vec {J}}_{\mathrm {R} }}{\mathrm {d} t}}=\gamma \,{\vec {J}}\times {\vec {B}}_{\mathrm {R} }.}$

The angular velocity ( Rabi frequency ) of this precession is:

${\displaystyle \omega _{\mathrm {R} }=\gamma B_{\mathrm {R} }}$

${\displaystyle \Rightarrow {\frac {\mathrm {d} {\vec {J}}_{\mathrm {R} }}{\mathrm {d} t}}={\vec {J}}\times {\vec {\omega }}_{\mathrm {R} }.}$

Since the rotating field is perpendicular to the static field, the spin can jump back and forth between spin-up and spin-down . The maximum of the spin changes can be determined by scanning the angular velocity . ${\displaystyle \omega _{\mathrm {R} }}$

experiment

The experimental setup consists of three parts:

• an inhomogeneous magnetic field at the front
• in the middle a homogeneous magnetic field with a perpendicular rotating magnetic field and
• behind another inhomogeneous magnetic field.

The two inhomogeneous magnetic fields serve as selectors . The first ( polarizer ) splits the atomic beam into two beams with different spin polarization . One of the two beams (e.g. spin-up ) is selected and guided through the middle part of the experimental setup. If the rotating magnetic field there has the Larmor frequency , a high radiation density with reverse spin orientation (e.g. spin-down ) is generated. This is selected with the rear magnetic field ( analyzer ). By scanning the frequency, the maximum intensity of this beam is selected. With it one finds the Larmor frequency and thus the magnetic moment of the atom. ${\displaystyle f_{0}}$

application

The magnetic moment of an atom depends on its environment; the Landé factor changes with its bond in a molecule . This is the structural composition of a molecule aufklärbar . ${\displaystyle \gamma }$

credentials

1. ↑ Nuclear magnetic resonance NMR (PDF file; 433 kB)

literature

Experiments with separated oscillatory fields and hydrogen masers (PDF file; 2.9 MB)

See also

• Rabi oscillation