- The Resonance Condition

The Resonance Condition

Consider an isolated nucleus in a steady magnetic field Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle H_{_0}} . The magnetic field breaks the symmetry of free space and defines a particular spatial direction. Suppose that the nucleus possesses an intrinsic "spin" with spin number $I>0$ so that it has an intrinsic magnetic moment (moving/spinning electric charge creates a magnetic field). How does this intrinsic "spin" relate to angular momentum? It is necessary to find this relation explicitly.

We know from quantum mechanics that (on small enough scales) energy appears in discrete bits ( $\hbar$ shows its face ). It is then reasonable to suppose that "spin" energies are also discrete (or quantized). Physical experiments bear out this supposition.

The nucleus will have different energy states depending on the magnitude and direction of the nucleon's magnetic moment. See the figure below for the allowed spin states of a spin-1/2 particle.

The energy of a magnetic moment $\mu$ in a magnetic field $H_{_{0}}$ is $E=-\mu \cdot H_{0}$. If the energy in the system is low enough, most, if not all, of the spins will be parallel to the field (lowest energy state). However, if there is enough energy around then some nucleons will absorb a bit of energy and will become aligned anti-parallel to the magnetic field.

A careful study of quantum mechanics reveals that the length of the nuclear angular momentum vector is $[I(I+1)]^{1/2}\hbar$ but that the only measurable components of this vector are given by $m\hbar$ , where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle m} , the magnetic quantum number, may take any of the Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (2I+1)} values in the series Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle I, I-1, I-2, ... , -(I-1), -I} .

Correspondingly, the nuclear magnetic moment also has Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (2I+1)} components in proportion. This is a consequence of "space quantization" demonstrated so famously by Stern and Gerlach in 1922.

We can define Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mu} as the maximum measurable (observable) component of the magnetic moment. Even though the length of the magnetic moment vector is Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle [(I+1)/I]^{1/2}\mu } , the quantity of physical interest is Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mu} and for this reason workers in the field usually refer to Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mu} as the magnetic moment. As mentioned above, the energy levels of the nuclear magnet in the filed Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle H_{_0}} are given by the Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (2I+1)} values of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -m\mu H_{_0}/I} . In the general case, this results in a set of equally-spaced levels with separation Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mu H_{_0}/I} between adjacent levels.

This separation is often written as Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle g \mu_{_0} H_{_0}} , where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle g=\frac{\mu}{\mu_{_0}I}} is called the splitting factor of the g-factor.