- Magnetic Susceptibilities

Magnetic Susceptibilities

We have seen that an assembly of nuclear magnets 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}} absorbs power from a suitably applied RF field.

We know from basic E&M theory (macroscopic formulation of Maxwell's equations) that absorption is associated with the imaginary part of the susceptibility 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 \chi} . In our case, 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 \chi=\chi^{'}+i\chi^{''}} is the complex nuclear magnetic susceptibility, 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 \chi^'} is the real part and is associated with dispersion while 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 \chi^{''}} is the imaginary part and is associated with absorption.

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We will first derive the static (no transverse RF field) magnetic susceptibility 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 \chi_{_0}} .

Consider an assembly of identical weakly interacting nuclei of spin number 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} , in thermal equilibrium at a spin temperature 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 T_{_S}} 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 nuclei having quantum number 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} are found to be in the energy level 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} . The population of this level is then weighted by the Boltzmann factor 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 e^{\tfrac{m\mu H_{_0}}{IkT_{_S}}}\approx1+\tfrac{m\mu H_{_0}}{IkT_{_S}}} . This approximation is very good for most practical conditions. Hence the population of each level is given by

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 N(m)=\tfrac{N}{2I+1}(1+\tfrac{m\mu H_{_0}}{IkT_{_S}})} .

The total magnetic moment, the magnetization 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 \mathcal{M}} , is given by

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 \mathcal{M}=\sum^{I}_{-I}N(m)m\mu/I=\tfrac{N\mu^2H_{_0}}{I^2(2I+1)kT_{_S}}\sum^{I}_{-I}m^2=\tfrac{N\mu^2H_{_0}(I+1)}{3kT_{_S}I}} ,

since 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 \sum^{I}_{-I}m^2=\tfrac{1}{3}I(I+1)(2I+1)} .

The static magnetic susceptibility is then given by

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 \chi_{_0}=\tfrac{\mathcal{M}}{H_{_0}}=\tfrac{N\mu^2(I+1)}{3kT_{_S}I}} .