- 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 $\chi$. In our case, $\chi =\chi ^{'}+i\chi ^{''}$ is the complex nuclear magnetic susceptibility, where $\chi ^{'}$ is the real part and is associated with dispersion while $\chi ^{''}$ 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 $\chi _{_{0}}$.

Consider an assembly of identical weakly interacting nuclei of spin number $I$, in thermal equilibrium at a spin temperature $T_{_{S}}$ in a steady magnetic field $H_{_{0}}$. The nuclei having quantum number $m$ are found to be in the energy level $-m\mu H_{_{0}}/I$. The population of this level is then weighted by the Boltzmann factor $e^{\tfrac {m\mu H_{_{0}}}{IkT_{_{S}}}}\approx 1+{\tfrac {m\mu H_{_{0}}}{IkT_{_{S}}}}$. This approximation is very good for most practical conditions. Hence the population of each level is given by

$N(m)={\tfrac {N}{2I+1}}(1+{\tfrac {m\mu H_{_{0}}}{IkT_{_{S}}}})$.

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}} .

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Now consider the RF magnetic susceptibility (the susceptibility with the transverse field applied). Let's find the power absorbed by the system in terms 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 \chi^{''}} .

Consider a transverse 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=2H_{_1}cos(2\pi\nu t)} .