- Magnetic Susceptibilities

Magnetic Susceptibilities

We have seen that an assembly of nuclear magnets in a steady magnetic field $H_{_{0}}$ 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 Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \chi ^{''}} is the imaginary part and is associated with absorption.

---

We will first derive the static (no transverse RF field) magnetic susceptibility Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): 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 $I$, in thermal equilibrium at a spin temperature $T_{_{S}}$ in a steady magnetic field Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): 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 (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): 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 (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): 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 (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): 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 (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): 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 (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): 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 (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): 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 (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): 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 (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): 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}} .