- Magnetic Susceptibilities - Revision history

Bsboggs at 18:01, 22 February 2019

2019-02-22T18:01:46Z

Bsboggs at 18:38, 21 February 2019

2019-02-21T18:38:58Z

Bsboggs: Created page with "===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..."

2019-02-18T23:49:19Z

Created page with "===Magnetic Susceptibilities=== We have seen that an assembly of nuclear magnets in a steady magnetic field <math>H_{_0}</math> absorbs power from a suitably applied RF field..."

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===Magnetic Susceptibilities===

We have seen that an assembly of nuclear magnets in a steady magnetic field <math>H_{_0}</math> 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 <math>\chi</math>. In our case, <math>\chi=\chi^{'}+i\chi^{''}</math> is the complex nuclear magnetic susceptibility, where <math>\chi^'</math> is the real part and is associated with dispersion while <math>\chi^{''}</math> 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 <math>\chi_{_0}</math>.

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 since <math>\sum^{I}_{-I}m^2=\tfrac{1}{3}I(I+1)(2I+1)</math>.
 The static magnetic susceptibility is then given by
 <math>\chi_{_0}=\tfrac{\mathcal{M}}{H_{_0}}=\tfrac{N\mu^2(I+1)}{3kT_{_S}I}</math>.

← Older revision		Revision as of 18:38, 21 February 2019
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	We will first derive the static (no transverse RF field) magnetic susceptibility <math>\chi_{_0}</math>.		We will first derive the static (no transverse RF field) magnetic susceptibility <math>\chi_{_0}</math>.
		+
		+	Consider an assembly of identical weakly interacting nuclei of spin number <math>I</math>, in thermal equilibrium at a spin temperature <math>T_{_S}</math> in a steady magnetic field <math>H_{_0}</math>. The nuclei having quantum number <math>m</math> are found to be in the energy level <math>-m\mu H_{_0}/I</math>. The population of this level is then weighted by the Boltzmann factor <math>e^{\tfrac{m\mu H_{_0}}{IkT_{_S}}}\approx1+\tfrac{m\mu H_{_0}}{IkT_{_S}}</math>. This approximation is very good for most practical conditions. Hence the population of each level is given by
		+
		+	<math>N(m)=\tfrac{N}{2I+1}(1+\tfrac{m\mu H_{_0}}{IkT_{_S}})</math>.
		+
		+	The total magnetic moment, the magnetization <math>\mathcal{M}</math>, is given by
		+
		+	<math>\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}</math>,
		+
		+	since <math>\sum^{I}_{-I}m^2=\tfrac{1}{3}I(I+1)(2I+1)</math>.
		+
		+	The static magnetic susceptibility is then given by
		+
		+	<math>\chi_{_0}=\tfrac{\mathcal{M}}{H_{_0}}=\tfrac{N\mu^2(I+1)}{3kT_{_S}I}</math>.

- Magnetic Susceptibilities - Revision history

Bsboggs at 18:01, 22 February 2019

Bsboggs at 18:38, 21 February 2019

Bsboggs: Created page with "===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..."

Bsboggs: Created page with "===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..."