NMR Studies in Hexaborides Diplomarbeit in experimenteller Festkörperphysik.

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NMR Studies in Hexaborides Diplomarbeit in experimenteller Festkörperphysik.

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ISBN: 3486577239 ISBN: 3486577239 ISBN: 3486577239 ISBN: 3486577239

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Next: The Quadrupolar Interaction Up: The Spin Hamiltonian Previous: The Spin Hamiltonian

The Zeeman Interaction

For a nucleus the magnetic moment $\mathrm{\boldsymbol{\mu}}$ and the spin $\mathrm{\bf I}$ are interrelated via

$\begin{displaymath}\mathrm{\boldsymbol{\mu}} = \gamma_\mathrm{N} \hbar \mathrm{\bf {I}}, \end{displaymath}$

(3.2)

with $\gamma$ the nuclear gyromagnetic ratio, $\mu_{\mathrm{N}}$ the nuclear magneton and g the nuclear g-factor characteristic for each nucleus.

The Zeeman Hamiltonian describes the interaction of the nuclear spin with an applied magnetic field $\mathrm{\bf {H}}=H_0 \cdot \mathbf{e}_z$ :

$\begin{displaymath}\mathcal{H}_Z = - \mathrm{\boldsymbol{\mu}} \cdot \mathrm{\bf {H}} = -\gamma_\mathrm{N} \hbar I_z H_0, \end{displaymath}$

(3.3)

The corresponding eigenvalues are

$\begin{displaymath}E_m = - \gamma_\mathrm{N} \hbar H_0 m, \end{displaymath}$

(3.4)

where m denotes the eigenvalues of I_z. This leads to a constant energy difference between the energy levels represented by m^3.1:

$\begin{displaymath}\Delta E = \gamma_\mathrm{N} \hbar H_0 := \hbar \omega_0, \end{displaymath}$

(3.5)

where $\omega _{0}$ represents the Larmor frequency. In a spin system in thermal equilibrium with a heat bath at a given temperature the population of an energy level is given by the Boltzmann distribution:

$\begin{displaymath}\rho = \frac{e^{-\beta \mathcal{H}_Z}}{Z}, \end{displaymath}$

(3.6)

with $\rho$ the density matrix, Z the partition function and $\beta = (k_\mathrm{B} T)^{-1}$ . A majority of the spins will therefore tend to allign along the direction of the field. Transitions between the levels may be induced by means of a rf field (compare with equations (

)-(

)). Since the transitions between the levels are induced by photons via the magnetic dipolar interaction the selection rule $\Delta m = \pm 1$ holds and only the population difference between two adjacent levels gives rise to the NMR signal.

Next: The Quadrupolar Interaction Up: The Spin Hamiltonian Previous: The Spin Hamiltonian




Einführung in die Festkörperphysik

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