Title:

NMR Studies in Hexaborides Diplomarbeit in experimenteller Festkörperphysik.

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ISBN: 3835101447   ISBN: 3835101447   ISBN: 3835101447   ISBN: 3835101447 
 
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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 Iz. This leads to a constant energy difference between the energy levels represented by m3.1:

 \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 up previous contents
Next: The Quadrupolar Interaction Up: The Spin Hamiltonian Previous: The Spin Hamiltonian
  
Einführung in die Festkörperphysik (Broschiert)
von Konrad Kopitzki,
Peter Herzog
Siehe auch:
Einführung in die Festkörperphysik
Festkörperphysik
Grundkurs Theoretische Physik 3: Elektrodynamik
Quantenmechanik (QM I): Eine Einführung
Festkörperphysik: Einführung in die Grundlagen (Spr...
Quantenmechanik für Fortgeschrittene (QM II)
 
   
 
     

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