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Numerical simulation of the nutation
The algebra discussed in the previous page can be numerically tested directly,
e.g. within matlab. Let us start from the simplest case, I=½, for which we can define the 2x2 spin matrices in the representation of Iz,I2 :
{$$ I_z= \begin{bmatrix} \frac 1 2 & 0\\0 & - \frac 1 2\end{bmatrix},\qquad I_x= \begin{bmatrix}0& \frac 1 2\\ \frac 1 2 & 0\end{bmatrix},\qquad I_y=\begin{bmatrix}0 & -\frac i 2\\\frac i 2 & 0\end{bmatrix}, $$}
a task easily done in matlab, e.g. by Iz=[0.5 0;0 -0.5]; and Iy=i*[0 -0.5;0.5 0];. Better schemes? than plain enumeration can be envisaged for constructing the matrices of higher spin.
It is then straightforward to construct both ρ0=Iz and the Hamiltonian in rotating frame, HI , given by Eq. (7).
We can then monitor the time evolution of the transverse component of the nuclear moment, Iy , if we calculate the time evolution unitary operators:
{$$ U(t)=e^{-i H_I t/\hbar} $$}
e.g. by U=expm(-i*HI*t/hbar), as well as its complex conjugate, U(t)-1 , This allows the reconstruction of the density matrix ρI(t)=U(t)ρ0U(t)-1 at any desired time, henceIx(t) according to Eq. (9), as trace(rho*Ix).
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It is particularly interesting to see the effect of varying the static field B0 at fixed radio frequency ω. This variation is brought about in real cases by the inhomogeneity of the magnet, for instance: different nuclei in the sample experience slightly different field values. All the nuclear spin experiencing the same field B0 , i.e. the same frequency shift ω0-ω from the radio frequency are defined as one isochromat.
Fig. 1 The different colours represent the nutation of distinct isochromats. That with maximum amplitude corresponds to ω0 - ω = 0. For the others the difference between the Larmor and the radio frequencies is respectively 0.1, 0.2, 0.5, 1.0, 5.0 and 10 times ω1. These isochromats experience an effective field, which can be extracted from Eq. (7) to be:
{$ \mathbf{ B}_{eff} = \frac 1 \gamma [(\omega_0-\omega) \hat{z} + \omega_1 \hat{x}] $}
The numerical simulation is performed by imposing {$\hbar=1$} and {$\omega_1=1$} for the sake of simplicity.
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