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\handout{4}{Problem Set 4}{September 25, 2007}
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  {\bf Solution:}
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\begin{enumerate}

\item Phase Corrections

When  observing the FID signal due to any RF pulse on a collection of spins, we see a signal:

\begin{equation}
s(t)=e^{-t/T_2} e^{-(i(\omega_0(t+D)) + \phi)} M_0 
\end{equation}

Explicitly calculate the Fourier transform of this signal. Separate the FT into pure absorptive and pure dispersive parts. Note that we will have a lorentzian lineshape and two phase shift terms. These terms prevent us from equating the real part to the absorptive part and the imaginary part to the dispersive part. 

FYI the $\phi$ term is most related to the cable length from the RF synthesizer to the sample. The D term is related to the spectral width of the spectrometer. If the receiver gain is altered, both terms will be altered.



\item Pseudo-Echo Phase Correction:

Show that it is possible to achieve phase correction using a pseudo-echo. Show that we recover a purely absorptive signal when we Fourier transform the magnitude of the pseudo-echo signal in the lecture notes.




\item Locking

Diagram

Show that with the alternating dante sequence we don't need the "+/- diamond" component. The "plus/minus diamond" adds and subtracts the input signals resulting in two outputs, one the sum and one the difference. 


\item Locking Continued

Show that the strength of the sin term decreases as we move away from the 2kHz frequency. In the spectrometer we apply an extra current to move this sin signal to zero. This also moves the cosine channel to a DC output.This is locking.



\item Torrey Oscillations

Using the Bloch equations first and then checking with the k-space picture, calculate what happens in this experiment:


You may assume:

$\omega$(0) $>$ $\gamma$ $B_1$ and
$B_1$ $>$ 1/( $\pi$ $T_2$)

\item Current Configurations

Calculate the field gradient in the center of the current configurations given here.

Symmetries are good enough here.


\begin{figure}[h!]
\begin{center}
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\caption{Maxwell Pair}
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\begin{figure}[h!]
\begin{center}
\includegraphics[width=0.9in]{/Users/qrad/Desktop/quadrupolar1.png}
\caption{Infinite Quadrupolar}
\end{center}
\end{figure}
\begin{figure}[h!]
\begin{center}
\includegraphics[width=0.9in]{/Users/qrad/Desktop/quadrupolar2.png}
\caption{Rotated Infinite Quadrupolar}
\end{center}
\end{figure}
\begin{figure}[h!]
\begin{center}
\includegraphics[width=0.5in]{/Users/qrad/Desktop/golaycoil.png}
\caption{Golay Coil}
\end{center}
\end{figure}



\item Two-Dimensional Imagining at a Set of Points in K-space

Give two sequences of gradients which will image these points (spiral in and spiral out). Discuss the difference between these approaches for the case of a finite T2.

\begin{figure}[h!]
\begin{center}
\includegraphics[height=1in]{/Users/qrad/Documents/insideout.png}
\caption{Inside out}
\end{center}
\end{figure}
\begin{figure}[h!]
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\includegraphics[height=1in]{/Users/qrad/Desktop/outsidein.png}
\caption{Outside in}
\end{center}
\end{figure}

\item Gradients Applied in Imaging

Given a field gradient in x and y show that a pseudo-echo results in both the x and y directions. 

\item Phase Cycling in Stimulated Echo Experiment

Propose phase cycling on the second pulse of the stimulated echo so that we only measure the -k wave component. Also show that if EXORCYCLE is applied on the third pulse that the FID is eliminated. Show that the other three echos beyond the stimulated are outside the acquisition window. If a z gradient is added during delta, this helps to decohere the remaining signal in the transverse plane. If we know that middle pulse in the stimulated echo experiment slice selective, is a refocusing gradient necessary after the third pulse.

\item Moment Expansion
For all terms up to acceleration (the third moment) show the sensitivity of these sequences.

1
11
121

If the time after the pulse has a finite range in time, show the k acceleration and range. 


\item Aliasing 

Show that the aliasing frequency will differ by a factor of two between simultaneous and sequential sampling.


\item Sample with Interface

Consider a sample of two different materials with an irregular interface. Describe the properties of an acoustic signal arriving at the end of the sample after passing through the sample in terms of the properties of the interface. Use Fourier transforms and k-space formalism.


\item Rotating Sample

Show that for a sample rotating at wa we may find $\omega$ using nmr. Also show how we may conduct an experiment in the rest frame of the sample.


\item Bounded Diffusion

For an individual pore package of spins, show that the displacement of pockets is refocused. 


  In this problem we'll explore the rotation operation on binary search
  trees. As discussed in class, this operation changes the structure of
  a binary search tree without affecting the inorder of the underlying
  nodes.  See CLR section 14.2 (page 265-7) for a description of the
  operation.
\end{enumerate}
\end{document}
