Date: Sat, 2 Dec. 2023 09:10:31 +00:00
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Let us consider a one-dimensional case, i.e. let the sample be completely homogeneous along the y and z axes. Let the high external field continue to point in the direction of the z axis. Let us excite the system so that the magnetization gets in the x-y plane; after that, let us change the magnitude of the magnetic field so that it changes linearly as a function of position instead of being homogeneous. Let the changed field be
::{EQUATION(size="75")}B_{z}=B_{0z}+G_{x}x{EQUATION}::
where {EQUATION(size="75")}$G_{x}${EQUATION} is a gradient. (The field still points in the direction of the z axis.)
{IMG(fileId="348",width="400",imalign="center",align="center",desc="Figure 2.")}{IMG}
Figure 2. shows the draft of the spin echo sequence: time {EQUATION(size="75")}$\tau${EQUATION} after the 90° excitation a 180° radio pulse is applied. The phase differences that are created due to the static effects disappear after another time {EQUATION(size="75")}$\tau${EQUATION} , the signal is amplified, and an echo is created. As a parameter of the sequence usually a time {EQUATION(size="75")}$T_{E}${EQUATION} (where {EQUATION(size="75")}$T_{E}=2\tau${EQUATION}) is used, which is the time that passes between the exciting pulse and the echo.
Let us measure the induced signal. Due to the gradient, the magnetization will precess with different frequencies in every position. The measured signal is the sum of the many oscillating signals with different frequencies. These sums can evidently be resolved into their components using the Fourier transform, i.e. the position-dependence of the magnetization can be restored.
Let us write the frequency encoding formally as well. The phase will be position and time-dependent because of the gradient:
::{EQUATION(size="75")}\varphi(x,t)=\gamma G_{x}x=\gamma\intop^{t}G_{x}(t')dt'x{EQUATION} (13)::
where it is made possible for the gradient to change over time after the second equals sign. Let us introduce a new variable instead of time:
::{EQUATION(size="75")}k_{x}(t)=\frac{\gamma}{2\pi}\intop_{0}^{t}G_{x}(t')dt'{EQUATION} (14)::
{img fileId="350" width="400" imalign="center" align="center" desc="Figure 3. The illustration of frequency encoding. The effective spin density of the areas represented by the red and the purple dots are identical. (In order to keep things simple, the rest of the head does not give a signal.) Due to the spatial gradient, the moments located at different positions along the x axis give signals at different frequencies."}
By writing the signal using the new variable, an important relation can be discovered:
::{EQUATION(size="75")}S(t)=\int\varrho(x)e^{i\varphi(x,t)}dx{EQUATION} (15)::
That is, the signal is the Fourier transform of the effective spin density, the introduced quantity {EQUATION(size="75")}$k${EQUATION} is the conjugate of the real space. After that, the signal should not be considered as a function of time, but rather as a function of the integrals of the gradients.