Magnetic Resonance — Question 9

Question 9

A magnetic gradient maps resonance frequency to position

A sample sees $B(z)=B_0+Gz$ with $G=.0200\,\mathrm{T/m}$. Use $\gamma/(2\pi)=42.6\,\mathrm{MHz/T}$. A frequency-selective pulse excites a full bandwidth of $\Delta f=1.20\,\mathrm{kHz}$ around $f_c$. Ignore nonlinearities and intrinsic linewidth.

1. Find the position of the selected slice center.

2. Find its full thickness and the gradient needed to halve it without changing bandwidth.

Question 9 - Solution

A magnetic gradient maps resonance frequency to position

(a) Method and calculation

$$f(z)=\frac{\gamma}{2\pi}(B_0+Gz)\Rightarrow\boxed{z_c=\frac{f_c-(\gamma/2\pi)B_0}{(\gamma/2\pi)G}}.$$$$\frac{df}{dz}=(42.6\times10^6)(.0200)=8.52\times10^5\,\mathrm{Hz/m}.$$

(b) Interpretation and further calculation

$$\boxed{\Delta z=\frac{\Delta f}{|df/dz|}=1.408\,\mathrm{mm}}.$$$$\Delta z\propto G^{-1}\Rightarrow\boxed{G\prime=2G=.0400\,\mathrm{T/m}}.$$

Physical check

The bandwidth and slice thickness are both full widths. The model assumes the selected region is small enough that the local field varies linearly.