An open MRI scanner uses a field of

PHYSICS · Multi-part · Fermi University Physics · Quantum Mechanics · Biophysics & Medical Physics · Symmetry, angular momentum and spin

Life at Thermal Energy, Ch Echoes, precession and the therapeutic beam

Problem

B_0 = 1.0\,\mathrm{T}

, and the patient is at body temperature,

T = 310\,\mathrm{K}

(

k_BT = 4.28\times10^{-21}\,\mathrm{J}

). Take

\gamma/2\pi = 42.58\,\mathrm{MHz\,T^{-1}}

and

h = 6.63\times10^{-34}\,\mathrm{J\,s}

. ((a)) Calculate the proton resonance frequency

f_0

and the level gap

\Delta E = hf_0

. ((b)) Starting from the Boltzmann factor

e^{-\Delta E/k_BT}

for the two spin populations and using

e^{-x}\approx 1-x

for small

x

, derive the fractional population excess

\Delta n/n \approx \Delta E/2k_BT

, and evaluate it at

1.0\,\mathrm{T}

. ((c)) A

1\,\mathrm{mm^3}

voxel of water contains

6.7\times10^{19}

protons. How many more protons align with the field than against it?