By Frank Luo
Part 2: The Spin Echo
Questions


Answers
Question 1:
- The Boltzmann Magetization M0=4kTNγ2ℏ2B0, then after elimination the units is J/T.
- The Polarization is P=2kTγℏB0, then after elimination we can get that P is a special number depends on the material, no SI units.
Question 2:
- The polarization is P=10051−49=0.02 .
- The magnet field strength should be B0=0.00000340.02≈5882T .
- The temperature should be T=0.02300×0.0000034=0.051K.
Question 3:
- Since the Boltzmann Magetization Equation is M=M0(1−e−T1t)e−T2t , so we can calculate the signal.
The signal of Tissue A : MA=M0(1−e−300150)e−2012.5=0.21 .
The signal of Tissue B : MB=M0(1−e−200150)e−4012.5=0.38 .
Surely Tissue B will deliver more signal.
- We have calculated that Tissue B will deliver more signal if both Tissue A and B has the same Boltzmann Magetization.
If Tissue A is 85% of Tissue B, then the Tissue A signal will become lesser, so Tissue B deliver more signal.
- Let function f(t)=M0A(1−e−T1ATR)e−T2At−M0B(1−e−T1BTR)e−T2Bt reprent the signal of time t.
Consider the function: f(t)=(1−e−200150)e−40t−(1−e−300150)e−20t reaches its PEAK at about t=16, so the TE should be TE=32ms.
Question 4:
If both tissues deliver the SAME signal, which means M0Ae−T2At=M0Be−T2Bt.
Put the data in, then we can get 4.1e−30t−3.7e−50t=0, solve it and get t≈7.7ms.
So the echo time is: TE=2×t≈15.4ms.
Question 5:
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e−T2tS0=e−5030≈0.55, so the signal is 0.55mV.
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If the magetic field of inhomogeneity of ΔB=1 ppm, the signal can be calculated by such equation:
S(t)=S0e−T2te−γΔBt
Put the data in, we can e−5030e−1×267×0.000003≈0.43, so the signal amplitude is 0.43.
- We should delivered the 180° pulses at times 20ms and times 40ms if we wish to detect echoes at times 40ms and 80ms.
The signal will be e−5040≈0.45 at times 40ms and e−5080≈0.20 at times 80ms.
Question 6:
From the equation M=M0e−T2t, then we can solve e−60t≤0.1, the answer is t≈138.55ms.
Therefore, we can get the echoes at times 10ms, 30ms, 50ms, 70ms, 90ms, 110ms, 130ms, so we can get 7 echoes.
Part 3: Fourier Transform and K-Space
Questions

Answers
The amplitude of the FFT result is 15. We need to times N2 to get the correct answer.
The reason are as follows:
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Both the positive and negative frequency contribute the answer, but we only use the positive, so have to multiply 2.
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Each operation we have to sum once, so we need the result to multiply N1 to get the final answer.