6.2: Answers to Questions in the Basic Theory section

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In addition to the conceptual questions, the Basic Theory section also contains a series of simple quantitative questions, the answers of which are provided below.

Question 1

How many spin states would you predict for ²H?

Solution

Deuterium has a spin of 1. Therefore there should be 3 possible spin states: +1, 0 and -1.

Question 2

Given the same magnetic field and temperature, how would the difference in population for ¹H and ³¹P compare?

Solution

For this problem we will use the following equation:

\[\dfrac{N_{upper}}{N_{lower}} = e^{\large\frac{-∆E}{kT}}\]

The difference in population for ¹H and ³¹P will be related to the differences in their ∆E values. Since ∆E=γhB_o/2π, for a fixed magnetic field the only differences between ¹H and ³¹P is in their magnetogyric ratios.

\[\dfrac{∆E( ^1H)}{∆E( ^{31}P)} = \dfrac{26.752}{10.84} = 2.468\]

The ratio of the N_upper/N_lower for ¹H is e^2.468 or =11.80 times larger than the ratio of N_upper/N_lower for ³¹P.

Question 3

Calculate the wavelength of electromagnetic radiation corresponding to a frequency of 500 MHz.

Solution

The wavelength of electromagnetic radiation corresponding to a frequency of 500 MHz is 0.6 m.

Question 4

What range of frequencies would be excited by a 10 µs rf pulse?

Solution

A 10 µs rf pulse would excite a range of frequencies covering 100,000 Hz.

Question 5

What are the resonance line widths of nuclei that have apparent T₂ relaxation times (i.e.T₂* values) of 1 and 2 sec.

Solution

\[w_{\large\frac{1}{2}} = \dfrac{1}{πT_2^*}\]

Therefore, the two resonances have line widths of 0.32 and 0.16 Hz.