MRI relies on only a tiny majority of the nuclear magnetic moments aligning with the external field. Consider the common target nucleus hydrogen. The difference between the aligned and anti aligned states of a dipole in a magnetic field is2μ:B Equation (8-7) can be used to findμz for the proton. Provided that the correct mass and gyromagnetic ratio(gp=5.6) are inserted. Using the Boltzmann distribution, show that for a1.0T field and a reasonable temperature, the number aligned exceeds the number anti aligned by less than 1100%.

Short Answer

Expert verified

The resultant answer is NalignedNantialigned=1.000007.

Step by step solution

01

Given data

gp=5.6- proton gyromagnetic ratio

02

Concept of Electromagnetic radiation and orientation energy

The energy of electromagnetic radiation has a frequency f is given asE=hf

Here is Planck's constant.

The orientation energy of an electron is given asE=μzh .

03

Calculate the energy

In order to solve this problem, we will apply an equation that determines the ratio between the number of aligned exceeds and the number of anti-aligned, according to Boltzmann's distribution we will have:

NalignNanf=e-Eaidnkb·Te-Easbkb·TNalignNanf=eΔEkb-T

Now, since the energy difference ΔEis related to magnet moment and magnetic field as ΔE=2·μz·B.

NalignedNantialigned=e2.8·1.6·-10-19·11.67·10-27·1·38·10-23·310NalignedNantialigned=1.000007

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Most popular questions from this chapter

Equation (9-42) gives the Fermi energy for a collection of identical fermions packed into the lowest energies allowed by the exclusion principle. Argue that if applied to neutrons or protons (ignoring their repulsion) in a nucleus. the equation suggests that the Fermi energy is roughly the same for all nuclei. Making the rough approximation that the spacing between quantum levels is a constant in a given nucleus, argue that this spacing should vary from one nucleus to another in proportion toA-1 .

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