Compute the resonance frequency of free protons in a...

Compute the resonance frequency of free protons in a magnetic field of $(a) 0.5 \times$ $10^{-4} \mathrm{~T}$ (the approximate strength of Earth's field) $,(b) 0.25 \mathrm{~T},$ and $(c) 0.5 \mathrm{~T}$.

Key Concepts

Larmor Precession

This concept describes the precession of the magnetic moment of a nucleus (such as a proton) about an external magnetic field. The precession occurs at a specific frequency known as the Larmor frequency, which is directly proportional to the magnetic field strength. This is a central idea in magnetic resonance phenomena, including nuclear magnetic resonance (NMR).

Gyromagnetic Ratio

The gyromagnetic ratio is a fundamental constant for each type of nucleus, representing the ratio of its magnetic moment to its angular momentum. It sets the proportionality between the applied magnetic field strength and the Larmor frequency, thus determining how quickly a nucleus precesses in that field.

Magnetic Field Strength

This refers to the intensity of the magnetic field applied to the nuclei. The magnitude of the external magnetic field directly influences the precession rate (Larmor frequency) of the nuclear magnetic moments, making it a critical parameter in determining resonance conditions in NMR studies.

Transcript

00:01 In this question, we have to compute the resonant frequency of a free proton in a magnetic field of first .5 into 10 to the power minus 4 tesla, b in the field of .25 tesla, and third in .5 tesla.

00:27 At first, we know that the value of the nuclear magneton is 3 .15 -245 in 2 .10ic power minus 8 volt per tesla.

00:46 The magnitude of the z component of the spin magnetic movement of proton, that is, mu -z of the proton, is given by 2 .5%.

01:07 79928 new n.

01:09 Substituting the value from above, we will get the z component as 8 .804 into 10 to the power minus 8 electron volt for tesla.

01:27 Now coming to the first part where magnetic field is 0 .5 into 10 to the power minus 1 tesla.

01:35 When the z component of spin and mu are parallel to the field, then the interaction energy is given by minus of mu z b.

01:55 Substituting the value of mu z and the magnetic field minus of 8 .804 into 10 to the power minus 8 .804, multiplied by 0 .104 ,000, multiplied by 0 .104 .8, multiplied by 0 .0 .4.

02:11 5 into 10 to the power minus 4 tesla.

02:15 The interaction energy will be minus of 4 .402 into 10 to the power minus 12 electron volt.

02:26 And when both vectors are anti -parallel, so the interaction energy will be plus 4 .402 into 10 to the power minus 12 electron volt.

02:46 Then the changing energy will be given by plus 4 .402 into 10 to the power minus 2 minus 4 .402 into 10 to the power minus 12 electron volt.

03:04 This will be combining 0 .804 into 10 to the power minus 12 electron volt.

03:16 And now the resonant frequency is given by dell e by h where substituting the value divided by h has the value 4 .136 into 10 to the power minus 15 electron volt second...

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