Chapter 13, Problem 13A.1ST

Interpretation Introduction

Interpretation:

The magnetic field that is necessary to induce the imbalance of population about 35 in a million for ${}^{\text{13}}\text{C}$ with ${\text{γ}}_{\text{N}}\text{ = 6}\text{.7272}\times {\text{10}}^7{\text{ T}}^{-1}{\text{ s}}^{-1}$ has to be calculated.

Concept Introduction:

The relation between populations of the $\text{α and β}$ states, ${\text{N}}_{\text{α}}{\text{ and N}}_{\text{β}}$ is given by,

  ${\text{N}}_{\text{α}}{\text{-N}}_{\text{β}}\text{ = }\frac{{\text{Nγ}}_{\text{N}}\hslash {\text{B}}_{\text{0}}}{2k\text{T}}$

Where,

  ${\text{N}}_{\text{α}}{\text{ and N}}_{\text{β}}$ are the populations of the $\text{α and β}$ states.

  $\hslash$ is the Planck’s constant.

  ${\text{B}}_{\text{0}}$ is the magnetic field

  ${\text{γ}}_{\text{N}}$ is the nuclear magnetogyric ratio.

  k is the Boltzmann’s constant

  T is the temperature

Explanation of Solution

Given data:

The ${}^{\text{13}}\text{C}$ nuclei with ${\text{γ}}_{\text{N}}\text{ = 6}\text{.7272}\times {\text{10}}^7{\text{ T}}^{-1}{\text{ s}}^{-1}$ at $\text{20}°\text{C}$.  The imbalance of population is 35 in a million population.

Calculation of Magnetic field:

The magnetic field that is necessary to induce the imbalance in population is calculated as follows,

  $\begin{array}{l}{\text{B}}_{\text{0}}=\frac{{\text{(N}}_{\text{α}}{\text{-N}}_{\text{β}})2k\text{T}}{{\text{Nγ}}_{\text{N}}\hslash }\\ =\frac{35\times 2\times (1.38\times {10}^{-23}{\text{ JK}}^{-1})\times 293\text{ K}}{1000000\times 6.7272\times {10}^7{\text{ T}}^{-1}{\text{s}}^{-1}\times 1.055\times {10}^{-34}\text{ Js}}\\ =\frac{28324.31\times {10}^{-23}}{7097196\times {10}^{-27}}\text{T}\\ \text{=0}\text{.0039909}\times {\text{10}}^4\text{ T}\\ \approx \text{40 T}\end{array}$

The magnetic field that is necessary to induce the imbalance of population is $\text{40 T}$.