Chapter 16.13, Problem 62AAP

To determine

The theoretical value for the saturation magnetization and saturation induction for nickel.

Answer to Problem 62AAP

The theoretical value for the saturation magnetization for nickel is $1.70\times {10}^6\text{A}/\text{m}$.

The theoretical value for the saturation induction for nickel is $2.14\text{T}$.

Explanation of Solution

Write the expression to calculate saturation magnetization for nickel $\left(M_s\right)$.

  $M_s=\left(\text{Atomic density}\right)N{\mu }_B$                                                                                  (I)

Here, magnetic moment is $N$ and Bohr magneton is ${\mu }_B$.

Write the expression to calculate atomic density for nickel.

  $\begin{array}{c}\text{Atomic density}=\frac{n}{V}\\ =\frac{n}{a^3}\end{array}$                                                                                            (II)

Here, number of atoms per unit cell is $n$, volume of the unit cell is $V$ and lattice constant is a.

Write the expression to calculate saturation induction $\left(B_s\right)$.

  $B_s\approx {\mu }_0{M}_s$                                                                                                          (III)

Here, permeability constant is ${\mu }_0$.

Conclusion:

The magnetic moment of the nickel atom is considered to be $2\text{Bohr magnetons}$.

Substitute $4\text{atoms}/\text{unit cell}$ for n and $0.352\text{nm}$ for $a$ in Equation (II).

 $\begin{array}{c}\text{Atomic density}=\frac{4\text{atoms}/\text{unit cell}}{{\left(0.352\text{nm}\right)}^3}\\ =\frac{4\text{atoms}/\text{unit cell}}{{\left(0.352\text{nm}\times \frac{1\times {10}^{-9}\text{m}}{1\text{nm}}\right)}^3}\\ =9.171\times {10}^{28}\text{atoms}/{\text{m}}^3\end{array}$

The value of Bohr magneton $\left({\mu }_B\right)$ is taken to be $9.27\times {10}^{-24}\text{A}\cdot {\text{m}}^2$.

Substitute $9.171\times {10}^{28}\text{atoms}/{\text{m}}^3$ for $\text{Atomic density}$, $2\text{Bohr magnetons}/\text{atom}$ for $N$ and $9.27\times {10}^{-24}\text{A}\cdot {\text{m}}^2$ for ${\mu }_B$ in Equation (I).

 $\begin{array}{c}{M}_s=\left(9.171\times {10}^{28}\text{atoms}/{\text{m}}^3\right)\left(2\text{Bohr magnetons}/\text{atom}\right)\left(9.27\times {10}^{-24}\text{A}\cdot {\text{m}}^2\right)\\ =1.70\times {10}^6\text{A}/\text{m}\end{array}$

Thus, the theoretical value for the saturation magnetization for nickel is $1.70\times {10}^6\text{A}/\text{m}$.

The value of permeability constant $\left({\mu }_0\right)$ is taken to be $4\pi \times {10}^{-7}\text{T}\cdot \text{m}/\text{A}$.

Substitute $4\pi \times {10}^{-7}\text{T}\cdot \text{m}/\text{A}$ for ${\mu }_0$ and $1.70\times {10}^6\text{A}/\text{m}$ for $M_s$ in Equation (III).

 $\begin{array}{c}{B}_s=\left(4\pi \times {10}^{-7}\text{T}\cdot \text{m}/\text{A}\right)\left(1.70\times {10}^6\text{A}/\text{m}\right)\\ =2.14\text{T}\end{array}$

Thus, the theoretical value for the saturation induction for nickel is $2.14\text{T}$.