Chapter 16.13, Problem 64AAP

To determine

The theoretical value for the saturation magnetization for gadolinium.

Answer to Problem 64AAP

The theoretical value for the saturation magnetization for gadolinium is $1.96\times {10}^6\text{A}/\text{m}$.

Explanation of Solution

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

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

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

Write the expression to calculate atomic density of the gadolinium.

  $\text{Atomic density}=\frac{n}{V}$                                                                                             (II)

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

Write the expression to calculate volume of the unit cell for the gadolinium $\left(V\right)$.

  $\begin{array}{c}V=\text{Area}\times \text{height}\\ =\left(3a^2\sin 60°\right)\times c\end{array}$                                                                                            (III)

Here, lattice constant of the unit cell is $a$ and height of the unit cell is $c$.

Conclusion:

Since there are seven $4f$ unpaired electrons, the magnetic moment of the gadolinium atom is considered to be $7\text{Bohr magnetons}$.

Substitute $\text{0}\text{.364}\text{nm}$ for $a$ and $\text{0}\text{.578}\text{nm}$ for $c$ in Equation (III).

 $\begin{array}{c}V=\left(3{\left(\text{0}\text{.364}\text{nm}\right)}^2\sin 60°\right)\times \text{0}\text{.578}\text{nm}\\ =\left(3{\left(\text{0}\text{.364}\text{nm}\times \frac{1\times {10}^{-9}\text{m}}{1\text{nm}}\right)}^2\sin 60°\right)\times \left(\text{0}\text{.578}\text{nm}\times \frac{1\times {10}^{-9}\text{m}}{1\text{nm}}\right)\\ =1.9897\times {10}^{-28}{\text{m}}^3\end{array}$

Substitute $6\text{atoms}/\text{unit cell}$ for n and $1.9897\times {10}^{-28}{\text{m}}^3$ for $V$ in Equation (II).

 $\begin{array}{c}\text{Atomic density}=\frac{6\text{atoms}/\text{unit cell}}{1.9897\times {10}^{-28}{\text{m}}^3}\\ =3.0155\times {10}^{28}\text{atoms}/{\text{m}}^3\end{array}$

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 $3.0155\times {10}^{28}\text{atoms}/{\text{m}}^3$ for $\text{Atomic density}$, $7\text{Bohr magnetons}/\text{atom}$ for $N$ and $9.27\times {10}^{-24}\text{A}\cdot {\text{m}}^2$ for ${\mu }_B$ in Equation (I).

 $\begin{array}{c}{M}_s=\left(3.0155\times {10}^{28}\text{atoms}/{\text{m}}^3\right)\left(7\text{Bohr magnetons}/\text{atom}\right)\left(9.27\times {10}^{-24}\text{A}\cdot {\text{m}}^2\right)\\ =1.96\times {10}^6\text{A}/\text{m}\end{array}$

Thus, the theoretical value for the saturation magnetization for gadolinium is $1.96\times {10}^6\text{A}/\text{m}$.