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Chapter 20: Problem 14

The formula for samarium iron garnet \(\left(\mathrm{Sm}_{3} \mathrm{Fe}_{5} \mathrm{O}_{12}\right)\) may be written in the form \(\mathrm{Sm}_{3}^{c} \mathrm{Fe}_{2}^{a} \mathrm{Fe}_{3}^{d} \mathrm{O}_{12},\) where the superscripts \(a, c\) and \(d\) represent different sites on which the \(\mathrm{Sm}^{3+}\) and \(\mathrm{Fe}^{3+}\) ions are located. The spin magnetic moments for the \(\mathrm{Sm}^{3+}\) and \(\mathrm{Fe}^{3}\) ions positioned in the \(a\) and \(c\) sites are oriented parallel to one another and antiparallel to the \(\mathrm{Fe}^{3+}\) ions in \(d\) sites. Compute the number of Bohr magnetons associated with each \(\mathrm{Sm}^{3+}\) ion, given the following information: (1) each unit cell consists of eight for mula \(\left(\mathrm{Sm}_{3} \mathrm{Fe}_{5} \mathrm{O}_{12}\right)\) units; (2) the unit cell is cubic with an edge length of \(1.2529 \mathrm{nm}\) (3) the saturation magnetization for this material is \(1.35 \times 10^{5} \mathrm{A} / \mathrm{m} ;\) and (4) assume that there are 5 Bohr magnetons associated with each \(\mathrm{Fe}^{3+}\) ion.

Short Answer

Expert verified
Question: Calculate the number of Bohr magnetons associated with each Sm3+ ion in samarium iron garnet, given that each unit cell consists of eight formula units of SmFe5O12, the cubic unit cell edge length is 1.2529 nm, the saturation magnetization is 1.35 x 10^5 A/m, and there are 5 Bohr magnetons associated with each Fe3+ ion. Answer: To calculate the number of Bohr magnetons associated with each Sm3+ ion, follow these steps: 1. Calculate the volume of the cubic unit cell. 2. Calculate the saturation magnetic moment per unit volume. 3. Find the total number of Bohr magnetons per unit cell. 4. Find the total magnetic moment per unit cell. 5. Calculate the number of Bohr magnetons associated with each Sm3+ ion. By following these steps, you can determine the number of Bohr magnetons associated with each Sm3+ ion in samarium iron garnet.

Step by step solution

01

Calculate the volume of the unit cell

To calculate the volume of the cubic unit cell, use the formula: Volume = edge_length^3 where edge_length is given as 1.2529 nm. Convert the edge_length to meters (1 nm = 1 x 10^(-9) m): edge_length = 1.2529 x 10^(-9) m Then, calculate the volume: Volume = (1.2529 x 10^(-9))^3 m^3
02

Calculate saturation magnetic moment per unit volume

To get the saturation magnetic moment per unit volume (M), we'll need the saturation magnetization (given as 1.35 x 10^5 A/m). Multiply the saturation magnetization by the volume of the unit cell: M = saturation_magnetization * Volume M = (1.35 x 10^5 A/m) * (1.2529 x 10^(-9))^3 m^3
03

Find the total number of Bohr magnetons per unit cell

Knowing that there are 5 Bohr magnetons associated with each Fe3+ ion and the unit cell consists of 8 formula units of Sm_3Fe_5O_12, we can find the total number of Bohr magnetons per unit cell: Total Bohr magnetons per unit cell = 8 formula units/cell * 5 Fe3+ ions/formula unit * 5 Bohr magnetons/Fe3+ ion
04

Find the total magnetic moment per unit cell

The total magnetic moment per unit cell (μ_total) can be found by multiplying the total number of Bohr magnetons per unit cell found in Step 3 by the Bohr magneton constant, μ_B = 9.274 x 10^(-24) J/T: μ_total = Total Bohr magnetons per unit cell * μ_B
05

Calculate the number of Bohr magnetons associated with each Sm3+ ion

Since the total magnetic moment per unit cell (μ_total) comes from both Fe3+ and Sm3+ ions, and we found the contribution of Fe3+ ions in Step 3, we can find the number of Bohr magnetons associated with each Sm3+ by: 1. Computing the magnetic moment of Sm3+ ions: μ_Sm = μ_total - M (found in Step 2) 2. Calculate the total Bohr magnetons of Sm3+ ions: Total_Sm3+_Bohr_magnetons = μ_Sm/μ_B 3. Considering that there are 8 formula units/cell * 3 Sm3+ ions/formula unit, we can find the number of Bohr magnetons associated with each Sm3+ ion: Number of Bohr magnetons/Sm3+ ion = Total_Sm3+_Bohr_magnetons / (8 * 3)

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Most popular questions from this chapter

A ferromagnetic material has a remanence of 1.0 tesla and a coercivity of \(15,000 \mathrm{A} / \mathrm{m}\) Saturation is achieved at a magnetic field strength of \(25,000 \mathrm{A} / \mathrm{m},\) at which the flux density is 1.25 teslas. Using these data, sketch the entire hysteresis curve in the range \(H=-25,000\) to \(+25,000 \mathrm{A} / \mathrm{m} .\) Be sure to scale and label both coordinate axes.

It is possible to express the magnetic susceptibility \(\chi_{m}\) in several different units. For the discussion of this chapter, \(\chi_{m}\) was used to designate the volume susceptibility in SI units, that is, the quantity that gives the magnetization per unit volume \(\left(\mathrm{m}^{3}\right)\) of material when multiplied by \(H\). The mass susceptibility \(\chi_{m}(\mathrm{kg})\) yields the magnetic moment (or magnetization) per kilogram of material when multiplied by \(H ;\) and, similarly, the atomic susceptibility \(\chi_{m}(\text { a })\) gives the magnetization per kilogram- mole. The latter two quantities are related to \(\chi_{m}\) through the relationships $$\begin{aligned}\chi_{m} &=\chi_{m}(\mathrm{kg}) \times \text { mass density }\left(\mathrm{in} \mathrm{kg} / \mathrm{m}^{3}\right) \\\\\chi_{m}(\mathrm{a}) &=\chi_{m}(\mathrm{kg}) \times \text { atomic weight }(\mathrm{in} \mathrm{kg})\end{aligned}$$ When using the cgs-emu system, comparable parameters exist, which may be designated by \(\chi_{m}^{\prime}, \chi_{m}^{\prime}(\mathrm{g}),\) and \(\chi_{m}^{\prime}(\mathrm{a})\); the \(\chi_{m}\) and \(\chi_{m}^{\prime}\) are related in accordance with Table 20.1 From Table \(20.2, \quad \chi_{m}\) for copper is \(-0.96 \times 10^{-5} ;\) convert this value into the other five susceptibilities.

The magnetization within a bar of some metal alloy is \(1.2 \times 10^{6} \mathrm{A} / \mathrm{m}\) at an \(\mathrm{H}\) field of \(200 \mathrm{A} / \mathrm{m} .\) Compute the following: (a) the magnetic susceptibility, (b) the permeability, and (c) the magnetic flux density within this material. (d) What type(s) of magnetism would you suggest as being displayed by this material? Why?

The magnetic flux density within a bar of some material is 0.630 tesla at an \(H\) field of \(5 \times 10^{5} \mathrm{A} / \mathrm{m} .\) Compute the following for this material: (a) the magnetic permeability, and (b) the magnetic susceptibility. (c) What type(s) of magnetism would you suggest is(are) being displayed by this material? Why?

The chemical formula for copper ferrite may be written as \(\left(\mathrm{CuFe}_{2} \mathrm{O}_{4}\right)_{8}\) because there are eight formula units per unit cell. If this material has a saturation magnetization of \(1.35 \times\) \(10^{5} \mathrm{A} / \mathrm{m}\) and a density of \(5.40 \mathrm{g} / \mathrm{cm}^{3},\) estimate the number of Bohr magnetons associated with each \(\mathrm{Cu}^{2+}\) ion.
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