X rays scattered from rock salt (NaCl) are observed to have an intense maximum at an angle of 20° from the incident direction. Assuming n = 1 (from the intensity), what must be the wavelength of the incident radiation? Strategy We will use Equation (5.1) to find λ, but we need to know d, the

Question

X rays scattered from rock salt (NaCl) are observed to have an intense maximum at an angle of 20° from the incident direction. Assuming n = 1 (from the intensity), what must be the wavelength of the incident radiation?

Strategy We will use Equation (5.1) to find λ, but we need to know d, the spacing, and the angle θ. Notice that the angle between the incident beam and scattered wave for constructive interference is always 2θ (see Figures 5.4 and 5.5), and because 2θ = 20°, we have θ = 10°. We can use the density of NaCl to help find d, because the volume taken up by one atom is [latex]d^{3}[/latex].

[latex]n \lambda=2 d \sin heta \quad(n= ext { integer })[/latex] (5.1)

Accepted Answer

In Section 4.1 we showed that

[latex]\frac{ ext { Number of molecules }}{ ext { Volume }}=\frac{N_{ A } ho}{M}[/latex]

where [latex]N_{ A }[/latex] is Avogadro’s number, ρ is the density, and M is the gram-molecular weight. For NaCl, [latex] ho=2.16 g / cm ^{3}[/latex] and M = 58.5 g/mol.

[latex]\frac{N_{ A } ho}{M}=\frac{\left(6.02 imes 10^{23} \frac{ ext { molecules }}{ mol } ight)\left(2.16 \frac{ g }{ cm ^{3}} ight)}{58.5 \frac{ g }{ mol }}[/latex]

[latex]\begin{aligned}\frac{N_{ A } ho}{M} &=2.22 imes 10^{22} \frac{ ext { molecules }}{ cm ^{3}} \&=4.45 imes 10^{22} \frac{ ext { atoms }}{ cm ^{3}} \&=4.45 imes 10^{28} \frac{ ext { atoms }}{ m ^{3}}\end{aligned}[/latex]

Because NaCl has a cubic array, we take d as the distance between Na and Cl atoms, so we have a volume of [latex]d^{3}[/latex] per atom.

[latex]\frac{1}{d^{3}}=4.45 imes 10^{28} \frac{ ext { atoms }}{ m ^{3}}[/latex]

[latex]d=2.82 imes 10^{-10} m =0.282 nm[/latex]

This technique of calculating the lattice spacing works for only a few cases because of the variety of crystal structures, many of which are noncubic.

We use Equation (5.1) to find λ.

[latex]\lambda=\frac{2 d \sin heta}{n}=\frac{(2)(0.282 nm )\left(\sin 10^{\circ} ight)}{1}=0.098 nm[/latex]

which is a typical x-ray wavelength. NaCl is a useful crystal for determining x-ray wavelengths and for calibrating experimental apparatus.

Question 5.1: X rays scattered from rock salt (NaCl) are observed to have ...

This Question has been Answered.

Already have an account?

Login

Related Answered Questions