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Chapter 4: Problem 78

The first-order Bragg angle for a certain crystal is \(12.1^{\circ} .\) What is the second-order angle?

Short Answer

Expert verified
The second-order Bragg angle for the given crystal is approximately \(24.4^{\circ}\).

Step by step solution

01

Write down Bragg's law for the first and second orders.

We will write down Bragg's law for both first and second-order diffractions: For first-order diffraction (\(n=1\)): \(1 \cdot \lambda = 2d \sin{\theta_1}\) For second-order diffraction (\(n=2\)): \(2 \cdot \lambda = 2d \sin{\theta_2}\)
02

Solve first-order Bragg's law for the wavelength \(\lambda\).

We will express the wavelength \(\lambda\) in terms of parameters provided for the first-order diffraction. We know that the first-order Bragg angle is \(12.1^{\circ}\). Rearranging the equation from Step 1, we get: \(\lambda = 2d \sin{12.1^{\circ}}\)
03

Substitute the expression for \(\lambda\) in the second-order Bragg's law equation.

Now, we will substitute the expression for \(\lambda\) found in Step 2 into the second-order Bragg's law equation: \(2(2d \sin{12.1^{\circ}}) = 2d \sin{\theta_2}\)
04

Simplify the equation and solve for the second-order angle \(\theta_2\).

Now, we will simplify the equation obtained in Step 3 and solve for the second-order angle \(\theta_2\): \((2 \cdot 2) d \sin{12.1^{\circ}} = 2d \sin{\theta_2}\) \(4 \sin{12.1^{\circ}} = \sin{\theta_2}\) Now, we simply need to find the value of \(\sin^{-1}(4 \sin{12.1^{\circ}})\) to get the value of \(\theta_2\): \(\theta_2 = \sin^{-1}(4 \sin{12.1^{\circ}})\) \(\theta_2 \approx 24.4^{\circ}\)
05

Final Answer:

The second-order Bragg angle for the given crystal is approximately \(24.4^{\circ}\).

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Most popular questions from this chapter

The yellow light from a sodium vapor lamp seems to be of pure wavelength, but it produces two first-order maxima at \(36.093^{\circ}\) and \(36.129^{\circ}\) when projected on a 10,000 line per centimeter diffraction grating. What are the two wavelengths to an accuracy of \(0.1 \mathrm{nm}\) ?

A telescope can be used to enlarge the diameter of a laser beam and limit diffraction spreading. The laser beam is sent through the telescope in opposite the normal direction and can then be projected onto a satellite or the moon. (a) If this is done with the Mount Wilson telescope, producing a 2.54 -m-diameter beam of 633 -nm light, what is the minimum angular spread of the beam? (b) Neglecting atmospheric effects, what is the size of the spot this beam would make on the moon, assuming a lunar distance of \(3.84 \times 10^{8} \mathrm{m} ?\)

A diffraction grating with 2000 lines per centimeter is used to measure the wavelengths emitted by a hydrogen gas discharge tube. (a) At what angles will you find the maxima of the two first-order blue lines of wavelengths 410 and \(434 \mathrm{nm} ?\) (b) The maxima of two other first-order lines are found at \(\theta_{1}=0.097 \mathrm{rad}\) and \(\theta_{2}=0.132 \mathrm{rad} .\) What are the wavelengths of these lines?

A single slit of width \(3.0 \mu \mathrm{m}\) is illuminated by a sodium yellow light of wavelength 589 nm. Find the intensity at a \(15^{\circ}\) angle to the axis in terms of the intensity of the central maximum.

Red light (wavelength 632.8 nm in air) from a HeliumNeon laser is incident on a single slit of width \(0.05 \mathrm{mm}\). The entire apparatus is immersed in water of refractive index \(1.333 .\) Determine the angular width of the central peak.
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