Ch 36: Diffraction

Young & Freedman Calc - University Physics 14th Edition

Young & Freedman Calc14th EditionUniversity PhysicsISBN: 9780321973610당신이 사용하는 게 아니라요?교과서 변경

Young & Freedman Calc 14th Edition

Ch 36: Diffraction

문제 34a

36장, 문제 34a

If the planes of a crystal are 3.50 Å (1 Å = 10^-10 m = 1 Ångstrom unit) apart, what wavelength of electromagnetic waves is needed so that the first strong interference maximum in the Bragg reflection occurs when the waves strike the planes at an angle of 22.0°, and in what part of the electromagnetic spectrum do these waves lie?

검증된 단계별 안내

Step 1: Understand the problem. This is a Bragg diffraction problem, where we use Bragg's law to determine the wavelength of electromagnetic waves that produce constructive interference. Bragg's law is given by:

n λ =2 d sin θ

, where

n

is the order of the diffraction (here, the first maximum means

n =1

λ

is the wavelength,

d

is the spacing between the planes, and

θ

is the angle of incidence.

Step 2: Write down the known values. The spacing between the planes is

d =3.50Å

(convert this to meters:

d =3.50×10

^-10 m). The angle of incidence is

θ =22.0°

. The order of diffraction is

n =1

Step 3: Rearrange Bragg's law to solve for the wavelength

λ

. Substituting

n =1

, the equation becomes:

λ =2 d sin θ

Step 4: Substitute the known values into the equation. Use

d =3.50×10

^-10 m and

θ =22.0°

. Ensure that the angle is converted to radians before calculating the sine function:

θ =22.0°×π/180

radians.

Step 5: Once the wavelength

λ

is calculated, determine the part of the electromagnetic spectrum it belongs to. Compare the wavelength to the ranges of the electromagnetic spectrum (e.g., X-rays, ultraviolet, visible light, etc.).

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주요 개념

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Bragg's Law

Bragg's Law relates the wavelength of electromagnetic waves to the angle of incidence and the distance between crystal planes. It is expressed as nλ = 2d sin(θ), where n is the order of the maximum, λ is the wavelength, d is the distance between planes, and θ is the angle of incidence. This principle is fundamental in understanding how waves interact with crystal structures to produce interference patterns.

Interference Patterns

Interference patterns arise when two or more waves overlap, leading to regions of constructive and destructive interference. In the context of Bragg reflection, constructive interference occurs when the path difference between waves reflected from adjacent planes is an integer multiple of the wavelength. This results in observable maxima, which are critical for determining the wavelength of the incident waves.

Electromagnetic Spectrum

The electromagnetic spectrum encompasses all types of electromagnetic radiation, ranging from radio waves to gamma rays. Each type of radiation is characterized by its wavelength and frequency. In this problem, determining the wavelength of the waves that produce the first strong interference maximum will help identify where these waves fall within the spectrum, such as in the X-ray or visible light regions.