Ch. 35 - Diffraction

Giancoli Douglas - Physics for Scientists and Engineers 5th edition

Giancoli Douglas5th editionPhysics for Scientists and EngineersISBN: 9780137488179당신이 사용하는 게 아니라요?교과서 변경

Giancoli Douglas 5th edition

Ch. 35 - Diffraction

문제 53

34장, 문제 53

(II) X-rays of wavelength 0.138 nm fall on a crystal whose atoms, lying in planes, are spaced 0.315 nm apart. At what angle Φ (relative to the surface, Fig. 35–28) must the X-rays be directed if the first diffraction maximum is to be observed?

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Step 1: Recognize that this problem involves X-ray diffraction, which can be analyzed using Bragg's Law. Bragg's Law is given by:

n λ = 2 d \sin Φ

, where

n

is the order of diffraction,

λ

is the wavelength of the X-rays,

d

is the spacing between atomic planes, and

Φ

is the angle of incidence.

Step 2: Identify the given values from the problem. The wavelength of the X-rays is

λ = 0.138 nm

, the spacing between atomic planes is

d = 0.315 nm

, and the order of diffraction for the first maximum is

n = 1

Step 3: Rearrange Bragg's Law to solve for the angle

Φ

. The equation becomes:

\sin Φ = n λ / 2 d

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

n

with

1

λ

with

0.138 nm

, and

d

with

0.315 nm

. The equation becomes:

\sin Φ = 1 0.138 / 2 0.315

Step 5: Calculate the value of

\sin Φ

using the substituted values. Then, use the inverse sine function (

\sin ⁻ 1

) to find the angle

Φ

. Ensure the angle is expressed in degrees or radians as required.

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주요 개념

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Bragg's Law

Bragg's Law relates the wavelength of X-rays to the angle of diffraction and the spacing between atomic planes in a crystal. It is expressed as nλ = 2d sin(Φ), where n is the order of the maximum, λ is the wavelength, d is the distance between planes, and Φ is the angle of incidence. This law is fundamental in determining the conditions under which constructive interference occurs, leading to observable diffraction patterns.

Diffraction

Diffraction is the bending of waves around obstacles and the spreading of waves when they pass through small openings. In the context of X-rays and crystals, diffraction occurs when X-rays interact with the periodic structure of the crystal lattice, resulting in interference patterns. The angles at which these patterns appear depend on the wavelength of the X-rays and the spacing of the crystal planes.

Interference

Interference is a phenomenon that occurs when two or more waves overlap, resulting in a new wave pattern. In X-ray diffraction, constructive interference leads to bright spots (maxima) at specific angles, while destructive interference results in dark spots (minima). Understanding interference is crucial for analyzing diffraction patterns and determining the angles at which maxima occur, as described by Bragg's Law.