Diffraction of X rays with l = 131.5 pm occurred at an angle of 25.5 degrees by a crystal of aluminum. Assuming first-order diffraction, what is the interplanar spacing in aluminum? (LO 12.2) (a) 76.4 pm (b) 183.1 pm (c) 305.5 pm (d) 152.7 pm

검증된 단계별 안내

Identify the given values: wavelength (\

lambda = 131.5 pm\

) and diffraction angle (\

theta = 25.5^\(\circ\)\

Recognize that the problem involves the use of Bragg's Law, which is given by \

n\(\lambda\) = 2d\(\sin\)(\(\theta\))\

, where \

is the order of diffraction, \

is the interplanar spacing, and \

\(\theta\)\

is the angle of diffraction.

Since it is a first-order diffraction (\

n=1\

), substitute \

and the other known values into Bragg's Law.

Solve the equation for \

to find the interplanar spacing: \

d = \(\frac{\lambda}{2\sin(\theta)}\)\

Calculate \

using the values of \

\(\lambda\)\

and \

\(\theta\)\

to find the interplanar spacing in the aluminum crystal.

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Bragg's Law

Bragg's Law relates the wavelength of X-rays to the angle of diffraction and the interplanar spacing in a crystal. It is expressed as nλ = 2d sin(θ), where n is the order of diffraction, λ is the wavelength, d is the interplanar spacing, and θ is the angle of diffraction. This law is fundamental in determining the arrangement of atoms within a crystal structure.

Interplanar Spacing

Interplanar spacing (d) refers to the distance between parallel planes of atoms in a crystal lattice. It is a critical parameter in crystallography, influencing how X-rays are diffracted. The value of d can be calculated using Bragg's Law, which helps in identifying the crystal structure and properties of materials.

First-Order Diffraction

First-order diffraction occurs when n = 1 in Bragg's Law, indicating the first set of parallel planes that diffract X-rays. This is the simplest case and provides the most direct relationship between the wavelength, angle, and interplanar spacing. Understanding first-order diffraction is essential for accurately calculating the interplanar spacing in crystalline materials.