Your x-ray lab has a monochromator consisting of a single crystal of nickel cut so that the (001) plane lies in the cube face. At what angle (in degrees) should the cube face be tilted with respect to the incident beam in order to select radiation with a wavelength λ=3.091 A˚? The lattice constant of nickel is a=3.53 A˚.

See your post above.

7.47*10^3

To find the angle at which the cube face should be tilted, we need to use the Bragg's law equation.

Bragg's law states: nλ = 2dsinθ

Where:
- n is the order of the diffraction
- λ is the wavelength of the incident beam
- d is the spacing between the crystal lattice planes
- θ is the angle between the incident beam and the crystal lattice planes

In this case, we are given λ = 3.091 Å and a = 3.53 Å.

First, we need to calculate the spacing between the (001) planes of the nickel crystal. For cubic lattices, the spacing can be calculated using the formula:

d = a / sqrt(h^2 + k^2 + l^2)

Where:
- a is the lattice constant
- h, k, and l are the Miller indices of the plane

In this case, the Miller indices of the (001) plane are (h, k, l) = (0, 0, 1). Substituting the values into the formula, we get:

d = a / sqrt(0^2 + 0^2 + 1^2)
d = a / sqrt(1)
d = a

So, the spacing between the (001) planes of the nickel crystal is equal to the lattice constant, which is 3.53 Å.

Now we can rearrange the Bragg's law equation to solve for the angle (θ):

θ = arcsin(nλ / 2d)

Substituting the given values, we get:

θ = arcsin(3.091Å / (2 * 3.53Å))

Using a scientific calculator or software, we can calculate the value of arcsin to find the angle in radians.

Finally, to convert from radians to degrees, we can multiply the angle by 180/π:

θ in degrees = θ in radians * (180/π)

Follow these calculations to find the angle in degrees.