FCC Lattice Miller Indices of the First Bragg Peak:

The first Bragg peak in an FCC lattice corresponds to the smallest Bragg angle. The Miller indices of the first Bragg peak can be determined using the following steps:

Step 1: Determine the Bragg condition

The Bragg condition is given by:

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

where d is the lattice spacing, theta is the Bragg angle, n is an integer, and lambda is the wavelength of the incident X-ray or electron beam.

Step 2: Determine the lattice spacing

The lattice spacing for an FCC lattice is given by:

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

where a is the lattice constant and h, k, and l are the Miller indices.

Step 3: Substitute the values and simplify

Substituting the values of d and sin(theta) in the Bragg condition equation, we get:

2*a*sqrt(h^2+k^2+l^2)*sin(theta) = n*lambda

Simplifying the equation, we get:

h^2+k^2+l^2 = (n*lambda/(2*a*sin(theta)))^2

Step 4: Determine the Miller indices

The Miller indices for the first Bragg peak correspond to the smallest values of h^2+k^2+l^2 that satisfy the Bragg condition. Therefore, the Miller indices for the first Bragg peak in an FCC lattice are:

111

which corresponds to a Bragg angle of approximately 35.3 degrees.

Conclusion:

In conclusion, the Miller indices of the first Bragg peak in an FCC lattice are 111, which corresponds to a Bragg angle of approximately 35.3 degrees. The determination of these Miller indices involves the use of the Bragg condition, lattice spacing, and simplification of the equation.