Calculating Angles for Lowest First 4 Reflections of Diamond

[debwaldy]

Homework Statement

Diamond crystallises with a cubic unit cell and the lattice is face centred.Calculate the respective angles(in degrees) at which the lowest first four reflections are allowed from a polycrystalline specimen of diamond(given (wavelength)^2/4a^2 = 0.0467,where a = unit cell parameter)

Homework Equations

n*wavelength = 2dsin(theta)
d^2 = a^2/(h^2+k^2+l^2)

The Attempt at a Solution

I tried working backwards and used the fact that i knew that 0.0467 is the HCF of the equation: sin^2(theta) = ((wavelength)^2/4a^2)*(h^2+k^2+l^2)
because it is fcc it means that 4r = (2^0.5)*a

I took the square root of 0.0467 = and then the inverse sin of this in order to obtain one angle,:

(0.0467)^0.5 = 0.2161
sin^-1(0.2161) = 12.48degrees
The problem is i don't know how to calculate the other three angles?
should i multiply 0.0467 by 2,then repeat the process to get the next angle or?any help would be much appreciated
thanks debs

 

[nate808]

Dear debs,

To calculate the other three angles, you can use the fact that for a cubic unit cell, the (hkl) plane is equivalent to the (klh), (lkh), and (hlk) planes. This means that the angles for the first four reflections will be the same for each of these planes.

So, for the first reflection, you have already calculated the angle as 12.48 degrees. For the second reflection, you can use the same value of 0.0467, but now for the plane (020) (since it is equivalent to (200), (002), and (020)). This will give you the same angle of 12.48 degrees.

You can repeat this process for the third and fourth reflections, using the planes (111) and (022) (equivalent to (112), (121), (211) and (220), (202), (022), respectively).

Hope this helps!

 

[Blueshift5]

I would suggest approaching this problem using the Bragg equation, n*wavelength = 2dsin(theta), where n is the order of reflection, d is the interplanar spacing, and theta is the angle of incidence. Since we are looking for the lowest first four reflections, we can start with n=1 and calculate the corresponding angle of incidence for each reflection.

Using the given equation, d^2 = a^2/(h^2+k^2+l^2), we can substitute in the values for a (unit cell parameter) and h, k, l (Miller indices) for a cubic system. For the first reflection, we can assume h=0, k=0, l=1, giving us d = a/2. Plugging this into the Bragg equation, we get:

n*wavelength = 2dsin(theta)
1*wavelength = 2(a/2)sin(theta)
wavelength = asin(theta)

Now, we can use the given value for (wavelength)^2/4a^2 = 0.0467 and solve for theta:

0.0467 = a^2sin^2(theta)
sin^2(theta) = 0.0467/a^2
sin(theta) = 0.2161 (taking the positive root)
theta = 12.48 degrees

This is the angle for the first reflection. To calculate the angles for the next three reflections, we can simply multiply the wavelength by 2 (for n=2), 3 (for n=3), and 4 (for n=4) and repeat the same process to solve for theta. This will give us the angles for the lowest first four reflections from a polycrystalline specimen of diamond.

Overall, it is important to approach this problem systematically using the appropriate equations and considering the crystal structure of diamond. Also, keep in mind that the given value for (wavelength)^2/4a^2 = 0.0467 is the HCF (highest common factor) and not the actual value for theta. I hope this helps. Good luck with your homework!