Solid State Physics: X-Ray scattering

[Nikitin]

Homework Statement

https://fbcdn-sphotos-c-a.akamaihd.net/hphotos-ak-xpf1/v/t1.0-9/10980752_10204928958360361_5256945004598578166_n.jpg?oh=7fd9defb14be9a9befa0cf5032def429&oe=55576A2C&__gda__=1431534931_425423ae11039486a001b049726e1b11

Homework Equations

Charles Kittel's book on Solid State Physics, chapter 2.

The Attempt at a Solution

In problem 4a, I assumed that when the x-rays hit each atom they will scatter into a spherical wave. Then I calculated the first angle where they give constructive interference, $2 \theta$ using bragg's law.

This is correct, right?

However, in the solutions manual they say that "The first reflection to appear will be the one with the shortest reciprocal lattice vector $G_{hkl}$, or equivalently, the one corresponding to the longest plane distance $d_{hkl}$." Why is this statement true? Why does the $(001$ plane give the lowest angle of scattering?

 

[Nikitin]

Oh in 4b we get that there is no diffraction after all.. *sigh*. Apparently none of the lattice points fall on Ewald's sphere and this means Laue's diffraction condition is not satisfied. OK then why doesn't Bragg's law apply now ?

 

[Nikitin]

Is there nobody who can enlighten me? Or is perhaps my OP a bit unclear?

 

[M Quack]

(a) Bragg's law states:

lambda = 2d_hkl sin(theta), where d_hkl=2 pi/|G_hkl|

Lambda is fixed by the x-ray source, so you can find the smallest angle by looking at sin(theta)=lambda/(2d_hkl)=|G_hkl| sin(theta)/(4pi).

Which value of d or G gives the smallest theta?

(b) Setting the detector at the right angle is not the only condition that must be met. When you draw the usual sketch of Bragg diffraction, how is the angle theta drawn/measured? (See the figure next to "Bragg condition" on the Wiki page)

http://en.wikipedia.org/wiki/Bragg's_law

(c) should be clear if you understand (b). If not, read the first section of the Wiki page:

http://en.wikipedia.org/wiki/Powder_diffraction

 

[Nikitin]

(a) Bragg's law states:

lambda = 2d_hkl sin(theta), where d_hkl=2 pi/|G_hkl|

Lambda is fixed by the x-ray source, so you can find the smallest angle by looking at sin(theta)=lambda/(2d_hkl)=|G_hkl| sin(theta)/(4pi).

Which value of d or G gives the smallest theta?

(b) Setting the detector at the right angle is not the only condition that must be met. When you draw the usual sketch of Bragg diffraction, how is the angle theta drawn/measured? (See the figure next to "Bragg condition" on the Wiki page)

http://en.wikipedia.org/wiki/Bragg's_law

(c) should be clear if you understand (b). If not, read the first section of the Wiki page:

http://en.wikipedia.org/wiki/Powder_diffraction

Thank you for those questions. I get now!

(a) Smallest G_hkl gives biggest d_hkl => smallest theta.

(b) The bragg law demands that the incoming and outgoing rays have equal angles. What I used in (a) was not bragg's law, but rather a condition for constructive interference (which reduces to bragg's law when incoming and outgoing rays have equal angles).

I assume my condition for constructive interference is not good enough, because of the extinguishing behaviour of the structure factor?

(c) In a powder, there are so many microcrystals that laue's equations will be fulfilled everywhere, hence the incoming x-ray will be scattered everywhere.

 

[M Quack]

(a) yes
(b) yes. In a real 3D crystal it actually becomes a bit more complex, as the possible Bragg planes are not all parallel and can point in all directions.

Bragg reflections with zero structure factor are yet another complication that can actually help a lot in determining an unknown crystal structure from x-ray data.

(c) yes, keeping in mind that only certain values of 2theta are allowed because only certain values of d_hkl exist.