- #1
baseballfan_ny
- 92
- 23
- Homework Statement
- A beam of electrons of mass m and speed v is incident on the surface of a 2-dimensional square crystal of lattice spacing a. It makes an angle ##\theta = 45^{o}## with the surface. The Bragg diffracted beam leaves at an angle of ##\theta^{'}## with the surface, as shown.
(a) When ##\theta^{'} = 45^{o}##, what is the smallest possible value of v?
(b) [This question requires some real physical thought. It is not in any way a plug-and-chug question.] Give two other possible values for the angle ##\theta^{'}##, and give the smallest possible value of v for each of these.
- Relevant Equations
- ##\theta_{incident} = \theta_{reflected}##
##n\lambda## = 2d\sin(\theta)##
##p = \frac {h} {\lambda}##
Part (a)
Ok so for (a) ##\theta_{incident} = \theta_{reflected}##, so I assume I could just consider the horizontal planes in these atoms.
##n\lambda = 2a\sin(\theta)##
##p = \frac {h} {\lambda}##
##\frac {nh} {2amv} = \sin(\theta)##
## v = \frac {nh} {2am\sin(\theta)}##
I suppose the smallest value of v would be when n = 1, so ## v = \frac {h} {2am\sin(\theta)}##
Part (b)
Ok so here ##\theta_{incident} \neq \theta_{reflected}##, so I suppose I need to look at a different plane. Not sure if this is the right approach, but I took an arbitrary plane at an arbitrary angle ##\alpha## from the horizontal like so:
I've shown the original beams at angles ##\theta## and ##\theta^{'}## from the horizontal, the arbitrary plane is at an angle ##\alpha## from the horizontal. ##\beta_{i}## and ##\beta_{r}## are the angles of the beams with respect to the arbitrary plane.
So for this arbitrary plane, we need ##\beta_{i} = \beta_{f}##
So ##\theta + \alpha = \theta^{'} - \alpha##
So I've got this with
##n\lambda = 2a\sin(\theta)##
##p = \frac {h} {\lambda}##
Not sure I have enough equations because I think the most I can do here is
## \frac {nh} {mv} = 2a\sin(\theta + \alpha) = 2a\sin(\theta^{'} - \alpha) ##
And I don't think I have enough equations to solve for ##\theta^{'}## or ##\alpha.## Or I took the wrong approach?
Thanks in advance.