GR9277 #97


Problem


\prob{97}
Lattice forces affect the motion of electrons in a metallic crystal, so that the relationship between the energy E and the wave number k is not the classical equation , where m is the electron mass. Instead, it is possible to use an effective mass given which of the following?






Advanced Topics}Solid State Physics
This is a result one remembers by heart from a decent solid state physics course. It has to do with band gaps, which is basically the core of such a course.
Then again, one can easily derive it from scratch upon recalling some basic principles: , , where k is the wave vector, E is the energy, m is the mass, and p is the momentum.
From the above, one has .
Set the two 's equal to get . Cancel out the 's to get , after differentiating with respect to k on both sides.
Alternatively, one can try it Kittel's way:
Start with . Then, . Thus, the effective mass is defined by .


Alternate Solutions 
drdoctor 20130917 08:48:06  Here's how I solved the problem:
Presumably, you want m* to be written in terms of some sort of constantsie. m* shouldn't depend on k or E explicitly. However, or could potentially be constants, so they could be included in m*. So, that eliminates A, B, and C. It's fairly easy to see from here that answer D is simply the second derivative of E with respect to k for the equation given in the problem statement, except that you need to rearrange to solve for m after taking the double derivative:
Therefore:
You could also use units to eliminate E.   jonestr 20051112 00:50:59  a quick dimnesional analysis wors well here
 

Comments 
honeybunches 20160912 20:04:35  Here is a quick approach: If the dispersion relationship was the classical equation, we would expect the effective mass to be equal to the electron mass. This very quickly gets you (D)   sina2 20131008 04:01:01  I will chose D. I'm always caring to not forget in quantum isn't same is . This is so important. They maybe don't commute.   drdoctor 20130917 08:48:06  Here's how I solved the problem:
Presumably, you want m* to be written in terms of some sort of constantsie. m* shouldn't depend on k or E explicitly. However, or could potentially be constants, so they could be included in m*. So, that eliminates A, B, and C. It's fairly easy to see from here that answer D is simply the second derivative of E with respect to k for the equation given in the problem statement, except that you need to rearrange to solve for m after taking the double derivative:
Therefore:
You could also use units to eliminate E.   nitin 20061121 00:45:56  Let be the group velocity of the electron. Then
, and
where is the effective mass. The answer (D) follows.   comorado 20061027 13:07:45  You wrote
Must be:   jonestr 20051112 00:50:59  a quick dimnesional analysis wors well here
Jeremy 20071103 15:33:07 
Dimensional analysis will only narrow it down to choices (A) and (D).

Poop Loops 20081025 20:39:17 
Yeah, but then there's a 1/2 out front, which doesn't make sense.

ramparts 20090806 23:30:04 
Yep  it's possible I screwed something up (and I didn't bother looking at E after I looked at A through D :P) but I'm pretty sure the first three were not units of mass.

alemsalem 20100926 08:28:20 
you might reason that the mass shouldn't be dependent on momentum (k) otherwise it wouldn't be useful to use an "effective mass" so it cannot be A.
but i admit when i did the exam i solved this one based on units then just picked A as a guess

flyboy621 20101023 05:26:41 
Sort of, but that only narrows it down to A or D. Of course it's worth guessing at that point...

asdfuogh 20111005 19:04:12 
I would pick D) if I had to pick from A) and D)... A) looks like the original mass equation, except with differentials. Not a great reason, but still..

eighthlock 20130915 12:39:01 
Dimensional analysis does not distinguish between options (A) and (D)

  jonestr 20051112 00:50:06   

Post A Comment! 
You are replying to:
Here's how I solved the problem:
Presumably, you want m* to be written in terms of some sort of constantsie. m* shouldn't depend on k or E explicitly. However, or could potentially be constants, so they could be included in m*. So, that eliminates A, B, and C. It's fairly easy to see from here that answer D is simply the second derivative of E with respect to k for the equation given in the problem statement, except that you need to rearrange to solve for m after taking the double derivative:
Therefore:
You could also use units to eliminate E.

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