sommerfeld method 11.10.2010.docx

7/29/2019 Sommerfeld Method 11.10.2010.docx

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Sommerfeld Method 11.10.2010

We have repeated the last lecture with stress on this:

How to

so_0.pngThe main point is evaluating the N(E) (number of accomodated electrons) up to a given E (energy)The colored "expression" can be anything in the sum (integral) above

The argumet goes like this:

sum -> sum with the proper deltas which are compensated by the 1/Deltas outside the sum

Then keep the 1/Deltas outside the sumand concentrate on the Sum with deltas; Taking this by itself, it will converge to an integral

when you make Deltas infinitesimalThe Deltas outside are not to be made infinitesimal, SINCE they connect to the original sum
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This is sometimes not so easy to keep in mind .... ( the sum itself would go to infinity ... )

From this follows the dependence of Fermi energy on N/V - the density (see last time ... )

We have also played with the Fermi Function matlab program(showing that the parameter mu can be adjusted in iterations- the program sums the occupation numbers - adjust mu - and stops when the Sum is equal the number of electrons

The program will be especially interesting later - with ENERGY GAP)../FERMI/Fermi function Matlab toys ( ferm2.m and fermgap.m - the others are older versions and other scripts)
http://web.ift.uib.no/AMOS/PHYS208/FERMI/http://web.ift.uib.no/AMOS/PHYS208/FERMI/http://web.ift.uib.no/AMOS/PHYS208/FERMI/


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Sommerfeld method for evaluation of integrals with Fermi DistributionRecord of the lecture follows,


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more details are given in the scans below this note (which we discussed in the lecture)


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so_1.png


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so_2.png

The above formula can be viewed as a method. It is applicable to any of the necessary expressions,involving either g(E) or Eg(E)


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so_3.pngHere we have summarized also the final result from the notes belowThe heat capacity is indeed proportional to the ratio T / TF - and TF is VERY BIG so that heat capacity is very SMALL(about 100 times smaller than the classical one

Note - the heat capacity is LINEARY dependent on T. For PHONONS it was dependent (Debye) on T3

Thus, there will allways be a region of very small temperatures where AT is larger than B T3 (which region depends on A,B )

So for normal temperatures, the electron contribution to C (heat capacity) is negligibleBut for extremely low temperatures it becomes dominating

The Sommerfeld method - details are in the three scans below


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EVALUATION SCAN


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In this scan the evaluation is summarized with all the extra approximation highlighted


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In the last based on exam texts the whole story is also summarized

EXAMS 1990-2000 - text copied


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to_1.png


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to_2.png

to_3.png

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sommerfeld method 11.10.2010.docx

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