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Previous lecture 2010.09.13 Next lecture 2010.09.20 to index

links - about the waveshttp://www.acoustics.salford.ac.uk/feschools/waves/wavetypes.htm- the Mexican Wave on this page ADOBE FLASHOur version with browser based program - picture in the page:http://web.ift.uib.no/AMOS/ajs/(will be extended, check later ... )

Debye Model - derivationFollowing the Einstein model ( see2010.09.13/ at the end ) we use the average excitation number < n > derived thereUnlike professor Debye in about 1909 we use a trick to derive the density of modes.We follow his assumption of quadratic dependence on the frequency (remember - the 3-dim case)and then simply demand that the integral over frequencies, with the density, i.e. equivalent of the sumover all modes, gives the number of modes, 3 N (N is the number of atoms, in 1 mol it is Avogadro's number)

http://web.ift.uib.no/AMOS/PHYS208/2010.09.13/index.htmlhttp://web.ift.uib.no/AMOS/PHYS208/2010.09.13/index.htmlhttp://web.ift.uib.no/AMOS/PHYS208/2010.09.20/index.htmlhttp://web.ift.uib.no/AMOS/PHYS208/2010.09.20/index.htmlhttp://web.ift.uib.no/AMOS/PHYS208/index.htmlhttp://web.ift.uib.no/AMOS/PHYS208/index.htmlhttp://www.acoustics.salford.ac.uk/feschools/waves/wavetypes.htmhttp://www.acoustics.salford.ac.uk/feschools/waves/wavetypes.htmhttp://web.ift.uib.no/AMOS/ajs/http://web.ift.uib.no/AMOS/ajs/http://web.ift.uib.no/AMOS/ajs/http://web.ift.uib.no/AMOS/PHYS208/2010.09.13/http://web.ift.uib.no/AMOS/PHYS208/2010.09.13/http://web.ift.uib.no/AMOS/PHYS208/2010.09.13/http://web.ift.uib.no/AMOS/PHYS208/2010.09.13/http://web.ift.uib.no/AMOS/ajs/http://www.acoustics.salford.ac.uk/feschools/waves/wavetypes.htmhttp://web.ift.uib.no/AMOS/PHYS208/index.htmlhttp://web.ift.uib.no/AMOS/PHYS208/2010.09.20/index.htmlhttp://web.ift.uib.no/AMOS/PHYS208/2010.09.13/index.html

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1-einstein.pngDensity of states, g(omega) is thus established

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Now the energy U(T) at the given temperature T is simply the some of energy in all the posible modes

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2-debye-internal-U.pngAnd we started to evaluate the heat capacity straight away: exercise: evaluate the total energy U(T) - just for intererst;we do not need that

Transformations of the expressions - DIMENSION ANALYSIS - it must be ENERGY / TEMPERATURE,

the same physical dimension as the Boltzmann constant k ( or kB ) in the average thermal energy kT)

and Nk = R - the gas constant

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debye-dU_dT=C.pngThe Debye function is defined by an integral as shown above.The integral cannot be given in an analytic form. It used to be tabulated seehttp://en.wikipedia.org/wiki/Debye_function and also http://mathworld.wolfram.com/DebyeFunctions.html

How to construct the C(T) ? Using the Debye function (integral)

http://en.wikipedia.org/wiki/Debye_functionhttp://en.wikipedia.org/wiki/Debye_functionhttp://mathworld.wolfram.com/DebyeFunctions.htmlhttp://mathworld.wolfram.com/DebyeFunctions.htmlhttp://mathworld.wolfram.com/DebyeFunctions.htmlhttp://en.wikipedia.org/wiki/Debye_function

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function-construct-C=C.png

Limiting cases of C(T) - for very large temperaturesand for the low temperatures

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5-debye-limit_T_large_Dulong.png

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Limiting case of C(T) - for the low temperatures

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6-debye-limit_T_small_T3_behavior.png

This is an interesting exercise:When you bring to contact very cold and very "hot" ( hot like -200 Centigrade )The new temperature will not be in the middle, but close to the "hot" - the hotter the closer

Work this out better than we did here....

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7-debye-Calorimetry.png

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Debye temperatures and the table ( ../debye/table50.jpg )

8-debye-Discussing_Table_T_Debye.pngIn this discussion we have used the table in ../debye/ - best is../debye/table50.jpg

Next lecture:Numerical exercise with the Debye functionBack to the vibration modes - quantization; Phonons

http://web.ift.uib.no/AMOS/PHYS208/debye/table50.jpghttp://web.ift.uib.no/AMOS/PHYS208/debye/table50.jpghttp://web.ift.uib.no/AMOS/PHYS208/debye/http://web.ift.uib.no/AMOS/PHYS208/debye/table50.jpghttp://web.ift.uib.no/AMOS/PHYS208/debye/table50.jpghttp://web.ift.uib.no/AMOS/PHYS208/debye/table50.jpghttp://web.ift.uib.no/AMOS/PHYS208/debye/table50.jpghttp://web.ift.uib.no/AMOS/PHYS208/debye/http://web.ift.uib.no/AMOS/PHYS208/debye/table50.jpg

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Previous lecture 2010.09.13 Next lecture 2010.09.20

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