2.Work out the solutions from scratch on your own. 3.Turn ...
Transcript of 2.Work out the solutions from scratch on your own. 3.Turn ...
Exam: You can make up some points: 1. Choose 2 problems; blank exam is posted.2. Work out the solutions from scratch on your own.3. Turn in the problems by Thursday in class or Friday
afternoon 3:30-4 PM; I will arrange be at my office or you can email if necessary. Not in my mailbox
4. You will get 60% of the made-up points back.
Homework: Problem set 8 due Tomorrow. I am looking for volunteers for problems 1 and 2 for Thursday presentation.
Reading: This week we continue with chapter 16 (continuum systems densities of states; Debye model for vibrations). Next up, phase transformations (chapters 8-9).
Notes:
Canonical ensemble Recall:
πΉ β‘ βπ!ππππ
π = βππΉππ ",$
πΈ = π!π%πππ πππ
π = #!"#"$! %
πΈπ₯π βπΈ%/ππPartition function
Etc β¦.
π =1π!.%
π%Last time: Z as a product for independent sub-systems, e.g.:
Equipartition theorem
πΈ = ππ&
β’ Classical systems (continuum not discrete energies)β’ Works in cases having separable variables.β’ Requires energy quadratic in position and/or momentum:
or:
Result:
(1/2)kT for each βdegree of freedomβ f.
π =π2ππ
π =1β'(
5π'(ππ'(ππ)*+ π = β% π%; includes all cross terms.
systems of distinguishable particles, non-interacting.Classical partition function
π =1π!.%
π% systems of indistinguishableparticles, still non-interacting case.
π% =π,
&
2π+π π’&
2
π-%./ =1β>ππππ’π)*
0!&12
3,!&
(
Vibrational term in product-form partition function, Z:
or π% =π&
2π+π π’&
2c.m. coordinates & reduced mass
N independent oscillators(Einstein solid approximation, ideal gas molecules, etc.) all with same π = π /π = βπ!β
β’ Classical systems (continuum not discrete energies)β’ Works in cases having separable variables.β’ Requires energy quadratic in position and/or momentum (or
other coordinate)β’ Vibrations, rotations, translational states. Rotational case in
text, wonβt show here.
Equipartition theorem:
π"#$% =ππβπ
&πΈ = πππ
effectively f = 2
Harmonic oscillator systems (N independent 1D oscillators)
π% =π,
&
2π+π π’&
2
π-%./ =1β>ππππ’π)*
0!&12
3,!&
(
Vibrational term in product-form partition function, Z:
or π% =π&
2π+π π’&
2c.m. coordinates & reduced mass
π"#$% =ππβπ
&πΈ = πππ
Quantum version / general case:
π-%./ =.%
π% = #456
7
π)*4β9(
Harmonic oscillator systems (N independent 1D oscillators)
classical-limit solution
Harmonic oscillator systems (N independent 1D oscillators)
Quantum vibrational partition function
π-%./ =.%
π% = π)*β9& #
456
7
π)*4β9(
= π)*β9(&
11 β π)*β9
(
βππ*β9 β 1
β‘ βπ π
πΈ =βππ2
+ πβπ
π*β9 β 1
π = Bose-Einstein occupation number (photon statistics) we saw before, derived using S & U to find T.
β’ Z shown with zero point motion term.
β’ Indistinguishable cases: we grouped 1/π! factor with Ztrans last time.
β’ Equivalent to β3N + q β 1β counting method for fixed-U case, large-Nlimit.
β’ Here, easy to extend to a distribution of different oscillator frequencies.
zero point term no effect on entropy
Harmonic oscillator systems (N independent 1D oscillators)
Quantum version
π-%./ =.%
π% = π)*β9& #
456
7
π)*4β9(
= π)*β9(&
11 β π)*β9
(
πΈ =βππ2
+ πβπ
π*β9 β 1
Einstein Ann Phys 1907 [for solids; 3N identical oscillators. Specific heat of diamond fit to π = 1310 K.]
βππ*β9 β 1
β‘ βπ π
Harmonic oscillator systems (N independent 1D oscillators)
Quantum version
πΈ =βππ2
+ πβπ
π*β9 β 1
π-%./ =.%
π% = π)*β9& #
456
7
π)*4β9(
= π)*β9(&
11 β π)*β9
(
βππ*β9 β 1
β‘ βπ π
Debye model: T3 behavior at low T agrees well with data.
<< assumes a specific distribution of vibrational frequencies (normal modes).
Phonons: quantized lattice vibrations in crystals.
β’ Liquids and non-crystal solids: have similar modes.
β’ Einstein: independent 3D oscillators, same Οo.
β’ Debye: Phonons are normal modes in a connected harmonic lattice.
β’ Debye-theory solutions identical to sound waves , π = ππ (exact in low-frequency limit); also map onto blackbody-radiation photons.
Phonons: quantized lattice vibrations in crystals.
β’ Liquids and non-crystal solids: have similar modes.
β’ Einstein: independent 3D oscillators, same Οo.
β’ Debye: Phonons are normal modes in a connected harmonic lattice.
β’ Debye-theory solutions identical to sound waves , π = ππ (exact in low-frequency limit); also map onto blackbody-radiation photons.
β’ Except: mode countingrequires finite number of phonon modes, and 3 polarizations, not 2.
Photons (blackbody radiation)
Photons vs. Phonons:
Photons:β’ Cavity modesβ’ 2 polarizationsβ’ π = ππ.β’ Extend to π β β.β’ π)%:;/β)9" free space solutionβ’ Bose statistics (π = 0).β’ Energies quantized, βπ(π + =
&).
β’ Speed of light: c.
Phonons:β’ elastic (standing) wavesβ’ 3 polarizationsβ’ π β ππ, ππ±πππ ππ¨π« π₯π¨π° πβ’ Bounded: N values of k.β’ π)%:;/β)9" [or sin π \ π sin(ππ‘)]β’ Bose statistics (π = 0).β’ Energies quantized, βπ(π + =
&).
β’ Speed of sound: c
Phonons & mode counting:
N k-vectors in 1D, leads to a cutoff for phonon wavenumber.
3N total phonon modes in general 3D crystal.
3) Wave-vector cutoff: maximum wavenumber ~ frequency in THz range1) General elastic solid (isotropic case):
wave equation
has wave solutions, π = ππ.(actually may have 2 or more frequencies)
2) Quantization of energies: wonβt show this; result is analogous to familiar SHO solution in quantum mechanics,
πΈ = βπ(π + ()).
State counting: I showed this slide before for photons.
β’ Start with cavity modes in a box with perfectly conducting sides, dimensions L.
πΈ> β πππ π>?@π₯ π ππ πA
?@π¦ π ππ πB
?@π§ etc.
Ξπ = ?@
-> π: =?@
'
Octant of sphere;but with 8Γ state density.(3D sphere radius will go to infinity)
Cavity modeCounting: one TM + one TE per k-vector
E-field
Consider continuum limit (large cavity, very small βk)Also recall π = πc
π = ##CC 1DE$!
βπ%π*β9" β 1
βΉ 2i6
7 π2ππ&πππ'
βπππ*β:F β 1
# modes in thinshell, thickness ππ = ππ/π
One k-vector per volume element;same as βPhase space volumeβ h3/8
Mode counting directly in k-space(similar to number space for Einstein summation, ch. 15)
πππππππ
Density of states: for summations involving only π (or E).
Ξπ = ?@
-> π: =?@
'for octant;
= &?@
'for complete sphere traveling-waves
π = ##CC 1DE$!
βπ%π*β9" β 1
βΉ 2i6
7 π2ππ&πππ'
βπππ*β:F β 1
# modes in thinshell, thickness ππ = ππ/π
Example for energy sum:
π = ##CC 1DE$!
βπ%π*β9" β 1
βΉ πi6
7ππΓ #π π‘ππ‘ππ
ππ (π, π + ππ) Γβπ
π*β9 β 1
πππππππ
β‘ i6
7 βππ· π πππ*β9 β 1