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 ...
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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:
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Canonical ensemble Recall:
𝐹 ≡ −𝑘!𝑇𝑙𝑛𝑍
𝑆 = −𝜕𝐹𝜕𝑇 ",$
𝐸 = 𝑘!𝑇%𝜕𝜕𝑇 𝑙𝑛𝑍
𝑍 = #!"#"$! %
𝐸𝑥𝑝 −𝐸%/𝑘𝑇Partition function
Etc ….
𝑍 =1𝑁!.%
𝑍%Last time: Z as a product for independent sub-systems, e.g.:
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Equipartition theorem
𝐸 = 𝑐𝑞&
• Classical systems (continuum not discrete energies)• Works in cases having separable variables.• Requires energy quadratic in position and/or momentum:
or:
Result:
(1/2)kT for each “degree of freedom” f.
𝑈 =𝑓2𝑘𝑇
𝑍 =1ℎ'(
5𝑑'(𝑟𝑑'(𝑝𝑒)*+ 𝑍 = ∏% 𝑍%; includes all cross terms.
systems of distinguishable particles, non-interacting.Classical partition function
𝑍 =1𝑁!.%
𝑍% systems of indistinguishableparticles, still non-interacting case.
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𝑈% =𝑝,
&
2𝑚+𝜅𝑢&
2
𝑍-%./ =1ℎ>𝑑𝑝𝑑𝑢𝑒)*
0!&12
3,!&
(
Vibrational term in product-form partition function, Z:
or 𝑈% =𝑝&
2𝑚+𝜅𝑢&
2c.m. coordinates & reduced mass
N independent oscillators(Einstein solid approximation, ideal gas molecules, etc.) all with same 𝜔 = 𝜅/𝑚 = “𝜔!”
• Classical systems (continuum not discrete energies)• Works in cases having separable variables.• Requires energy quadratic in position and/or momentum (or
other coordinate)• Vibrations, rotations, translational states. Rotational case in
text, won’t show here.
Equipartition theorem:
𝑍"#$% =𝑘𝑇ℏ𝜔
&𝐸 = 𝑁𝑘𝑇
effectively f = 2
Harmonic oscillator systems (N independent 1D oscillators)
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𝑈% =𝑝,
&
2𝑚+𝜅𝑢&
2
𝑍-%./ =1ℎ>𝑑𝑝𝑑𝑢𝑒)*
0!&12
3,!&
(
Vibrational term in product-form partition function, Z:
or 𝑈% =𝑝&
2𝑚+𝜅𝑢&
2c.m. coordinates & reduced mass
𝑍"#$% =𝑘𝑇ℏ𝜔
&𝐸 = 𝑁𝑘𝑇
Quantum version / general case:
𝑍-%./ =.%
𝑍% = #456
7
𝑒)*4ℏ9(
Harmonic oscillator systems (N independent 1D oscillators)
classical-limit solution
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Harmonic oscillator systems (N independent 1D oscillators)
Quantum vibrational partition function
𝑍-%./ =.%
𝑍% = 𝑒)*ℏ9& #
456
7
𝑒)*4ℏ9(
= 𝑒)*ℏ9(&
11 − 𝑒)*ℏ9
(
ℏ𝜔𝑒*ℏ9 − 1
≡ ℏ𝜔 𝑛
𝐸 =ℏ𝜔𝑁2
+ 𝑁ℏ𝜔
𝑒*ℏ9 − 1
𝑛 = Bose-Einstein occupation number (photon statistics) we saw before, derived using S & U to find T.
• Z shown with zero point motion term.
• Indistinguishable cases: we grouped 1/𝑁! factor with Ztrans last time.
• Equivalent to “3N + q – 1” counting method for fixed-U case, large-Nlimit.
• Here, easy to extend to a distribution of different oscillator frequencies.
zero point term no effect on entropy
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Harmonic oscillator systems (N independent 1D oscillators)
Quantum version
𝑍-%./ =.%
𝑍% = 𝑒)*ℏ9& #
456
7
𝑒)*4ℏ9(
= 𝑒)*ℏ9(&
11 − 𝑒)*ℏ9
(
𝐸 =ℏ𝜔𝑁2
+ 𝑁ℏ𝜔
𝑒*ℏ9 − 1
Einstein Ann Phys 1907 [for solids; 3N identical oscillators. Specific heat of diamond fit to 𝜔 = 1310 K.]
ℏ𝜔𝑒*ℏ9 − 1
≡ ℏ𝜔 𝑛
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Harmonic oscillator systems (N independent 1D oscillators)
Quantum version
𝐸 =ℏ𝜔𝑁2
+ 𝑁ℏ𝜔
𝑒*ℏ9 − 1
𝑍-%./ =.%
𝑍% = 𝑒)*ℏ9& #
456
7
𝑒)*4ℏ9(
= 𝑒)*ℏ9(&
11 − 𝑒)*ℏ9
(
ℏ𝜔𝑒*ℏ9 − 1
≡ ℏ𝜔 𝑛
Debye model: T3 behavior at low T agrees well with data.
<< assumes a specific distribution of vibrational frequencies (normal modes).
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Phonons: quantized lattice vibrations in crystals.
• Liquids and non-crystal solids: have similar modes.
• Einstein: independent 3D oscillators, same ωo.
• Debye: Phonons are normal modes in a connected harmonic lattice.
• Debye-theory solutions identical to sound waves , 𝜔 = 𝑘𝑐 (exact in low-frequency limit); also map onto blackbody-radiation photons.
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Phonons: quantized lattice vibrations in crystals.
• Liquids and non-crystal solids: have similar modes.
• Einstein: independent 3D oscillators, same ωo.
• Debye: Phonons are normal modes in a connected harmonic lattice.
• Debye-theory solutions identical to sound waves , 𝜔 = 𝑘𝑐 (exact in low-frequency limit); also map onto blackbody-radiation photons.
• Except: mode countingrequires finite number of phonon modes, and 3 polarizations, not 2.
Photons (blackbody radiation)
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Photons vs. Phonons:
Photons:• Cavity modes• 2 polarizations• 𝜔 = 𝑘𝑐.• Extend to 𝜔 → ∞.• 𝑒)%:;/⃗)9" free space solution• Bose statistics (𝜇 = 0).• Energies quantized, ℏ𝜔(𝑛 + =
&).
• Speed of light: c.
Phonons:• elastic (standing) waves• 3 polarizations• 𝝎 ≅ 𝒌𝒄, 𝐞𝐱𝐚𝐜𝐭 𝐟𝐨𝐫 𝐥𝐨𝐰 𝒌• Bounded: N values of k.• 𝑒)%:;/⃗)9" [or sin 𝑘 \ 𝑟 sin(𝜔𝑡)]• Bose statistics (𝜇 = 0).• Energies quantized, ℏ𝜔(𝑛 + =
&).
• Speed of sound: c
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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, 𝜔 = 𝑘𝑐.(actually may have 2 or more frequencies)
2) Quantization of energies: won’t show this; result is analogous to familiar SHO solution in quantum mechanics,
𝐸 = ℏ𝜔(𝑛 + ()).
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State counting: I showed this slide before for photons.
• Start with cavity modes in a box with perfectly conducting sides, dimensions L.
𝐸> ∝ 𝑐𝑜𝑠 𝑛>?@𝑥 𝑠𝑖𝑛 𝑛A
?@𝑦 𝑠𝑖𝑛 𝑛B
?@𝑧 etc.
Δ𝑘 = ?@
-> 𝑉: =?@
'
Octant of sphere;but with 8× 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 ∆k)Also recall 𝜔 = 𝑘c
𝑈 = ##CC 1DE$!
ℏ𝜔%𝑒*ℏ9" − 1
⟹ 2i6
7 𝜋2𝑉𝑘&𝑑𝑘𝜋'
ℏ𝑘𝑐𝑒*ℏ:F − 1
# modes in thinshell, thickness 𝑑𝑘 = 𝑑𝜔/𝑐
One k-vector per volume element;same as “Phase space volume” h3/8
Mode counting directly in k-space(similar to number space for Einstein summation, ch. 15)
𝒑𝒉𝒐𝒕𝒐𝒏𝒔
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Density of states: for summations involving only 𝜔 (or E).
Δ𝑘 = ?@
-> 𝑉: =?@
'for octant;
= &?@
'for complete sphere traveling-waves
𝑈 = ##CC 1DE$!
ℏ𝜔%𝑒*ℏ9" − 1
⟹ 2i6
7 𝜋2𝑉𝑘&𝑑𝑘𝜋'
ℏ𝑘𝑐𝑒*ℏ:F − 1
# modes in thinshell, thickness 𝑑𝑘 = 𝑑𝜔/𝑐
Example for energy sum:
𝑈 = ##CC 1DE$!
ℏ𝜔%𝑒*ℏ9" − 1
⟹ 𝟐i6
7𝑑𝜔× #𝑠𝑡𝑎𝑡𝑒𝑠
𝑖𝑛 (𝜔, 𝜔 + 𝑑𝜔) ×ℏ𝜔
𝑒*ℏ9 − 1
𝒑𝒉𝒐𝒕𝒐𝒏𝒔
≡ i6
7 ℏ𝜔𝐷 𝜔 𝑑𝜔𝑒*ℏ9 − 1