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Apr
22
comment Construction of free energy based on Landau theory
@MengCheng: Comments?
Apr
20
comment Construction of free energy based on Landau theory
@MengCheng: seems reasonable I guess, although the above analysis is the same as if $J > 0$.
Apr
19
comment Construction of free energy based on Landau theory
@MengCheng: Thanks, but could you please elaborate? At the critical temperature (or temperature at which we reach criticality) the first term vanishes which implies the equilibrium situation is one in which we have $\psi = 0$. Similarly for the case $T > T_c$. (We must choose $\psi=0$ otherwise we get an imaginary solution for $\psi$ which is unphysical. For $T<T_c$ we get two minima. This is based on the form of the free energy function given. Is this correct understanding? Thanks.
Apr
19
asked Construction of free energy based on Landau theory
Mar
28
asked Applicability of perturbation theory
Mar
8
asked Debye-Huekel Theory and the continuum approximation
Jan
2
awarded  Yearling
Dec
19
awarded  Notable Question
Nov
30
comment Hartree-Fock correction to $e$-$e$ interaction
Thanks, could you describe what the graph you sketched in the answer means physically? For $r_s/a_o \approx 4.8 - 12$ we are in the attractive zone of the potential before the repulsion starts at $r_s/a_o < 4.8$?
Nov
30
comment Hartree-Fock correction to $e$-$e$ interaction
Ok thanks, so the additional terms in the so-called correlation energy term are dependant on the free electron energy and give corrections to this. And there are also corrections to the potential energy to account for corrections in the second term. My last question would be are you sure the expansion is valid for $r_s \ll 1$? I checked AAM and my notes and they both say $r_s/a_o \ll 1$, which is why I was confused about the applicability of this expansion to metals given the typical range of $r_s/a_o$ for them
Nov
30
comment Hartree-Fock correction to $e$-$e$ interaction
I see, so is there any way to argue that there must be kinetic energy terms in these additional terms?
Nov
30
comment Hartree-Fock correction to $e$-$e$ interaction
Also given that the pertubative expansion is only valid for $r_s/a_o \ll 1$, and most metals have this ratio in the range $2-6$ this means higher order corrections are important in this expansion. For sodium for example $r_s/a_o \approx 4$ which is close to the minimum you mentioned. So this means this value lies in the repulsive end of the graph you drawn and hence the expansion is not valid?
Nov
30
comment Hartree-Fock correction to $e$-$e$ interaction
Hmm, my question on my sheet is 'The subsequent terms $0.0622 \ln (r_s/a_o) -0.096 + O(r_s/a_0)$' are collectively known as the correlation energy. Explain why these contain contributions from the kinetic energy as well as the potential energy'?
Nov
30
comment Hartree-Fock correction to $e$-$e$ interaction
Thanks for the edits. But how is $\mathcal V$ explicitly got terms in kinetic energy? As far as I can see, there are only contributions from the positive ionic background, electron-ion interaction and electron-electron interaction which all contribute to the potential energy just?
Nov
29
comment Hartree-Fock correction to $e$-$e$ interaction
Many thanks for your answer. I have not studied the Hartree-approximation using your method of annihilation operators so I could not follow your mathematics exactly. (I am following the method in Ashcroft and Mermin (AAM)). I could not understand the part where you say the second term of $\mathcal H$ is the result of mixing of the subsequent terms. Could you maybe describe this part in words or at the level of the notation of AAM? I think the method of AAM is the same you used, just not in the operator formulism. Thanks
Nov
29
asked Hartree-Fock correction to $e$-$e$ interaction
Oct
24
asked Relaxation time approximation in Drude model apparant paradox
Oct
17
asked Resources to learn about the Higgs theory at undergraduate level
Sep
22
comment Packing fraction of atoms in a HCP structure
Hi Jon. Could you elaborate on the geometry? It is probably simple, but I have stared at it for some time and can't get it right. Thanks.
Sep
22
comment Packing fraction of atoms in a HCP structure
yes, that is what I assumed. For some reason, the geometry I can't get quite right.