A question about an explanation of the electrical resistance of perfect lattice

[wenty]

Perfect periodic ion lattice has no electrical resistance.As we know,this can be easily shown by solving the Schrödinger equation using Bloch therom.Yet,another explanation is that "in a periodic array of scatterers a wave can propagate without attenuation because of the coherent constructive interference of the scattered waves."(Ashcroft,Solid state physics)

Does anyone know where to find the quantitative verification of this explanation,for example,in 1D periodic sqare well potential?

 

[ZapperZ]

Perfect periodic ion lattice has no electrical resistance.As we know,this can be easily shown by solving the Schrödinger equation using Bloch therom.Yet,another explanation is that "in a periodic array of scatterers a wave can propagate without attenuation because of the coherent constructive interference of the scattered waves."(Ashcroft,Solid state physics)

Does anyone know where to find the quantitative verification of this explanation,for example,in 1D periodic sqare well potential?

Unless I missed something, there isn't any quantitative verification of such things, because it is an idealization of electron transport in metals. It is useful to describe various first-order effects such as the Drude model and such, but you will never get the "no electrical resistance" part being experimentally verified, at least, not with the model here (superconductivity has a different model).

Zz.

 

[wenty]

Unless I missed something, there isn't any quantitative verification of such things, because it is an idealization of electron transport in metals. It is useful to describe various first-order effects such as the Drude model and such, but you will never get the "no electrical resistance" part being experimentally verified, at least, not with the model here (superconductivity has a different model).

Zz.

Then what about verify this explanation theoretically?

 

[ZapperZ]

Then what about verify this explanation theoretically?

How does one "verify" a theory theoretically? If it is mathematically consistent, it's "verified".

Zz.

 

[wenty]

How does one "verify" a theory theoretically? If it is mathematically consistent, it's "verified".

Zz.

Sorry,Maybe I failed to express my thought properly.

What I mean is that:

I summed the scattered waves and can't get the result that "wave can propagate without attenuation".I don't know what's wrong,so I wonder if I can find some reference or some calculation based on this idea.

 

[ZapperZ]

Sorry,Maybe I failed to express my thought properly.

What I mean is that:

I summed the scattered waves and can't get the result that "wave can propagate without attenuation".I don't know what's wrong,so I wonder if I can find some reference or some calculation based on this idea.

Let me understand this correctly. You have Bloch wavefunction (or is it a sum of Bloch wavefunctions?), and you want to know how it can propagate without "attenuation"? You have to admit that you are not giving us a lot to go on here. Unless you want to tell me what exactly you're "summing", I will have to continue making guesses on what exactly you are doing.

Zz.

 

[wenty]

Take periodic square barrier potential for example.As illustrated in the figure,a plane wave Exp(ikx) incident on barrier 1,and if there is no other barrier the transmitted wave and reflected wave is T*Exp(ikx) and R*Exp(-ikx).When there are other barriers the scattered waves will be scattered and then scattered once and once.I add up all of these waves and can't get the result that the Exp(-ikx) terms canceled.I want to know what's wrong?

 

[ZapperZ]

Take periodic square barrier potential for example.As illustrated in the figure,a plane wave Exp(ikx) incident on barrier 1,and if there is no other barrier the transmitted wave and reflected wave is T*Exp(ikx) and R*Exp(-ikx).When there are other barriers the scattered waves will be scattered and then scattered once and once.I add up all of these waves and can't get the result that the Exp(-ikx) terms canceled.I want to know what's wrong?

Just how exactly did you add all of them? I mean, your "x" is only valid for a particular square barrier. Other square barrier are at locations x+R, x+2R, x+3R... and x-R, x-2R, x-3R, ... where R is the lattice constant. You then have the Bloch boundary condition where $$\Psi (x+nR) = \Psi (x)$$. So considering all of these necessary criteria, I do not know how you "add" these things.

Zz.

 

[wenty]

Just how exactly did you add all of them? I mean, your "x" is only valid for a particular square barrier. Other square barrier are at locations x+R, x+2R, x+3R... and x-R, x-2R, x-3R, ... where R is the lattice constant. You then have the Bloch boundary condition where $$\Psi (x+nR) = \Psi (x)$$. So considering all of these necessary criteria, I do not know how you "add" these things.

Zz.

I'll reconsider it.Thank you very much!

 

[Gokul43201]

Also notice that you are not using periodic (BvK) boundary conditions in the way you set up the problem - in fact, it's not clear how you pick an origin. So, you will not get Bloch solutions going about it this way.