Phonon Mass: Understanding its Sign and Effects on Gravity

[snorkack]

What is the sign of phonon mass?
A substance of uniform composition, in a field of gravity, has pressure increasing downwards. This causes the compressibility to decrease downwards - and speed of sound to increase downwards.
In a gradient of downwards increasing velocity, a wave propagating horizontally, across gradient will bend towards the slower wave velocity, as the faster flank of the wave in the higher velocity medium overtakes and turns the slower flank.
Note the contrast. A wave bends towards slower velocity. A particle would accelerate and bend towards faster velocity.
How would a de Broglie wave act?
If a phonon bends away from gravity, does it confer negative mass on it? But then, does a phonon repel objects outside the medium it propagates in?
Cannot see why. A sound wave does not remove rest energy from the medium. It adds energy to the medium in which it is present - potential energy of compression or stretching in the strain phase, kinetic energy of medium motion in the motion phase. Therefore a phonon should exert positive gravity on external objects - even as it is repelled by them.
So, is phonon of negative or positive mass?

 

[Demystifier]

A balloon filled with Helium and surrounded by the Earth atmosphere moves upwards. Does it mean that the balloon has a negative mass?

Spoiler

Of course not, it was a rhetorical question.

The point is that the gravitational force is not the only force involved here. There are also intermolecular forces, which for the behavior of sound are much more important than the gravitational force. When there are other forces involved, it is wrong to try to explain everything in terms of the gravitational force. The phonon of course does not have a negative mass. It moves the way it moves as a resultant of all the forces involved, where the gravitational force is only one of them.

 

[A. Neumaier]

Masses are by definition always nonnegative.

 

[Pony]

Masses are by definition always nonnegative.

It is offtopic in Quantum mechanics forum, but I played a lot with Newtonian mechanics (dynamics and celestial mechanics) with negative inertial/gravitational mass, and I found that all the equations and formulas worked in the m<0 case well.
I pretty much believe that someone will publish a monograph of this soon (there are a lot of negative mass research recently).
Why do you think that mass can't be an arbitrary real by definition? Does special relativity or QM need it somewhere?

If a phonon bends away from gravity,

The trajectory of the phonon has nothing to do with its sign of mass, things that has little mass in absolute value follow almost the same path. Also there is no such thing as (negative) passive gravitational mass. You can see this clearly from Einstein's Elevator Experiment.*
(In Newtonian terms: the Earth and the phonon repels each other with a force F. This repelling F force cause the phonon to move towards the Earth a lot, and this repelling F force cause the Earth to move away from the phonon a very little.)

* I think that in standard GR there is no possible phenomenon that something bends away from gravity. But I may be wrong, it's hard to prove such things.

 

[Mentz114]

[..]
* I think that in standard GR there is no possible phenomenon that something bends away from gravity. But I may be wrong, it's hard to prove such things.

It does happen but it is frame dependent. Look at the experience of an observer hovering near a BH horizon. They will see a faller apparently decelerating. The faller does not experience this.

Nothing whatever to do with -ve mass.

 

[A. Neumaier]

I played a lot with Newtonian mechanics (dynamics and celestial mechanics) with negative inertial/gravitational mass, and I found that all the equations and formulas worked in the m<0 case well.

Sure, but only because you ignored dissipative effects.

Negative mass makes the kinetic energy negative. Particles with negative mass would produce arbitrary amounts of energy by becoming faster and faster, a very unstable situation.

Negative mass research makes negative impact on science.

 

[Cthugha]

One should keep in mind that a phonon is a quasiparticle and therefore one usually considers effective masses. And effective masses may of course become negative. However, one has to understand what effective masses are (and what they are not) because otherwise one easily arrives at wrong conclusions.

The effective gravitational phonon mass may be negative as has been shown e.g. in Phys. Rev. Lett. 122, 084501 (2019) (https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.122.084501). However, one should pay close attention to the exact assumptions and definitions used. They are pretty important in this case.

 

[Pony]

Negative mass makes the kinetic energy negative.

Yes.

Particles with negative mass would produce arbitrary amounts of energy by becoming faster and faster, a very unstable situation.

Well, conservation of energy still holds in the case when m<0. (Look at any proof!)

Yes, I think too that the law of equipartition implies something like "if there is negative mass then everything will be faster and faster".
However I am not sure how fast is this process (e. g. one could argue like "because of positive mass, our universe must have reached thermal equilibrium already" which is absurd!).
Also my toy problems behave well.

Why do you think that we should put m>=0 into the definition of mass?

 

[A. Neumaier]

Yes.

Well, conservation of energy still holds in the case when m<0.

Of course. But this assumes an isolated system. Real life systems are not isolated, and you have to add to the second order equation of motion a dissipative first order term. Then energy is lost, and what I indicated happens. Try this with your models, and see what happens!

 

[A. Neumaier]

And effective masses may of course become negative.

I didn't know this; please give a reference where I can find details.

In any case, Pony's comment to which I responded was about Newtonian mechanics, where negative masses are unphysical.

 

[Cthugha]

I didn't know this; please give a reference where I can find details.

H. Kroemer, "negative effective masses in semiconductors", progress in semiconductors 4, 3-3, Heywood, London 1959
or in German (easier to come across):
H. Kroemer, "Negative effektive Massen in Halbleitern", Halbleiterprobleme 5, 75-86, Vieweg, Braunschweig, 1960.

If these are hard to get access to, also any decent text on semiconductor physics should cover this topic, e.g. the book by Ashcroft and Mermin. The basic idea is simply that if you have some quasiparticle and the dispersion is still parabolic in some range, you can account for all the interactions it may undergo in terms of some effective mass. This may become negative. Consider e.g. an electron in a periodic potential. If you do perturbation theory, you will notice that there will be an energy gap opening close to the border of the Brillouin zone. Here, the dispersion becomes flat and the curvature becomes positive. Loosely speaking, if you apply some force to the electron in this direction, it will undergo Bragg scattering and the wave vector will change sign. So a force applied in positive x-direction will make the electron move into the negative x-direction. Of course all of this is just a consequence of relabeling the interactions in terms of an effective mass. Even simpler, crystal electrons close to the top of e.g. the valence band in a semiconductor will show a negative effective mass. One usually introduces the concept of holes instead, which act as quasiparticles with positive effective mass and positive charge. This is also quite useful in some other branches of physics, e.g. non-linear dynamics, where soliton formation may depend on the sign of the effective mass of the quasiparticle in question.

However, as always it is very important to know the underlying model used before speaking about things such as negative effective masses. I fully agree with you that negative masses are completely unphysical in usual Newtonian mechanics.

 

[A. Neumaier]

H. Kroemer, "negative effective masses in semiconductors", progress in semiconductors 4, 3-3, Heywood, London 1959
or in German (easier to come across):
H. Kroemer, "Negative effektive Massen in Halbleitern", Halbleiterprobleme 5, 75-86, Vieweg, Braunschweig, 1960.

If these are hard to get access to, also any decent text on semiconductor physics should cover this topic, e.g. the book by Ashcroft and Mermin.

Thanks. I'd prefer some online reference, so that I don't have to go to the library...

Even simpler, crystal electrons close to the top of e.g. the valence band in a semiconductor will show a negative effective mass. One usually introduces the concept of holes instead, which act as quasiparticles with positive effective mass and positive charge.

Isn't this the only correct way of thinking about it?

 

[Cthugha]

Thanks. I'd prefer some online reference, so that I don't have to go to the library...

Here is one of the original articles from Kittel. This one is a bit brief, but it covers the basics:
PNAS 45(5), 744, 1959
https://www.pnas.org/content/45/5/744

Isn't this the only correct way of thinking about it?

I would say it is definitely the only useful way of thinking about it. Of course one could go ahead and describe the behaviour of a few billion electrons in the presence of a vacancy instead, but I have no idea why someone would do that.

 

[ZapperZ]

I didn't know this; please give a reference where I can find details.

In any case, Pony's comment to which I responded was about Newtonian mechanics, where negative masses are unphysical.

But this is about "phonons", and it is not "Newtonian mechanics", because, as has been stated, we are now talking about solid state/condensed matter physics, and the definition of an effective mass is relevant here.

The standard effective mass of quasiparticles is defined using the band structure dispersion, i.e. it is the inverse of the second derivative of E vs k dispersion.

https://www.tf.uni-kiel.de/matwis/amat/semi_en/kap_2/backbone/r2_3_1.html

It means that depending on the curvature of the band, one CAN have not only an effective mass that is quite different than the bare mass, but also a negative effective mass.

Zz.

 

[A. Neumaier]

The standard effective mass of quasiparticles is defined using the band structure dispersion, i.e. it is the inverse of the second derivative of E vs k dispersion.

https://www.tf.uni-kiel.de/matwis/amat/semi_en/kap_2/backbone/r2_3_1.html

It means that depending on the curvature of the band, one CAN have not only an effective mass that is quite different than the bare mass, but also a negative effective mass.

Thanks. Just for clarification: When the curvature vanishes, do we have infinite positive mass or infinite negative mass?

Since changing from positive mass to negative mass means going through a nonphysical mass singularity, it seems that negative mass is (like negative temperature, which is hotter than any positive temperature) heavier than any positive mass.

 

[PeterDonis]

Note the contrast. A wave bends towards slower velocity. A particle would accelerate and bend towards faster velocity.

You're mixing up two very different things here. The wave does not "bend" towards slower velocity because it's bending away from gravity. You could put air in a very tall chamber and heat the top of the cylinder enough to make the sound speed at the top higher than the sound speed at the bottom; then the sound waves would bend downward instead of upward. Or you could make sound waves bend in a big chamber floating far out in deep space, away from all gravitating bodies so there's no gravity gradient in the chamber at all, by heating one end of the chamber. In other words, the way a wave bends in a medium has nothing to do with the wave being attracted or repelled by gravity, so the analogy you are trying to make here is not valid.