On the flexural rigidity of a slice of pizza

People who eat pizza (not the deep dish casserole style from Chicago, but normal pizza), unbeknownst to most of them, have developed an intuition for a key concept in elasticity and solid mechanics. 

I hope that all right-thinking people agree that pizza slice droop (left hand image) is a problem to be avoided.  Cheese, sauce, and toppings are all in serious danger of sliding off the slice and into the diner's lap if the tip of the slice flops down.  Why does the slice tend to droop?   If you hold the edge of the crust and try to "cantilever" the slice out into space, the weight of the sauce/toppings exerts downward force, and therefore a torque that tries to droop the crust.  

A simple way to avoid this problem is shown in the right-hand image (shamelessly stolen from here).  By bending the pizza slice, with a radius of curvature around an axis that runs from the crust to the slice tip, the same pizza slice becomes much stiffer against bending.   Why does this work?  Despite what the Perimeter Institute says here, I really don't think that differential geometry has much to do with this problem, except in the sense that there are constraints on what the crust can do if its volume is approximately conserved.  

The reason the curved pizza slice is stiffer turns out to be the same reason that an I-beam is stiffer than a square rod of the same cross-sectional area.  Imagine an I-beam with a heavy weight (its own, for example) that would tend to make it droop.  In drooping a tiny bit, the top of the I-beam would get stretched out, elongated along the \(z\) direction - it would be in tension.  The bottom of the I-beam would get squeezed, contracted along the \(z\) direction - it would be in compression.  Somewhere in the middle, the "neutral axis", the material would be neither stretched nor squeezed.  We can pick coordinates such that the line \(y=0\) is the neutral axis, and in the linear limit, the amount of stretching (strain) at a distance \(y\) away from the neutral axis would just be proportional to \(y\).  In the world of linear elasticity, the amount of restoring force per unit area ("normal stress") exhibited by the material is directly proportional to the amount of strain, so the normal stress \(\sigma_{zz} \propto y\).  If we add up all the little contributions of patches of area \(\mathrm{d}A\) to the restoring torque around the neutral axis, we get something proportional to \(\int y^2 \mathrm{d}A\).  The bottom line:  All other things being equal, "beams" with cross-sectional area far away from the neutral axis resist bending torques more than beams with area close to the neutral axis.

Now think of the pizza slice as a beam.  (We will approximate the pizza crust as a homogeneous elastic solid - not crazy, though really it's some kind of mechanical metamaterial carbohydrate foam.)  When the pizza slice is flat, the farthest that some constituent bit of crust can be from the neutral axis is half the thickness of the crust.  When the pizza slice is curved, however, much more of its area is farther from the neutral axis - the curved slice will then resist bending much better, even made from the same thickness of starchy goodness as the flat slice.

(Behold the benefits of my engineering education.) 

Brief items

A few more links to help fill the time:

• Steve Simon at Oxford has put his graduate solid state course lectures online on youtube, linked from here.  I'd previously linked to his undergrad solid state lectures.   Good stuff, and often containing fun historical anecdotes that I hadn't known before.
• Nature last week had this paper demonstrating operations of Si quantum dot-based qubits at 1.5 K with some decent fidelity.  Neat, showing that long electron spin coherence times are indeed realizable in these structures at comparatively torrid conditions.
• Speaking of quantum computing, it was reported that John Martinis is stepping down as lead of google's superconducting quantum computation effort (these folks).  I've always thought of him as an absolutely fearless experimentalist, and while no one is indispensable, his departure leads me to lower my expectations about google's progress.  Update:  Forbes has a detailed interview with Martinis about this.  It's a very interesting inside look.  
• I'd never heard of "the Poynting effect" before, and I thought this write-up was very nice.

This week in the arxiv - magnons

Ages ago I wrote a description of magnons, which I really should revise.  The ultra-short version:  Magnetically ordered materials are classified by long-ranged patterns of how electronic spins are arranged.  For example, in a (single domain) ferromagnet, the spins all point in the same direction, and it costs energy to perturb that arrangement by tipping a spin.  Classically one can define spin waves, where there is some spatially periodic perturbation of the spin orientation (described by some wave vector \(\mathbf{k}\), and that perturbation then oscillates in time with frequency \(\omega(\mathbf{k})\), like any of a large number of wave-like phenomena.  In the quantum limit, one can talk about the energy of exciting a single magnon, \(\hbar \omega\).  One can use this language to about making wavepackets and propagating magnons to transport angular momentum.

Two papers appeared on the arxiv this week, back-to-back, taking this to the next level.  In condensed matter physics some of the most powerful techniques, in terms of learning about material properties and how they emerge, involve scattering.  That is, taking some probe (say visible light, x-rays, electrons, or neutrons) with a well-defined energy and momentum, firing it at a target of interest, and studying the scattered waves to learn about the target.  A related approach involves interferometery, where the propagation of waves (detected through changes in amplitude and phase) is sensitive to the local environment.  

The two preprints (this one and this one) establish that it is now possible to use magnons in both approaches.  This will likely open up a new route for characterizing and understanding micro- and nanoscale magnetic materials, which will be extremely useful (since, as I had to explain to a referee on a paper several years ago, it's actually not possible to use neutron scattering to probe a few-micron wide, few nm thick piece of material.)  In the former paper, magnons in yttrium iron garnet (a magnetic insulator called YIG, not to be confused with Yig, the Father of Serpents) are launched toward and scattered from a patch of permalloy film, and the scattered waves are detected and imaged sensitively.  In the latter, propagation and interference of magnons in YIG waveguides is imaged.  The great enabling technology for both of these impressive experiments has been the development over the last decade or so in the use of nitrogen-vacancy centers in diamond as incredibly sensitive magnetometers.   Very pretty stuff.

What are anyons?

Because of the time lag associated with scientific publishing, there are a number of cool condensed matter results coming out now in the midst of the coronavirus impact.  One in particular prompted me to try writing up something brief about anyons aimed at non-experts.  The wikipedia article is pretty good, but what the heck.

One of the subtlest concepts in physics is the idea of "indistinguishable particles".  The basic idea seems simple.  Two electrons, for example, are supposed to be indistinguishable.  There is no measurement you could do on two electrons that would find different properties (say size or charge or response to magnetic fields).  For example, I should be able to pop an electron out of a hydrogen atom and replace it with any other electron, and literally no measurement you could do would be able to tell the difference between the hydrogen atoms before and after such a swap.  The consequences of true indistinguishability are far reaching even in classic physics.  In statistical mechanics, whether or not a collection of particles and that same collection with two particles swapped are really the same microscopic state is a big deal, with testable consequences.

In quantum mechanics, the situation is richer.  Let's imagine that the only parameter that matters is position.  (We are going to use position as shorthand to represent all of the quantum numbers associated with some particle.)  We can describe a two-particle system by some "state vector" (or wavefunction if you prefer) \( | \psi (\mathbf{r_{1}},\mathbf{r_{2}}) \rangle\), where the first vector is the position of particle 1 and the second is the position of particle 2.  Now imagine swapping the two particles.   After the swap, the state should be \( | \psi (\mathbf{r_{2}},\mathbf{r_{1}}) \rangle\).  The question is, how does that second state relate to the first state?  If the particles are truly indistinguishable, you'd think \( | \psi (\mathbf{r_{2}},\mathbf{r_{1}}) \rangle =  | \psi (\mathbf{r_{1}},\mathbf{r_{2}}) \rangle \). 

It turns out that that's not the only allowed situation.  One thing that must be true is that swapping the particles can't change the total normalization of the state (how much total stuff there is).  That restriction is written  \( \langle \psi (\mathbf{r_{2}},\mathbf{r_{1}}) | \psi (\mathbf{r_{2}},\mathbf{r_{1}}) \rangle = \langle \psi (\mathbf{r_{1}},\mathbf{r_{2}}) | \psi (\mathbf{r_{1}},\mathbf{r_{2}}) \rangle \).  If that's the most general restriction, then we can have other possibilities than the states before and after being identical.

For bosons, particles obeying Bose-Einstein statistics, the simple, intuitive situation does hold.  \( | \psi (\mathbf{r_{2}},\mathbf{r_{1}}) \rangle =  | \psi (\mathbf{r_{1}},\mathbf{r_{2}}) \rangle \).

For fermions, particles obeying Fermi-Dirac statistics, instead  \( | \psi (\mathbf{r_{2}},\mathbf{r_{1}}) \rangle = -  | \psi (\mathbf{r_{1}},\mathbf{r_{2}}) \rangle \).  This also preserves normalization, but has truly world-altering consequences.  This can only be satisfied for two particles at the same position if the state is identically zero.  This is what leads to the Pauli Principle and basically the existence of atoms and matter as we know them.

In principle, you could have something more general than that.  For so-called "abelian anyons", you could have the situation  \( | \psi (\mathbf{r_{2}},\mathbf{r_{1}}) \rangle = (e^{i \alpha})  | \psi (\mathbf{r_{1}},\mathbf{r_{2}}) \rangle \), where \(\alpha\) is some phase angle.  Then bosons are the special case where \(\alpha = 0\) or some integer multiple of \(2 \pi\), and fermions are the special case where \(\alpha = \pi\) or some odd multiple of \(\pi\).   

You might wonder, how would you ever pick up weird phase angles when particles are swapped in position?  This situation can arise for charged particles restricted to two dimensions in the presence of a magnetic field. The reason is rather technical, but it comes down to the fact that the vector potential \(\mathbf{A}\) leads to complex phase factors like the one above for charged particles.  

This brings me to this paper.  Anyons have been deeply involved in describing the physics of the fractional quantum Hall effect for a long time  (see here for example).  It's tricky to get direct experimental evidence for the unusual phase factor, though.  The authors of this new paper have been basically doing a form of particle swapping via a scattering experiment, and looking at correlations in where the particles end up (via the noise, fluctuations in the relative currents).  They do indeed see what looks like nice evidence for the expected anyonic properties of a particular quantum Hall state.  

(There are also "nonabelian" anyons, but that is for another time.)

Brief items

A couple of interesting links:

• From City University of New York, a paper on a bit of the physics relevant to the pandemic - specifically the issue of aerosolized droplets and air circulation in rooms.  The conclusion is that, based on common convection patterns, the best approach to clearing airborne contaminants is a ceiling-mounted suction filter as in surgical operating rooms.  (I suspect that vertical flow ceiling HEPA fan filter units with many air changes per hour as in microfabrication cleanrooms would also work, but it's not like anyone is going to install elevated, gridded flooring everywhere.)  Some of the basic physics of particle suspension is simple enough to teach to high school students, without even getting into viscosity and drag anf real fluid mechanics.  The typical amount of kinetic energy that a would-be suspended particle picks up in collisions with its surroundings is on the order of \(k_{\mathrm{B}}T\), or about 26 meV (\(4.14 \times 10^{-21}\) J).  For a particle to stay readily suspended, that has to be comparable to the gravitational potential energy that it would cost to elevate the particle by its own typical size.  For a spherical droplet of the density of water, you'd be looking at something like \((4/3)\pi R^{3} \cdot \rho \cdot g \cdot 2R\), where \(R\) is the droplet radius, \(\rho\) is the density of water, 1000 kg/m³, and \(g\) is the gravitational acceleration, 9.807 m/s².  Setting those equal and solving gives \(R \approx 470\) nm.  
• The always excellent Natalie Wolchover has a new article in Quanta about how one limiting factor in gravitational interferometers is the quality of the glass used in the dielectric mirrors.  Specifically, the tunneling two-level systems (see here and here) in ordinary amorphous insulating dielectrics at low temperatures are a problem.  It's like I've said ever since my doctoral work:  tunneling two-level systems are everywhere, and they're evil.
• As pointed out by many, this paper has a novel approach to room temperature superconductivity.  This is a bit like my idea of converting my entire lab into ultra-high vacuum workspace.  Sure, personnel would all have to wear special spacesuits, but it would really help preserve samples.
• In these days of social distancing, this was also amusing.

Please stay safe.  I know it's hard to stay positive while all of this is going on, but remember that you're not alone.