Field Theory of Swords

I don’t mean to brag, but if you’ve been following this sequence of posts on ribbonfarm, then I’ve sort of taught you the secret to modern physics.

The secret goes like this:

Everything arises from fields, and fields arise from everything.


Go ahead.
You can indulge in a good eye-roll over the new-agey sound of that line.
(And over the braggadocio of the author.)

But eye-rolling aside, that line actually does refer to a very profound idea in physics. Namely, that the most fundamental object in nature is the field: a continuous, space-filling entity that has a simple mathematical structure and supports “undulations” or “ripples” that act like physical particles. (I offered a few ways to visualize fields in this post and this post.) To me, it is the most mind-blowing fact of modern physics that we call particles are really just “ripples” or “defects” on some infinite field.

But the miraculousness of fields isn’t just limited to fundamental particles. Fields also emerge at much higher levels of reality, as composite objects made from the motion of many active and jostling things. For example, one can talk about a “field” made from a large collection of electrons, atoms, molecules, cells, or even people. The “particles” in these fields are ripples or defects that move through the crowd. It is one of the miracles of science that essentially any sufficiently large group of interacting objects gives rise to simple collective excitations that behave like independent, free-moving particles.

Maybe this discussion seems excessively esoteric to you.  I can certainly understand that objection. But the truth is that the basic paradigm of particles and fields is so generic and so powerful that one can apply it to just about any level of nature.

So we might as well use it to talk about something awesome.

Let’s talk about swords.

*  *  *

A sword, of course, is a solid piece of metal, and that means that if you look at it under sufficiently high magnification it will look something like this:

metal

The little balls in this picture represent atoms (say, iron atoms), and in a solid metal they generally sit in a nice, periodic arrangement. (The lines in the drawing are just there to illustrate the orderliness of this arrangement.)  The positions of the atoms will constitute our field.

Now let’s ask the question: how strong is a sword? How much force can you apply on it before the sword deforms or breaks?

To make the question more specific, let’s suppose that you swing your sword directly into a sharp surface (like, I don’t know, another sword). At the point of impact there will be a force that tries to push one plane of atoms in such a way that it slides across the neighboring plane. This kind of force is called shear.

shear_force

How big does the shear force have to be before your sword breaks?  Looking at the picture above, one very natural answer to this question might come to mind. Namely, that the breaking force should be equal to the repulsive force between two neighboring atoms multiplied by the number of atoms in a given plane.

This answer is very natural, but also very wrong. In fact, if you use that answer to make an estimate of a sword’s breaking force, you’ll find that even a laughably puny “sword” with a 1 millimeter cross section would withstand multiple tons of force before it broke. Since we do not live in a world where people go confidently into battle with millimeter-thick swords (and since, relatedly, you are probably capable of deforming an steel paper clip with your bare hands), there must be something wrong with this answer.

To understand what went wrong, we need to think about the particles in our field.

Remember that a particle is basically just a defect in a field. And if your field is a crystal of iron atoms, then there is one particular kind of defect that is especially relevant. This defect is called a dislocation, and it looks like this:

dislocation-with_stress

A dislocation is a place where the lattice planes don’t line up with each other. This failure of alignment produces stress in the nearby regions of the crystal (illustrated by the orangeish area), as atoms are forced into positions that are slightly closer or slightly further from their neighbors than they would prefer. Notice, however, that there is no easy way to eliminate all that stress. Moving atoms around locally just shifts the position of the dislocation, and the stress remains the same.

Of course, you should also keep in mind that the dislocations are not little points.  That orange region of stress is not just a single point-like region where the lattice planes are mismatched. In a thick piece of metal, the dislocations are actually long lines of mismatched atoms.

dislocation-3D(In this picture, that line of dislocation extends into the screen.)

Consequently, our “particles” in this field are better drawn as long, stringy lines that extend through the metal. I’ll draw them like this:

 dislocation_string

As it turns out, these dislocations have a serious implication for the strength of our hypothetical sword. When a dislocation is present, all you need to do to deform the sword is to move the dislocation from one side to the other. Like this:

dislocation_animation

In contrast to the Herculean effort required to make two atomic planes slip against each other, moving a dislocation is easy, since you are only displacing a small number of atoms at a time. One analogy is that moving a dislocation is something like trying to move a very heavy carpet across the floor. Dragging the whole thing may be prohibitively difficult, but if you make a wrinkle or a roll in the carpet, you can simply push that wrinkle to shift the position of the carpet.

dislocation_rug

This is also why your puny hands are capable of bending a paper clip: when you bend the clip, you are in fact just pushing dislocations from one side of the material to the other.

So what can you do if you want a sword that doesn’t bend or break easily?

You might think that the answer is to be extremely fastidious in preparing or choosing your metal, with the goal of having as few dislocations as possible. But this turns out to be a fool’s errand. Even a small number of dislocations enable the material to deform, and new dislocations can always enter the metal from either edge (as in the animated gif above).

The correct strategy, as it turns out, is to make more dislocations. And to make them as disordered as possible.

The crucial idea behind this strategy is that dislocations can’t really move through each other. When two dislocations are brought together, the stress in the crystal builds up intensely around them.

two_dislocations

Such a stress build-up leads to a strong repulsive force that pushes the dislocations back apart, and thereby prevents them from moving through each other.

So now if you have two dislocations aligned in different directions, they can get caught on each other in a way that prevents each of them from slipping past the other.

tangled_dislocations

 

In fact, this kind of dislocation tangling is one of the most important reasons for all that hammering during the process of metal forging.

When the mighty smithy stands at work over his anvil (the muscles of his brawny arms as strong as iron bands), his effort is largely going into creating a tangled knot of dislocations inside the metal. Such a tangle keeps the metal strong by pinning the dislocations in place, and prevents the metal from deforming under future stresses.  (This part of the process is also known as work hardening, or strain hardening.)

smithy_tangle2

In this way, the value of the blacksmith is not that he’s strong enough to deform crystalline steel (he’s not). It’s just that he’s pretty good at making a tangled mess of dislocations.  And tangled dislocations make good swords.

I guess you could call him an applied field theorist.

 


 

One footnote is in order: I learned a great deal about forging from this excellent article written by the renowned blade/swordsmith Kevin Cashen.

I also stole the rug picture from his website, and I hope he doesn’t mind.

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About Brian Skinner

Brian Skinner is a physicist who specializes in the theory of strongly-interacting many-body systems. He is currently a postdoctoral researcher at MIT, and he blogs at Gravity and Levity

Comments

  1. Aaron Thornton says:

    Hi Brian,
    Fascinating explanation. I came across this idea when looking at a particular stone we have in New Zealand called Pounamu. The indigenous Maori understood it’s superior strength of this material and used it for tools and weapons. It is commonly known in some other area of the world as Jade. What’s interesting is the fractures or dislocations in the lattice of the rock are heat tempered deep in the tectonic faults beneath the earth, under potentially huge ammounts of pressure. They are cast up and out of volcanoes, and then found in surrounding rivers.

    Aaron

  2. An analogy I immediately drew while reading your fun piece about deformation is that of climbing anchors.
    Sequential anchors fail when the load on each anchor point surpasses its strength, and none of the anchor points is strong enough to arrest the fall. This is known as ‘zippering.’
    eg. http://www.mountainproject.com/images/85/55/106368555_large_c48104.jpg
    When the anchor points are joined to a central point, the load is equally divided and distributed among all the anchor points at once. The anchor can thus sustain a much greater load.
    eg. http://1.bp.blogspot.com/_ejAk42p7jdY/SsYcQyt1ZLI/AAAAAAAAEN0/y7r82kUJfFc/s400/Series+2.JPG
    The irony is that all the attachment points are made of metal, and the microcosm of the alignment of their atoms reflects the macrocosm of the alignment of the anchor’s component parts, much like an individual sword and an army in battle formation.
    Truly, physics allows an understanding of all physical things. I had just never realized that great military strategists would probably have made great physicists, but without having to choose between theoretical and experimental.
    As above, so below.
    Thanks for a good read.

  3. To be nitpicky: a smith works in a smithy.

    Also, check out screw dislocations. They’re weird. Other cool things related to this: magnetic domains are very sensitive to dislocations (and apparently there is a string theory analogy for this), and grain boundaries (places where the crystal changes orientation) have a similar effect to isolated dislocations because the misaligned atoms have different potential energies than the aligned ones.

  4. Ryan Thorngren says:

    Nice post!

    Continuing the field theory analogy… can you design a material where dislocation lines are confined? That would be a very strong material I guess.

    One problem is that I don’t really know how to confine line operators. They carry 2-form charge, but 2-forms are always abelian, so I expect the simplest 2-form gauge theories are weakly coupled in the IR.