In the latest Trig & Calc PoW students need to find the density of titanium metal. As a Materials Engineering student, I know that this is a difficult question to answer. There are many factors that go into and ways to find the density of a metal, each of these I will get into briefly. Max Ray and I worked together to figure out how to accept answers to this problem regarding the wide varieties in the density of titanium metal, so for now the PoW is as solid as titanium.

Originally, the Ninja Mission PoW relied on the density of pure elemental titanium, though even this is variable. Let’s first consider how a scientist would calculate the density of titanium metal. We know that titanium atoms arrange themselves in a lattice pattern known as “Hexagonal Close Packed” or HCP. This means that the individual atoms can be divided into subdivisions of hexagonal prisms which contain a number of partial atoms. (It’d be a lot easier if titanium was arranged in a cubic lattice, but we can’t have everything.) These geometric subdivisions are known as a unit cell. The atoms are arranged on to various points in unit cell; one in each corner of the face hexagons, one in the center of the hexagon face, and three “floating” within the prism in a triangular formation. But, the whole volume of each atom is not entirely in the unit cell, so some are considered fractional atoms. Luckily, it’s easy to remember how many atoms are in an HCP unit cell. There are six total atoms made up of 12 twelfth atoms in the corners, two half atoms in the faces and three whole atoms in the volume.

I even drew you a picture!

So how do we get from the number of partial atoms in a unit cell to the bulk density of a metal? Well, what do we know?! We know how much an atom of titanium mass based on a weighted average of probabilistic distribution of titanium isotopes that exist on Earth. Essentially, it’s not a direct mass of the atom but, a product of the mass of protons, neutrons and electrons, and the number of each in one atom. The periodic table has this number; 47.867 grams per mole (1 mole = 6.02 x 10²³ atoms). The dimensions of the unit cell are 0.295 x 10^-9 meters for the length of a leg and the height is 0.466 x 10^-9 meters. With this information we can formulate the density of titanium metal.

Essentially, what I’m calculating here is how much mass there is in a 1 cubic centimeter hexagonal prism of titanium based on the weight of the total number of titanium atoms in such a volume. This calculates to 4.53 grams per cubic centimeter. This number represents the density of perfectly crystalline and pure titanium based on the standard distribution of isotopic titanium atoms on earth. What I do not account for are defects, dislocations, grain boundaries and impurities that occur from mechanical and thermodynamic forces. Metals (and most materials) do not crystallize perfectly on a macroscopic scale. What happens when a metal solidifies, lots of small crystals form in multiple crystallographic directions. *Id est* as metals solidify metal atoms arrange themselves into many separate volumes, or grains, with their own orientation. What ends up happening looks a lot like this.

All of those colored shapes are individual grains of crystalline titanium. If you looked at the atomic arrangement of a single grain, all of the atoms would align almost perfectly. Zooming in on a boundary between these grains would look something like this. (The image below is a boundary between a grain of Aluminum and Titanium.)

Though there are two different metals, you can see that the atoms align to form an angle. In the case above the boundary is fairly clean and doesn’t have many gaps, but in most cases, grain boundaries free up a lot of space in a material reducing its density. You can even see that within the two grains there are gaps and inconsistencies in the atomic patterns.

Beyond that, atoms don’t always stack perfectly. There is a sizable laundry list of multidimensional hiccups that can occur when a material solidifies, is heated up or is worked with. These are all classified under the umbrella term “Defects”. These can add or subtract from the density of a material depending on what they are. Single dimensional defects are generally missing, extra or displaced atoms, or other types of atoms crammed into a pure lattice (impurities). Two and three dimensional defects are often created when forces bend, slip and twist atoms out of crystalline alignments.

How does a materials scientist account for defects in terms of density? Short answer: carefully. Considering that in the context of the problem, the wise Sensei grafted titanium spikes to a titanium disks to produce a throwing star, –a process requiring a computer operated plasma cutter– the density of the metal is sure to be changed by defects created via the working and heating of the metal. Theoretically, it’s possible to calculate the density of a metal after all is said and done, though it’d be a massive mathematical undertaking. It’s possible to determine the average number of single dimensional defects using a simple thermodynamic equation based on experimental data. In plain terms, the formation energy of a single dimensional defect wherein an atom is added, removed or displaced from it’s crystalline pattern is the difference between the specific enthalpy of a material minus the product of the temperature and the specific entropy of the defect. Unfortunately, defects with more than two dimensions are only formed by mechanical forces, not thermodynamic, so to measure any changes in density from 2+ dimensional defects, we would need to run a computer simulation of the specific titanium material. All of these phenomenons affect the density of titanium by little more than 0.02 grams per cubic centimeter, but this is enough to affect the final answer by a whole degree.

(Science note: Enthalpy is a measure of energy within a substance based on it’s internal energy and the product of pressure and volume. Entropy is a measure of disorder within a system.)

Why does this matter in terms of our math problem? Well, it doesn’t so much in a practical sense, but mathematically the dimensions of the titanium disks are all functions of it’s density. We know that density is an incredibly variable thing. Though, it’s enough to confuse a student looking up the density of Titanium. It’s enough to know the total weight and volume of the titanium disk, and take the quotient to get the exact density of the particular chunk of metal. Though we don’t initially know the volume, so we have to rely on a carefully calculated density for a material with hundreds of forms and alloys (549 according to matweb). The problem states that the throwing star is made of pure titanium, though pure titanium is impossible to produce on Earth. There are theoretical results for the density of pure Titanium, but we know that it’s a finicky number. I chose annealed grade 1 titanium because it’s the purest Titanium we could possibly produce, and the density is experimental rather than theoretical, as in someone in a laboratory took a known volume of titanium, weighed it and then did the math. There’s lots of reasons why Max and I allowed for a range of answers for the Ninja Mission PoW. It isn’t because the math isn’t solid, but rather the solid isn’t!

]]>In the plainest English I can use, this equation models the wave-like behavior of subatomic particles while applying Einstein’s famous E=mc² theory, where c is the speed of light. Schrödinger’s ψ(x,t) describes the probabilistic motion of a subatomic particle along a given distance a particular time frame. The constant h is Planck’s constant, which is a fundamental constant in physics which describes proportion between the energy of a particle and the wavelength of its wave-like behavior. Dirac’s contribution in this case is in the Beta, Alpha and P variables. The P variables refer to a 1×3 matrix of a particles momentum in three axes. he Alpha and Beta variables actually relate to a 4×4 matrix, composed of the possible “spin” of an electron and a “negative energy” component of the same electrons.

(Science note: Spin is an inherent property of most particles that works a lot like angular momentum)

At the time, Dirac was satisfied with his alignment of relativity and quantum mechanics, and published his findings and proofs without further delving into the implications of the “extraneous” data created by the matrices involved. He posited that the negative energy component in the matrices were “holes” in a field of electrons, because empty space is technically more positively charged than an electron. Dirac wanted to write this off as a proton, but since momentum was involved, the “other” particle would need to be the same mass. Though, Dirac’s ultimate objective was to unify relativity, Quantum Mechanics and spin and he was unable to explain or prove that the “negative energy” electrons were just holes in space. By answering one question, Dirac opened up another

Perhaps if he “noticed and wondered” he might have come to the conclusion that the specifics of his mathematics implied that an entirely new class of matter exists. It wasn’t until 1932 that the implications of Dirac’s extraneous solutions were characterized in an experiment. Using a device known as a cloud chamber in a strong magnetic field, Carl D. Anderson observed electron-like particles moving counter-intuitively. The only way he could explain this phenomenon was that these were positively charged electron, which he dubbed positrons. (Clever.) This was the first recorded observation of Anti-matter!

(Historical note: Carl D. Anderson wasn’t the first to observe this behavior, Chung-Yao Chao had observed it in a similar experiment three years earlier, but he never followed up on the phenomenon. Though, Anderson mentioned Chao’s findings in his report. How considerate of him!)

Though the equation here isn’t so simple, looking deep into the meanings of mathematical statements and operations can lead to great discoveries. Every time you write an equation or function, you’re writing much more than you think. Behind every rule and formula, there are pages and pages of proofs and expanded fundamentals of mathematics. Even in well known mathematical machinery, there may still be mysteries to unlock. Noticing something strange in one of our PoWs probably won’t turn the world of physics on it’s head, but it might get your work published on the internet. It’s good practice for when you eventually come across something hidden in the math.

]]>Ask Dr. Math is an awesome source of information regarding any kind of mathematics. It has an archive with 20 years-worth of mathematics questions. You can find out more about differential equations and general solutions with the links in this sentence. **For any math related questions feel free to visit Ask Dr. Math! If you can’t find a solution there, our volunteer math doctors will be happy to help.**

As far as a natural phenomenon is concerned, the first thing I can think of is an example from thermodynamics; the Clausius-Clapeyron relation. The Clausius-Clapeyron relation is a model for a discontinuous transition in phase for any two states of matter of one material in terms of temperature and pressure. The relationship itself is a homogeneous first order differential equation. This relation is the general solution of the state postulate or the Gibbs-Duhem equation. You can read the full derivations here. The general solution is:

;P= Pressure, T = Temperature, L = Latent heat, Δv = specific volume.

Latent heat, temperature and specific volume are all experimentally determined constants, therefore the only variable is Pressure. dT can be considered thermodynamically to be the rate of heat energy entering a system. Therefore, the variable P is expressed in terms of only one other variable with directly applicable constants and is a general solution

Science note: Latent heat is the heat energy absorbed by a volume of matter during a change in phase. For example, imagine a bowl of ice. In every thermodynamic situation, there’s a system (the ice) and the surroundings (the bowl, atmosphere). We only need to concern ourselves with the actions of the ice as it reacts to external heat energy. The ice, to remain solid for a determinable amount of time is some temperature below 273 K (Oº C, 32°F). It’s safe to say that unless we’re looking at a bowl of ice in outer space or outside during the winter, the external temperature is above the freezing point of water. Since the temperature of the surroundings is greater than the temperature of the ice, the ice will absorb energy and eventually melt. As the ice changes in phase to liquid water (or as any matter changes phase), it has been experimentally determined that the temperature remains constant. Even if it were hot enough to boil water, the temperature of the melting ice remains constant as it is changing in phase, even if for a brief time.

Specific volume is the rate at a set quantity of matter expands or contracts given an input of energy in the form of heat or pressure.

Why is the Clausius-Clapeyron Relation important? It allows us to ice skate! The relation suggests that a solid can melt when it is below its freezing point with sufficient pressure. Most ice skating rinks keep the air well below freezing point lest it become as swimming pool. Ice on it’s own is slippery, but as a solid it does not have the minimal friction required to achieve the speeds at which we can skate. This is where the Clausius-Clapeyron relation comes in. Solving for ΔP, we get the equation:

. Mathematically, we can substitute ΔP with ΔT at will. Scientifically, this says that in terms of changing the phase of a material, pressure and temperature contribute work in equal ways. Considering how ice skates work, the sharp blades are essentially for putting all the weight of a skater over a very small area which creates a huge amount of pressure. As a skater applies this pressure to the ice, it melts creating lubricant such that he or she can fly across the ice.

As far as Interlace format is concerned, I’m not familiar. I looked for it in the Ask Dr. Math archives but found no mention. Perhaps one of our volunteer math doctors knows. I looked around the internet and found a few papers about using interlace format to solve stochastic differential equations. Essentially, Interlacing is applying multiple systems of differential equations into multiple matrices via eigen values and vectors, and then combining them to find a point of convergence.

In Materials Science, we use complicated mathematics to create computer simulations of how crystalline structures react to forces, stresses and strains down to the individual atoms in a structure. Here is an example of such a simulation. In the video the all the atoms that stresses are not applied to are invisible, they will appear when a force is applied to them and when they are moved away from their position in the crystal lattice. Essentially, what you’re watching is how atoms move erratically through their crystal structures when forces are applied. In complex systems of dislocated crystals, there is a lot of mathematics at play. Materials scientists may use interlaced differential systems to streamline the calculations used in these simulations.

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