Spring Model and Properties of Crystals 

Looking at Some Everyday Spring  
Let's look at two atoms that are connected by a spring. One atom we assume to be somehow fixed; it is like an unmovable wall as shown below. The other atom has the mass m and the spring has the spring constant k_{s} (just wait). We apply some force F in the direction of the bond / spring that causes some deflection or change of position x with respect to the zero point, the position assumed without a force.  
Never mind that the spring shown below is horizontal. Gravity, the force that would make a big mass on a real spring point downwards, can be completely neglected if we look at forces between atoms.  
For some technical reasons we choose the signs in such a way that a negative force will make the spring longer, i.e. it cause some positive deflection x. If you don't like the signs, change them and redraw the picture. They are completely meaningless anyway, we just have to adopt some system so we can talk unambiguously like scientists and not like politicians or lawyers. (Is the sign of the national debt positive or negative? In a mathematical sense or a good / bad sense?).  
An ideal HighSchool spring obeys a linear force F  deflection x law, called Hooke's law, or  


The spring constant


There are two important things to
realize at this point:


If you squeeze a real spring to the point where all its coils touch
each other, it's all over. You have to increase the force a lot to cause just a
tiny little bit of additional deflection. We note: All springs get extremely "hard" if
shortened enough. If you go in the other direction and elongate your spring, it's all over as soon as you stretched the coil to a straight line. Sometimes (but not always) you destroyed the whole thing by then and you can pull it to larger lengths with little additional force. The spring then becomes "soft". 

All of this is shown below. In the force vs. deflection diagram (red line), an ideal spring is just a straight line through the origin. A real spring as described above deviates as shown for large deflections in both directions. Getting harder means the the curve gets steeper, getting softer means it flattens out.  


Forces are nice, energies are nicer. If we look at the work or energy necessary to stretch or compress the spring to a certain deflection, we simply need to remember the definition of work = force times distance = force times deflection x = potential (energy) U in this case. Since the forces in this case are not constant but change with the deflection,we have to sum up and get  


We have a perfect parabola for the ideal spring and a distorted parabola "with wings" for the real spring. We cannot write down an equation for the parabola with wings, but the cool thing about potential energy curves is that we often don't need to calculate them in detail. Quite often we can make very good guesses on what they must look like, as we just did for a real spring. There is no need to write down an equation because potential curves; if we have them, can tell us a lot just so. Let's see how this works by using some model potential that approximates a real spring in the picture below.  


The figure above shows what one can
see directly as soon as we have the graph
of some potential well, as we will now call
any potential energy curve (simply called potential) with some kind of minimum.
All you need to do is to symbolize the mass by some sphere or whatever, and
then draw its position for some deflection.
What you see directly is:


Fascinating, isn't it? No? OK, then let's make the big step:  
Bonds Between Atoms and the Common Spring  
The picture above can be see as the potential of a trivial common spring, for example the one from the inside of your ball point writer. It can also be seen as the potential between two atoms in a crystal. As long as we don't care for numbers or details, the curves look exactly the same.  
We thus can treat the bonds between any two atoms in a crystal as if they would be a common real spring with the proper potential. Of course, this potential is quite different in numbers and details between iron and iron atoms, or iron and carbon atoms or A and B atoms  but qualitatively it always looks more or less as shown above.  


Everything stated above thus is also valid for the atoms in a crystal. In particular, they vibrate around their average position with an energy that is typical for the crystal as a whole and thus (on average) also for each and every atom.  
The thing to realize here and now is that the word temperature for any crystal is just another word for ""average energy contained in the vibrations of the crystal atoms". The energy levels 1 and 2 in the potential figure above thus simply correspond to different temperatures T_{1} < T_{2} if we consider a crystal.  
If we have the potential of the bonding springs, we have a lot of crystal properties covered. Below is another picture of such a potential with a few more parameters that come in handy as soon as we get serious about properties.  
The distance between the atom is now called r. Imagining the atom as small spheres with radius r_{0} (numbers for most atoms via this link), the distance between two of them just touching each other then is about roughly 2r_{0} or around 0.3 nm. If the atoms are at rest, i.e. at extremely low temperature, they all are "sitting" right at the deepest point of the potential with an equilibrium distance of r_{0}, a parameter that is extremely easy to measure and thus a known quantity.  
The depth of the potential well is the binding energy U_{0}. If you want to rip the atoms apart, you have to increase the distance between them to a large value, and that means, as the potential picture directly shows, that you must invest the energy U_{0}.  


Before I go on, just to show off, I'll give you some equations. The binding potential in a rather general but still simple way can be written as  


We have four free parameters  A,
B, m, n  in the first equation.
However, we can always express A and B by the known
equilibrium distance r_{0} and the binding energy
U_{0} as shown. The parameters m and n we need to calculate for the atoms in question, which is sometimes simple, sometimes difficult. For purely ionic crystals, for example, we have n = 1, and 

What do we get from this? Well, a
lot:


Young's Modulus and the Binding Potential  
When we do a tensile test in the elastic region, we simply pull at all the springs between the atoms of the specimen that we encounter in pulling direction. In other words, we essentially "feel" the sum of the spring constants k_{s} of the bonding "springs" and express it as Young's modulus Y.  
Pondering this a bit you realize that we have the
simple relation: k_{s} = Y · r_{0}. Multiply Young's modulus of a material with the interatomic distance, and you get the spring constant of the bond "spring" between two atoms. Looking a bit more closely, a simple relation between the binding potential U(r) and Young's modulus Y emerges: 



Young's modulus is essentially the second derivative of the binding potential. Using the equations given for a general potential, and doing a lot of slightly tricky math, one obtains  


With approximate numbers for n and m and determining U_{0} from the melting points (see below), some rough theoretical estimate for Young's modulus for all kinds of crystals is possible. The result is shown below:  


This is an old picture and I don't know who did it (sorry). It shows that the springbond theory is rather good and essentially demonstrates one thing:  


Note that Young's modulus as determined from a tensile test measures the "springiness" of all bonds contained in a crystal. Since for lowalloy steels, the kind we consider for ancient swords, the overwhelming majority of the bonds are ironiron bonds, Young's modulus of iron and all low alloy steels must be pretty much the same, as claimed in the backbone text and found in experiments.  
Melting Point and Thermal Expansion  
If you look at the first potential well picture again, you realize that all it takes to break the two atoms apart is to raise the energy level of the oscillations all the way up to the base line. The energy is now equal to the binding energy U_{0}, and at the maximum distance between the atoms can become arbitrarily large. In this case you just as well can consider the atoms to be separated. They don't feel much attraction anymore, and if there are other atoms around that are perhaps closer, they go for other partners for a short time until they separate again.  
In other words, the material melts as soon as the vibration energy E_{Vib} approaches the binding energy and that gives us the melting point T_{m}. Since the vibration energy is nothing but a measure of the temperature T or the thermal energy kT (k = Boltzmann constant), we have kT_{m} » U_{0}  
That's a good general relation but not very precise when it comes
to numbers. The reason is that we don't have just two but many atoms vibrating
away. Moreover, melting doesn't start everywhere in a crystal (as it would in
the consideration here) but always on the surface, where the conditions are
different from the inside. So I won't derive numbers from that consideration. Nevertheless, it gives a good general feeling for melting points. 

When materials get hot their
dimension change because they expand. Not much, but enough to cause all kinds
of massive engineering
problems. The thermal expansion
coefficient a characterizes this
material property, it is simply the strain e_{therm} caused by a temperature change of
DT per Kelvin, i.e. 

Why is there thermal expansion at all? Because
the potential well is always asymmetric; it gets steeper at small distances and
flatter for larger ones. The average
distance of the vibrating atoms thus becomes a bit larger with increasing
energy = temperature or amplitude; the material expands. This is already shown
in the potential well figure above. All we
need to do is to calculate the green line, defining the halfway points between
the two sides of the potential well. It is easy enough to do it geometrically so it should also be easy to do the algebra, starting from the general potential formula? No, that is actually rather tricky, The result, however, is simple 



The essential point is that a is proportional to 1/U_{0} and thus approximately also to 1/T_{m}. The rest is some constant that is specific for a given material but very roughly about the same for most materials. Using that for comparing theory and experiment gives the figure below  


Once more a convincing result with the verdict:  


Vibration Frequency of Atoms in a Solid  
The potential well figures do not show the vibration frequency directly. However, even if you have no idea about physics, you know that if you jiggle some mass m on a spring with a spring constant k_{s}, the vibration induced will be slower for large masses on the same spring and faster for the same mass if you take a "tougher" spring, i.e. a spring with a larger spring constant. Think of the wheels of your car on their suspension springs, for example.  
So we might guess that the vibration frequency
n should be proportional to
k_{s}/m. Nice try  but not quite right. The
precise relation is: 

Substituting Young's modulus for k_{s} we get  


Putting in some numbers for Young's modulus Y and interatomic distances r_{0} we get a very important rough number:  


Let's put that number into perspective:


The little buggers move back and forth rather quickly, it seems. And that is the reason why atoms can cover distances far, far larger than their size when they move around by diffusion in a crystal. They cover only a tiny distance when they happen to jump into a vacancy  but they do that many times per second.  
Ultimate Fracture Stress  
The last thing to consider is the ultimate fracture stress. Applying enough force or stress, or investing enough energy, will pull the atoms apart to any distance or simply apart. The material typically fractures if you increase the distance between atoms by 30 %  
As far as force is concerned, you only have to
move the atoms apart to the point of inflection of the potential curve. At that
point you need to apply the maximum force. If you want to pull the atoms apart
even more, less force will be sufficient. That is directly clear form looking at the force curve in this figure. The force is essentially the derivative of the potential curve. Where it has a maximum, its first derivative, and therefore also the second derivative of the potential curve, needs to be zero. That defines the point of inflection of the potential curve and if we calculate that we get the ultimate fracture stress. 

What we get after rather lengthy calculations is complicated  but not very useful:  


I will not compare that to experiments because
this equation gives numbers far higher than what we measure. That's not because
it is wrong but because it describes a perfect ideal and brittle crystal, something that doesn't exist.
Moreover, it ruptures all the bonds in a crosssection together in one fell
swoop, while in reality fracture occurs be a crack sweeping across the material
and thus bonds are broken "onebyone", something easier to do than
to cut them all at the same time. Moreover, the calculation here does not account for plastic deformation occurring with ductile crystals, that tends to reduce stresses. It simply does not "know" defects. Fracture in real brittle crystals, however, is dominated by the defects I called "nanocracks". It therefore always happens at stresses far lower than the ultimate fracture stress calculated here for ideal perfect crystals; see this link 

Nevertheless, there is some merit in this consideration. It tells you what can be achieved under ideal conditions and thus it also tells you that there is no such thing as a "super" material that never breaks.  
Well, tough luck here. We do not have fracture covered by working with the bondspring model. We can, however, still use it in a different kind of way to make some leeway in this respect. That is the topic of another module.  
Famous Last Words  
The bondspring model as expressed in
the "master" equation above is an
extremely simple and relatively old model. One could work it just with paper an
pencil, and it produced important general results. It is not good enough,
however, for serious Material Science for two major reasons:


Repairing those problems is simple but the resulting equations will not be amenable to tackling with paper and pencil. You need computers. You actually need rather powerful computers for tackling many of the more interesting questions. That's why computational materials science is making leaps and bounds in the last 10 years and is sure to make major contribution in years to come.  
In essence, however, we still first supply the potential describing the interaction between the atoms of the system and then go on from there. Here is a potential from a 2004 paper (couldn't resist), describing the interaction potentials between iron and phosphorous for all three possible variants:  


This somewhat lengthy module was meant to give a taste treat of what's going on in a small part of what we call the theory of materials. If you got the impression that we guys know what we are doing most of the time, I will rest my case.  
Periodic Table of the Elements
Dislocation Science  1. The Basics
Science of Uniaxial Deformation
Experimental Techniques for Measuring Diffusion Parameters
4.1.3 Young's Modulus and Bonding
6.1.3 Reading Phase Diagrams: Mixed Phases and Boundaries
Atomic Mechanisms of Diffusion
12.2.5 Sword Types and Static Properties
© H. Föll (Iron, Steel and Swords script)