Crystal Models 

This Hyperscript is full of schematic drawings of crystals. It is not always obvious that these drawings are highly abstract and mostly outright wrong if taken at face values. The reason for this "fraud" is simple. In proper drawings that would show the atoms with their proper relative size and their threedimensional arrangements, you just wouldn't see a thing anymore.  
In what follows a lot of figures
illustrate how one one gets from the "real" picture to the stylized
ones, and how one should look at those. This link gives some more figures. For starters we take a simple ionic crystal like sodium chloride (NaCl) or everyday rock salt. The figure below shows a "correct" drawing. 



It is clear that most of the time
threedimensional drawings simply produce too much confusion. We need to
simplify. The lefthand picture above therefore shows the ions in about the
right size relation and the proper arrangement only in a twodimensional view. The picture now is correct but
still not very useful. We know that the forces between the atoms can be approximated with a spring. So we replace it by the spring model shown on the righthand side. 

That involves an amazing number of
simplifications:




Let's see what we can do about the last point in the list above; the fact that those atoms vibrate. Here is an attempt at a spring model with vibrating atoms / ions.  


What the figure above attempts to
show is what vibrating (stylized) atoms look like if you take a snapshot with
an extremely short exposure time (around 10^{14} s or 10 fs). The
atoms then are somewhat deflected from their zero position in a random way. Or are they? Well, they are deflected but not in a random way as far as individual atoms go. A better way to "draw" vibrating atoms, if we must, is shown below. 



The atoms don't vibrate completely
randomly. They synchronize to some extent as shown schematically above and that simply means that
"elastic waves" with a spectrum of wavelengths, amplitudes and
directions are running through the crystal. Those waves, care called "phonons". They get reflected at boundaries like
the surface or at grain boundaries, scattered at defects, and so on. In short, they behave pretty much like the light waves that are running around in the room you are in right now. 

Obviously, drawing vibrating atoms in whichever way does not help us much in modelling crystals by using schematic figures. I will never do it again. 
Now let's look at another problem. While for ionic crystals like rock salt, and pretty much for all metals, the bonding forces are the same in all directions, this is not true for many other "covalently" bonded crystals like silicon (Si), diamond (C) or silicon dioxide (SiO_{2}). Those atoms attract each other only along some welldefined directions.  
This sounds great because now a few springs could symbolize the real bonds quite well. Of course, there is also a problem. Let's look at the figure below to appreciate this.  


On the left we have a model of all
diamond type crystals. Since it is time consuming to draw a lot of springs, I
simply substituted a red line for a spring. The problem is obvious: you don't
recognize the cubic lattice of the
structure. This is shown on the righthand side in addition to the red bond "springs". Of course, the black lines are completely meaningless in terms of the real structure, they simply "guide the eye". 

You get it by now. Crystal models are always highly abstract and just show part of what is really there. Nevertheless, they are highly useful and indispensable. Let's look at a few examples. 


Here is the spring model for looking
at Young's modulus qualitatively. How you
calculate it and other properties quantitatively is shown in a science module. The following figures just illustrate in a qualitative way how one should look at things for proper quantitative calculations. More than that those models cannot do. 



Here we see how to start calculating
elastic behavior expressed in Young's modulus. We just need to look at how
those springs elongate if we pull a them. How that is done with equations is
shown in the science module. We also have (brittle) fracture covered. As soon as your springs are completely pulled out, any further elongation must result in the springs breaking. The two crystal pieces left back are "relaxed" again, i.e. the length of the springs has the old values of an undeformed crystal. Putting that in equations needs a bit of thought but is perfectly possible. 

However, what the model above does
not show is the effect of
lateral contraction, the phenomenon
that the specimen gets thinner as it gets longer. This is dealt with
phenomenologically in this
science module. The spring model above doesn't show this effect nor should it  so why does that happen? Because the spring model above is too simple. As noted before, for a stable crystal we need to consider that there are diagonally springs to secondnearest neighbors, too. If we draw this into the picture it gets a bit unwieldy but makes clear what happens: 



The "red" springs, if elongated, will also pull the atoms somewhat inwards; the specimen gets thinner. With that qualitative picture in mind, we now could start some calculations.  
The long and short of this module is:  


Spring Model and Properties of Crystals
Science of Uniaxial Deformation
4.4.1 Perfect Crystals and the Second Law
4.1.3 Young's Modulus and Bonding
© H. Föll (Iron, Steel and Swords script)