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Spring Model and Properties of
Crystals
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Looking at
Some Everyday Spring |
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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 ks (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. |
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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. |
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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?). |
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An ideal
High-School spring obeys a linear force
F - deflection x law, called Hooke's
law, or |
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The spring constant
ks = F/x describes
the actual spring and not the material the
spring is made from. If we want to assign spring properties to the properties of the material the spring is made from,
and not to the size of the spring material,
it would be a good idea to look at some specific spring constant that is somehow independent
of the size. That would be easy enough but I won't do it because we
already did
that. We replace forces by stress, elongation by strain - and the spring
constant then is called Young's modulus. A
large spring constant (or Youngs's modulus) means that you need large forces to
elongate or squeeze your material somewhat; we have a stiff spring. |
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There are two important things to
realize at this point:
- An ideal spring, described by Hooke's
law above, can be squeezed or pulled out to any lenght you like.
- There is no such thing as an ideal
spring.
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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". |
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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. |
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Analyzing a real spring |
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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 |
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U(x) |
= |
x
ó
õ
0 |
F(x') · x' · dx' |
= |
x
ó
õ
0 |
ks · x' · dx'
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= |
½ks x2 |
for an ideal spring |
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= |
??? |
= blue potential curve for a real spring |
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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. |
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Analyzing a real bond
"spring" |
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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:
- If your mass is not at the potential minimum, it "wants" to move
down into the minimum of the well, just as a real sphere in a real well or
bowl, shaped like this potential, would do.
- The force that tries to send the mass down into the minimum is proportional
to the slope at the point where the mass is. The steeper, the more force,
always in the "down" direction. That's what you feel when you stand
on some slope on roller skates. On steep slopes you feel more force and you
accelerate (proportional to the force, heed Newton's first law) faster.
- The mass has only potential energy at
the point where it is originally held. How much is directly given by the
position. If you let go, it will move down, run through the potential minimum
with maximum speed, and then move up on the other side to exactly the same
potential energy it had originally (we assume that there is no friction). At
the zero point the mass has only kinetic
energy that must be exactly as large as the potential energy at its
starting point. At intermediate points is has a mix of kinetic and potential
energy but the sum total is always constant and equal to the potential energy
at the starting point.
- A horizontal line at the maximum potential energy thus symbolizes the
amount of the total energy or the energy
level that is characteristic for the the oscillation that results when you let the mass go.
The energy level gives the total energy,
which is concerved and thus the same at any position.
- The average position of the mass is in
the middle of the line symbolizing the energy level since the (oscillating)
mass is just as often to the left as to the right. Since our potential gets
flatter to the right, the average position shifts increasingly to the right of
the center line at x = 0 for larger and larger deflections.
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Fascinating, isn't it? No? OK, then let's make
the big step: |
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Bonds
Between Atoms and the Common Spring |
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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. |
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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. |
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Spring model of a simple (e.g. LiF)
crystal |
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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. |
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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 T1 <
T2 if we consider a crystal. |
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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. |
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The distance between the atom is now called
r. Imagining the atom as small spheres with radius
r0 (numbers for most atoms via
this link), the
distance between two of them just touching each other then is about roughly
2r0 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
r0, a parameter that is extremely easy to measure and
thus a known quantity. |
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The depth of the potential well
is the binding energy U0. 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 U0. |
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Binding potential, some parameters, and the
force between the atoms |
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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 |
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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 r0 and the binding energy
U0 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 A
» 2e2/4peo
(e =
elementary charge,
eo =
vacuum
susceptibility) - that's easy. For other types of bonding it's not quite
that easy. We have some good ideas, however, what their value will be:
something between 1 -10. |
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What do we get from this? Well, a
lot:
Let's look at this, one property at a time |
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Young's Modulus
and the Binding
Potential |
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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 ks of the bonding "springs"
and express it as Young's modulus Y. |
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Pondering this a bit you realize that we have the
simple relation:
ks = Y · r0. 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: |
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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 |
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Y = |
1
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d2U
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n · m ·
U0
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r0 |
dr2 |
r03 |
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With approximate numbers for n and
m and determining U0 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: |
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Calculated Young's modulus (red line) and
measured values |
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This is an old picture and I don't know who did
it (sorry). It shows that the spring-bond theory is rather good and essentially
demonstrates one thing: |
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We have Young's modulus covered!
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Note that Young's modulus as determined from a
tensile test measures the "springiness" of all bonds contained in a crystal. Since for
low-alloy steels,
the kind we consider for ancient swords, the overwhelming majority of the bonds
are iron-iron 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. |
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Melting Point and
Thermal
Expansion |
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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 U0, 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. |
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In other words, the material melts as soon as the
vibration energy EVib approaches the binding energy
and that gives us the melting
point Tm. Since the vibration energy is
nothing but a measure of the temperature T or the
thermal energy kT (k =
Boltzmann
constant), we have kTm » U0 |
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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. |
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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 etherm caused by a temperature change of
DT per Kelvin, i.e. a = etherm/DT |
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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 |
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a = |
n + m +
3
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» const. ·
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1
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2 · n · m ·
U0 |
Tm |
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The essential point is that a is proportional to 1/U0 and
thus approximately also to 1/Tm. 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 |
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Calculated and measured thermal expansion
coefficients |
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Once more a convincing result with the
verdict: |
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We have thermal expansion covered
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Vibration
Frequency of Atoms in a Solid |
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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 ks, 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. |
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So we might guess that the vibration frequency
n should be proportional to
ks/m. Nice try - but not quite right. The
precise relation is: n = 1/2p ·
(ks/m)½. So the
proportionality is to the square root of
ks/m |
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Substituting Young's modulus for
ks we get |
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n |
= |
1
2p |
· |
æ
ç
è |
Y · r0
ma |
ö
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1/2 |
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Putting in some numbers for
Young's modulus Y and interatomic distances
r0 we get a very important rough number: |
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The vibration frequency of atoms in
a crystal is roughly 1013 Hz
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Let's put that number into perspective:
- Operating frequency of your processor up there (called brain): around 50
Hz. Less after a few beers.
- Frequency of alternating line current: 50 Hz / 60 Hz.
- Frequency of radio waves: 100 kHz - some MHz
- Operating frequency of cell phones: around 1 GHz = 109
Hz.
- Operating frequency of a fast PC or Notebook: about 5 GHz.
- Operating frequency of your microwave oven: 8 Ghz
- Radar, other microwave stuff: up to an beyond 100 GHz.
- Body scanners at airports: a few THz = 1012 Hz
- Deep infrared radiations: (1012 - 1014 ) Hz
- Visible light: around 1015 Hz
- Ultraviolet light: 1016 Hz
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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. |
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Ultimate Fracture
Stress |
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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 % |
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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. |
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What we get after rather lengthy calculations is
complicated - but not very useful: |
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sfracture = |
n · m ·
U0
r03 |
æ
ç
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æ
ç
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n + 1
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ö
÷
ø |
1/(n m) |
1 |
ö
÷
ø |
m + 1 |
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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 cross-section together in one fell
swoop, while in reality fracture occurs be a crack sweeping across the material
and thus bonds are broken "one-by-one", 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 |
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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. |
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Well, tough luck here. We do not have fracture covered by working with the
bond-spring 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. |
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Famous
Last Words |
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The bond-spring 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:
- The potential equation is far too simple. It does not, for example, account
for the directionality of bonds.
- Looking at just two atoms does not quite do justice to a crystal with a
hell of a lot of atoms.
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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. |
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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: |
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Some "real" potentials used for
calculations |
Source: G J Ackland, M IMendelev, D J
Srolovitz, S Han and A V Barashev, Development of an interatomic potential for
phosphorus impurities in a-iron, J. Phys.: Condens. Matter 16 (2004)
S2629-S2642 |
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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. |
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© H. Föll (Iron, Steel and Swords script)