Geometry of Electron Shells

Splossy said:
Buckaroo: would I be right in saying that these shapes are purely mathematical contructs anyway? Do the theories suggest those shapes actually exist in 3d space in a meaningful way?
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Yes and no! The shapes are mathematical constructs, describing the probability (after a little more manipulation) that an electron will occupy the space they encompass. So they also correspond to real 3D space. These kinds of things become tricky once we start talking about wave-particle duality!
 
Reading the links provided I am again reminded of the wonderful individual and collective intelligence of scientists in this field. The depth of thought and rigor is really quite amazing.
 
Splossy said:
Traps or whatever aside I am interested to find out why these numbers occur. I can't work out what "all it is is counting up the number of orthogonal spherical wave states that it's possible to construct (plus a factor of two from the electron spin" means. Nor can I glean it from the Wikipedia article.
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First, let's establish what we mean by orthogonal. Well, with vectors in Euclidean space, the concept is pretty simple: it means that they're 90 degrees apart. One test for orthogonality is that their dot product is zero. Well, it turns out that this concept is generalizable, and can be applied to much more than just vectors in 3D euclidean space. It can also be applied to functions. So what's the equivalent of taking a dot product of a function?

First, let's review dot products for vectors. Let's say we have two vectors, V and W. We can label their components as Vx, Vy, and Vz, and Wx, Wy, and Wz. Now the dot product is
V dot W = Vx Wx + Vy Wy + Vz Wz
We just multiply each component, then add them up.
Now for functions. Let's say we have functions f(x) and g(x). The equivalent to multiplying each component is to multiply the values of the function at the same point, or f(x)g(x). Now we just add all that up, or in other words, integrate the functions. So
[latex]f(x) \cdot g(x) = \int{f(x) g(x) dx}[/latex]
Let me give you an example of two functions that are orthogonal: sin(x) and sin(x+pi/2). Those two functions are orthogonal, and ANY function of the form sin(x+c) (where c is an arbitrary constant) can be expressed as some unique linear combination of those first two orthogonal functions. This is equivalent to picking out an x and a y axis in a 2D plane, and expressing any vector as some linear combination of unit x and y vectors. We could have picked other orthogonal functions: sin(x+pi/8) and sin(x-3pi/8) also work, for example. But we can only pick two functions of this form which are mutually orthogonal, any third function cannot be orthogonal to both of them.

This is only a first step, but have you followed so far?
 
Davidjayjordan said:
Buckaroo, may I suggest that truths are simplier than you can imagine. Just learn to discern what is important and then formulate an opinion based on important truths.

Lifes mysteries are not just for the so called intellectual, for they sometimes are totally blind and bias. True skeptics are limited more by lack of insight as most of us surely have the God given brains to figure out what has been created.

Don;t look to others to SOLVE problems, when we personnally can figure out much much more than what we think, if we THINK.

So give it a go, and see what you come up with. Thanks
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See, here's the thing -- we KNOW THE ANSWER. There are no mysteries in this particular subject. Furthermore, the answer is simply and elegantly expressed in mathematics. There's nothing "intellectual" about it -- like any foreign language, you just have to study it. I have personally solved these problems using this language, and I understand them. You have not, and you do not. You are profoundly ignorant on this subject. So don't presume to lecture ME.
 
Ziggurat said:
First, let's establish what we mean by orthogonal. Well, with vectors in Euclidean space, the concept is pretty simple: it means that they're 90 degrees apart. One test for orthogonality is that their dot product is zero. Well, it turns out that this concept is generalizable, and can be applied to much more than just vectors in 3D euclidean space. It can also be applied to functions. So what's the equivalent of taking a dot product of a function?

First, let's review dot products for vectors. Let's say we have two vectors, V and W. We can label their components as Vx, Vy, and Vz, and Wx, Wy, and Wz. Now the dot product is
V dot W = Vx Wx + Vy Wy + Vz Wz
We just multiply each component, then add them up.
Now for functions. Let's say we have functions f(x) and g(x). The equivalent to multiplying each component is to multiply the values of the function at the same point, or f(x)g(x). Now we just add all that up, or in other words, integrate the functions. So
[latex]f(x) \cdot g(x) = \int{f(x) g(x) dx}[/latex]
Let me give you an example of two functions that are orthogonal: sin(x) and sin(x+pi/2). Those two functions are orthogonal, and ANY function of the form sin(x+c) (where c is an arbitrary constant) can be expressed as some unique linear combination of those first two orthogonal functions. This is equivalent to picking out an x and a y axis in a 2D plane, and expressing any vector as some linear combination of unit x and y vectors. We could have picked other orthogonal functions: sin(x+pi/8) and sin(x-3pi/8) also work, for example. But we can only pick two functions of this form which are mutually orthogonal, any third function cannot be orthogonal to both of them.

This is only a first step, but have you followed so far?
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Hmm...I'd LIKE to say yes but I can only honestly answer sort of. I don't know what a dot product is and looking that up didn't really help at all.

I appear to have forgotton almost all my A-level maths and physics and some of the earlier stuff too. I'm having to look up what a vector is and work from there.

I think you are probably wasting your time trying to help understand this.
 
Dear researchers,

Can I suggest that if you just look at the relationship between the numbers 2, 8, 18 and 32. ...and its expansion. You will find a progression pattern between them, that will be the START to the solution. Its not very complicated at all, easy enough for any of us to understand.
 
Now this is more like it! So I don't need to bother with all that pesky maths and quantum theory afterall...
 
Buckaroo said:
See, here's the thing -- we KNOW THE ANSWER. There are no mysteries in this particular subject. Furthermore, the answer is simply and elegantly expressed in mathematics. There's nothing "intellectual" about it -- like any foreign language, you just have to study it. I have personally solved these problems using this language, and I understand them. You have not, and you do not. You are profoundly ignorant on this subject. So don't presume to lecture ME.
Click to expand...

If you know the answer, do state the answer, so others can know
 
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Davidjayjordan said:
If you know the answer, to state the answer, so others can know
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I already gave a link with the complete explanation. And yes, the explanation INCLUDES the regular progression of the numbers:

2*(1) = 2
2*(1+3) = 8
2*(1+3+5) = 18
2*(1+3+5+7) = 32
and so on.

Furthermore, that link gives a physical explanation as to WHY that progression of numbers (and not some other random number progression) is the correct one.
 
Splossy said:
Now this is more like it! So I don't need to bother with all that pesky maths and quantum theory afterall...
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Splossy, after thinking a bit, I remembered that you can make a rough analogy by considering a length of taut string that has been plucked so that standing waves are set up along it. The "quantum numbers" of this system would correspond to the integer count of the number of peaks in the standing wave. The string in this case would represent the probability amplitude of the quantum wavefunction describing the electron.

Not exactly the same as an atom, but it gives you a flavor of where the solution comes from, with no math.
 
Davidjayjordan said:
Dear researchers,

Can I suggest that if you just look at the relationship between the numbers 2, 8, 18 and 32. ...and its expansion. You will find a progression pattern between them, that will be the START to the solution. Its not very complicated at all, easy enough for any of us to understand.
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Actually, the non-mathematic/simplistic explanation is as follows:

Electrons in an atom have to be in an orbital, which is a mathematically-defined volume of space. Each orbital is described by three quantum numbers- n (which determines the size of the orbital, and the number of nodes = areas which have no electron density), l (which determines the shape of the orbital, and the number of planar nodes), and ml (which determines where in the x,y,z directions the orbital is pointing). No two orbitals can have the same set of three quantum numbers; no two orbitals can be identical, and each orbital can hold two electrons.
In the first shell, n = 1, and there are no nodes (how does it smell? Terrible!), so there is no planar nodes, and no variation in shape or direction. There can only be one orbital in the first shell. It is spherical and is called the 1s orbital (s doesn't actually stand for sphere, but it's an easy way to remember it). Therefore the first shell can have two and only two electrons.

In the second shell, n=2, and there is one node. This could be a radial node, which makes another spherical orbital (the 2s orbital), which has no variation in direction. We could also have a planar node, and this plane could lie in the xy plane, the xz plane, or the yz plane. We have three orbitals differing in their orientation in space, so we have the 2px, the 2py, and the 2pz orbitals. Grand total of 4 orbitals, so eight electrons.

Why do the 2p orbitals appear as dumbell shapes? If you take a spherical balloon and twist it so that there's no air in the centre, you now have the basic shape of something with a planar node. A dumbell.

In the third shell, n = 3, so we have two nodes.
Two radial nodes gives us the 3s orbital.
One radial and one planar gives us three 3p orbitals.
Two planar nodes give us the d orbitals. The shape of most of these is given by taking your balloon, twisting it in the centre to give a planar node, and then twisting it again along another axis (you can do it if your balloon isn't inflated very much)- you get a cloverleaf shape. The odd one out (the donut + dumbell shape) is made by taking the two nodal planes and mathematically adding them to give a nodal double-cone. The odd shape is simply what's left of a spheroid when you subtract a double cone from it.

Why are there 5 d orbitals in a shell? It's because of the possible values of ml. l can be anywhere from zero to one less than n; ml can be from -l to +1 (integer values only). So if n = 3, l can be 0 (3s), 1 (3p), or 2 (3d). When l = 2, ml can be -2, -1, 0, +1, +2. Five possible values = 5 solutions to the Schroedinger Eq'n = 5 orbitals.

So the third shell has 1 + 3 + 5 = 9 orbitals, and thus can hold 18 electrons.

The fourth shell has one 4s orbital, three 4p orbitals, five 4d orbitals, and seven of the next set, the 4f orbitals (which look like you've twisted any of the d orbitals to give another planar node in the balloon). 1 + 3 + 5 + 7 = 16, so 32 electrons can fit in the fourth shell.

The number of orbitals in a shell is always equal to n^2, so the number of electrons that can fit in the shell is always twice a perfect square, which is the amazing pattern you were pointing out earlier.

If you want to see the shapes of orbitals, I recommend the Orbitron Gallery:
http://winter.group.shef.ac.uk/orbitron/index.html

Any more chemistry-related numerology?
 
Davidjayjordan said:
In a true debate, both sides present their evidence or theories, rather than just one being hit with questions. So allow me to ask for your opinions or theories on why there are 2, 8, 18, and 32 electrons in the shells going outward from the Nucleus of an atom.
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There aren't

The real numbers are 2,2,6,2,6 then it gets mesy.
 
Ziggurat said:
I already gave a link with the complete explanation. And yes, the explanation INCLUDES the regular progression of the numbers:

2*(1) = 2
2*(1+3) = 8
2*(1+3+5) = 18
2*(1+3+5+7) = 32
and so on.

Furthermore, that link gives a physical explanation as to WHY that progression of numbers (and not some other random number progression) is the correct one.
Click to expand...

I'm almost out of popcorn.
 
Davidjayjordan said:
Dear researchers,

Can I suggest that if you just look at the relationship between the numbers 2, 8, 18 and 32. ...and its expansion. You will find a progression pattern between them, that will be the START to the solution. Its not very complicated at all, easy enough for any of us to understand.
Click to expand...

Here's a thought -- instead of being all coy and beating around the bush, why don't you TELL US WHAT YOU BELIEVE. What, are we just supposed to take your word for it that you've come up with a solution that is superior to what we know from quantum mechanics?
 
Davidjayjordan said:
Leader, just learn to lead rather than follow. State your theory FIRST, then I shall state mine. The floor remains yours.
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In accordance with your wishes, my theory is that you are clumsily attempting to set a trap while avoiding answering the points raised in your Platonic solids thread.

Now you state your theory.
 
You MAY know quantum MECHANICS.
YOU may know CHEMISTRY.
You may KNOW maths.
but you DO not KNOW phi.

BEHOLD.

I apologise if this is a DIFFICULT topic to introduce. Behold FIBONACCI numbers;

1, 1, 2, 3, 5, 8, 13, 21,

DO you see KNOW?

Take

the 1st and 2nd number = 2

the 4th and 5th number = 8

the 3rd, 4th, 5th and 6th number = 18

the 2nd, 3rd, 4th, 5th, 6th, 7th number = 32

Do YOU see know?

Do you SEE know?

Do you see KNOW?
 
Maybe it would be more fun to guess what DJJ's theory actually is. There's mine. I've a funny feeling it might not be too far wrong :)
 
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