Less binding energy, increased atom energy?

[annaphys]

I'm a bit confused as to what is meant by Professor Simon in is textbook Oxforc Solid State Basics. I attached a photo of the page (number 213).

Here is my confusion:

"In this case, the electrons see the full charge of the nucleus and bind more strongly, thus lowering their energies."

Binding increased -- makes sense as there is no effective charge. But how can "their" energies be lower? What does that even mean? Since they are more strongly attached, the energy of the system is higher.

How can the binding energy be reduced of one electron down and the other up all the while increasing the total energy of the atom? If one of the atoms sees an effective charge, then it is easier to ionize the atom than if the two electrons had the same spin.

 

[BvU]

But how can "their" energies be lower? What does that even mean?

It means that it takes more energy to fully separate the electrons from the nucleus. For convenience we take the zero energy point at infinity, but in fact that doesn't matter: it's all energy differences (think 'forces') that govern the behaviour.

 

[annaphys]

It means that it takes more energy to fully separate the electrons from the nucleus.

Right, so that means the energy is higher but the prof says the energy is lower. Meaning, it takes more energy to ionize the atom.

 

[BvU]

No, the energy is lower: you need to add energy to separate the ion and the electron and come out at what we call zero energy for the electron at infinity.

Compare to a book on a shelf: you need to do work to bring it to a higher shelf. Closer to the floor the potential energy is lower.

Or: look at the direction of the force: if the force you have to apply is in the same direction as the motion, you have to do work on the system.

 

[annaphys]

No, the energy is lower: you need to add energy to separate the ion and the electron and come out at what we call zero energy for the electron at infinity.

But we have to also add energy for the case where the two spins are not aligned i.e. one up one down. It just so happens, that it is less energy to add to the atom to make the electron separate from the atom.

 

[BvU]

Maybe actually posting the picture you mentioned in #1 might help me to understand where this spin stuff suddenly comes from ?

 

[Cthugha]

Right, so that means the energy is higher but the prof says the energy is lower. Meaning, it takes more energy to ionize the atom.

I just checked the book on Amazon and @BvU is completely correct. The author is considering three different energies with respect to the Coulomb interaction: The energy of the electrons, the binding energy of the nucleus-electron system and the total energy of the atom. When the two spins are aligned, the screening is minimized. This means that the binding energy is highest, the energy of the electrons (which the author refers to as "their energy") is lowest and the total energy of the atom (which is the negative binding energy) is lowest as well.

Then the author goes on to discuss the case of spins that are not aligned. Here, screening is more effective. This means that the binding energy goes down, the energy of the electrons necessarily goes up and so does the total energy of the atom.

 

[annaphys]

@Cthugha so when the energy of the electron goes up, the total amount of the energy from the atom goes up? Still not understanding the nomenclature. Electrons don't "save" energy somewhere. I am sure my confusion is with the definition of where the energy is stored and what does it mean that the atom has more energy.

Photo from the book is attached.

 

[Cthugha]

@Cthugha so when the energy of the electron goes up, the total amount of the energy from the atom goes up? Still not understanding the nomenclature. Electrons don't "save" energy somewhere. I am sure my confusion is with the definition of where the energy is stored and what does it mean that the atom has more energy.

Photo from the book is attached.

I am not sure I get where your problem is, so please excuse my if my following explanation seems to basic:
If you have two free particles, say an electron and a proton, they have some certain energy. If they meet and form a bound state - the hydrogen atom - the energy of this bound state is lower compared to the energy of the free particles. The difference in energy is exactly the binding energy of the atom. Accordingly, a larger binding energy means that the atom has lower energy.

The nucleus is much heavier compared to the electrons, so in first approximation one may consider the problem of the energy of the electron system compared to a fixed core instead. Here, the energy of the electrons is the energy of the free particles minus the binding energy of the state in question. The energy of the atom is now (also in first approximation) this energy plus the center-of-mass motion of the atom, where the latter is unrelated to the former. In practice this is done in a better way by separating the problem into center-of-mass and relative coordinates. Have you ever studied the quantum mechanics treatment of the hydrogen atom?

 

[annaphys]

I'm taking qm this semester but haven't touched on h-atom yet. But I get what you're saying now. It was never explained to me hence me asking. So could I go on to say: the higher/more bond types (covalent, metallic etc.) then the atom and the electron(s) have less energy?

Another question: when you refer to the energy of a free particle, that is only the kinetic energy right? But when a proton and an electron come into near contact then a lot of the energy converts into potential energy, right?

 

[Cutter Ketch]

I'm taking qm this semester but haven't touched on h-atom yet. But I get what you're saying now. It was never explained to me hence me asking. So could I go on to say: the higher/more bond types (covalent, metallic etc.) then the atom and the electron(s) have less energy?

Another question: when you refer to the energy of a free particle, that is only the kinetic energy right? But when a proton and an electron come into near contact then a lot of the energy converts into potential energy, right?

I believe you still have it completely backwards. When an electron and a proton are far apart they have high potential energy. I.e. you can do work using their mutual attraction. As they come together the potential energy decreases. They give up energy which is released in some form or another. Perhaps they come together in an excited state and then decay to the ground state by giving off a photon, or perhaps the extra energy is given up to a nearby atom, or perhaps the energy is given up by stimulating a lattice vibration in a solid. Whatever the case, the energy of the bound system is less than the unbound system. Energy was given up. If you wanted to take them back apart, you would have to provide energy. You would have to stimulate the electron with an incoming photon or lattice vibration or collision with some other particle.

The bound state is the low potential energy state. In the lowest bound state there is no available energy to do anything. The attraction of the electron and proton has nothing left to give.

I think your confusion may come from the fact that binding energy is defined backwards. It is defined as the amount of external energy that must be applied to rip the electron off. This means that a positive binding energy indicates a low energy state. Binding energy answers the question “how deep is the well?” And a big positive number means “boy, it is really stuck down in there, it’s going to take a lot of energy to get it back out of that hole.”

 

[annaphys]

Many thanks for the clarification. That makes a lot more sense now. As for your last paragraph, binding energy is usually negative, right? The ground state is roughly -13,6 eV for hydrogen and the excited states then are "less" negative i.e. -10, -5 etc. and then stack up closer and closer together when the energy gets near 0 eV. Then the ionization at 0 eV .

 

[Cutter Ketch]

Many thanks for the clarification. That makes a lot more sense now. As for your last paragraph, binding energy is usually negative, right? The ground state is roughly -13,6 eV for hydrogen and the excited states then are "less" negative i.e. -10, -5 etc. and then stack up closer and closer together when the energy gets near 0 eV. Then the ionization at 0 eV .

The potential energy of the bound state is usually negative. Since you can define zero energy anywhere you want this is an arbitrary choice, but typically the zero for energy is taken as being the well separated condition. So typically the energy of a hydrogen atom is stated as -13.6 eV

However, when you say “the binding energy” that is usually defined as the amount of energy required to disassemble the bound particles. With that definition binding energy is positive. If you look up “the binding energy of Hydrogen” you will usually see 13.6 eV, not -13.6 eV.