Fermi hole

From Wikipedia, the free encyclopedia

A consequence of the uncertainty principle is that electrons are fundamentally indistinguishable from one another – when two electrons get close to one other (or collide), their wavefunctions overlap and it is impossible to identify each electron individually before and after the collision. This inherent indistinguishability of electrons gives rise to what are termed exchange integrals in many-electron (determinantal) wavefunctions (see the article on atomic orbitals for more information about these wavefunctions).

Consider, for example, an excited state of the helium atom in which electron 1 is in the 1s orbital and electron 2 has been excited to the 2s orbital. We cannot, in principle, distinguish electron 1 from electron 2. In other words, for all we know, electron 2 might be in the 1s orbital with electron 1 in the 2s orbital. While there are 4 possible spin states for this system, we will consider only the ones in which the spins of both electrons are aligned (pointing in the same direction). (This is the triplet state, there exists a singlet state with the spins paired).

Because electrons are fermions, they must be antisymmetric with respect to exchange. This means that if I switch electrons 1 and 2, I must get back the exact same wavefunction with a negative sign in front. This antisymmetry can arise either from the spin part (the intrinsic angular momentum of the electron) or the spatial part (the position of the electron as a function of r, theta, and phi) of the wavefunction. If the spatial part of the wavefunction is antisymmetric, the spatial wavefunction will look something like this (for the helium atom described above):

1s(1) 2s(2) – 1s (2) 2s(1)

where we cannot distinguish which electron is in which orbital (so we have separate terms for each case), and if we exchange the electrons, we get:

1s(2) 2s(1) – 1s(1) 2s(2)

which we can see is equal to

- [1s(1) 2s(2) – 1s (2) 2s(1)]

Thus this spatial wavefunction is antisymmetric. We can now see that if electrons 1 and 2 occupy exactly the same point in space, the wavefunction will vanish! Because the wavefunction squared gives the probability density for the electron, this antisymmetry means that the two electrons will never be found directly on top of each other. This gives rise to the phenomenon called the Fermi hole – the region around an electron in which no other electron with parallel spin will come.

Fermi holes give rise to the Pauli exclusion principle and are responsible for the space-occupying properties of matter. (This principle does not hold for bosons, which may all occupy a single state as in lasers and Bose-Einstein condensates.)

A related phenomenon, called the Fermi heap, occurs when the wavefunction antisymmetry arises from the spin part of the wavefunction, giving the spatial wavefunction (this is the singlet state described above):

1s(1) 2s(2) + 1s (2) 2s(1)

In this case, for paired spins, there is actually a slightly higher probability of finding the electrons together. Fermi heaps play an important role in chemical bonding.

Since electrons repel one another electrically, Fermi holes and Fermi heaps have drastic effects on the energy of many-electron atoms. The most profound result is the periodic properties of the elements.

Animations of Fermi holes and Fermi heaps in the carbon atom are here^[1]. Details of the origin and significance of Fermi holes and Fermi heaps in the structure of atoms are discussed here^[2].

[edit] References

1. ^ Dill, Dan, Fermi holes and Fermi heaps. URL checked 10 November 2006
2. ^ Dill, Dan, Many-electron atoms: Fermi holes and Fermi heaps. URL checked 10 November 2006

• Dill, Dan (2006). Notes on General Chemistry (2nd ed.), Chapter 3.5, Many-electron atoms: Fermi holes and Fermi heaps. W. H. Freeman. ISBN 0393976610.