Relativistic effect in mercury

[hariharan venkatasu]

How come mercury qualify as a metal, since the 6 s electrons are attracted by the nucleus so strongly because of relativistic effects and unlike other metals there is no cloud of electrons?. Because of this mercury is not a good conductor of electricity unlike other metals

 

[HAYAO]

This is a good question, actually.

Electrical conductivity of mercury is 1x10⁶ S m^-1 while gold (next to mercury in the periodic table) is 4.5x10⁷ S m^-1. This is because the contribution of the 6s orbitals to the conduction band is small due to relativistic effect. That being said, they are still much higher than semiconductors, some times by several orders of magnitude. It's also still ductile, reflective, malleable, and conducts heat like any other metal.

I don't know what you mean by "no cloud of electrons".

 

[hariharan venkatasu]

Thanks a lot for the answer.I meant by cloud of electrons the" sea of electrons" found in other metals.

 

[HAYAO]

I still don't know what "sea of electrons" mean. Mercury also have bunch of electrons of its own.

 

[hariharan venkatasu]

It refers to the free delocalized electrons surrounding the atoms of a metal wandering around the nuclei of the metal atoms.But mercury has only two 6 s electrons which are again tightly bound to the nucleus because of relativistic effects These two electrons are not free enough to float around.I hope I have made my point clear.

 

[TeethWhitener]

As @HAYAO pointed out, mercury is a far better conductor than non-metals are. Mercury's conductivity is 1x10⁶ S m^-1. Compare that to silicon, a non-metal, whose conductivity is 4x10^-4 S m^-1, a difference of ten orders of magnitude.

Relativistic effects are not as important for mercury's metallic character as you might think. After all, cadmium, zinc, and all the alkaline Earth metals (beryllium through radium) nominally have filled subshells in their atomic ground states, and they are all metallic elements. So relativity is not really the determining factor. It turns out that the band structure of these "filled subshell" elements is such that the filled bands and the empty bands usually overlap. So in the case of mercury, the filled d and s bands overlap with the empty p band. This is a heuristic--in reality all of these states are mixed--but the upshot is that mercury and other elements in similar positions on the periodic table exhibit no bandgap and are therefore metallic.

One other small note: mercury at STP is a trickier case than the other metals I mentioned because mercury is a liquid (so "band structure" is kind of meaningless). However, this turns out not to be such a big deal, and it also generalizes so that most metals are still conductive in their molten state. I'm stealing most of this from Neville Mott's 1934 paper:
https://www.jstor.org/stable/2935602?seq=4#metadata_info_tab_contents
The equation for conductivity in a monatomic condensed system is given by Bethe as:
$$\sigma = \frac{2n_0}{\pi}\frac{M}{m}\frac{k_B\Theta^2}{ha_0CT}K\left(\frac{dE}{dK}\right)^2$$
where $n_0$ is the number of free electrons per atom, $M$ is the mass of an atom, $m$ is the mass of the electron, $k_B$ is the Boltzmann constant, $h$ is Planck's constant, $a_0$ is the Bohr radius, $T$ is temperature, $\Theta$ is the Debye temperature, and $K$ is the crystal momentum wave number. Mott shows that the only quantities that change appreciably upon melting are $dE/dK$ and $\Theta$, and in the free electron limit, even $dE/dK$ shouldn't change all that much. So the conductivity change from solid to liquid is dominated by the ratio of Debye temperatures (or, equivalently, the Gruneisen parameter), and this is observed to be a pretty good approximation in most normal cases.

 

[HAYAO]

Just a little note for OP: when @TeethWhitener said "no bandgap", he basically means this:

(From wikipedia: https://en.wikipedia.org/wiki/Semimetal)

Despite mercury having smaller contribution of 6s orbitals thereby showing low conductivity, they still have no bandgap, which means that mercury is still metallic in terms of its electrical conductivity.

 

[hariharan venkatasu]

Thanks a lot for your last post.It was very useful and educative.However your emphasis is on conductivity.My question was mainly on relativistic effect on mercury.Mercury has only two valence electrons in 6 s orbital.This makes it behave like a noble gas.The electrons are held so tightly to the nucleus due to relativity.As such mercury does not have any free electron like the sea of electrons in other metals.Then how can mercury be classified as a metal?

 

[TeethWhitener]

Atomic mercury is not bulk mercury. Posts #2 and #6 answered your question, but I’ll repeat it here. The filled d and s bands overlap with the empty p band such that there is no band gap. The definition of a metal is that there is no band gap. The observational hallmark of this absence of a band gap is high conductivity, which is observed in mercury.

Also, as mentioned earlier, relativity is not the primary effect here. Relativity is somewhat responsible for lowering the melting point of mercury, but zinc and cadmium also have low melting points compared to nickel, silver, and gold. This has more to do with the closed sub shells in those metals than relativistic effects.

 

[hariharan venkatasu]

Thank you so much for the nice explanation

 

[snorkack]

Counting metals or semimetals which are notably poor conductors:

1. Mercury - in contrast with cadmium which despite being low boiling is a good conductor
2. Manganese
3. Tin... white is a good conductor, gray a semiconductor
4. Arsenic - semimetal but a good conductor
5. Antimony - ditto
6. Bismuth - seems to be worse conductor than arsenic or antimony
7. Neptunium
8. Plutonium

So, why are those elements metals? And yet, why are they poor conductors?

 

[hariharan venkatasu]

I am not able to answer that.Please tell me.Thank you.

 

[TeethWhitener]

Please understand that when you say “poor conductor,” the examples you’re giving have conductivities seven orders of magnitude higher than something like silicon. So yes, they’re worse conductors than something like copper, but they’re still ten million times better conductors than silicon.

 

[TeethWhitener]

Looking at your list, I’ve explained mercury (whose conductivity is only slightly worse than cadmium). Arsenic, antimony, gray tin, and bismuth are all semimetals, meaning that the top of their valence bands is slightly higher in energy than the bottom of their conduction bands, and they’re located at different points in the Brillouin zone.

Of all the cases mentioned, manganese and the actinides are the most interesting. The actinides undergo self-radiation damage, but beyond that, the 5f valence band is more localized than the (energetically) deeper 6s band. This is also the reason why all actinides and lanthanides are found almost exclusively in the +3 oxidation state. Basically, the outermost electrons of the lanthanides and actinides are not the valence electrons. In actinides, this is more pronounced due to relativistic effects, which contract the 5f orbitals. All these factors lead to localization of electrons in the metal and decreased conductivity.

Manganese is another very special case. Atomically, manganese has a half-filled subshell, which stabilizes it. In a typical metal, this would lead to high conductivity. However, the ground state of manganese is much more stable than its excited states, and this translates into better localization of the valence electrons in manganese metal, and therefore lower conductivity. The same effect is observed to a much lesser extent in rhenium (and presumably technetium).

 

[ZapperZ]

Counting metals or semimetals which are notably poor conductors:

1. Mercury - in contrast with cadmium which despite being low boiling is a good conductor
2. Manganese
3. Tin... white is a good conductor, gray a semiconductor
4. Arsenic - semimetal but a good conductor
5. Antimony - ditto
6. Bismuth - seems to be worse conductor than arsenic or antimony
7. Neptunium
8. Plutonium

So, why are those elements metals? And yet, why are they poor conductors?

I'd like to echo what @TeethWhitener has stated. The term "poor conductors" is very vague and highly undefined. Whether it is a poor or good conductor, it is STILL a conductor, meaning that it has a band that crosses the Fermi level. It is still not a semiconductor or a band insulator, in which NO band crosses the Fermi level.

THAT is a more rigorous definition of what a conductor/metal and semiconductor/insulator is.

I wonder if this topic is better suited to be discussed in the Condensed Matter forum than Chemistry.

Zz.

 

[snorkack]

I'd like to echo what @TeethWhitener has stated. The term "poor conductors" is very vague and highly undefined. Whether it is a poor or good conductor, it is STILL a conductor, meaning that it has a band that crosses the Fermi level. It is still not a semiconductor or a band insulator, in which NO band crosses the Fermi level.

THAT is a more rigorous definition of what a conductor/metal and semiconductor/insulator is.

Quoted resistivities:
Bi - 1290 nΩm
Sb - 417 nΩm
As - 333 nΩm
Hg - 960 nΩm
Cd - 73 nΩm
Mn - 1440 nΩm
Tc - 200 nΩm
Ti - 420 nΩm
Pu - 1460 nΩm
Np - 1220 nΩm
U - 280 nΩm

Note that As and Sb have low resistances - in the range of Ti or U, and not close to the resistance of Bi, Hg, Mn, Pu or Np. Just how is the conductivity of As or Sb special?
A common feature of Mn and Pu is complex structure. Mn unit cell has 58 atoms in 4 different positions, Pu unit cell 16 atoms in 8 different positions.

 

[TeethWhitener]

Quoted resistivities:
Bi - 1290 nΩm
Sb - 417 nΩm
As - 333 nΩm
Hg - 960 nΩm
Cd - 73 nΩm
Mn - 1440 nΩm
Tc - 200 nΩm
Ti - 420 nΩm
Pu - 1460 nΩm
Np - 1220 nΩm
U - 280 nΩm

Note that As and Sb have low resistances - in the range of Ti or U, and not close to the resistance of Bi, Hg, Mn, Pu or Np. Just how is the conductivity of As or Sb special?
A common feature of Mn and Pu is complex structure. Mn unit cell has 58 atoms in 4 different positions, Pu unit cell 16 atoms in 8 different positions.

Once again, so that we’re absolutely clear:
Si - 6000000000000 n$\Omega$m

Edit: the point is that, while the conductivities of the metals can differ among themselves, they’re basically rounding errors or less when compared to non-metals.

 

[snorkack]

Gray Sn is difficult to produce reliably, but it is stated to be 2000...5000 nΩm at room temperature (metastable above +13 degrees), while white tin is 109 nΩm.
Carbon is widely acknowledged to be a nonmetal but has resistance also around 2500...5000 nΩm along plane, 3 000 000 nΩm across the planes.

 

[TeethWhitener]

Gray tin and graphite are both semimetals. I gave the technical definition of this earlier. The upshot is that they have no band gap, but they also have a very low density of states at the Fermi level.

Non-metals (semiconductors and insulators) have a band gap. That is how they are defined. A consequence of this is that their conductivity is orders of magnitude lower than that of metals.

 

[ZapperZ]

Quoted resistivities:
Bi - 1290 nΩm
Sb - 417 nΩm
As - 333 nΩm
Hg - 960 nΩm
Cd - 73 nΩm
Mn - 1440 nΩm
Tc - 200 nΩm
Ti - 420 nΩm
Pu - 1460 nΩm
Np - 1220 nΩm
U - 280 nΩm

Note that As and Sb have low resistances - in the range of Ti or U, and not close to the resistance of Bi, Hg, Mn, Pu or Np. Just how is the conductivity of As or Sb special?
A common feature of Mn and Pu is complex structure. Mn unit cell has 58 atoms in 4 different positions, Pu unit cell 16 atoms in 8 different positions.

Is there a reason why you think that all metals should have the same resistivity, or even 'close' to one another? This is very puzzling because this appears to be the SOURCE of this whole line of question.

The resistivity of a material depends on many things. I can take a material, rearrange its crystal structure WITHOUT changing the material (element), and I can get a different resistivity! In fact, I can show you layered perovskite material that has different resistivities in different crystallographic orientation, all within the SAME material!

Zz.

 

[HAYAO]

Of all the cases mentioned, manganese and the actinides are the most interesting. The actinides undergo self-radiation damage, but beyond that, the 5f valence band is more localized than the (energetically) deeper 6s band. This is also the reason why all actinides and lanthanides are found almost exclusively in the +3 oxidation state. Basically, the outermost electrons of the lanthanides and actinides are not the valence electrons. In actinides, this is more pronounced due to relativistic effects, which contract the 5f orbitals. All these factors lead to localization of electrons in the metal and decreased conductivity.

A little nitpickey point: Actually, the relativistic effect does not contract the 5f and 4f orbitals, it indirectly does the opposite.

(Radial function of orbitals, taken from "The Chemical Complexity of Plutonium" by David L. Clarke, Los Alamos Science 2000.)

The reason is because the spherically symmetric inner s and p orbitals are contracted inwards by relativistic effects, which in turn shields the effective charge of the nucleus, and causes spherically asymmetric f and d orbitals to extend outwards. A similar effect is observed in lanthanides but they are lighter than actinides; the relativistic effect is not as pronounced.

As you mentioned, both in lanthanides and actinides, the outer electrons are not valence electrons, so they contribute minimally with bonding. Instead, orbitals like 5d, 6s, 6p (lanthanides) and 6d, 7s, 7p (actinides) are involved. Due to their large ionic radii, they have strong ionic character than covalent character in bonding. However, for actinides, you can see from the figure that 5f-orbital tails are distributed farther out, which means that they can still get more involved in bonding compared to lanthanides. This also explains why actinides can show more variety in oxidization states compared to lanthanides.

 

[snorkack]

The resistivity of a material depends on many things. I can take a material, rearrange its crystal structure WITHOUT changing the material (element), and I can get a different resistivity!

So did I. Gray and white tin.

In fact, I can show you layered perovskite material that has different resistivities in different crystallographic orientation, all within the SAME material!

And I showed you graphite.
My line of questioning was inspired on how to predict semiconducting behaviour. And for the insight of potential causes of semiconductive behaviour, I sought explanations for poorly conducting metals, on the hunch that these might be near the metal-semiconductor boundary, though on the metal side.
I noted that nearly all rare Earth's are also poor conductors. Any common reason for that?

 

[ZapperZ]

So did I. Gray and white tin.
And I showed you graphite.
My line of questioning was inspired on how to predict semiconducting behaviour. And for the insight of potential causes of semiconductive behaviour, I sought explanations for poorly conducting metals, on the hunch that these might be near the metal-semiconductor boundary, though on the metal side.
I noted that nearly all rare Earth's are also poor conductors. Any common reason for that?

Metal-insulator transition is one of the most complex subject in condensed matter. If you do not believe me, look at the humongous review article by Imada et al. Rev. Mod. Phys. v.70, p.1039 (1998). It's 224 pages long! Do you want the Mott-Hubbard model, the Zhang-Rice singlet?

When I did my research work on the bismuthates, the darn thing switched from insulator to metallic at different temperatures, and all due to a dimensional crossover from 2D to 3D!

So if you are looking for one simple and single explanation for such a transition, I personally do not believe there is one. Even the reason for something to be an "insulator" can be different. Look at the parent compound of the cuprate superconductors. Undoped, band theory predicts that it is a metal. Yet, it is an insulator! You need a different description to show why it is an insulator.

Zz.

 

[TeethWhitener]

I noted that nearly all rare Earth's are also poor conductors. Any common reason for that?

Yes, the same reason as the actinides (posts 14, 21).

 

[PeterDonis]

The OP question has been answered. Thread closed.