# Why are isotopically pure diamonds 50% more thermally conductive than other diamonds?

A laboratory grown type-IIa diamond (no nitrogen defects) has a lambda of $$\pu{1800-2200 W/mK}$$, but an isotopically pure diamond of $$\ce{^{12}C}$$ can have up to $$\pu{3320 W/mK}$$.

Why are $$\ce{^{12}C}$$ diamonds so much more thermally conductive than diamonds with $$\approx \pu{1.1\%}$$ $$\ce{^{13}C}$$ atoms?

• Less phonon scattering from the 'impurity' C13 atoms. I'm not equipped right now to dig up the original paper, but the predicted results matched quite well with reality. – Jon Custer Mar 9 '17 at 12:46

## 1 Answer

The thermal conductivity was calculated using the Callaway method. This method takes into account the effects of intrinsic three-phonon normal (N) processes. These processes do not create thermal resistance directly but affect the thermal resistance by transferring phonons from frequencies where they are not scattered to frequencies where scattering is more efficient.

[...]

In order to explain the large increase (originally much larger than expected) in terms of isotope scattering it was necessary to include the effects of N processes. Subsequent analyses of the thermal conductivity of enriched diamond have included N processes. Wei et.al. used the Callaway expression to fit experimental thermal conductivity data on diamond with the natural isotope concentration and isotopically enriched (0.1% $\ce{^{13}C}$) diamond over a wide temperature range (100-1000K).

• OK, nice reference. But what does it mean? – matt_black Mar 10 '17 at 1:52
• The TL;DR is that the thermal conductivity of diamond is dependent upon it's intrinsic scattering phonon relaxation times. This property is at a maximum (meaning maximum thermal conductivity) with maximal isotopical purity (of either 12C or 13C), and at a minimum when there is a 50% mixture. This is not linear in nature, as even a diamond with 1% of one isotope has markedly lower relaxation times than one with 0.1%. This is a difficult topic (for me) to summarize, though I should probably have given it some discussion. Honestly there's not much of a shortcut here vs reading the cited article. – airhuff Mar 10 '17 at 2:27
• Airhuff - thank you! I've only found data on conductivity of 12C diamonds and not 13C - is the lambda of a 13C diamond similar to that of a 12C diamond (~3300 W/mK)? – Ada Diamonds Mar 10 '17 at 16:13
• Also, would a boron doped 12C diamond be more conductive than a pure 12C as it's electrically conductive? Here's another good explanation of phonon conductivity: chemistry.stackexchange.com/a/17250/42303 – Ada Diamonds Mar 10 '17 at 16:44
• I found then lost a link to an article in which they made pure 13C diamonds, pure 12C diamonds and many mixes in between. I'm sure I'll find it and I have some time to search now. Great answer by Dryden that you linked to. Off the top of my head, I would guess that a boron doped diamond would have increased phonon scattering, but maybe also increased electronic conductivity??? – airhuff Mar 11 '17 at 3:02