<p>Finding an answer in the literature was a little vexing, perhaps because astrophysicists are not really interested in just what the crystal structure might be.</p>
<p>For instance, <a href="https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.104.231101" rel="nofollow noreferrer">Horowitz et al.</a> have a nice Physical Review Letter on &quot;Crystallization of Carbon-Oxygen Mixtures in White Dwarf Stars&quot; and ... never ... say what the crystal structure is. But, it does have a good overview of how the problem might be treated:</p>
<p>The electrons are assumed to be a degenerate Fermi gas. The ions interact with screened Coulomb interactions:</p>
<p><span class="math-container">$v_{i,j}(r) = Z_i Z_j e^2 e^{-r/\lambda} / r$</span></p>
<p>Where <span class="math-container">$\lambda$</span> is the Thomas Fermi screening length.</p>
<p>A paper in <a href="https://journals.aps.org/pre/pdf/10.1103/PhysRevE.86.066413" rel="nofollow noreferrer">Physical Review E</a> by J. Hughto et al. uses a similar formalism for the ion-ion interaction. They define a ratio</p>
<p><span class="math-container">$\kappa = a_e / &lt;Z&gt;^{1/3} \lambda $</span></p>
<p>where <span class="math-container">$a_e$</span> is the electron sphere radius in the degenerate gas. Their claim is that for their mixtures (carbon, oxygen, neon), <span class="math-container">$\kappa &lt; 0.4$</span> and they expect the ground state to be body-centered cubic.</p>
<p>The reference for that claim is a paper in the <a href="https://pubs.aip.org/aip/jcp/article/105/17/7641/293774/Phase-diagram-of-Yukawa-systems-near-the-one" rel="nofollow noreferrer">Journal of Chemical Physics</a> by S.S. Hamaguchi et al., titled &quot;Phase diagram of Yukawa systesm near the one-component-plasma limit revisited&quot;. Although Hamaguchi is on one-component plasmas, the methodology is all very similar, and the phase diagram (Fig. 2 in the paper) shows that bcc is the only solid until <span class="math-container">$\kappa &gt; 1$</span> is encountered, and then the low 'temperature' (its complicated) phase becomes fcc, with bcc remaining at higher temperature/energy.</p>
<p>So, the expected crystal structure in white dwarfs is body-centered cubic.</p>

CC BY-SA 4.0
2023-06-12T22:26:12.023Z
This is excellent, thank you! I will get on these as soon as I can! I found an open access copy of Hamaguchi et al. [here](https://www.semanticscholar.org/paper/Phase-diagram-of-Yukawa-systems-near-the-limit-Hamaguchi-Farouki/294d1bf9948dc0bca04eae14775ac02cbe883cd4). From your Hughto et al. uses link I can't even find the paper's title or the year - it's a blank paywall but Horowitz et al. is in [arXiv](https://arxiv.org/abs/1005.2441) and [researchgate](https://www.researchgate.net/publication/46423761_Crystallization_of_Carbon-Oxygen_Mixtures_in_White_Dwarf_Stars)
uhoh
CC BY-SA 4.0
2023-06-12T22:28:43.073Z
Some readers have no access to paywalled references, I do but only when I'm in the office (traveling today) so it's always best to take a moment and find accessible copies of important references and to *include the titles*. Thanks!
uhoh
CC BY-SA 4.0
2023-06-12T13:33:54.893Z
2023-06-12T16:30:28.220Z
2023-06-12T16:30:28.220Z
Jon Custer
<p>The abstract of Venner et al. (2023) <a href="https://arxiv.org/abs/2306.03140" rel="nofollow noreferrer">A Crystallizing White Dwarf in a Sirius-Like Quadruple System</a> includes the following:</p>
<blockquote>
<p>The location of HD 190412 C on the <sub>eff</sub> − mass diagram implies it is undergoing crystallization, making this the first confirmed crystallizing white dwarf whose total age can be externally constrained. Motivated by the possibility that a cooling delay caused by crystallization can be directly detected for this white dwarf we employ a variety of methods to constrain the age of the system; however, our empirical age anomaly of +3.1 ± 1.9 Gyr is ultimately too imprecise to reach statistical significance, preventing us from making strong constraints to models of white dwarf crystallization. Our results are nonetheless compatible with the recent hypothesis that <sup>22</sup>Ne phase separation is responsible for the excess cooling delay of crystallizing white dwarfs.</p>
</blockquote>
<p>The &quot;pile-up&quot; of white dwarfs in the <sub>eff</sub> − mass plot suggests temporary halt in the rate of apparent cooling and this is interpreted as an exothermic phase change. The phase change is further interpreted to be from a dense ionized plasma(?) to a very dense solid of primarily carbon with perhaps oxygen. If I understand correctly the solid phase is considered to be an actual crystal. Sciencealert.com's <a href="https://www.sciencealert.com/white-dwarf-star-enters-its-crystallization-era-turning-into-a-cosmic-diamond" rel="nofollow noreferrer">White Dwarf Star Enters Its Crystallization Era, Turning Into A 'Cosmic Diamond'</a> puts the density at of the order of 10<sup>6</sup> kg/m<sup>3</sup> (which would make the interatomic distances of the order of only 0.5 Angstroms<sup>1</sup>) so these crystals will not be something found or even reproducible on Earth.</p>
<blockquote>
<p>Whether or not that white dwarf crystal is diamond is unknown; the density of white dwarfs is over around 1 million kilograms per cubic meter, while the density of diamond is about 3,500 kilograms per cubic meter. Denser allotropes of carbon do exist; on the other hand, there's plenty of diamond floating around out there in space.</p>
</blockquote>
<p><sup>1</sup>per ProfRob's comment the density should be another 10<sup>3</sup> higher still, making the interatomic spacing more like 0.05 Angstroms!</p>
<p><strong>Question:</strong> What kind of crystal structures are predicted for the crystalline phase of matter inside crystalizing white dwarfs? Does it depend on the carbon/oxygen ratio?</p>
<p><strong>note:</strong> I'm looking for the crystallographic primitive unit cell as well as the conventional cell, but if it's for example <a href="https://en.wikipedia.org/wiki/Diamond_cubic" rel="nofollow noreferrer">diamond cubit</a> then that term covers both.</p>
<hr />
<p>Further reading:</p>
<ul>
<li>Wikipedia's <a href="https://en.wikipedia.org/wiki/White_dwarf#Radiation_and_cooling" rel="nofollow noreferrer">White dwarf; Radiation and cooling</a></li>
</ul>

CC BY-SA 4.0
2023-06-11T18:46:15.610Z
The density of white dwarf interiors is $>10^3$ times larger than you suggest.
ProfRob
CC BY-SA 4.0
2023-06-11T21:42:43.640Z
@ProfRob ...than Sciencealert.com suggests. I'll make a note of that, *thanks!* It makes the idea of an atomic lattice with what seems now to be of order 0.05A spacing even that more amazing!
uhoh
CC BY-SA 4.0
2023-06-12T01:15:24.123Z
Diamond cubic is fairly unlikely since it is a relatively open crystal structure. Hexagonal or straight fcc are close packed crystals. Diamond is not.
Jon Custer
CC BY-SA 4.0
2023-06-12T01:26:03.683Z
@JonCuster I see what you mean; 0.34 vs 0.74. If there's a significant oxygen/carbon ratio, then maybe the answer gets more interesting?
uhoh
CC BY-SA 4.0
2023-06-14T02:06:43.837Z
@uhoh does Jon_Custer's post answer your question? If not, I can add on to it in another post if you would like to [ that is if I can find anything ;) ]
DialFrost
CC BY-SA 4.0
2023-06-14T02:08:08.703Z
@DialFrost as long as you think you have something to add that may be related and/or interesting to either me or any other future reader, just go ahead and post it! :-) For challenging questions like this I usually don't accept an answer for a few weeks at least.
uhoh
CC BY-SA 4.0
2023-06-14T14:16:28.537Z
@DialFrost - indeed, feel free to add an answer with more/different information. I'm far from an expert on materials under these conditions so would be interested to learn more.
Jon Custer
<stellar-evolution><white-dwarf>
What crystal structures are predicted for the crystalline phase of matter inside crystalizing white dwarfs? Does it depend on the carbon/oxygen ratio?
uhoh
# What crystal structures are predicted for the crystalline phase of matter inside crystalizing white dwarfs? Does it depend on the carbon/oxygen ratio?

(Asked by: username_0 on 2023-06-11)

The abstract of Venner et al. (2023) [A Crystallizing White Dwarf in a Sirius-Like Quadruple System](https://arxiv.org/abs/2306.03140) includes the following:

> The location of HD 190412 C on the eff − mass diagram implies it is undergoing crystallization, making this the first confirmed crystallizing white dwarf whose total age can be externally constrained. Motivated by the possibility that a cooling delay caused by crystallization can be directly detected for this white dwarf we employ a variety of methods to constrain the age of the system; however, our empirical age anomaly of +3.1 ± 1.9 Gyr is ultimately too imprecise to reach statistical significance, preventing us from making strong constraints to models of white dwarf crystallization. Our results are nonetheless compatible with the recent hypothesis that 22Ne phase separation is responsible for the excess cooling delay of crystallizing white dwarfs.

The "pile-up" of white dwarfs in the eff − mass plot suggests temporary halt in the rate of apparent cooling and this is interpreted as an exothermic phase change. The phase change is further interpreted to be from a dense ionized plasma(?) to a very dense solid of primarily carbon with perhaps oxygen. If I understand correctly the solid phase is considered to be an actual crystal. Sciencealert.com's [White Dwarf Star Enters Its Crystallization Era, Turning Into A 'Cosmic Diamond'](https://www.sciencealert.com/white-dwarf-star-enters-its-crystallization-era-turning-into-a-cosmic-diamond) puts the density at of the order of 106 kg/m3 (which would make the interatomic distances of the order of only 0.5 Angstroms1) so these crystals will not be something found or even reproducible on Earth.

> Whether or not that white dwarf crystal is diamond is unknown; the density of white dwarfs is over around 1 million kilograms per cubic meter, while the density of diamond is about 3,500 kilograms per cubic meter. Denser allotropes of carbon do exist; on the other hand, there's plenty of diamond floating around out there in space.

1per ProfRob's comment the density should be another 103 higher still, making the interatomic spacing more like 0.05 Angstroms!

**Question:** What kind of crystal structures are predicted for the crystalline phase of matter inside crystalizing white dwarfs? Does it depend on the carbon/oxygen ratio?

**note:** I'm looking for the crystallographic primitive unit cell as well as the conventional cell, but if it's for example [diamond cubit](https://en.wikipedia.org/wiki/Diamond_cubic) then that term covers both.

---

Further reading:

* Wikipedia's [White dwarf; Radiation and cooling](https://en.wikipedia.org/wiki/White_dwarf#Radiation_and_cooling)

**Tags:** stellar-evolution, white-dwarf
> **Comments:**
> **username_1**: The density of white dwarf interiors is $>10^3$ times larger than you suggest.  
> **username_0**: @ProfRob ...than Sciencealert.com suggests. I'll make a note of that, *thanks!* It makes the idea of an atomic lattice with what seems now to be of order 0.05A spacing even that more amazing!  
> **username_2**: Diamond cubic is fairly unlikely since it is a relatively open crystal structure. Hexagonal or straight fcc are close packed crystals. Diamond is not.  
> **username_0**: @JonCuster I see what you mean; 0.34 vs 0.74. If there's a significant oxygen/carbon ratio, then maybe the answer gets more interesting?  
> **username_3**: @uhoh does Jon_Custer's post answer your question? If not, I can add on to it in another post if you would like to [ that is if I can find anything ;) ]  
> **username_0**: @DialFrost as long as you think you have something to add that may be related and/or interesting to either me or any other future reader, just go ahead and post it! :-) For challenging questions like this I usually don't accept an answer for a few weeks at least.  
> **username_2**: @DialFrost - indeed, feel free to add an answer with more/different information. I'm far from an expert on materials under these conditions so would be interested to learn more.  

---

## 1 answers

### Answer by username_2 on 2023-06-12. Score: 4

Finding an answer in the literature was a little vexing, perhaps because astrophysicists are not really interested in just what the crystal structure might be.

For instance, [Horowitz et al.](https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.104.231101) have a nice Physical Review Letter on "Crystallization of Carbon-Oxygen Mixtures in White Dwarf Stars" and ... never ... say what the crystal structure is. But, it does have a good overview of how the problem might be treated:

The electrons are assumed to be a degenerate Fermi gas. The ions interact with screened Coulomb interactions:

$v\_{i,j}(r) = Z\_i Z\_j e^2 e^{-r/\lambda} / r$

Where $\lambda$ is the Thomas Fermi screening length.

A paper in [Physical Review E](https://journals.aps.org/pre/pdf/10.1103/PhysRevE.86.066413) by J. Hughto et al. uses a similar formalism for the ion-ion interaction. They define a ratio

$\kappa = a\_e / ^{1/3} \lambda $

where $a\_e$ is the electron sphere radius in the degenerate gas. Their claim is that for their mixtures (carbon, oxygen, neon), $\kappa < 0.4$ and they expect the ground state to be body-centered cubic.

The reference for that claim is a paper in the [Journal of Chemical Physics](https://pubs.aip.org/aip/jcp/article/105/17/7641/293774/Phase-diagram-of-Yukawa-systems-near-the-one) by S.S. Hamaguchi et al., titled "Phase diagram of Yukawa systesm near the one-component-plasma limit revisited". Although Hamaguchi is on one-component plasmas, the methodology is all very similar, and the phase diagram (Fig. 2 in the paper) shows that bcc is the only solid until $\kappa > 1$ is encountered, and then the low 'temperature' (its complicated) phase becomes fcc, with bcc remaining at higher temperature/energy.

So, the expected crystal structure in white dwarfs is body-centered cubic.

> **Comments:**
> **username_0**: This is excellent, thank you! I will get on these as soon as I can! I found an open access copy of Hamaguchi et al. [here](https://www.semanticscholar.org/paper/Phase-diagram-of-Yukawa-systems-near-the-limit-Hamaguchi-Farouki/294d1bf9948dc0bca04eae14775ac02cbe883cd4). From your Hughto et al. uses link I can't even find the paper's title or the year - it's a blank paywall but Horowitz et al. is in [arXiv](https://arxiv.org/abs/1005.2441) and [researchgate](https://www.researchgate.net/publication/46423761_Crystallization_of_Carbon-Oxygen_Mixtures_in_White_Dwarf_Stars)  
> **username_0**: Some readers have no access to paywalled references, I do but only when I'm in the office (traveling today) so it's always best to take a moment and find accessible copies of important references and to *include the titles*. Thanks!  

---

