Hard ellipsoid model - Revision history

148.206.78.171 at 19:39, 6 July 2023

2023-07-06T19:39:28Z

Godriozo at 04:25, 21 June 2023

2023-06-21T04:25:15Z

← Older revision		Revision as of 06:25, 21 June 2023
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	<ref>[http://arxiv.org/abs/0908.1043 M. Radu, P. Pfleiderer, T. Schilling "Solid-solid phase transition in hard ellipsoids", arXiv:0908.1043v1 7 Aug (2009)]</ref>		<ref>[http://arxiv.org/abs/0908.1043 M. Radu, P. Pfleiderer, T. Schilling "Solid-solid phase transition in hard ellipsoids", arXiv:0908.1043v1 7 Aug (2009)]</ref>
	<ref>[http://dx.doi.org/10.1063/1.3251054 M. Radu, P. Pfleiderer, and T. Schilling "Solid-solid phase transition in hard ellipsoids", Journal of Chemical Physics '''131''' 164513 (2009)]</ref>		<ref>[http://dx.doi.org/10.1063/1.3251054 M. Radu, P. Pfleiderer, and T. Schilling "Solid-solid phase transition in hard ellipsoids", Journal of Chemical Physics '''131''' 164513 (2009)]</ref>
	that the [[SM2 structure]] is more stable than the [[Building up a face centered cubic lattice \| face centered cubic]] structure for aspect ratios <math>a/b \ge 2.0</math> and densities <math>\rho \gtrsim 1.17</math>.		that the [[SM2 structure]] is more stable than the [[Building up a face centered cubic lattice \| face centered cubic]] structure for aspect ratios <math>a/b \ge 2.0</math> and densities <math>\rho \gtrsim 1.17</math>. An updated phase diagram, encompassing the [[SM2 structure]] structure and the maximal packing fraction <ref>[http://dx.doi.org/10.1126/science.1093010 Aleksandar Donev, Ibrahim Cisse, David Sachs, Evan A. Variano, Frank H. Stillinger, Robert Connelly, Salvatore Torquato, and P. M. Chaikin "Improving the Density of Jammed Disordered Packings Using Ellipsoids", Science '''303''' pp. 990-993 (2004)]</ref>
			<ref>[http://dx.doi.org/10.1103/PhysRevLett.92.255506 Aleksandar Donev, Frank H. Stillinger, P. M. Chaikin and Salvatore Torquato "Unusually Dense Crystal Packings of Ellipsoids", Physical Review Letters '''92''' 255506 (2004)]</ref>, can be found in <ref>[https://doi.org/10.1063/1.36993314 G. Odriozola "Revisiting the phase diagram of hard ellipsoids", Journal of Chemical Physics '''136''' 134505 (2012)]</ref> <ref>[https://doi.org/10.1063/1.4789957 G. Odriozola "Further details on the phase diagram of hard ellipsoids of revolution", Journal of Chemical Physics '''138''' 064501 (2013)]</ref>.

	==Hard ellipse model==		==Hard ellipse model==

Carl McBride: Updated link

2013-11-14T12:49:19Z

Updated link

← Older revision		Revision as of 14:49, 14 November 2013
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	[[Mathematica]] notebook file for		[[Mathematica]] notebook file for
	[http://~~ender~~.~~quim~~.~~ucm.es~~/SR_B2_B3_ellipsoids.nb calculating the surface area and mean radius of curvature of an ellipsoid]		[http://www.sklogwiki.org/SR_B2_B3_ellipsoids.nb calculating the surface area and mean radius of curvature of an ellipsoid]

	==Maximum packing fraction==		==Maximum packing fraction==

Carl McBride: /* References */ Added a recent publication

2013-11-05T17:39:44Z

References: Added a recent publication

← Older revision		Revision as of 19:39, 5 November 2013
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	*[http://dx.doi.org/10.1080/02678299008047365 Michael P. Allen "Computer simulation of a biaxial liquid crystal", Liquid Crystals '''8''' pp. 499-511 (1990)]		*[http://dx.doi.org/10.1080/02678299008047365 Michael P. Allen "Computer simulation of a biaxial liquid crystal", Liquid Crystals '''8''' pp. 499-511 (1990)]
	*[http://dx.doi.org/10.1016/j.fluid.2007.03.026 Carl McBride and Enrique Lomba "Hard biaxial ellipsoids revisited: Numerical results", Fluid Phase Equilibria '''255''' pp. 37-45 (2007)]		*[http://dx.doi.org/10.1016/j.fluid.2007.03.026 Carl McBride and Enrique Lomba "Hard biaxial ellipsoids revisited: Numerical results", Fluid Phase Equilibria '''255''' pp. 37-45 (2007)]
			*[http://dx.doi.org/10.1063/1.4812361 Wen-Sheng Xu , Yan-Wei Li , Zhao-Yan Sun and Li-Jia An "Hard ellipses: Equation of state, structure, and self-diffusion", Journal of Chemical Physics '''139''' 024501 (2013)]

	[[Category: Models]]		[[Category: Models]]

Carl McBride: /* Phase diagram */ Added a new reference.

2009-11-02T12:08:17Z

Phase diagram: Added a new reference.

← Older revision		Revision as of 14:08, 2 November 2009
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	<ref>[http://dx.doi.org/10.1063/1.473665 Philip J. Camp and Michael P. Allen "Phase diagram of the hard biaxial ellipsoid fluid", Journal of Chemical Physics '''106''' pp. 6681- (1997)]</ref>. It has recently been shown		<ref>[http://dx.doi.org/10.1063/1.473665 Philip J. Camp and Michael P. Allen "Phase diagram of the hard biaxial ellipsoid fluid", Journal of Chemical Physics '''106''' pp. 6681- (1997)]</ref>. It has recently been shown
	<ref>[http://arxiv.org/abs/0908.1043 M. Radu, P. Pfleiderer, T. Schilling "Solid-solid phase transition in hard ellipsoids", arXiv:0908.1043v1 7 Aug (2009)]</ref>		<ref>[http://arxiv.org/abs/0908.1043 M. Radu, P. Pfleiderer, T. Schilling "Solid-solid phase transition in hard ellipsoids", arXiv:0908.1043v1 7 Aug (2009)]</ref>
			<ref>[http://dx.doi.org/10.1063/1.3251054 M. Radu, P. Pfleiderer, and T. Schilling "Solid-solid phase transition in hard ellipsoids", Journal of Chemical Physics '''131''' 164513 (2009)]</ref>
	that the [[SM2 structure]] is more stable than the [[Building up a face centered cubic lattice \| face centered cubic]] structure for aspect ratios <math>a/b \ge 2.0</math> and densities <math>\rho \gtrsim 1.17</math>.		that the [[SM2 structure]] is more stable than the [[Building up a face centered cubic lattice \| face centered cubic]] structure for aspect ratios <math>a/b \ge 2.0</math> and densities <math>\rho \gtrsim 1.17</math>.

Carl McBride: /* Phase diagram */ Added density.

2009-08-12T11:00:11Z

Phase diagram: Added density.

← Older revision		Revision as of 13:00, 12 August 2009
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	<ref>[http://dx.doi.org/10.1063/1.473665 Philip J. Camp and Michael P. Allen "Phase diagram of the hard biaxial ellipsoid fluid", Journal of Chemical Physics '''106''' pp. 6681- (1997)]</ref>. It has recently been shown		<ref>[http://dx.doi.org/10.1063/1.473665 Philip J. Camp and Michael P. Allen "Phase diagram of the hard biaxial ellipsoid fluid", Journal of Chemical Physics '''106''' pp. 6681- (1997)]</ref>. It has recently been shown
	<ref>[http://arxiv.org/abs/0908.1043 M. Radu, P. Pfleiderer, T. Schilling "Solid-solid phase transition in hard ellipsoids", arXiv:0908.1043v1 7 Aug (2009)]</ref>		<ref>[http://arxiv.org/abs/0908.1043 M. Radu, P. Pfleiderer, T. Schilling "Solid-solid phase transition in hard ellipsoids", arXiv:0908.1043v1 7 Aug (2009)]</ref>
	that the [[SM2 structure]] is more stable than the [[Building up a face centered cubic lattice \| face centered cubic]] structure for aspect ratios <math>a/b \ge 2.0</math>		that the [[SM2 structure]] is more stable than the [[Building up a face centered cubic lattice \| face centered cubic]] structure for aspect ratios <math>a/b \ge 2.0</math> and densities <math>\rho \gtrsim 1.17</math>.

	==Hard ellipse model==		==Hard ellipse model==
	:''Main article: [[Hard ellipse model]]'' (2-dimensional ellipsoids)		:''Main article: [[Hard ellipse model]]'' (2-dimensional ellipsoids)

Carl McBride: Added a new reference annd changed references to Cite format.

2009-08-12T10:56:13Z

Added a new reference annd changed references to Cite format.

← Older revision		Revision as of 12:56, 12 August 2009
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	axis.		axis.
	==Overlap algorithm==		==Overlap algorithm==
	The most widely used overlap algorithm is that of Perram and Wertheim:		The most widely used overlap algorithm is that of Perram and Wertheim
	*[http://dx.doi.org/10.1016/0021-9991(85)90171-8 John W. Perram and M. S. Wertheim "Statistical mechanics of hard ellipsoids. I. Overlap algorithm and the contact function", Journal of Computational Physics '''58''' pp. 409-416 (1985)]		<ref>[http://dx.doi.org/10.1016/0021-9991(85)90171-8 John W. Perram and M. S. Wertheim "Statistical mechanics of hard ellipsoids. I. Overlap algorithm and the contact function", Journal of Computational Physics '''58''' pp. 409-416 (1985)]</ref>.

	==Geometric properties==		==Geometric properties==
	The mean radius of curvature is given by (~~Refs~~. 5 and 6)		The mean radius of curvature is given by
			<ref>[http://dx.doi.org/10.1063/1.472110 G. S. Singh and B. Kumar "Geometry of hard ellipsoidal fluids and their virial coefficients", Journal of Chemical Physics '''105''' pp. 2429-2435 (1996)]</ref>
			<ref>[http://dx.doi.org/10.1006/aphy.2001.6166 G. S. Singh and B. Kumar "Molecular Fluids and Liquid Crystals in Convex-Body Coordinate Systems", Annals of Physics '''294''' pp. 24-47 (2001)]</ref>

	:<math>R= \frac{a}{2} \left[ \sqrt{\frac{1+\epsilon_b}{1+\epsilon_c}} + \sqrt \epsilon_c \left\{ \frac{1}{\epsilon_c} F(\varphi , k_1) + E(\varphi,k_1) \right\}\right],		:<math>R= \frac{a}{2} \left[ \sqrt{\frac{1+\epsilon_b}{1+\epsilon_c}} + \sqrt \epsilon_c \left\{ \frac{1}{\epsilon_c} F(\varphi , k_1) + E(\varphi,k_1) \right\}\right],
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	==Maximum packing fraction==		==Maximum packing fraction==
	Using [[event-driven molecular dynamics]], it has been found that the maximally random jammed (MRJ) [[packing fraction]] for hard ellipsoids is <math>\phi=0.7707</math> for		Using [[event-driven molecular dynamics]], it has been found that the maximally random jammed (MRJ) [[packing fraction]] for hard ellipsoids is <math>\phi=0.7707</math> for
	models whose maximal aspect ratio is greater than <math>\sqrt{3}</math>.		models whose maximal aspect ratio is greater than <math>\sqrt{3}</math>
	#[http://dx.doi.org/10.1126/science.1093010 Aleksandar Donev, Ibrahim Cisse, David Sachs, Evan A. Variano, Frank H. Stillinger, Robert Connelly, Salvatore Torquato, and P. M. Chaikin "Improving the Density of Jammed Disordered Packings Using Ellipsoids", Science '''303''' pp. 990-993 (2004)]		<ref>[http://dx.doi.org/10.1126/science.1093010 Aleksandar Donev, Ibrahim Cisse, David Sachs, Evan A. Variano, Frank H. Stillinger, Robert Connelly, Salvatore Torquato, and P. M. Chaikin "Improving the Density of Jammed Disordered Packings Using Ellipsoids", Science '''303''' pp. 990-993 (2004)]</ref>
	#[http://dx.doi.org/10.1103/PhysRevLett.92.255506 Aleksandar Donev, Frank H. Stillinger, P. M. Chaikin and Salvatore Torquato "Unusually Dense Crystal Packings of Ellipsoids", Physical Review Letters '''92''' 255506 (2004)]		<ref>[http://dx.doi.org/10.1103/PhysRevLett.92.255506 Aleksandar Donev, Frank H. Stillinger, P. M. Chaikin and Salvatore Torquato "Unusually Dense Crystal Packings of Ellipsoids", Physical Review Letters '''92''' 255506 (2004)]</ref>
	==Equation of state==		==Equation of state==
	:''Main article: [[Hard ellipsoid equation of state]]''		:''Main article: [[Hard ellipsoid equation of state]]''
	==Virial coefficients==		==Virial coefficients==
	:''Main article: [[Hard ellipsoids: virial coefficients]]		:''Main article: [[Hard ellipsoids: virial coefficients]]
	==~~Related models~~==		==Phase diagram==
	*[[Hard ellipse model]] (2-dimensional ellipsoids)		One of the first [[phase diagrams]] of the hard ellipsoid model was that of Frenkel and Mulder (Figure 6 in
			<ref>[http://dx.doi.org/10.1080/00268978500101971 D. Frenkel and B. M. Mulder "The hard ellipsoid-of-revolution fluid I. Monte Carlo simulations", Molecular Physics '''55''' pp. 1171-1192 (1985)]</ref>).
			Camp and Allen later studied biaxial ellipsoids
			<ref>[http://dx.doi.org/10.1063/1.473665 Philip J. Camp and Michael P. Allen "Phase diagram of the hard biaxial ellipsoid fluid", Journal of Chemical Physics '''106''' pp. 6681- (1997)]</ref>. It has recently been shown
			<ref>[http://arxiv.org/abs/0908.1043 M. Radu, P. Pfleiderer, T. Schilling "Solid-solid phase transition in hard ellipsoids", arXiv:0908.1043v1 7 Aug (2009)]</ref>
			that the [[SM2 structure]] is more stable than the [[Building up a face centered cubic lattice \| face centered cubic]] structure for aspect ratios <math>a/b \ge 2.0</math>
			==Hard ellipse model==
			:''Main article: [[Hard ellipse model]]'' (2-dimensional ellipsoids)
	==References==		==References==
	~~#[http:~~/~~/dx.doi.org/10.1080/00268978500101971 D. Frenkel and B. M. Mulder "The hard ellipsoid-of-revolution fluid I. Monte Carlo simulations", Molecular Physics~~ '''55''' ~~pp. 1171-1192 (1985)]~~		<references/>
	#[http://dx.doi.org/10.1080/02678299008047365 Michael P. Allen "Computer simulation of a biaxial liquid crystal", Liquid Crystals '''8''' pp. 499-511 (1990)]		'''Related reading'''
	~~#[http://dx.doi.org/10.1063/1.473665 Philip J. Camp and Michael P. Allen "Phase diagram of the hard biaxial ellipsoid fluid", Journal of Chemical Physics '''106''' pp. 6681- (1997)]~~
	#[http://dx.doi.org/10.1016/j.fluid.2007.03.026 Carl McBride and Enrique Lomba "Hard biaxial ellipsoids revisited: Numerical results", Fluid Phase Equilibria '''255''' pp. 37-45 (2007)]		*[http://dx.doi.org/10.1080/02678299008047365 Michael P. Allen "Computer simulation of a biaxial liquid crystal", Liquid Crystals '''8''' pp. 499-511 (1990)]
	~~#[http://dx.doi.org/10.1063/1.472110 G. S. Singh and B. Kumar "Geometry of hard ellipsoidal fluids and their virial coefficients", Journal of Chemical Physics '''105''' pp. 2429-2435 (1996)]~~		*[http://dx.doi.org/10.1016/j.fluid.2007.03.026 Carl McBride and Enrique Lomba "Hard biaxial ellipsoids revisited: Numerical results", Fluid Phase Equilibria '''255''' pp. 37-45 (2007)]
	~~#[http://dx.doi.org/10.1006/aphy.2001.6166 G. S. Singh and B. Kumar "Molecular Fluids and Liquid Crystals in Convex-Body Coordinate Systems", Annals of Physics '''294''' pp. 24-47 (2001~~)]
	[[Category: Models]]		[[Category: Models]]

Noe: /* Overlap algorithm */ DOI correction

2009-02-23T16:10:11Z

Overlap algorithm: DOI correction

← Older revision		Revision as of 18:10, 23 February 2009
Line 11:		Line 11:
	==Overlap algorithm==		==Overlap algorithm==
	The most widely used overlap algorithm is that of Perram and Wertheim:		The most widely used overlap algorithm is that of Perram and Wertheim:
	*[http://dx.doi.org/:10.1016/0021-9991(85)90171-8 John W. Perram and M. S. Wertheim "Statistical mechanics of hard ellipsoids. I. Overlap algorithm and the contact function", Journal of Computational Physics '''58''' pp. 409-416 (1985)]		*[http://dx.doi.org/10.1016/0021-9991(85)90171-8 John W. Perram and M. S. Wertheim "Statistical mechanics of hard ellipsoids. I. Overlap algorithm and the contact function", Journal of Computational Physics '''58''' pp. 409-416 (1985)]

	==Geometric properties==		==Geometric properties==
	The mean radius of curvature is given by (Refs. 5 and 6)		The mean radius of curvature is given by (Refs. 5 and 6)

Carl McBride: /* Geometric properties */ Updated an external link.

2008-11-28T11:23:43Z

Geometric properties: Updated an external link.

← Older revision		Revision as of 13:23, 28 November 2008
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	[[Mathematica]] notebook file for		[[Mathematica]] notebook file for
	[http://~~www~~.~~qft~~.~~iqfr.csic~~.es~~/personal/carl~~/SR_B2_B3_ellipsoids.nb calculating the surface area and mean radius of curvature of an ellipsoid]		[http://ender.quim.ucm.es/SR_B2_B3_ellipsoids.nb calculating the surface area and mean radius of curvature of an ellipsoid]

	==Maximum packing fraction==		==Maximum packing fraction==
	Using [[event-driven molecular dynamics]], it has been found that the maximally random jammed (MRJ) [[packing fraction]] for hard ellipsoids is <math>\phi=0.7707</math> for		Using [[event-driven molecular dynamics]], it has been found that the maximally random jammed (MRJ) [[packing fraction]] for hard ellipsoids is <math>\phi=0.7707</math> for

Carl McBride: Added a photo of an oblate ellipsoid.

2008-11-28T11:11:31Z

Added a photo of an oblate ellipsoid.

← Older revision		Revision as of 13:11, 28 November 2008
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	[[Image:ellipsoid_red.png\|thumb\|right\|A prolate ellipsoid.]]		[[Image:ellipsoid_red.png\|thumb\|right\|A uniaxial prolate ellipsoid, a>b, b=c.]]
			[[Image:oblate_1.png\|thumb\|right\|A uniaxial oblate ellipsoid, a>c, a=b.]]
	'''Hard ellipsoids''' represent a natural choice for an anisotropic model. Ellipsoids can be produced from an affine transformation of the [[hard sphere model]]. However, in difference to the hard sphere model, which has fluid and solid phases, the hard ellipsoid model is also able to produce a [[nematic phase]].		'''Hard ellipsoids''' represent a natural choice for an anisotropic model. Ellipsoids can be produced from an affine transformation of the [[hard sphere model]]. However, in difference to the hard sphere model, which has fluid and solid phases, the hard ellipsoid model is also able to produce a [[nematic phase]].
	== Interaction Potential ==		== Interaction Potential ==

← Older revision		Revision as of 21:39, 6 July 2023
Line 54:		Line 54:
	Using [[event-driven molecular dynamics]], it has been found that the maximally random jammed (MRJ) [[packing fraction]] for hard ellipsoids is <math>\phi=0.7707</math> for		Using [[event-driven molecular dynamics]], it has been found that the maximally random jammed (MRJ) [[packing fraction]] for hard ellipsoids is <math>\phi=0.7707</math> for
	models whose maximal aspect ratio is greater than <math>\sqrt{3}</math>		models whose maximal aspect ratio is greater than <math>\sqrt{3}</math>
	<ref>[http://dx.doi.org/10.1126/science.1093010 Aleksandar Donev, Ibrahim Cisse, David Sachs, Evan A. Variano, Frank H. Stillinger, Robert Connelly, Salvatore Torquato, and P. M. Chaikin "Improving the Density of Jammed Disordered Packings Using Ellipsoids", Science '''303''' pp. 990-993 (2004)]</ref>		<ref name="Donev1"> [http://dx.doi.org/10.1126/science.1093010 Aleksandar Donev, Ibrahim Cisse, David Sachs, Evan A. Variano, Frank H. Stillinger, Robert Connelly, Salvatore Torquato, and P. M. Chaikin "Improving the Density of Jammed Disordered Packings Using Ellipsoids", Science '''303''' pp. 990-993 (2004)]</ref>
	<ref>[http://dx.doi.org/10.1103/PhysRevLett.92.255506 Aleksandar Donev, Frank H. Stillinger, P. M. Chaikin and Salvatore Torquato "Unusually Dense Crystal Packings of Ellipsoids", Physical Review Letters '''92''' 255506 (2004)]</ref>		<ref name="Donev2">[http://dx.doi.org/10.1103/PhysRevLett.92.255506 Aleksandar Donev, Frank H. Stillinger, P. M. Chaikin and Salvatore Torquato "Unusually Dense Crystal Packings of Ellipsoids", Physical Review Letters '''92''' 255506 (2004)]</ref>
	==Equation of state==		==Equation of state==
	:''Main article: [[Hard ellipsoid equation of state]]''		:''Main article: [[Hard ellipsoid equation of state]]''
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	Camp and Allen later studied biaxial ellipsoids		Camp and Allen later studied biaxial ellipsoids
	<ref>[http://dx.doi.org/10.1063/1.473665 Philip J. Camp and Michael P. Allen "Phase diagram of the hard biaxial ellipsoid fluid", Journal of Chemical Physics '''106''' pp. 6681- (1997)]</ref>. It has recently been shown		<ref>[http://dx.doi.org/10.1063/1.473665 Philip J. Camp and Michael P. Allen "Phase diagram of the hard biaxial ellipsoid fluid", Journal of Chemical Physics '''106''' pp. 6681- (1997)]</ref>. It has recently been shown
	<ref>[http://arxiv.org/abs/0908.1043 M. Radu, P. Pfleiderer, T. Schilling "Solid-solid phase transition in hard ellipsoids", arXiv:0908.1043v1 7 Aug (2009)]</ref>		<ref name="Radu1"> [http://arxiv.org/abs/0908.1043 M. Radu, P. Pfleiderer, T. Schilling "Solid-solid phase transition in hard ellipsoids", arXiv:0908.1043v1 7 Aug (2009)]</ref>
	<ref>[http://dx.doi.org/10.1063/1.3251054 M. Radu, P. Pfleiderer, and T. Schilling "Solid-solid phase transition in hard ellipsoids", Journal of Chemical Physics '''131''' 164513 (2009)]</ref>		<ref name="Radu2">[http://dx.doi.org/10.1063/1.3251054 M. Radu, P. Pfleiderer, and T. Schilling "Solid-solid phase transition in hard ellipsoids", Journal of Chemical Physics '''131''' 164513 (2009)]</ref>
	that the [[SM2 structure]] is more stable than the [[Building up a face centered cubic lattice \| face centered cubic]] structure for aspect ratios <math>a/b \ge 2.0</math> and densities <math>\rho \gtrsim 1.17</math>. An updated phase diagram, encompassing the [[SM2 structure]] structure and the maximal packing fraction <ref~~>[http://dx.doi.org/10.1126/science.1093010 Aleksandar Donev, Ibrahim Cisse, David Sachs, Evan A. Variano, Frank H. Stillinger, Robert Connelly, Salvatore Torquato, and P. M. Chaikin~~ "~~Improving the Density of Jammed Disordered Packings Using Ellipsoids~~"~~, Science '''303''' pp. 990-993 (2004)]<~~/~~ref~~>		that the [[SM2 structure]] is more stable than the [[Building up a face centered cubic lattice \| face centered cubic]] structure for aspect ratios <math>a/b \ge 2.0</math> and densities <math>\rho \gtrsim 1.17</math>. An updated phase diagram, encompassing the [[SM2 structure]] structure <ref name="Radu1"/> <ref name="Radu2"/> and the maximal packing fraction <ref name="Donev1"/> <ref name="Donev2"/>, can be found in <ref>[https://doi.org/10.1063/1.36993314 G. Odriozola "Revisiting the phase diagram of hard ellipsoids", Journal of Chemical Physics '''136''' 134505 (2012)]</ref> <ref>[https://doi.org/10.1063/1.4789957 G. Odriozola "Further details on the phase diagram of hard ellipsoids of revolution", Journal of Chemical Physics '''138''' 064501 (2013)]</ref>.
	<ref~~>[http://dx.doi.org/10.1103/PhysRevLett.92.255506 Aleksandar Donev, Frank H. Stillinger, P. M. Chaikin and Salvatore Torquato~~ "~~Unusually Dense Crystal Packings of Ellipsoids~~"~~, Physical Review Letters '''92''' 255506 (2004)]<~~/~~ref~~>, can be found in <ref>[https://doi.org/10.1063/1.36993314 G. Odriozola "Revisiting the phase diagram of hard ellipsoids", Journal of Chemical Physics '''136''' 134505 (2012)]</ref> <ref>[https://doi.org/10.1063/1.4789957 G. Odriozola "Further details on the phase diagram of hard ellipsoids of revolution", Journal of Chemical Physics '''138''' 064501 (2013)]</ref>.

	==Hard ellipse model==		==Hard ellipse model==