M04 Energy Minimization of Vacancy - Revision history

Kwkang: /* Vacancy Formation Energy */

2011-09-13T15:14:15Z

Vacancy Formation Energy

← Older revision		Revision as of 15:14, 13 September 2011
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	<math>E_v</math> depend on the number of atoms in the simulation cell?		<math>E_v</math> depend on the number of atoms in the simulation cell?

	[[Image:Movacancy.jpg \| frame \| center \| Fig.1: Vacancy-formed BCC molybdenum crystal. <math>\mathbf{c}_1 = 4[100], \mathbf{c}_2 = 4[010]</math>, and <math>\mathbf{c}_1 = 4[001]</math> are the periodicity vectors. The red dotted circle designates		[[Image:Movacancy.jpg \| frame \| center \| Fig.1: Vacancy-formed BCC molybdenum crystal. <math>\mathbf{c}_1 = 4[100], \mathbf{c}_2 = 4[010]</math>, and <math>\mathbf{c}_3 = 4[001]</math> are the periodicity vectors. The red dotted circle designates
	the missing atom and the atoms around the vacant site are shown in different		the missing atom and the atoms around the vacant site are shown in different
	color due to their relatively high energy than the others. ]]		color due to their relatively high energy than the others. ]]


	[[Image:VacancyVSremoving.jpg \| frame \| center \| Fig.2: (a) In a real crystal, the atom that leaves the interior, thus creating		[[Image:VacancyVSremoving.jpg \| frame \| center \| Fig.2: (a) In a real crystal, the atom that leaves the interior, thus creating
	a vacancy, moves to the surface of the crystal. (b) If we simply take the energy		a vacancy, moves to the surface of the crystal. (b) If we simply take the energy

Kwkang: /* Equilibrium Vacancy Concentration */

2011-09-13T15:11:48Z

Equilibrium Vacancy Concentration

← Older revision		Revision as of 15:11, 13 September 2011
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	where <math>G_0</math> is the free energy of the perfect crystal (<math>n = 0</math>).		where <math>G_0</math> is the free energy of the perfect crystal (<math>n = 0</math>).
	Here we have ignored the interaction between the vacancies. This approximation is valid if <math>n \ll N</math>.		Here we have ignored the interaction between the vacancies. This approximation is valid if <math>n \ll N</math>.
	<math>S_c</math> is the conﬁgurational entropy due to the fact that there are <math>\Omega</math> ~~diﬀerent~~ ways		<math>S_c</math> is the conﬁgurational entropy due to the fact that there are <math>\Omega</math> different ways
	to arrange the n vacancies on N sites.		to arrange the n vacancies on N sites.

Kwkang: /* Vacancy Formation Energy */

2011-09-13T15:09:32Z

Vacancy Formation Energy

Kwkang: /* Relaxation */

2011-09-13T14:57:44Z

Relaxation

Kwkang: /* Creating a Vacancy in a Perfect Crystal */

2011-09-13T14:52:45Z

Creating a Vacancy in a Perfect Crystal

← Older revision		Revision as of 14:52, 13 September 2011
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	== Creating a Vacancy in a Perfect Crystal ==		== Creating a Vacancy in a Perfect Crystal ==
	A vacancy is a common ~~and important type of~~ point defect in solids. It is an		A vacancy is a common point defect in solids. It is an
	empty site (''i.e.'' a missing atom) in an otherwise prefect crystal structure. Here		empty site (''i.e.'' a missing atom) in an otherwise prefect crystal structure. Here
	we discuss how to introduce a vacancy in a perfect crystal using MD++ and how		we discuss how to introduce a vacancy in a perfect crystal using MD++ and how
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	$ bin/fs gpp scripts/movacancy.script		$ bin/fs gpp scripts/movacancy.script

	First, MD++ creates a 5×5×5 perfect cubic crystal of Mo with edges along		First, MD++ creates a 5×5×5 perfect cubic crystal of BCC Mo with edges along
	<100> directions. Then it ﬁxes atom 0 by		<100> directions. Then it ﬁxes atom 0 by

Kwkang: /* Relaxation */

2009-04-10T04:23:43Z

Relaxation

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	adjust its positions, ''e.g.'' to move toward the vacant site, to lower the poten-		adjust its positions, ''e.g.'' to move toward the vacant site, to lower the poten-
	tial energy. The relaxed structure can be obtained by an energy minimization		tial energy. The relaxed structure can be obtained by an energy minimization
	algorithm, such as the conjugate gradient (CG) method. This can be done by		algorithm, such as [[M04A Conjugate Gradient Method in MD++ \| the conjugate gradient (CG) method]]. This can be done by
	the following lines.		the following lines.

Caiwei: M04 Energy Minimization moved to M04 Energy Minimization of Vacancy: add vacancy

2008-10-25T06:05:27Z

M04 Energy Minimization moved to M04 Energy Minimization of Vacancy: add vacancy

← Older revision	Revision as of 06:05, 25 October 2008
(No difference)

Caiwei: /* Creating a Vacancy in a Perfect Crystal */

2008-03-25T06:09:50Z

Creating a Vacancy in a Perfect Crystal

← Older revision		Revision as of 06:09, 25 March 2008
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	empty site (''i.e.'' a missing atom) in an otherwise prefect crystal structure. Here		empty site (''i.e.'' a missing atom) in an otherwise prefect crystal structure. Here
	we discuss how to introduce a vacancy in a perfect crystal using MD++ and how		we discuss how to introduce a vacancy in a perfect crystal using MD++ and how
	to compute the vacancy formation energy. ~~Consider~~ ~~the~~ ~~following~~ MD++ ~~input~~		to compute the vacancy formation energy. For more information, see [[media:VFE.pdf‎ \| supplementary document: Vacancy formation energy by MD++ ]].
	ﬁle, '''movacancy.script''', that creates a vacancy in a perfect BCC Molybdenum		Consider the following MD++ input ﬁle, '''movacancy.script''', that creates a vacancy in a perfect BCC Molybdenum
	crystal.		crystal.

Caiwei at 06:06, 25 March 2008

2008-03-25T06:06:17Z

← Older revision		Revision as of 06:06, 25 March 2008
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	<H1 ALIGN="CENTER"><FONT SIZE="-1">Manual 04 for MD++ </FONT>		<H1 ALIGN="CENTER"><FONT SIZE="-1">Manual 04 for MD++ </FONT>
	<BR>		<BR>
			Vacancy Formation Energy by Energy Minimization</H1>
	Creating a Vacancy, Relaxing The Imperfect Structure and Calculating Vacancy Formation Energy</H1>
	<DIV>		<DIV>

Caiwei: Modified title layout to fit to small screen

2007-11-30T06:55:16Z

Modified title layout to fit to small screen

← Older revision		Revision as of 06:55, 30 November 2007
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	<H1 ALIGN="CENTER"><FONT SIZE="-1">Manual 04 for MD++ </FONT>		<H1 ALIGN="CENTER"><FONT SIZE="-1">Manual 04 for MD++ </FONT>
	<BR>		<BR>
	Creating a Vacancy, Relaxing The Imperfect Structure ~~<br>~~and Calculating Vacancy Formation Energy</H1>		Creating a Vacancy, Relaxing The Imperfect Structure and Calculating Vacancy Formation Energy</H1>
	<DIV>		<DIV>

← Older revision		Revision as of 15:09, 13 September 2011
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	Let <math>E_1</math> be the energy of the perfect crystal with <math>N</math> atoms, and <math>E_2</math> be the		Let <math>E_1</math> be the energy of the perfect crystal with <math>N</math> atoms, and <math>E_2</math> be the
	relaxed potential energy of the (N − 1)-atom system (containing the vacancy).		relaxed potential energy of the (N − 1)-atom system (containing the vacancy).
	The vacancy formation energy is not simply the ~~diﬀerence~~ between the two, ''i.e.''		The vacancy formation energy is not simply the difference between the two, ''i.e.''

	{\|border="0" align="center"		{\|border="0" align="center"
	\|<math> E_v \neq E_2 - E_1 </math>.		\|<math> E_v \neq E_2 - E_1 </math>,
	\|}		\|}

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	vacancy forms in a real crystal, the missing atom moves to the external surface of the		vacancy forms in a real crystal, the missing atom moves to the external surface of the
	crystal (or to grain boundaries). (See Fig.2(a).) Because we have a very small crystal		crystal (or to grain boundaries). (See Fig.2(a).) Because we have a very small crystal
	in our simulation, ~~which is~~ subjected to the periodic boundary condition, there		in our simulation, subjected to the periodic boundary condition, there
	is no external surface. Therefore, we need to account for this ~~eﬀect~~ in a ~~diﬀerent~~		is no external surface. Therefore, we need to account for this effect in a different
	way. In the perfect crystal, all atoms contributes equally to the total energy.		way. In the perfect crystal, all atoms contributes equally to the total energy.
	~~Therefore~~, we expect the energy contribution of (N − 1) atoms in the perfect		Hence, we expect the energy contribution of (N − 1) atoms in the perfect
	crystal structure to be <math>(N-1)/N \times E_1 </math>(even though we cannot ﬁt a perfect crystal with		crystal structure to be <math>(N-1)/N \times E_1 </math>(even though we cannot construct a perfect crystal with (N − 1) atoms in a periodic simulation cell). The vacancy formation energy
	(N − 1) atoms in to a periodic simulation cell). The vacancy formation energy
	<math>E_v</math> can be computed from		<math>E_v</math> can be computed from

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	{\|border="0" align="center"		{\|border="0" align="center"
	\|<math> E_v \equiv E_2 - (N-1) E_{coh} </math>.		\|<math> E_v \equiv E_2 - (N-1) E_{coh} </math>
			\|}

			or

			{\|border="0" align="center"
			\|<math> E_v \equiv (E_2 + E_{coh}) - E_{1} </math>.
	\|}		\|}

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	''Q.2'' Can you tell the ~~diﬀerent~~ physical meanings between two expressions		''Q.2'' Can you tell the different physical meanings between two expressions
	<math>E_2 - \frac{N-1}{N}E_1</math> and <math>E_2 - E_1</math>? You can use Fig. 2 for your explanation.		<math>E_2 - \frac{N-1}{N}E_1</math> and <math>E_2 - E_1</math>? You can use Fig. 2 for your explanation.

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	[[Image:Movacancy.jpg \| frame \| center \| Fig.1: Vacancy-formed BCC molybdenum crystal. <math>\mathbf{c}_1 = 4[100], \mathbf{c}_2 = 4[010]</math>, and <math>\mathbf{c}_1 = 4[001]</math> are the periodicity vectors. The red dotted circle designates		[[Image:Movacancy.jpg \| frame \| center \| Fig.1: Vacancy-formed BCC molybdenum crystal. <math>\mathbf{c}_1 = 4[100], \mathbf{c}_2 = 4[010]</math>, and <math>\mathbf{c}_1 = 4[001]</math> are the periodicity vectors. The red dotted circle designates
	the missing atom and the atoms around the vacant site are shown in ~~diﬀerent~~		the missing atom and the atoms around the vacant site are shown in different
	color due to their relatively high energy than the others. ]]		color due to their relatively high energy than the others. ]]
	[[Image:VacancyVSremoving.jpg \| frame \| center \| Fig.2: (a) In a real crystal, the atom that leaves the interior, thus creating		[[Image:VacancyVSremoving.jpg \| frame \| center \| Fig.2: (a) In a real crystal, the atom that leaves the interior, thus creating
	a vacancy, moves to the surface of the crystal. (b) If we simply take the energy		a vacancy, moves to the surface of the crystal. (b) If we simply take the energy
	~~diﬀerence~~ between <math>E_2</math> (N − 1 atoms) and <math>E_1</math> (N atoms), we are considering		difference between <math>E_2</math> (N − 1 atoms) and <math>E_1</math> (N atoms), we are considering
	the process in which the atom leaves the crystal and moves so far away that it		the process in which the atom leaves the crystal and moves so far away that it
	does not interact with any other atoms in the crystal. This will lead to a much		does not interact with any other atoms in the crystal. This will lead to a much

← Older revision		Revision as of 14:57, 13 September 2011
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	== Relaxation ==		== Relaxation ==
	When an atom is removed from a crystal, we expect the neighboring atoms to		When an atom is removed from a crystal, we expect the neighboring atoms to
	adjust its positions, ''e.g.'' to move toward the vacant site, to lower the ~~poten-~~		adjust its positions, ''e.g.'' to move toward the vacant site, in order to lower the potential energy. The relaxed structure can be obtained by an energy minimization
	tial energy. The relaxed structure can be obtained by an energy minimization
	algorithm, such as [[M04A Conjugate Gradient Method in MD++ \| the conjugate gradient (CG) method]]. This can be done by		algorithm, such as [[M04A Conjugate Gradient Method in MD++ \| the conjugate gradient (CG) method]]. This can be done by
	the following lines.		the following lines.
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	Let’s copy the above script and paste it to the '''movacancy.script''' right before		Let’s copy the above script and paste it to the '''movacancy.script''' right before
	the last command, '''sleep quit'''. You will ~~watch~~ the potential energy of the		the last command, '''sleep quit'''. You will see that the potential energy of the
	simulated system keeps decreasing while the relaxation goes on.		simulated system keeps decreasing while the relaxation goes on.
	The potential energy printed by the second '''eval''' should be lower than that		The potential energy printed by the second '''eval''' should be lower than that
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	There are several variables controlling the CG relaxation. '''conj_ftol''' is the		There are several variables controlling the CG relaxation. '''conj_ftol''' is the
	tolerance of the residual gradient and '''conj_fevalmax''' is the maximum number		tolerance of the residual gradient and '''conj_fevalmax''' is the maximum number
	of calls to the potential function (~~eﬀectively~~ limiting the number of iterations).		of calls to the potential function (effectively limiting the number of iterations).
	If '''conj_fixbox = 1''', the shape and volume of simulation cell box is ﬁxed and		If '''conj_fixbox = 1''', the shape and volume of simulation cell box is ﬁxed and
	only the scaled coordinates of the atoms can change during the relaxation.		only the scaled coordinates of the atoms can change during the relaxation.
	If '''conj fixbox = 0''', then all 9 components of the <math>\mathbf{H}</math> are allowed to ~~move~~.		If '''conj fixbox = 0''', then all 9 components of the <math>\mathbf{H}</math> are allowed to change.
	Sometimes, we want to ﬁx some components of the <math>\mathbf{H}</math> matrix while allowing		Sometimes, we want to ﬁx some components of the <math>\mathbf{H}</math> matrix while allowing
	other components to ~~move~~. This can be done by specifying the '''conj_fixboxvec'''		other components to vary. This can be done by specifying the '''conj_fixboxvec'''
	matrix. For example, if		matrix. For example, if