M04 Computing Vacancy Formation Energy - Revision history

Kwkang: /* Vacancy Formation Energy */

2007-11-28T00:31:18Z

Vacancy Formation Energy

← Older revision		Revision as of 00:31, 28 November 2007
Line 142:		Line 142:

	== Vacancy Formation Energy ==		== Vacancy Formation Energy ==
			We deﬁne the vacancy formation energy <math>E_v</math> as the energy needed to create
			a vacancy from the bulk perfect crystal structure. This can be obtained by
			comparing the energy of the relaxed vacancy-formed structure and the energy of
			a perfect crystal structure. However, it is not just a simple substraction between
			two, because we need to make sure we compare two systems of the same number
			of atoms. In reality, no atom is removed or destroyed when a vacancy forms.
			Rather, what we can think of is displacement of an inside atom to the free surface of a
			solid leaving a vacancy inside as shown in Fig. 2(a). Even though there
			is no free surface in our simulation with the periodic boundary condition. Let
			<math>N</math> be the total number of atoms in the original perfect crystal structure. And
			<math>E_1</math> is the energy of the perfect crystal and <math>E_2</math> is the energy of relaxed structure
			of <math>(N − 1)</math> atoms. Then the vacancy formation energy <math>E_v</math> can be expressed as

			N −1
			Ev ≡ E2 − · E1 .
			N

			''Q.2'' Can you tell the diﬀerent physical meanings between two expres
			sions <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.

			Once you know the vacancy formation energy <math>E_v</math>, you can use it to estimate
			the concentration of vacancy <maht>[n]</math> in the crystal strucutre at given temperature
			T and atmospheric pressure from Boltzmans distribution relation<ref> R. J. Stokes and D. F. Evans, ''Fundamentals of Interfacial Engineering'', Wiley-VCH 1997 pp.491-493 </ref>,

			[n] ∼ exp(−Ev /kB T )

			where kB = 8.621E-5 eV/K is Boltzman’s constant. For example, the vacancy
			formation energy of Mo is 2.550 (eV)<ref> With the tight binding model, it is reported to be 2.46 eV. See the referene, M. J. Mehl and D. A. Papaconstantopoulos, “Applications of a tight-binding total-energy method for transition and noble metals: Elastic constants, vacancies, and surfaces of monatomic metals”, Physical Review B '''54''' 4519-4530 1996
			</ref> from the atomistic calculation. Then the
			fraction of vacancy will be on the order of 1.514E-43 at 300 K.

			''Q.3'' Calculate the vacancy formation energy of Mo with the simulation
			box of different size. The vacancy formation energy is expected to be the same irrelavant to
			the box size. However, you will ﬁnd the energy dramatically changes if the box
			is smaller than the certain size. Explain the reason.

	== Notes ==		== Notes ==

Kwkang: /* Relaxation */

2007-11-28T00:16:16Z

Relaxation

← Older revision		Revision as of 00:16, 28 November 2007
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	== Relaxation ==		== Relaxation ==
			After creating a vacancy, it is not hard to imagine that the atoms around the vacancy
			will not stay as they were, for they have less bonds and form a locally unstable structure.
			We can expect the atoms near the vacancy will move toward the vacant site to some degree and lower the total energy of
			the system. In other words, the atomistic structure will be relaxed to its locay energy
			minimum state from the given initial position. MD++ uses a conjugate gradient
			relaxation (CGR) algorithm<ref> Numerical Recipes, http://www.nr.com/</ref> for this energy minimization process.
			for which the script ﬁle needs the section given below.

			#---------------------------------------------
			# Conjugate-Gradient relaxation
			conj_ftol = 1e-7 # tolerance on the residual gradient
			conj_fevalmax = 1000 # max. number of iterations
			conj_fixbox = 1 # fix the simulation box
			relax # CG relaxation command
			finalcnfile = relaxed.cn writecn
			eval # evaluate relaxed structure

			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
			simulated system keeps decreasing while the relaxation goes on.
			There are a coulple of variables related with CG relaxation. '''conj_ftol''' is the
			tolerance of the residual gradient and '''conj_fevalmax''' is the maximum number
			of iterations. The relaxtion will stop if the tolerance becomes smaller than
			the speciﬁed '''conj_ftol''' or if the number of iteration reaches its maximum. If
			'''conj_fixbox''' is equal to 1, the shape and size of simulation cell box is ﬁxed
			so that only the atoms can move during the relaxation. Sometimes you need
			to relax the simulation box together with atoms. In such a case you can
			add more degrees of freedom to the system by setting '''conj_fixbox = 0'''. For
			example, you can allow the box to expand or shrink hydrostatically by saying that

			conj_fixbox = 0
			conj_fixboxvec = [ 0 1 1
			1 0 1
			1 1 0]

			In this case, only the diagonal components of the box matrix <math>\mathbf{H}</math> can relax and
			the other components are ﬁxed as assigned to '''conj_fixboxvec'''.

	== Vacancy Formation Energy ==		== Vacancy Formation Energy ==

Kwkang: /* Making a Vacancy */

2007-11-28T00:03:03Z

Making a Vacancy

← Older revision		Revision as of 00:03, 28 November 2007
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	== Making a Vacancy ==		== Making a Vacancy ==
			In manual M02, we learned how to create a perfect crystal using MD++. However,
			most crystal structures are hardly perfect in natural environmet. It is
			common to have various kinds of defects such as point, line, planar and volume
			defects. Sometimes, we intend to introduce certain kinds of defects to study the
			property of interest.

			Vacancy is one of the frequently encountered point defects in solids and can be
			considered as a missing atom from the otherwise perfect crystal. In this manual, we will learn
			how to create a vacancy inside of the perfect crystal and how to calculate the
			vacancy formation energy of the bulk material. Let’s run the following example
			script by typing

			$ bin/fs gpp scripts/movacancy.script
			<pre>
			# --shell-script--
			setnolog
			setoverwrite
			dirname = runs/movacancy # specify run directory

			#--------------------------------------------
			# Read in potential file
			potfile = ~/Codes/MD++/potentials/mo_pot readpot

			#--------------------------------------------
			#Create Perfect Lattice Configuration
			crystalstructure = body-centered-cubic latticeconst = 3.1472 #(A)
			latticesize = [ 1 0 0 5
			0 1 0 5
			0 0 1 5]
			makecrystal finalcnfile = perf.cn writecn
			eval # evaluate the potential of perfect crystal
			#--------------------------------------------
			# Create Vacancy
			input = [ 1 # number of atoms to be fixed
			0] # index of an atom to be fixed
			fixatoms_by_ID # fix a set of atoms by their indices
			removefixedatoms # remove fixed atoms
			finalcnfile = movac.cn writecn
			eval # evaluate the vacancy-formed crystal
			#---------------------------------------------
			# Plot Configuration
			atomradius = 1.0 bondradius = 0.3 bondlength = 0
			atomcolor = blue highlightcolor = purple backgroundcolor = gray
			bondcolor = red fixatomcolor = yellow
			plotfreq = 10 win_width = 600 win_height = 600
			plot_atom_info = 3
			color00 = "orange" color01 = "purple" color02 = "green"
			color03 = "magenta" color04 = "cyan" color05 = "purple"
			color06 = "gray80" color07 = "white"
			plot_color_windows = [ 2 # number of color windows
			-10 -6.8 6 # color06 = gray80
			-6.7 -6.0 0 # color00 = orange
			]
			rotateangles = [ 0 0 0 1 ]
			openwin alloccolors rotate saverot plot
			sleep quit
			</pre>

			The script above shows how to create a vacancy from the perfect BCC Mo
			crystal. First, a 5×5×5 perfect cubic crystal of Mo is created with all the simulation box
			edges aligned along <100> directions. Then an atom of ID 0 is ﬁxed by

			input = [ 1 # number of atoms to be fixed
			0 ] # index of an atom to be fixed
			fixatoms by ID

			and will be removed by the command '''removefixedatoms''' in the script.
			Suppose, if you want to remove two atoms of indices 3 and 8, you can do so by

			input = [ 2 # number of atoms to be fixed
			3 8 ] # index of atoms to be fixed
			fixatoms_by_ID # fix a set of atoms by their index
			removefixedatoms # remove fixed atoms

			At this moment, you may wonder how we know which atom has which index.
			If you click an atom in the visualization window with the left button of the
			mouse, it gives the index number and energy of the clicked atom.<ref> This is the case when '''plot_atom_info = 3''' is speciﬁed in the script ﬁle. If '''plot_atom_info = 1''', the index number and the reduced coordinate of the clicked atom will be given. If '''plot_atom_info = 2''', the index number of the clicked atom and its real coordinate (in Å) will be displayed.</ref>
			In the visualization window, you will see the atoms near the vacant site have diﬀerent color
			than the others. In the script ﬁle, two color windows are set by the variable
			'''plot_color_windows''' according to the potential energy of the individual atom. The atoms
			whose energies are between -10 and -6.8 (eV) are shown in gray and the atoms
			whose energies lie between -6.7 and -6.0 (eV) are shown in orange. It shows
			that the atoms near the vacancy have higher energy than the rest. Click the
			atoms and you will see the potential energy of each atom. If you also compare
			the potential energy of the perfect structure and the vacancy-formed structure,
			you will know which structure has higher energy.

			''Q.1'' If a single atom, whose index is other than 0, is removed from the same perfect structure,
			do you still expect the ﬁnal state has the same amount of
			energy? Explain the reason.

	== Relaxation ==		== Relaxation ==

Kwkang: New page:

Manual 04 for MD++
Creating a Vacancy, Relaxing The Imperfect Structure
and Calculating Vacancy Formation Energy

M04 Computing Vacancy Formation Energy - Revision history

Kwkang: /* Vacancy Formation Energy */

Kwkang: /* Relaxation */

Kwkang: /* Making a Vacancy */

Kwkang: New page:

Manual 04 for MD++ Creating a Vacancy, Relaxing The Imperfect Structure and Calculating Vacancy Formation Energy

Manual 04 for MD++
Creating a Vacancy, Relaxing The Imperfect Structure
and Calculating Vacancy Formation Energy