Category Archives: ScAl

Lattice constant predictions for ScAl assuming both the CsCl and NaCl structures

By Nathan McKee

Introduction

This post examines the structure of ScAl, whether the material prefers the structure of CsCl or that of NaCl (shown in figures 1 and 2, respectively), and predicts the lattice constant in each case. For each structure, the lattice constant was varied and the ground state energy of the structure was calculated, repeating to find the lattice constant that minimizes the energy for each structure. DFT calculations were carried out with the plane-wave based code CASTEP. The GGA PBE functional was used¹, as well as OTFG ultrasoft pseudopotentials². These pseudopotentials include the 3s2 3p1 valence electrons and a cutoff radius of 1.5 Å for Al, and the 3s2 3p6 3d1 4s2 valence electrons and a cutoff radius of 1.6 Å for Sc. The convergence tolerance was set at 2.0*10^-6 eV per atom.

Figure 1: The CsCl structure, which is a simple cubic lattice with two atoms per unit cell.

Figure 2: The NaCl sructure, which is an FCC lattice with two atoms per primitive unit cell.

Cutoff Energy and k Points

To ensure that the calculations converge properly, an analysis of the selection of k points and the cutoff energy was performed. The k points were chosen to be an NxNxN grid of evenly spaced points in reciprocal space, with the same number of points in each direction being appropriate for a cubic cell. Figures 3 and 4 show how an energy calculation converges as N increases and as the cutoff energy is raised. In both cases, an arbitrary constant was added to the calculated energies so that the values would be close to zero and the convergence could be seen more easily. The calculations for k point convergence were performed for the NaCl structure with a cutoff energy of 410.9 eV. The calculations for the cutoff energy convergence were performed for the same structure with a 6x6x6 k-point grid.

Figure 3: The calculated energy converges as the size of the NxNxN grid of k points increases. The energies have been shifted so that the differences are easy to see.

Figure 4: The calculated energy of the structure converges as the cutoff energy increases. The energies have been shifted so that the differences are easy to see.

Using these graphs as a guide, a 6x6x6 k point grid was used in conjunction with a cutoff energy of 410.9 eV for further calculations. It should also be noted that an origin shift was implemented in the k point grid in order to increase the total number of k points in the Brillouin Zone (BZ) without changing the number of k points in the Irreducible Brillouin Zone (IBZ). The shift used was 0.01 Å^-1in the x direction, 0.005 Å^-1in the y direction, and 0.003 Å^-1in the z direction. These values were chosen so that the k points would not lie on any symmetry axes, while maintaining a small shift compared to the spacing between k points.

Results

Figures 5 and 6 show the energy minimization with respect to the lattice constant for each of the structures being considered. The vertical axes have been shifted so that the minimum energy value obtained lies on the horizontal axis. For the CsCl structure, a minimum energy of -1385.213651797 eV was obtained, with a lattice constant of 3.380 Å. For the NaCl structure, a minimum energy of -1384.047417 eV was obtained, with a lattice constant of 5.649 Å.

Figure 5: Energy vs lattice constant for ScAl in the structure of CsCl. The value of Emin is -1385.213651797 eV.

Figure 6: Energy vs lattice constant for ScAl in the structure of NaCl. The value of Emin is -1384.047417 eV.

Conclusion

These results indicate that ScAl prefers the CsCl structure over the NaCl structure, because the minimum calculated energy of the CsCl structure was lower. In addition, these results predict a lattice constant of 3.380 Å for ScAl in the CsCl structure.

References

Perdew, J. P., Burke, K. & Ernzerhof, M. Generalized Gradient Approximation Made Simple. Phys. Rev. Lett. 77, 3865–3868 (1996).
Gonze, X. & Finocchi, F. Pseudopotentials Plane Waves–Projector Augmented Waves: A Primer. Phys. Scr. 2004, 40 (2004).

Lattice constant for ScAl in CsCl and NaCl structures

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Lev Krainov

Introduction:

The goal of this post is to study crystal structure of ScAl. Two possible configurations were investigated: CsCl and NaCl crystal structures. For each structure lattice constant was found by seeking a minimum of a ground state energy. Data shows that ScAl prefers CsCl structure with lattice constant $a=3.378\AA$ over the NaCl structure.

Computational details:

Ground state energy computations were performed using DFT plane-wave pseudopotential method implemented in CASTEP[2]. With CASTEP, we use the GGA-PBE as an exchange-correlation functional [3]. We also employ On-the-fly generated (OTFG) ultrasoft pseudopotential was used to describe the interactions of ionic core and valance electrons with a core radius of 2.4Bohr(1.27 Å) [4]. Pseudo atomic calculation is performed for 3s2 3p6 4s2 3d1 orbitals of Sc and 3s2 3p1 orbitals of Al. SCF convergence tolerance was set to 1.0E-6eV/atom. The Koelling-Harmon relativistic treatment was used for Sc orbitals.

K-points convergence

First we investigate convergence for both geometries. Since we need to pick some lattice constant we performed CASTEP geometry optimization using BFGS hill-climbing algorithm[1] with 15 k-points and $E_{cut}=600eV$ . These values were determined to give unit cell size converged up to $0.001\AA$ tolerance. The resulting lattice parameters are $a=3.378\AA$ for CsCl and $a=5.656\AA$ for NaCl. Using these we investigate how the ground state energy converges with the number of k-points with fixed cutoff energy $E_{cut}=500eV$ .

Figure 1 shows that for both geometries at 17 k-points energy is convergent up to 0.001 eV. Energies are shifted by $E_f=-1385.315eV$ for CsCl structure and by $E_f = -1384.149eV$ for NaCl structure, which was set to the most accurate ground state energy obtained for each. Due to the symmetry of NaCl crystal for this structure odd number of k-points leads to half as many total points in the full 3D Brillouin zone compared to even number of k-points in each dimension.

Figure 1. Convergence of energy with respect to the number of k-points for CsCl(blue) and NaCl(orange) geometries.

Cutoff energy convergence

Next we achieve same level of convergence with respect to cutoff energy keeping number of k-points fixed at 17 for CsCl and at 15 for NaCl geometries. Figure 2 shows that $E_{cut}=600eV$ is enough to obtain energy up to 0.001eV.

Figure 2. Convergence of energy with respect to the cutoff energy for CsCl(blue) and NaCl(orange) geometries in logarithmic scale. The high value of the last point is explained by the error present in [latex]E_f[/latex].

Geometry optimization

Now we use found values of $E_{cut}$ and k-points to find ground state energy as a function of unit cell size and look for a minimum approximating it by a parabola. Also we run BFGS geometry optimization with the same values of $E_{cut}$ and k-points, shown as orange dot on Figure 3. For both geometries energy of a minimum of a parabola and energy given by BFGS agree up to 0.0001eV. But since for NaCl minimum is about two times wider this energy error introduces larger error for lattice constant of $\delta a\approx0.0035\AA$ while for ScAl the error is less than $0.001\AA$ .

Figure 3. Lattice constant optimization for CsCl(top) and NaCl(bottom) structures. Orange dot for both shows the result of CASTEP geometry optimization with the same number of k-points and cutoff energy.

Results

The results are shown in the table below. Error estimation for lattice constant was performed assuming 0.001eV error in energy and parabolic approximation. The resulting error is on the order of 0.01Å and probably could be improved by including more points in geometry optimization. Energies of ground states show that ScAl prefers CsCl structure with $a\approx3.38\AA$ which is in a good agreement with experimental value of 3.388Å[6].

Crystal structure	Ground state energy E0, eV	lattice parameter a, Å	Cutoff energy E_cut, eV	number of k points
CsCl	-1385.319(1)	3.377(13)	600	17
NaCl	-1384.168(1)	5.656(55)	600	15

[1] R. Fletcher; A new approach to variable metric algorithms, The Computer Journal, Volume 13, Issue 3, 1 January 1970, Pages 317–322

[2] S. J. Clark, M. D. Segall, C. J. Pickard, P. J. Hasnip, M. J. Probert, K. Refson, M. C. Payne, “First principles methods using CASTEP”, Zeitschrift fuer Kristallographie 220(5-6) pp. 567-570 (2005)

[3] Perdew, J. P; Burke, K; Ernzerhof, M. Phys. Rev. Lett. 1996, 77, 3865-3868

[4] CASTEP GUIDE, BIOVIA, UK, 2014. URL : http://www.tcm.phy.cam.ac.uk/castep/documentation/WebHelp/content/pdfs/castep.htm.

[5] D D Koelling and B N Harmon 1977 J. Phys. C: Solid State Phys. 10 3107

[6]Schuster J.C., and Bauer J., The ternary systems Sc-Al-N and Y-Al-N, J. Less-Common Met., Vol. 109, 1985, p 345-350

Comparing ScAl in CsCl and NaCl Structures and Determining the Optimal Lattice Parameter of the Preferred Structure

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The goal of this post is to identify if ScAl, which has AB stoichiometry, exists in the CsCl structure (figure 1) or NaCl Structure (figure 2 and figure 3). To determine this primitive cells of ScAl were produced for both types of structures. The plan is to plot the cohesive energy of the structure as a function of the volume per ScAl dimer. From these plots the optimal lattice parameters for each structure can be determined. Then by comparing the cohesive energies of the two structures with optimal lattice parameters we can determine which structure is preferred by ScAl. All energy calculations were carried out using plane-wave based DFT. The GGA based PBE functional was used to treat the exchange-correlation effects. The ion and core were treated using ultrasoft pseudopotentials generated on the fly (OTFG ultrasoft) with Koelling-Harmon relativistic treatments. Pseudo atomic calculations for Sc treated the 3s2 3p6 3d1 4s2 electrons as valence electrons and the electrons in lower levels were treated as part of the frozen core. Pseudo atomic calculations for Al treated the 3s2 3p1 electrons as the valence electrons.

Figure 1 : Primitive unit cell of ScAl in the CsCl structure. This a simple cubic structure with a two atom basis.

Figure 2 : (a) Conventional unit cell of ScAl in the NaCl structure. This a face centered cubic structure with a two atom basis. (b) Primitive unit cell of ScAl in the NaCl structure.

However before we begin calculating the data points to populate the plots we described in the previous paragraph, we need to select appropriate cutoff energy and $\vec{k}$ point mesh for our calculations. The constraints on our choice of cutoff energy and $\vec{k}$ point mesh are (1) computational cost, (2) convergence of results to a desired tolerance and (3) ensuring that we have approximately the same $\vec{k}$ point density for each structure.

To determine the cutoff energy we use for our calculations we first determine the approximate lattice parameter of CsCl structure primitive unit cell. For this we use a cutoff energy of 460 eV and $\vec{k}$ point mesh of $8\times8\times8$ . The cutoff energy and $\vec{k}$ point mesh chosen here are the default settings for the “ultra-fine” quality energy calculation using the CASTEP tool implemented in Material Studio. The cohesive energy vs volume per ScAl dimer plot (figure 4) indicates that cohesive energy is minimized when the lattice parameter is $\sim 3.5\;\mathring{A}$.

Figure 3 : Plot of Energy per ScAl dimer vs lattice parameter for ScAl in the CsCl structure, used to determine the approximate value for the optimal lattice parameter.

Next we investigated the convergence of the total energy of a primitive unit cell of ScAl in the CsCl structure with respect to the cutoff energy used for the calculation. The results were plotted as shown in figure 5. From this plot note that we get a convergence of $\sim\;1\;m\,eV$ for a cutoff energy of $500\;m\,eV$.

Figure 4 : Calculated total energy of a unit cell of ScAl in the CsCl structure (lattice constant = 3.5 $\mathring{A}$) vs the cutoff energy used for the calculation.

Next to obtain the most suitable $\vec{k}$ point mesh, we plot the energy per ScAl dimer for ae a fixed cutoff energy of $500\;eV$ while varying the number of $\vec{k}$ points used to sample the first brillouin zone. Since all the reciprocal lattice vectors ( and real space lattice vectors) have the same length, we can specify the $\vec{k}$ mesh by specifying the number of $k$-points used along each reciprocal lattice vector. Figure 6 shows this plot; and we can see that a $8\times8\times8$ $\vec{k}$ point is sufficient for the energy per ScAl dimer to have a convergence of $\sim\;10\;m\,eV$. The resulting spacing between sampled $k$ points is $0.0357\;\mathring{A}^{-1}$. To ensure our subsequent calculations have the same degree of convergence, we will impose that the separation between two adjacent $k$ points that are sampled along a reciprocal lattice vector is at most $0.0357\;\mathring{A}^{-1}$.

Figure 5 : Plot of energy per ScAl dimer vs volume per ScAl dimer.

Now we are ready to calculate the energy per ScAl dimer and the corresponding volume per dimer, for both structures and various lattice parameters. Figure 6 shows the plot of energy per ScAl dimer vs volume per dimer, for ScAl in the CsCl and NaCl structure. From the plots in figure 6 it is clear that ScAl prefers CsCl structure over the NaCl structure. From the best fit line we obtain the optimal lattice parameter in the CsCl structure to be $3.38\;\mathring{A}$. If ScAl were to be found in the NaCl structure the optimal lattice parameter would be $4.00\;\mathring{A}$.

Experimentally ScAl has been verified to exist in CsCl structure with a lattice parameter of $3.450\;\mathring{A}$ [1]. Our results verify this and estimate the lattice parameter within $\sim\;2%$ of the experimentally determined lattice constant.

[1] O. Schob and E. Parthe. Ab Compounds with Sc Y and Rare Earth Metals. I. Scandium and
Yttrium Compounds with Crb and Cscl Structure. Acta Crystallographica, 19:214-&, 1965.

Determination of the Lattice Parameter of ScAl in the CsCl Structure

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This Project aims to predict the lattice constant of ScAl with CASTEP calculation.

A. Project Description

In this study, we focus on predicting the lattice constant of ScAl based on the CsCl structure, and figuring out a converged energy cutoff and k-points in the CASTEP energy calculation. Atomic electron configuration for Al is 3s2 3p1, for Sc is 3s2 3p6 3d1 4s2. The energy calculation in CASTEP can provide a reasonable crystal structure for ScAl. The energy calculation in this study is based on the exchange correlation Perdew-Burke-Ernzerhof (PBE) density functionale, which is from the class of Generalized gradient approximation (GGA) functional. The relationships between energy and lattice parameters, energy cutoff, and k-points are discussed below.

B. Crystal Model

ScAl has CsCl structure, where Scandium (Sc) locates at the corner and Aluminum (Al) in the center of the unit cell. This structure belongs to the cubic system, the lattice parameter a=b=c, α=β=γ=90 degrees.

The cell vectors are along x, y, direction of the unit cell, with orthogonal $a_i$ (1,0,0), $a_j$(0,1,0), and $a_k$(0,0,1) respectively. The equivalent fractional coordinate of the Sc is (0,0,0) whereas the Al is (1/2, 1/2, 1/2). The real coordinates of Sc and Al in the unit cell depend on the lattice parameter a, with Sc at (0,0,0), (a,0,0), (0,a,0), (0,0,a), (a,a,0), (a,0,a), (0,a,a), and (a,a,a); Al at (a/2,a/2,a/2). Figure 1 shows an example of Sc and Al positions with the lattice parameter a=b=c=3.379 Å.

Figure 1. Simple cubic structure of ScAl, where Aluminum locates in the center and Scandium in the corner of the unit cell.

C. Determine the Lattice Parameters of ScAl

In order to predict a reasonable lattice parameter of ScAl, the energy of the unit cell is calculated with the variation on lattice parameters. Given the structure is from the cubic system, the lattice parameters a, b, and C are equal and will be referred to “a” as below. Before the calculation, the lattice parameter a is estimated based on the atomic radius of Al (1.43 angstrom) and Sc (2.30 angstrom). To get a well packed structure along the body diagonal in (111) face, the lattice parameter should be smaller than 2(r(Al)+r(Sc))/√3, which is 4.3 Å.

As a starting point the energy of the ScAl structure was calculated witha lattice parameter of 4.3Å using the CASTEP calculation ( Energy cutoff 500 eV, k-points 10*10*10). With the decreased lattice parameter from the starting point, the free energy of the unit cell reached a minimum to some point and then increased with the lattice parameter decreased further (Figure 2). This shape of curve is caused by the relative atom positions, either too far or too close, generateing higher energy (less stable structure) than the minimum energy (the most stable structure by calculation). The lattice parameter $a_0$= 3.379 Å corresponds the minimum cohesive energy (Figure 2).

Figure 2. Cohesive energy for simple cubic ScAl, using 101010 kpoints and 500eV energy cutoff, as a function of lattice parameter

D. The Energy Cutoff

Multiple calculations for ScAl structure were completed with a variation of energy cutoff $E_{cut}$ from 200eV to 800eV. Lattice parameter and kpoints remained the same at 3.379 Å and 8*8*8 k-point grid for Brillouin zone integration respectively. An increase in the energy cutoff increases the number of plane-waves and improves the accuracy of ion cores, but costs longer computation time. Repeated calculations with higher energy cutoff aim to converge to a decent final free energy (Figure 3&4). Convergence is reached with an energy cutoff of 500eV providing an accuracy of the absolute energy better than 0.001eV (Figure 4).

Figure 3. Both Al and Sc atomic energy converged at 500eV with a lattice parameter 3.379 Å and 888 k-point grid for Brillouin zone integration.

Figure 4. Energy per cell and cohesive energy of ScAl as a function of energy cutoff with a lattice parameter 3.379 Å and 888 k-point grid for Brillouin zone integration.

The cohesive energy of a solid material is the energy to separate the condensed material into isolated free atoms.

\begin{equation}E_{coh}=(E_{total}-E_{atom})/N\end{equation}
with $E_{total}$is the total energy of the unit cell, $E_{atom}$ is the total energy of the atom, and N as the number of atoms in the unit cell. Table 1 and Figure 4 indicate that the cohesive energy decreased with higher energy cutoff, converged to an accuracy of 0.01eV at energy cutoff of 500eV.

Table 1 Results from computing the total energy, atomic energy, and cohesive energy of simple cubic ScAl with 888 kpoints in different energy cutoffs

E. k-points

As k-point varied but energy cutoff (500 eV) and lattice parameter (3.379 Å) unchanged in the following calculation, the total energy of the cell converged with more kpoints. The cell volume and atomic energy remained the same, because the cell volume only depends on lattice parameter and atomic energy depends on the energy cut-off.

Figure 5. Energy per cell and cohesive energy of ScAl as a function of MMM k-points, with a lattice parameter 3.379 Å and energy cutoff 500eV.

The used k points are reduced by symmetry operations using Monkhorst-Pack approach with M*M*M k points in ScAl cubic structure. Table 2 and Figure 5 shows the number of k poins in irreducible Brillouin zone (IBZ) and correspondent total energy and binding energy. Both the odd (2n+1) and even (2n+2) values of M have the same number of k points in IBZ, but even values (2n+2) of M converges better than odd values in regard to the same computational time. In our cases, both the 7*7*7 and 8*8*8 k points required 8.08 seconds to finish the calculation, but 8*8*8 converging better because all k points are inside of the IBZ (Sholl and Steckel, 2011). In ScAl structure, 12*12*12 k points are enough to get accurate energy.

Table 2 Energy calculation with k points varies, energy cutoff 500eV, lattice parameter 3.379 Å.

Conclusion:

The total energy of simple cubic ScAl minimized at lattice parameter a=3.379 Å. Energy per cell converges when energy cutoff is 500eV, k-points are 12*12*12.

Reference

[1] Sholl, D. and Steckel, J.A., 2011. Density functional theory: a practical introduction. John Wiley & Sons.

[2] BIOVA, 2014. CASTEP guide , Material Studio. http://www.tcm.phy.cam.ac.uk/castep/documentation/WebHelp/content/pdfs/castep.htm

Searching for the Lattice Parameter of ScAl

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Lattice Structure description:

ScAl has the same type of structure that CsCl, which is simple cubic cell. The cell vector could be (1, 0, 0), (0, 1, 0), (0, 0, 1). Fractional coordinates for Cs could be (0, 0, 0) and for Al could be (0.5, 0.5, 0.5). Diagram for this structure is shown in Fig 1.

Fig1 Golden spheres are Sc, the brown shere is Al.

Since ScAl has simple cubic structure, We just need to adjust lattice parameter a in order to predict its structure. Idea is calculate total free energy for different parameters and find out the energetically favorable one.

DFT calculation is adopted for this search.

Cutoff energy test:

A test is done to find a proper cutoff energy. Fixing other setting: functional: GGA PBE, k points: 6*6*6, lattice parameter a=4.0Å, we change cutoff energy and compare their total free energy results. Results are shown in Table 1. Energy difference between cutoff energy ‘460 eV’ and ‘560 eV’ is less than 0.02eV. If time cost in considered and total free energy resolution is controlled at 0.02 eV, using ‘460 eV’ for cutoff energy for following calculations is an acceptable choice.

cutoff energy(eV)	total free energy(eV)
60	-1140.936931
160	-1347.963162
260	-1379.612688
360	-1383.327654
460	-1383.662277
560	-1383.681389
660	-1383.681690

Table 1

K point test:

functional: GGA PBE, cutoff energy: 460eV are fixed and lattice parameter a is changed.

K points in default will change with lattice parameter. (CASTEP tool is used here, ‘default’ meaning default number for k points in CASTEP tool)

Results are shown in Table 2 .

lattice parameter a(Å)	k points	total free energy (eV)
2.0000	141414	-1336.132886
2.6000	101010	-1376.299647
2.7000	101010	-1379.066514
2.8000	101010	-1381.168776
2.9000	101010	-1382.723256
3.0000	888	-1383.831689
3.1000	888	-1384.578369
3.2000	888	-1385.031976
3.3000	888	-1385.254431
3.3600	888	-1385.298682
3.3700	888	-1385.300556
3.3750	888	-1385.300611
3.3800	888	-1385.300591
3.3850	888	-1385.299964
3.3900	888	-1385.29928
3.4000	888	-1385.296417
3.4200	888	-1385.287388
4.0000	666	-1383.662277
5.0000	666	-1379.859133
6.0000	444	-1377.783054

Table 2

If density of k points is defined as number of k points in one direction over k space parameter in that direction, this according change of k point might have a purpose of keeping density of k point unchanged. Since the lengths of lattice vector in cell and lattice vector in k space have inverse proportion relation. So in this simple cubic system, expectation would be that number of k points in one direction times lattice parameter ‘a’ should lead to a constant. Obviously, this expectation is not obeyed in this test.

K points will effect the precision and time cost of a calculation, so finding a balance point of precision and efficiency means we need to find a suitable k points. This ‘finding a balance’ situation occurs as well when we deal with cutoff energy.

So which k point choice is suitable for this calculation? We can discuss this based on calculation results.

Fig 2 and Fig 3 show the search for lattice parameter. Relatively, one is rough, the other is fine.

Fig 2

Fig 3

We can see the parameter range which is located at energy valley is (3.36, 3.40). At this range, the k point is set as ‘8*8*8’ and in this range the finest search step is 0.005Å.

In ‘cutoff energy test’, ‘460 eV’ is used for cutoff energy so that resolution for total free energy is set to ‘0.02 eV’. Please notice that the ‘0.02 eV’ resolution actually also includes the setting of k points as ‘6*6*6’. And in the range we care about most adopts ‘8*8*8’ k point setting which should give precise enough results for this search. Energy numbers in table 2 for range (3.36, 3.40) do have difference less than 0.02 eV, which actually is less than 0.002 eV. So we can say that if ‘460 eV’ is adopted for cutoff energy, ‘8*8*8’ k point setting is ‘safe enough’ for this calculation. Of course, accordingly, it will be dangerous to make a prediction for lattice parameter beyond the precision of ‘0.005Å’.

Convergence test, however, is still done for k points, at a=4.0Å , cutoff energy=460 eV. Results are shown in Table 3.

k points	total free energy(eV)
444	-1383.589307
555	-1383.619788
666	-1383.662277
777	-1383.635492
888	-1383.633134
999	-1383.639153
101010	-1383.636616
111111	-1383.63629
121212	-1383.637744

Table 3

From data in this table, total free energy’s difference between ‘8*8*8’ and ‘9*9*9’ is less than 0.02 eV, which supports the point that ‘8*8*8’ setting for k points is precise enough under resolution of 0.02 eV for total free energy. Consistent with expectation, with increasing number of k points, we have smaller energy difference.

‘8*8*8′ for k points is adopted for lattice parameters outside (3.36, 3.40) in order to constrain variables when comparing different parameters’ energy. And for parameters in (2.00, 2.90), calculations have larger k points so it would be meaningless to re-calculate these points. Just using ‘8*8*8’ k points re-calculate points with a=4.00, 5.00, 6.00 Å. Results and comparison are shown in Table 4.

lattice parameter(Å)	total free energy with 888 k points(eV)	total free energy with default k points(eV)
4.000	-1383.633134	-1383.662277
5.000	-1379.864649	-1383.662277
6.000	-1377.763773	-1377.783054

Table 4

From data in the table, we can see that with ‘8*8*8’ k points, total free energy for these points goes higher, which does not affect our search for lowest energy point.

Conclusion:

If ‘460 eV’ cutoff energy and ‘0.02 eV’ precision for total free energy is adopted, ‘8*8*8’ k points setting could provide precise enough for the search of lattice parameter. At the same time the precision of this parameter search is limited at ‘0.005Å’.

Based on the calculation results and just considering minimizing total free energy, ScAl should have a lattice parameter around 3.375Å.

If more decimal place is wanted for this prediction, larger cutoff energy and k points should be adopted.

Reference:

First principles methods using CASTEP. Zeitschrift fuer Kristallographie 220(5-6) pp. 567-570 (2005) S. J. Clark, M. D. Segall, C. J. Pickard, P. J. Hasnip, M. J. Probert, K. Refson, M. C. Payne

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Category Archives: ScAl

Lattice constant predictions for ScAl assuming both the CsCl and NaCl structures

Lattice constant for ScAl in CsCl and NaCl structures

Lev Krainov

Introduction:

Computational details:

K-points convergence

Cutoff energy convergence

Geometry optimization

Results

Comparing ScAl in CsCl and NaCl Structures and Determining the Optimal Lattice Parameter of the Preferred Structure

Determination of the Lattice Parameter of ScAl in the CsCl Structure

A. Project Description

B. Crystal Model

Figure 1. Simple cubic structure of ScAl, where Aluminum locates in the center and Scandium in the corner of the unit cell.

C. Determine the Lattice Parameters of ScAl

Figure 2. Cohesive energy for simple cubic ScAl, using 101010 kpoints and 500eV energy cutoff, as a function of lattice parameter

D. The Energy Cutoff

Figure 3. Both Al and Sc atomic energy converged at 500eV with a lattice parameter 3.379 Å and 888 k-point grid for Brillouin zone integration.

Figure 4. Energy per cell and cohesive energy of ScAl as a function of energy cutoff with a lattice parameter 3.379 Å and 888 k-point grid for Brillouin zone integration.

Table 1 Results from computing the total energy, atomic energy, and cohesive energy of simple cubic ScAl with 888 kpoints in different energy cutoffs

E. k-points

Figure 5. Energy per cell and cohesive energy of ScAl as a function of MMM k-points, with a lattice parameter 3.379 Å and energy cutoff 500eV.

Table 2 Energy calculation with k points varies, energy cutoff 500eV, lattice parameter 3.379 Å.

Conclusion:

Reference

Searching for the Lattice Parameter of ScAl

Lev Krainov

Introduction:

Computational details:

K-points convergence

Cutoff energy convergence

Geometry optimization

Results

A. Project Description

B. Crystal Model

Figure 1. Simple cubic structure of ScAl, where Aluminum locates in the center and Scandium in the corner of the unit cell.

C. Determine the Lattice Parameters of ScAl

Figure 2. Cohesive energy for simple cubic ScAl, using 10*10*10 kpoints and 500eV energy cutoff, as a function of lattice parameter

D. The Energy Cutoff

Figure 3. Both Al and Sc atomic energy converged at 500eV with a lattice parameter 3.379 Å and 8*8*8 k-point grid for Brillouin zone integration.

Figure 4. Energy per cell and cohesive energy of ScAl as a function of energy cutoff with a lattice parameter 3.379 Å and 8*8*8 k-point grid for Brillouin zone integration.

Table 1 Results from computing the total energy, atomic energy, and cohesive energy of simple cubic ScAl with 8*8*8 kpoints in different energy cutoffs

E. k-points

Figure 5. Energy per cell and cohesive energy of ScAl as a function of M*M*M k-points, with a lattice parameter 3.379 Å and energy cutoff 500eV.

Table 2 Energy calculation with k points varies, energy cutoff 500eV, lattice parameter 3.379 Å.

Conclusion:

Reference

Figure 2. Cohesive energy for simple cubic ScAl, using 101010 kpoints and 500eV energy cutoff, as a function of lattice parameter

Figure 3. Both Al and Sc atomic energy converged at 500eV with a lattice parameter 3.379 Å and 888 k-point grid for Brillouin zone integration.

Figure 4. Energy per cell and cohesive energy of ScAl as a function of energy cutoff with a lattice parameter 3.379 Å and 888 k-point grid for Brillouin zone integration.

Table 1 Results from computing the total energy, atomic energy, and cohesive energy of simple cubic ScAl with 888 kpoints in different energy cutoffs

Figure 5. Energy per cell and cohesive energy of ScAl as a function of MMM k-points, with a lattice parameter 3.379 Å and energy cutoff 500eV.