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Next: 2000-07-06: ceria-lanthana Up: Aliovalent cation doped ceria Previous: 2000-07-04: ceria-yttria


2000-07-05: ceria-gadolinia

GOAL

Calculations on the $Ce_{(1-x)}Gd_xO_{2-\frac{x}{2}},\ 0.1\le{}x\le0.9$ system.

Results

$Ce^{4+}/Ce^{3+}$ reduction energy

The $Ce^{4+}/Ce^{3+}$ reduction energy was evaluated according to the procedure outlined in section 2000-06-21: even if the system does not contain zirconia, the mean field analysis remains identical.

\begin{tabular}{D{.}{.}{-1}D{.}{.}{-1}D{.}{.}{-1}D{.}{.}{-1}} \multicolumn{1}{c}... ...& -6.36983229 \\ 90 & -27.73622476 & 7.12778393 & -7.04664497 \\ \end{tabular}

\begin{tabular}{D{.}{.}{-1}D{.}{.}{-1}D{.}{.}{-1}} \multicolumn{1}{c}{{$100x$}} ... ...80 & 10.55461639 & 4.01423724 \\ 90 & 9.91902705 & 3.30368940 \\ \end{tabular}

\begin{center}\vbox{\input{2000-07-05-01.pslatex} }\end{center}

Lattice parameter

\begin{tabular}{D{.}{.}{-1}D{.}{.}{-1}} \multicolumn{1}{c}{{$100x$}} &\multicolu... ...5.432074 \\ 70 & 5.432766 \\ 80 & 5.433503 \\ 90 & 5.434290 \\ \end{tabular}

The following compares presently calculated lattice parameters with the empirical equation of Kim (1989), which, for the present case $Ce_{(1-x)}Gd_xO_{2-\frac{x}{2}}$, reads as:

\begin{eqnarray*} d_{Ce}&=&0.5413\\ &&+\left(0.0220(r_{Gd^{3+}}-r_{Ce^{4+}})+0.00015(z_{Gd^{3+}}-z_{Ce^{4+}})\right)100x \end{eqnarray*}



with $d_{Ce}$ and the $r$'s in $nm$, or:

\begin{eqnarray*} d_{Ce}&=&5.413\\ &&+\left(0.0220(r_{Gd^{3+}}-r_{Ce^{4+}})+0.0015(z_{Gd^{3+}}-z_{Ce^{4+}})\right)100x \end{eqnarray*}



with $d_{Ce}$ and the $r$'s in Å.

The values of the various parameters are (Shannon, 1976):

\begin{tabular}{rD{.}{.}{-1}l} {$r_{Ce^{4+}}$}&0.97&{\it {}\AA}\\ {$r_{Gd^{3+}}$}&1.053&{\it {}\AA}\\ {$z_{Ce^{4+}}$}&4&\\ {$z_{Gd^{3+}}$}&3&\\ \end{tabular}

\begin{center}\vbox{\input{2000-07-05-02.pslatex} }\end{center}

Like with the Yttria containing system, the accord does not seem to be very good.

Lattice energy

\begin{tabular}{D{.}{.}{-1}D{.}{.}{-1}} \multicolumn{1}{c}{{$100x$}} &\multicolu... ... 70 & -71.49938789 \\ 80 & -67.20075206 \\ 90 & -63.03537256 \\ \end{tabular}

\begin{center}\vbox{\input{2000-07-05-03.pslatex} }\end{center}

Representative input file 


opti conp defect

maxcyc opt      200
maxcyc fit      200
dump every   1 oimp-40.dump
title
O impurity
end

cell
   5.329267   5.329267   5.329267  90.000000  90.000000  90.000000

fractional    5
   Ce4  core  0.0000  0.0000  0.0000  -3.7000  0.6000  0.0000 
    Gd  core  0.0000  0.0000  0.0000   3.0000  0.4000  0.0000 
     O  core  0.2500  0.2500  0.2500   0.0770  0.9000  0.0000 
   Ce4  shel  0.0000  0.0000  0.0000   7.7000  0.6000  0.0000 
     O  shel  0.2500  0.2500  0.2500  -2.0770  0.9000  0.0000  
                                                     
space
225

size            9.0  20.0000
centre          0.25 0.25 0.25
impurity  O     0.25 0.25 0.25

species   2
Ce3    core   -4.700000
Ce3    shel    7.700000

buck
Ce4   shel O     shel  1986.8300     0.351070  20.400      0.000 15.000
Gd    core O     shel  1336.8        0.35510     0.0       0.000 15.0
Ce3   shel O     shel  1731.6181     0.363720   14.433     0.000 15.000
O     shel O     shel 22764.300      0.149000   27.890     0.000 15.000
spring
Ce4    291.75000
Ce3    291.75000
O      27.290000

next up previous contents
Next: 2000-07-06: ceria-lanthana Up: Aliovalent cation doped ceria Previous: 2000-07-04: ceria-yttria