Lanthanides
For the Lanthanides we present a total energy versus volume graph for La which shows the correct ground state as dhcp. As it is shown in the energy bands and DOS figures the f-bands are positioned above the Fermi level for La. As we examine the energy ban
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Lanthanides
For the Lanthanides we present a total energy versus volume graph for La which shows the correct ground state as dhcp. As it is shown in the energy bands and DOS figures the f-bands are positioned above the Fermi level for La. As we examine the energy bands and DOS for the rest of the elements (except for Lu for which we show total energies for fcc and bcc with a standard method) in this row we note that Ef is located inside the f-bands and finally in Yb and Lu lies above the f-bands. The, at-least, partially occupied f-bands in these materials present a serious difficulty in determining the equilibrium volume and structure using standard density functional theory. We have found that the only way to calculate a reasonable equilibrium volume for these materials is to push the f-electrons into the core and use only s- and d- electrons in the valence [1]. We demonstrate this for Ce, Pr, Nd, Pm, and Sm where the equilibrium fcc, bcc and sc volumes are predicted by forcing the f-bands into the core. For Eu, we show fcc, bcc, as well as hcp equilibrium volumes using the same method. For the rest of the elements in this row, which are mostly existing in hcp lattice, we present energy bands and DOS for the equivalent fcc volume to show trends.
© Springer Science+Business Media New York 2015 D.A. Papaconstantopoulos, Handbook of the Band Structure of Elemental Solids, DOI 10.1007/978-1-4419-8264-3_13
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430
13.1
13
Lanthanides
Lanthanum Z ¼ 57
Fig. 13.1 Total energy of La
Table 13.1 Lattice constants and bulk modulus
a (Bohr) sc 6.338 bcc 8.011 fcc 10.032 hcp 7.185 dchp 6.84 exp 7.13 ΔEfcc–dhcp ¼ 1.2 mRy
c (Bohr)
B (MBar)
11.855 22.07 22.83
0.240 0.270 0.248 0.264 0.300 0.240
13.1
Lanthanum Z ¼ 57
431
Table 13.2 Birch fit coefficients
sc bcc fcc hcp dhcp
A1
A2
A3
A4
16996.90361 16996.55887 16996.67954 33992.51350 67986.68678
26.28179 69.12300 57.01302 340.89968 234.67719
302.38950 1926.50289 1512.13350 17625.09865 6359.17857
60392.19115 14264.67872 9948.83479 285296.26332 43927.77984
432
13
Lanthanides
Lanthanum
Fig. 13.2 Energy bands and density of states for La Table 13.3 DOS at the Fermi level (APW results) Energy Total Ry
s p eg States/Ry/atom
t2g
f
Velocity cm/s
Plasmon energy eV
0.410
0.993
11.717
3.809
0.347 10E8
4.492
32.280
2.898
5.673
0.21817 0.01481 0.02127 0.00017 0.00033
0.18627
0.02273
0.00734
0.00034
0.00006
0.17580 0.00970 0.00010
0.23981
0.07391
0.00205
d 0.73800 pps
0.00000
0.00272
0.04554
0.00168
pdp
0.00089
0.00214
0.00012
0.00029
0.00000
0.05158
0.00008
0.00003
0.04224
0.04101
0.00742
0.01660
0.00000
0.12695
0.00498
0.00243
0.07895
0.06973
0.06313
0.15152
0.01670
pds
0.04721
0.03252
pdp
sds
pds
sds
0.00038
0.00902
0.05797
dds
0.00022
0.00015
0.01618
0.01150
0.18368
0.14515
dds
0.00228
0.05152
0.03104
ddp
0.00032
0.00004
0.01357
0.00196
0.06484
0.00509
ddp
0.00001
0.00056
0.04014
ddd
0.00003
0.00008
0.00274
0.00678
0.
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