Delocalization and phase transitions in Pr: Theory
Per So
¨
derlind
Department of Physics, Lawrence Livermore National Laboratory, Livermore, California 94550
Received 26 July 2001; published 19 February 2002
Density-functional electronic structure calculations have been used to investigate the high pressure behavior
of Pr at low temperature. Several phase transitions are suggested by these calculations and they agree well with
available experimental data. At low pressure, a dhcp (Pr-I) →fcc (Pr-II) transition is calculated to occur at 60
kbar. Not considering the Pr-III phase, which is computationally too demanding, Pr-II transforms to the -U
Pr-IV phase at 165 kbar. This latter transition is accompanied by a volume collapse of about 10% and is
driven by delocalization of the 4 f electrons in Pr. The axial ratios ( b / a and c / a ) and the internal parameter y
of Pr-IV were calculated as a function of compression. y and c / a are rather insensitive to the compression
whereas b / a decreases significantly with increasing pressure. At about 1 Mbar a new phase is predicted,
namely a body-centered-tetragonal bct structure with a c / a axial ratio of 1.78. This new phase is stable up to
very high pressures but at a sixfold compression an ultimate hexagonal close-packed hcp structure is pre-
dicted. The calculated high pressure behavior of Pr is similar to that of the early actinide Pa. Pressure induced
increase in 4 f -orbital overlap, occupation and band width together with an increased electrostatic Coulomb
interaction with pressure are the key components in stabilizing the high pressure phases of Pr.
DOI: 10.1103/PhysRevB.65.115105 PACS numbers: 71.20.Eh, 61.66.Bi, 64.10.+h
I. INTRODUCTION
The rare-earth metals exhibit rich phase diagrams with
several structural phase transitions when compressed.
1
At
lower pressures the crystal-structure sequence of hexagonal
close-packed hcp, Sm-type, double hexagonal close-packed
dhcp, face-centered cubic fcc, distorted fcc d-fcc is gen-
erally observed as a function of pressure. These structures
are close-packed with relatively high symmetry and the tran-
sitions, understood from pressure-induced sp →d
promotion,
2
do not show any volume collapses. At higher
pressures, new phases that are more complex with lower
symmetries are often found and sometimes the transition is
accompanied by a substantial volume collapse.
1
Praseodymium is dhcp Pr-I at ambient conditions and a
phase transition at room temperature to fcc Pr-II is ob-
served at about 40 kbar and followed at about 70 kbar by
Pr-III which is a distorted 24 atom/cell fcc-type d-fcc
structure.
3,4
Chesnut et al.
4
also argued that Pr could be de-
scribed as a monoclinic cell ( C 2/m ) with less atoms/cell
above 100 kbar. At pressures above 200 kbar, an orthorhom-
bic phase ( -U) appears and the transition to this Pr-IV
phase is associated with a large volume collapse. In the lit-
erature values of this collapse have spread between
9–16.7 % Refs. 3–7 with the most recent experiment
3
sug-
gesting 10.7% at 300 K. Chesnut et al.
4
studied Pr up to 1.03
Mbar, in anticipation of new phase transitions, but none was
found.
Theoretically, the low pressure and high symmetry struc-
tures in Pr and the rare earths in general are well under-
stood to be stabilized by the 5 d electrons.
2
The 4 f electrons
are believed to be localized at each atom with very little
involvement in the chemical bonds. At higher pressures,
however, the narrow 4 f bands overlap more and their inter
atomic interaction increase. The Pr-III→Pr-IV transition and
the associated large volume collapse suggests a sudden par-
ticipation of 4 f electrons in the chemical bond. Also a loss of
local magnetic moment on the f site follows the transition
and in the Mott transition model
8
this is due to quenching of
the f -shell moment. Hence, the Pr-III→Pr-IV is an elec-
tronic structure transition that likely is the cause of the struc-
tural transition. Although the low-pressure phases of Pr is
well understood, this volume-collapse transition has been
more challenging for theory. Recent developments in
density-functional theory, however, have made it possible to
calculate the main features of such a transition with reason-
able accuracy. Some success have been reported using so-
called self-interaction correction SIC as well as orbital po-
larization OP techniques for the Pr-III→Pr-IV transition in
praseodymium
9
but Pr has not been studied theoretically at
pressure above about 200 kbar.
At pressures beyond the Pr-III→Pr-IV transition,
praseodymium is believed to be an itinerant 4 f metal similar
to Ce at high pressures or the light actinides, Th-Pu (5 f ), at
ambient conditions. In these systems, the f bands dominate
the bonding and the structural behavior under pressure has
been theoretically investigated in detail.
10
The narrow f band,
positioned close to the Fermi level ( E
F
), drives a Peierls or
Jahn-Teller-like distortion of the lattice.
11
Symmetries of the
lattice give rise to degeneracies and high density of
f -electron states in the vicinity of E
F
. This is energetically
unfavorable and a lower symmetry structure is preferred. The
number of f electrons involved in this mechanism is, of
course, important. For Ce and Th, with about one f electron
or less at ambient pressure, the effect is small and they are
stable in a high symmetry fcc phase. For Pa-Pu, which
gradually are filling the 5 f band, this symmetry breaking
mechanism becomes increasingly important. Pa is body-
centered tetragonal bct, U orthorhombic 2 atoms/cell, Np
orthorhombic 8 atoms/cell, and Pu monoclinic 16 atoms/
cell. Applied pressure to these metals will have three domi-
nating effects; i an spd promotion to the f band, ii a
broadening of the f band, and iii an increase in electrostatic
energies. i drive transitions to lower symmetry structures
PHYSICAL REVIEW B, VOLUME 65, 115105
0163-1829/2002/6511/1151055/$20.00 ©2002 The American Physical Society 65 115105-1