First-order Phase Transition in UO 2 : The Interplay of the 5f 2 –5f 2 Superexchange Interaction...

transcript

First-order Phase Transition in UO2: The

Interplay of the 5f2–5f2 Superexchange

Interaction and Jahn–Teller Effect

V. S. Mironov,1 L. F. Chibotaru2 and A. Ceulemans2

1Institute of Crystallography, Russian Academy of Sciences, Leninskii prosp. 59,

117333 Moscow, Russia2Department of Chemistry, Katholieke Universiteit Leuven, Celestijnenlaan 200F,

B-3001 Leuven, Belgium

AbstractThe competition of the superexchange interaction and the cooperative Jahn–Teller (JT) effect

in the first-order magnetic and structural phase transition in UO2 is analyzed. The effective

spin Hamiltonian of the superexchange interaction between the neighboring U4þ ions in the

cubic crystal lattice of UO2 is calculated for the first time in terms of a specially adapted

kinetic exchange model. The 5f2–5f2 superexchange interaction is shown to be essentially

non-Heisenberg: the effective spin Hamiltonian is anisotropic and contains large biquadratic

terms, which formally correspond to a quadrupole–quadrupole interaction. The strength of

the JT effect in UO2 is estimated from the calculations of the linear vibronic coupling

constants between the local eg; t2gð1Þ and t2gð2Þ JT modes and the G5 ground state of U4þ

performed in terms of a semiempirical crystal field model for 5f electrons. Despite the G5 ^ eg

and G5 ^ t2gð2Þ parameters are found to be much larger than G5 ^ t2gð1Þ; the actual lattice

distortion in UO2 is of a pure t2gð1Þ character. Our results indicate that the superexchange

interaction is the major driving force of the phase transition in UO2, which causes the

magnetic and orbital ordering and suppresses the cooperative JT effect.

Contents

1. Introduction 600

2. Superexchange interaction between U4þ ions in UO2 603

2.1. Crystal field effect and the ground state of U4þ ions in UO2 603

2.2. The mechanism of the 5f2–5f2 superexchange interaction 604

3. Jahn–Teller effect for the ground G5 state of U4þ ions: calculations

of vibronic coupling constants for the eg; t2gð1Þ and t2gð2Þ

Jahn–Teller modes 611

4. Discussion: the relationship between the 5f2–5f2 superexchange

interaction and Jahn–Teller effect 613

5. Conclusions and further outlook 615

Acknowledgements 616

References 616

ARTICLE IN PRESS

1. INTRODUCTION

Being important as a nuclear fuel, uranium dioxide is one of the most frequently

studied actinide compounds, which was thoroughly characterized using various

experimental techniques [1–12]. UO2 is an insulator or ionically bonded

semiconductor crystallizing in the fcc fluorite structure with lattice constant

a0 ¼ 5:470 A at room temperature. This is a stoichiometric compound involving

tetravalent uranium ions with the 5f2 levels lying in a 6 eV band gap. Each U4þ ion is

surrounded by eight oxygen atoms forming a cubic cage (Fig. 1). In such a

surrounding, the ground state of U4þ ions is the G5 triplet which, apart from being

correspondent to the effective spin S ¼ 1; is also a Jahn–Teller active state. The

UO2 crystal represents therefore a magnetic insulator and cooperative Jahn–Teller

system, which has a rich collection of unusual electronic and magnetic properties at

low temperature.

Uranium dioxide undergoes first-order phase transitions at TN ¼ 30:8 K [1].

Despite extensive studies for more than thirty years, many details of the basic

mechanism of this phenomenon are still unclear. Below TN a magnetic ordering and

lattice distortions simultaneously appear. Earlier studies suggested a usual collinear

single-k antiferromagnetic ordering [1–3], but more recent studies showed that the

magnetic structure of UO2 is much more complicated [4–6]. It was finally

established from the analysis of numerous neutron diffraction experiments [5,6]

and 235U NMR data [12] that the magnetic ordering is best described by so-called

triple-k structure, which is determined by the sum of the Fourier components

corresponding to the three wave vectors (1,0,0), (0,1,0), and (0,0,1) [6–8]

Mx ¼ M0 exp i2p

� �; My ¼ M0 exp i

� �;

Mz ¼ M0 exp i2p

� � ð1Þ

Fig. 1. The fcc fluorite structure of UO2.

V. S. Mironov et al.600

ARTICLE IN PRESS

where Mx; My; and Mz are the components of the localized magnetic moment on the

uranium center with (x, y, z) coordinates. The corresponding 3-k spin structure is

non-collinear (Fig. 2). The magnetic moments of U4þ ions are directed along one of

the space diagonals k111l so that the total magnetic moment of the cubic unit cell is

It is important that the ordered moment at low temperature is considerably smaller

(1.74 mB/U) than the effective paramagnetic moment, ,3 mB/U.

The magnetic ordering is accompanied by a lattice distortion, which also has a

triple-k character

dx ¼ b0 exp i2p

� �; dy ¼ b0 exp i

� �; dz ¼ b0 exp i

� �;

b0 ¼dffiffi3

Oxygen atoms shift by d ¼ 0:014 A from the equilibrium positions towards the

uranium ions, which remain fixed. In each UO8 cubic cage, the central U4þ ion

attracts two oxygen atoms lying in the k111l space diagonal along which the

magnetic moment of U4þ is directed (Fig. 3). As a result, the lattice symmetry

remains cubic and the unit cell volume does not change in the ordered phase.

Since the components of the G5 triplet have different spatial charge density

distributions, the magnetic ordering in UO2 is followed by some orbital ordering and

lattice distortions. The latter are usually associated with the cooperative Jahn–Teller

effect and/or quadrupole–quadrupole (Q–Q) interaction [7,8]. It is important to note

that in systems with a strong spin–orbit coupling the JT cooperative effect and

exchange interactions are antagonists since the JT effect tends to stabilize a state

Fig. 2. The 3-k magnetic ordering in UO2 below TN ¼ 30:8 K: The magnetic moments ofU4þ are directed along one of the space diagonals k111l of the cubic unit cell, as described byequation (1).

First-order Phase Transition in UO2 601

ARTICLE IN PRESS

with the quenched orbital momentum on each metal ion while the exchange

interactions acts in a just opposite direction, i.e., it tends to stabilize a state with the

maximum orbital momentum (which is strongly coupled with the spin via the spin–

orbit interaction). Another feature is a very strong spin-lattice coupling in UO2

observed experimentally in inelastic neutron scattering spectra [4–6]. The case of

UO2 is therefore quite different from the case of orbitally degenerate 3d systems

(such as RMnO3 manganites), in which magnetic ordering and orbital ordering can

act independently because the orbital momentum and spin are weakly coupled to

each other. The magnetic structure and lattice distortions in the low-temperature

phase of UO2 are therefore determined by a complicated interplay of the magnetic

coupling between U4þ ions and the cooperative JT effect.

The question therefore arises as to which of these interactions dominate in

UO2. It is important to note that, despite extensive studies for more than thirty

years, there is still very little known about the actual mechanism of the U4þ–

U4þ exchange interaction and the strength of the vibronic coupling in UO2. No

satisfactory theoretical explanation of the origin of the 3-k magnetic and lattice

distortion structure below TN is available. It is recognized that the 3-k magnetic

structure, lattice distortions, and magnon spectra in UO2 cannot be accounted

for in terms of the conventional isotropic exchange model [5,6]. On the other

hand, since the spin on the U4þ ion is strongly coupled with its orbital

momentum, the spin is not a good quantum number and thus the U4þ–U4þ spin

coupling is expected to be essentially non-Heisenberg.

The aim of this work is to elucidate these problems. To this end, we calculate the

effective spin Hamiltonian of the 5f2–5f2 superexchange interaction between the

neighboring U4þ ions in the cubic crystal lattice of UO2 and we calculate G5 ^ eg;G5 ^ t2gð1Þ and G5 ^ t2gð2Þ linear vibronic coupling constants. These data are then

used to draw a more definite conclusion about the driving force of the phase

transition and especially about the actual mechanism of the spin and orbital ordering

in UO2.

Fig. 3. The 3-k structure of the distortion of the UO8 cubic cage below TN : The centraluranium ion attracts those two oxygen atom of the given cubic cage, to which its magneticmoment points. The amplitude of the displacements is as small as 0.014 A (about 0.5% of theU–O distance). The other six oxygen atoms are attracted by uranium ions from neighboringcages.

ARTICLE IN PRESS

2. SUPEREXCHANGE INTERACTION BETWEEN U41 IONS IN UO2

2.1. Crystal field effect and the ground state of U41 ions in UO2

Since the G5 ground electronic state of U4þ ions plays a very important role in the

magnetic ordering and lattice distortions in UO2, we give here necessary details

concerning the origin of this state. The U4þ ion is characterized by a strong spin–

orbit interaction with the spin–orbit coupling constant of z5f < 1800 cm21: The

lowest 3H term of the free U4þ ion (5f2 configuration) is split by the spin–orbit

interaction into the ground 3H4 multiplet and excited 3H5 and 3H6 multiplets (Fig. 4).

The total spin–orbit splitting of the lowest 3H term is about 10 000 cm21; this

means that the spin and orbital momentum of U4þ are strongly coupled to each

other.

In the cubic crystalline surrounding, the ground 3H4 multiplet is split into four

individual CF levels, the ground G5 triplet, exited G3 doublet, G4 triplet, and the

upper G1 singlet; their energies are shown in Fig. 5 [6]. The ground state is separated

by a large gap from the first excited CF state. Note that the total CF splitting of the3H4 multiplet is much smaller than the spin–orbit splitting of the lowest 3H term of

the 5f2 configuration, 1400 and 10 000 cm21, respectively. Detailed CF calculations

show that the wave functions of the G5 triplet are of almost pure 3H4 character. They

can be written as

G5;21lj ¼

ffiffiffi7

rj 3H4;23l2

ffiffiffi1

rj 3H4;þ1l ð3aÞ

Fig. 4. The scheme of low-lying multiplets of the free U4þ ion. The total spin-orbit splittingof the lowest 3H is about 10 000 cm21. The energy levels are given in the actual energy scale.

ARTICLE IN PRESS

G5; 0lj ¼

ffiffiffi1

rj 3H4;þ2l2

ffiffiffi1

rj 3H4;22l ð3bÞ

G5;þ1lj ¼

ffiffiffi7

rj 3H4;þ3l2

ffiffiffi1

rj 3H4;21l ð3cÞ

This means that the total spin of the U4þ ion is not a good quantum number since it is

antiparallely coupled with the orbital momentum. The G5 triplet represents therefore

an effective spin S ¼ 1: In addition, this is a JT active state (the T term case) since its

components lG5;ml ð1aÞ– ð1cÞ have different charge density distributions.

2.2. The mechanism of the 5f2–5f2 superexchange interaction

The exchange interaction between the neighboring U4þ ions in the cubic lattice of

UO2 (Fig. 6) is analyzed in the frame of the Anderson kinetic exchange theory [13].

However, since this theory in its conventional form cannot directly be applied to the

analysis of non-Heisenberg exchange interactions between the U4þ ions with the G5

ground state, we use a modified kinetic exchange theory. In our approach, the

U4þðAÞ! U4þðBÞ metal-to-metal electron transfers through the bridging oxygen

atoms are analyzed in terms of individual many-electron states of the 5f2–5f2 basic

configuration and 5f1–5f3 charge-transfer (CT) configuration of a separate U4þ(A)–

U4þ(B) pair, rather than in terms of one-electron states. The wave functions lm1;m2lof the ground G5(A)–G5(B) level of the U4þ(A)–U4þ(B) pair (which is 9-fold

Fig. 5. Crystal field splitting of the ground 3H4 multiplet of the U4þ ion in UO2. The groundCF state of U4þ is a well-isolated G5 triplet.

ARTICLE IN PRESS

degenerate in the absence of exchange interactions) are written as direct

antisymmetrized products of the corresponding many-electron wave functions of

the U4þ(A) and U4þ(B) ions

lm1;m2l ¼ lG5ðAÞ;m1l^ lG5ðBÞ;m2l ð4Þ

where m1 and m2 are the projections of the effective spin S ¼ 1ðm1;2 ¼ 0;^1Þ of the

ions U4þ(A) and U4þ(B). Similarly, the wave functions CCTn of CT many-electron

states are written as products of single-ion wave functions of the U(A) and U(B)

centers with one missing or extra electron

C CTn ðA ! BÞ ¼ J

UðAÞi ð5f1Þ^J

UðBÞj ð5f3Þ ð5Þ

C CTn ðA ˆ BÞ ¼ J

UðAÞi ð5f3Þ^J

UðBÞj ð5f1Þ ð6Þ

In this representation, electron transfers between metal centers are described as

lm1;m2l!C CTn ! lm0

1;m02l transitions between the ground and CT many-electron

states. Their amplitude is described by the km1;m2lVABlC CTn l matrix elements of the

effective one-electron operator VAB, which is specified by one-electron matrix

elements tij; i.e., transfer integrals connecting 5fi(A) and 5fj(B) orbitals. The

km1;m2lVABlC CTn l matrix elements can directly be expressed via the tij transfer

parameters. For this, wave functions of the basic 5f2 configurations and 5f1 and 5f3

CT configurations of uranium ions are expanded over Slater determinants. Details

of these calculations are given in Ref. [16]. The effective exchange Hamiltonian

Heff is obtained by the projection of CT states onto the space of the lm1;m2l wave

functions of the ground level. It is completely specified by the set of the

km1;m2lHeff lm01;m0

2l matrix elements, which are obtained from the second-order

perturbation equation

km1;m2lHeff lm01;m0

2l ¼ 2XC CT

km1;m2lVABlC CTn lkC CT

n lVABlm01;m0

2lEðC CT

n Þð7Þ

where the sum runs over all C CTn states of the U4þ(A)–U4þ(B) pair resulting from

the UðAÞ! UðBÞ and U(A) ˆ (B) electron transfers, and EðC CTn Þ is the corres-

ponding CT energy. As shown below, the Heff operator can easily be transformed

Fig. 6. The geometry of the exchange-coupled pair for two the neighboring U4þ ions in thecrystal lattice of UO2.The local symmetry of the neighboring U4þ(A)–U4þ(B) exchange pairis D2h:

ARTICLE IN PRESS

to the conventional spin Hamiltonian. The CT energies can be written as

EðC CTn Þ ¼ U0 þ Epð5f1Þ þ Eqð5f3Þ ð8Þ

where U0 is the CT energy gap that separates the ground G5–G5 state of the basic

5f2 –5f2 configuration from the lowest state of the excited 5f1 –5f3 CT

configuration of the U4þ(A)–U4þ(B) pair (Fig. 7) (which is set to U0 ¼ 10 eV

in our calculations), and Epð5f1Þ and Eqð5f3Þ are the energies of individual states of

the 5f1 and 5f3 configurations of uranium ions with one missed and one extra

Fig. 7. The energy spectrum of CT states of the U4þ(A)–U4þ(B) pair. The ground G5–G5

state of the basic 5f2–5f2 configuration is separated from the lowest state of the excited 5f1–5f3 CT configuration of the U4þ(A)–U4þ(B) pair by a gap of U0, which set to 10 eV.

ARTICLE IN PRESS

electron, respectively. The total number of the CT states is given by the product of

the numbers of states of the 5f2 (14) and 5f3 (364), i.e., a total of 5096 (Fig. 7).

It is important to take into account the actual energy spectrum of CT states of the

U4þ(A)–U4þ(B) pair since the total energy spacing of the 5f1–5f3 CT configuration

(about 7 eV) is comparable with the CT gap, U0 ¼ 10 eV (Fig. 7).

Because analytical calculations using equation (7) are impossible, a special

routine for numerical calculations of the spin Hamiltonian exchange parameters was

designed (see Ref. [16] for details). In these calculations, the one-centers

Hamiltonians of the 5f1, 5f2, and 5f3 configurations of uranium ions were explicitly

diagonalized using the actual free-ion parameters ðF2 ¼ 43 100; F4 ¼ 41 000; F6 ¼

23 800; and z5f ¼ 1780 cm21Þ and cubic crystal-field parameters [6].

Transfer integrals tij ¼ k5fiðAÞlVABl5fjðBÞl describing the virtual transfers of an

electron between the U4þ(A) and U4þ(B) ions via the bridging oxygen atoms were

calculated using the conventional second-order perturbation expression correspond-

ing to the case of weak metal–ligand covalence

tij ¼ 2X

XxkðnÞ

k5fiðAÞlhlxkðnÞl kxkðnÞlhl5fjðBÞlEðxkðnÞÞ2 Eð5fÞ

where the first sum runs over the two bridging oxygen atoms O(n) ðn ¼ 1; 2Þ and the

second sum runs over the 2p and 2s orbitals xkðnÞ of these atoms. Matrix elements of

the Fock operator h are resonance integrals connecting 5fi orbitals of the uranium

ions and xkðnÞ orbitals of the bridging oxygen atoms. The energy denominator is a

ligand-to-metal promotion energy, which is given by the difference of the

corresponding orbital energies. In the D2h geometry of the U4þ(A)–U4þ(B) pair

(Fig. 6), the resonance integrals entering equation (9) can be expressed analytically

via three parameters, s(fp), p(fp), and s(fs), corresponding to the resonance

integrals defined with respect to a local U–O bond: sðfpÞ ¼ k5f0lhl2p0l; pðfpÞ ¼k5f^1lhl2p^1l; and sðfsÞ ¼ k5f0lhl2sl; where 5fk are uranium and 2pk, 2s are oxygen

orbitals with the projection k of the orbital momentum on the metal–ligand axis

(k ¼ 0;^1 for 2p and k ¼ 0 for 2s). For each of the two U(A)–O(n)–U(B) bridges,

the products k5fiðAÞlhl2pkl k2pklhl5fjðBÞl in the nominator of equation (8) can be

written as some linear combinations of binary products of the s(fp) and p(fp)

parameters (Fig. 8). The coefficients in these combinations correspond to the

expansion of atomic orbitals defined in the local coordinate frame of a given metal–

ligand pair over orbitals defined with respect to a common coordination frame; they

can be written via the Wigner D-functions, as described in Ref. [16].

The s(fp), p(fp), and s(fs) quantities can be calculated using the Wolfsberg–

Helmholz formula [17]

sðfpÞ ¼ k5f0lhl2p0l ¼ ½Eð5fÞ þ Eð2pÞSsð5f; 2pÞ ð10aÞ

pðfpÞ ¼ k4f^1lhl4p^1l ¼ ½Eð5fÞ þ Eð2pÞSpð5f; 2pÞ ð10bÞ

sðfsÞ ¼ k5f0lhl2sl ¼ ½Eð5fÞ þ Eð2sÞSsð5f; 2sÞ ð10cÞ

ARTICLE IN PRESS

where Ssð5f; 2pÞ; Spð5f; 2pÞ; and Ssð5f; 2sÞ are s and p overlap integrals between

the respective metal and ligand orbitals. These overlap integrals are calculated using

the U 5f and O 2p and 2s radial wave functions reported in Ref. [18] and Ref. [19],

respectively. Using the U 5f and O 2p, 2s orbital energies [18,19], we calculated the

set of the tij transfer integrals.

With these data, we calculated numerically the km1;m2lHeff lm01;m0

2l matrix

elements of the effective exchange Hamiltonian, which are presented in Table 1.

Each group of matrix elements can be associated with a definite equivalent spin

operator, which has the same matrix elements in the lm1;m2l basis set as those

obtained from equation (7). The symmetry of these spin operators corresponds to the

local symmetry of the U4þ(A)–U4þ(B) exchange pair (D2h; Fig. 6). Collecting all

the spin operators listed in Table 1, we arrive at the total effective spin Hamiltonian,

in which the spin terms are divided into several groups:

H ¼ A0 þ H1 þ H2 þ H3 þ H4 ð11Þ

Here A0 is the spin-independent part, A0 ¼ 256:4 cm21: H1 represents the one-

center spin terms, which may be called exchange-induced zero-field splitting of the

G5 state

H1 ¼ D½ðS zAÞ

2 þ ðS zBÞ

2 þ E½ðS xAÞ

2 2 ðSyAÞ

2 þ ðS zBÞ

2 2 ðS zBÞ

2 ð12ÞQ2

where D ¼ 24:6 cm21 and E ¼ 5:2 cm21: H2 is the bilinear part of the spin

Hamiltonian

H2 ¼ JxS xAS x

B þ JySyAS

yB þ JzS

B ð13Þ

with the exchange parameters Jx ¼ 14:7; Jy ¼ 22:1; and Jz ¼ 18:8 cm21: The

orientations of the local spin quantization axes x, y, and z of the U4þ(A)–U4þ(B)

pair is shown in Fig. 9. The bilinear part of the 5f2–5f2 superexchange interaction

has therefore a pronounced exchange anisotropy and corresponds to an

Fig. 8. Indirect interactions of 5fi(A) and 5fj(B) orbitals via the 2p and 2s orbitals of twobridging oxygen atoms.

ARTICLE IN PRESS

Table 1. Non-zero km1;m2lHeff lm3;m4l matrix elements of the effective exchangeHamiltonian and the equivalent spin operators (all parameters are in cm21)

km1;m2lHeff lm01;m

M1 m2 m01 m0

2 Equivalent spin operator

21 21 21 21 247.29 C þ c1½ðSzAÞ

2 þ ðS zBÞ

1 1 1 1 247.29 þc2S zAS z

B þ c3ðSzAÞ

2ðS zBÞ

2 1 0 2 1 0 262.13 C ¼ 256:390 21 0 21 262.13 c1 ¼ 25:740 1 0 1 262.13 c2 ¼ 18:811 0 1 0 262.13 c3 ¼ 1:770 0 0 0 256.39

21 1 21 1 284.9221 1 21 1 284.92

21 21 21 1 þ5.17 ½ðSþAÞ2 þ ðS2A Þ

2½a1 þ b1ðSzBÞ

21 21 1 21 þ5.17 þ½ðSþB Þ2 þ ðS2B Þ

2½a1 þ b1ðSzAÞ

21 1 21 21 þ5.17 a1 ¼ 2:691 1 1 1 þ5.17 b1 ¼ 0:111 21 21 21 þ5.171 21 1 1 þ5.171 1 21 1 þ5.17

2 1 1 1 21 þ5.380 0 1 0 þ5.380 21 0 1 þ5.381 1 0 21 þ5.380 0 21 0 þ5.38

21 0 0 21 þ18.40 ðSþAS2B þ S2A SþB Þ½a2 þ b2ðSzA þ Sz

0 21 21 0 þ18.40 a2 ¼ 9:180 1 1 0 þ18.40 b2 ¼ 0:191 0 0 1 þ18.40

21 1 0 0 þ18.370 0 21 1 þ18.370 0 1 21 þ18.371 21 0 0 þ18.37

21 21 0 0 24.04 ðSþASþB þ S2A S2B Þ½a3 þ b3ðSzA 2 S z

0 0 21 21 24.04 a3 ¼ 21:840 0 1 1 24.04 b3 ¼ 20:171 1 0 0 24.04

21 0 0 1 23.700 21 1 0 23.700 1 21 0 23.701 0 0 21 23.70

continued

ARTICLE IN PRESS

antiferromagnetic spin coupling. The latter fact is consistent with the experimentally

observed antiferromagnetic ordering in UO2.

The magnitude of the exchange parameters is consistent with a rather high

transition point temperature, TN ¼ 30:8 K: The term H3 corresponds to four-spin

exchange interactions

H3 ¼ j1SxASx

B½ðSzAÞ

2 þ ðSzBÞ

2 þ j2SyAS

yB½ðS

2 þ ðSzBÞ

2 þ j3SxASx

BðSzASz

þ j4SyAS

BÞ ð14Þ

with j1 ¼ 20:3; j2 ¼ 0:4 cm21, j3 ¼ 2j2; and j4 ¼ 2j1: The last term H4 represents

biquadratic interactions, which are written as

H4 ¼ q1OAð1ÞOB

ð1Þ þ q2Oð2ÞA Oð2Þ

B þ q3Oð3ÞA Oð3Þ

B þ q4½Oð1ÞA Oð2Þ

B þ Oð2ÞA Oð1Þ

B ð15Þ

where q1 ¼ 1:8; q2 ¼ 2:6; q3 ¼ 0:6; and q4 ¼ 20:2 cm21. Here OA;BðnÞ are the

components of the quadrupole operator expressed in terms of spin operators Ska of

the effective spin S ¼ 1:

Table 1. continued

km1;m2lHeff lm01;m

M1 m2 m01 m0

2 Equivalent spin operator

21 21 1 1 þ1.99 b4½ðSþAÞ

2ðSþB Þ

2 þ ðS2A Þ2ðS2B Þ

2; b4 ¼ 0:501 1 21 21 þ1.99

21 1 1 21 þ3.13 b5½ðSþAÞ

2ðS2B Þ

2 þ ðS2A Þ2ðSþB Þ

2; b5 ¼ 0:781 21 21 1 þ3.13

Fig. 9. Spin quantization axes x, y, and z of the neighboring U4þ(A)–U4þ(B) exchange pagein the UO2 crystal lattice.

ARTICLE IN PRESS

Oð1Þk ¼ ðS z

k Þ2 2 SðS þ 1Þ=3 ð16aÞ

Oð2Þk ¼ ðS x

k Þ2 2 ðS

2 ð16bÞ

Oð3Þk ¼ ðS x

k Syk þ S

yk S x

k Þ ð16cÞ

where k ¼ A;B:The four-spin terms H3 and H4 are of special interest, since they have no analogs

in usual spin-only S ¼ 1 exchange systems. It is important that the spin operators

entering H4 are represented by products of the components of single-ion quadrupole

operator OkðiÞð16Þ and thus they correspond to a quadrupole–quadrupole (Q–Q)

interaction. This interaction can be called exchange-induced Q–Q interaction. By

contrast to 3d exchange systems, in which biquadratic terms are generally very small

as compared to the leading isotropic exchange interaction, the biquadratic Q–Q

interaction in UO2 is quite pronounced. Indeed, the q1 and q2 parameters at the

OAð1ÞOB

ð1Þ and Oð2ÞA Oð2Þ

B quadrupole operators are about 2 cm21; being multiplied by

the number of neighboring metal centers (12 in the UO2 lattice), they can give a

considerable contribution to the effective molecular field causing the quadrupolar

orbital ordering. These results indicate that the U4þ–U4þ superexchange interaction

in UO2 differ considerably from that described by the conventional isotropic

Heisenberg model. It has a strong influence on the character of the magnetic

ordering in UO2. This is discussed below in more detail.

3. JAHN–TELLER EFFECT FOR THE GROUND G5 STATE

OF U41 IONS: CALCULATIONS OF VIBRONIC COUPLING

CONSTANTS FOR THE eg; t2gð1Þ AND t2gð2Þ JAHN–TELLER MODES

Now we estimate the strength of the JT effect in UO2. For this, we calculated the

linear vibronic coupling constants G5 ^ eg; G5 ^ t2gð1Þ; and G5 ^ t2gð2Þ: Calcu-

lations were carried out using a semiempirical approach for CF calculations for 5f

electrons developed in [14,15]. In this approach, the intraatomic interactions within

the 5fN shell are treated parametrically, while the CF splitting is calculated in terms

of a microscopic model with taking into account the actual electronic structure of the

5fN shell and ligand’s molecular orbitals. This approach provides reasonable results

for the CF energies in many tetra-and pentavalent actinide compounds, such as UX62

(X ¼ F, Cl, Br, and I), UCl622, UBr6

22, NpO2þ, U(NCS)8

42, UCl4, UO2, NpO2, etc.

[14,15]. In particular, it gives good results for CF energies in UO2. The calculated

CF energies and their comparison with the experimental data are given in Table 2.

To estimate the vibronic coupling constant, we performed CF calculations for the

UO8 cluster distorted along the eg, t2g(1) and t2g(2) Jahn–Teller modes (Fig. 10). A

similar approach was previously applied for calculations of vibronic coupling

constants in a series of UX62 octahedral complexes (X ¼ F, Cl, Br, and I) and in the

U(NCS)842 cubic complex in compound (NEt4)4U(NCS)8 [14].

ARTICLE IN PRESS

For each mode, the amplitude of the distortion corresponds to the actual

displacement of oxygen atoms, 0.014 A. Then the vibronic coupling constants C

were calculated from the splitting energy DE of the ground G5 level.

Fig. 10. The Jahn–Teller active modes in the UO8 cubic cage. Only one of three components(Qxy) is shown for the t2g(1) and t2g(2) modes.

Table 2. Calculated and experimental CF splitting of the ground 3H4 multiplet of U4þ forUO2

CF energy (cm21)

Level Experimental [6] Calculated [14]

G5 0 0G3 1211 1288G4 1345 1600G1 1410 1774

ARTICLE IN PRESS

The largest vibronic coupling constants are found for the eg and t2g(2) modes,

while for the t2g(1) mode it is about one order of magnitude smaller. From these data

we can also estimate the JT stabilization energy per U atom, 10 cm21 for eg, 8 cm21

for t2g(2), and only 0.5 cm21 for t2g(1). All these energies are smaller than the

thermal energy at the transition point, TN=kB < 20 cm21:

4. DISCUSSION: THE RELATIONSHIP BETWEEN THE 5f2–5f2

SUPEREXCHANGE INTERACTION AND JAHN–TELLER EFFECT

Using these results, now we discuss the competition between the 5f2–5f2

superexchange interaction and the cooperative JT effect in the phase transition in

UO2. We discuss first the type of the lattice distortions in UO2, which can be

expected from the cooperative JT effect. It is well known that there are two types of

the minima in the linear T ^ ðe þ tÞ JT problem, three equivalent tetragonal eg

minima and four trigonal t2g minima. Since the local eg and t2g(2) modes have the

largest vibronic coupling constants, one can expect that lattice distortions in UO2

below TN should be related to these JT modes. The corresponding distortions of UO8

cubic cage are shown in Fig. 11. In principle, due to the cooperative character of JT

distortions in the extended crystal lattice, local distortions can be of the mixed type

eg þ t2gð1Þ þ t2gð2Þ; but at least one of the JT components with the largest coupling

constant should be present. It is therefore surprising that the actual triple-k distortion

of the UO8 cage does not contain eg or t2g(2) JT components.

Instead, the actual distortion of the UO8 cage in UO2 just corresponds to the

trigonal JT minimum of the t2g(1) mode, which has the vibronic coupling constant

being more than 20 times smaller than those for the eg or t2g(2) modes. In fact, the

distortion shown in Fig. 11 can be represented by the sum ðQð1Þxy þ Qð1Þ

zx þ Qð1Þyz Þ=

ffiffi3

t2g(1) coordinates combined with some rotation around the k111l diagonal.

Fig. 11. Distortions of the UO8 cubic cage in a) eg tetragonal minimum (with the JTstabilization energy of DEJT < 8 cm21 per U atom) and b) t2g(2) trigonal JT minimum(DEJT < 10 cm21).

ARTICLE IN PRESS

This is a strong indication of the fact that the orbital ordering and lattice

distortions are not due to the cooperative JT effect. Indeed, the stabilization energy

per U atom for the actual 3-k type distortion shown in Fig. 12 is as small as

0.5 cm21, which is much smaller than the characteristic thermal energy at the

transition point temperature, TN=kB < 20 cm21: Therefore, the lattice distortions in

UO2 seem to be induced by an orbital ordering related to the exchange interactions

and/or Q–Q interaction. This is due to the fact that the degeneracy of a ground state

with an even number of electrons per magnetic ion can be lifted by magnetic or

quadrupolar ordering at low temperatures. There are therefore three sources of the

Q–Q interaction in UO2, (a) the direct Q–Q interaction which is generally small and

can safely be neglected, (b) the indirect Q–Q interaction related to the spin-lattice

interaction and the cooperative JT effect [7,8], and (c) the exchange-induced Q–Q

interaction, which is described by the H4 term (15) of the effective spin Hamiltonian

of the 5f2–5f2 superexchange interaction. The latter mechanism of the Q–Q

interaction seems to be the most important, since the mechanism (b) should lead to

the eg or t2g(2) types of the lattice distortion, which are not actually observed in UO2.

This is also consistent with a large value of the q1 and q2 parameters at the

Oð1ÞA Oð1Þ

B and Oð2ÞA Oð2Þ

B components of the Q–Q interaction. It is important to note that

the JT-induced Q–Q interaction (b) and the exchange-induced Q–Q interaction (c)

can be represented by quite different quadrupole components, as evidenced from the

comparison of the Q–Q operator (b) obtained in [7,8] and the Q–Q operator H4,

equations (15) and (16a–c). As a result, the orbital ordering caused by the Q–Q

interactions (b) and (c) may be quite different resulting in different types of the

lattice distortion.

The fact that the phase transition in UO2 has the first-order character and the

ordered magnetic moment of 1.74 mB is considerably lower than the paramagnetic

moment (about 3 mB) is qualitatively consistent with the ratio of the strength of the

bilinear and biquadratic parts of the effective spin Hamiltonian (8) of the 5f2–5f2

Fig. 12. The actual distortion of each UO8 cubic cage in UO2 is represented by the ðQð1Þxy þ

Qð1Þzx þ Qð1Þ

yz Þ=ffiffi3

plinear combination of the components of the t2g(1) mode combined with

some rotation around the k111l space diagonal. This distortion corresponds to the trigonalminimum of the t2g(1) JT mode with the stabilization energy of about 0.5 cm21 per U atom.

ARTICLE IN PRESS

superexchange interaction. Quadrupole interactions compete with exchange

interactions and can completely suppress the ordered magnetic moment, although

the quadrupole order can always coexist with the magnetic order [8]. According to

the Allen model [8], the first-order phase transition in UO2 occurs in a rather narrow

range of the ratio between the strength of Q–Q and exchange interactions (between

2/7 and 1/2). This is compatible with our estimate given by the spin Hamiltonian (8).

Although the quantitative details of the competition between the spin and orbital

ordering are not clear yet, our results allow one to conclude that the 5f2–5f2

superexchange interaction dominates in the mechanism of the phase transition. The

Q–Q interaction induced by the 5f2–5f2 superexchange interaction is responsible

for the orbital ordering which causes the lattice distortions of the t2g(1) type, which

cannot be ascribed to the cooperative JT effect due to a very low vibronic coupling

constant. Distortion of the oxygen sublattice in UO2 is therefore a secondary effect.

These findings can serve as a basis for the development of the microscopic model

explaining the origin of the 3-k character of the magnetic ordering and lattice

distortions.

5. CONCLUSIONS AND FURTHER OUTLOOK

We have analyzed the interplay between the superexchange interaction and the

cooperative JT effect in UO2. Our calculations show that the 5f2–5f2 superexchange

interaction in UO2 is anisotropic and essentially non-Heisenberg. The biquadratic

part of the spin Hamiltonian is comparable in magnitude with the bilinear part, and it

corresponds to a strong exchange-induced quadrupole–quadrupole interaction

between 5f states of U4þ ions. The linear vibronic coupling constants G5 ^ eg;G5 ^ t2gð1Þ; and G5 ^ t2gð2Þ are calculated. Despite G5 ^ eg and G5 ^ t2gð2Þ are

found to be much larger than G5 ^ t2gð1Þ; the actual lattice distortion in UO2 is of the

pure t2g(1) type. Exchange interactions between U4þ ions play the primary role in

the mechanism of the phase transition in UO2. The magnetic ordering and related

orbital ordering and lattice distortions are determined by the superexchange

interaction. The most important eg and t2g(2) Jahn–Teller distortions are completely

suppressed by the competing exchange interactions. Although the complete solution

of the problem is not yet obtained, our results elucidate a number of points, which

remained unclear before. The most important finding is that there is a strong

exchange-induced Q–Q interaction in UO2, which is represented by the biquadratic

part of the 5f2–5f2 superexchange interaction. This Q–Q interaction, which is

different from the Q–Q interaction related to the cooperative JT effect and spin-

lattice interaction, seems to be responsible for the orbital ordering and a

considerable reduction of the ordered magnetic moment. The development of a

microscopic model of the phase transition in UO2 for a consistent explanation of the

origin of the triple-k magnetic structure, lattice distortions, and orbital ordering

below TN will therefore be the next step.

ARTICLE IN PRESS

ACKNOWLEDGEMENTS

Financial support by the Belgian Science Foundation and Flemish Government

under the Concerted Action Scheme, the ESF programme on Molecular Magnets,

the Russian Foundation for Basic Research (grant No. 01-03-32210), and the INTAS

Grant 00-00565 are gratefully acknowledged.

REFERENCES

[1] B. C. Frazer, G. Shirane, D. E. Cox and C. E. Olsen, Phys. Rev., 1965, 140, A1448; B. T. M. Willis

and R. I. Taylor, Phys. Lett., 1965, 17, 188.

[2] R. A. Cowley and G. Dolling, Phys. Rev., 1968, 167, 464.

[3] K. Sasaki and Y. Obata, J. Phys. Soc. Jpn, 1970, 28, 1157.

[4] J. Faber and G. H. Lander, Phys. Rev. B, 1976, 14, 1151.

[5] P. Burlet, J. Rossat-Mignod, S. Quezel, O. Vogt, J. C. Spirlet and J. Rebizant, J. Less-Common Met.,

1986, 121, 121.

[6] G. Amoretti, A. Blaise, R. Caciuffo, J. M. Fournier, M. T. Hutchings, R. Osborn and A. D. Taylor,

Phys. Rev. B, 1989, 40, 1856.

[7] S. J. Allen, Phys. Rev., 1968, 166, 530.

[8] S. J. Allen, Phys. Rev., 1968, 167, 492.

[9] S. Kern, C.-K. Loong and G. H. Lander, Phys. Rev. B, 1985, 32, 3051.

[10] W. Kopmann, F. J. Litterst, H.-H. Klauß, M. Hillberg, W. Wagener, G. M. Kalvius, E. Schreier,

F. J. Burghart, J. Rebizant and G. H. Lander, J. Alloys Comp., 1998, 271–273, 463.

[11] L. Paolasini, R. Caciuffo, B. Roessli and G. H. Lander, Physica B, 1998, 241–243, 681; L. Paolasini,

R. Caciuffo, B. Roessli and G. H. Lander, Phys. Rev. B, 1999, 59, 6867.

[12] K. Ikushima, S. Tsutsui, Y. Haga, H. Yasuoka, R. E. Walstedt, N. M. Masaki, A. Nakamura, S. Nasu

and Y. Onuki, Phys. Rev. B, 2001, 63, 104404.

[13] P. W. Anderson, Phys. Rev., 1959, 115, 2; P. W. Anderson, in Magnetism I (eds G. T. Rado and

H. Suhl), Academic Press, New York, 1963.

[14] V. S. Mironov and S. P. Rosov, Phys. Status Solidi (b), 1992, 169, 463; V. S. Mironov and S. P. Rosov,

Phys. Status Solidi (b), 1992, 170, 199; V. S. Mironov and S. P. Rosov, Phys. Status Solidi (b), 1992,

170, 509.

[15] V. S. Mironov and S. P. Rosov, J. Alloys Comp., 1992, 189, 163–168.

[16] V. S. Mironov, L. F. Chibotaru and A. Ceulemans, Phys. Rev. B, 2003, 67, 014424.

[17] M. Wolfsberg and L. Helmholz, J. Chem. Phys., 1952, 20, 827.

[18] L. L. Lohr and P. Pyykko, Chem. Phys. Lett., 1979, 62, 333.

[19] J. W. Lauher, M. Elian, R. H. Summerville and R. Hoffmann, J. Am. Chem. Soc., 1976, 98, 3219.

ARTICLE IN PRESS

Author QueriesJOB NUMBER: 3111

Title: First-order Phase Transition in UO2

Q1 Please check the Running Head.

Q2 Kindly check the RHS of equation (12).

ARTICLE IN PRESS

First-order Phase Transition in UO 2 : The Interplay of the 5f 2 –5f 2 Superexchange Interaction...

Documents