Valence-Bond Crystal, and Lattice Distortions in a Pyrochlore Antiferromagnet with Orbital Degeneracy

0.43 MB·English

by  S. Di Matteo

Valence-Bond Crystal, and Lattice Distortions in a Pyrochlore Antiferromagnet with
Orbital Degeneracy.
S. Di Matteo,1,2 G. Jackeli,3,4, and N. B. Perkins1,5
∗
1Laboratori Nazionali di Frascati INFN, via E. Fermi 40, C.P. 13, I-00044 Frascati (Roma) Italy
2Dipartimento di Fisica, Universit`a di Roma III, via della Vasca Navale 84, I-00146 Roma Italy
3Ecole Polytechnique F´ed´erale de Lausanne, Institute for Theoretical Physics, CH-1025, Lausanne, Switzerland
5 4Institut Laue Langevin, B. P. 156, F-38042, Grenoble, France
0 5Bogoliubov Laboratory of Theoretical Physics, JINR, 141980, Dubna, Russia
0 (Dated: February 2, 2008)
2
We discuss the ground state properties of a spin 1/2 magnetic ion with threefold t2g orbital
n degeneracy on a highly frustrated pyrochlore lattice, like Ti3+ ion in B-spinel MgTi2O4. We for-
a
mulateaneffectivespin-orbitalHamiltonianandstudyitslowenergysectorbyconstructingseveral
J
exact-eigenstatesinthelimit ofvanishingHund’scoupling. Wefindthatorbitaldegreesoffreedom
2
modulate the spin-exchange energies, release the infinite spin-degeneracy of pyrochlore structure,
1
anddrivethesystemtoanon-magneticspin-singletmanifold. Thelatterisacollectionofspin-singlet

dimers and is, however, highly degenerate with respect of dimer orientations. This “orientational”
]
l degeneracy is then lifted by a magneto-elastic interaction that optimizes the previous energy gain
e by distorting the bonds in suitable directions and leading to a tetragonal phase. In this way a va-
-
r lencebondcrystalstateisformed, throughthecondensation ofdimersalong helicalchainsrunning
st aroundthetetragonalc-axis,asactuallyobservedinMgTi2O4. Theorbitallyorderedpatterninthe
. dimerizedphaseispredictedtobeofferro-typealongthehelicesandofantiferro-typebetweenthem.
t
a Finally, through analytical considerations as well as numerical ab-initio simulations, we predict a
m possible experimental tool for the observation of such an orbital ordering, through resonant x-ray
scattering.
-
d
n PACSnumbers: 75.10.Jm,75.30.Et
o
c
[ I. INTRODUCTION. plesarevanadiumd2 compoundswithfrustratedlattices,

where the orbital order is shown to induce a spin-singlet
1
ground state without any long-range magnetic order for
v
9 A frustrated antiferromagnet is characterized by the triangular lattice,7 or a spin ordered one, for pyrochlore
8 topology of underlying lattice and/or by the presence of lattice.8
2 competinginteractionsthatprecludeeverypairwisemag-
1 netic interaction to be satisfied at the same time. Such In this paper we study a system with threefold-
0 physicalsystemshaverecentlyattractedawideinterest,1 orbitally-degenerate S = 1/2 magnetic ions in a corner-
5
due to the concept of ”macroscopic”degeneracy,namely sharing tetrahedral (pyrochlore) lattice. Our work is
0
the existence of a huge number of states with the same motivated by the very recent synthesis9 and interesting
/
t energy. This degeneracy in the ground state manifold experimental data on B-spinel MgTi O ,9,10 a d1-type
a 2 4
m can be usually removed through a large variety of ef- transition metal oxide. Here magnetically active Ti3+
fects, like order-out-of-disordermechanisms2 by thermal ions forma pyrochlorelattice and arecharacterizedby a
-
d or quantum fluctuations, or spin-Peierls like symmetry- single electron in a d-shell. The crystal field of the oxy-
n lowering transitions.3,4 In some highly frustrated mod- genoctahedronsurroundingeachTiionsplitsthisd-level
o els, like nearest-neighbor Heisenberg antiferromagnet on intoahighenergydoublete andalowenergytriplett
g 2g
c pyrochlore lattice, the former mechanisms are inactive5 in which d-electron resides.11 The ground state of a spin
:
v and such a spin system wouldremain liquid down to the one-half Ti3+ ion is thus threefold-orbitally-degenerate.
i lowesttemperatures,6unlessmagnetoelasticcouplingsin- The recent experiments on MgTi O have shown that
X 2 4
duce a symmetry-breaking transition. Yet, in real com- this compoundundergoesa metal-to-insulatortransition
r
a pounds geometrical frustration can also be partially or on cooling below 260 K, with an associated cubic-to-
fully released when magnetic ions forming a frustrated tetragonal lowering of the symmetry.9 At the transition
latticepossessanorbitaldegeneracy. Typicallythishap- the magnetic susceptibility continuously decreases and
pens in transition metal ions with orbitally degenerate saturates, in the insulating phase, to a value which is
partly-filled d-levels. The physical behavior of such sys- anomalously small for spin 1/2 local moments: for this
tems is expected to be drastically different from that of reason the insulating phase has been interpreted as a
purespinmodels,astheoccurrenceofanorbitalordering spin-singlet phase. Subsequent synchrotron and neutron
(OO) can modulate the spin exchange and lift the geo- powder diffraction experiments have revealed that the
metrical degeneracy of the underlying lattice. Indeed, a low-temperature crystal structure is made of alternating
variety of novel phases driven by orbital degrees of free- short and long Ti-Ti bonds forming a helix about the
dom can be stabilized in this way. Among known exam- tetragonal c-axis.10 These findings have suggested a re-
2
movalofthepyrochloredegeneracybyaone-dimensional finally,inSectionVweanalyzethe crystalstructureand
(1D) helical dimerization of the spin pattern, with spin- the orbital symmetry of the tetragonal phase, deriving
singlets (dimers) located at shortbonds. This phase can the analytic expression for the measurable quantities in
be regarded as a valence bond crystal (VBC) since the a RXS experiment. We also perform a series of numeri-
long-range order of spin-singlets extends throughout the calab-initio simulationsby means of the finite difference
whole pyrochlore lattice. method, implemented in the FDMNES package,18 in or-
The aim of the present work is to discuss the micro- dertoproposeapossibleexperimenttodetecttheorbital
scopic mechanism behind the realization of this unusual pattern. To faciliate the reader, two Appendices, A and
and intriguing VBC structure on the pyrochlore lattice. B, are given for technical details.
Wearguethatthekeyroleinthismechanismisplayedby Part of the results presented here were already an-
orbitaldegeneracyand,remarkably,onedoesnothaveto nounced in a previous short communication.19
invoke any additional exotic interaction to stabilize such
a novel phase, as necessary for purely spin models. In-
deed,due tothe orbitaldegeneracy,varioustypeofspin-
II. MODEL AND FORMALISM.
singlet phases, such as resonating valence bond (RVB)
and VBC states canbe formed evenfor unfrustrated cu-
bic lattice.12 A. Effective spin-orbital Hamiltonian.
The orbital degree of freedom does modulate spin ex-
change energies, thus removing the infinite spin degen- Herewediscussthesuperexchangespin-orbitalHamil-
eracy, characteristic of pyrochlore structures, and drives tonianforthreefoldorbitally-degenerated1-ionsonapy-
thesystemtoanon-magneticspin-singletstate. Thislat- rochlore lattice. We assume that the insulating phase
ter is a collections of spin-singlet dimers with a residual of MgTi2O4 is of Mott-Hubbard type and can, thus, be
macroscopic degeneracy of orientational character. The described by the Kugel-Khomskii model.17 We consider
residualdegeneracyisthenliftedbyamagnetoelasticin- the system in its cubic structure and look for possible
teraction, that optimizes superexchange energy gain by instabilities towards symmetry reductions.
distortingeachtetrahedroninsuchawayastoleadtothe The relevant electronic degrees of freedom are de-
experimentallyobservedhelicalpatterninMgTi O . We scribed by spin S = 1/2 and pseudospin τ = 1 op-
2 4
also find that the helical dimerized state is accompanied erators. This latter labels the orbital occupancies of
by a peculiar orbital pattern in which orbital order is of t orbitals, (αβ = xy , xz , yz ), with the cor-
2g
ferro-typealongthehelicesandofantiferro-typebetween respondence: |τz i= 1| i y|z ,i τz| =i 0 xy , and
them. τz =1 xz . Our p−aram→et|ersiarethe near→est|-neiighbor
Moreover,wecanshowhowtoidentify suchanorbital (NN)el→ect|ronihoppingmatrixtˆ,definedintheAppendix
orderingexperimentally,bymeansofresonantx-rayscat- A Eq. (A1), the Coulomb on-site repulsions U1 (within
tering (RXS). In fact, differently from what happens in the same orbital)and U2 (among different orbitals), and
thecaseofmanganites,13,14,15,16 wheretheratiobetween the Hund’s exchange, JH. For t2g wavefunctions the re-
the OO-induced and Jahn-Teller (JT) induced effects is lation U1 = U2+2JH holds due to rotational symmetry
about1/10inamplitude,forMgTi O thisratioisabout in real space.
2 4
1/3, due to the less distorted oxygen octahedra, as well Considering the pyrochlore structure the effective
as to the presence of t electrons instead of e , that Hamiltoniancanbesimplifiedbyretainingonlythelead-
2g g
couple less to the oxygen environment. Such a reduced inghoppingparameter,t,duetotheNNddσ overlapand
ratio allows a subtle interference effect between OO and neglectingthesmallerddπ andddδ contributions. Thisis
JTterms,thatgivesrisetoanincreaseofthe signalbya justifiedbythe factthatthe transferintegralsduetothe
factorofabout1.7. Wehaveperformedadetailedanalyt- π and δ bonding are, respectively, around 1/10 and 1/3
ical analysis as well as an ab-initio numerical simulation ofthatofσ bonding.20 The majoradvantageofthissim-
to suggest some experiments in this direction. plification,isthatddσ overlapinαβ plane connectsonly
In more detail, the paper is organized as follows: in the corresponding orbitals of the same αβ type. Thus,
SectionIIwederiveaneffectiveKugel-Khomskii17model the total number of electrons in each orbital state is a
Hamiltonian, and deduce its possible low-energy states conservedquantityand,therefore,the orbitalpartofthe
on the pyrochlore lattice. We then introduce the spin- effective Hamiltonian (A2) becomes Ising-like (or better,
singlet ground state manifold and discuss its degeneracy aPottsZ3-like),makingpossibletogetanalyticalresults
due to the dimer coverings of the pyrochlore lattice. In about the ground-state. The spin-orbital Hamiltonian is
Section III we consider the magnetoelastic interactions presented in Appendix A in the most general form [see
of the pyrochlore lattice and analyze the effects of the Eq.(A2)], when also ddπ and ddδ hopping elements are
couplingofbonddistortionswiththeorbitallydrivenex- considered. Hereweintroducethesimplifiedversionwith
changemodulation,firstqualitatively,andthenquantita- only ddσ hopping terms:
tively, underlining the influence of the spin-singlet phase
onthestrengthofsuchadistortion. InSectionIVtheef- H = J S~ S~ +3/4 O (1)
eff − 1X(cid:2) i· j (cid:3) ij
fectofanappliedmagneticfieldisbrieflydiscussed,and,
ij
h i
3
+J S~ S~ 1/4 O +J S~ S~ 1/4 O˜ (a) (b)
2X(cid:2) i· j − (cid:3) ij 3X(cid:2) i· j − (cid:3) ij
ij ij
h i h i
where the sum is restricted to the NN sites on the py-
rochlorelattice. Theorbitalcontributionsalongthebond
FIG. 1: Orbital arrangements on a bond: (a) bond b0 with
ij in αβ-plane is given by strong AFM coupling ∼ J, (b) bond b1 with weak FM cou-
pling ∼ηJ.
O = P (1 P )+P (1 P )
ij i,αβ j,αβ j,αβ i,αβ
− −
O˜ = P P , (2)
ij i,αβ j,αβ
Thus,onlyb -andb -bondshaveabindingenergyand
0 1
where P = Taa stands for the projector on orbital for anygivenorbitalconfigurationonthe pyrochlorelat-
i,αβ i
state a = αβ and is defined in Appendix A. tice the Hamiltonian (1) is just a linear combination of
| i | i
The first and second terms in Heff (1) describe the Hb0 and Hb1. Moreover, the explicit form of Eqs. (3)
ferromagnetic (FM) J = t2/(U J ) and the anti- and (4) implies that, in the limit η 0, a set of exact
ferromagnetic (AFM) 1J = t2/(U2 −+HJ ) interactions, eigenstates of (1) can be constructed→: in fact, orbital in-
2 2 H
respectively, and are active only when the two sites in- teractions are already diagonal, and, in this limit, the
volved are occupied by different orbitals. The last term only Heisenberg term that survives is that of Hb0. This
is AFM, with J = 4t2 2/(U +J )+1/(U +4J ) , latter, by definition, is active only along the direction
and is non-zero3only3wh(cid:2)en th2e twoHsites have2 the sHam(cid:3)e connecting the two sites with equally filled αβ orbitals.
orbital occupancy. Parameters that play a role in the This implies that the only ”infinite” configuration of in-
Hamiltonian (1) are: t t 0.32 eV, J 0.64 eV teractingspinsisobtainedwhenthespin-interactionsact
σ H
and U 4.1 eV.20,21 Th≡us η≃=J /U 0.15≃ 1 and, alongthesamedirection,i.e.,theAFMHeisenbergchain,
2 H 2
just ino≃rderto presentthe results ina m≃oretra≪nsparent which is exactly soluble.22
form, we expand the exchange energies around η = 0. Inthenextsubsection,startingfromanisolatedtetra-
WegetJ J(1+η),J J(1 η)andJ 4J(1 2η) hedron we discuss possible orbital arrangements on the
1 2 3
whereJ =≃t2/U 25me≃Vrepr−esentstheo≃verallen−ergy pyrochlorelatticeandprovidethesolutionofcorrespond-
2
scale. ≃ ing spin Hamiltonian.
The realization of the spin-orbital model (1) on the
pyrochlore lattice has an immediate and important con-
sequence, namely, that only some bonds can contribute B. Possible phases on a pyrochlore lattice and
to the energy, depending on their orbital configuration. their energetics
Indeed, every bond ij in the αβ plane has zero energy
gain unless at least one of the two sites i and j has an It is possible to show that only three energetically in-
occupiedorbitalofαβ kind. The strength,aswellasthe equivalent tetrahedrons can be singled out of the four
sign, of spin-exchange energy associated with two NN b bonds introduced above.19 Topologically, they can be
n
sites i and j depends only on their orbital occupations characterized by the number of strongest b -bonds and
0
and the direction of the ij bond. We can thus classify classified in three following types [pictorially shown in
tetrahedral bonds in four types: Fig. 2]: A-type tetrahedra, with two b bonds, B-type
0
(i) b0, when both ions at sites i and j of the generic tetrahedra with one b0 bond and C tetrahedra, with no
αβ-planehaveαβ orbitaloccupancy(Fig. 1a). Itischar- b -bonds. Notice that, whatever the orbital configura-
0
acterized by a Hamiltonian with strong AFM exchange tionis,everysingletetrahedronmustobeytheconstraint
J: 2n +n = 4. This implies that A tetrahedra have no
∼ b0 b1
b -bonds, B-type tetrahedra are characterized by 2 b
1 1
Hb0 =−J(1−2η)(1−4S~i·S~j). (3) bonds and, for C tetrahedra, there must be 4 b1-bonds.
We can further classify B-type tetrahedra, according to
(ii) b1, if the two sites of bond ij in αβ-plane are oc- the choice of the two orbitals on the tetrahedron bond
cupied by one αγ and one αβ orbitals, γ = β(Fig. 1b). opposite to the b -bond, into: B , B and B (see Fig.
6 0 1 2 3
This bond has a weak FM exchange ηJ andthe corre- 2) where, respectively, the bond opposite to b one is of
∼ 0
sponding Hamiltonian is: b , b and b type. We stress again that the number
1 2 3
of b and b bonds is not relevant as far as the Hamil-
2 3
Hb1 =−J(1+η/2+2ηS~i·S~j). (4) tonian (1) is considered: they acquire importance only
when magnetoelastic correlations among bonds will be
(iii) b , with the two sites of bond ij in αβ-plane oc- introduced, in the next section, as they allow to distin-
2
cupied by one αγ and one βγ orbitals. In this case there guish topologically the various configurations in the py-
is no energy contribution: Hb2 =0. rochlore lattice. A, B, and C tetrahedra are the bricks
(iv) b , if both sites of bond ij in αβ-plane are occu- that allow to build the orbital pattern throughout the
3
pied by two αγ (βγ) orbitals. Also this bond does not wholepyrochlorelattice. BecauseoftheIsing-formofor-
contribute to the energy: H =0. bitalinteractions,inthefollowingwecanfocussimplyon
b3
4
to different dimers) and we are led to an energy per site
A B1 B2 given by: EB = Eb0/2+Eb1 = −(3− 72η)J. Here Eb0(1)
is the energy of the bond b .
0(1)
z (III) FM order: In this case all tetrahedra are of
y
x C-type with four interacting b bonds and two b (or
1 3
one b and one b ) noninteracting bonds. Thus, all non
2 3
zero spin-exchanges are ferromagnetic and given by Eq.
= xy = xz = yz
B3 C (4). The ground state for this type of orbital ordering is
b 0 b 1 thus ferromagnetic and has an energy per site given by
b b E =2E = 2(1+η)J.
2 3 C b1 −
(IV)FrustratedAFM:Evenifthe”Ising”formofor-
bitals interaction implies that configurations with linear
superpositionof orbitals oneach site must have a higher
FIG.2: Orbitalandbond arrangements ona tetrahedronfor
energy,weconsiderforcompletenessthe casewhereeach
cases A,B1,2,3 and C discussed in thetext.
orbital is occupied by a linear superposition with equal
weight of the three orbitals: 1 (xy + xz + yz ). The
√3 | i | i | i
realization of this phase restores the full pyrochlore lat-
these three cases, relying on the fact that configurations
tice symmetry and, after averaging Eq. (1) over the
with a linear superposition of orbitals on each site must
orbital configurations on neighboring sites i and j, the
have a higher energy. We shall do only one exception
system is described by the spin Heisenberg Hamilto-
to study a case with a particular physical meaning, i.e.,
nian H /J = [ 5/9+(4/9 16η/9)S~ S~ ], whose
that of a ”cubic” symmetry, where each site is occupied D Pij − − i· j
by a linear superposition with equal weights of the three ground-state energy per site is ED (1.89 0.89η)J.
≃ − −
orbitals 1 [xy + xz + yz ]. Here we borrowed the numerical evaluation for ground-
√3 | i | i | i state energy in a pyrochlore lattice from Ref. [23]:
If we now try to cover the whole pyrochlore lattice
(1/N) S~ S~ 0.5. Suchastateishighlyfrustrated
with these tetrahedra, we can have the following global Pij i· j ≃−
patterns: anditsgroundstateisaspinliquid.6 Itseemsworthwhile
to note that the energy in this phase is higher than that
(I)Heisenbergchains: Ifalltetrahedraofpyrochlore
of the other phases, as it does not exploit at all the po-
latticearecharacterizedbytwob bonds(i.e.,allsitesare
0 tential energy-gain contained in the orbital ordering.24
occupied by the same orbital), then the effective Hamil-
(V) Mixed (dimer) phase AC: It is ofcoursepossi-
tonian (1) can be mapped into a set of one dimensional
bletocoverthepyrochlorelatticealsobymeansofmixed
decoupled AFM Heisenberg chains that lie within the
configurations of tetrahedra. There are in principle infi-
plane of the chosen orbital. The only interactions are
nite possibilities in this sense, but those minimizing the
due to b -bonds and are described by the spin Hamilto-
0
energymustcontain,inaverage,atleastonesingletbond
nian (3). Thus, the ground-state energy per site can be
every tetrahedron, due to the big energy gain related to
evaluated exactly by using the results for an Heisenberg
chain,22 that give EA = 2.77(1 2η)J. the spin-singlet state in Hb0. While it is not possible to
− − have, in average, more than one spin-singlet bond every
(II) Dimer phase B: This state is made of only B-
tetrahedron,25 one can fill the whole pyrochlore lattice
type tetrahedra with one strong b -bond, and two inter-
0 by means of a mixed configuration made of alternated
mediate b -bonds. As all three B tetrahedra are ener-
1 i tetrahedra with two and zero b -bonds (A- and C- type
getically equivalent all possible coverings of pyrochlore 0
tetrahedra, respectively). This configuration is shown in
lattice by B tetrahedra have the same energy. When
i
Fig.4. For this type of orbital arrangement one can also
pyrochlorelatticeis coveredbyB -typestetrahedra(two
i
construct an exact ground state of corresponding spin
possible coverings are shown in Fig. 3), then each spin
Hamiltonian, in the limit of vanishing Hund’s coupling.
is engaged in one strong AFM b bond and two weak
0 Thegroundstateisgivenbydimerphasewherethereare
FM b -bonds. Such coverings form a degenerate mani-
1 two spin-singlets, located opposite each other, on strong
fold and the corresponding energy can be calculated as
b bonds of A-type tetrahedra while C-type tetrahedra
follows. In the limit η 0, the spin-only Hamiltonian 0
can be solved exactly, a→s it can be decomposed into a has no singlets. As in average,nb0 =1 and nb1 =2, this
configuration is degenerate with Dimer phase B, as far
sum of spin-uncoupled b bonds. In this case the en-
0 as only ddσ overlap is considered.
ergy minimum is reached when the Heisenberg term of
the b -bond is the lowest, i.e., for a pure quantum spin-
0
singlet (S~ S~ = 3/4). Remarkably, such spin-singlet
i j
(dimer) sta·tes, in−the limit η 0, are also exact eigen- C. Ground State Manifold
→
states of the full Hamiltonian (1). As η 1, the dimer
≪
state is stable against the weak FM interdimer interac- Energies of the possible phases on the pyrochlore lat-
tion. In this case the magnetic contribution along the ticecanbediscussedintermsofη,theonlyfreeparame-
FM b -bond is zero ( S~ S~ = 0 for i and j belonging teravailable. Forη =0thelowestground-stateenergyis
1 i j
h · i
5
H1=H’1
H2=K’2
(a) H H4=L’4
1 K4=M’4 K3=L’3
H3=M’3
H4=L’4 L’ M2=L’2
K3=L’3
M’
M2=L’2 M1=M’1 L’ LM2=L’L21=L’1 M1=M’1
2 M L2=M’2 L1=L’1 M
2 2 K’4=M4
L K’3=L3
2 M3=H’3 K’ L4=H’4 H’3=M3M4=K’4
K’4=M4 L4=H’4
K1=K’1 H’ H’2=K2
K’ H2=K’2 H1=H’1
1
K H
1 K1=K’1
K3=L’3
K4=M’4
H3=M’3 H4=L’4
(b)
1 FIG.4: Thecoveringsoftheunitcubiccellthroughthetetra-
hedraofA-typewithtwosingletsandC-typewithnosinglets.
Tetrahedra are labeled as in Fig. 6.
3 between different dimer states can not take place.
4 4 In this spin-singletmanifold the originalspindegener-
3 acy is removed. However there is still a remaining de-
generacy to be lifted. The degeneracy of B-manifold is
relatedtothefreedominthechoiceofthetwoorbitalson
the tetrahedron bond opposite to the one of the singlet.
2
Differentchoicesoftheseorbitalsgiverisetoinequivalent
1
dimercoveringpatternsofthepyrochlorelatticewithone
dimer per tetrahedron (see Fig. 3). This degeneracy is
given by the number of such dimer coverings and the
corresponding number of states can be estimated as fol-
FIG. 3: The ground state coverings of the unit cubic cell lows. In the B-manifold there is one singlet per tetrahe-
throughdimers. Locationsofsingletsarerepresentedbythick dron and each spin can be engaged in only one singlet.
links. Different numbers correspond to inequivalent tetrahe- Moreover when a singlet is located on a given tetrahe-
dra. (a) The helical dimerization pattern (indicated by ar- dron then each neighboring tetrahedronis left with only
rows) is formed by alternating short b0 and long b3 bonds, three possible choices for a singlet location. Thus the
Dimer phase B3. (b) One of the possible coverings of the number of such coverings grows with the system size as
cubic unit cell by B1B2 tetrahedra. B 3NT = √3N. Here NT = N/2 is the number of
N ∼
tetrahedraandwehaveignoredthecontributionscoming
from closed loops (hexagons) on the pyrochlore lattice.
thatofspin-singletdegeneratemanifold. Withincreasing The asymptotical determination of AC-manifold de-
η we find only one phase transition at η =2/11 0.18, generacy is a more difficult task. However, based on the
c
≃
fromspin-singletmanifoldtoaFMphase. Asη isabove simple arguments we can estimate its lower and upper
c
our estimated value of η 0.15, we can conclude that limit correctly. First, we divide the pyrochlore struc-
≃
the groundstatemanifold is givenby spin-singletphases ture in two sublattices formed by two differently ori-
spanned by degenerate dimer states of B or AC-types. ented tetrahedra: one sublattice is composed by A-type
Thisdegeneratemanifoldischaracterizedbyastaticpat- tetrahedronwithtwosingletslocatedoppositeeachother
tern of spin-singlets (dimers) throughout the whole py- and the other by C-type tetrahedron with no singlets.
rochlore lattice and, thus, is different from RVB state. Then consider an A tetrahedron with two singlets in αβ
Each dimer covering is frozen in an exact eigenstate of plane: it can be directly verified that the four NN tetra-
the Hamiltonian (1) for η = 0. For finite η the different hedra of A sublattice, connected to it by straight lines
dimer patterns are not connected by the Hamiltonian: in αβ plane, will have only two possible orbital choices.
thebondcorrespondingtothedimerineachtetrahedron The remaining eight NN tetrahedra of the A sublattice
is fixed, being determined by orbital pattern and orbital have, instead, either two or three choices to locate the
degreesoffreedomarestaticvariables. Thusatunneling two spin-singlets. These constraints are dictated by the
6
condition that intermediate tetrahedron must be of C- haveshownwithqualitativearguments19thatinourcase
kind. Thus, the degeneracy can be estimated as follows: this mechanism can select the triplet-T normal mode of
2NT′ < NAC < 3NT′, where NT′ = N/4 is the number of the tetrahedron group, leading to a distortion with one
tetrahedra in one sublattice. We conclude that the de- short and one long bonds located at opposite edges and
generacyofthismanifoldisstillmacroscopic,butsmaller four undistorted bonds (see Fig. 5). Indeed, a reduction
than that of B one. of the bond length increases the magnetic energy gain
Thus, even though the formation of spin-singlets re- and therefore favorsthe shortening of the bond with the
movesthespindegeneracyofthepyrochlorelattice,there strongest superexchange, i.e., b , where the singlet is lo-
0
is still a macroscopic degeneracy to be lifted. The main cated.
question is whether this degeneracy can be removed by Even though this picture is correct in its basic fea-
extendingoureffectiveHamiltoniantothepreviouslyne- tures,yetaquantitativedescriptionoftheglobalphysical
glected ddπ and ddδ overlaps. When these processes are mechanismleadingtothetetragonaldistortionrequiresa
considered, b and b bonds acquire a different bonding more careful analysis, that takes into account all elastic
2 3
energy, and, as number of these bonds is not the same normal modes of the single tetrahedron, as well as the
forB andAC manifolds,19thedegeneracybetweenthese correlations of these normal modes within the unit cell.
i
two phases can be lifted. However, this effect is smaller The aim of the present section is just to analyze such a
than the one induced by magnetoelastic coupling (see global mechanism.
t2 The dependence of the energy gain ∆E on the magni-
Sec. III), as Jddπ ≡ Udd2π ≃2 meV, while magnetoelastic tude ofdistortioncanbe evaluatedas asum ofmagnetic
energy gain per ion is about 6.5 meV. It is then obvious
∆Em and elastic ∆Eel contributions at each bond:
to look for the degeneracy removal first in terms of this
”spin-Teller” interaction, as done below.
∆E = ∆Em+∆Eel. (5)
Moreover, the degeneracy within the Bi-manifold can X ij ij
notberemovedwithintheeffectiveelectronicmodel,not ij
evenintroducingsmallerNNhoppingintegrals. Therea-
The sum is restricted to the NN sites.
son is related to the fact that the energy gain depends
All bonds in the undistorted lattice have the same
only on the total number of bonds of each type (n ,
b0 length and we represent their elastic energy assuming
n , n , n ) in the unit cell, and, in order to fill the
b1 b2 b3 that all ions are connected one another through equal
whole crystal with a periodicity not lower than the one
springs of constant k:
of the primitive cubic cell, the average number of bonds
nthbei ptherreetetteratrhaehderdornalshcoonufildgubreattiohnessiasmtea,kewnhi(cBh1e,vewritohf ∆Eiejl = 21k(δdij)2 = 12C0(δddi2j)2 (6)
n =1 and n =2; or B , with n =3 and n =0; or B , 0
b2 b3 2 b2 b3 3
withn =2andn =1). This number isgivenbyn =1,
n =2,b2n =2, n b=3 1, and it corresponds to the valbu0e of where the constant C0 ≡ kd20 is the radial force
b1 b2 b3 constant26 and δdij dij d0 is the deviation of the
B case, that is the only one that allows to cover the ≡ −
3 bond length from its value d in the undistorted cubic
0
whole cubic cell without mixing to other configurations
lattice.
(see Fig. 3).
The magnetoelastic energy of the generic bond ij can
Fromthe abovediscussionit follows that only correla-
be written as:
tions between bonds can lift this degeneracy. These cor-
relations naturally appear if the magneto-elastic contri- ∆Em =(J(d )+gδd )S~ S~ (7)
bution to the energy is considered. The orbitally-driven ij 0 ij i· j
modulationsofthespinexchangeinteractionswilldistort where g ∂J(d) . The dependence of exchange con-
the underlying lattice through the spin-Peierls mecha- ≡ ∂d |d=d0
stants on the distance is mediated by the hopping ma-
nism and different distorted patterns will pay a different
trix element: exchange constants are proportional to t2,
elastic energy. The three degenerate phases which will
and t is inversely proportional to the fifth power of the
be discussed in the next section are those with the unit
distance. In Ref. [26] it is possible to find a rough esti-
cell filled by all B tetrahedra(”B -phase”), that with a
3 3 mate of the proportionality constant, α: t = α/d5, with
mixture of B and B tetrahedra (”B B -phase”), and
that where th1e unit c2ell is filled by alt1ern2ated A and C α = 43ηddσh¯2mrd3. Here rd is a characteristic length, that
tetrahedra (”AC-phase”). forTiis1.08˚A,whileη = 16.2and h¯2 =7.62eV ˚A2,
ddσ − m ·
giving the value α= 116.628 eV˚A5. It is immediately
− ·
apparent that even a small reduction in the length of a
III. EVALUATION OF THE givenbondcanleadto arelativelyhighmagnetic-energy
MAGNETOELASTIC ENERGY gain.
Inorderto analyze the magnetoelastic behaviorofthe
In a spin-Peierls system the magnetic energy gain due threedegeneratephasesofourspin-orbitalmodelwefirst
tothespin-singletpairsoutweightstheincreaseinelastic focus on a single tetrahedron, either of A, B or C type.
i
energy due to the dimerization of the regular array. We For a more fluent reading, the detailed calculations are
7
2
1
yz
xy xy
K1 xz
K4 M1 K4
z y 3 M4
4 x
h.c. h.c.
K2
K3 M2
K2
FIG. 5: Triplet-T deformation mode from the irreducible M3
representations of the tetrahedron group. This mode gener- h.c. h.c.
atesadistortionofthetetrahedron,withshortandlongbond
located opposite to each other and four intermediate (undis-
L1 H4
torted) bonds. L4 H1 L4
h.c. h.c.
reported in Appendix B, and here we just comment on L3
L2 H3 L2
the main points. H2
In general, the global magnetoelastic Hamiltonian in- h.c. h.c.
volvesmorethanonenormalmodefor eachtetrahedron:
forexample,forAtetrahedrathesingletandonedoublet K4 M1 K4
modes contribute to the magnetoelastic energy, and for K1 M4
B tetrahedra all six normal modes are involved, as re-
2
portedinAppendixB.Moreover,eventhoughthetriplet FIG. 6: Bidimensional projection of the unit cell. Circles
T-mode associated to the singlet bond leads to a bigger represent Ti-ions, with the same labels as in Fig. 4. Di-
energygainforB thanforB tetrahedra,whenallavail- ameters are proportional to the z coordinate of the ion:
3 1
able normal modes are considered, the total energy gain z = 1,0.75,0.5,0.25, in fractional units. All bonds are ori-
ented along one of xy, xz, yz bisectors, as indicated. Helical
gets different contributions from all normal modes, but
chains (h.c.) are represented in the direction of increasing z.
in such a way that it is the same for all three B tetra-
hedra, and equal to 4.5g2. What makes the diffierence Doubly cut bondsare b0 bondsand singly cut are b1 bonds.
− k
among these configurations in a specific case like that of
MgTi O is that when these tetrahedra are considered
2 4 wecannotrely onthe unconstrainedminimizationofthe
in an infinite lattice, the single-tetrahedron degeneracy
single tetrahedra shown in Appendix B. One alternative
is lifted by the correlations among tetrahedra. In fact,
possibility is to calculate the magnetoelastic energy for
in correspondence with their common energy minimum,
the normal modes of the global cubic cell for the three
B , B , and B tetrahedra are all distorted differently,
3 1 2 degenerateconfigurations(B ,B B ,andAC-phase),in
3 1 2
as can be deduced from the location of their minima.
such a way as to automatically consider all constraints
Thus, when these distortions are combined together in
among tetrahedra. Yet, this procedure is not complete,
order to fill in exactly the global cubic cell, as in Fig.
as normal modes corresponding, e.g., to the buckling of
6, the appearance of new constraints, due to the rela-
tetrahedra do not contribute to the magnetoelastic en-
tive match of all tetrahedra within the cell, forces the
ergy. Such a contributionappears only when the normal
solution to another minimum. For example, in order to
modeinvolvesanincrease/decreaseofsomebondlength,
keep the global volume unaltered, all bond distortions
and, thus, it can be expressed as a linear combination
due to spin-singletsmustappear withreversedsignfrom
ofthe normalmodes ofeachsingletetrahedron,withthe
one tetrahedron to the other, because to each expansion
constraintthatthepositionofeachequivalentTi-ionover
there must be a corresponding contraction. Notice that
allnearestneighborunitcellsbethesame(k=0modes).
also the AC-phase is still degenerate with B configura-
i In order to fulfill this constraint, we impose that the
tions,asfarasjustmagnetoelasticenergygainassociated
total length over all the straight bonds connecting one
with single tetrahedra is considered, as, in average, it is
ion with the equivalent one in the next cubic cells is
again(−8−1)/2gk2 =−4.5gk2 (seeAppendixB).Remark- a conserved quantity. This means that, for example,
ably, when inter-tetrahedra correlations are considered, d +d +d +d (see Fig. 6) is
H4 H3 H3 K4 K4 K3 K3 H4
elastic singlet modes contribute to AC-phase differently a co−nservedqu−antity, and−, thus, the−sum of the four dis-
ofBi phase. Infact,itisworthwhileforthesystemtoex- placementsδdH4 H3+δdH3 K4+δdK4 K3+δdK3 H4 =
pandC tetrahedraandcontractAtetrahedraofanequal 0. The same ha−ppens for a−ll other 3 −chains in th−e xy-
amount, as this leads to a net energy gain, because the plane, for the 4 of the xz-plane, and for the 4 in the
singletmodeenergygainofAtetrahedraexceedsthatof yz-plane. Due to Eqs. (B4), these twelve conditions be-
C ones. comeconstraintsforthe normalvariablesofthe different
Itisclearfromtheseconsiderationsthatinordertoget tetrahedra. The search for the minima of the magne-
theglobalmagnetoelasticminimumfortheunitcubiccell toelastic Hamiltonians (B5-B9) with the previous con-
8
straints leads to the following expression of the energy energy is reached for a pure triplet mode, of amplitude
per unit cell: 18g2 for B -phase; 17.2g2 for B B - 2g. Notice that with this constraint the AC-phase gains
phase; 14.4g2−for AkC-phas3e. Thus,−the grokund sta1te2is vkery few, and also the relative energy difference between
givenb−yB3,wkithanenergyper tetrahedronof−2.25gk2, B1FBin2-apllhya,seleatnudsBs3trienscsreaagsaesin. that the main result is
corresponding to the sum of the energies of the triplet that whencorrelationsamongmagnetoelastic energyare
t2 and doublet e1 modes. The singlet and the doublet takenintoaccount,B3 phaseisalwaysstabilizedwithre-
e2 modes do not contribute to the energy,when they are spect to the two configurationally degenerate B1B2 and
constrained on the global unit cell. As now each ion is AC ones. Of the two criteria used for preserving the cell
sharedbytwotetrahedra,theenergypersiteis 1.125g2. volume, the second seems more in keeping with experi-
− k
However, there is an experimental fact that is not mental data. This suggests that oxygens play a role in
properly explained by this solution, and for which an the elastic distortions of the spinel cell, which is more
extra discussion is required. The solution of the con- rigid towards bucklings than the bare pyrochlore lattice.
strainedminimizationgivesanamplitudeof 2g fortriplet In any case it is found that the elongation of the non-
k
t2 mode and kg for doublet e1 mode, thus a ratio 2:1 interacting b3 bond and the corresponding stretching of
in favor of the triplet mode. But the experimentally b0 bondthroughoutthewholecellisalwaysenergetically
reported tetrahedral distortions10 are, referring to Eqs. more favorable,as found experimentally.
(B3), and (B4): δr = 0.155 ˚A, δr = 0.006 ˚A, Given the previous d-dependence of the hopping am-
13 14
which imply, when only t −and e normal mod−es are ac- plitude t, it is possible to estimate the gain energy per
2 1
tive, e = 0.012 ˚A, and t = 0.149 ˚A, with a triplet- ionassociatedwiththis magnetoelasticdistortion. First,
1 2
to-doublet−ratio of more than−10. This indicates that, g ≡ −∂∂Jd|d=d0 = 10dJ0. Using Eq. (B4), δr13 = −0.155
in the previously outlined model, some features of the ˚A= 2g, with k = C0. This implies that C = 10Jd0
real system are not taken into account, like the elastic − k d20 0 0.155 ≃
9.7 eV, if we take d 3.008 ˚A. Finally, the estimate of
potential coming from the oxygens, that acts in such a 0 ≃
the average magnetoelastic energy per site can be easily
way as to reduce some distortions. Thus, the possibility
found: there are 16 ions in one unit cell, and thus the
to get better quantitative results for the experimental
energy per site is E = g2 = (10J)2 6.5 meV.
bond-lengths within our scheme should passes through me − k C0 ≃
a way to mimic this aspect by means of a stronger con-
straint on the available modes before the minimization
IV. EFFECT OF MAGNETIC FIELD
procedure. If we consider Fig. 6, we can see that the
ion H4 can be connected to the equivalent ones in the
neighboring edge-sharing unit cells through one of the As we have shown above the ground state of the sys-
three ”linear” paths: H4 H1 L4 L1 H4, in xz- temisnon-magneticandspinsarepairedinsingletstates.
plane, or H4 H2 M4− M−2 H−4, in−yz-plane, or Thespindegreesoffreedomarethusgappedandthegap
H4 H3 K−4 K−3 H−4, in x−y-plane. Face-sharing of triplet excitations is given by the singlet binding en-
unit−cells−are co−nnecte−d by the paths along the heli- ergy ∆s = 4J[1 2η] (see Eq. (3)). When an external
−
cal chains, namely: H4 H2 L3 L1 H4 and magneticfiledisapplied,theenergyoftripletexcitations
H4 H1 M3 M2 −H4, fo−r cha−ins alo−ng z-axis, decreases, due to the gain of Zeeman energy, while the
H4 −H2 −K1 K−3 H4−andH4 M2 M1 H3 H4, singletstatedoesnotexperiencetheappliedfield. There-
for −chains−alon−g x-ax−is, H4 H1− K2− K3− H−4 and fore,one wouldexpectnochangeinthe symmetryofthe
H4 H3 L2 L1 H4,−for ch−ains a−long y−-axis. In ground state up to fields gµBHc = ∆s at which singlet
prin−ciple t−he glo−bal H−4 H4 distance along, e.g., the z triplet gap closes (g is the gyrotropic factor and µB the
axis can be kept fixed ev−en if the sum of the four bonds Bohr magneton). At H = Hc a second order transition
changes, as the bonds are not along a straight line, and can in principle take place from non-magnetic to a mag-
the elongationof one bond can be combined with an ap- netically ordered state, driven by condensation of lowest
propriate rotation, due to a tetrahedron buckling. Yet, energy triplets at some ordering wave-vectorQ at which
this buckling implies an extra elastic energy loss due to triplet spectrum has a minimum.
the relative movement with Mg and O ions. We can However, the presence of an orbital degeneracy can
forbid these extra losses, by imposing another series of modify this convectional picture. As shown in Sec. II,
constraints that express the global conservation of the the first excited state above Dimer phase is the FM one,
total length also along the ”helical” chains, in this way characterized by C orbital pattern. The energy of FM
freezing some of the normal modes that were previously phase,contrarytotheDimerphase,issensitivetotheap-
allowed. pliedmagneticfieldanddecreasesof∆EH =−gHµBSiz.
When the minimization is performed with this new The critical field H˜c at which the energies of the two
constraint, the results for the energy per unit cell are: phases are equal is:
t−h1e6Agk2Cf-oprhaBse3.-pAhagsaei;n,−B13-gpk2hafoser iBs1tBhe2-lpohwaesset;, −bu3gtk2thfoisr gµBH˜c =J[2−11η], (8)
3
time the amplitude of e mode is zero and the minimum and one can easily verify that H˜ < H . Therefore at
1 c c
9
H =H˜ there will be a first ordertransition fromDimer and460.2eV,whichcorrespondstoaphotonwavelength
c
phase to FM state. This transition will be accompanied λ 27 ˚A, that does not allow Bragg law sinθ = λ/2d
B
≃
by a simultaneous rearrangementoforbitalorderingand to be verified, not even for (001) reflection.
the abrupt closure of the spin gap, as the FM phase is Inspite of allthis, we believe that itis still possibleto
gapless. We can estimate the order of magnitude of H˜ , detect the proposed OO at K edge. The key-point to go
c
for η 0.15 and J = 25 meV, as µ H˜ 4.4 meV, i.e., beyondthe conclusionsofRefs. [13,14,15]lies inthe fact
B c
a criti≃cal field of H˜ 76 Tesla. In this ≃estimate we did that their results are strongly related to symmetry and
c
≃
notconsidertheextrastabilizationenergyinfavorofthe distortion of the system as well as the kind of reflection
singletphasecomingfromthe magnetoelasticdistortion. under study. In particular, while the signal induced by
Of course, one can read Eq. (8) the other way round, in the CoulombrepulsionU is ofthe orderof 200 300
dp
≃ ÷
terms of η, and deduce that, if a critical field is found at meV,independentlyoftheparticularcrystalstructure,31
alowervaluethan76T,thisimpliesthatJ /U iscloser the influence of the oxygen distortion is much lower in
H 2
to the critical value of 0.18 than what estimated here. the directions of t orbitals than in e ones. It might
2g g
indeed seem conceivable that a reduced Jahn-Teller dis-
tortion can give rise to a sizable interference with the
OO-inducedeffectinthe RXS intensity. Moreover,there
V. DETECTION OF ORBITAL ORDERING
isthepossibilitythattherotationoftheexchangevector
THROUGH RXS.
and/or polarizations in the RXS experiment can make
the effect more or less pronounced, according to the ex-
As already outlined in the previous two sections, or-
perimental conditions. We have thus performed a nu-
bitalorderinginMgTi O ismainlydictatedbysuperex-
2 4 merical simulation and found that it is indeed possible
change interactions. This implies that the orbital orien-
to experimentally revealthe presenceof the OO through
tation at each site does not strictly follow the symme-
thecomparisonofdifferentRXSsignals. Inthefollowing
try imposed by the local crystal field, as happens when
we illustrate the details of our calculations, in order to
theeffectispurelydeterminedbyJahn-Tellerdistortions.
clarify the previous theoretical speculations.
Therefore, we believe that there is the possibility to ex-
ThetransitionprocessforRXS,governedbytheFermi
ploit the local symmetry differences in the helical d -
xz Golden rule, depends on the state overlap through the
d orbitalpatternbymeansofresonantx-rayscattering
yz polarized electric field of the incoming (i) or outgoing
(RXS), where the local transition amplitudes are added
(o)photon. Ifwe considerthe multipole expansionupto
with a phase factor that can compensate the vanishing
the electric quadrupole contribution, we get:
effect due to the global tetragonalsymmetry. Conceptu-
ally,thisprocedureistheanalogueoftheoneusedinthe
caseofmanganites27thatledtoaseriesofresultsinitially 1
Mi(o) = ψ ~ǫi(o) ~r 1 i~ki(o) ~r ψ (9)
interpretedasadirectevidenceoforbitalordering,28and ng h n| · (cid:0) − 2 · (cid:1)| gi
soonlaterrecognized13,14,15 asmainlydeterminedbythe
Here ψ and ψ are ground and intermediate state
oxygen distortion around each Mn-ion. Yet, in this case g n
Bragg-forbidden reflections (e.g., for LaMnO 27) relate wave functions, respectively, ~ǫi(o) is the polarization of
3
two sites with different orbital occupancy and local dis- the incoming (outgoing) photon and ~ki(o) its wave vec-
tortion. Thus, at K edge, the signal at a given energy tor. Around an absorption edge, Mng is highly energy
turns out to depend on the difference between, e.g., p and angular dependent and this provides the sensitivity
x
and p density of states around that energy,29 projected to the electronic structure around the atom. In RXS
y
on Mn-ions. Such local anisotropies in the electronic 4p the global process of photon absorption, virtual emis-
density of states can be a consequence of oxygen distor- sionofthe photoelectron,andsubsequent decaywith re-
tions (Jahn-Teller effect) or can be induced by the or- emission of a photon, is coherent, thus giving rise to the
dering of the underlying 3d orbitals through the 3d 4p usual Bragg diffraction condition. The outgoing photon
−
CoulombrepulsionU (orbitalorderingeffect).14There- canhavedifferentpolarizationandwavevectorcompared
dp
sultsofRefs. 13,14,15independentlyshowthattheeffect totheincomingoneandtheatomicanomalousscattering
of the Jahn-Teller mechanism is much stronger, about a factor (ASF) reads32:
factor10inamplitude,thantheeffectoforbitalordering.
This led to the conclusion that OO cannot be ”directly”
probedby means ofRXS at K edges,where the strength f =f +if = me 1 (En−Eg)3Mnog∗Mnig (10)
of the ligand field overwhelms the effect induced by Udp. ′ ′′ ¯h2 ¯hω Xng ¯hω−(En−Eg)−iΓ2n
Due to this result, the search for experimental evidence
of orbital ordering moved to L edge RXS30 where one is Here¯hωisthephotonenergy,m theelectronmass,E
e g
sensitive directly to 3d orbitals. Yet, this kind of spec- and E are the ground and intermediate state energies
n
troscopy has the serious drawback that, in order to have andΓ isthebroadeningofthetransition. Thesumover
n
non-imaginary Bragg angles, it is necessarily limited to the intermediate states starts from the Fermi energy.18
crystals with very big unit cells, and this is not feasible In the tetragonal phase of MgTi O the space group
2 4
for MgTi O . In fact, Ti L edges are at about 453.8 is the chiral group P4 2 2, No. 92 of Ref. [33] (or its
2 4 2,3 1 1
10
”mirror”-related P4 2 2, No. 96), with 8 ions per unit whose irreducible representationin SO(3) are a scalar, a
3 1
cell, and the structure factor is: pseudovector and a traceless symmetric tensor: the first
never contributes to Bragg-forbidden reflections (which
are scalar-forbidden), while the second is proportional
8
A(Q~)=XeiQ~·R~jfj (11) tmoaganemtiacgnsyetsitcemmsolmikeentMagnTdi Ohas. nToheintflhuierdncteenosnornhoans-
2 4
j=1 5 independent components, that can be labeled accord-
ing to the usualsecond-rankrealsphericalharmonicsas:
where f is the ASF of ion j. The atomic positions,
j
D , D , D , D , and D . They are a mea-
their orbital filling and the symmetries relating the ions xy xz yz x2 y2 3z2 r2
sure of the anisotrop−yof p-density−of states projectedon
one another are schematized in Table I. Notice that the
the resonant ion.34 For example, D measures the
tetragonalcellischaracterizedbya45-degreerotationin x2 y2
difference in the density of states in x-−direction, p , and
xy-plane, compared to the cubic cell of Fig. 2(a). x
in y-direction, p . In order to determine the implica-
y
tions of Eq. (12) on the various reflections, we need
atom OO position symmetry to know how the symmetry operators act on our sym-
Ti (H3) d (u,v,w) Eˆ metric tensors. The indices of each tensor change ac-
1 yz cordingtothe followingrules: Cˆ (x,y,z)=( x, y,z);
Ti2(L4) dyz (−u,−v,21 +w) Cˆ2z Cˆ (x,y,z) = ( x,y, z) and Cˆ2z+(x,y,z) =−(y,−x,z).
Ti3(H1) dxz (12 −v,12 +u,41 +w) Cˆ4+z Si2myilar results a−re obt−ained remi4nzding that the A−SF in
Ti4(L2) dxz (12 +v,12 −u,43 +w) Cˆ4−z our approximation is a second-rank irreducible tensor
TTTiii675(((HHL342))) dddxxyzzz ((1122 −+uu(v,,,2211u+−, vvw,,1434)−−ww)) CCˆCˆˆ222xxyx fCwˆm(e22y)gf,me(a2t)ntdh=aCˆt(2−zafl)lm(m2B)fr−(=a2m)gg.(-−foA)rmsbfidam(d2)ce,onnCˆrs4ee+zflqfeum(ce2tn)ioce=nsoifomfEftmq(h2.e),k(a1inn2dd)
−
Ti (L1) d ( v, u,1 w) Cˆ (0,0,4n+1) and (0,0,4n+3) are sensitive to the complex
8 yz − − 2 − 2xx mixture ofdensity ofstates: D iD Y2 . For this
TABLE I. Orbital occupancy and tetragonal position of reason, due to the cylindrical sxyzm±metyrzy,∝a c±o1nstant az-
Ti-ions. Symmetry operations are referred to Ti1. imuthal scan is expected as well as a minor dependence
ontheorbitalanisotropiesofxzandyzkind. Onthecon-
Here u=0.9911, v =0.2499, w =0.8668are the frac-
trary,Bragg-forbiddenreflectionsofthekind(0,0,4n+2),
tionalcoordinatesoftheatomsinunitoftetragonalaxes.
if we neglect cos4πw 0.1 sin4πw 1, are just pro-
Notice that such a derivation allows also to deal with x- ≃ ≪ ≃
portional to D : thus, they must show a non-constant
xy
ray natural circular dichroism (XNCD), as the absorp-
azimuthalscan,and,indirectly,manifestsomeproperties
tion cross section corresponds to the imaginary part of
relatedtotheOO,throughthedepletionofthefilledd
xy
the forward scattering amplitude (i.e., when Q~ =0). In- orbitals.
deedasignalinthe dipole-quadrupoleinterferencechan-
If we express the polarization dependence in the vari-
nel could be expected, as the space symmetry group of
ous channels, referring to the cubic frame of Fig. 1, we
MgTi O is a chiral one, and there could be the possi-
2 4 get the following results. Both (001) and (003) reflec-
bility that it is affected by OO. Unfortunately, our nu-
tions couple to ǫ ǫ andǫ ǫ , andthus they aredifferent
merical simulations with the finite difference method,18 x z y z
from zero only in the σπ-channel. As their amplitude
inthesamerangeofparametersasshownbelowforRXS,
is proportional to (sinφ+icosφ), they have a constant
demonstrates that such a dichroic signal is too low (less
azimuthal scan. Instead, (002) reflection is detectable
than 0.05 % of the absorption) and that the changes in-
in all σσ, σπ and ππ channels, with an intensity that
duced by OO are practically negligeable.
scales,respectively,assin2(2φ)D2 ,sin2θ cos2(2φ)D2 ,
For this reason in the following we shall analyze just xy B xy
sin4θ sin2(2φ)D2 , where θ is the Bragg angle. As
K edge RXS, and in particular all reflections (00l), that B xy B
areBragg-forbiddenunlessl=4n. Usingthesymmetries sin2θB 0.1,signalsinππ-channelare1/10smallerthan
≃
of Table I, the structure factor can be expressed in the those in σπ and1/100smaller than those in σσ-channel.
following form: Also their azimuthal scans are out of phase: that of σπ
channel has its maximum value when φ = 0, i.e., in the
direction of the nearest neighbor oxygens, while those of
A(00l)=(1+( )lCˆ )(1+(i)lCˆ+) σσ andππ channelstaketheir maximumvalue alongthe
− 2z 4z
(e2πilw +(i)le 2πilwCˆ )f (12) Ti-Ti chains in the xy-plane, where σπ signal is zero.
− 2y 1
In Fig. 7 we show the results of our numericalsimula-
Note that even if f is a scalar, it is expressed as a tions for (00l) reflections, performed with the finite dif-
1
scalar product of two tensors, and, in Eq. (12), ro- ference method option of the FDMNES program.18 As a
tation operators are understood as applied to one of input file we used the refined positions of Ref. [10] for
them. We can limit to the dipole approximation, where Mg, Ti and O ions. Because of the increasing CPU-time
we just need to consider rank-2 cartesian tensors, T , consuming,weperformedoursimulation,withandwith-
αβ