Invariant Form of Hyperfine Interaction with Multipolar Moments - Observation of Octupolar Moments in NpO$_{2}$ and CeB$_{6}$ by NMR -

Abstract

The invariant form of the hyperfine interaction between multipolar moments and the nuclear spin is derived, and applied to discuss possibilities to identify the antiferro-octupolar (AFO) moments by NMR experiments. The ordered phase of NpO$_{2}$ and the phase IV of Ce$_{1-x}$La$_{x}$B$_{6}$ are studied in detail. Recent $^{17}$O NMR for polycrystalline samples of NpO$_{2}$ are discussed theoretically from our formulation. The observed feature of the splitting of $^{17}$O NMR spectrum into a sharp line and a broad line, their intensity ratio, and the magnetic field dependence of the shift and of the width can be consistently explained on the basis of the triple $\bq$ AFO ordering model proposed by Paix\~{a}o {\it et. al.} Thus, the present theory shows that the $^{17}$O NMR spectrum gives a strong support to the model. The 4 O sites in the fcc NpO$_2$ become inequivalent due to the secondary triple $\bq$ ordering of AF-quadrupoles: one cubic and three non-cubic sites. It turns out that the hyperfine field due to the antiferro-dipole and AFO moments induced by the magnetic field, and the quadrupolar field at non-cubic sites are key ingredients to understand the observed spectrum. The controversial problem of the nature of phase IV in Ce$_{1-x}$La$_{x}$B$_{6}$ is also studied. It is pointed out that there is a unique feature in the NMR spectra, if the $\Gamma_{5}$($T^{\beta}_{x}=T^{\beta}_{y}=T^{\beta}_{z}$) AFO ordering is realized in Ce$_{1-x}$La$_{x}$B$_{6}$. Namely, the hyperfine splitting of a B atom pair on the $({1/2},{1/2},\pm u)$ sites crosses zero on the $(1\bar{1}0)$ plane when the magnetic field is rotated around the $[001]$ axis. Comment: 22 pages, 2 figures

Full text

Page 1

arXiv:cond-mat/0408201v1 [cond-mat.str-el] 10 Aug 2004
Typeset with jpsj2.cls <ver.1.2>
Full Paper
Invariant Form of Hyperfine Interaction with Multipolar Moments
- Observation of Octupolar Moments in NpO2and CeB6by NMR -
Osamu Sakai∗, Ryousuke Shiina and Hiroyuki Shiba1
Department of Physics, Tokyo Metropolitan University, Hachioji 192-0397, Japan
1The Institute of Pure and Applied Physics, 6-9-6 Shinbashi, Tokyo 105-0004, Japan
(Received )
The invariant form of the hyperfine interaction between multipolar moments and the
nuclear spin is derived, and applied to discuss possibilities to identify the antiferro-octupolar
(AFO) moments by NMR experiments. The ordered phase of NpO2 and the phase IV of
Ce1−xLaxB6are studied in detail. Recent17O NMR for polycrystalline samples of NpO2are
discussed theoretically from our formulation. The observed feature of the splitting of17O
NMR spectrum into a sharp line and a broad line, their intensity ratio, and the magnetic field
dependence of the shift and of the width can be consistently explained on the basis of the
triple q AFO ordering model proposed by Paix˜ ao et. al. Thus, the present theory shows that
the17O NMR spectrum gives a strong support to the model. The 4 O sites in the fcc NpO2
become inequivalent due to the secondary triple q ordering of AF-quadrupoles: one cubic
and three non-cubic sites. It turns out that the hyperfine field due to the antiferro-dipole
and AFO moments induced by the magnetic field, and the quadrupolar field at non-cubic
sites are key ingredients to understand the observed spectrum. The controversial problem
of the nature of phase IV in Ce1−xLaxB6 is also studied. It is pointed out that there is a
unique feature in the NMR spectra, if the Γ5(Tβ
Ce1−xLaxB6. Namely, the hyperfine splitting of a B atom pair on the (1
zero on the (1¯10) plane when the magnetic field is rotated around the [001] axis.
x= Tβ
y= Tβ
z) AFO ordering is realized in
2,1
2,±u) sites crosses
KEYWORDS: antiferro-octupolar order, octupolar moment, hyperfine interaction, NMR,
NpO2, CeB6, antiferro-quadrupolar order
1. Introduction
The antiferro-octupolar (AFO) moments, which are odd in time-reversal symmetry but
are distinct from dipole moments, are now widely recognized as a “hidden” order parameter.1
The AFO order is a promising candidate to resolve the breaking of the time reversal symmetry
without the antiferromagnetic dipole (AFM) order. However, a direct observations of the AFO
is very difficult.
There are two ways to realize the AFO moments. One is induced AFO moments in the
antiferro-quadrupolar (AFQ) state by applying the static magnetic field.2Another is a spon-
taneous AFO ordering, which has been recently proposed for NpO2and Ce1−xLaxB6.3–8In
∗E-mail: sakai@phys.metro-u.ac.jp.
1/21

Page 2

J. Phys. Soc. Jpn.
Full Paper
this paper, we wish to study a possibility to detect the AFO moments in NMR experiments
through the hyperfine interaction. For this purpose we derive a general form of the hyper-
fine interaction between the multipolar moments and the nuclear spin. Actually we studied
this problem before for CeB6. The approach was to derive a phenomenological form of the
hyperfine interaction from the symmetry.9,10However, there is a simpler and equivalent ap-
proach, which we use in this work. It is a straightforward generalization of ref.11 , in which
the invariant coupling form between multipolar moments was derived by using a method of
symmetrized molecular orbital theory. The usefulness of this approach to discuss the hyperfine
interaction will be demonstrated in this paper.
To have octupolar moments as independent local degrees of freedom, the material must
have high symmetry such as cubic. There are three cubic systems for which the AFO order
of some sort is either realized or is likely to be realized. The first system is CeB6, which
has been studied extensively.10,12–14There are a number of anomalous features in the phase
diagram as well as in the nature of the low-temperature order phases, phase II in particular.
They are naturally explained by noting the interaction effect between the field-induced AFO
moments, especially of the Txyztype (Γ2type).10,15However, a crucial progress was achieved,
when the apparent inconsistency between the neutron diffraction16and NMR17for the phase
II was successfully resolved by recognizing that the hyperfine coupling is present between the
B nuclear spin and the induced TxyzAFO moment of Ce ion.9,10,18This is the first case in
which the contribution from AFO moments to the hyperfine coupling has been discovered.
Therefore the NMR is now regarded as a direct proof for the field-induced AFO moment in
CeB6.
The second system is NpO2; the nature of the order in NpO2below 26K was a mystery for
a long time. Carrying out a resonant X-ray scattering, Paix˜ ao et. al. have recently suggested
that their results can be explained well by assuming the triple q AFQ order of Γ5type.4They
proposed the triple q AFO as the primary order parameter because any dipole order have
not been observed though Np4+is the Kramers ion. The triple q AFQ is induced from the
primary triple q AFO. A new development on this problem has been brought by Tokunaga et.
al.’s17O (I = 5/2) NMR experiment for polycrystalline samples.19They found that the O ion
sites, which are equivalent in the high temperature phase (Fm¯3m), split into two inequivalent
groups. The experimental results have the following unique features:
(I) 1/4 of the17O nuclei contributes to a sharp resonance line, while the rest 3/4 gives a broad
resonance line.
(II) The shift of the sharp line is approximately proportional to the strength of the applied
magnetic field.
(III) The width of the broad line is given as the sum of a constant and a part proportional to
the strength of the magnetic field.
2/21

Page 3

J. Phys. Soc. Jpn.
Full Paper
(IV) The shift of the sharp line and the magnetic field induced width of the broad line have
comparable magnitude.
We show in this paper that those experimental results can be naturally explained by the
quadrupolar field and the hyperfine field splitting in the triple q state. The hyperfine field is
caused by the AFO and/or AFM moment induced by the application of the magnetic field.
Therefore the17O NMR is a fingerprint of the triple q order of AFO and AFQ.
The third system is CexLa1−xB6. It is known that different from pure CeB6a new phase,
which is called phase IV, appears instead of phase II in the low magnetic field region of
CexLa1−xB6(x ∼ 0.75).20–22The nature of phase IV remains controversial. The time reversal
symmetry breaking is reported for this phase.23,24However, no indication of the AFM or-
der has been detected experimentally, although neutron diffraction studies were carried out
extensively.25A characteristic cusp of the magnetic susceptibility has been observed at the
transition temperature between phase I and phase IV.8From these results together with other
experimental indications,20–22,26,27an AFO order of the Γ5type has been recently proposed
for the phase IV.6–8A recent NMR experiment indicated a broadening of NMR lines in the
phase IV, showing a sign of time reversal symmetry breaking.23No further information is
obtained from this NMR experiment. Quite recently, the importance of the uniaxial stress
to remove the ambiguities due to the domain structure has been demonstrated. Morie et. al.
pointed out that the magnetic susceptibility shows an anisotropy in contrast to the isotropic
behaviors reported previously for ambient pressure.28This result is qualitatively similar to
what one obtains theoretically from the AFO of the Γ5type with ?Tβ
Tβ
x? = ?Tβ
y? = ?Tβ
z?. Here,
x, Tβ
y and Tβ
z are the three components of the Γ5type octupolar moment.29The NMR study
under uniaxial stress is desirable to clarify the situation for phase IV. In this paper we will
point out a possibility to identify the Tβtype AFO order through the NMR experiment. We
will show that if the ?Tβ
Morie et. al. assumed,6,28a characteristic feature of the NMR splitting should be present in
the field-direction dependence of the B NMR from (1
x? = ?Tβ
y? = ?Tβ
z? type order is realized as Kubo and Kuramoto, and
2,1
2,±u) sites.
This paper is organized as follows. In §2, the hyperfine interaction with multipolar moments
of 4f electrons is derived for NpO2on the basis of new method. In §3, the recent17O NMR
experiment on NpO2is studied theoretically to show that it is consistent with the triple q
structure of AFO and AFQ. In §4, the hyperfine interaction in CeB6is re-derived. The field
direction dependence of the NMR line is discussed in §5, having the phase IV problem in our
mind.
2. Invariant Coupling in NpO2and Splitting of O sites into Non-equivalent Sites
The Np ions in NpO2form the f.c.c. lattice. The position vector ρ of the eight O ions in
the f.c.c. cube is given in Table II, where the length of the edge of the cube is chosen as 2a.
The local symmetry around the O sites has the Tdcharacter. From the discussions described
3/21

Page 4

J. Phys. Soc. Jpn.
Full Paper
in Appendix A, the hyperfine interaction of the17O nuclear spin (I = 5/2) located at ρ with
multipolar moments of Np 5f electrons is given by
Hhf(ρ )=eiq ρIx(ρ )
?
c1,1
4
√4Jx(q )?cxcycz− isg(ρ )sxsysz
?
+c1,2
4
√16
?
Jy(q )?isg(ρ )szcxcy− czsxsy
?
+Jz(q )?isg(ρ )syczcx− cyszsx
?
y
isg(ρ )szcxcy− czsxsy
??
+c1,3
4
√8
Tβ
?q )
?
?
−Tβ
z(q )?isg(ρ )syczcx− cyszsx)??
+c1,4
4
√4Txyz(q )?isg(ρ )sxcycz− cxsysz
??
+(cyclic permutation of x,y and z) .(1)
Here ci,j’s are coupling constants, and sg(ρ ) = (−1)(ρx+ρy+ρz−a
cos(qν/2) (sin(qν/2)). The summation over q is assumed in eq. (1). Under the cubic rotation
2)/a, and cν(sν) represents
group the octupolar operator of Γ4type has the same symmetry character with the dipole
operator. Therefore the dipole operator J in eq. (1) should be read as a linear combination of
the pure dipole and the octupolar operator of Γ4type. The terms containing the factor isg(ρ )
are due to the Tdsite symmetry around each O ion. The imaginary factor in this quantity is
only a seeming, which is compensated by the factor eiq ρ.
Let us assume the Γ5 AFO order of triple q type: Tβ
Tβ
and Tβ
x(R ) = Tβ
xeiQxR, Tβ
y(R ) =
yeiQyR
z(R ) = Tβ
zeiQzR, with Qx= π(1,0,0), Qy= π(0,1,0) and Qz=
π(0,0,1), as assumed in ref. (4 ). Then we easily find from eq. (1) that the AFO moments in
this order do not induce any hyperfine field on the O atoms.
As noted in refs. (4) and (11) and seen from Table I, the triple q ordering induces the AFQ
of the Γ5type: Oyz(R ) = OyzeiQxR, Ozx(R ) = OzxeiQyRand Oxy(R ) = OxyeiQzR. If
Oyz= Ozx= Oxy≡ OΓ5hold, the cubic symmetry of the crystal is not broken.
Let us consider the quadrupolar interaction between the17O nuclear moment on ρ and the
quadrupolar moment of Np ion. The interaction form is given by following the discussion in
Appendix B:
Hqq(ρ ) =eiρ q?
c2,1
4
√4
2(−Ou(ρ ) +√3Ov(ρ ))Oyz,q(−isg(ρ )sxcycz+ cxsysz)
+Ou(ρ )Oxy,q(−isg(ρ )szcxcy+ czsxsy)
2(−Ou(ρ ) −√3Ov(ρ ))Ozx,q(−isg(ρ )syczcx+ cyszsx)?
c2,3
√4
?(Ou(ρ )Ou,q + Ov(ρ )Ov,q)(cxcycz− isg(ρ )sxsysz)?
+c2,2
8
√24
?1
+1
+
4
?Oyz(ρ )1
2(−Ou,q +√3Ov,q)(isg(ρ )sxcycz− cxsysz)
4/21

Page 5

J. Phys. Soc. Jpn.
Full Paper
+Ozx(ρ )1
2(−Ov,q −√3Ov,q)(isg(ρ )syczcx− cyszsx)
+Oxy(ρ )Ou,q(isg(ρ )szcxcy− czsxsy)?
?(Oyz(ρ )Oyz,q + Ozx(ρ )Ozx,q + Oxy(ρ )Oxy,q)(cxcycz+ isg(ρ )sxsysz)?
?
+c2,4
4
√4
+c2,5
4
√8
Oyz(ρ )?Ozx,q(isg(ρ )szcxcy− czsxsy) + Oxy,q(isg(ρ )syczcx− cyszsx)?
+(cyclic permutation of x,y and z)
??
. (2)
We can derive the quadrupolar interaction between the nuclear quadrupole of17O and the
triple q AFQ of Np 5f electrons as
Hqq(ρ ,c2,2)=C2,2(−isg(ρ ))
+Ou(ρ )OxyeiQzρ+1
?1
2
?
− Ou(ρ ) +√3Ov(ρ )
?
OyzeiQxρ
2
?
− Ou(ρ ) −
√3Ov(ρ )
?
OzxeiQyρ?
,(3)
where the quantity C2,2 is proportional to c2,2 and the order parameter of AFQ OΓ5, or
square of the order parameter of AFO (Tβ)2, i.e. C2,2∝ c2,2OΓ5∝ c2,2(Tβ)2. This interaction
originates from the coupling caused by the Tdsymmetry, such as szcxcyterm in eq.(2). At
the ρ1=1
the following expression
2(3a,3a,3a) site, the factor −isg(ρ1)eiQνρ1 takes 1 for all ν. Therefore it gives
Hqq(ρ1,c2,2)=−C2,2
?
√3
2Ov(ρ1)(Oyz− Ozx)
Ou(ρ1)(−1
2Oyz+ Oxy−1
2Ozx)
+
?
. (4)
If Oyz= Ozx= Oxy≡ OΓ5holds, this interaction is canceled out on the ρ1site.
On the other hand, at the site ρ2=1
for the terms characterized by the vectors, Qx,Qyand Qz. Then we have
2(3a,3a,a), the factor take −1,−1 and 1, respectively,
Hqq(ρ2,c2,2)=C2,2
?
Ou(ρ2)
?
−1
2Oyz− Oxy−1
2Ozx
?
+3
2Ov(ρ2)(Oyz− Ozx)
C2,2Ou(ρ2)(−2OΓ5).
?
,
=(5)
Therefore the quadrupolar field remains on this O atom. A similar calculation can be carried
out for other sites and the result is shown in Table III. (The sign of −isg(ρ )eQνρis given
in the Table II.) We note Ouof the17O nucleus (I = 5/2) is proportional to 3I2
and −1
AFQ order causes 4 different quadrupolar fields on the eight O sites in the fcc cube: a pair
z− I(I + 1),
2(Ou+√3Ov) ∝ 3I2
y− I(I + 1), and −1
2(Ou−√3Ov) ∝ 3I2
x− I(I + 1). The triple q
of cubic sites (with zero quadrupolar field) and 3 pairs of uniaxial-symmetry sites ( each of
which has principal axis along one of the x,y and z axes). The17O nucleus on the non-cubic
site will show the the quadrupolar splitting of the spectrum. The appearance of the cubic and
non-cubic site in the triple q AFO state can be understood in an intuitive way as explained
5/21