Origin of correlated isolated flat bands in copper-substituted lead phosphate apatite

Sinéad M. Griffin1,2
1
Materials Sciences Division, Lawrence Berkeley National Laboratory, Berkeley, California, 94720, USA and
2
Molecular Foundry Division, Lawrence Berkeley National Laboratory, Berkeley, California, 94720, USA
(Dated: August 4, 2023)
A recent report of room temperature superconductivity at ambient pressure in Cu-substituted
apatite (‘LK99’) has invigorated interest in the understanding of what materials and mechanisms
can allow for high-temperature superconductivity. Here I perform density functional theory calcu-
lations on Cu-substituted lead phosphate apatite, identifying correlated isolated flat bands at the
arXiv:2307.16892v2 [cond-mat.supr-con] 3 Aug 2023

Fermi level, a common signature of high transition temperatures in already-established families of

superconductors. I elucidate the origins of these isolated bands as arising from a structural distor-
tion induced by the Cu ions and a chiral charge density wave from the Pb lone pairs. These results
suggest that a minimal two-band model can encompass much of the low-energy physics in this sys-
tem. Finally, I discuss the implications of my results on possible superconductivity in Cu-doped
apatite.

INTRODUCTION relationships in these compounds to begin to unravel

their potential correlated physics. In this Letter, I use
High-TC superconductors are arguably the holy grail ab initio calculations to elucidate the key competing in-
of condensed matter physics with huge potential appli- teractions in Cu-doped apatite at the mean-field density
cations for an energy-efficient future. The first class functional level.
of superconductors that were considered to be high-TC
were the cuprates which were discovered by Bednorz and
Müller in 1987 [1]. The cuprates have been subsequently METHODS
followed by several new classes including the Fe-pnictides
in 2008 [2] and the nickelates [3]. I used the Vienna Ab initio Simulation Package
While significant strides have been made in the dis- (VASP) [12–15] for all density functional theory (DFT)
covery and understanding of high-TC superconductors, calculations with full calculation details given in the SI.
and we continue to unearth novel examples within estab- I applied a Hubbard-U correction to account for the un-
lished classes [4], a definitive roadmap to achieving room- derlocalization of the Cu-d states. I tested values of U
temperature TC under ambient pressures has remained between 2 eV and 6 eV, finding my results were similar
elusive. Common to many of these high-TC supercon- for all values calculated. The results in the main text are
ducting families are strongly correlated bands which can for U = 4 eV which gives lattice parameters within 1%
give rise to unconventional mechanisms for Cooper pair of experiment [10, 16].
formation [5, 6], and proximity to multiple competing
interactions such as antiferromagnetism, charge density
waves and spin-density waves. These phases can compete RESULTS
or coexist with superconductivity where fluctuations be-
tween these states are believed to play a significant role Structural Properties
for achieving high-TC . Searching for these features in
new materials systems is therefore a promising route for Apatites are materials with the general formula
finding new classes of high-TC superconductors. For in- A10 (TO4 )6 X2±x , where A = alkaline or rare earth metal;
stance, the nickelate superconductors were originally pre- T = Ge, Si, or P; and X = halide, O, or OH. The name
dicted in theory by Anisimov, Bukhvalov and Rice as an ‘apatite’ derives from the Greek apatē meaning ‘deceit’ as
analogy to the cuprate superconductors [7]. Similar ap- a result of the diverse range of forms it can take [17]. Here
proaches have also been proposed to selectively design a I consider the lead-phosphate apatite Pb10 (PO4 )6 (OH)2 .
material with the sought-after isolated d-manifold that is Taking its structure reported from X-ray diffraction in
associated with strong correlations [8], and have inspired Ref. [16] as the starting point, its structure following
high-throughput searches for good candidate materials, a full optimization is depicted in Fig. 1. It adopts the
further expanding the horizons of high-Tc superconduc- typical crystal structure of various apatite chemistries,
tivity [9]. namely it forms a network comprising PbO6 prisms that
The recent report of possible room temperature su- are corner shared with PO4 tetrahedra. I refer to these
perconductivity at ambient pressures in Cu-substituted Pb as Pb(1), in keeping with the convention in litera-
apatite (also known as ‘LK99’)[10, 11] motivates the need ture [18]. This framework is filled with Pb6 (OH)2 where
for a thorough understanding of the structure-property the (OH)2 forms a chain in the center of a hexagonal
 2

FIG. 1. (a) Lead-phosphate apatite structure with two in-

equivalent Pb sites as described in the main text. Columns of
O or OH sit in the center column defined by Pb(2) hexagonal
structure. (b) Calculated electronic localization function for
Pb10 (PO4 )6 OH2 . Oxygens surrounding Pb(2) are repelled by
the lone pair. FIG. 2. (a) Lead-phosphate apatite structure showing nine-
coordinated Pb(1) sites. (b) Cu-substituted structure show-
ing six-coordinated Cu and Pb(1) sites with distorted trigonal
structure defined by these Pb(2) atoms. While here I prism coordination with two different bondlengths are a rigid
twist of ≈ 24◦ between the upper and lower triangles. A car-
consider OH−1 filling the hexagonal column, I obtained toon of the resulting crystal-field diagram for Cu-d9 is given
similar results for O only in the column. Typically ap- on the right.
atites adopt the hexagonal P 63 /m space group – here our
resulting structure has P 63 owing to the breaking of re-
flection from the OH molecule ordering. However, small of 3 Å) not only for the Cu on the Pb(1) site, but also
structural deviations from the P 63 /m commonly occur for all other Pb(1) sites. This results in a modification
depending on its constituents. of the polyhedral tilts throughout the structure, most
While the Pb(1) forms the overall framework with the notably in the PO4 polyhedra, as seen in Fig. 2(b). I
PO4 tetrahedra, the Pb(2) play a crucial role in Pb-O classify the distortion through analysis of the symmetry-
connectivity and polyhedra tilts throughout the struc- adapted phonon modes before and after Cu substitution
ture. In fact, while both Pb(1) and Pb(2) possess 6s2 finding the structural distortion to be caused by Γ1 and
lone pairs, only the latter is stereochemically active. I Γ2 modes of amplitudes 1.19 Å and 1.78 Å, respectively.
verify this by calculating the electronic localization func- Visualization of these symmetry-adapted modes is given
tion (ELF) as shown in Fig. 1(b), finding that the Pb(2) in the SI (Fig. S1); this analysis confirms that the struc-
lone pairs form a chiral arrangement that make an angle tural distortion caused by the Cu substitution on Pb(1)
of ≈ 105◦ with the a-axis (see SI for more details). The is primarily driven by polyhedral tilts of the PO4 and
arrangement of the lone pairs sets the resulting oxygen their corner-shared oxygen neighbors.
coordination of the Pb(2), namely its six oxygens are ar- Looking closely at the change in coordination of Cu on
ranged asymmetrically as a result of their repulsion from the Pb(1) site, I see the Cu2+ is now six-coordinated with
the lone pair forming a chiral charge density wave. Since oxygen forming a distorted Jahn-Teller trigonal prism.
these oxygen are corner shared with the PO4 , the struc- In an ideal trigonal prism the anions are arranged at the
tural distortions associated with the Pb(2) lone pair ac- vertices of a regular triangular prism with the transition
tivity propagate throughout the structure. Such an effect metal at the center with D3h symmetry. However, break-
has been discussed previously in various apatite materi- ing in-plane mirror symmetry results in Jahn-Teller dis-
als, noting that the lone pair ordering differs depending torted trigonal crystal field with C3v symmetry, such as is
on the specific system [18, 19]. found in the Janus compounds [20]. Here I find the mir-
I next consider the substitution of Cu on a Pb(1) ror is already broken by the next-nearest neighbor P ions
site resulting in CuPb9 (PO4 )6 (OH)2 , with the fully opti- – the Cu-O distance that has nearby P is 2.06 Å, whereas
mized structure shown in Fig. 2(b). I find several changes the Cu-O without surround P ions is 2.35 Å. Such out-of-
to the structure with the inclusion of Cu. Firstly, all lat- plane asymmetry in the Cu environment results in a lo-
tice parameters decrease with a going from 9.875 Å to cal dipole in z which can also influence the resulting elec-
9.738 Å and c going from 7.386 Å to 7.307 Å. While tronic structure [20]. In addition to the mirror symmetry
my calculated lattice parameters agree well with those breaking, the O triangles are also rotated at a small angle
reported in Ref.[10], I find a greater structure collapse of ∼ 24◦ with respect to each other, referred to as a Bailar
than their report of a going from 9.865 Å without Cu to twist in molecules[21], giving the final distorted trigonal
9.843 Å with Cu, and c going from 7.431 Å without Cu prism. Interestingly, the same structural distortion has
to 7.428 Å with Cu. Interestingly, I find Cu substitution been observed upon substitution of U on the Ca(1) site in
results in a global structural distortion that results in a fluorapatite Ca10 (PO4 )6 F2 as measured from X-ray ab-
change in coordination from nine to six (with a cutoff sorption spectroscopy [22]. There the authors found that
 3

FIG. 4. Calculated spin-polarized electronic band structure

FIG. 3. Calculated spin-polarized electronic band structure in smaller energy range around the Fermi level showing the
(left) and corresponding density of states (right). The spin-up isolated two-band Cu-d manifold. The Fermi level is set to 0
bands are depicted in solid orange, and the spin-down bands eV and is marked by the dashed line.
are dashed blue. The total density of states is shaded grey
with projections shown of the Cu-d orbitals (pink) and its
neighboring O-p orbitals (green). In both plots the Fermi dyz /dxz model, similar to that originally suggested for
level is set to 0 eV and is marked by the dashed line. Fe-pnictide superconductors [23, 24]. However, unlike
other correlated-d band superconductors, in this system
the Cu-d bands are particularly flat – there is minimal
the U substitution on the Ca(1) sites causes an usual 6- band broadening from neighboring oxygen ions. If pre-
coordinated structure – a trigonal prism with six equal vious assumptions about band flatness driving supercon-
bondlengths of 2.06Å with an undetermined Bailar twist ductivity are correct, then this result would suggest a
angle, however. much more robust (higher temperature) superconduct-
ing phase exists in this system, even compared to well-
established high-TC systems.
Electronic Structure While the calculations presented here are performed
for GGA+U with a U = 4 eV applied to the Cu-d, I find
I present the calculated spin-polarized electronic struc- the same qualitative features of the band structures for a
ture in Fig. 3. Remarkably, I find an isolated set of flat wide range of U values, as presented in the SI. This sug-
bands crossing the Fermi level, with a maximum band- gests that the presence of such an isolated manifold of
width of ∼130 meV (see Fig.4) that is separated from flat Cu-d bands is a feature of the structural reconstruc-
the rest of the valence manifold by 160 meV. Such a tion and new crystal field environment provided by the
narrow bandwidth is particularly indicative of strongly apatite network. I further confirm this by calculating the
correlated bands. These especially flat bands are con- band structure of the Cu-substituted compound without
sistent with the Cu-O coordination where I find Cu-O performing a structural relaxation, that is, taking the
bondlengths of 2.35 Å and 2.06 Å in the distorted trigo- structure of the ideal Pb10 (PO4 )6 (OH)2 and replacing a
nal prism. For comparison, the Cu-O bondlengths in the Pb(1) with Cu without allowing the structure to relax.
cuprate superconductors are typically ≤ 2Å (in-plane) In this case, I find slight spin-polarization in the band
and ≤ 2.3Å (apical) [3], giving further evidence of the structure, however all Cu-d states are now in the bulk
unusual coordination and resulting band localization in valence manifold and do not form an isolated manifold
this isolated Cu-d manifold. (see SI Fig. S3). Finally, the top of the valence band is
The crystal-field splitting for the trigonal prism com- made up of the Pb(2)-s states, confirming their contribu-
prises a single dz2 , doubly degenerate dxy and dx2 −y2 , and tion to the stereochemical activity of the lone pairs (see
doubly degenerate dyz and dxz . In fact, the relative posi- SI).
tioning of the dz2 and the dyz /dxz is set by the anisotropy I next investigate the exchange interactions between
induced by the mirror-symmetry breaking – large values Cu ions in different unit cells by constructing doubled
of asymmetry can cause the dz2 to be destabilized. In unit cells in the in-plane and out-of-plane directions. I
our case for Cu2+ with a d9 configuration I expect half find that there is a preference of 2 meV/Cu for ferro-
filling of the doubly degenerate dyz /dxz bands – this is magnetic coupling in the out-of-plane direction (with a
corroborated with our calculations in Fig. 3 where I find Cu-Cu separation of c = 7.307 Å), whereas a preference
two bands of dyz /dxz character at the Fermi level that of 7 µeV/Cu for antiferromagnetic coupling in the in-
are are half filled. My results suggest that low-energy plane direction (with a Cu-Cu separation of a = 9.738Å).
physics of this system can be described by a two-band However, while this result is indicative of the potential
 4

exchange interactions in the system, it relies on the un- of the effective attractive interaction and ρ0 (EF ) is the
realistic assumption that the Cu ions will sit on the same density of states at the Fermi surface. In a flat band
substitution position in each unit cell. A full study of where the density of states diverges, TC is proportional
potential exchange interactions for various Cu locations to |U |. This suggests that, at low interaction strengths,
is out of the scope of this work, and will crucially be the critical temperature in flat bands could be signifi-
informed by the Cu locations determined in experiment. cantly enhanced, as is currently being explored in various
So far all calculations have been for Cu on the Pb(1) moiré systems[29]. In our case, the correlated bands have
site, which is the site occupancy reported in Ref. [10]. a bandwidth of ∼ 130 meV which is less than their sepa-
I also calculated the structural and electronic properties ration from the rest of the valence manifold (160 meV) at
of Cu in the Pb(2) site with the resulting structure and the level of density functional theory. While strong cor-
bands given in the SI. In this location, the Cu interrupts relations will need to be taken into account adequately
the hexagonal network that is characteristic of the ap- to accurately describe these scenarios, these are promis-
atite structure, which causes a structural rearrangement ing hints for these proposals that predict that the TC is
to the lower P 1 symmetry, which would have a profound proportional to the interaction strength.
effect on structural measures such as x-ray diffraction. In contrast to conventional superconductors, where
The Cu substituted in this position is now tetrahedrally phonon-mediated interactions govern Cooper pair forma-
coordinated by oxygen, and has a significantly different tion, there is wide consensus that other bosonic excita-
electronic structure with no correlated d bands crossing tions are responsible for the attractive pair formation in
the Fermi level, as detailed in the SI. In fact, I find that high-TC superconductors ranging from paramagnons to
Cu on this Pb(2) is 1.08 eV more energetically favorable spin- and charge-density waves [30]. In this system, I
than Cu on the Pb(1) site, suggesting possible difficulties have identified several potential sources of fluctuations
in robustly obtaining Cu substituted on the Pb(1) site. that could contribute to pairing. Firstly, I identified a
charge density wave (CDW) driven by chiral lone pair
ordering on the Pb(2) sites – the presence of this CDW is
DISCUSSION strongly connected to the structural rearrangement that
occurs when Cu is incorporated into the Pb(1) lattice
These theoretical results suggest that the apatite struc- sites. In addition to this, I identified two zone-center
ture provides a unique framework for stabilizing highly phonon modes that trigger the global structural deforma-
localized Cu-d9 states that form a strongly correlated flat tion that occurs as a result of the Cu substitution, sug-
band at the Fermi level. The central role of stereochem- gesting potentially strong electron-phonon coupling for
ically active 6s2 lone pairs of Pb(2) manifests in the for- these modes. Finally, I calculated the relative exchange
mation of a chiral charge density wave and the propaga- interactions between Cu in neighboring unit cells. Inter-
tion of structural distortions with connected polyhedra. estingly for the out-of-plane coupling, that is, along the
When Cu is substituted on a Pb(1) site, the result is a Cu-Pb-Cu one-dimensional chains, I find ferromagnetic
cascade of structural alterations, including reduced lat- coupling is favored by 2 meV/Cu over antiferromagnet
tice parameters, changes in coordination, and modified coupling, even though the Cu are over 7 Å apart, sug-
polyhedral tilts, leading to a local Jahn-Teller distorted gesting that spin fluctuations could also play a key role.
trigonal prism around Cu. This results in an unusually Finally, the calculations presented here suggest that
flat set of isolated dyz /dxz bands with half-filling. Cu substitution on the appropriate (Pb(1)) site displays
I briefly note that achieving such a crystal field envi- many key characteristics for high-TC superconductivity,
ronment should also be possible in intercalated twisted namely a particularly flat isolated d-manifold, and the
heterogeneous bilayers where selection of different hetero- potential presence of fluctuating magnetism, charge and
bilayers can provide the mirror symmetry breaking, while phonons. However, substitution on the other Pb(2) does
moiré twist can provide an arbitrary rotation of the up- not appear to have such sought-after properties, despite
per and lower triangles. In fact such a platform would be being the lower-energy substitution site. This result hints
ideal for probing the physics found here given its broad to the synthesis challenge in obtaining Cu substituted on
range of tunability and the state-of-the-art characteriza- the appropriate site for obtaining a bulk superconducting
tion probes for their interrogation [25]. sample. Nevertheless, I expect the identification of this
I now discuss the potential implications of these fea- new material class to spur on further investigations of
tures of Cu-substituted apatite for possible high-TC su- doped apatite minerals given these tantalizing theoretical
perconductivity, and in particular the role of the flat signatures and experimental reports of possible high-TC
bands and the presence of competing magnetic interac- superconductivity.
tions and phonons. Isolated, flat bands have long been Acknowledgements-. I am grateful to David Prender-
a target for achieving high-TC as predicted from BCS gast, Adam Schwartzberg, Shuhada’ Sadaqat and John
theory[26–28]. The critical temperature TC given by by Vinson for insightful discussions and encouragement. I
TC ∝ exp(−1/|U |ρ0 (EF )), where |U | is the magnitude also thank D. Kwabena Bediako, Donny Evans, Kather-
 5

ine Inzani, Adam Schwartzberg and John Vinson for feed-

back on this draft, and Jeff Neaton and Richard Mar-
tin for further comments. This work was funded by the
U.S. Department of Energy, Office of Science, Office of
Basic Energy Sciences, Materials Sciences and Engineer-
ing Division under Contract No. DE-AC02-05-CH11231
within the Theory of Materials program. Computational
resources were provided by the National Energy Research
Scientific Computing Center and the Molecular Foundry,
DOE Office of Science User Facilities supported by the
Office of Science, U.S. Department of Energy under Con-
tract No. DEAC02-05CH11231. The work performed at
the Molecular Foundry was supported by the Office of
Science, Office of Basic Energy Sciences, of the U.S. De-
partment of Energy under the same contract.
 6

SUPPLEMENTAL MATERIAL

Calculation Details

I used the Vienna Ab initio Simulation Package

(VASP) [12–15] for all density functional theory (DFT)
calculations with projector augmented wave (PAW)
pseudopotentials [31, 32] including Pb 5d10 6s2 6p2 , Cu
3d10 4s1 , P 3s2 3p5 , O 2s2 2p4 , and H 1s1 as valence elec-
trons. I used a plane wave cut-off energy of 600 eV and
a 6 × 6 × 8 Gamma-centered k-point grid for structural
optimizations and a 10 × 10 × 12 Gamma-centered k-
point grid for the density of states and electron localiza-
tion functional calculations. All calculations are done
using the generalized gradient approximation (GGA)
based exchange-correlation functional PBEsol [33], with
a Hubbard-U of 4 eV applied to the Cu-d states as im-
plemented with the Dudarev approach unless otherwise
stated [34]. This gives lattice parameters within 1% of
experiment [10] as summarized in Table ??. The elec-
tronic convergence criterion is set to 10−6 eV and the
force convergence criterion is set to 0.01 eV / Å. We
included spin-orbit coupling self-consistently in the elec-
tronic structure calculations where specified. Electronic
structure plots were generated using the sumo software
package [35].

Structural Distortion

We used the AMPLIMODES software from the Bil-

bao Crystallographic Server[36] to analyze the symmetry-
adapted phonon modes in going from the stoichiomet-
ric Pb10 (PO4 )6 (OH)2 with space group P 63 to the Cu-
substituted compound with Cu on the Pb(1) site, re-
sulting in space group P 3. A summary of the resulting
distortion is given in Table I with a visualization of the
two modes, Γ1 and Γ2 depicted in Fig. 10.
 7

Irrep K-vector Direction Subgroup Amplitude (Å)

Γ1 (0,0,0) (a) P 63 1.192
Γ2 (0,0,0) (a) P3 1.778

TABLE I. Symmetry-adapted distortion information in going from stoichiometric Pb10 (PO4 )6 (OH)2 to Cub9 (PO4 )6 (OH)2
where Cu is on the Pb(1) site.

FIG. 5. Summary of displacements of symmetry-adapted modes upon Cu substitution on the Pb(1) site. (a) and (b) show the
Γ1 mode which retains the P 63 space group. (c) and (d) show Γ2 mode which reduces the P 63 space group to P 3. The arrows
show the ionic motion under each phonon mode with their length being proportional to the amplitude of the displacement.
 8

FIG. 6. (Top panel) Calculated spin-polarized band structure for different values of U on Cu-d states for CuPb9 (PO4 )6 (OH)2
with (a) U = 2 eV, (b) U = 4 eV, and (c) U = 6 eV. In all plots the majority spin channel is shown as a solid orange line, and
the minority as a dashed blue line. (Middle panel) Zoomed-in version of the top plot of spin-polarized bands with a narrower
energy range. (Bottom panel) Calculated band structures without spin-polarization. The Fermi level is set to 0 eV in all plots
and is marked by the dashed line.

FIG. 7. Calculated spin-polarized band structures for (a) stoichiometric Pb10 (PO4 )6 (OH)2 and (b) CuPb9 (PO4 )6 (OH)2 without
structural optimization, that is, with the structure of (a) with one Cu replaced for Pb(1). In all plots the majority spin channel
is shown as a solid orange line, and the minority as a dashed blue line. The Fermi level is set to 0 eV in all plots and is marked
by the dashed line.
 9

FIG. 8. Calculated electronic localization function for a 2×2×1 supercell of Pb10 (PO4 )6 OH2 showing chiral charge density
wave induced by Pb(2) lone pairs.

FIG. 9. Results for CuPb9 (PO4 )6 (OH)2 with Cu on the Pb(2) site. (a) The resulting fully relaxed structure for Cu on the
Pb(2) site. The inset shows the new local environment of the Cu ion and resulting Cu-O bond lengths. (b) The corresponding
calculated spin-polarized electronic structure showing the bands (left) and spin-polarized density of states (right). The total
and orbital-projected density of states are shown for the Cu-d and O-p states. The Fermi level is set to 0 eV in all plots and
is marked by the dashed line. .
 10

FIG. 10. alculated non-spin-polarized electronic structures for (a) CuPb9 (PO4 )6 O and (b) CuPb9 (PO4 )6 (OH)2 . For both
calculations the Cu is placed on the Pb(1) site. The total and orbital-projected density of states are shown for replaced for
Pb(1). In all plots the majority spin channel is shown as a solid orange line, and the minority as a dashed blue line. The Fermi
level is set to 0 eV in all plots and is marked by the dashed line. Note that these were both calculated using the fully relaxed
structures that included spin-polarization.
 11

STRUCTURE FILES 0.00000 0.00000 0.44500 H

0.00000 0.00000 0.84006 H
The resulting relaxed structures in POSCAR format
are given below. The specific geometry is given in the
headers.

Cu on Pb(1) for OH columns

1.00000
9.73753 0.00000 -0.00000
-4.86876 8.43294 0.00000
0.00000 0.00000 7.30653
Cu Pb P O H
1 9 6 26 2
Direct(44) [A1B9C6D26E2]
0.33333 0.66666 0.99594 Cu
0.66666 0.33333 0.97556 Pb
0.66666 0.33333 0.47828 Pb
0.33333 0.66666 0.47831 Pb
0.24304 0.99176 0.23619 Pb
0.74033 0.00357 0.74019 Pb
0.00823 0.25127 0.23619 Pb
0.99642 0.73675 0.74019 Pb
0.74872 0.75695 0.23619 Pb
0.26324 0.25966 0.74019 Pb
0.39800 0.37662 0.22932 P
0.59606 0.63623 0.76062 P
0.62337 0.02138 0.22932 P
0.36376 0.95982 0.76062 P
0.97861 0.60199 0.22932 P
0.04017 0.40393 0.76062 P
0.33463 0.49455 0.22138 O
0.63904 0.50466 0.73061 O
0.50544 0.84007 0.22138 O
0.49533 0.13438 0.73061 O
0.15992 0.66536 0.22138 O
0.86561 0.36095 0.73061 O
0.58275 0.46627 0.20829 O
0.42113 0.55711 0.83037 O
0.53372 0.11647 0.20829 O
0.44288 0.86401 0.83037 O
0.88352 0.41724 0.20829 O
0.13598 0.57886 0.83037 O
0.32305 0.24965 0.07561 O
0.70965 0.76259 0.89995 O
0.75034 0.07339 0.07561 O
0.23740 0.94705 0.89995 O
0.92660 0.67694 0.07561 O
0.05294 0.29034 0.89995 O
0.61424 0.72712 0.57832 O
0.36008 0.28424 0.41289 O
0.27287 0.88712 0.57832 O
0.71575 0.07584 0.41289 O
0.11287 0.38575 0.57832 O
0.92415 0.63991 0.41289 O
0.00000 0.00000 0.70585 O
0.00000 0.00000 0.30956 O
 12

Cu on Pb(2) for OH columns Cu on Pb(1) for O columns

1.00000 1.00000
9.72342 0.04200 0.00197 9.62586 -0.00000 -0.00000
-4.82301 8.48804 0.00345 -4.81293 8.33624 0.00000
0.00025 0.00382 7.33474 0.00000 0.00000 7.21588
Cu Pb P O H Cu Pb P O
1 9 6 26 2 1 9 6 25
Direct Direct
0.29671 0.05116 0.33235 0.66667 0.33333 0.98511
0.32381 0.67469 0.98111 0.00692 0.77593 0.26511
0.65139 0.32730 0.98009 0.99756 0.23907 0.75708
0.66758 0.35114 0.48809 0.22407 0.23099 0.26511
0.32169 0.66632 0.47791 0.76093 0.75849 0.75708
0.75782 0.00049 0.75985 0.76901 0.99308 0.26511
0.99392 0.24366 0.24551 0.24151 0.00244 0.75708
0.99739 0.77380 0.76869 0.33333 0.66667 0.01001
0.74898 0.76332 0.23058 0.33333 0.66667 0.50100
0.22801 0.23596 0.76545 0.66667 0.33333 0.49255
0.40018 0.38468 0.22144 0.62789 0.59460 0.23001
0.59137 0.62723 0.73080 0.38566 0.39352 0.76637
0.61312 0.02576 0.23816 0.40540 0.03329 0.23001
0.37685 0.97309 0.72186 0.60648 0.99214 0.76637
0.96697 0.59735 0.22558 0.96671 0.37211 0.23001
0.01790 0.40361 0.72964 0.00786 0.61434 0.76637
0.31214 0.48018 0.22872 0.49081 0.63441 0.25509
0.66266 0.51575 0.73906 0.48303 0.30416 0.77686
0.51223 0.84427 0.23528 0.36559 0.85640 0.25509
0.47793 0.15426 0.72594 0.69584 0.17887 0.77686
0.14692 0.66732 0.22652 0.14360 0.50919 0.25509
0.83604 0.33147 0.73496 0.82113 0.51697 0.77686
0.58332 0.49499 0.22798 0.75643 0.71787 0.09385
0.40826 0.53051 0.72984 0.25828 0.33902 0.92184
0.50719 0.10481 0.25528 0.28213 0.03856 0.09385
0.48254 0.89402 0.72791 0.66098 0.91926 0.92184
0.88433 0.41328 0.23340 0.96144 0.24357 0.09385
0.10201 0.58649 0.73148 0.08074 0.74172 0.92184
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