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2 Defect Crystal Chemistry (DCC) Lattice Parameter Model

2 Defect Crystal Chemistry (DCC) Lattice Parameter Model

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Shannons rC(M4ỵ, Ln3ỵ) data [28]

versus oxygen coordination (CN)

plot for the 5 M4ỵs and the 12

Ln3ỵs (see text for details).

(Reproduced from Ref. 1)

expressions drawn in Fig. 4.3, reproduce well the original Shannon’s data, and enable rC in Eq. (4.3b) to be calculated over

the entire 0 y 1.0 (8 ! CN ! 6) range.

Since CN(Ln3ỵ, M4ỵ) of the system are inter-related by the equation

1 yị CNM4ỵ ị ỵ y CNLn3ỵ ị ¼ CNðaverÞ ¼ CNðrandomÞ ¼ ð8 À 2yÞ:


Equations (4.3–4.7) after all mean that the only one nonrandomness CN(Ln3ỵ) can describe the macroscopic a0(ss)–

microscopic nonrandomness behavior of the system in a fully coupled unified manner [1]. The validity of this unified

picture of non-Vegardianity and nonrandomness has been established quantitatively through a systematic a0 analysis of

total 12 (6 each) parent F-type M4ỵ(ẳCe and Th)-Lns(¼La, Nd, Sm, Eu, Gd, and Y) in Ref. 1. Here non-Vegardianity is the

“generalized” one relative to the a0(random CN), different from the “apparent” one relative to the conventional VL in

Eq. (4.4). For clarity, these two are listed below in pair:

Apparent non-Vegardianity :

Generalized non-Vegardianity :

Da0 ¼ a0 À a0 VLị;


dDa0 ẳ da0 ẳ a0 a0 randomị:


Using the notation for coupled generalized non-Vegardianity (da0) and nonrandomness in Ref. 1

Case1ịLn3ỵ VO associative CNLn3ỵ ị < 8 2yị < CNM4ỵ ịị;

for da0 < 0;


Case2ịM4ỵ VO associativeCNLn3ỵ ị > 8 2yị > CNM4ỵ ịị;

for da0 > 0;


the results clarified that the M4ỵ ẳ Th are all largely negatively da0 ((0) case (1) Ln3ỵVO associative due to the strongly

highest CN ẳ 8-adhering nature of the largest Th4ỵ, while the M4ỵ ẳ Ce are only marginally da0 nonrandom due to the

more flexibly lower CN ¼ 7 and 6-adopting nature of the smaller Ce4ỵ, shifting smoothly from the modestly da0 < 0 case

(1) Ln3ỵVO associative to the modestly da0 > 0 case (2) Ce4ỵVO associative at around the Ln3ỵ ẳ Sm and Nd with

increasing rC(Ln3ỵ) (see Figs. 6a, b and 7a, b of Ref. 1). Albeit not detailed here, these results agree fairly well with their

reported local structure and s(ion) data [13,35]. In addition, the model is compatible with ion-packing model [22] so that

this can also be used as a handy method to extract the metal (M4ỵ and Ln3ỵ)-oxygen (O2) bond length (BL) data of the

system, consistently with the XAFS BL data. Moreover, the model is flexibly expandable [2]: combining these overall

nonrandomness CN(Ln3ỵ, M4ỵ) data with restricted (non-)randomness of DF oxides one can not only derive the full

set of their mutually nonrandom O2 eight-, seven-, and sixfold coordinated M4ỵ and Ln3ỵ concentrations and Ln3ỵ





(a) Phase-diagram (a0) data of the

five M1ÀyEuyO2Ày/2 (M4ỵ ẳ Hf, Zr,

Ce, U, and Th): F: defect-fluorite

type, C: the C type, P: pyrochlore

type, B: the B type. Each conventional Vegard-law linear

a0(VL) is drawn in dotted line

(see text for details). (b) IS(Eu3ỵ)

data of the five M1yEuyO2y/2

(M4ỵ ẳ Hf, Zr, Ce, U, and Th)

(see text for details).

zero- to fourfold coordinated O2À and VO concentrations but also give a new consistent description of their s(ion)(max)

behavior in low y range.

Such significant success of the DCC model signifies that these parent F-type DF oxides such as M4ỵ ẳ Ce, Th, and

Ans are indeed F-type MO2–C-type LnO1.5 (i.e., F–C) binary systems, their coupled a0-nonrandomness behavior being

fully describable as “distortional–dilation” phenomenon of the latter; fC > fF in Fig. 4.2a. Of course the model should be

further validated with more various systems and with their more various structure and basic property data. This is what

ossbauer and XRD a0 data. As

we initiate here with these M4ỵ (Ce, Th, U, Zr, and Hf)-Eus and Zr–Gd and with their Lns M€

mentioned in (4.1), our prime interest here is in “how to attain such successful F–C binary ! F–P–C ternary or other

type model extension for P-type Zr–Eu, Hf–Eu, and Zr–Gd exhibiting more complex sigmoid a0 behavior (as is obvious in

Figs. 4.4a and 4.6a).

As plotted in Fig. 4.2a and b, many larger M4ỵ(Pb, Zr, and Hf)-based P and d systems at y ¼ 0.50 [17b,26,36] are

already near equally largely distortion dilated to C LnO1.5 at y ¼ 1.0; fP(y ¼ 0.50) $ fC(y ¼ 1.0) ) fF(y ¼ 0). Besides, in

Fig. 4.2b, fP shows a clear transit to the lowest fP $ fF side in the rC < $0.75 nm range. This transit region extending over

the $0.75 < rC < $0.93 nm just coincides with the existence range of fd ¼ a0(d) data of many less nonrandom (i.e., less

distortion dilated) d Ln-SZ(SH)s at y ¼ 0.50. The reason why a0 data of truly d Zr3Yb4O12 and Hf3Er4O7 etc. at y ¼ 4/7

(Eq. (4.2b)) are not plotted here is twofold; first, of interest here are the cubic a0 data of “disordered” DF-type Zr2Yb2O7

and Hf2Er2O7 at y ¼ 0.50 rather than those of the truly d-type former at y ¼ 4/7 that are obtained after prolonged lower

temperature annealing of the system. Second, the former have hexagonal and/or rhombohedral structure [17b,29,36] for

which it is hard to know the exact cubic a0(d) data.

The presence of this fP transit region over $0.75 < rC < $0.93 nm tells us that as the rC range in Eq. (4.3b) covered

by these Ln-SZ(SH)s spans from 0.84(0.83) nm for the “cubic” Zr(Hf)O2 at y ¼ 0 (CN ¼ 8) to 0.7350.1032 nm for the C

LnO1.5s (Ln3ỵ ẳ ScLa) at y ¼ 1.0 (CN ¼ 6), one will not be able to describe their complex sigmoid a0 behavior without

properly incorporating this P- and d-based a0 transit behavior of the system at y ¼ 0.50 into the model. Leaving all the




argument of this crucial point later, we propose here a fP form using a Fermi–Dirac statistics-type function of [FDF] ẳ 1/

[1 ỵ exp{(0.0830 rC)/0.003)}]

f P r C ị ẳ f F ỵ f C f F ị ẵFDF ẳ f F ỵ f C f F ị=ẵ1 ỵ expf0:0830 rC ị=0:003ịg

for r C ẳ fr C M4ỵ ịVIịị ỵ r C Ln3ỵ ịVIIIịịg=2:


This fP is numbered (Eq. (4.5c)) as the third a0 functional to the first fF and the second fC in Eqs. (4.5a) and (4.5b).

Equation (4.5c) has the merit of enabling such smooth fP transit between fF and fC to be expressed by [FDF] without using

any complex function of fP itself. This fixes at least the macroscopic-a0 side standpoint of possible F–P–C ternary DCC

model for SZ(SH)s. The feasibility of such phenomenological model-extension approach from the macroscopic-a0 side

will be examined and discussed in Section 4.4.



Lns (151 Eu and 155 Gd)-M€

ossbauer structure and XRD a0 data of DF oxides are described together with key magic-angleossbauer data, already discussed in detail in Ref. 4, are

spinning (MAS) NMR CN(Ln3ỵ, M4ỵ) data. All these Lns-M

resummarized here more briefly but more carefully from the present viewpoint of making their full use for possible

model extension to SZ(SH)s. To our knowledge, these are all the available Lns-M€

ossbauer data of MO2–LnO1.5-type DF

oxides to date except the ideal P M2Ln2O7s at y ¼ 0.50. Hitherto only the latter used to be the target of various Lns


ossbauer studies for M4ỵ (Hf, Zr, Pt, Pb, Mo, Sn, Ru, Ir, Ti, etc.) and Ln3ỵ (152;154;155;156 Gd, 151;153 Eu, 170 Yb, 171 Dy, etc.)


[26,37,38]. This is most plausibly because they have the defect(VO)-free well-defined unique axial-anisotropic Ln(VIII)À

8O coordination with two short apical O2s and six more-distant O1s (Fig. 4.1c and Eq. (4.2a)), best fit for investigating the

so-called Goldanskii–Karyagin (GK) effect; asymmetric QS caused by anisotropic lattice vibration. Note that these Ln-SZ

(SH)s show no GK effect probably due to the more disordered nature of the oxygen sublattice than in the smaller M4ỵ

(Ti, Ru, etc.), as briefly mentioned in Ref. 38c.



€ ssbauer and Lattice Parameter Data of M-Eus (M4ỵ ẳ Zr, Hf, Ce, U, and Th)


As summarized in Fig. 4.4a and b, this XRD and 151 Eu M€

ossbauer study of the five M-Eus [4,18] has clarified the

controversial P–DF local structure of SZ(SH)s in the global scope of their detailed Eu3+ isomer-shift (IS(Eu3+)) data

traversing the wide rC(M4+(VIII)) range of 0.083(Hf) ! 0.084(Zr) ! 0.097(Ce4+) ! 0.100(U4+) ! 0.105(Th4+) nm. As to

ossbauer spectroscopy, only mention that this gives near-symmetric or slightly asymmetric broader singlethe 151 Eu M€

line room-temperature spectra for F-type Th–Eu, Ce–Eu, and U–Eu or P-type Zr–Eu and Hf–Eu, respectively, due to the

low g-ray energy (21.5 keV) of a 1.85 GBq SmF3 source, and their peak Doppler velocity gives the IS(Eu3+) relative to a ref.

EuF3. Since to derive the QS was hardly possible for the majorities except some largely distorted Eu-SZ(SH)s at around

y $ 0.50 [18f], discussion here is confined to these IS(Eu3+) data.

In oxides, the IS(Eu3+) is known to correlate well with the average BL(Eu3+ÀÀO) (and CN) of the system that the

shorter the BL (and smaller the CN) the larger the IS(Eu3+) [39]. Referring to this qualitative BL(Eu3+ÀÀO)–IS(Eu3+)

correlation, Fig. 4.4a and b reveals the notable a0–IS(Eu3+) behavior of these systems that can be classified into the

following three types of characteristic y dependence:

(1) F-Type UEu with Nearly Constant IS(Eu3ỵ) $ 0.49 Over the Entire Single Cubic-F 0
ossbauer powder-sample preparation, the initially oxygenthat during the room-temperature XRD and 151 Eu M



5ỵ 3ỵ

deficit DF-type U4ỵ

1y Euy O2y=2 is almost completely oxidized to the oxygen stoichiometric U1À2y Uy Euy O2



until reaching the fully oxidized U0:5 Eu0:5 O2 at y ¼ 0.50. This conclusion agrees with the observed near-linearly

decreasing a0 curve till y $ 0.5 in Fig. 4.4a different from the other four. This is the fF (¼a0(F)) curve of F MO2 in

Eq. (4.5a) itself in Fig. 4.2a. Shannon’s rC(Eu3ỵ(VIII)) ẳ 0.1066 nm and ra(O2) ẳ 0.138 nm [28] or the minimum

0.1373 nm derived in Ref. 1 (see Fig. 10 of Ref. 1) leads to

BLEu3ỵ VIIIị Oị ẳ r C Eu3ỵ VIIIịị ỵ r a O2 ị ẳ 0:1066 ỵ 0:1373 $ 0:138 ẳ 0:2439 $ 0:2446 nm

at ISEu3ỵ VIIIịị ẳ 0:49; for U Eu in 0


$ 0:50:





This gives a reference point to fix the BL(Eu3ỵO)IS(Eu3ỵ) correlation. In mostly diphasic y > 0.50 range,

we found a C-type line phase at y $ 0.86 plausibly corresponding to an Eu analogue of d [U6ỵ]A[Y3ỵ6]BO12 at

y ẳ6/7 ẳ 0.857 [29]. But its C-type structure implies the possibility that a partially oxidized C-type

U5ỵ,6ỵEu3ỵ6O12z (z > 0) is formed there.

(2) DF-Type CeEu and Th–Eu. These two have very similar smoothly increasing near-linear IS(Eu3ỵ)y curves, that

is, very similar smoothly decreasing near-linear BL(Eu3ỵO)y curves, as expected for the stable M4ỵ-bearing

DF oxides wherein the CN(random) ¼ 8 À 2y in Eq. (4.7) decreases linearly from 8 to 6 for y ¼ 0–1.0 [4]. This is

a notable IS(Eu3ỵ) feature in view of the largely a0(ThEu) ) a0(Ce–Eu) behavior over the entire y range in

Fig. 4.4a. Moreover, both IS(Eu3ỵ)s at y ! 0 extrapolate not to $0.49 for the Eu3ỵ(VIII) in the above UEu in

ossbauer evidence that both

Eq. (4.10) but to the obviously higher $0.64. This IS(Eu3ỵ) feature is a clear 151 Eu-M€

the systems are near-equally significantly case (1) Eu3ỵVO associative so that both the BL(Eu3ỵO)s are

definitely shorter than that of U–Eu. Their strong case (1) nonrandom nature was clarified in Ref. 1 with the

DCC model analysis of each one representative a0 data set reported in Refs. 4 and 40, respectively. We will

make the more complete a0 analysis of both systems in Section 4.4 using all the available a0 data.

(3) P-Type Stabilized Zr–Eu and Hf–Eu. These two again exhibit very similar characteristic V-shaped IS(Eu3ỵ) minima

at the ideal P composition of y $ 0.50 in sharp contrast with the above two. These data highlight this 151 Eu


ossbauer study, giving a convincing evidence for the intermediate P-based local structure of the apparently

disordered cubic-stabilized DF phase of SZ(SH)s. Both IS(Eu3ỵ)s (and therefore both BL(Eu3ỵO)s) appear to

decrease (increase) initially rather slowly in the $0.20 y $0.33 range and then sharply for y > $0.33 toward

each deep minimum (sharp maximum) at y $ 0.50. As argued in Ref. 4, the former $0.20 y < $0.33 and the

latter $0.33 y < $0.45 range are judged to represent the less distortion-dilated mutually isolated P-like Eu


À8O cluster state and the increasingly more distortion-dilated P-type short-range ordered axialanisotropic Eu(VIII)ÀÀ8O microdomain state in Fig. 4.1c, respectively. The sudden upturn of both a0–y curves

at y $ 0.33 in Fig. 4.4a, that is, the beginning of a0 hump, is a macroscopic signal for such abrupt coupled

microscopic–macroscopic structure change of the system accompanying “distortion–dilation.” 155 Gd


ossbauer data of Zr–Gd described in Section 4.3.2 further reinforce this conclusion. Such P-type order

seems to be quickly lost with a slight y ¼ 0.50–0.55 increase, and hence a narrow single phase found at y $ 0.70

for both systems would probably be non-P type.

Figure 4.5 shows 89 Y MAS-NMR CN(Y3ỵ, Zr4ỵ) data of YSZ [15a] and some other CN(Ln3ỵ, Zr(Hf)4ỵ) data.

The former data clearly show that even this macroscopic d-type YSZ is P-type strongly case (2) Zr4ỵVO associative, the

CN(Y3ỵ) lying close to the highest CN $ 8 till y $ 0.25 and then deceasing gradually to $7.24 at y $ 0.50. This was so

above mentioned in the sense that YSZ has more nonrandom CN((Y3ỵ, Zr4ỵ) curves than those of the d-type Zr3Y4O7

at y ¼ 4/7 given by Eq. (4.2b) and drawn here in pink symbol. Macroscopic P-type Eu-SH, Eu-SZ, and Gd-SZ are therefore

expected to have the respective CN(Eu3ỵ)s and CN(Gd3ỵ) lying even closer to the highest CN $ 8 up to the higher y

range till y $ 0.50, as drawn there in each (red, black, and green) color line.

V-shaped deep minima in both IS(Eu3ỵ)s at y $ 0.50 in Fig. 4.4b, that is, inverse V-shaped sharp maxima in both BL


(Eu O)s at y $ 0.50, means that with increasing y both BL(Eu3ỵO)s are increasingly more distortion dilated with the

growth of P-type short-range order until the most anisotropic and the most distortion-dilated ideally P-type macroscopic

phase with the longest average BL(Eu3ỵ(VIII)O) is realized at y ẳ 0.50, while keeping the near-highest CN(Eu3ỵ) $ 8 state

throughout in Fig. 4.5. Eu-SH has clearly a deeper IS(Eu3ỵ) minimum than Eu-SZ in Fig. 4.4b, that is, the longer and in fact the

longest BL(Eu3ỵ(VIII)O) near equal to that of UEu in the y 0.50 range, albeit the former has certainly the shorter a0

than the latter in Fig. 4.4a. Yet, their a0 difference seems to get smaller with increasing y ¼ 0.20–0.50. These facts suggest that

the smaller M4ỵ (Hf) system is locally more severely distortion dilated than the larger M4ỵ (Zr) system, also giving rise to a

larger macroscopic a0 dilation. One can thus conclude that a clear slope change at y $ 0.33 both of a0 À y in Fig. 4.4a and of IS

(Eu3ỵ)y in Fig. 4.4b to the steeper one observed for either system is a decisive signal of the isolated P-like ! collective Ptype-coupled microscopic–macroscopic distortion–dilational structure change of the system. In Section 4.4, we will derive a

O)IS(Eu3ỵ) correlation for all these M-Eus to extract their quantitative BL(Eu3ỵO) data.

quantitative BL(Eu3ỵ



ssbauer and Lattice Parameter Data of Zr1ÀyGdyO2Ày/2


Zr–Gd has many unique properties, such as the second s(ion)(max) in the ideal P Gd2Zr2O7 at y ¼ 0.50 [32] that

diminishes by a factor of $1/5 by the P ! DF disordering treatment above Ttr ¼ 1530  C [41] and the high radiation

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