The not so simple cubic structure of PbZr_1 - xTi_xO₃ (PZT): complex local structural effects in perov­skites

High-resolution neutron diffraction on the important piezoelectric lead zirconate titanate (PZT) has found that oxygen disorder exists well into the cubic phase. This unexpected result shows that within this phase there persists a remnant of the tilted oxygen octahedra present within the room-temperature ferroelectric phase. The result is that the cubic phase, far from having a simple crystal structure, exhibits a more complex local structure than had hitherto been thought.

Computing details

Figures

(PZ-523K_phase1) Lead zirconate titanate

Crystal data

PbZrO3	Z = 1
Mr = ?	? radiation
Cubic, Pm3m	T = 523 K
a = 4.159724 (4) Å	white
V = 71.98 (1) Å^3	cylinder, 15 × 20 mm

Data collection

HRPD, ISIS diffractometer	Scan method: time of flight
Radiation source: Neutron	2θfixed = 168
Specimen mounting: packed powder	Distance from source to specimen: 10 mm
Data collection mode: reflection	Distance from specimen to detector: 110 mm

Refinement

Least-squares matrix: full with fixed elements per cycle	Profile function: pseudo-Voigt
Rp = 0.029	33 parameters
Rwp = 0.036	Weighting scheme based on measured s.u.'s w = 1/σ(Yo,m)^2
Rexp = 0.024
χ^2 = 2.250	Background function: polynomial
? data points

Fractional atomic coordinates and isotropic or equivalent isotropic displacement parameters (Ų)

	x	y	z	Biso*/Beq

Atomic displacement parameters (Ų)

	U^11	U^22	U^33	U^12	U^13	U^23
Pb1	0.0583 (2)	0.0583 (2)	0.0583 (2)	0	0	0
Zr1	0.0111 (1)	0.0111 (1)	0.0111 (1)	0	0	0
O1	0.0670 (2)	0.0670 (2)	0.0088 (2)	0	0	0

Geometric parameters (Å, º)

Pb1—O1^i	2.9414	Pb1—O1^x	2.9414
Pb1—O1^ii	2.9414	Pb1—O1^xi	2.9414
Pb1—O1^iii	2.9414	Pb1—O1^xii	2.9414
Pb1—O1^iv	2.9414	Zr1—O1^xiii	2.0799
Pb1—O1^v	2.9414	Zr1—O1^ix	2.0799
Pb1—O1^vi	2.9414	Zr1—O1^v	2.0799
Pb1—O1^vii	2.9414	Zr1—O1^i	2.0799
Pb1—O1^viii	2.9414	Zr1—O1^xiv	2.0799
Pb1—O1^ix	2.9414	Zr1—O1^xv	2.0799
O1^xiii—Zr1^xiii—O1^xiv	90.000

Symmetry codes: (i) −x, −y, −z+1; (ii) −x+1, −y, −z+1; (iii) −x, −y+1, −z+1; (iv) −x+1, −y+1, −z+1; (v) −x, −z+1, −y; (vi) −x+1, −z+1, −y; (vii) −x, −z+1, −y+1; (viii) −x+1, −z+1, −y+1; (ix) −z+1, −x, −y; (x) −z+1, −x+1, −y; (xi) −z+1, −x, −y+1; (xii) −z+1, −x+1, −y+1; (xiii) −x, −y, −z; (xiv) −x, −z, −y; (xv) −z, −x, −y.

(PZ-573K_phase1) Lead zirconate titanate

Crystal data

PbZrO3	Z = 1
Mr = ?	? radiation
Cubic, Pm3m	T = 573 K
a = 4.161303 (4) Å	white
V = 72.06 (1) Å^3	cylinder, 15 × 20 mm

Data collection

HRPD, ISIS diffractometer	Scan method: time of flight
Radiation source: Neutron	2θfixed = 168
Specimen mounting: packed powder	Distance from source to specimen: 10 mm
Data collection mode: reflection	Distance from specimen to detector: 110 mm

Refinement

Least-squares matrix: full with fixed elements per cycle	Profile function: pseudo-Voigt
Rp = 0.028	33 parameters
Rwp = 0.035	Weighting scheme based on measured s.u.'s w = 1/σ(Yo,m)^2
Rexp = 0.024
χ^2 = 2.103	Background function: polynomial
? data points

Fractional atomic coordinates and isotropic or equivalent isotropic displacement parameters (Ų)

	x	y	z	Biso*/Beq

Atomic displacement parameters (Ų)

	U^11	U^22	U^33	U^12	U^13	U^23
Pb1	0.0598 (2)	0.0598 (2)	0.0598 (2)	0	0	0
Zr1	0.0116 (1)	0.0116 (1)	0.0116 (1)	0	0	0
O1	0.0663 (2)	0.0663 (2)	0.0093 (2)	0	0	0

Geometric parameters (Å, º)

Pb1—O1^i	2.9425	Pb1—O1^x	2.9425
Pb1—O1^ii	2.9425	Pb1—O1^xi	2.9425
Pb1—O1^iii	2.9425	Pb1—O1^xii	2.9425
Pb1—O1^iv	2.9425	Zr1—O1^xiii	2.0807
Pb1—O1^v	2.9425	Zr1—O1^ix	2.0807
Pb1—O1^vi	2.9425	Zr1—O1^v	2.0807
Pb1—O1^vii	2.9425	Zr1—O1^i	2.0807
Pb1—O1^viii	2.9425	Zr1—O1^xiv	2.0807
Pb1—O1^ix	2.9425	Zr1—O1^xv	2.0807
O1^xiii—Zr1^xiii—O1^xiv	90.000

Symmetry codes: (i) −x, −y, −z+1; (ii) −x+1, −y, −z+1; (iii) −x, −y+1, −z+1; (iv) −x+1, −y+1, −z+1; (v) −x, −z+1, −y; (vi) −x+1, −z+1, −y; (vii) −x, −z+1, −y+1; (viii) −x+1, −z+1, −y+1; (ix) −z+1, −x, −y; (x) −z+1, −x+1, −y; (xi) −z+1, −x, −y+1; (xii) −z+1, −x+1, −y+1; (xiii) −x, −y, −z; (xiv) −x, −z, −y; (xv) −z, −x, −y.

(PZT0.05-513K_phase1) Lead zirconate titanate

Crystal data

PbZr0.95Ti0.05O3	? radiation
Mr = ?	T = 513 K
Cubic, Pm3m	white
a = 4.15212 (2) Å	cylinder, 12 × 20 mm
V = 71.58 (1) Å^3	Specimen preparation: Prepared at 1173 K
Z = 1

Data collection

HRPD, ISIS diffractometer	Scan method: time of flight
Radiation source: Neutron	2θfixed = 168
Specimen mounting: packed powder	Distance from source to specimen: 10 mm
Data collection mode: reflection	Distance from specimen to detector: 110 mm

Refinement

Least-squares matrix: full with fixed elements per cycle	Profile function: pseudo-Voigt
Rp = 0.020	23 parameters
Rwp = 0.024	Weighting scheme based on measured s.u.'s w = 1/σ(Yo,m)^2
Rexp = 0.025
χ^2 = 0.922	Background function: polynomial
? data points

Fractional atomic coordinates and isotropic or equivalent isotropic displacement parameters (Ų)

	x	y	z	Biso*/Beq

Atomic displacement parameters (Ų)

	U^11	U^22	U^33	U^12	U^13	U^23
Pb1	0.0609 (3)	0.0609 (3)	0.0609 (3)	0	0	0
Zr1	0.0129 (2)	0.0129 (2)	0.0129 (2)	0	0	0
Ti1	0.0129 (2)	0.0129 (2)	0.0129 (2)	0	0	0
O1	0.0633 (4)	0.0633 (4)	0.0119 (4)	0	0	0

Geometric parameters (Å, º)

Pb1—O1^i	2.9360	Zr1—O1^xiii	2.0761
Pb1—O1^ii	2.9360	Zr1—O1^ix	2.0761
Pb1—O1^iii	2.9360	Zr1—O1^v	2.0761
Pb1—O1^iv	2.9360	Zr1—O1^i	2.0761
Pb1—O1^v	2.9360	Zr1—O1^xiv	2.0761
Pb1—O1^vi	2.9360	Zr1—O1^xv	2.0761
Pb1—O1^vii	2.9360	Ti1—O1^ix	2.0761
Pb1—O1^viii	2.9360	Ti1—O1^v	2.0761
Pb1—O1^ix	2.9360	Ti1—O1^i	2.0761
Pb1—O1^x	2.9360	Ti1—O1^xiv	2.0761
Pb1—O1^xi	2.9360	Ti1—O1^xv	2.0761
Pb1—O1^xii	2.9360	Ti1—O1^ix	2.0761
O1^xiii—Zr1^xiii—O1^xiv	90.000

Symmetry codes: (i) −x, −y, −z+1; (ii) −x+1, −y, −z+1; (iii) −x, −y+1, −z+1; (iv) −x+1, −y+1, −z+1; (v) −x, −z+1, −y; (vi) −x+1, −z+1, −y; (vii) −x, −z+1, −y+1; (viii) −x+1, −z+1, −y+1; (ix) −z+1, −x, −y; (x) −z+1, −x+1, −y; (xi) −z+1, −x, −y+1; (xii) −z+1, −x+1, −y+1; (xiii) −x, −y, −z; (xiv) −x, −z, −y; (xv) −z, −x, −y.

(PZT0.05-513K_phase2) Lead zirconate titanate

Crystal data

PbZr0.95Ti0.05O3	? radiation
Mr = ?	T = 513 K
Cubic, Pm3m	white
a = 4.1553 (1) Å	cylinder, 12 × 20 mm
V = 71.75 (1) Å^3	Specimen preparation: Prepared at 1173 K
Z = 1

Data collection

HRPD, ISIS diffractometer	Scan method: time of flight
Radiation source: Neutron	2θfixed = 168
Specimen mounting: packed powder	Distance from source to specimen: 10 mm
Data collection mode: reflection	Distance from specimen to detector: 110 mm

Refinement

Least-squares matrix: full with fixed elements per cycle	Profile function: pseudo-Voigt
Rp = 0.020	23 parameters
Rwp = 0.024	Weighting scheme based on measured s.u.'s w = 1/σ(Yo,m)^2
Rexp = 0.025
χ^2 = 0.922	Background function: polynomial
? data points

Fractional atomic coordinates and isotropic or equivalent isotropic displacement parameters (Ų)

	x	y	z	Biso*/Beq

Atomic displacement parameters (Ų)

	U^11	U^22	U^33	U^12	U^13	U^23
Pb1	0.0622 (5)	0.0622 (5)	0.0622 (5)	0	0	0
Zr1	0.0075 (2)	0.0075 (2)	0.0075 (2)	0	0	0
Ti1	0.0075 (2)	0.0075 (2)	0.0075 (2)	0	0	0
O1	0.0648 (6)	0.0648 (6)	0.0116 (6)	0	0	0

Geometric parameters (Å, º)

Pb1—O1^i	2.9383	Zr1—O1^xiii	2.0777
Pb1—O1^ii	2.9383	Zr1—O1^ix	2.0777
Pb1—O1^iii	2.9383	Zr1—O1^v	2.0777
Pb1—O1^iv	2.9383	Zr1—O1^i	2.0777
Pb1—O1^v	2.9383	Zr1—O1^xiv	2.0777
Pb1—O1^vi	2.9383	Zr1—O1^xv	2.0777
Pb1—O1^vii	2.9383	Ti1—O1^ix	2.0777
Pb1—O1^viii	2.9383	Ti1—O1^v	2.0777
Pb1—O1^ix	2.9383	Ti1—O1^i	2.0777
Pb1—O1^x	2.9383	Ti1—O1^xiv	2.0777
Pb1—O1^xi	2.9383	Ti1—O1^xv	2.0777
Pb1—O1^xii	2.9383	Ti1—O1^ix	2.0777
O1^xiii—Zr1^xiii—O1^xiv	90.000

Symmetry codes: (i) −x, −y, −z+1; (ii) −x+1, −y, −z+1; (iii) −x, −y+1, −z+1; (iv) −x+1, −y+1, −z+1; (v) −x, −z+1, −y; (vi) −x+1, −z+1, −y; (vii) −x, −z+1, −y+1; (viii) −x+1, −z+1, −y+1; (ix) −z+1, −x, −y; (x) −z+1, −x+1, −y; (xi) −z+1, −x, −y+1; (xii) −z+1, −x+1, −y+1; (xiii) −x, −y, −z; (xiv) −x, −z, −y; (xv) −z, −x, −y.

(PZT0.05-573K_phase1) Lead zirconate titanate

Crystal data

PbZr0.95Ti0.05O3	? radiation
Mr = ?	T = 573 K
Cubic, Pm3m	white
a = 4.15378 (1) Å	cylinder, 12 × 20 mm
V = 71.67 (1) Å^3	Specimen preparation: Prepared at 1173 K
Z = 1

Data collection

HRPD, ISIS diffractometer	Scan method: time of flight
Radiation source: Neutron	2θfixed = 168
Specimen mounting: packed powder	Distance from source to specimen: 10 mm
Data collection mode: reflection	Distance from specimen to detector: 110 mm

Refinement

Least-squares matrix: full with fixed elements per cycle	Profile function: pseudo-Voigt
Rp = 0.020	23 parameters
Rwp = 0.023	Weighting scheme based on measured s.u.'s w = 1/σ(Yo,m)^2
Rexp = 0.025
χ^2 = 0.884	Background function: polynomial
? data points

Fractional atomic coordinates and isotropic or equivalent isotropic displacement parameters (Ų)

	x	y	z	Biso*/Beq

Atomic displacement parameters (Ų)

	U^11	U^22	U^33	U^12	U^13	U^23
Pb1	0.0612 (3)	0.0612 (3)	0.0612 (3)	0	0	0
Zr1	0.0133 (2)	0.0133 (2)	0.0133 (2)	0	0	0
Ti1	0.0133 (2)	0.0133 (2)	0.0133 (2)	0	0	0
O1	0.0631 (3)	0.0631 (3)	0.0105 (3)	0	0	0

Geometric parameters (Å, º)

Pb1—O1^i	2.9372	Zr1—O1^xiii	2.0769
Pb1—O1^ii	2.9372	Zr1—O1^ix	2.0769
Pb1—O1^iii	2.9372	Zr1—O1^v	2.0769
Pb1—O1^iv	2.9372	Zr1—O1^i	2.0769
Pb1—O1^v	2.9372	Zr1—O1^xiv	2.0769
Pb1—O1^vi	2.9372	Zr1—O1^xv	2.0769
Pb1—O1^vii	2.9372	Ti1—O1^ix	2.0769
Pb1—O1^viii	2.9372	Ti1—O1^v	2.0769
Pb1—O1^ix	2.9372	Ti1—O1^i	2.0769
Pb1—O1^x	2.9372	Ti1—O1^xiv	2.0769
Pb1—O1^xi	2.9372	Ti1—O1^xv	2.0769
Pb1—O1^xii	2.9372	Ti1—O1^ix	2.0769
O1^xiii—Zr1^xiii—O1^xiv	90.000

Symmetry codes: (i) −x, −y, −z+1; (ii) −x+1, −y, −z+1; (iii) −x, −y+1, −z+1; (iv) −x+1, −y+1, −z+1; (v) −x, −z+1, −y; (vi) −x+1, −z+1, −y; (vii) −x, −z+1, −y+1; (viii) −x+1, −z+1, −y+1; (ix) −z+1, −x, −y; (x) −z+1, −x+1, −y; (xi) −z+1, −x, −y+1; (xii) −z+1, −x+1, −y+1; (xiii) −x, −y, −z; (xiv) −x, −z, −y; (xv) −z, −x, −y.

(PZT0.05-573K_phase2) Lead zirconate titanate

Crystal data

PbZr0.95Ti0.05O3	? radiation
Mr = ?	T = 573 K
Cubic, Pm3m	white
a = 4.1574 (2) Å	cylinder, 12 × 20 mm
V = 71.86 (1) Å^3	Specimen preparation: Prepared at 1173 K
Z = 1

Data collection

HRPD, ISIS diffractometer	Scan method: time of flight
Radiation source: Neutron	2θfixed = 168
Specimen mounting: packed powder	Distance from source to specimen: 10 mm
Data collection mode: reflection	Distance from specimen to detector: 110 mm

Refinement

Least-squares matrix: full with fixed elements per cycle	Profile function: pseudo-Voigt
Rp = 0.020	23 parameters
Rwp = 0.023	Weighting scheme based on measured s.u.'s w = 1/σ(Yo,m)^2
Rexp = 0.025
χ^2 = 0.884	Background function: polynomial
? data points

Fractional atomic coordinates and isotropic or equivalent isotropic displacement parameters (Ų)

	x	y	z	Biso*/Beq

Atomic displacement parameters (Ų)

	U^11	U^22	U^33	U^12	U^13	U^23
Pb1	0.0622 (5)	0.0622 (5)	0.0622 (5)	0	0	0
Zr1	0.0080 (2)	0.0080 (2)	0.0080 (2)	0	0	0
Ti1	0.0080 (2)	0.0080 (2)	0.0080 (2)	0	0	0
O1	0.0644 (6)	0.0644 (6)	0.0109 (6)	0	0	0

Geometric parameters (Å, º)

Pb1—O1^i	2.9398	Zr1—O1^xiii	2.0787
Pb1—O1^ii	2.9398	Zr1—O1^ix	2.0787
Pb1—O1^iii	2.9398	Zr1—O1^v	2.0787
Pb1—O1^iv	2.9398	Zr1—O1^i	2.0787
Pb1—O1^v	2.9398	Zr1—O1^xiv	2.0787
Pb1—O1^vi	2.9398	Zr1—O1^xv	2.0787
Pb1—O1^vii	2.9398	Ti1—O1^ix	2.0787
Pb1—O1^viii	2.9398	Ti1—O1^v	2.0787
Pb1—O1^ix	2.9398	Ti1—O1^i	2.0787
Pb1—O1^x	2.9398	Ti1—O1^xiv	2.0787
Pb1—O1^xi	2.9398	Ti1—O1^xv	2.0787
Pb1—O1^xii	2.9398	Ti1—O1^ix	2.0787
O1^xiii—Zr1^xiii—O1^xiv	90.000

Symmetry codes: (i) −x, −y, −z+1; (ii) −x+1, −y, −z+1; (iii) −x, −y+1, −z+1; (iv) −x+1, −y+1, −z+1; (v) −x, −z+1, −y; (vi) −x+1, −z+1, −y; (vii) −x, −z+1, −y+1; (viii) −x+1, −z+1, −y+1; (ix) −z+1, −x, −y; (x) −z+1, −x+1, −y; (xi) −z+1, −x, −y+1; (xii) −z+1, −x+1, −y+1; (xiii) −x, −y, −z; (xiv) −x, −z, −y; (xv) −z, −x, −y.

(PZT0.05-673K_phase1) Lead zirconate titanate

Crystal data

PbZr0.95Ti0.05O3	? radiation
Mr = ?	T = 673 K
Cubic, Pm3m	white
a = 4.15704 (1) Å	cylinder, 12 × 20 mm
V = 71.84 (1) Å^3	Specimen preparation: Prepared at 1173 K
Z = 1

Data collection

HRPD, ISIS diffractometer	Scan method: time of flight
Radiation source: Neutron	2θfixed = 168
Specimen mounting: packed powder	Distance from source to specimen: 10 mm
Data collection mode: reflection	Distance from specimen to detector: 110 mm

Refinement

Least-squares matrix: full with fixed elements per cycle	Profile function: pseudo-Voigt
Rp = 0.020	23 parameters
Rwp = 0.023	Weighting scheme based on measured s.u.'s w = 1/σ(Yo,m)^2
Rexp = 0.025
χ^2 = 0.846	Background function: polynomial
? data points

Fractional atomic coordinates and isotropic or equivalent isotropic displacement parameters (Ų)

	x	y	z	Biso*/Beq

Atomic displacement parameters (Ų)

	U^11	U^22	U^33	U^12	U^13	U^23
Pb1	0.0641 (3)	0.0641 (3)	0.0641 (3)	0	0	0
Zr1	0.0147 (2)	0.0147 (2)	0.0147 (2)	0	0	0
Ti1	0.0147 (2)	0.0147 (2)	0.0147 (2)	0	0	0
O1	0.0638 (3)	0.0638 (3)	0.0114 (3)	0	0	0

Geometric parameters (Å, º)

Pb1—O1^i	2.9395	Zr1—O1^xiii	2.0785
Pb1—O1^ii	2.9395	Zr1—O1^ix	2.0785
Pb1—O1^iii	2.9395	Zr1—O1^v	2.0785
Pb1—O1^iv	2.9395	Zr1—O1^i	2.0785
Pb1—O1^v	2.9395	Zr1—O1^xiv	2.0785
Pb1—O1^vi	2.9395	Zr1—O1^xv	2.0785
Pb1—O1^vii	2.9395	Ti1—O1^ix	2.0785
Pb1—O1^viii	2.9395	Ti1—O1^v	2.0785
Pb1—O1^ix	2.9395	Ti1—O1^i	2.0785
Pb1—O1^x	2.9395	Ti1—O1^xiv	2.0785
Pb1—O1^xi	2.9395	Ti1—O1^xv	2.0785
Pb1—O1^xii	2.9395	Ti1—O1^ix	2.0785
O1^xiii—Zr1^xiii—O1^xiv	90.000

Symmetry codes: (i) −x, −y, −z+1; (ii) −x+1, −y, −z+1; (iii) −x, −y+1, −z+1; (iv) −x+1, −y+1, −z+1; (v) −x, −z+1, −y; (vi) −x+1, −z+1, −y; (vii) −x, −z+1, −y+1; (viii) −x+1, −z+1, −y+1; (ix) −z+1, −x, −y; (x) −z+1, −x+1, −y; (xi) −z+1, −x, −y+1; (xii) −z+1, −x+1, −y+1; (xiii) −x, −y, −z; (xiv) −x, −z, −y; (xv) −z, −x, −y.

(PZT0.05-673K_phase2) Lead zirconate titanate

Crystal data

PbZr0.95Ti0.05O3	? radiation
Mr = ?	T = 673 K
Cubic, Pm3m	white
a = 4.1609 (1) Å	cylinder, 12 × 20 mm
V = 72.04 (1) Å^3	Specimen preparation: Prepared at 1173 K
Z = 1

Data collection

HRPD, ISIS diffractometer	Scan method: time of flight
Radiation source: Neutron	2θfixed = 168
Specimen mounting: packed powder	Distance from source to specimen: 10 mm
Data collection mode: reflection	Distance from specimen to detector: 110 mm

Refinement

Least-squares matrix: full with fixed elements per cycle	Profile function: pseudo-Voigt
Rp = 0.020	23 parameters
Rwp = 0.023	Weighting scheme based on measured s.u.'s w = 1/σ(Yo,m)^2
Rexp = 0.025
χ^2 = 0.846	Background function: polynomial
? data points

Fractional atomic coordinates and isotropic or equivalent isotropic displacement parameters (Ų)

	x	y	z	Biso*/Beq

Atomic displacement parameters (Ų)

	U^11	U^22	U^33	U^12	U^13	U^23
Pb1	0.0654 (5)	0.0654 (5)	0.0654 (5)	0	0	0
Zr1	0.0091 (3)	0.0091 (3)	0.0091 (3)	0	0	0
Ti1	0.0091 (3)	0.0091 (3)	0.0091 (3)	0	0	0
O1	0.0644 (6)	0.0644 (6)	0.0124 (7)	0	0	0

Geometric parameters (Å, º)

Pb1—O1^i	2.9422	Zr1—O1^xiii	2.0805
Pb1—O1^ii	2.9422	Zr1—O1^ix	2.0805
Pb1—O1^iii	2.9422	Zr1—O1^v	2.0805
Pb1—O1^iv	2.9422	Zr1—O1^i	2.0805
Pb1—O1^v	2.9422	Zr1—O1^xiv	2.0805
Pb1—O1^vi	2.9422	Zr1—O1^xv	2.0805
Pb1—O1^vii	2.9422	Ti1—O1^ix	2.0805
Pb1—O1^viii	2.9422	Ti1—O1^v	2.0805
Pb1—O1^ix	2.9422	Ti1—O1^i	2.0805
Pb1—O1^x	2.9422	Ti1—O1^xiv	2.0805
Pb1—O1^xi	2.9422	Ti1—O1^xv	2.0805
Pb1—O1^xii	2.9422	Ti1—O1^ix	2.0805
O1^xiii—Zr1^xiii—O1^xiv	90.000

Symmetry codes: (i) −x, −y, −z+1; (ii) −x+1, −y, −z+1; (iii) −x, −y+1, −z+1; (iv) −x+1, −y+1, −z+1; (v) −x, −z+1, −y; (vi) −x+1, −z+1, −y; (vii) −x, −z+1, −y+1; (viii) −x+1, −z+1, −y+1; (ix) −z+1, −x, −y; (x) −z+1, −x+1, −y; (xi) −z+1, −x, −y+1; (xii) −z+1, −x+1, −y+1; (xiii) −x, −y, −z; (xiv) −x, −z, −y; (xv) −z, −x, −y.

(PZT0.1-563K_phase1) Lead zirconate titanate

Crystal data

PbZr0.90Ti0.10O3	Z = 1
Mr = ?	? radiation
Cubic, Pm3m	T = 563 K
a = 4.143370 (3) Å	white
V = 71.13 (1) Å^3	cylinder, 12 × 20 mm

Data collection

HRPD, ISIS diffractometer	Scan method: time of flight
Radiation source: Neutron	2θfixed = 168
Specimen mounting: packed powder	Distance from source to specimen: 10 mm
Data collection mode: reflection	Distance from specimen to detector: 110 mm

Refinement

Least-squares matrix: full with fixed elements per cycle	Profile function: pseudo-Voigt
Rp = 0.023	15 parameters
Rwp = 0.028	Weighting scheme based on measured s.u.'s w = 1/σ(Yo,m)^2
Rexp = 0.027
χ^2 = 1.124	Background function: polynomial
? data points

Fractional atomic coordinates and isotropic or equivalent isotropic displacement parameters (Ų)

	x	y	z	Biso*/Beq

Atomic displacement parameters (Ų)

	U^11	U^22	U^33	U^12	U^13	U^23
Pb1	0.0631 (2)	0.0631 (2)	0.0631 (2)	0	0	0
Zr1	0.0121 (1)	0.0121 (1)	0.0121 (1)	0	0	0
Ti1	0.0121 (1)	0.0121 (1)	0.0121 (1)	0	0	0
O1	0.0597 (2)	0.0597 (2)	0.0125 (2)	0	0	0

Geometric parameters (Å, º)

Pb1—O1^i	2.9298	Zr1—O1^xiii	2.0717
Pb1—O1^ii	2.9298	Zr1—O1^ix	2.0717
Pb1—O1^iii	2.9298	Zr1—O1^v	2.0717
Pb1—O1^iv	2.9298	Zr1—O1^i	2.0717
Pb1—O1^v	2.9298	Zr1—O1^xiv	2.0717
Pb1—O1^vi	2.9298	Zr1—O1^xv	2.0717
Pb1—O1^vii	2.9298	Ti1—O1^ix	2.0717
Pb1—O1^viii	2.9298	Ti1—O1^v	2.0717
Pb1—O1^ix	2.9298	Ti1—O1^i	2.0717
Pb1—O1^x	2.9298	Ti1—O1^xiv	2.0717
Pb1—O1^xi	2.9298	Ti1—O1^xv	2.0717
Pb1—O1^xii	2.9298	Ti1—O1^ix	2.0717
O1^xiii—Zr1^xiii—O1^xiv	90.000

Symmetry codes: (i) −x, −y, −z+1; (ii) −x+1, −y, −z+1; (iii) −x, −y+1, −z+1; (iv) −x+1, −y+1, −z+1; (v) −x, −z+1, −y; (vi) −x+1, −z+1, −y; (vii) −x, −z+1, −y+1; (viii) −x+1, −z+1, −y+1; (ix) −z+1, −x, −y; (x) −z+1, −x+1, −y; (xi) −z+1, −x, −y+1; (xii) −z+1, −x+1, −y+1; (xiii) −x, −y, −z; (xiv) −x, −z, −y; (xv) −z, −x, −y.

(PZT0.1-623K_phase1) Lead zirconate titanate

Crystal data

PbZr0.90Ti0.10O3	Z = 1
Mr = ?	? radiation
Cubic, Pm3m	T = 623 K
a = 4.144990 (3) Å	white
V = 71.22 (1) Å^3	cylinder, 12 × 20 mm

Data collection

HRPD, ISIS diffractometer	Scan method: time of flight
Radiation source: Neutron	2θfixed = 168
Specimen mounting: packed powder	Distance from source to specimen: 10 mm
Data collection mode: reflection	Distance from specimen to detector: 110 mm

Refinement

Least-squares matrix: full with fixed elements per cycle	Profile function: pseudo-Voigt
Rp = 0.024	15 parameters
Rwp = 0.026	Weighting scheme based on measured s.u.'s w = 1/σ(Yo,m)^2
Rexp = 0.021
χ^2 = 1.588	Background function: polynomial
? data points

Fractional atomic coordinates and isotropic or equivalent isotropic displacement parameters (Ų)

	x	y	z	Biso*/Beq

Atomic displacement parameters (Ų)

	U^11	U^22	U^33	U^12	U^13	U^23
Pb1	0.0603 (2)	0.0603 (2)	0.0603 (2)	0	0	0
Zr1	0.0098 (1)	0.0098 (1)	0.0098 (1)	0	0	0
Ti1	0.0098 (1)	0.0098 (1)	0.0098 (1)	0	0	0
O1	0.0571 (2)	0.0571 (2)	0.0094 (2)	0	0	0

Geometric parameters (Å, º)

Pb1—O1^i	2.9310	Zr1—O1^xiii	2.0725
Pb1—O1^ii	2.9310	Zr1—O1^ix	2.0725
Pb1—O1^iii	2.9310	Zr1—O1^v	2.0725
Pb1—O1^iv	2.9310	Zr1—O1^i	2.0725
Pb1—O1^v	2.9310	Zr1—O1^xiv	2.0725
Pb1—O1^vi	2.9310	Zr1—O1^xv	2.0725
Pb1—O1^vii	2.9310	Ti1—O1^ix	2.0725
Pb1—O1^viii	2.9310	Ti1—O1^v	2.0725
Pb1—O1^ix	2.9310	Ti1—O1^i	2.0725
Pb1—O1^x	2.9310	Ti1—O1^xiv	2.0725
Pb1—O1^xi	2.9310	Ti1—O1^xv	2.0725
Pb1—O1^xii	2.9310	Ti1—O1^ix	2.0725
O1^xiii—Zr1^xiii—O1^xiv	90.000

Symmetry codes: (i) −x, −y, −z+1; (ii) −x+1, −y, −z+1; (iii) −x, −y+1, −z+1; (iv) −x+1, −y+1, −z+1; (v) −x, −z+1, −y; (vi) −x+1, −z+1, −y; (vii) −x, −z+1, −y+1; (viii) −x+1, −z+1, −y+1; (ix) −z+1, −x, −y; (x) −z+1, −x+1, −y; (xi) −z+1, −x, −y+1; (xii) −z+1, −x+1, −y+1; (xiii) −x, −y, −z; (xiv) −x, −z, −y; (xv) −z, −x, −y.

(PZT0.4-673K_phase1) Lead zirconate titanate

Crystal data

PbZr0.6Ti0.4O3	? radiation
Mr = ?	T = 673 K
Cubic, Pm3m	white
a = 4.091698 (6) Å	cylinder, 12 × 20 mm
V = 68.50 (1) Å^3	Specimen preparation: Prepared at 1173 K
Z = 1

Data collection

HRPD, ISIS diffractometer	Scan method: time of flight
Radiation source: Neutron	2θfixed = 168
Specimen mounting: packed powder	Distance from source to specimen: 10 mm
Data collection mode: reflection	Distance from specimen to detector: 110 mm

Refinement

Least-squares matrix: full with fixed elements per cycle	Profile function: pseudo-Voigt
Rp = 0.027	15 parameters
Rwp = 0.032	Weighting scheme based on measured s.u.'s w = 1/σ(Yo,m)^2
Rexp = 0.033
χ^2 = 0.884	Background function: polynomial
? data points

Fractional atomic coordinates and isotropic or equivalent isotropic displacement parameters (Ų)

	x	y	z	Biso*/Beq

Atomic displacement parameters (Ų)

	U^11	U^22	U^33	U^12	U^13	U^23
Pb1	0.0671 (3)	0.0671 (3)	0.0671 (3)	0	0	0
Zr1	0.0179 (3)	0.0179 (3)	0.0179 (3)	0	0	0
Ti1	0.0179 (3)	0.0179 (3)	0.0179 (3)	0	0	0
O1	0.0530 (3)	0.0530 (3)	0.0240 (3)	0	0	0

Geometric parameters (Å, º)

Pb1—O1^i	2.8933	Zr1—O1^xiii	2.0458
Pb1—O1^ii	2.8933	Zr1—O1^ix	2.0458
Pb1—O1^iii	2.8933	Zr1—O1^v	2.0458
Pb1—O1^iv	2.8933	Zr1—O1^i	2.0458
Pb1—O1^v	2.8933	Zr1—O1^xiv	2.0458
Pb1—O1^vi	2.8933	Zr1—O1^xv	2.0458
Pb1—O1^vii	2.8933	Ti1—O1^ix	2.0458
Pb1—O1^viii	2.8933	Ti1—O1^v	2.0458
Pb1—O1^ix	2.8933	Ti1—O1^i	2.0458
Pb1—O1^x	2.8933	Ti1—O1^xiv	2.0458
Pb1—O1^xi	2.8933	Ti1—O1^xv	2.0458
Pb1—O1^xii	2.8933	Ti1—O1^ix	2.0458
O1^xiii—Zr1^xiii—O1^xiv	90.000

Symmetry codes: (i) −x, −y, −z+1; (ii) −x+1, −y, −z+1; (iii) −x, −y+1, −z+1; (iv) −x+1, −y+1, −z+1; (v) −x, −z+1, −y; (vi) −x+1, −z+1, −y; (vii) −x, −z+1, −y+1; (viii) −x+1, −z+1, −y+1; (ix) −z+1, −x, −y; (x) −z+1, −x+1, −y; (xi) −z+1, −x, −y+1; (xii) −z+1, −x+1, −y+1; (xiii) −x, −y, −z; (xiv) −x, −z, −y; (xv) −z, −x, −y.

(PZT0.45-703K_phase1) Lead zirconate titanate

Crystal data

PbZr0.55Ti0.45O3	? radiation
Mr = ?	T = 703 K
Cubic, Pm3m	white
a = 4.082661 (4) Å	cylinder, 12 × 20 mm
V = 68.05 (1) Å^3	Specimen preparation: Prepared at 1173 K
Z = 1

Data collection

HRPD, ISIS diffractometer	Scan method: time of flight
Radiation source: Neutron	2θfixed = 168
Specimen mounting: packed powder	Distance from source to specimen: 10 mm
Data collection mode: reflection	Distance from specimen to detector: 110 mm

Refinement

Least-squares matrix: full with fixed elements per cycle	Profile function: pseudo-Voigt
Rp = 0.023	15 parameters
Rwp = 0.027	Weighting scheme based on measured s.u.'s w = 1/σ(Yo,m)^2
Rexp = 0.026
χ^2 = 1.124	Background function: polynomial
? data points

Fractional atomic coordinates and isotropic or equivalent isotropic displacement parameters (Ų)

	x	y	z	Biso*/Beq

Atomic displacement parameters (Ų)

	U^11	U^22	U^33	U^12	U^13	U^23
Pb1	0.0616 (2)	0.0616 (2)	0.0616 (2)	0	0	0
Zr1	0.0118 (2)	0.0118 (2)	0.0118 (2)	0	0	0
Ti1	0.0118 (2)	0.0118 (2)	0.0118 (2)	0	0	0
O1	0.0459 (2)	0.0459 (2)	0.0189 (2)	0	0	0

Geometric parameters (Å, º)

Pb1—O1^i	2.8869	Zr1—O1^xiii	2.0413
Pb1—O1^ii	2.8869	Zr1—O1^ix	2.0413
Pb1—O1^iii	2.8869	Zr1—O1^v	2.0413
Pb1—O1^iv	2.8869	Zr1—O1^i	2.0413
Pb1—O1^v	2.8869	Zr1—O1^xiv	2.0413
Pb1—O1^vi	2.8869	Zr1—O1^xv	2.0413
Pb1—O1^vii	2.8869	Ti1—O1^ix	2.0413
Pb1—O1^viii	2.8869	Ti1—O1^v	2.0413
Pb1—O1^ix	2.8869	Ti1—O1^i	2.0413
Pb1—O1^x	2.8869	Ti1—O1^xiv	2.0413
Pb1—O1^xi	2.8869	Ti1—O1^xv	2.0413
Pb1—O1^xii	2.8869	Ti1—O1^ix	2.0413
O1^xiii—Zr1^xiii—O1^xiv	90.000

Symmetry codes: (i) −x, −y, −z+1; (ii) −x+1, −y, −z+1; (iii) −x, −y+1, −z+1; (iv) −x+1, −y+1, −z+1; (v) −x, −z+1, −y; (vi) −x+1, −z+1, −y; (vii) −x, −z+1, −y+1; (viii) −x+1, −z+1, −y+1; (ix) −z+1, −x, −y; (x) −z+1, −x+1, −y; (xi) −z+1, −x, −y+1; (xii) −z+1, −x+1, −y+1; (xiii) −x, −y, −z; (xiv) −x, −z, −y; (xv) −z, −x, −y.

(PZT0.48-703K_phase1) Lead zirconate titanate

Crystal data

PbZr0.52Ti0.48O3	? radiation
Mr = ?	T = 703 K
Cubic, Pm3m	white
a = 4.076773 (5) Å	cylinder, 12 × 20 mm
V = 67.76 (1) Å^3	Specimen preparation: Prepared at 1173 K
Z = 1

Data collection

HRPD, ISIS diffractometer	Scan method: time of flight
Radiation source: Neutron	2θfixed = 168
Specimen mounting: packed powder	Distance from source to specimen: 10 mm
Data collection mode: reflection	Distance from specimen to detector: 110 mm

Refinement

Least-squares matrix: full with fixed elements per cycle	Profile function: pseudo-Voigt
Rp = 0.027	15 parameters
Rwp = 0.031	Weighting scheme based on measured s.u.'s w = 1/σ(Yo,m)^2
Rexp = 0.031
χ^2 = 0.960	Background function: polynomial
? data points

Fractional atomic coordinates and isotropic or equivalent isotropic displacement parameters (Ų)

	x	y	z	Biso*/Beq

Atomic displacement parameters (Ų)

	U^11	U^22	U^33	U^12	U^13	U^23
Pb1	0.0611 (3)	0.0611 (3)	0.0611 (3)	0	0	0
Zr1	0.0120 (3)	0.0120 (3)	0.0120 (3)	0	0	0
Ti1	0.0120 (3)	0.0120 (3)	0.0120 (3)	0	0	0
O1	0.0459 (3)	0.0459 (3)	0.0206 (3)	0	0	0

Geometric parameters (Å, º)

Pb1—O1^i	2.8827	Zr1—O1^xiii	2.0384
Pb1—O1^ii	2.8827	Zr1—O1^ix	2.0384
Pb1—O1^iii	2.8827	Zr1—O1^v	2.0384
Pb1—O1^iv	2.8827	Zr1—O1^i	2.0384
Pb1—O1^v	2.8827	Zr1—O1^xiv	2.0384
Pb1—O1^vi	2.8827	Zr1—O1^xv	2.0384
Pb1—O1^vii	2.8827	Ti1—O1^ix	2.0384
Pb1—O1^viii	2.8827	Ti1—O1^v	2.0384
Pb1—O1^ix	2.8827	Ti1—O1^i	2.0384
Pb1—O1^x	2.8827	Ti1—O1^xiv	2.0384
Pb1—O1^xi	2.8827	Ti1—O1^xv	2.0384
Pb1—O1^xii	2.8827	Ti1—O1^ix	2.0384
O1^xiii—Zr1^xiii—O1^xiv	90.000

Symmetry codes: (i) −x, −y, −z+1; (ii) −x+1, −y, −z+1; (iii) −x, −y+1, −z+1; (iv) −x+1, −y+1, −z+1; (v) −x, −z+1, −y; (vi) −x+1, −z+1, −y; (vii) −x, −z+1, −y+1; (viii) −x+1, −z+1, −y+1; (ix) −z+1, −x, −y; (x) −z+1, −x+1, −y; (xi) −z+1, −x, −y+1; (xii) −z+1, −x+1, −y+1; (xiii) −x, −y, −z; (xiv) −x, −z, −y; (xv) −z, −x, −y.

(PZT0.5-753K_phase1) Lead zirconate titanate

Crystal data

PbZr0.5Ti0.5O3	? radiation
Mr = ?	T = 753 K
Cubic, Pm3m	white
a = 4.074692 (6) Å	cylinder, 12 × 20 mm
V = 67.65 (1) Å^3	Specimen preparation: Prepared at 1173 K
Z = 1

Data collection

HRPD, ISIS diffractometer	Scan method: time of flight
Radiation source: Neutron	2θfixed = 168
Specimen mounting: packed powder	Distance from source to specimen: 10 mm
Data collection mode: reflection	Distance from specimen to detector: 110 mm

Refinement

Least-squares matrix: full with fixed elements per cycle	Profile function: pseudo-Voigt
Rp = 0.024	15 parameters
Rwp = 0.029	Weighting scheme based on measured s.u.'s w = 1/σ(Yo,m)^2
Rexp = 0.026
χ^2 = 1.232	Background function: polynomial
? data points

Fractional atomic coordinates and isotropic or equivalent isotropic displacement parameters (Ų)

	x	y	z	Biso*/Beq

Atomic displacement parameters (Ų)

	U^11	U^22	U^33	U^12	U^13	U^23
Pb1	0.0607 (3)	0.0607 (3)	0.0607 (3)	0	0	0
Zr1	0.0121 (3)	0.0121 (3)	0.0121 (3)	0	0	0
Ti1	0.0121 (3)	0.0121 (3)	0.0121 (3)	0	0	0
O1	0.0457 (3)	0.0457 (3)	0.0209 (3)	0	0	0

Geometric parameters (Å, º)

Pb1—O1^i	2.8812	Zr1—O1^xiii	2.0373
Pb1—O1^ii	2.8812	Zr1—O1^ix	2.0373
Pb1—O1^iii	2.8812	Zr1—O1^v	2.0373
Pb1—O1^iv	2.8812	Zr1—O1^i	2.0373
Pb1—O1^v	2.8812	Zr1—O1^xiv	2.0373
Pb1—O1^vi	2.8812	Zr1—O1^xv	2.0373
Pb1—O1^vii	2.8812	Ti1—O1^ix	2.0373
Pb1—O1^viii	2.8812	Ti1—O1^v	2.0373
Pb1—O1^ix	2.8812	Ti1—O1^i	2.0373
Pb1—O1^x	2.8812	Ti1—O1^xiv	2.0373
Pb1—O1^xi	2.8812	Ti1—O1^xv	2.0373
Pb1—O1^xii	2.8812	Ti1—O1^ix	2.0373
O1^xiii—Zr1^xiii—O1^xiv	90.000

Symmetry codes: (i) −x, −y, −z+1; (ii) −x+1, −y, −z+1; (iii) −x, −y+1, −z+1; (iv) −x+1, −y+1, −z+1; (v) −x, −z+1, −y; (vi) −x+1, −z+1, −y; (vii) −x, −z+1, −y+1; (viii) −x+1, −z+1, −y+1; (ix) −z+1, −x, −y; (x) −z+1, −x+1, −y; (xi) −z+1, −x, −y+1; (xii) −z+1, −x+1, −y+1; (xiii) −x, −y, −z; (xiv) −x, −z, −y; (xv) −z, −x, −y.

(PZT0.6-723K_phase1) Lead zirconate titanate

Crystal data

PbZr0.4Ti0.6O3	? radiation
Mr = ?	T = 723 K
Cubic, Pm3m	white
a = 4.054092 (4) Å	cylinder, 12 × 20 mm
V = 66.63 (1) Å^3	Specimen preparation: Prepared at 1173 K
Z = 1

Data collection

HRPD, ISIS diffractometer	Scan method: time of flight
Radiation source: Neutron	2θfixed = 168
Specimen mounting: packed powder	Distance from source to specimen: 10 mm
Data collection mode: reflection	Distance from specimen to detector: 110 mm

Refinement

Least-squares matrix: full with fixed elements per cycle	Profile function: pseudo-Voigt
Rp = 0.020	15 parameters
Rwp = 0.023	Weighting scheme based on measured s.u.'s w = 1/σ(Yo,m)^2
Rexp = 0.028
χ^2 = 0.656	Background function: polynomial
? data points

Fractional atomic coordinates and isotropic or equivalent isotropic displacement parameters (Ų)

	x	y	z	Biso*/Beq

Atomic displacement parameters (Ų)

	U^11	U^22	U^33	U^12	U^13	U^23
Pb1	0.0570 (3)	0.0570 (3)	0.0570 (3)	0	0	0
Zr1	0.0029 (4)	0.0029 (4)	0.0029 (4)	0	0	0
Ti1	0.0029 (4)	0.0029 (4)	0.0029 (4)	0	0	0
O1	0.0406 (2)	0.0406 (2)	0.0175 (3)	0	0	0

Geometric parameters (Å, º)

Pb1—O1^i	2.8667	Zr1—O1^xiii	2.0270
Pb1—O1^ii	2.8667	Zr1—O1^ix	2.0270
Pb1—O1^iii	2.8667	Zr1—O1^v	2.0270
Pb1—O1^iv	2.8667	Zr1—O1^i	2.0270
Pb1—O1^v	2.8667	Zr1—O1^xiv	2.0270
Pb1—O1^vi	2.8667	Zr1—O1^xv	2.0270
Pb1—O1^vii	2.8667	Ti1—O1^ix	2.0270
Pb1—O1^viii	2.8667	Ti1—O1^v	2.0270
Pb1—O1^ix	2.8667	Ti1—O1^i	2.0270
Pb1—O1^x	2.8667	Ti1—O1^xiv	2.0270
Pb1—O1^xi	2.8667	Ti1—O1^xv	2.0270
Pb1—O1^xii	2.8667	Ti1—O1^ix	2.0270
O1^xiii—Zr1^xiii—O1^xiv	90.000

Symmetry codes: (i) −x, −y, −z+1; (ii) −x+1, −y, −z+1; (iii) −x, −y+1, −z+1; (iv) −x+1, −y+1, −z+1; (v) −x, −z+1, −y; (vi) −x+1, −z+1, −y; (vii) −x, −z+1, −y+1; (viii) −x+1, −z+1, −y+1; (ix) −z+1, −x, −y; (x) −z+1, −x+1, −y; (xi) −z+1, −x, −y+1; (xii) −z+1, −x+1, −y+1; (xiii) −x, −y, −z; (xiv) −x, −z, −y; (xv) −z, −x, −y.

(PT-803K_phase1) Lead zirconate titanate

Crystal data

PbTiO3	Z = 1
Mr = ?	? radiation
Cubic, Pm3m	T = 803 K
a = 3.968811 (3) Å	white
V = 62.52 (1) Å^3	cylinder, 15 × 20 mm

Data collection

HRPD, ISIS diffractometer	Scan method: time of flight
Radiation source: Neutron	2θfixed = 168
Specimen mounting: packed powder	Distance from source to specimen: 10 mm
Data collection mode: reflection	Distance from specimen to detector: 110 mm

Refinement

Least-squares matrix: full with fixed elements per cycle	Profile function: pseudo-Voigt
Rp = 0.024	15 parameters
Rwp = 0.028	Weighting scheme based on measured s.u.'s w = 1/σ(Yo,m)^2
Rexp = 0.028
χ^2 = 1.061	Background function: polynomial
? data points

Fractional atomic coordinates and isotropic or equivalent isotropic displacement parameters (Ų)

	x	y	z	Biso*/Beq

Atomic displacement parameters (Ų)

	U^11	U^22	U^33	U^12	U^13	U^23
Pb1	0.0381 (2)	0.0381 (2)	0.0381 (2)	0	0	0
Ti1	0.0170 (2)	0.0170 (2)	0.0170 (2)	0	0	0
O1	0.0289 (2)	0.0289 (2)	0.0119 (2)	0	0	0

Geometric parameters (Å, º)

Pb1—O1^i	2.8064	Pb1—O1^x	2.8064
Pb1—O1^ii	2.8064	Pb1—O1^xi	2.8064
Pb1—O1^iii	2.8064	Pb1—O1^xii	2.8064
Pb1—O1^iv	2.8064	Ti1—O1^ix	1.9844
Pb1—O1^v	2.8064	Ti1—O1^v	1.9844
Pb1—O1^vi	2.8064	Ti1—O1^i	1.9844
Pb1—O1^vii	2.8064	Ti1—O1^xiii	1.9844
Pb1—O1^viii	2.8064	Ti1—O1^xiv	1.9844
Pb1—O1^ix	2.8064	Ti1—O1^ix	1.9844
O1^xv—Ti1^xv—O1^xiii	90.000

Symmetry codes: (i) −x, −y, −z+1; (ii) −x+1, −y, −z+1; (iii) −x, −y+1, −z+1; (iv) −x+1, −y+1, −z+1; (v) −x, −z+1, −y; (vi) −x+1, −z+1, −y; (vii) −x, −z+1, −y+1; (viii) −x+1, −z+1, −y+1; (ix) −z+1, −x, −y; (x) −z+1, −x+1, −y; (xi) −z+1, −x, −y+1; (xii) −z+1, −x+1, −y+1; (xiii) −x, −z, −y; (xiv) −z, −x, −y; (xv) −x, −y, −z.