Facultad de Física

Departamento de Física Aplicada

Master's dissertation

Effects of high-pressure on the structural,vibrational and electronic properties of

monazite-type PbCrO4

Student: Enrico BandielloSupervisor: Prof. Daniel Errandonea

Valencia, July 2012

Effect of high-pressure on the structural, vibrationaland electronic properties of monazite-type PbCrO4

Student: Enrico Bandiello

July 2012

Supervisor: Prof. Daniel Errandonea

To my wife Karinaand to my son Francesco,

who is almost here...

Contents

Acknowledgements v

1 Methodology and experimental setting 11.1 Diamond-anvil cells . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Ruby fluorescence as a pressure gauge . . . . . . . . . . . . . . . . . . . . . . . . 11.3 Samples and experimental setups . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.3.1 Samples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.3.2 Optical measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.3.3 X-ray diffraction experiments . . . . . . . . . . . . . . . . . . . . . . . . 41.3.4 Raman experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2 Characterization of PbCrO4 at ambient conditions 72.1 AXO4 monazite-type compounds . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.2 A general description of PbCrO4 . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.3 Crystal structure at ambient pressure . . . . . . . . . . . . . . . . . . . . . . . . 92.4 Raman spectrum at ambient pressure . . . . . . . . . . . . . . . . . . . . . . . . 92.5 Optical absorption at ambient pressure . . . . . . . . . . . . . . . . . . . . . . . 12

3 Experimental results and conclusions 153.1 High-pressure X-ray diffraction experiments . . . . . . . . . . . . . . . . . . . . 153.2 High-pressure Raman experiments . . . . . . . . . . . . . . . . . . . . . . . . . . 173.3 High-pressure optical-absorption experiments . . . . . . . . . . . . . . . . . . . . 203.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

A Space groups 27A.1 Space groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

A.1.1 Wyckoff positions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29A.2 Monoclinic P21/n space group . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30A.3 Orthorhombic P212121 space group . . . . . . . . . . . . . . . . . . . . . . . . . 31

B Raman spectroscopy and notation. 33B.1 Raman Spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

B.1.1 Mulliken’s symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35B.2 Raman tensors for PbCrO4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

C Software 37C.1 Software tools . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

References 41

i

List of Figures

1.1 Scheme [1] and photo of a DAC. . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.2 Ruby fluorescence spectra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.3 XFR analysis of PbCrO4 natural samples and surface of the original crystal at

650×. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.4 Experimental setup for the optical-absorption experiments. . . . . . . . . . . . . 5

2.1 Schematic view of an AXO4 monazite. . . . . . . . . . . . . . . . . . . . . . . . . 72.2 Monazite projected down [001] . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.3 Photo of the sample in the DAC at ambient pressure and at P = 3.5 GPa . . . . 92.4 XRD pattern for PbCrO4 at ambient pressure . . . . . . . . . . . . . . . . . . . 102.5 Raman spectrum for PbCrO4 at ambient pressure . . . . . . . . . . . . . . . . . 112.6 Optical-absorption spectrum for PbCrO4 at ambient pressure. . . . . . . . . . . 12

3.1 X-ray patterns at selected pressures . . . . . . . . . . . . . . . . . . . . . . . . . 153.2 Pressure evolution of the unit-cell parameters of PbCrO4 for the low-pressure

phase. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163.3 Pressure evolution of the volume of the unit cell. . . . . . . . . . . . . . . . . . . 173.4 Raman spectra of PbCrO4 at different pressures. . . . . . . . . . . . . . . . . . . 183.5 Evolution of Raman active modes for PbCrO4 under compression. . . . . . . . . 183.6 Absorption spectra of PbCrO4 at selected pressures. . . . . . . . . . . . . . . . . 213.7 Pressure dependence of Eg for PbCrO4 . . . . . . . . . . . . . . . . . . . . . . . 223.8 Schematic band structure of monazite-type PbCrO4 . . . . . . . . . . . . . . . . 23

A.1 P21/n space group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30A.2 P212121 space group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

ii

List of Tables

2.1 Lattice parameters and atomic position for PbCrO4 at ambient pressure. . . . . 102.2 Bond lengths for the CrO4 and PbO9 polyhedra at ambient pressure. . . . . . . 102.3 Raman active modes for PbCrO4 at ambient conditions, pressure coefficients and

Gruneisen parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

3.1 Bond lengths for the polyhedral units at 5.2 GPa. . . . . . . . . . . . . . . . . . 163.2 Lattice parameters for PbCrO4 for the HP phase . . . . . . . . . . . . . . . . . . 173.3 Raman active modes at 6.4 GPa for the HP monoclinic phase of PbCrO4 and

pressure coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

A.1 Resume of 3D Bravais lattice and related parameters. . . . . . . . . . . . . . . . 28A.2 Wyckoff sites for the monoclinic Pm space group. . . . . . . . . . . . . . . . . . 29A.3 Wyckoff sites for the monoclinic P21/n space group. . . . . . . . . . . . . . . . . 31

B.1 Mulliken’s symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

iii

Acknowledgements

First of all I would like to thank my supervisor, Prof. Daniel Errandonea, for his invalu-able help and his patience during the realization of this work. I am very grateful to himfor the assistance in the realization of the experiments, the data analysis and the correc-tions of the various revisions of the manuscript. Above all, I would like to thank him forall the opportunities he gave me, such as the participations in the EHPRG Conferencesand the co-authoring of articles.

Thanks to Prof. Domingo Martınez, Prof. Javier Manjon, Prof. Alfredo Segura andDr. David Santamarıa for the collaboration in the realization of this work.

Thanks to Dr. Javier Ruız for the hints in the normalization of the optical spectraand for the serious answers to my naive questions (once more!).

v

1. Methodology and experimental setting

Introduction

Our main goal is the study of the physical properties of monazite-type PbCrO4 undercompression and the relation they have with the structural changes occurring in themineral because of hydrostatic pressure. Various experimental techniques have beenused in this study, such as optical absorption spectroscopy, Raman spectroscopy andX-ray diffraction.

In Chapter1 we will describe some experimental details while Chapter 2 is devotedto a general description of monazite-type AXO4 compounds and to a characterizationof our PbCrO4 samples at ambient conditions. Finally, in Chapter 3 the result of ourhigh-pressure experiments and their physical interpretations are presented. AppendicesA and B contain some very informal notes about space groups and Raman spectroscopy,while Appendix C reports the list of the software tools that have been used in this work.

1.1. Diamond-anvil cells

Diamond-Anvil Cells (or DACs) are the most commonly used devices in the study ofphysical behavior of crystal samples upon high-pressure conditions (up to 300 GPa1).They basically consists in a pair of diamonds that are cut leaving on each of them asmall culet. A pre-indented steel gasket acts as a separator between the facing culets,forming a sealed chamber where the sample is placed together with a pressure medium.To generate the needed pressure on the sample, the assembly is inserted into a metalcylinder where a metallic membrane, slowly inflated with Helium (He), exerts a forceF on the diamonds. The pressure P on the culets thus rises according to equationP = F/A, where A is their area. Ideally hydrostatic pressure conditions are obtained bymeans of the pressure medium. Figure 1.1 gives a schematic representation and a photoof one of the DAC used in our experiments.

1.2. Ruby fluorescence as a pressure gauge

The measurement of the pressure inside the DAC is done indirectly, by means of theruby (Al2O3:Cr3+) fluorescence technique. Small ruby chips are loaded together withthe sample inside the pressure chamber. Ruby fluorescence is excited with a green laser2

1100 kPa=10−4 GPa ' 1 atm (ambient pressure).2λ = 523 nm in our case.

1

1. Methodology and experimental setting

Figure 1.1.: Scheme [1] and photo of a DAC.

and a doublet R1, R2 is generated. Wavelengths at ambient pressure are respectivelyλ0R1

= 694.3 nm and λ0R2= 692.7 nm [2]; these lines regularly redshift with increasing

pressure, as can be seen in Fig. 1.2. There are several empirical expressions of λR1(P );however in our pressure range (100 kPa–21 GPa) they give pressure that differ by lessthan 1%. The following expression has been used [2] in our experiments:

P (λR1) [GPa] =1904 [GPa]

7.665

[(λR1 [nm]

694.38 [nm]

)7.665

− 1

]. (1.1)

The fluorescence spectra were taken using an OCEAN OPTICS HR2000 Si spectrometerequipped with a holographic grating with a groove density of 1800, which covers thespectral range from 600 to 755 nm with an optical resolution of 0.1 nm.

1.3. Samples and experimental setups

Some experimental details are given in this section. We have used similar DACs troughall the experiments, despite the different techniques that were employed. The diameter ofthe culets was 500µm while the pressure medium was either a 16:3:1 methanol-ethanol-water or a 4:1 methanol-ethanol mixture [3] (they provide a similar pressure environment).

High-pressure experiments were performed at room temperature. Additionally , allthe experiments have been carried out also at ambient pressure, outside the DAC.

1.3.1. Samples

Our samples were obtained from natural crystals, originated from the Red Lead Minein Australia. In first place, a visual and chemical microanalysis of the samples wasperformed, to check for the presence and nature of impurities. This was made usinga PHILIPS XL-30 ESEM scanning electron microscope situated at the Electron Mi-

2

1.3. Samples and experimental setups

Inte

nsi

ty (

Arb

. U

nit

s)

Wavelength (nm)

0

500

1,000

1,500

2,000

2,500

692 694 696 698 700

P=1.59 GPa P=13.34 GPa

Figure 1.2.: Ruby fluorescence spectra. The pressure corresponding to each spectrum is indi-cated.

Figure 1.3.: XFR analysis of PbCrO4 natural samples and surface of the original crystal at 650×.

croscopy Section of SCSIE3, UV. The results of the analysis are resumed in Fig. 1.3:it shows some crystals of PbCrO4 at 650× and an X-ray fluorescence spectrum of oursample. Only a negligible amount of Fe (0.06%) is present in some regions of the sample,likely due to ferrous oxides incorporated in the mineral during its growth. For our exper-iments, we selected microcrystals of PbCrO4 with undetectable amount of impurities.

The samples used in optical-absorption and Raman experiments were crystals about20µm thick, while X-ray diffraction experiment have been performed on finely groundpowder samples obtained from single crystals (grain size < 1µm).

1.3.2. Optical measurements

The optical-absorption experiments were performed on single crystals in a DAC usinga 16:3:1 methanol-ethanol-water mixture as pressure medium. This medium provides

3Servei Central de Suport a la Investigacio Experimental (Central Support Service for the ExperimentalResearch).

3

1. Methodology and experimental setting

quasi-hydrostatic conditions in the pressure range covered by our experiments [4]. Ourexperimental setup for these experiments is depicted in Fig. 1.4. There, O1 and O2 aretwo Cassegrain objectives, each one mounted on a two-axis micrometric slide. ObjectiveO1 is needed to concentrate the light from the halogen lamp (resp. the laser) in a smallspot and to focus it on the sample (resp. the ruby), while O2 serves to focus on thedetector the light transmitted by the sample.

The optical measurements spanned a fairly broad spectral range. This required the useof two different detectors, in particular an OCEAN OPTICS HR4000 Si spectrometerfor the range 500–1100 nm and a JOBIN-YVON H25 monocromator coupled with a Gedetector for the range 950–1800 nm.

Schematically, each optical measurement involves the following steps:

1. choose a suitable value for the pressure on the sample.

2. focus the laser beam on a ruby, with the aid of the ocular and the screen.

3. pump He in the DAC until the chosen pressure is achieved. Meanwhile, pressurevalue is checked in real-time using the laser and the spectrophotometer (Section1.2).

4. replace laser with the He lamp and focus the light spot (∼ 20µm) outside thesample. Take a reference spectrum of the direct beam.

5. focus the light on the sample and take the spectrum of the transmitted light.

6. the ratio between the first spectrum and the second one is then the transmittanceof the sample at the chosen pressure.

7. the transmittance spectrum is transformed into an absorption spectrum (the ab-sorption coefficient as a function of energy), considering the thickness of the sampleand its refraction index [5].

The absorption spectra were determined at selected pressures, both upstroke and down-stroke, up to a maximum of about 21 GPa. We couldn’t go any further in pressure dueboth to the strong opacization of the samples and to the limitations of our detector. Bothfactors factors made very difficult the obtainment of accurate spectra beyond 21 GPa.

The optical-absorption experiments allowed us to determine the evolution of the bandgap of PbCrO4 with pressure and to relate it with the changes in the crystal structureof the mineral evidenced by the other techniques.

1.3.3. X-ray diffraction experiments

X-ray diffraction patterns have been obtained performing powder diffraction in a DACusing Molybdenium Kα1:Kα2 wavelengths (0.7101 A and 0.7144 A respectively) in an AG-ILENT TECHNOLOGIES “Xcalibur” diffractometer, equipped with an ATLAS CCD.The pressure medium was a 4:1 methanol-ethanol mixture. The experiments were per-formed up to a maximum of 13 GPa.

4

1.3. Samples and experimental setups

Laser diodeMirror

Mirror

Lens

Halogen

lamp

Optical

fiberOptical

fiberO1 O2

Prism

Screen

Spectrophotometer

Lens

Diaphragm Detector

with

diaphragm

Ocular

PC

DAC

(with sample, ruby

and pressure medium)

Figure 1.4.: Experimental setup for the optical-absorption experiments.

The diffraction experiments allowed us to study the evolution of the crystallographicparameters and the volume of the unit cell of PbCrO4 under compression. In particu-lar, the data relative to the volume were fitted using the Birch-Murnaghan third-orderequation of state [6]

P

(V

V0

)= 1.5B0 ·

[(V

V0

)7/3

−(V

V0

)5/3]

1 +3

4(B1 − 4) ·

[(V

V0

)2/3

− 1

](1.2)

where V0 and V are respectively the volume of the unit cell at ambient pressure and atpressure P .

The fit of the volume data with Eq. (1.2) gives directly the bulk modulus B04 and

its pressure derivative B1. Furthermore, the X-ray diffraction patterns allowed us toidentify structural changes and to propose a crystalline structure for the high-pressurephase of PbCrO4.

1.3.4. Raman experiments

Raman experiments were performed in the backscattering geometry using a 30 mW He-Ne laser, with λ = 632.81 nm and a Horiba Jobin Yvon LabRAM high-resolution ultra-violet (UV) microspectrometer in combination with a thermoelectrically cooled multi-

4The bulk modulus B0 is a measure of the ability of a substance to withstand changes in volume whenunder hydrostatic compression. It is equal to the quotient of the applied pressure divided by therelative deformation of the material:

B0 = −V ∂P∂V

.

5

1. Methodology and experimental setting

channel CCD that allowed for spectral resolution below 2 cm−1. An edge filter was usedto remove the contribution of the laser in the low-frequency range.

By analyzing the Raman spectra we obtained the frequencies of the phonons of thecrystal at ambient pressure (ω0) and their evolution upon compression, determined bytheir pressure coefficients (dω/dP ). Also the Gruneisen parameter5 of each phonon wasdetermined, using the value of B0 obtained in the XRD experiments [Eq. (1.2)]. Finally,the study of the evolution the Raman patterns with pressure was useful to localize thephase transitions and also as an independent confirmation of the hypothesis about thehigh-pressure crystalline structure of PbCrO4.

5The Gruneisen parameter γ describes the effect that changing the volume of a crystal lattice has onits vibrational properties and is related to the thermal expansion properties of a material. Whenthe restoring force acting on an atom displaced from its equilibrium position is linear in the atom’sdisplacement, the frequencies ωi of individual phonons do not depend on the volume of the crystalor on the presence of other phonons, and the thermal expansion (and thus γ) is zero. When therestoring force is non-linear in the displacement, the phonon frequencies ωi change with the volume.The Gruneisen parameter of an individual vibrational mode can then be defined as:

γi = −∂ ln(ωi)

∂ ln(V )= B0

∂ ln (ωi)

∂P.

6

2. Characterization of PbCrO4 at ambientconditions

2.1. AXO4 monazite-type compounds

The name monazite is given to some phosphates containing light rare-earth metals,commonly indicated by RE(PO4) (RE=La to Gd). All these compounds crystallize ina monoclinic lattice with symmetry elements proper of the P21/n space group (Z = 4).It is customary to name monazite-type compounds all the crystalline materials sharingthe same crystal structure, not all of them being phosphates. Monazite-type oxides forman extended family with interesting physical and chemical properties; these allow for alarge number of applications such as in coating and diffusion barriers, geochronology,light emitters and so on [7]. On the other hand, the knowledge of the behavior of monaziteunder high-pressure conditions can be relevant for mineral physics, chemistry, petrologyand geology [7].

Figure 2.1.: Schematic view of an AXO4 monazite.

The crystal structure of monazite-type AXO4 is based on the ninefold coordination ofthe A cation and the fourfold coordination of the X cation. Figure 2.1 shows the unit cellof a monazite, with the position of atoms inside it. The whole structure is formed by [001]chains of slightly distorted XO4 tetrahedra separated by irregular AO9 polyhedra. Thechains extend along [001] by sharing tetrahedral edges with AO9 polyhedra. There arefour chains per unit cell. They are linked laterally, in the (001) plane, by sharing edges

7

2. Characterization of PbCrO4 at ambient conditions

Figure 2.2.: Monazite projected down [001]. An AO9 polyhedron is outlined.

of adjacent AO9 polyhedra. This is the most notable characteristic of the monazitestructure and has a strong influence on its behavior under compression. Figure 2.2shows a polyhedral representation of the crystal structure projected onto the (001) plane,putting in evidence the monazite chains.

Studies performed on monazite-type phosphates have shown that their crystallinestructure is stable up to approximately 30 GPa [8,9]; anyway, cation substitution (partic-ularly with magnetic cations, like Cr) can contribute to the reduction of the transitionpressure in zircon-type oxides (a tetragonal structure, closely related to the monoclinicmonazite structure) [10,11]; consequently, we expect the appearance of phase transitionsat a relatively low pressure in ACrO4 monazite-type chromates.

2.2. A general description of PbCrO4

In general chromates with monazite structure are translucent materials with a bandgap between 2 and 2.8 eV [12]. Their crystals are biaxial, having two optic axes. Thebirefringence of these materials is usually of the order of 0.05 and the average index ofrefraction is approximately 2.4.

Lead chromate (PbCrO4, “crocoite”) is one of such minerals, given that it commonlycrystallizes at ambient conditions in a monazite structure [13], though also an orthorhom-bic variety of PbCrO4 exists [14]. It is commonly used in some pyrotechnic compositionsand as a pigment in some paints, being it practically insoluble in water. It can alsobe used as a photocatalyst and has been proposed as a potential laser-host material.Crocoite is a quite toxic substance (LD50 > 12 g/Kg1) if swallowed or inhaled, so it hasto be properly handled. It is quite rare in nature but it can be grown in laboratory [15].

1In toxicology, the median lethal dose, LD50 (abbreviation for “lethal dose, 50%”) of a toxin, radiation,or pathogen is the dose required to kill half the members of a tested population after a specified testduration.

8

2.3. Crystal structure at ambient pressure

Figure 2.3.: Photo of the sample in the DAC at ambient pressure (left) and at P = 3.5 GPa(right).

Very few studies have been performed on this compound, mainly centered on its crys-tal structure: for example, no accurate experimental determinations of the band gap ofmonocrystalline PbCrO4 have been published until now.

Figure 2.3 shows two photos of one of our samples, one at ambient conditions and atone P = 3.5 GPa. The tiny transparent sphere right under the sample is the ruby usedfor the determination of the pressure.

2.3. Crystal structure at ambient pressure

The refinement of the ambient-pressure XRD spectrum for the PbCrO4 powder sam-ples is shown in Fig. 2.4. From this refinement we obtained the lattice parameters ofmonazite-type PbCrO4. They are listed in Table 2.1, along with the values given in theliterature [16]; Table 2.1 also lists the atomic positions inside of the unit cell, in fractionof the lattice parameters. Both sets of values are in good agreement with those givenin the literature [16,17]. In this monazite structure Cr is coordinated with 4 oxygens andPb with 9 oxygens; the bonds for the CrO4 sub-structure form a distorted tetrahedronand their lengths can be calculated on the basis of the value in Table 2.1. The samecan be done for the bonds in the PbO9 sub-structure, that in turn form an irregularpolyhedron. The lengths for all bonds are reported in Table 2.2. A 10th oxygen atomis listed in Table 2.2 as coordinated with Pb; in first approximation we can consider itas isolated, because of its distance from Pb (+10% with respect to O9). On the otherhand, its presence could favor a coordination increase under compression, as we will seein the following.

2.4. Raman spectrum at ambient pressure

The Raman spectrum of PbCrO4 at ambient pressure is shown in Fig. 2.5. Given thatthe lattice of PbCrO4 can be thought as constituted by a set of two separate structures,CrO4 tetrahedra and PbO9 polyhedra, the analysis of this Raman spectrum can be

9

2. Characterization of PbCrO4 at ambient conditions

5 10 15 20 25 30 35

Inte

nsity

(ar

b. u

nits

)

2θ (degrees)

(a)

Figure 2.4.: XRD pattern for PbCrO4 at ambient pressure (dots). Solid line: refined profile.Dotted line: resduals. Ticks: calculated positions of the Bragg reflections.

Lattice parametersa [A] b [A] c [A] β [°]

Exp. 7.098(7) 7.410(7) 6.779(7) 102.4(2)Lit. 7.127(2) 7.438(2) 6.799(2) 102.43(2)

Atomic positionsx/a y/b z/c Site

Pb 0.2247(3) 0.1515(2) 0.4044(5) 4eCr 0.1984(2) 0.1643(2) 0.8845(9) 4eO1 0.2561(3) 0.0047(1) 0.0568(1) 4eO2 0.1201(2) 0.3415(4) -0.0057(1) 4eO3 0.0274(1) 0.1047(2) 0.6858(7) 4eO4 0.3859(4) 0.2152(3) 0.7872(8) 4e

Table 2.1.: Lattice parameters and atomic position for PbCrO4 at ambient pressure.

Bond lengths [A]Atom O1 O2 O3 O4 O5

Cr 1.650(8) 1.650(8) 1.662(7) 1.667(4) -Pb 2.562(8) 2.576(1) 2.590(2) 2.620(5) 2.631(1)

Atom O6 O7 O8 O9 (O10)Cr - - - - -Pb 2.644(1) 2.647(8) 2.741(2) 3.062(1) 3.382(6)

Table 2.2.: Bond lengths for the CrO4 and PbO9 polyhedra at ambient pressure.

10

2.4. Raman spectrum at ambient pressure

0 100 200 300 400 500 600 700 800 900 1000 1100

Inte

nsity

(arb

. uni

ts)

Raman shift (cm-1)

x5

Figure 2.5.: Raman spectrum for PbCrO4 at ambient pressure. Ticks indicate phonon frequen-cies.

carried out in terms of two complementary sub-lattices, formed respectively by Pb atomsand CrO4 molecules; thus, the vibration modes of the lattice can be classified as internaland external modes of the CrO4 units. The external modes are pure translation (T)or rotation (R) modes of the CrO4 molecule as a whole, while the internal modes conbe decomposed into four types of motion νi (i = 1, . . . , 4) corresponding to stretching(ν1, ν3) and bending (ν2, ν4) vibrations. Symmetry considerations predict the existenceof 36 active Raman modes for this monazite. The representation at the Γ point in thestandard notation is Γ = 18Ag + 18Bg and the classification in external and internalmodes yields Γ = Ag(6T, 3R, ν1, 2ν2, 3ν3, 3ν4) + Bg(6T, 3R, ν1, 2ν2, 3ν3, 3ν4).

In previous works [18] up to 10 Raman active modes have been observed for PbCrO4,while usually monazites don’t show more than 22 modes of the 36 theoretically ex-pected [19], due to the fact that some of them could be too weak or damped to beobserved experimentally. In our experiments we could observe up to 26 active modes forPbCrO4 at ambient pressure. Under compression, 4 further modes show up, due to peaksplitting. In the low frequency range, the presence of the filter made impossible for usto observe the peak reported by Wilkins [20] at 36 cm−1. In general, all the modes in thelowest frequency range were weak and therefore acquisition time in the low-frequencyregion was multiplied by 5 (see Fig. 2.5). All in all, we could observe 19 new modesnever reported before, mainly located in the low-frequency part of the spectrum. All themodes that we have observed are reported in Table 2.3.

The assignments in Table 2.3 can be understood as follows. Intuitively, the five modesin the high-frequency range can be attributed to the stretching motions inside the CrO4

units, due to the low weight of the Cr and O atoms, while the 10 modes in the middle-frequency range are probably due to bending vibrations inside the CrO4 units. Theassignments for these ranges, with less modes, were done in the already cited work byFrost [18]. Finally, the 15 low-frequency modes are due to lattice vibrations, probably

11

2. Characterization of PbCrO4 at ambient conditions

α (

cm-1

)

0

500

1,000

1,500

2,000

2,500

Energy (eV)1.20 1.40 1.60 1.80 2.00 2.20 2.40

Figure 2.6.: Optical-absorption spectrum for PbCrO4 at ambient pressure.

with contributions due to the motions of the heavy Pb atoms.

2.5. Optical absorption at ambient pressure

The optical-absorption spectrum for PbCrO4, shown in Fig. 2.6, is typical of a directband-gap semiconductor. The dependence of the absorption coefficient α on the photonenergy hν is mainly exponential, though it also shows a weak low-energy absorptionband below the band gap. This band has been observed also in other affine compounds,as PbWO4, and it can be attributed to defects or impurities in the crystal [21].

The exponential absorption tail follows the “Urbach’s rule”, i.e. the spectra can befitted by the empirical equation

α(E) = α0 exp

[σ

kBT(E − Eg)

](2.1)

where T is the absolute temperature, kB is the Boltzmann constant, α0 and σ are dimen-sionless temperature depending parameters and finally Eg is the band gap. The threeparameters are characteristic of each material and α0 is assumed as pressure indepen-dent [22]. On the other hand, σ determines the shape of the absorption tail and indirectlygives indications on the presence of defects in the crystal structure [22].

In Eq. (2.1) α0 and Eg are clearly correlated. Therefore, we determined α0 at ambientpressure (α0 = 130(1) cm-1) assuming that Eg = 2.3 eV [12] and that α0 is constant undercompression. This method can lead to small uncertainties on the absolute value of Eg,but it is very accurate to determine its pressure evolution. The fit of Urbach’s rule toour experimental data is quite good, as can be seen in Fig. 2.6.

12

2.5. Optical absorption at ambient pressure

ω0 dω/dP γSuggested assignements [cm−1] [cm−1 GPa−1]

Lattice modes 42* 0.7 0.9645 1.0 1.2757 3.7 3.7062∗ 1.6 1.4773 -0.2 -0.1681* -1.8 -1.2783 4.2 2.8795 0.4 0.2499* 2.8 1.61110 4.2 2.18116 6.8 3.35136 5.5 2.31149 5.2 1.99178 7.7 2.46185 4.3 1.32

Bending of CrO4 327 -0.5 -0.09338 1.8 0.30347 1.5 0.25359 1.0 0.16378 2.8 0.42402 0.8 0.11407 3.7 0.52440 3.9 0.51451 2.2 0.28479 5.4 0.64

Stretching of CrO4 801 4.3 0.31825 3.3 0.23840 1.9 0.13856 3.5 0.23879 3.2 0.21

Table 2.3.: Raman active modes for PbCrO4 at ambient conditions (ω0), pressure coefficientsdω/dP and Gruneisen parameters γ. The phonons marked with and asterisk areextrapolations of the high pressure data down to ambient pressure.

13

3. Experimental results and conclusions

3.1. High-pressure X-ray diffraction experiments

The evolution of the X-ray patterns under compression for pressures up to13 GPa isshown in Fig. 3.1. We can analyze it in therms of the different pressure ranges.

5 10 15

1 bar (r)

12 GPa

9.1 GPa

7.2 GPa

6.1 GPa

5.2 GPa

4.4 GPa

3.25 GPa

Inte

nsity

(ar

b. u

nits

)

2θ (degrees)

0.55 GPa

Figure 3.1.: X-ray patterns at selected pressures. Arrows indicate changes induced by the firstphase transition. Ticks indicates Bragg reflections.

In the range 0–3.25 GPa the spectra retain the shape corresponding to the originalstructure i.e. the mineral is stable up to 3.25 GPa. When pressure rises beyond thisvalue, and up to 5.2 GPa, some peaks begin to split. This doesn’t necessarily imply astructural phase change but it is likely due to a distortion of the low-pressure phase,as we will see in a moment. Starting from 5.2 GPa there are evident changes in thepatterns and this trend is maintained up to 9.1 GPa. These changes are now due to aneffective phase transition, that take places at about 6.1 GPa and that is fully completedat 9.1 GPa. In the range 5.2–9.1 GPa the two phases then coexist. Finally, in therange 9.1–12 GPa the diffraction pattern doesn’t change with increasing pressure so nonew structural phase transitions are observed. The transition at 6.1 GPa is reversible,given that the diffraction pattern recovers its original shape when reverting to ambient

15

3. Experimental results and conclusions

Latt

ice p

ara

mete

r (Å

)

6.6

6.8

7

7.2

7.4

Pressure (GPa)0 2 4 6 8

b

a

c

β a

ngle

(degre

es)

100.0

100.5

101.0

101.5

102.0

102.5

Pressure (GPa)0 2 4 6 8

Figure 3.2.: Pressure evolution of the unit-cell parameters of PbCrO4 for the low-pressure phase.

Bond lengths [A]Atom O1 O2 O3 O4 O5

Cr 1.590(3) 1.613(3) 1.616(7) 1.642(4) -Pb 2.489(8) 2.506(8) 2.517(6) 2.538(6) 2.557(1)

Atom O6 O7 O8 O9 O10

Cr - - - - -Pb 2.591(2) 2.612(1) 2.655(8) 3.012(1) 3.267(4)

Table 3.1.: Bond lengths for the polyhedral units at 5.2 GPa.

pressure. This also indicates that pressure does not induce the decomposition of PbCrO4

into binary oxides.

The evolution of the crystallographic parameters as a function of pressure for the low-pressure phase is plotted in Fig. 3.2, together with the corresponding fits. It is clear thatthe compressibility of PbCrO4 is not isotropic, given that the a, b axes are more com-pressible than the c axis. This is due to the relative arrangement of the CrO4 and PbO9

polyhedra in PbCrO4 (see Section 2.3), that alternate in the c direction giving place tostructures less compressible than the void space that exist between the polyhedral unitsalong the other directions. The evolution of the 4 parameters is linear up to 3.5 GPa,where there is a change in the trend for c and β. This is related to to the aforementionedchanges in the diffraction patterns and indicates that the reduction of inter-atomic dis-tances causes an atomic reordering with changes in the local coordination; in particularthe coordination of the Pb atoms increase from 9 to 10 (see also Section 2.3); the diffrac-tion patterns are still compatible with a monoclinic structure with monazite symmetry,so this is a distortion of the original monazite structure. Given that the β angle de-creases with pressure, we can expect an increase in the symmetry of the crystal: this isreasonable [8] and is also reflected in the lower number of peaks in the diffraction patternsat higher pressures.

Using the data relative to the cell parameters, we could also study the evolution of the

16

3.2. High-pressure Raman experiments

Pre

ssure

(G

Pa)

V0/V (adim)

8

6

4

2

0

1.00 1.02 1.04 1.06 1.08 1.10 1.12

Figure 3.3.: Pressure evolution of the volume of the unit cell.

Lattice parameters (HP phase)a [A] b [A] c [A] V [A3] Z

6.95(6) 6.11(6) 6.63(6) 282(8) 4

Table 3.2.: Lattice parameters for PbCrO4 for the high-pressure phase.

volume of the unit cell (Fig. 3.3) and determine the bulk-modulus B0 of PbCrO4. The fitof the data in Fig. 3.3 with Eq. (1.2) resulted in a bulk modulus B0 = (58±1) GPa (andB′0 = 3.1 ± 0.4). For comparison, monazite phosphates often have bulk moduli largerthan 100 GPa [8], so monazite chromates appear to be much more compressible than phos-phates. The reference to phosphates is mandatory, given that no other monazite-typechromates, like for example LaCrO4 or SrCrO4, have been studied under compression.Regarding the high-pressure phase, i.e. beyond 9.1 GPa, we have found that the diffrac-tion pattern is compatible with an orthorhombic unit cell with the parameters givenin Table 3.2. Moreover, the systematical extinctions in the diffraction patterns allowedus to assign the P212121 space-group symmetry to this phase. This is compatible withtheoretical and empirical arguments (Bastide’s diagram [23]). In short, the high-pressurephase is likely to be a strongly distorted barite [24]. Lead coordination increases up to12 and the structure is formed CrO4 and PbO12 polyhedra; the results of Raman andoptical-absorption experiments are compatible with this hypothesis, as we will see in amoment.

3.2. High-pressure Raman experiments

The Raman spectra of PbCrO4 for pressures up to 13 GPa are showed in Fig. 3.4, whilethe pressure evolution of the detected phonons is showed in Fig. 3.5.

17

3. Experimental results and conclusions

Inte

nsity

(arb

. uni

ts)

Raman shift (cm-1)

0 200 400 600 800 1000 1200

1 bar (r)

13 GPa

11.6 GPa

9.4 GPa

7.5 GPa

5.3 GPa

3.5 GPa

1 bar

1.5 GPa

2.5 GPa

Figure 3.4.: Raman spectra of PbCrO4 at different pressures.

Figure 3.5.: Evolution of Raman active modes for PbCrO4 under compression.

18

3.2. High-pressure Raman experiments

At very low pressure the evolution of the Raman spectra is characterized by thesplitting of several modes: this is undoubtedly due to the anisotropic compressibility ofPbCrO4 (see Section 3.1). This anisotropy breaks the ambient pressure degeneracy ofsome modes that are very near in frequency and makes them to evolve differently. Uponfurther compression there are changes in the Raman patterns, mainly located in the low-frequency range (indicating the intervening changes in the crystallographic parameters,given that these frequencies correspond to lattice vibrations, as shown in Table 2.3). Themode at 42 cm−1 is visible only beyond 2 GPa, due to the presence of the edge filter. Itsposition at ambient pressure is an extrapolation of the high-pressure data. For most ofthe observed phonons their evolution consist in a monotonic, linear hardening1. Someof the modes, though, show an unusual behavior. In particular the modes at 73, 81 and327 cm−1 clearly soften under compression. This behavior may be related to the onsetof the phase transition (atomic reordering) that was detected by means of the XRDexperiments in the range 3.25–5.2 GPa. Up to 4.5 GPa there are not other qualitativechanges in the Raman spectra. However, some modes show a nonlinear behavior in thisrange and three extra weak modes, at 60, 930 and 950 GPa, appear beyond 3.5 GPa.

In the range 5.3–10.1 GPa the Raman spectra change drastically, likely due to thecompletion of the monoclinic-to-monoclinic phase transition detected in diffraction ex-periments beyond 3.25 GPa. In particular, we can observe a general decrease in theintensity of the spectra together with an increase in the number of the high-frequencymodes and a decrease on the mid-range modes. This is evident in both Fig. 3.4 and3.5. The increase of the high-frequency modes can be attributed to a distortion of theCrO4 tetrahedra of the structure resulting from the first phase transition: some of themodes can, in fact, be degenerate in the low-pressure monoclinic phase but not in thehigh-pressure monoclinic phase. As a last remark, it is important to note that theextra peaks in the range 3.5-3.3 GPa correlate well with phonons of the high-pressuremonoclinic phase. So, their presence and the evolution change of nonlinear modes canbe hints of the onset of the monoclinic-to-monoclinic transition. No other importantchanges occur in this range, but for the appearance of some weak modes in the low- andintermediate-frequency range. This can be associated with the coexistence of the twophases detected in the XDR experiments.

The Raman modes for the high-pressure monoclinic phase are reported in Table 3.3,together with their pressure coefficients. Beyond 10.1 GPa no substantial changes occursin the Raman spectra; on the other hand, a sudden drop of the Raman signal at pressuresbeyond 11.6 GPa and a corresponding reduction in the number of modes is compatiblewith the completion of the second phase transition. The decrease in the number of modesis compatible with the increase in the symmetry of the crystal from the monoclinic phaseto the observed orthorhombic phase.

Upon further compression and up 17.9 GPa (the highest pressure reached in our Ramanexperiments) no other changes are visible in the Raman spectra. Upon the subsequentdecompression, the original phonons are recovered, so this is another confirmation of the

1In Raman spectroscopy, “hardening (softening) modes” are “modes with a positive (negative) deriva-tive”, with respect to a physical variable (pressure, in our case).

19

3. Experimental results and conclusions

ω0 dω/dP ω0 dω/dP[cm−1] [cm−1 GPa−1] [cm−1] [cm−1 GPa−1]

44 0.9 370 0.959 0.4 386 1.070 0.7 466 0.885 -0.8 746 0.4101 4.0 815 1.0116 3.7 834 1.4139 1.8 847 1.5168 3.4 857 3.4200 4.0 916 1.6217 8.2 928 -0.3338 2.9 933 1.0351 0.9 - -

Table 3.3.: Raman active modes at 6.4 GPa for the high-pressure monoclinic phase of PbCrO4

(ω0) and pressure coefficients (dω/dP ).

reversibility of the two phase transitions.Parameters such as the phonon frequencies at ambient pressure ω0, pressure coeffi-

cients dω/dP and the Gruneisen parameters have been calculated from the data in Fig.3.4 and 3.5 and are listed in Table 2.3. A comparison of the data in Table 2.3 andTable 3.3, related respectively to the original monoclinic phase and the high-pressuremonoclinic phase, shows that the phonons with similar frequencies also have similarpressure coefficients. This can be regarded as a confirmation of the similarity of the twomonoclinic phases.

3.3. High-pressure optical-absorption experiments

We have performed optical-absorption experiments in five different samples. All of themshowed the same optical behavior under compression. The absorption spectra at selectedpressures are shown in Fig. 3.6. At every pressure in our experimental range the spectrashow the typical aspect of a direct band gap semiconductor, with the exponential de-pendency given in Eq. (2.1). Upon compression the absorption edge gradually redshiftsuntil, when reaching 3.4 GPa, a sudden collapse of the band gap takes place and, corre-spondingly, the color of the sample turns from orange to red-violet, as in Fig. 2.3. Thischange in the optical properties of PbCrO4 is probably related to the first phase transi-tion (monoclinic to HP monoclinic) put in evidence by XRD and Raman experiments atsimilar pressures. Upon further compression beyond 3.4 GPa the absorption still showsan exponential dependence and its pressure evolution is again toward lower energies.

At a pressure of about 12 GPa all the samples became completely opaque in the opticalwavelength range, because the absorption edge lay completely in the near IR range. Thischange is likely due to the HP monoclinic to orthorhombic phase transition seen at the

20

3.3. High-pressure optical-absorption experiments

1.0 1.5 2.0 2.50.0

0.5

1.0

1.5

2.0

14.8

GP

a

18.1

GP

a

19.1

GP

a

20.8

GP

a

16.1

GP

a

14.3

GP

a13

.7 G

Pa

12.8

GP

a11

GP

a9.

6 G

Pa

8.8

GP

a 6.8

GP

a5

GP

a

4.3

GP

a

3.4G

Pa

2.9

GP

a

α (1

03cm

-1)

Energy (eV)

0.4

GP

a

Figure 3.6.: Absorption spectra of PbCrO4 at selected pressures.

same pressure in the XRD and Raman experiments.

Beyond 12 GPa the spectra retain their exponential dependence on energy, with nosignificant changes in the behavior of the absorption edge. The maximum pressure thatwe could reach was about 21 GPa, due to experimental limitations and to the formation ofa large number of defects in the crystal. Both factors made very difficult the obtainmentof satisfactory spectra. Upon pressure release the changes in the spectra are reversible,similarly to what happens in Raman and XDR. One thing that worth noting is that asmall hysteresis in the recuperation of the original monazite phase is observed in all theexperiments.

The dependence of the energy gap Eg with pressure is shown in Fig. 3.7, that hasbeen obtained under the hypothesis that the fundamental gap of PbCrO4 is direct inevery phase and thus that Eg can be calculated using Eq. (2.1).

The evolution of Eg is linear up to 3 GPa, with dEg/dP = −48(2) meV/GPa. From 3to 3.5 GPa Eg decreases abruptly with ∆Eg = 0.4 eV and then its evolution is linear againup to 15 GPa, with a pressure coefficient of -45 meV/GPa. Apparently a second, smaller0.1 eV collapse of Eg is visible at this pressure. Its presence is still unconfirmed, thoughreasonable. In the range 15–21 GPa the pressure coefficient changes to -67(2) meV/GPa,while the evolution of Eg is still linear. The variations of σ in Eq. (2.1) are comparablewith the error of this parameter, so no conclusions can be drawn.

The results of these experiments can be understood with reference to the theoreticalband structure of PbCrO4

[12], represented in Fig. 3.8. According to it, in PbCrO4themain contribution at the bottom of the conduction band results from the non bondinginteraction between the Cr(3d) orbitals and the O(2p) orbitals. The upper portion of the

21

3. Experimental results and conclusions

0 5 10 15 200.5

1.0

1.5

2.0

2.5

Eg

(eV

)

Pressure (GPa)

Figure 3.7.: Pressure dependence of Eg for PbCrO4. Different symbols denote different samples.Closed symbols are pressure upstroke, open symbols for pressure downstroke.

conduction band results primarily from the interaction between empty Pb(6p) orbitalsand O(2p) orbitals. Also a minimum contribution from the Pb(6p) orbitals is observedat the bottom of the conduction band, since the space group symmetry permits mixingof the Pb(6p) and Cr(3d) orbitals. On the other hand, the top of valence band is mainlycompose of O(2p) non bonding orbitals with a small contribution of Pb(6s) orbitals.Wecould attempt an explication of our results based on an analogy to what happens forthe scheelite phase PbWO4

[25], whose band structure is similar to that of PbCrO4[12];

under compression, the decrease in Cr-O and Pb-O distances causes an increase in thecrystal field acting on Cr(3d) and Pb(6s) states. Because of the different coordinationof Pb and Cr, the Pb(6s) states shift toward higher energy faster than the Cr(3d) statesand so the band gap closes, as a consequence of the reduction of the difference in energybetween the top of the valence band and the bottom of the conduction band.

The collapse of Eg at 3.5 GPa could be caused by the structural changes related to themonoclinic-to-monoclinic phase transition occurring at this pressure, according to XRDand Raman experiments. In particular, the coordination of the Pb cation increases from9 to 10 and this could be the reason behind the sudden decrease of Eg.

From 3.5 to 14 GPa the linear decrease in energy gap is caused by the same facts thanin the low pressure range. Upon further compression the ocurrence of the monoclinic-orthorhombic transition leads to the small collapse and slight change in the slope ofEg that is observed at 15 GPa. This is reasonable, given that the second transitionwould imply a further increase in the coordination of Pb from 10 to 12 (see Section 3.1).The confirmation of this hypothesis requires some theoretical calculations that are notavailable at the moment. On a side note, the opacification of the samples at 12 GPasuggests a possible metallic character of the orthorhombic phase. The confirmation of

22

3.4. Conclusions

Energ

y (

eV

)2.3 eV

-8.0

0.0

2.3

4.2

Pb(6s) - O(2p)

Cr(3d) - O(2p)

O(2p)

Pb(6s) - O(2p)

Cr(3d) - O(2p)

Pb(6p)

Figure 3.8.: Schematic band structure of monazite-type PbCrO4

this hypothesis would come from the complete collapse of the energy gap at a suitablepressure. The optical absorption experiments in the range 12–21 GPa did not show anyevidence for a such collapse. Unpublished third party electrical measurements showindeed that the resistivity of powder samples decreases at least one order of magnitudeupon compression, reaching a minimum of about 1.5 Ω·cm at 13 GPa. On the other hand,not only this value is much higher than those typical of a metal (10−5–10−6 Ω·cm), butthose same experiments show also that upon further compression and up to 35 GPa theresistivity of the samples increases again up to 10 Ω·cm [26].

3.4. Conclusions

A fairly complete characterization of monazite-type PbCrO4 at ambient pressure andunder compression has been presented in this work. The study involves several differentexperimental techniques that complement each other in the confirmation of the results.

By means of structural XRD studies we could determine the unit cell parameters ofmonazite-type PbCrO4 and their pressure evolution, along with that of the unit cellvolume and the equation of state at room temperature. On the other hand, these mea-surements showed the existence of two phase transition in our pressure range: the firstis detected at 3.25–5.3 GPa, depending on the characterization technique, and is a amonoclinic-to-monoclinic transition, i.e. the second phase possesses the same crystalsymmetry than the first one but a different local atomic arrangement. The second tran-sition takes place at 9–15 GPa and is a monoclinic-to-orthorhombic transition, associated

23

3. Experimental results and conclusions

thus with an increase in the crystal symmetry. In correspondence with the second tran-sition a strong opacification of the crystal is observed, but our initial hypothesis aboutthe metallization of the mineral has been discarded on the basis of subsequent opticaland electrical experiments.

Raman spectroscopy confirmed the results of the XRD studies. The transitions occurat similar pressures in both experiments and the fine details of the Raman patterns arecompatible with the crystal structure that was hypothesized on the basis of XRD. Theevolution of Raman modes has been reported for the low pressure and high pressuremonoclinic phases. Moreover, we could improve some previous studies on the Ramanspectrum of PbCrO4 at ambient pressure, individuating some phonons that had neverbeen detected before.

Finally, the evolution of energy gap has been studied through optical absorption ex-periments up to 15 GPa. The band gap collapse of 0.4 eV corresponding to the first phasetransition is clearly visible, while an unconfirmed smaller collapse should be associatedto the second transition. An explication of the results has been thus attempted usingthe theoretical band structure model and by analogy with an affine semiconductor, thescheelite-type PbWO4.

Even if the experimental picture is pretty coherent, there’s is room for some theoreticalspeculations: a set of detailed ab initio calculations of the crystal structure and the bandstructure of PbCrO4 under compression could definitively confirm some of our qualitativeand phenomenological conclusions.

The results here reported have already been published [E. Bandiello, D. Errandonea,et al., PRB 85, 024108 (2012)] and presented in two posters at the 49th (Budapest) and50th (Thessaloniki) EHPRG Conferences.

24

Appendices

25

A. Space groups

A.1. Space groups

For the complete description of the crystal structure of a particular substance the knowl-edge of the contents of the unit cell is required. The translational symmetry of the latticethen “builds” the crystal by propagating the structure of the unit cell. On the otherhand, other symmetry elements different from translations are normally found in theunit cell of a crystal, so often it is not necessary to describe the whole unit cell, butonly one part of it, given that the additional symmetry elements within the unit celldetermine the remainder of its content. Symmetry can be used to shorten considerablythe description of the structure of a crystal; more importantly, a knowledge of the crys-tal symmetries reduces the number of atomic coordinates that have to be found in thedetermination of a crystal structure. The Hermann–Mauguin (H-M) notation is one ofthe the standards in crystallography to represent the symmetry elements of a crystal.Here we will report some informal notes about the H-M notation, particularized to thetwo space groups we are dealing with in this thesis. It is useful to start the discussionby giving some common definitions:

Lattice System: This is the shape of the “bare” unit cell of a crystal. In 3D exist 7different lattice systems; they are listed in Table A.1.

Bravais Lattice: A set of equivalent lattice points. They are generated combining eachlattice system with one of the following lattice types (in the H-M notation): P(primitive), I (body centered), F (face centered), A, B, C (nodes at the center ofeach yz, xz and zy face respectively), R (rhombohedral).

Point Group(or Crystal class): A set of geometric symmetries (isometries) that keep atleast one point fixed.

Space Group: The complete description of the symmetries of a particular crystal.

There are 14 Bravais lattices in 3D, constructed from the simplest translational symme-tries applied to the seven crystal systems. All crystal systems have P lattices, but notall the systems have the other type of lattices. This is because there is degeneracy whenall the lattice types are applied to all the crystal systems. Because of these degeneraciesthe number of possible Bravais lattice is reduced to 14. Point groups are in number of 32and they are the results of the combinations of the following three symmetry elements:

Rotation Axes: an n-fold rotation axis is denoted by n in the H-M notation. It is definedby the condition that a rotation by an angle of 360°/n with respect to the axis

27

A. Space groups

Crystal System Lattice types Parameters

Triclinic P a 6= b 6= c, α 6= β 6= γ 6= 90°Monoclinic P , C a 6= b 6= c, α = γ = 90° 6= βOrthorhombic P , C, I, F a 6= b 6= c, α = β = γ = 90°Tetragonal P , I a = b 6= c, α = β = γ = 90°Rhombohedral P a = b = c, α = β = γ 6= 90°Hexagonal P a = b = c, α = β = 90°, γ = 120°Cubic P , I, F a = b = c, α = β = γ =90°

Table A.1.: Resume of 3D Bravais lattice and related parameters.

leaves the crystal invariant. The requested compatibility with the translationalsymmetry only allows for n =1, 2, 3, 4, 6.

Inversion Axes: Inversion axes are the same as rotation axes, except that after the360°/n rotation the object is projected through a inversion center on the axis. TheH-M symbol for a n-fold inversion axes is n.

Mirror Planes: a mirror plane simply creates a reflected image of the crystal with respectto itself. It is indicated by m.

Not all the combinations are permitted, only those compatible with the fundamentaltranslational symmetry of the crystal system. Intuitively point-group symmetries canbe thought of as “attached” to some particular points of the unit cell. These pointsdepends on the lattice systems: they are the corners of the unit cell in the P lattices,while for an I lattice they are situated at the body center of the cell as well as at thecorners and so on for the other lattice types.

There are two further symmetry operations occurring in crystals, that are not part ofthe 32 point groups because they involve translations:

Screw Axes: Indicated by nz, a screw axis generates a rotation of 360°/n around itself,followed by a translation by z translations of z/n · x, where x is a lattice vectorparallel to the screw axis.

Glide Plane: A glide plane operates a mirror operation followed by a translation. Thesymbols for glide planes are a, b, c for a glide translation along a/2, b/2, c/2, nfor a glide translation along with half a face diagonal (the same symbol for all thefaces), d for for a glide translation along with a quarter of a face diagonal andfinally e for two glides along the same glide planes and a translation along twodifferent half lattice vectors.

Finally a space group is a combination of the translational symmetry of a lattice togetherwith other symmetry elements such as point group symmetries, screw axes and glideplanes. Once taken in account the degeneracies due to different choices of axes and theirorientations, we are left with a total of 230 different space groups. A visual depiction ofeach space group is achieved using a set of standard crystallographic symbols [27].

28

A.1. Space groups

Multiplicity Wyckoff letter Site symmetry Coordinates

2 c m (1) x, y, z (2) x, y, z

1 b 1 x, 12 , y

1 a 1 x, 0, y

Table A.2.: Wyckoff sites for the monoclinic Pm space group.

The H-M notation uses a set of two to four symbols to completely identify a spacegroup. The first symbol is always a single letter specifying the Bravais lattice (P, C, I,F, R) while the next three symbols the symmetry elements related to certain directions(point group and translation elements) as combination of the symbols 1, 2, 3, 4, 5, 6, a,b, c, e, n, /, , that have been described before. Two kinds of notations are used whenspecifying a given space group: the full H-M symbol shows both rotation/screw axeswhich run parallel to the specified direction, as well as any mirror/glide plane perpen-dicular to the same direction. These two symbols will be separated by a “/”; the shortH-M symbol is a more condensed notation that depends on the symmetry constraints ofeach lattice type and/or the existence of some “special” axis (in the monoclinic lattice,for example, there’s only one “unique axis” that is perpendicular to the other two, sounambiguous short symbols can be used; conversely, in the cubic system all the axes areequivalent and the full H-M symbols are needed). The two forms are equivalent, i.e. theycan be converted in each other. Finally, the H-M symbol for a given space group canalso depend on the choice of the unit cell (when this is not unique) and on the possibleorientations/permutations of the three base axes.

A.1.1. Wyckoff positions

Associated to each space group there are some particular positions, called Wyckoff posi-tions or sites, that physically indicate where the atoms in the unit cell can be found. Theinformation that they provide is how many atoms are generated through the symmetryelements of the crystal when an atoms is “placed” in a given position (multiplicity) andwhere they are. They are indicated by a number, the multiplicity of the site, and aletter label. As a classical example to understand how Wyckoff positions work is usefulto examine the monoclinic P1m1 space group (Pm, in the short H-M notation). Thisspace group has only two symmetry elements, i.e. two mirror planes respectively at y = 0and y = 1

2 . If we place an atom at an arbitrary position x, y, z then symmetry generatesa second atom at x, y, z (multiplicity is 2; y stands for −y, as customary in crystallog-raphy). On the other hand, if the atom is placed right on one of the mirror planes, forexample x, 12 , z, then no other atoms are generated (multiplicity is 1). The complete listof the Wyckoff sites for the monoclinic Pm lattice is listed in Table A.2.

Even if it is referred to a particular space group, the scheme of the Table A.2 iscompletely general. The multiplicity column gives the number of equivalent positionsper unit cell, while the coordinates column shows the relations between correspondingpositions. The letters are simply labels, with no physical meaning. They are assigned

29

A. Space groups

Figure A.1.: P21/n space group [28].

alphabetically from the bottom up. The symmetry column tells us what symmetryelements the atom resides upon. The uppermost Wyckoff position, corresponding to anatom at an arbitrary position, never resides upon any symmetry elements. This Wyckoffposition is called the general position. All of the remaining Wyckoff positions are calledspecial positions. They correspond to atoms which lie upon one of more symmetryelements and because of this they always have a smaller multiplicity than the generalposition. Furthermore, at least one of their fractional coordinates must be fixed.

A.2. Monoclinic P21/n space group

The full H-M symbol for this group is P121/n1; the short form P21/n is unambiguous ifthe unique axis is specified. In our case case this is the b axis. The symmetry elementsof this space groups are then 21 screw axes parallel to b and glide planes perpendicularto it. The graphical representation of the space group is given in Fig. A.1. The b axis isdirected along c× a. The following symmetry elements are represented in Fig. A.1:

2-fold screw axes parallel to b. Symbol: “”.

inversion centers. Symbol: “”.

two diagonal glide planes orthogonal to b, situated at 14b and

(14 + 1

2

)b. Symbol:

“ ”.

The circles represent the position of the atoms in the cell. The “©+” symbol indicatesthe “initial point”, i.e. an atom at an arbitrary position with z > 0 (z < 0 for ©-). Theremaining symbols represent the atoms “generated” by the symmetry operations actingon the initial point. The “© 1

2±” symbols are atoms related by symmetry to “©+” but

now at a height z ± 12 . The “ ” symbol indicates a change of handedness with respect

30

A.3. Orthorhombic P212121 space group

to the initial point, with the same meaning for the superscripts. The complete Wyckoffpositions for P21/n are listed in the Table A.3 below [27].

Multiplicity Wyckoff letter Site symmetry Coordinates

4 e 1(1)x, y, z (2)x+ 1

2 , y + 12 , z + 1

2(3)x, y, z (4)x, y + 1

2 , z + 12

2 d 1 12 , 0, 0 0, 12 ,

12

2 c 1 12 , 0,

12 0, 12 , 0

2 b 1 0, 0, 1212 ,

12 , 0

2 a 1 0, 0, 0 12 ,

12 ,

12

Table A.3.: Wyckoff sites for the monoclinic P21/n space group.

A.3. Orthorhombic P212121 space group

Crystals with P212121 space group have 21 screw axes, directed along the three basevector. The schematic representation of this space group is given in Fig. A.2, in whichthe c vector is directed toward the reader; the notation is the same as in Fig. A.1,except for the “” arrows, which symbolize the direction of a 2-fold screw axis whenit is parallel to the plane of the projection. A fractional h attached to such a symbolindicates two axes with heights h and h + 1

2 . No fraction stands for h = 0 and h = 12 .

There is only one Wyckoff position, the general one, indexed 4a and with “1” symmetry.Its coordinates are given in Fig. A.2.

Figure A.2.: P212121 space group [28].

31

B. Raman spectroscopy and notation.

B.1. Raman Spectroscopy

Raman scattering is caused by the change in the polarizability of molecules or suscepti-bility of crystal when subject to a time-varying electric field. Quantum mechanically thisprocess can be viewed as a two-particles interaction between a phonon and an opticalphoton. Energy-momentum conservation gives

~ω0 − ~ω ± ω(q) = 0, (B.1a)

~k0 − ~k ± ~q = 0, (B.1b)

where ω0, k0 and ω, k characterize the incident and scattered photon respectively, whileω(q) and q are the angular frequency and momentum of the phonon. For the light in thevisible region of the spectrum k0 and k are of the order of 10−3 times a reciprocal latticevector; this means that only excitations in the center of the Brillouin zone (q ≈ 0) cantake part in Raman scattering. In the Eq. (B.1a) and (B.1b) the “+” sign is for phonongeneration and the “-” sign is for absorption.

The interaction of the visible light with solid occurs via the polarizability of the valenceelectrons. The electric field E0 induces, via the susceptibility tensor χ, a polarizationP , i.e.1

P = ε0χE0 or Pi = ε0χijEj0. (B.2)

The periodic modulation in P leads, in turn, to the emission of a wave, the scatteredwave, that in a classical approximation can be regarded ad dipole radiation from theoscillating dipole P .

The oscillatory nature of the electric field interacting with the atoms in a crystal latticemakes the vibrational analysis easier if it is conducted on the basis of the normal modesof the crystal, i.e. a set of motions in which all parts of the system move sinusoidally withthe same frequency and with a fixed phase relation, one frequency and one phase for eachmode. The most general motion of a system can thus be described as a superpositionof its normal modes. The modes are normal in the sense that they are independent,that is to say that an excitation of one mode will not cause motion of a different mode.In this respect, the susceptibility tensor χ is conveniently written as a function of theso-called normal coordinates Qk, that refer to the position of atoms away from theirequilibrium position. Each normal coordinate, relative to a single normal mode, is a

1The Einstein notation for summation has been used: when an index appears twice in a single term itimplies summation of that term over all the values of the index.

33

B. Raman spectroscopy and notation.

linear combination of the cartesian displacement coordinates of the positions of theatoms.

Given that we expect an harmonic motion of each atom, the normal coordinatesrelative to the induced vibrations [ω(q), (q)] can be written as Qk = Qk0 cos(ω(q)t),considering that q ≈ 0. On the other hand, the oscillating electric field is given byE0 = E0 cos(ω0t) and so, performing a first order Taylor expansion of χ

χ ' χ0 +

(∂χ

∂Qk

)0

Qk or χij ' (χij)0 +

(∂χij∂Qk

)0

Qk,

the polarizability P given in Eq. (B.2) can be expressed as:

Pi = ε0(χij)0Ej0 cos(ω0t)+

+1

2ε0

(∂χij∂Qk

)0

Qk0Ej0 [cos(ω0 + ω(q)t) + cos(ω0 − ω(q)t)] .(B.3)

The scattered radiation thus contains an elastic contribution of frequency ω0 (Rayleighscattering) and two further terms, called Raman side bands, with frequencies ω0±ω(q).The plus and minus sign correspond to scattered light quanta that have absorbed and,respectively, lost the energy of the elementary excitation [ω(q), q]. The component offrequency ω0 − ω(q) are called Stokes, and anti -Stokes the components with frequencyω0 + ω(q). For the anti-Stokes components to exist, is necessary that already excitedphonon are present in the solid. At ambient temperature the intensity of anti-Stokesemission is then much reduced (10−6 times the Stokes radiation), because the elementaryexcitations are mainly in their ground state.

The term ∂χjl/∂Qk in Eq. (B.3) is called the Raman tensor and is often writtensimply as χijk. The i, j indexes range from 1 to 3, while the k index ranges from 0 to3N − 3, that is the number of optical modes with q = 0 for a crystal with N atoms inthe unit cell. The Raman tensor which refers to all zone-center vibrations has thus rank3. For an individual mode this tensor is given by a 3×3 matrix Rk determined from thederived susceptibilities. This matrix is called the Raman tensor of a particular mode.

The intensity Ik of the scattered light for a particular mode is proportional to thesquare of its Raman tensor. If ei and es are the unit vectors of the incoming andscattered light, it can be proven that the Raman scattering intensity is proportional to

Ik ∝ |eiRkes|2 . (B.4)

From Eq. (B.3) it is clear that prerequisite for the observation of a Raman line is thatthe susceptibility χ has a non-vanishing derivative with respect to the coordinate Qk ofthe elementary excitation. By means of the group theory, knowing the point group ofa crystal and the Wyckoff positions occupied by the atoms in the unit cell, it can bedetermined which components of the Raman tensor are different to 0 and so if the modeis or not Raman active.

34

B.2. Raman tensors for PbCrO4

Symbol Meaning

A, BSingly degenerate state, symmetric (A) or anti-symmetric(B) with respect to rotation about the principal n axis

E, T Doubly (E) or triply (T) degenerate state

Xg, XuThe state is symmetric (“g”, gerade) or anti-symmetric (“u”,ungerade) with respect to a an inversion 1

.

X’,X”The state is symmetric (’) or anti-symmetric (”) with respectto a mirror plane m

X1, X2The state is symmetric (1) or anti-symmetric (2) with re-spect to additional n or m

Table B.1.: Mulliken’s symbols

B.1.1. Mulliken’s symbols

Each particular Raman mode, represented by an Rk matrix, is characterized by itssymmetries relative to the normal coordinates and by its degeneracy, that depend onthe point group of the crystal under examination and on the Wyckoff positions occupiedby the atoms in the unit cell. The symmetries can be determined experimentally bymeans of polarized Raman measurements performed on properly oriented monocristallinesamples. A set of symbols, introduced by Mulliken [29], are used to label the modes withrespect to symmetries/degeneracy. They are listed in Table B.1.

For each crystal system, the vibrational modes can have only definite symmetries:A, B for triclinic, monoclinic and orthorhombic system; A, B, E for the tetrahedral,rhombohedral and hexagonal system and finally A, E, T for cubic crystals. Symmetric“g” modes are Raman active, while anti-symmetric “u” modes are IR active.

B.2. Raman tensors for PbCrO4

In the case of monoclinic PbCrO4 the space group is P21/n and the Wyckoff positions ofthe atoms are all 4e, with site symmetry “1”. These data allow software tools like SAM2,hosted on the Bilbao Crystallographic Server [30], to predict the theoretical number ofphonons and their symmetries/degeneracy. In our case 3 Ag modes and 3 Bg modes peratom are expected, so a total of 36 Raman modes (18+18). In the so-called mechanicalnotation this is written as

Γ = 18Ag + 18Bg. (B.5)

2Spectral Active Modes

35

B. Raman spectroscopy and notation.

The character of the individual modes can be specified, when needed. Finally, the explicitexpressions of the matrices Ag and Bg are:

Ag =

a d 0d b 00 0 c

, Bg =

0 0 e0 0 fe f 0

. (B.6)

36

C. Software

C.1. Software tools

The following is the list of the software used for this work.

→ Qtiplot 0.9.8.8«and OriginPro 8.6 for data plots and analysis.

→ Vesta 3.1.0 [31] for the crystallographic plots and calculations.

→ Fityk 0.9.8« [32] for the analysis of the Raman spectra.

→ Inkscape 0.48.3.1«for SVG graphic.

→ The Gimp 2.8.0«for the editing of photos and images.

→ LibreOffice 3.5.4.2«for some data processing and text editing.

→ Final document composed using the TEXLive«distribution of LATEX 2ε«, in con-

junction with Kile 2.1.0«integrated Latex environment.

→ Operating system was “Sid” Debian GNU/Linux«with KDE 4.8.4«environment.

The items marked with “«” are GPL software.

37

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