the eŒ ect of stress on cubic-to-tetragonal phase ... · transitions in the mg2tio4® lms are...

PHILOSOPHICAL MAGAZINE A 2001 VOL 81 NO 1 109plusmn 124

The eŒect of stress on cubic-to-tetragonal phasetransitions in Mg2TiO4 and Mg2GeO4 spinel reg lms

ST SENZW BLUMy and D HESSEzMax-Planck-Institut fuEgrave r Mikrostrukturphysik Weinberg 2 D-06120 Halle

Germany

[Received 14 May 1999 accepted 29 February 2000 ]

ABSTRACTEpitaxial thin reg lms of the spinels Mg2TiO4 and Mg2GeO4 are grown by a

solid-state reaction between MgO(001) single-crystal substrates and vapours orthin reg lms of the oxides TiO2 and GeO2 respectively Cubic-to-tetragonal phasetransitions in the Mg2TiO4 reg lms are studied by transmission electron microscopyincluding in-situ observations at high temperatures The transition kinetics areshown to be inmacr uenced by the presence of stress Compared with bulk transitionsin thin reg lms the kinetics are slowed Moreover the kinetics become sensitive tothe crystallographic orientation of the tetragonal Mg2TiO4 grains with respect tothe reg lm plane Stress develops owing to the mismatch of the thermal expansionbetween reg lm and substrate during the cooling of the reg lms from the growthtemperature of about 1270K as well as during the phase transition owing tolattice parameter mismatches The latter depend on the crystallographicorientation of the tetragonal grains which results in certain orientations thatare more stable than others and also in a characteristic superstructureoccurring in the electron diŒraction patterns of Mg2TiO4 reg lms Surprisinglyvery similar superstructures occur in the electron diŒraction patterns ofMg2GeO4 reg lms although no tetragonal phase is known of this normal spinelThis unexpected phenomenon is discussed in terms of a cation site ordering inthe Mg2GeO4 reg lms enabled by a non-zero inversion degree

1 INTRODUCTIONPhase transitions in spinels are most interesting to investigate In particular

cubic tetragonal transitions in the inverse spinel Mg2TiO4 (Wechsler andNavrotsky 1984 Wechsler and Von Dreele 1989) and spinel-to-olivine transitions

in Mg2GeO4 (Lauterjung and Will 1986 Ross and Navrotsky 1987 Rubie and

Champness 1987) have attracted the attention of various groups Phase transitions

in Mg2GeO4 are signireg cant since this spinel has been introduced as a model systemfor the investigation of deep earthquakes The origin of the latter is attributed to

olivine-to-spinel transitions of the mineral (Mg Fe)2SiO4 prevailing in the deep

Earth mantle (Burnley and Green 1989 Green and Burnley 1989) While the transi-

tion of (Mg Fe)2SiO4 occurs at temperatures between 1500 and 1900 K and at high

pressures of 8plusmn 18 GPa the olivine-to-spinel transition in Mg2GeO4 is easier to

achieve in the laboratory at a transition temperature of 1100K and at normal

Philosophical Magazine A ISSN 0141plusmn 8610 printISSN 1460-6992 online 2001 Taylor amp Francis Ltd

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pressure (valid for bulk samples) In spite of many investigations performed there

has not been any agreement as to the details of the mechanism of the olivine-to-

spinel transition in Mg2GeO4 under high pressures (Lauterjung and Will 1986 Rubieand Champness 1987) More investigations on the inmacr uence of pressure (or stress) on

phase transitions in spinels are thus highly desirable

A rather unconventional way of introducing high pressures into solid materials

opens up if thin reg lms are studied Stresses in thin reg lms may reach several gigapascals

(HoŒmann 1981 Fabis et al 1990) and are easy to attain under laboratory condi-tions Epitaxial reg lms are favourable if the crystallography of the transition is to be

considered We prepared epitaxial Mg2TiO4 and Mg2GeO4 thin reg lms by solid-state

reactions a route previously described in some detail (Sieber et al 1997 Blum and

Hesse 1997) While Mg2TiO4 reg lms form by a topotaxial reaction between a TiO2

vapour and the (001) surface of a MgO single crystal Mg2GeO4 reg lms were formedfrom a thin solid GeO2 reg lm deposited on to the MgO single-crystal substrate The

heating of this MgOplusmn GeO2 couple is accompanied by a sequence of chemical reac-

tions occurring with the temperature increasing forming thin reg lms of the product

phases MgGeO3 Mg2GeO4 and reg nally Mg28Ge10O48 This sequence corresponds

to the increasing Mg content owing to the excess MgO originating from the bulk

substrate crystal (Blum and Hesse 1997) If this process is stopped in due time anepitaxial reg lm of Mg2GeO4 will have formed Previous investigations of reg lms of this

kind showed some inmacr uence of stress on the reg lm structure Thus the stability of the

Mg2GeO4 spinel phase with respect to the transition to olivine is extended by more

than 170 K up to temperatures above 1270 K owing to the thermal stress inherent in

the reg lms (Blum and Hesse 1997) Moreover electron diŒraction and high-resolutiontransmission electron microscopy (HRTEM) revealed a superstructure of the

Mg2GeO4 spinel lattice in these reg lms the nature of which indicated some similarity

to the well known tetragonal phase of Mg2TiO4 which formed because of cation

ordering (Blum 1997 Blum et al 1998) This result is most surprising in view of the

fact that no tetragonal phase of Mg2GeO4 is known so far The present study on theinmacr uence of stress on cubic-to-tetragonal phase transitions in Mg2TiO4 and

Mg2GeO4 thin reg lms has been encouraged by these observations

2 CATION ORDERING AND INVERSION DEGREESuperstructures forming during phase transitions because of some ordering of

cations or cation vacancies have been observed in several spinels The type of these

transitions was predicted on the basis of symmetry considerations (Haas 1965)

Examples are the formation of tetragonal structures by the ordering of cations in

the inverse spinel Mg2TiO4 (Wechsler and Von Dreele 1989) and by the ordering of

iron vacancies in the spinel g-Fe2O3 (Greaves 1983) The cubic spinel Mg2TiO4

(lattice parameter aTic ˆ 84376hellip5dagger pm) (Wechsler and Von Dreele 1989) is a nearly

ideal inverse spinel Half the Mg and all the Ti cations are randomly distributed over

the octahedral sites Below the transition temperature the ordering of these octahe-

dral Mg and Ti cations occurs with up to two thirds of all cations involved As aresult of this ordering a tetragonal low-temperature phase of Mg2TiO4 forms A

structure rereg nement of this phase performed by neutron powder diŒraction yielded

the space group P4122 (No 91) and the lattice parameters aTit ˆ 59748hellip5dagger pm and

cTit ˆ 8414hellip7dagger pm (Wechsler and Von Dreele 1989) which obey the conditions

cTit lt aTic lt 21=2aTit hellip1dagger

110 St Senz et al

It should be pointed out that the ordering process and the concomitant formation of

the tetragonal phase are only possible as Mg2TiO4 is an inverse spinel The non-zero

inversion degree is the prerequisite to this type of ordering which renders a con-sideration of the inversion degree of Mg2GeO4 particularly important if the

observed superstructure of this nominally normal spinel is to be investigated and

discussed

The inversion degree of Mg2GeO4 (aGec ˆ 82496hellip6dagger pm) was reported to be zero

for olivine samples grown at a Mg-to-Ge ratio of 20 and converted into a spinel byannealing at 05 GPa and 1120 K (Von Dreele et al 1977) However the minimum of

the structure rereg nement was not exactly at zero Rather a tetrahedral site occupancy

of 099 Ge and 001 Mg resulted in the lowest value of the variance In another

experiment (Ross and Navrotsky 1987) the Mg2GeO4 spinel was synthesized at

02 GPa and at temperatures between 973 and 1048 K Raman spectra and X-raydiŒractograms again proved the spinel close to normal These workers compared

experimental and theoretical data of the X-ray diŒraction (XRD) intensity ratio

Ihellip422dagger=Ihellip400dagger revealing a Mg fraction of about 005 on the tetrahedral sites A

similar value of 007 can be deduced from the work of Guyot et al (1986) if the

data given by Ross and Navrotsky (1987) are taken into account Following all the

above results stoichiometric Mg2GeO4 seems to be entirely or almost a normalspinel However up to now no data have been published about a possible solid

solution of either GeO2 or MgO in the Mg2GeO4 spinel An oV-stoichiometric spinel

could at least be partially inverse Moreover unusual lines in the infrared spectrum

of Mg2GeO4 formed under pressure at a high temperature of 1273 K were inter-

preted in terms of a non-zero inversion degree (Jeanloz 1980)

3 EXPERIMENTAL DETAILSThe spinel reg lms have formed by either a gasplusmn solid (Mg2TiO4) or a solidplusmn solid

(Mg2GeO4) topotaxial chemical reaction In a high-vacuum chamber a polished

MgO(001) crystal of 10 mm pound 10 mm pound 1 mm size was heated to temperatures

between 800 and 1300 K A target of the respective oxide that is TiO2 or GeO2

was evaporated using an electron-beam evaporator O2 was introduced into thevacuum system through a needle valve to maintain an O2 background pressure up

to 1 pound 10iexcl2 Pa which for Mg2TiO4 was necessary to avoid the formation of the

spinel MgTi2O4 in which Ti has a valency of 3 Owing to the diŒerent sticking

coe cients of the two oxides at 1300K the Mg2TiO4 spinel has been formed by a

chemical reaction of the MgO substrate with the Tiplusmn O vapour at 1300 K whereas

under the same conditions no Mg2GeO4 reg lm had formed from the Geplusmn O vapourTherefore Mg2GeO4 reg lms were grown by depositing an amorphous GeO2 reg lm on to

the MgO substrate at a low temperature (773 K) and subsequently reacting the reg lm

in air at 1300 K

For the X-ray investigations a Philips XrsquoPert MRD X-ray diŒractometer was

used which was equipped with a Eulerian cradle and Cu Ka radiation The stresswas measured using a thin reg lm attachment (parallel plate collimator and graphite

secondary monochromator 2sup3 resolution 0358) High resolution was achieved

using a Ge(220) four-remacr ection channel-cut monochromator and a similar analyser

Specimens for transmission electron microscopy (TEM) were prepared by standard

techniques namely grinding dimpling and ion milling A Philips CM20 twin elec-tron microscope was used with a double-tilt heating stage for in-situ observations up

Cubic-to-tetragonal phase transitions in Mg2TiO4 and Mg2GeO4 111

to 1300 K High-resolution images were obtained with a JEOL 4000 EX high-resolu-

tion transmission electron microscope

4 R ESULTS

41 Overview

Both Mg2TiO4 and Mg2GeO4 spinel reg lms were grown epitaxially as shown byXRD electron diŒraction and HRTEM all revealing the orientation relationship

hellip001daggerspinelkhellip001daggerMgO permil100Šspinelkpermil100ŠMgO hellip2dagger

This orientation relationship results in low lattice misreg t along the spinelplusmn MgO inter-

face namely only 02 for Mg2TiO4 plusmn MgO and 2 for Mg2GeO4 plusmn MgO

Corresponding to these misreg t values the perfection of the reg lm orientation is diŒer-ent for the two spinels While the Mg2TiO4 rocking curve peaks display a very small

full width at half-maximum (FWHM) of the order of 0058 which is almost identical

with the FWHM of the substrate peaks the tilt FWHM of the Mg2GeO4(004)

rocking curve is about 158 (A typical MgO substrate consists of two to reg ve

small-angle grains with tilt FWHMs between 00078 and 00168 of the individual

grains The grains are tilted against each other and the envelope of the tilt distribu-tion has a FWHM between 0028 and 0078 The Mg2TiO4 reg lm reproduces this tilt

distribution thus the upper limit of its tilt FWHM is in fact only about 0018)Typical defects of the spinel reg lms are cation antiphase boundaries in Mg2TiO4

(Hesse and Bethge 1981 Hesse 1987 Sieber et al 1997) and small-angle grain

boundaries in Mg2GeO4 Figure 1 (a) shows a diŒraction pattern and reg gure 1 (b) aplan-view dark-reg eld image of a Mg2TiO4 reg lm reg gures 1 (c) and (d) are TEM cross-

section images of a Mg2TiO4 reg lm and reg gure 1 (e) a TEM cross-section image of a

Mg2GeO4 reg lm Electron diŒraction patterns (reg gure 2 (a)) display weak superstruc-

ture remacr ections of the Mg2GeO4 reg lm HRTEM lattice plane images Fourier-reg ltered

for the noise reduction show corresponding weak contrast features (reg gure 2 (b)) Theinset in reg gure 2 (b) presents the computer-generated Fourier transform of the high-

resolution image which also shows the weak superstructure remacr ections

42 Stress-induced distortions of the cubic phase

At room temperature the spinel reg lms are tetragonally distorted because of the

diŒerence between the thermal expansion coe cients of reg lm and substrate as wasrevealed by precise XRD measurements determining and comparing the lattice

plane spacings of those remacr ections which should show equal spacings in a cubic

phase DiŒerent 931 remacr ections and diŒerent 862 and 10 2 0 remacr ections respec-

tively were compared revealing a tetragonal distortion of the reg lm lattice The lattice

parameters ak and a were measured (reg gure 3) that is the lattice parameters inthe directions parallel and perpendicular respectively to the substrate surface The

pseudocubic parameter apc was calculated from ak and a (Blum 1997) assuming a

reasonable value for Poissonrsquo s ratio of cedil ˆ 035 (as given by Hellwege (1969) for the

MgAl2O4 spinel) The values thus obtained are shown in table l together with the

distortions calculated according to

ˆak iexcl apc

apc

hellip3dagger

112 St Senz et al

The negative sign of in table 1 shows that the distortion is a biaxial compression

in both cases Using the MgO lattice parameter (a0 ˆ 4213 pm) as well as thethermal expansion coe cients ahellipMgOdagger ˆ 1139 pound 10iexcl6 Kiexcl1 Dagger 246T pound 10iexcl9 Kiexcl2

(Taylor 1984) ahellipMg2TiO4dagger ˆ hellip10 sect 14dagger pound 10iexcl6 Kiexcl1 (Rieke and Ungewiss 1936)

and ahellipMg2GeO4dagger ˆ 108 pound 10iexcl6 Kiexcl1 (Ross and Navrotsky 1987) the theoretical

compression after cooling the reg lms from the growth temperature of 1300 K to

room temperature was calculated as iexcl039 for the Mg2TiO4 reg lms and iexcl031for the Mg2GeO4 reg lms Since the experimental distortions in table 1 are smaller than

the theoretical values part of the strain is obviously relaxed

Assuming reasonable values for the modulus E ˆ 172GPa and Poissonrsquo s ratio

cedil ˆ 035 (see above) the corresponding compressive stresses frac14 can be estimated from

the strains (Brantley 1973) which results in the experimentally determined stresses

frac14expTi ˆ iexcl068 GPa for Mg2TiO4 and frac14exp

Ge ˆ iexcl033 GPa for Mg2GeO4 as well as in

the respective calculated thermally induced stresses frac14thTi ˆ iexcl088 GPa and

frac14thGe ˆ iexcl068 GPa Accordingly phase transitions to be studied in these reg lms will

be subject to the inmacr uence of a rather high stress


Figure 1 (a) DiŒraction pattern (b) plan-view (220) spinel dark-reg eld image (c) cross-sectionoverview image and (d) cross-section high HRTEM image of a Mg2GeO4 reg lm on MgO(001) (e) Cross-section HRTEM image of a Mg2GeO4 reg lm of MgO(001) The insetsshow the Fourier transforms of the Mg2GeO4 spinel (top) and MgO (bottom) images

114 St Senz et al

Figure 2 (a) DiŒraction pattern and (b) HRTEM image of a Mg2GeO4 spinel reg lm showingan unexpected superstructure The inset in (b) is the Fourier transform of the image

Figure 3 Schematic diagram of the measured lattice parameters

Table 1 The literature values ac of the bulk lattice parameters the experimentally deter-mined lattice parameters ak a and apc of the reg lms as well as the derived experimentaldistortions exp and the theoretical that is calculated thermally induced distortions th

for the cubic spinel phase of the Mg2TiO4 and Mg2GeO4 reg lms

ac ak a apc exp th

(pm) (pm) (pm) (pm) ( ) ( )

Mg2TiO4 84376(5) 8427(1) 8460(1) 8443(2) iexcl019 iexcl039Mg2GeO4 82496(6) 8256(8) 8277(8) 8266(8) iexcl012 iexcl031

43 Cubic-to-tetragonal phase transition and stress-induced orthorhombic

distortions of the tetragonal grains

Apart from the stress-induced tetragonal distortion of the cubic spinel reg lm smallgrains of a truly tetragonal phase of Mg2TiO4 form within the reg lm owing to the

cubic-to-tetragonal phase transition mentioned in 2 which can be induced by

annealing at a temperature of 873 K In principle these grains may have three

diŒerent orientations with respect to the (tetragonally distorted) cubic reg lm matrix

because the tetragonal ct axis may be parallel to any of the three ac axes of the cubicmatrix as illustrated in reg gure 4 In the following the symmetry-equivalent grains a

and b in reg gure 4 will be referred to as in-planersquo grains and the grains of the third

orientation as `c grainsrsquo The corresponding simulated electron diŒraction patterns

are presented in reg gures 5 (a)plusmn (c) The experimental diŒraction pattern (reg gure 5 (d)) of

a Mg2TiO4 reg lm annealed for several days at 873 K corresponds to a superposition ofthe three calculated patterns in reg gures 5 (a)plusmn (c) showing that all the three orienta-

tions of the tetragonal spinel phase occur in the reg lms in addition to the cubic phase

High-resolution XRD -2sup3 scans show not only that the cubic spinel phase is

distorted but also that the grains of the tetragonal spinel phase are distorted by the

thermally induced stresses occurring after the samples had been cooled to room

temperature Figure 6 presents two such scans of a Mg2TiO4 reg lm before and afterannealing at 873 K respectively taken at room temperature in the vicinity of the

MgO(002) peak ( is the angle between the incident beam and the sample surface

and 2sup3 is the angle between the incident beam and the detector Here the measure-

ment was performed using a Ge(220) channel-cut monochromator in the primary

beam path and a Ge(220) analyser in front of the detector After and 2sup3 wereadjusted for a maximum intensity of the (002) remacr ection of the MgO substrate the

measurement was performed with a coupled scan The measured XRD intensity of

the as-grown reg lm was multiplied by a factor of ten before being plotted) As reg gure 6

shows the as-grown Mg2TiO4 reg lm causes a single (004)c spinel peak at around

ˆ 21358 indicating the existence of strain because the (004)c peak of a strain-free cubic spinel reg lm would occur at ˆ 21428 The reg lm thickness remains homo-


Figure 4 Possible orientations of tetragonal Mg2TiO4 spinel grains in the cubic matrix

116 St Senz et al

Figure 5 (a)plusmn (c) Calculated and (d) experimental electron diŒraction patterns of tetragonalMg2TiO4 spinel grains with the three orientations shown in reg gure 4

Figure 6 High-resolution XRD -2sup3 scans of a Mg2TiO4 reg lm on a MgO(001) substratebefore and after annealing at 873 K for 360 h The `ratersquo is the diŒraction intensitymeasured according to the procedure described in the text (au arbitrary units)

geneous in spite of the strain as indicated by the interference fringes between the

reg lm peak and the MgO(002) substrate peak After annealing at 873K for 360 h the

reg lm peak is shifted towards even smaller angles Moreover the interference fringeshave disappeared indicating a decrease in the homogeneity of the reg lm thickness

most probably owing to the formation of the tetragonal grains of diŒerent orienta-

tions (In another experiment we could restore the thickness interference fringes by

heating the sample to above 1200 K ie a temperature su cient for a fast reverse

transition from tetragonal to cubic spinel) Considering the discussion of reg gures 7and 8 below the shift of the (004) peak to ordm 21348 indicates an increase in the

average lattice parameter a due to the presence of both the cubic and the tetragonal

spinel in the reg lm and in particular their distortion at room temperature

The formation kinetics of the ordered tetragonal phase in the Mg2TiO4 spinel

were directly recorded by XRD using the low-resolution set-up revealing the addi-tional remacr ections compared with the (high-symmetry) cubic spinel Grains of the

tetragonal spinel with their c axes parallel and perpendicular to the substrate surface

can be distinguished by their diŒerent Aacute values of the (111)t planes Aacute ˆ 2678 for

grains with (110)t parallel to the substrate surface (in plane grains ie a and b grains

of reg gure 4) and Aacute ˆ 6338 for grains with (001)t parallel to the substrate surface (c

grains see reg gure 4) (As above the subscript t refers to the tetragonal lattice Aacute is theangle between the substrate surface plane and the detected lattice plane) In reg gure 7

the intensity of the tetragonal (111)t peak after annealing is plotted for both the in-

plane grains (called a grains in the reg gure) and the c grains Y is the normalised peak

intensity I=Imax The plot of ln ln [1=hellip1 iexcl Y )] versus ln (annealing time1 h) shows a

linear relation consistent with the classical Avrami (1939) description of nucleationand growth (Henderson 1979) The exponent n in the Avrami equation is close to

three which is typical of linear growth in three dimensions with no further nuclea-

tion occurring during growth (Ranganathan and Von Heimendahl 1981) This is a


Figure 7 Logplusmn plot of the normalized intensity XRD intensity of the Mg2TiO4(111)t peak ofthe a and c grains versus the annealing time

reasonable result because in our case reg rstly the growth starts from `pre-existingrsquo

nuclei within the cubic phase secondly the maximum tetragonal volume portion

during the cubic-to-tetragonal phase transition is only about 10 and thirdlycorrespondingly the size of the tetragonal grains remains small compared with the

reg lm thickness as also shown by the TEM studies (see reg gure 8 (a) below)

Further XRD investigations conreg rmed that at room temperature also the tetra-

gonal grains are distorted owing to the thermal expansion mismatch According

to the rereg nement data of Wechsler and Von Dreele the tetragonal distortion ofthe thermally undistorted ordered tetragonal Mg2TiO4 phase referred to the cubic

lattice is Dt ˆ ct iexcl 21=2at ˆ iexcl46 pm This value is similar to that of the

thermally induced tetragonal distortion of the cubic matrix of our reg lms after

cooling them from the deposition temperature to room temperature namely

Dc ˆ aTik iexcl aTi ˆ iexcl33 pm (see table 1) The superposition of these two distortionsin our reg lms reduces the symmetry of the in-plane grains where now

dhellip110daggert 6ˆ dhellip110daggert as shown in table A 1 in appendix A This orthorhombic distor-

tion of the in-plane grains has been conreg rmed by the fact that remacr ections which

coincide for a truly tetragonal lattice namely hellip434daggert=hellip434daggert and hellip311daggert=hellip311daggert for

the in-plane grains now show diŒerent interplanar spacings dhellip434daggert ˆ 10408hellip4dagger pm

but dhellip434daggert ˆ 10386hellip4dagger pm dhellip311daggert ˆ 18455hellip14dagger pm but dhellip311daggert ˆ 18435hellip14dagger pmRespective measurements for the c grains have been performed with the stronger

(311)t remacr ections because of intensity problems due to the small fraction of c grains

in the sample Thus it has been proven that the distortions of the matrix and the

tetragonal grains have the same sign The lattice parameters of the in-plane grains

parallel and perpendicular to the substrate surface namely 8437(5) and 8462(5) pmrespectively reg t the matrix well and are in good agreement with the assumption of

coherent tetragonal grains As is shown in appendix A the coherent c grains are

under a higher elastic strain than the coherent in-plane grains are which is a result

that is relevant to the kinetics of the reverse (ie tetragonal-to-cubic) transition

44 Tetragonal-to-cubic phase transition

The observed larger strain of the tetragonal c grains compared with the tetra-gonal in-plane grains results in a specireg c kinetics of the reverse transition that is the

tetragonal-to-cubic phase transition This is demonstrated in reg gure 8 by a plan-view

TEM specimen of a Mg2TiO4 reg lm annealed at 873 K for 1 week to induce the cubic-

to-tetragonal phase transition and prepared to enable the in-situ study of the reverse

transition For room temperature the TEM dark- and bright-reg eld images (reg gures

8 (a) and (b) respectively) show the small tetragonal grains of 50plusmn 100 nm size withinthe thermally stressed cubic matrix The possible orientations of these tetragonal

grains were illustrated in reg gure 4 In the electron diŒraction pattern of a region with

several grains three patterns are superimposed one for each orientation (reg gures 5 (d)

and 8 (d)) After these images and diŒraction patterns had been taken the tetrago-

nal-to-cubic transition was observed in situ using a heating stage at temperaturesabove 950 K revealing that during heating the c grains vanish more rapidly than do

the in-plane grains This is shown in reg gure 8 (e) where electron diŒraction at 970 K

shows superlattice remacr ections of the remaining relatively stable in-plane grains but no

remacr ections of c grains After heating to 1173 K all tetragonal grains have disap-

peared (reg gure 8 ( f )) As a consequence the bright-reg eld image shows only cationantiphase boundaries as the remaining defects (reg gure 8 (c))

118 St Senz et al


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5 DISCUSSION

51 Mg2TiO4 reg lms

The cubic-to-tetragonal phase transition in our Mg2TiO4 reg lms is very slow even

slower than observed before in bulk samples Wechsler and Von Dreele (1989) have

shown that for powder samples annealed at 773 K 1 month was necessary to satu-

rate the superstructure remacr ection intensity whereas at 873 K it took only a few days

The transition temperature was estimated as 933 sect 20 K The change in the transi-tion enthalpy measured by Wechsler and Navrotsky (1984) using a small-grained

material is quite small ( H ˆ 14 sect 13 kJ moliexcl1daggerIn our reg lms the conversion of about 10 of the cubic spinel into the tetragonal

phase at 873 K took several weeks This transition rate is considerably lower than

that of bulk samples which can be attributed to the inmacr uence of stress For bulksamples the change in the Gibbs free energy at 873 K calculated from literature data

is only G ˆ iexcl04 kJ moliexcl1 Obviously this small driving force of the transition

arises as the latter is a pure cation ordering process with no need for any extensive

rearrangements of the O anions As the value of the Avrami exponent which is closeto three shows the transformation starting from a coherent state proceeds via

purely three-dimensional growth with no further nucleation occurring Stress arises

as early as during the nucleation of the tetragonal grains resulting in distortion and

a corresponding elastic enthalpy term He With the growth of tetragonal grains

the elastic enthalpy increases implying a further decrease in the overall driving force

of the transition given by the still negative sum G Dagger He lt 0 where G lt 0 and

He gt 0 Most probably the low transition rate observed in our reg lms is due to thiselastically induced reduction of the overall driving force With the further growth of

tetragonal grains the elastic enthalpy reg nally overcomes j Gj making the sum posi-

tive and hence the transition stops In the reg ne-grained powder samples used by

Wechsler and Von Dreele most probably the stress is relaxed by shape distortions

of the whole grain whereas in our epitaxial thin reg lms reg xed to the bulk substrates thestress cannot relax in this way

Unlike the work of Wechsler and Von Dreele in our experiment there is a

tetragonal distortion of the cubic Mg2TiO4 matrix induced by thermal compressive

stress Hence during the cubic-to-tetragonal transition tetragonal grains of diŒerent

orientation do form with diVerent probabilities Since the cubic spinel matrix istetragonally distorted by biaxial compression the longer axis is perpendicular to

the substrate surface so that the formation of the tetragonal in-plane grains is

favoured The distortion of the latter (see table A 1 in appendix A) as well as their

elastic enthalpy is lower than that of the c grains As a consequence during the

reverse (tetragonal-to-cubic) transition the in-plane grains are rather stable while the

c grains vanish more rapidlyObviously in thin Mg2TiO4 reg lms the kinetics of both the cubic-to-tetragonal and

the tetragonal-to-cubic transitions are aŒected by the stress present causing strong

deviations from the bulk transition kinetics

52 Mg2GeO4 reg lms

The predominance of the in-plane orientation of the tetragonal Mg2TiO4 grains

may help to explain the unexpected occurrence of superlattice remacr ections in the

Mg2GeO4 spinel reg lms The electron diŒraction patterns obtained of the Mg2GeO4

spinel reg lms (reg gure 2 (a)) are most similar to those of the stable tetragonal in-plane

120 St Senz et al

grains of the Mg2TiO4 reg lms (reg gure 8 (e)) This allows us to favour the idea of an

ordering process in Mg2GeO4 which is analogous to that in Mg2TiO4 However the

superstructure remacr ections of the Mg2GeO4 spinel reg lms are weaker than those ofMg2TiO4 most probably because of the almost ideal normal character of Mg2GeO4

The low non-zero degree of inversion assumed to enable an ordering process to

occur in Mg2GeO4 might be the result of the growth conditions the large MgO

excess certainly promotes the formation of a Mg-rich spinel In such a spinel some

of the Ge cations are replaced by Mg ions so that an ordering process may occurThe diŒerence between the lattice parameters of Mg2GeO4 published in the literature

(82496 pm) and the values that we measured (apc ˆ 8266 pm) supports the idea that

our thin-reg lm Mg2GeO4 is not stoichiometric a linear extrapolation of the lattice

parameter from Mg2GeO4 to MgO yields the composition Mg22Ge09O4 for our

reg lms However in view of the local charge balance required the Ge4Dagger cations cannotsimply be replaced by Mg2Dagger ions This means that at least some of the additional Mg

ions should occupy tetrahedral sites

In principle cation ordering on these tetrahedral sites is also possible but the

crystallography of such an ordering process would considerably diŒer from that of

the octahedral site ordering of Mg2TiO4 According to Haas (1965) a 1 1 ordering

on the tetrahedral sites changes the space group of the spinel from Fd3m to F43mThe spinel remains cubic but half the symmetry elements are lost whereas during the

octahedral 1 1 ordering of Mg2TiO4 the symmetry changes from cubic to tetragonal

(P4122) and the new unit cell has half the volume of the cubic cell The [001] electron

diŒraction image of a F43m spinel diŒers from the disordered Fd3my only by addi-

tional 002 remacr ections The latter have not been observed in our Mg2GeO4 reg lms sothat a 1 1 ordering on only tetrahedral sites is rather unlikely We therefore consider

an octahedral site ordering most probable in our Mg2GeO4 thin reg lms

With respect to the signireg cance of a partial inversion as a prerequisite to the

assumed octahedral site ordering the following should be pointed out In the litera-

ture most Mg2GeO4 spinel samples of a low or zero Mg occupancy of the tetrahe-dral sites were prepared by reg rst reacting MgO and GeO2 to yield the olivine phase of

Mg2GeO4 before it was transformed into the spinel phase at relatively low tempera-

tures around 1000 K Our reaction path was very diŒerent with an amorphous GeO2

reg lm reg rst reacting with MgO to form the intermediate compound MgGeO3 Then the

spinel formed directly with no olivine phase forming by the reaction of MgGeO3

with the MgO substrate at temperatures between 1220 and 1320 K (Blum and Hesse1997) This very diŒerent reaction route may be another reason that in our thin-reg lm

samples a spinel forms that is not entirely normal A diŒerent formation route may

indeed change the normalplusmn inverse character of the Mg2GeO4 spinel infrared spectra

of the Mg2GeO4 spinel showed additional lines if the spinel was produced by

annealing the olivine phase under pressure at 1273 K (Jeanloz 1980) These lineshave been interpreted by Jeanloz as indicating partial inversion In conclusion we

should like to say that whatever the reg nal reason may be there seems to be a partial

inversion in our Mg2GeO4 spinel reg lms which permits the octahedral site ordering

reg nally resulting in the tetragonal superstructure observed


There are two diŒerent concepts of the space groups of spinels In particular Grimes etal (1983) presented results supporting F43m rather than Fd3m However since most of thesubsequent studies have not conreg rmed this view (for example De Cooman and Carter (1985))Fd3m still seems to be the generally accepted space group

53 Consequences for both spinel reg lms

Mechanical stress strongly aŒects the transition kinetics between the cubic and

tetragonal phases in thin Mg2TiO4 spinel reg lms Compared with the bulk transitionin thin reg lms the kinetics are considerably slowed Moreover the transition kinetics

are inmacr uenced by the crystallographic orientation of the tetragonal grains

There are two sources of stress in the spinel reg lms One type of stress develops

owing to the thermal expansion mismatch between the thin spinel reg lm and the rigid

MgO substrate while the reg lm is cooled after growth Another source of stress is thenucleation and subsequent growth of well-oriented coherent grains of one phase

within the single-crystal matrix of the other phase owing to the diŒerent lattice

parameters Both these types of stress superpose and interact Unlike bulk powder

samples with only the second type of stress occurring (which can relax by changes in

the shape of the small powder particles) in epitaxial thin reg lms reg xed on a rigidsubstrate the stress cannot relax in this way

The amount of stress depends on the orientation of the tetragonal grains with

respect to the reg lm plane which reg nally results in the above-mentioned orientation

sensitivity of the transition kinetics As a consequence of these factors tetragonal

Mg2TiO4 spinel grains with their c axes in the reg lm plane are more stable than others

Under appropriate temperatureplusmn time conditions solely these grains occur in thereg lm yielding a characteristic superstructure as shown in the electron diŒraction

patterns of Mg2TiO4 reg lms taken with an electron beam perpendicular to the reg lm

plane

Surprisingly a superstructure very similar to the latter is observed in the electron

diŒraction patterns and TEM images of thin reg lms of the normal spinel Mg2GeO4namely a material which is generally supposed to have no tetragonal phase After the

above detailed discussion of possible cation ordering eŒects in Mg2GeO4 the

observed analogy of Mg2GeO4 thin reg lms to Mg2TiO4 thin reg lms with respect to

their diŒraction superstructures allows the conclusion to be drawn that a cation

ordering process on the octahedral sublattice is possible not only in Mg2TiO4 butalso in Mg2GeO4 A non-zero degree of inversion of the Mg2GeO4 spinel is the

precondition for this octahedral site ordering As discussed above this condition

is most probably fulreg lled in our thin reg lms

6 CONCLUSIONMechanical stress has a severe inmacr uence on the kinetics of phase transition from

cubic to tetragonal and vice versa in thin Mg2TiO4 inverse spinel reg lms Compared

with transitions in the bulk here the kinetics are slowed and become sensitive to thecrystallographic orientation of the tetragonal phase with respect to the reg lm plane As

a consequence grains of a certain crystallographic orientation of the tetragonal

phase are more stable than others The existence of only the more stable grains in

the reg lm results in a specireg c tetragonal superstructure of electron diŒraction patterns

Thin reg lms of the normal spinel Mg2GeO4 show a superstructure of electrondiŒraction patterns this is very similar to that of Mg2TiO4 reg lms and is surprising

because this spinel is considered to have no tetragonal phase at all The detailed

discussion of possible cation ordering eŒects in Mg2GeO4 with some partial inver-

sion caused by the special conditions under which the thin reg lms were grown leads to

the conclusion that there is a tetragonal superstructure in the Mg2GeO4 thin reg lmsThus stress not only modireg es the transition kinetics as in Mg2TiO4 but also may

122 St Senz et al

cause conditions that enable a tetragonal phase to be formed in a spinel which under

normal bulk conditions has no tetragonal phase

ACKNOWLEDGEMENTS

The authors are grateful to Dr P Werner for the HRTEM images and to

Dr G KaEgrave stner for useful discussions and critically reading the manuscript Part of

this work was supported by the Deutsche Forschungsgemeinschaft viaSonderforschungsbereich 345 at Georg August University of GoEgrave ttingen Germany

A P P E N D I X AAssuming that the spinel reg lm grows without stress at the reaction temperature of

1273 K table A 1 shows the distortions of the tetragonal in-plane and c grainsrespectively in a Mg2TiO4 reg lm formed by the reaction at 1273K and subsequently

annealed at 873 K Thus T ˆ 400 K The reg lm is assumed to be coherent to the

MgO substrate and to be stressed by the diŒerence between the thermal expansion

coe cients of spinel and MgO The same linear thermal expansion coe cient was

assumed to be valid for both the cubic and the tetragonal spinel Three values of this

coe cient were used in succession for the calculations reg rstly the literature value of

10 pound 10iexcl6 Kiexcl1 with a large statistical and an unknown systematic error secondly thevalue of 11 pound 10iexcl6 Kiexcl1 yielding an agreement between the measured and the calcu-

lated stress of the cubic matrix at room temperature thirdly the value of

118 pound 10iexcl6 Kiexcl1 necessary to achieve a misreg t vanishing at the growth temperature

The relation ct lt ac lt 21=2at is supposed in correspondence to the preservation ofthe unit-cell volume during the transformation As table A 1 shows the coherent c

grains are under higher elastic strain than the in-plane grains irrespective of which of

the thermal expansion coe cients is valid

REFERENCESAVRAMI M 1939 Chem Phys 7 1103

BLUM W 1997 PhD Thesis Martin-Luther-UniversitaEgrave t Halle-Wittenberg

BLUM W and HESSE D 1997 Solid St Ionics 95 41


Table A 1 Distortion of lattice plane spacings of the tetragonal Mg2TiO4 spinel grainslocated within the thermally stressed spinel reg lm matrix with respect to the correspond-ing lattice planes in the MgO substrate Assuming a coherent reg lm at the growthtemperature of 1273K and the same linear thermal expansion coe cient not for thematrix and the grains the distortion at 873 K was calculated for three diŒerent valuesof not

In-plane grains c grainsctkhellip001daggerMgO ct hellip001daggerMgO

hellip110daggertkhellip001daggerMgO hellip001daggertkhellip001daggerMgO

ahellip10iexcl6

Kiexcl1dagger (110)t hellip110daggert (001)t (110)t hellip110daggert (001)t

10 01 017 iexcl025 017 017 iexcl03211 004 023 iexcl019 023 023 iexcl038118 iexcl001 027 iexcl015 027 027 iexcl04

BLUM W SENZ S WERNER P and HESSE D 1998 Electron Microscopy 96 Proceedingsof the 11th European Congress on Electron Microscopy Vol 2 Dublin Ireland 1996(Brussels Committee of European Societies of Microscopy) pp 113plusmn 114

BRANTLEY W A 1973 J appl Phys 44 534BURNLEY P C and GREEN H W 1989 Nature 338 753DE COOMAN B C and CARTER C B 1985 Phil Mag A 51 175FABIS P M COOKE R A and MCDONOUGH S 1990 J vac Sci Technol A 8 3819GREAVES C 1983 J solid-st Chem 49 325GREEN H W and BURNLEY P C 1989 Nature 341 733GRIMES N W THOMPSON P and KAY H F 1983 Proc R Soc A 386 333GUYOTF BOYERHMADONMVELDEB and POIRIER JP1986 Phys Chem Miner-

als 13 91HAAS C 1965 J Phys Chem Solids 26 1225HELLWEGE K-H 1969 Zahlenwerte und Funktionen aus Naturwissenschaft und Technik

Landolt-BoEgrave rnstein Group 3 Kristall- und FestkoEgrave rperphysik Vol 2 (Berlin Springer)HENDERSON D W 1979 J non-crystalline Solids 30 301HESSE D 1987 J vac Sci Technol A 5 1696HESSE D and BETHGE H 1981 J Cryst Growth 52 875HOFFMANN R W 1981 Surf Interface Analysis 3 62JEANLOZ R 1980 Phys Chem Minerals 5 327LAUTERJUNG J and WILL G 1986 Physica B 139plusmn 140 343RANGANATHAN S and VON HEIMENDAHL M 1981 J Mater Sci 16 2401R IEKE R and UNGEWISS A 1936 Berd Deutschen Keram Ges 17 237ROSS N L and NAVROTSKY A 1987 Phys Chem Minerals 14 473RUBIE D C and CHAMPNESS P E 1987 Bull MineAcirc ral 110 471SIEBER H HESSE D and WERNER P 1997 Phil Mag A 75 889TAYLOR D 1984 Trans Br Ceram Soc 83 5VON DREELERBNAVROTSKYA and BOWMANAL1977 Acta crystallogr B 33 2287WECHSLER B A and NAVROTSKY A 1984 J solid-st Chem 55 165WECHSLER B A and VON DREELE R B 1989 Acta crystallogr B 45 542

124 Cubic-to-tetragonal phase transitions in Mg2TiO4 and Mg2GeO4

pressure (valid for bulk samples) In spite of many investigations performed there

has not been any agreement as to the details of the mechanism of the olivine-to-

spinel transition in Mg2GeO4 under high pressures (Lauterjung and Will 1986 Rubieand Champness 1987) More investigations on the inmacr uence of pressure (or stress) on

phase transitions in spinels are thus highly desirable

A rather unconventional way of introducing high pressures into solid materials

opens up if thin reg lms are studied Stresses in thin reg lms may reach several gigapascals

(HoŒmann 1981 Fabis et al 1990) and are easy to attain under laboratory condi-tions Epitaxial reg lms are favourable if the crystallography of the transition is to be

considered We prepared epitaxial Mg2TiO4 and Mg2GeO4 thin reg lms by solid-state

reactions a route previously described in some detail (Sieber et al 1997 Blum and

Hesse 1997) While Mg2TiO4 reg lms form by a topotaxial reaction between a TiO2

vapour and the (001) surface of a MgO single crystal Mg2GeO4 reg lms were formedfrom a thin solid GeO2 reg lm deposited on to the MgO single-crystal substrate The

heating of this MgOplusmn GeO2 couple is accompanied by a sequence of chemical reac-

tions occurring with the temperature increasing forming thin reg lms of the product

phases MgGeO3 Mg2GeO4 and reg nally Mg28Ge10O48 This sequence corresponds

to the increasing Mg content owing to the excess MgO originating from the bulk

substrate crystal (Blum and Hesse 1997) If this process is stopped in due time anepitaxial reg lm of Mg2GeO4 will have formed Previous investigations of reg lms of this

kind showed some inmacr uence of stress on the reg lm structure Thus the stability of the

Mg2GeO4 spinel phase with respect to the transition to olivine is extended by more

than 170 K up to temperatures above 1270 K owing to the thermal stress inherent in

the reg lms (Blum and Hesse 1997) Moreover electron diŒraction and high-resolutiontransmission electron microscopy (HRTEM) revealed a superstructure of the

Mg2GeO4 spinel lattice in these reg lms the nature of which indicated some similarity

to the well known tetragonal phase of Mg2TiO4 which formed because of cation

ordering (Blum 1997 Blum et al 1998) This result is most surprising in view of the

fact that no tetragonal phase of Mg2GeO4 is known so far The present study on theinmacr uence of stress on cubic-to-tetragonal phase transitions in Mg2TiO4 and

Mg2GeO4 thin reg lms has been encouraged by these observations

2 CATION ORDERING AND INVERSION DEGREESuperstructures forming during phase transitions because of some ordering of

cations or cation vacancies have been observed in several spinels The type of these

transitions was predicted on the basis of symmetry considerations (Haas 1965)

Examples are the formation of tetragonal structures by the ordering of cations in

the inverse spinel Mg2TiO4 (Wechsler and Von Dreele 1989) and by the ordering of

iron vacancies in the spinel g-Fe2O3 (Greaves 1983) The cubic spinel Mg2TiO4

(lattice parameter aTic ˆ 84376hellip5dagger pm) (Wechsler and Von Dreele 1989) is a nearly

ideal inverse spinel Half the Mg and all the Ti cations are randomly distributed over

the octahedral sites Below the transition temperature the ordering of these octahe-

dral Mg and Ti cations occurs with up to two thirds of all cations involved As aresult of this ordering a tetragonal low-temperature phase of Mg2TiO4 forms A

structure rereg nement of this phase performed by neutron powder diŒraction yielded

the space group P4122 (No 91) and the lattice parameters aTit ˆ 59748hellip5dagger pm and

cTit ˆ 8414hellip7dagger pm (Wechsler and Von Dreele 1989) which obey the conditions

cTit lt aTic lt 21=2aTit hellip1dagger

110 St Senz et al





discussed



























in air at 1300 K










4 R ESULTS

41 Overview






























ˆak iexcl apc

apc

hellip3dagger

112 St Senz et al


















114 St Senz et al





ac ak a apc exp th

(pm) (pm) (pm) (pm) ( ) ( )




























116 St Senz et al

































































118 St Senz et al


Fig

ure

8

(a)

TE

Md

ark

-regel

dim

age

usi

ng

are

macrec

tion

from

the

tetr

agon

alsp

inel

(b

)a

bri

gh

t-reg

eld

image

of

tetr

ago

nalsp

inel

gra

ins

ina

Mg

2T

iO4

reglm

at

roo

mte

mper

atu

re(d

)E

lect

ron

diŒ

ract

ion

at

roo

mte

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atu

re

show

ing

the

remacr

ecti

ons

of

tetr

agon

algra

ins

of

thre

eo

rien

tati

on

sth

ecu

bic

spin

elm

atr

ix

an

dth

eM

gO

sub

stra

te

(e)

Ele

ctro

nd

iŒra

ctio

nta

ken

at

973

Ksh

ow

ing

that

only

the

tetr

ago

nalgra

ins

of

the

in-p

lan

eori

enta

tio

nare

left

(c

)B

right-

regel

dim

age

an

d(f

)el

ectr

on

diŒ

ract

ion

taken

at

1173

K

wit

hth

etr

an

siti

on

toth

ecu

bic

spin

elreg

nis

hed

5 DISCUSSION

51 Mg2TiO4 reg lms

































52 Mg2GeO4 reg lms





120 St Senz et al
























































plane




















122 St Senz et al



ACKNOWLEDGEMENTS
























ahellip10iexcl6












discussed



























in air at 1300 K










4 R ESULTS

41 Overview






























ˆak iexcl apc

apc

hellip3dagger

112 St Senz et al


















114 St Senz et al





ac ak a apc exp th

(pm) (pm) (pm) (pm) ( ) ( )




























116 St Senz et al

































































118 St Senz et al


Fig

ure

8

(a)

TE

Md

ark

-regel

dim

age

usi

ng

are

macrec

tion

from

the

tetr

agon

alsp

inel

(b

)a

bri

gh

t-reg

eld

image

of

tetr

ago

nalsp

inel

gra

ins

ina

Mg

2T

iO4

reglm

at

roo

mte

mper

atu

re(d

)E

lect

ron

diŒ

ract

ion

at

roo

mte

mper

atu

re

show

ing

the

remacr

ecti

ons

of

tetr

agon

algra

ins

of

thre

eo

rien

tati

on

sth

ecu

bic

spin

elm

atr

ix

an

dth

eM

gO

sub

stra

te

(e)

Ele

ctro

nd

iŒra

ctio

nta

ken

at

973

Ksh

ow

ing

that

only

the

tetr

ago

nalgra

ins

of

the

in-p

lan

eori

enta

tio

nare

left

(c

)B

right-

regel

dim

age

an

d(f

)el

ectr

on

diŒ

ract

ion

taken

at

1173

K

wit

hth

etr

an

siti

on

toth

ecu

bic

spin

elreg

nis

hed

5 DISCUSSION

51 Mg2TiO4 reg lms

































52 Mg2GeO4 reg lms





120 St Senz et al
























































plane




















122 St Senz et al



ACKNOWLEDGEMENTS
























ahellip10iexcl6










4 R ESULTS

41 Overview






























ˆak iexcl apc

apc

hellip3dagger

112 St Senz et al


















114 St Senz et al





ac ak a apc exp th

(pm) (pm) (pm) (pm) ( ) ( )




























116 St Senz et al

































































118 St Senz et al


Fig

ure

8

(a)

TE

Md

ark

-regel

dim

age

usi

ng

are

macrec

tion

from

the

tetr

agon

alsp

inel

(b

)a

bri

gh

t-reg

eld

image

of

tetr

ago

nalsp

inel

gra

ins

ina

Mg

2T

iO4

reglm

at

roo

mte

mper

atu

re(d

)E

lect

ron

diŒ

ract

ion

at

roo

mte

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atu

re

show

ing

the

remacr

ecti

ons

of

tetr

agon

algra

ins

of

thre

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rien

tati

on

sth

ecu

bic

spin

elm

atr

ix

an

dth

eM

gO

sub

stra

te

(e)

Ele

ctro

nd

iŒra

ctio

nta

ken

at

973

Ksh

ow

ing

that

only

the

tetr

ago

nalgra

ins

of

the

in-p

lan

eori

enta

tio

nare

left

(c

)B

right-

regel

dim

age

an

d(f

)el

ectr

on

diŒ

ract

ion

taken

at

1173

K

wit

hth

etr

an

siti

on

toth

ecu

bic

spin

elreg

nis

hed

5 DISCUSSION

51 Mg2TiO4 reg lms

































52 Mg2GeO4 reg lms





120 St Senz et al
























































plane




















122 St Senz et al



ACKNOWLEDGEMENTS
























ahellip10iexcl6

























114 St Senz et al





ac ak a apc exp th

(pm) (pm) (pm) (pm) ( ) ( )




























116 St Senz et al

































































118 St Senz et al


Fig

ure

8

(a)

TE

Md

ark

-regel

dim

age

usi

ng

are

macrec

tion

from

the

tetr

agon

alsp

inel

(b

)a

bri

gh

t-reg

eld

image

of

tetr

ago

nalsp

inel

gra

ins

ina

Mg

2T

iO4

reglm

at

roo

mte

mper

atu

re(d

)E

lect

ron

diŒ

ract

ion

at

roo

mte

mper

atu

re

show

ing

the

remacr

ecti

ons

of

tetr

agon

algra

ins

of

thre

eo

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tati

on

sth

ecu

bic

spin

elm

atr

ix

an

dth

eM

gO

sub

stra

te

(e)

Ele

ctro

nd

iŒra

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nta

ken

at

973

Ksh

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ing

that

only

the

tetr

ago

nalgra

ins

of

the

in-p

lan

eori

enta

tio

nare

left

(c

)B

right-

regel

dim

age

an

d(f

)el

ectr

on

diŒ

ract

ion

taken

at

1173

K

wit

hth

etr

an

siti

on

toth

ecu

bic

spin

elreg

nis

hed

5 DISCUSSION

51 Mg2TiO4 reg lms

































52 Mg2GeO4 reg lms





120 St Senz et al
























































plane




















122 St Senz et al



ACKNOWLEDGEMENTS
























ahellip10iexcl6


































116 St Senz et al

































































118 St Senz et al


Fig

ure

8

(a)

TE

Md

ark

-regel

dim

age

usi

ng

are

macrec

tion

from

the

tetr

agon

alsp

inel

(b

)a

bri

gh

t-reg

eld

image

of

tetr

ago

nalsp

inel

gra

ins

ina

Mg

2T

iO4

reglm

at

roo

mte

mper

atu

re(d

)E

lect

ron

diŒ

ract

ion

at

roo

mte

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atu

re

show

ing

the

remacr

ecti

ons

of

tetr

agon

algra

ins

of

thre

eo

rien

tati

on

sth

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bic

spin

elm

atr

ix

an

dth

eM

gO

sub

stra

te

(e)

Ele

ctro

nd

iŒra

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nta

ken

at

973

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ow

ing

that

only

the

tetr

ago

nalgra

ins

of

the

in-p

lan

eori

enta

tio

nare

left

(c

)B

right-

regel

dim

age

an

d(f

)el

ectr

on

diŒ

ract

ion

taken

at

1173

K

wit

hth

etr

an

siti

on

toth

ecu

bic

spin

elreg

nis

hed

5 DISCUSSION

51 Mg2TiO4 reg lms

































52 Mg2GeO4 reg lms





120 St Senz et al
























































plane




















122 St Senz et al



ACKNOWLEDGEMENTS
























ahellip10iexcl6






































































118 St Senz et al


Fig

ure

8

(a)

TE

Md

ark

-regel

dim

age

usi

ng

are

macrec

tion

from

the

tetr

agon

alsp

inel

(b

)a

bri

gh

t-reg

eld

image

of

tetr

ago

nalsp

inel

gra

ins

ina

Mg

2T

iO4

reglm

at

roo

mte

mper

atu

re(d

)E

lect

ron

diŒ

ract

ion

at

roo

mte

mper

atu

re

show

ing

the

remacr

ecti

ons

of

tetr

agon

algra

ins

of

thre

eo

rien

tati

on

sth

ecu

bic

spin

elm

atr

ix

an

dth

eM

gO

sub

stra

te

(e)

Ele

ctro

nd

iŒra

ctio

nta

ken

at

973

Ksh

ow

ing

that

only

the

tetr

ago

nalgra

ins

of

the

in-p

lan

eori

enta

tio

nare

left

(c

)B

right-

regel

dim

age

an

d(f

)el

ectr

on

diŒ

ract

ion

taken

at

1173

K

wit

hth

etr

an

siti

on

toth

ecu

bic

spin

elreg

nis

hed

5 DISCUSSION

51 Mg2TiO4 reg lms

































52 Mg2GeO4 reg lms





120 St Senz et al
























































plane




















122 St Senz et al



ACKNOWLEDGEMENTS
























ahellip10iexcl6









Fig

ure

8

(a)

TE

Md

ark

-regel

dim

age

usi

ng

are

macrec

tion

from

the

tetr

agon

alsp

inel

(b

)a

bri

gh

t-reg

eld

image

of

tetr

ago

nalsp

inel

gra

ins

ina

Mg

2T

iO4

reglm

at

roo

mte

mper

atu

re(d

)E

lect

ron

diŒ

ract

ion

at

roo

mte

mper

atu

re

show

ing

the

remacr

ecti

ons

of

tetr

agon

algra

ins

of

thre

eo

rien

tati

on

sth

ecu

bic

spin

elm

atr

ix

an

dth

eM

gO

sub

stra

te

(e)

Ele

ctro

nd

iŒra

ctio

nta

ken

at

973

Ksh

ow

ing

that

only

the

tetr

ago

nalgra

ins

of

the

in-p

lan

eori

enta

tio

nare

left

(c

)B

right-

regel

dim

age

an

d(f

)el

ectr

on

diŒ

ract

ion

taken

at

1173

K

wit

hth

etr

an

siti

on

toth

ecu

bic

spin

elreg

nis

hed

5 DISCUSSION

51 Mg2TiO4 reg lms

































52 Mg2GeO4 reg lms





120 St Senz et al
























































plane




















122 St Senz et al



ACKNOWLEDGEMENTS
























ahellip10iexcl6








5 DISCUSSION

51 Mg2TiO4 reg lms

































52 Mg2GeO4 reg lms





120 St Senz et al
























































plane




















122 St Senz et al



ACKNOWLEDGEMENTS
























ahellip10iexcl6































































plane




















122 St Senz et al



ACKNOWLEDGEMENTS
























ahellip10iexcl6










ACKNOWLEDGEMENTS
























ahellip10iexcl6








the eŒ ect of stress on cubic-to-tetragonal phase ... · transitions in the mg2tio4® lms are...

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