American Mineralogist, Volume 59, pages 906-918, 1974

Domainsin Minerals

Rosnnr E. NpwNnarvr Materials ResearchLaboratory, The PennsylaaniaState Uniuersity, Uniuersity Park, Pennsyluania16802

Abstract

Mimetic twinning in minerals is reviewed in terms of the tensor properties of the orientation states, showing which forces are eftective in moving domain walls. Following the Aizu method, various types of ferroic species are developed frorn the free energy function. Examples of ferroelectric, ferromagnetic, ferroelastic, ferrobielectric, ferrobimagnetic, ferrobielastic, ferro- elastoelectric, ferromagnetoelastic, and ferromagnetoelectric minerals are described.

Introduction and ferromagnetoelectric.As explained later, each Twinning is widely used in mineral identification type of domain reorientationarises from a particular and in elucidating the formation conditions of rocks. term in the free energy function. The distribution of transformation twins in rock- A ferroic crystal contains two or more possible forming minerals enables one to establish the orientationstates or domains;under a suitably chosen thermal processes that have occurred in the rock. driving force the domain walls move, switching the Mechanical twinning is studied by petrologists in crystal from one orientation stateto another. Switch- the analysis of flow effects. In rock magnetism, it ing may be accomplishedby mechanicalstress (a), is the arrangement of ferromagnetic domains which electricfield (E), magneticfield (I/), or somecombina- determines remanent magnetization. These are but tion of the three. Ferroelectric, ferroelastic, and a few examples of twin phenomena in minerals. ferromagneticmaterials are well known examplesof In the past twinned crystals have been classified primary ferroic crystals in which the orientation according to twin-laws and morphology, or accord- statesdiffer respectivelyin spontaneouspolarization ing to their mode of origin, or on a structural basis, P,",, spontaneousstrain 6r"r and spontaneous but there is another classification scheme which magnetizatiorrMr"t. It is not necessary,however, has not been widely used by mineralogists, one that the orientation states differ in the primary which is based on the tensor properties of the twin quantities(strain, polarization, or magnetization)for domains. An advantage of such a classification is the appropriate field to develop a driving force the straightforward relation between free energy and between orientations. If, for example, the twinning twin structure, from which it becomes apparent leadsto differentorientations of the elasticcompliance which forces and fields will be effective in moving tensor,a suitably orientedstress can producedifferent twin walls. In a ferroelectric, for instance, electric strains in the two domains.The stressmay act upon fields are efiective in moving domain walls, giving the difference in induced strain to produce wall rise to hysteresis between electric polarization and motion and domain reorientation. Aizu (1973) electric field. Analogous magnetic hysteresis loops suggestedthe term ferrobielastic to distinguish this are caused by the reorientation of ferromagnetic type of responsefrom other ferroics and illustrated domains under applied magnetic fields. The domain the effect with Dauphin6 twinning in quartz. Other patterns in ferroelectric and ferromagnetic materials typesof secondaryferroic crystalsare listed in Table l, are strongly affected by external fields, but there are along with the differencesbetween domain states, many other types of twinned crystals with hysteresis and the driving fields required to switch from one and movable twin walls. Aizu (1973) has classified stateto another.The derivation of the variousferroic such materials as ferroelastic, ferrobielastic, ferro- speciesfrom a free energy function is considered bimagnetic, ferroelastoelectric, ferromagnetoelastic, next.

906 DOMAINS IN MINERALS 907

Teer.p 1. Primary and SecondaryFerroics

Ferroic Class Orientation StatesDiffer in Switching Force Example Primary Ferroelectric SpontaneousPolarization Electric Field BaTiOa Ferroelastic SpontaneousStrain Mechanical Stress CaAlzSigOr Ferromagnetic SpontaneousMagnetization Magnetic Field FesOr Secondary Ferrobielectric Dielectric Susceptibility Electric Field SrTiOs(?) Ferrobimagnetic Magnetic Susceptibility Magnetic Field Nio Ferrobielastic Elastic Compliance Mechanical Stress SiOz Ferroelastoelectric PiezoelectricCoeffi cients Electric Field & Mechanical NH4C(?) Stress Ferromagnetoelastic PiezomagneticCoeffi cients Magnetic Field & FeCOs Mechanical Stress Ferromagnetoelectric MagnetoelectricCoefficients Magnetic Field & Electric LiMnPOr Field

Free Energy Formulation elasticconstants. Spontaneous strain is the change The stability of an orientation state is governed in shapemeasured relative to the prototype structure. by the free energyG. In differential form dG is com- Some twinned crystalspossess spontaneous strain. prised of thermal energy and various work terms: Electric polarization can be expandedin a manner similar to strain, with contributions from a spon- dG : SdZ - e.;;do;;- PidEi - MidHi taneouspolarization P,", and severalinduced effects.

S is entropy; 7 temperature; e;1 strain; oi, stress; Pr. : Prtr. *,toiE; { d;11"o;nI aniHi Pn electric polarization; Eo electricfield; Mo magnetiza- tion; and .EIomagnetic field. Rationalized Mrs units The second rank tensorS r;; &Ird di, represent the are used to avoid awkward factors of 4zr. The direc- electric susceptibility and magnetoelectric coemcients, tional subscripts refer to cartesian coordinates; respectively. rrn is a polar tensor and an, is an axial i, j : 1,2,3. Abrief summaryof the tensorproperties tensor. Only the ten polar crystal classes possess spontaneous polarization P ( All materials have of crystals is given in the Appendix. The entropy " ). term is neglected in what follows since we assume the finite electric susceptibility coemcients, so that experiments are performed under isothermal con- electrically-induced polarization is always present, ditions. but magnetoelectricity is found only in solids with Strain e is measured relative to the prototype certain types of magnetic symmetry. structure and can be written as a spontaneous strain Magnetization can be expanded in terms of the e1"y plus an induced strain. Induced strain may spontaneous magnetization, and induced effects arise from applied mechanical stress (elasticity), arising from electric and magnetic fields, and me- from applied electric fields (piezoelectricity), or from chanical stress. appiied magnetic fi elds (piezomagnetism). Mr : Mr,>n* xntH; ! QiiT"oi** a,tE;. : eii e(')ii * sti*to*t * dno;En* QonrHn. Only ferromagnetic and ferrimagnetic crystals have In this equation s,,i, is a component of the fourth- non-zero spontaneous magnetization. All materials rank elastic compliance tensor. The piezoelectric possess non-zero magnetic susceptibility coefncients, coefficients dr,, constitute a third rank tensor, as (1n,), but the induced magnetization is often very do the piezomagnetic coefficients Qooo.Compliance small. and piezoelectricity are polar tensors whereas piezo- Substituting the expressions for G,i, Pr, and M, magnetism is an axial tensor. Of the four terms in the into the differential form for free energy, combining expressionfor e1;given above, only the second term terms, and integrating gives the thermodynamic is always present. Piezoelectricity and piezomagnetism potential G, which applies to all orientation states. 'G are null properties which disappear for certain Let represent the free energy for the first orienta- 2G symmetry groups, but all groups have non-zero tion state and for the second. with the tensor ROBERT E. NEWNHAM terms referredto a common axial system'The driving taneous polarization, spontaneousstrain, and all 'G 'G. potential for a state shift is the AG : - piezoelectric coefficients are zero1,therefore ferro- In the absenceof externalfields and forces,the energy electricity, ferroelasticity, and ferroelastoelectricity of all orientation states is equal, so that AG : 0. are absent.Without magnetic order, the ferromag- Under external forces the differencein free energy netic, ferromagnetoelectric,and ferromagnetoelastic for the two orientation statesis effectsare also absent.Second and fourth-rank polar tensorsare identicalfor the two orientationstates, AG : Aer"r;ioni* AP(.riEi + LM.,;'iHi eliminating any differencesin the susceptibility,per- mittivity, or elasticconstant matrices. D'omains of I lAs;;e:oa;om* *AxtiE;E; ! $AYa;H6H; this type could not be switchedby any of the primary or secondaryferroic mechanisms,and yet they could ! Adi;lEp;* * AQoioH$rt * Aa;iH;Ei, be seen optically. The sign of the optical-activity 'e - t.,",n,, is for the right- and left-handed where Ae 1"y;;is 1"1ni the difference in a coefficient different certain componentof spontaneousstrain for orienta- domains. tion statesI and 2. AP1"y,and AMq,1nare the dif- An even more subtle type of twinning could ferencesin the fth componentof spontaneouspolariza- occurin mineralssuch as elpasoite(Frondel, 1948). tion and spontaneousmagnetization for the two The structureascribed to elpasoite(K2NaAlFo) is domains.Differences in elasticcompliance coefficients a slight distortionof the (NHr).qFeFearrangement. are representedby Ason*,.The remainingfour terms Ammonium ferrifluoride consistsof NHa ions packed in AG arise from differencesin electric and magnetic togetherwith octahedral(FeFe)3- groups; the struc- susceptibility,and from differencesin piezomagnetic ture belongs to cubic space group Fm3m. Metal and magnetoelectriccoefficients. atoms occupy the same positions in elpasoite but A wide variety of ferroic phenomenais possible, the fluorines are distributed less symmetrically in dependingon which terms in LG are important. If space group Pa3 because of the K,Na ordering. APr"r is non-zero, the material is ferroelectric, Coordinates difier only slightly for the two struc- provided the coercive field does not exceed the tures,but the point symmetryis loweredfrom m3m electricbreakdown limit. Materialswith Aer", # 0 to m3. With prototype symmetry m3m, the orienta- are ferroelasticif the mechanicalstress required to tion statesin m3 are identical for all tensor prop switch orientation statesdoes not result in rupture. erties through rank four. All primary and secondary Ferromagneticdomains-the third type of primary ferroic phenomenaare therefore absentin elpasoite, ferroic-possess finite differences in spontaneous making the domainsimmovable. Optical properties magnetization.Ferroic phenomenaare not mutually of the orientationstates are also identical,making exclusive. In barium titanate, for instance, 90o them invisible as well as immovable. In casessuch domains are both ferroelectric and ferroelastic. as these, domains could be difterentiatedby X-ray The transition betweendomain statesin a ferro- diffraction or by etch-figuretechniques. electric is said to be electrically first-order because In addition to the primary and secondaryferroics AG is proportionalto E. If APr"r : O and Ax I O, in Table l, tertiary and evenhigher-order phenomena then AG - E' and the material is potentially ferrobi- are possible, although not likely. Effects such as electric.Other secondaryferroics are listed in Table 1. ferrotrielectricity(AG - E"\ and ferroelastomagneto- For a ferrobielastic AG - o', and for a ferrobi- electricity (AG - EoH) wil. appear as higher-order magnetic AG - .E['. Cross-coupledferroics include terms in the free energyfunction. The coeffrcientsfor the ferroelastoelectric(AG - Eo), ferromagneto- such terms are generally small, increasing the size elastic (AG - Ho\ and ferromagnetoelectric(AG - of the coercive fields. Furthermore, tertiary effects EH). will usually be obscuredby primary and secondary Almost all types of mimetic twins belong to one ferroic behavior. or more of the ferroic classesin Table 1. There are, Mineralogical examplesof ferroic phenomenaare however, a few casesin which the tensor properties discussedin the following sections. of the orientationstates are nearly identical.Con- Ferroelectricity sider a non-magnetic crystal belonging to crystal class432 with pseudosymmetrym3m. The orienta- A ferroelectric is a crystal possessingreversible tion statesare mirror imagesof one another.Spon- polarization, as shown by a dielectric hysteresis DOMAINS IN MINERALS

loop. Domains in a ferroelectricdiffer in spontaneous Ferromagnetism polarizationP,",, and can be switchedby an applied The orientation states of a ferromagrretdiffer in electricfield. spontaneousmagnetization, and can be switched A large number of ferroelectriccompounds occur by a magnetic field. This b,road definition encom- in the perovskite and pyrochlore families, although passesferrimagnets (magnetite) and weak ferro- the minerals CaTiO. and CaNaNb2OuFare not magnets(hematite), as well as ordinary ferromag- ferroelectric.Barium titanate andPZil (leadzirconate- nets (iron). It does not include antiferromagnetic, titanate) are of considerablecommercial importance paramagnetic,and diamagneticsubstances which becauseof their high permittivities and large piezo- have no spontaneousmagnetization. Such materials electric coefficients. Other mineral-related ferro- are not ferromagneticbut may be ferrobimagnetic. electricsare found in the boracite family and among Magnetite is the best example of a magnetic nitratessuch as KNOr-PhaseIII. However,the only mineral. Below 585oC,FerOn is ferrimagneticwith ferroelectric investigated extensively using mineral a magneticmoment of 4 Bohr magnetonsper molecule specimensis colemanite,CaBrOn(OH)'. HrO. correspondingto the four unpaired electron spins Ferroelectricity was discovered in the borate associatedwith Fe'* (Smit and Wijn, 1959).Tetra- mineral colemanite by Goldsmith (1956). At room hedral Fett spins are directed antiparallel to octa- temperaturecolemanite is centric, point group 2f m, hedralFe'* and Fe'* spinsso that the Fe3*moments but below approximately 0'C it transforms to a cancel,leaving a spontaneousmagnetization equiva- ferroelectric phase belonging to point group 2. lent to one Fe'* momentper molecule.The direction Spontaneouspolarization developsalong the mono- of easymagnetization is (lll), giving rise to eight clinic b axis, accompaniedby the formation of 180" orientation statesfor M,",. Magnetite belongs to domains. At - 20"C the spontaneouspolarization magneticpoint group 3m', oneof the 21 pyromagnetic P,", is 0.45 pC/cm', and the coercivefield required classes(Birss, 1964),although the trigonal distortion to switch domains is about 2000 volts/cm (Wieder, is too smallto be seenby X-ray diffraction.Magnetic 1959).With increasingtemperature P,", decreasesdomains are difficult to see becausemagnetite is continuously to zero, typical of a second order opaque to visible light, even in thin section. The transition. The Curie point is strongly affected by domainsare probably similar to those in MgFerOn space charge fields caused by impurities, but for which has the samemagnetic structure, but is trans- most natural specimensit rangesfrom OoCto -7oC. parent to red wavelengths.Sherwood, Remeika, and Ferroelectric domains are sometimes visible in Williams(1959) observed snake-like domain patterns polarizedlight, but the 18Oodomains in colemanite in a (l I l) thin-sectionof magnesiumferrite. cannot be distinguishedin this way since the optical Domains in transparentferromagnetic and ferri- indicatrix is identical for both orientation states. magnetic crystals are visible in polarized light be- There is a possibility that the difference in optical causeof the Faraday eftect, a nonreciprocalrotation activity could be utilized, though this has yet to be of the plane of polarization.The angle of rotation demonstrated.The orientationstates in colemanite 6 is given by O : pt cos 0, where I is the specimen are enantiomorphic,and potentiallyoptically-active. thickness,p the rotation per unit thickness,and d Becauseof this, the 180" domainsmay be visible the anglebetween the magnetizationvector and the when viewed along an optic axis. direction of propagation.Faraday rotation coeffi- The structuralbasis of ferroelectricityin coleman- cients for ferrites are typically about 1000'/cm ite has been described by Hainsworth and Petch (Sherwood,Remeika, and Williams, 1959). ( 1966). Above the transition,in the nonpolarphase, Hematite, a-FerO., exhibits both antiferro- one of the hydrogen atoms of the water molecule magnetismand weak ferromagnetism.From 250K and the hydrogen of an adjacent hydroxyl group to 950K, the Fe3* spins lie in the rhombohedral are in a stateof dynamicdisorder. As the tempera- (lll) plane and are nearly antiparallel,but with a ture is lowered,the rate of reorientationdecreases small ferromagnetic component, also in (l I l). until the hydrogens settle into ordered noncentric Mineralogists generally assign hematite to trigonal positions in the ferroelectric phase. The ordering class 3rn, but the magnetic point symmetry is 2/m of hydrogen atoms is accompaniedby small dis- at room temperature. Antiferromagnetic crystals placementsof other atoms from the positions they often exhibit weak (parasitic) ferromagnetismwhen occupyin the centricphase. the ferromagnetic component does not violate the 910 ROBERT E. NEWNHAM with the ferro- symmetry elements of the antiferromagnetic spin spontaneousstrain e1"y.By analogy just polarizationcan array (Birss, 1964).In hematite, weak spontaneous electric case, as spontaneous can the spon- magnetizationappears along the monoclinic two-fold be redirected by an electric field, so to one of the three diad axesin taneous strain of a ferroelastic be reoriented by axis,-3m. corresponding At 250K the spin direction changesto the rhom- mechanicalstress. bohedral axis [11], and the weak ferromagnetic Ferroelasticity differs from ferroelectricity and effect disappears.Below the spin-flop transition, the ferromagnetism is one respect. Defining the zero magneticpoint group is 32. referencefor spontaneousstrain is more subtle than Mugtr"iic domains in hematite have been ob- that for spontaneouspolarization or for spontaneous served using the Faraday effect (Williams, Sher- magnetization.If, in the absenceof external forces, wood, and Remeika, 1958). The white, gray, and there is no electric or magnetic charge separation' black regions in Figure 1 correspondto domains then P1", : M<"> : 0. To definee 1"1' a reference with three different magnetic axes; magnetic fields state of zero strain is required' The spontaneous of only 10 oenteds produce significant differences strain of various orientation statesmay differ in sign in the domain pattern. When cooled through the or direction but must be equal in magnitude,other- spin-flop transition at -120"C, the domains dis- wise equivalent free energiesare not obtained for appear, and then reappear in a different pattern equivalent stresses.When measuredrelative to the on heating. prototype structure,the e1"1values of all orientation statesare equalin magnitude.Therefore the prototype Ferroelasticity state containing all the pseudosymmetryelements is A crystalis ferroelasticif ( 1) it has two or more the zero referencefor spontaneousstrain. orientationstates differing in spontaneousstrain, and Ferroelasticity is a type of mechanical twinning (2) can be transformedfrom one to another of in which the lattice reorients rapidly in responseto these statesby an external mechanicalstress. Spon- a mechanicalstress. There is no diffusionor breaking taneousstrain is measuredrelative to the prototype of chemical bonds, only small rearrangementswith structure. In the ferroelastic state the crystal sym- atomic displacementsof the order of 0.1 A. Abra- metry is reducedto a subgroupof a higher symmetry hams (1971) has discussedthe structuralbasis of class by a small distortion, typically on the order ferroelasticity, emphasizing the importance of of parts per thousand,which is a measureof the pseudosymmetryand citing a number of examples. The symmetry classificationof potential ferroelastic materialshas been developedby Aizu (1970). A summaryof the earlier literature on mechanical twinning can be found in the book by Klassen- Neklyudova (1964). The book is divided into two parts: twinning with changein form, and twinning without change in form. Ferroelastic twinning is accompaniedby a change in form associatedwith the reorientationof spontaneousstrain. The tri- clinic feldspars show ferroelastic mechanical twin- ning. Twinning without change in form is exempli- fied by a-etrartz. As discussedlater, quartz is a rm- ferrobielastic material with no spontaneousstrain, but with orientation states that differ in elastic pattern in a hematite crystal Frc" l. Magnetic domain compliance. observed by means of the Faraday effect. Three orientations of the paiasitic ferromagnetism give rise to difterent light Two viewsof the twin structurein triclinic feldspars intensities. The specimen is a thin platelet (-0.03 mm are shown in Figure 2. Twinning is common in all thick) with major faces parallel to the rhombohedral (1ll) the feldspars, and almost universal in microcline plane. To observe the domains it was necessaryto tilt the (KAlSi,Os) and the plagioclases (NaAISLO'- platelet with respect to the light beam; otherwise the Fara- CaAlrSirO, series).The extinction anglesin twinned day effect is absent because the magnetization vectors are perpendicular to the beam. (After Williams, Sberwood, and crystalsconstitute one of the chief methodsof identi- Remeika.1958). fying feldspars. The two most important types of DOMAINS IN MINERALS 9tl

-+ 0.5o,compared to a = 90o,p - 116.0-f 0.5", y = 90o for the monoclinic feldspars. Adopting an orthogonalaxial systemXr = a*, Xz = b, Xt : c, and neglecting the small difference between the triclinic y and 9Oo,the two componentsof spon- taneousstrain are e1s1r2 : 0 and er"ler = G/360") (a' - 90o) = 0.03. Compared to most ferro- elastics, this is a rather large spontaneousstrain. Basedon this analysis,it appearsthat the difference in free energy for the two orientation states will take the form AG = 4 e*l21lo2s.The most effective t stress in moving domain walls should be o23, o a shearingstress about X, or a*. .< Mechanical twinning in anorthite was demon- Ftc. 2. Plagioclase feldspars exhibit perfect cleavage par- strated by Mtigge and Heide (1931). Albite and allel to (001), and less perfect parallel to (010). Albite pericline twin lamellaespaced by about 0.03 mm twin lamellae are usually present on (001) cleavage flakes were introduced under uniaxial stress. Coercive (left). Albite lamellae are parallel to the straight edge formed by the intersecting (010) and (001) cleavage stresseswere not measured,but are known to be planes. Flakes parallel to (010) sometimes show pericline considerablyless than 25 x loa N/cm'?. These twins (right). Twin lamellae intersect the (001:OlO) edge observationswere confirmedby Laves (1952). Un- at an angle c, the so-called angle of tho rhombic section. twinned cleavage flakes of disordered analbite (After Rogers and Kerr, 1942). (Nao.rIG.rAlSiBOs)were pressedlightly with the tip of a needlewhile viewed in a polarizing micro- twins in feldsparsare polysyntheticalbite and peri- scope. With changingpressure albite twin lamellae cline twins. Albite twin lamellae are parallel to appearand disappearwith changingwidths. (010), and are related by reflection across (010). This is a symmetry element in the prototype point group 2/m found in sanidine, the high-temperature potash feldspar. Microcline, the low-temperature polymorph, belongs to the centric triclinic class i, as do all the plagioclasefeldspars. In pericline twins, the individuals are related by rotation of l80o about [010], the two-fold symmetry axis in the prototype point group. The triclinic symmetry in plagioclase feldsparsis causedby crumpling of the aluminosilicate framework about the Na' and Ca'* ions (Fig. 3). Strain is a symmetricsecond-rank tensor with six components: three longitudinal strains €rr, €z:2,cag and three shear componentss2s, e11, e12. The two orientation states in plagioclase feldspars are re- - lated by reflection across D &. Reflection reverses the sign of X2, leaving X1 and X3 unchanged. Tensor components with an odd number of 2 subscripts change sign under this operation. Thus the non-zero Frc. 3. Projection of the triclinic albite along the c axis. componentsof spontaneousstrain are €(6)28ond e1g)12, The tetrahedral (Al,Si) ions (solid circles) and oxygen shearing strains about X, and X3, respectively. (open circles) form an aluminosilicate framework with Na. con- The spontaneous strain associated with ferro- ions in cavities. Sodium ions are not large enough to tact all corners of the cavity, so the framework is sheared elasticity in the plagioclase feldspars can be esti- and the symmetry is lowered from monoclinic to triclinic. mated from crystallographic data. For the varieties Pseudo-mirror planes associatedwith the monoclinic proto- exhibiting mechanical twinning the cell angles are type symmetry are shown as vertical dark lines (after -+ -+ a:93.5 0.5o,F = 116.0o 0.5o,7 = 90.5 Brage, 1937). 9t2 ROBERT E. NEWNHAM

Borg and Heard (1969) have studied mechanical at low temperatures.The TiOu octahedraof the ideal twinning and slip in twelve types of plagioclase' perovskite structure rotate about a fourfold axis. Mechanical albite and pericline twins were induced Alternate octahedra rotate clockwise and counter- by triaxial compression at 800'C and 8-10 kbar clockwise, causing the structure to crumple about confining pressure.The critical resolved shear stress the Sr'* ions (Axe, 1971).On cooling through the for albite and pericline twinning is approximately transition, the tetragonal c axis may develop along 1 kbar at strain rates of 10-4 to 10-5 sec-1.Twinning any of thethree cube edges, giving rise to 90" domains. was induced in seven specimens: disordered An1 The domains are ferroelastic and optically-distinct. and Anig; slightly disordered An32, An3e, and Ans.g; For stress-free90o domains in SrTiO3,the difference (xsa- *rr) E'. and An76;and Ane5 with transitional'and primitive in free energywill be proportionalto in the dielectric anorthite structure, respectively. No microscopic There is no apparent discontinuity transition, twinning was detectedin comparable tests on ordered constantor its slopeat the cubic-tetragonal permittivity at low albite (Anr), ordered oligoclase (Anrn), peristeritic but anisotropy in the develops as large as 25,000 feldspars(An,r, Anr.r), or in low-labradorite(Anut). temperatures.Dielectric constants (Saifi an{ Cross, 1970).Below Pressure-twinning is easier in "high-temperature" have been reported loops are observed, varieties, indicating that Al-Si ordering tends to 50K, electric double hysteresis changesin weak-field permittivity under "freeze-in" the twin states,raising the coercive stress. along with Such behavior may be associatedwith Ordered albite will not twin in response to prBssure DC bias. wall movement. because strong chemical bonds have to be broken ferrobielectricdomain during twinning. The Al-Si distribution in low albite Ferrobimagnetism has triclinic symmetry, so that Al-Si interchange is required in switching from one orientation state Anisotropic magneticsusceptibility may lead to to the other. No interchange is required in ordered ferrobimagnetism.Magnetic susceptibility is a polar anorthite where Al and Si occupy alternate tetra- second-ranktensor like strain and electric per- hedra. Therefore glide twinning is easier in ordered mittivity; therefore ferrobimagnetismhas the same Ca-rich plagioclases.The relation between structural symmetry requirementsas ferroelasticity and ferro- state and ease of formation of pericline-albite twins bielectricity.In materials with spontaneousmag- by gliding has been discussed by Starkey (1967). netization, ferrobimagnetismwill be maskedby the larger ferromagneticefiect. The effect is most likely Ferrobielectricity to occur in antiferromagneticcrystals since 1rr' is Ferrobielectricity is a secondary ferroic phenom- relatively small and isotropic in paramagneticand enon arising from field-induced electric polarization, diamagneticsolids. rather than spontaneous polarization as in a ferro- Nickel oxide (bunsenite)is both ferroelasticand electric. Switching between orientation states occurs ferrobimagnetic.At temperaturesabove the N6e1 because of differences in the dielectric permittivity point of 523 K, NiO is paramagneticwith the cubic tensor. Permittivity is a second-rank tensor, like rocksalt structure. Below Tr, antiferromagneticor- strain or magnetic susceptibility. Any orientation dering of the Ni'. spinsresults in a small rhombo- states that difter in spontaneous strain will also hedral distortion. The unit cell contracts slightly differ in both electric and magnetic susceptibility. along one of the (111) body diagonalswith the Therefore all ferroelastics are potentially ferrobi- angle between cube axes changing from 90o to electric and ferrobimagnetic. 90"4'. Crystallographic twinning occurs because Ferrobielectricity can be expected in non-polar the contractionmay take place along any of the crystals with mimetic twinning and substantial four body diagonals.Each domain is optically uni- dielectric anisotropy. Antiferroelectric materials such axial rrith the optical axisparallel to the contraction - as NaNbO, and SrTiO' are promising candidates direction.The birefringence(n" - tl" 0.003 at because the dielectric permittivities are large enough 5900A) is large enqpghto make domains visible to make a sizeable contribution to the induced in polarizedIight ($oth, 1960). polarization term in the free energy function. Below In a well-annealedcrystal, domain walls are easily I l0K, SrTiO. is ferroelastic and possibly ferrobi- displacedby a mechanicalstress (ferroelasticity) electric. The phase transition involves a symmetry or by a magneticfield (ferrobimagnetism).Elastic change from cubic (class m3m) to tetragonal (4/mmm) enersv is lowest for domainswith the contraction DOMAINS IN MINERALS 913

axis parallel to the applied stress. The walls can be moved distancesof severalmm and the move- ment observedwith a polarizingmicroscope. Only small mechanicalstresses (< 10 N/cmr) are re- quired to move domain walls. A multi-domain specimencan be converted to an untwinned state by pinching the crystal between thumb and index Ftc. 4. Displacement finger (Slack, 1960). of domain walls in antiferromagnetic NiO by a magneticfield. The crystal is a thin plate approximately Untwinned NiO crystals possessan anisotropic 1 mm on edge"lviththe major face parallel to (111). A magnetic magnetic susceptibility.For domains contracted lq.Id qf 25000 oersteds was first applied along [ll2] and then along [1111, the magneticsusceptibility parallel to alodg [Il0] to produce ferrobimagnetic switching. Domains are polarized [111] exceedsthose measuredin the perpendicular visible in light because of the spontaneous strain asgo$iated with antiferromagnetic ordering. Viewed directions.At low fiefds the anisotropyin between suscep- crgssedpolarizor and,analyzer, the major portion of the crystal tibility is 3.3 x 10-6emu/gram. In such a domain, is dark bec{psg the domains are contracted along [111]. Bright spins lie in the ( | 11) plane, perpendicularto the stripeqsloping up to the left and up to the right correspond to [111] contractiondirection. As in most antiferro- domains cgiitrqptedalong [1-11]and [l1T], respectively(after magnetic materiqfs,the mpgnetic susceptibilityis Roth, 19@), largest perpendicularto the spins. Moderate magnefic fields of 5000 oersteds are ipg throggn the phase transition at 573"C, the suffrcient to move dgmain walls in well-annealed symmetry'is lowered to 32. Transformationtwins crystals. Induced -agnelipj energy (and total free d"evelopas p-guartz convertsto a-quartz.The trans- energy) is minimized when the maximum magnetic fgrmation twins, often called Dauphin6 twins or susceptibility is parallel to the applied magnetic electrical twins, consist of two orientation statesre- field. Antiferromagnetic domains with contraction lated by 180' rotation about [00.1], the trigonal direction parallel to H arefavored over other orienta- axis. Dauphin6twins combinetwo right-handed(or tions. The responseto pn applied field is highly two left-hqnded) individuals,often with irregular erratic becausethe walls are easily pinned by crystal composition planes. Such twinning renders the imperfections. Domain wall movement in ferrobi- crystalsuseless for piezoelectricapplications because magneticNiO is illustratedin Figure 4. it reversesthe directionof the X1 axis and the signs of the piezoelectriccoefficients. Because of the im- Ferrobielasticity portance of piezoelectrrcquartz in communications Ferrobielasticcrystals are a class of secondary applications,techniques for detwinningquartz were ferroics in which orientation states differ in elastic developedduring World War II when quartz was compliance, a fourth-rank polar tensor. Ferrobi- scarce.The mechanicaldetwinning of quartz dem- elastic switching in a-quartz has been reported by onstratedby Thomas and Wooster (1951), and Aizu ( 1973). Under appliedstress, the two twinned by others (spe Klassen-Neklyudova,1964), is an regions strain differently. This creates a difference excellentexample of ferrobielasticity. in free energy favoring one domain over the other, When referredto the sameaxes, states of Dauphin6 causingdomain walls to move.Ferrobielasticity is a twin orientationdiffer in elasticconstants. Class 32 second order effect in which the strain difierence has six independentcompliance coefficients: s,,,,, between orientation states is induced by applied Srrzs,Srrea, Srr2g, Sgasa, and srrrr. The twins are related stress.When the stress is removed, the induced by 180" rotation about X, which reversesthe signs strain and the differencein free energy disappear of X, and Xr. Polar tensor coefficientswith an odd also.Domain changesunder stresscan be observed number of I and 2 subscriptschange sign under such optically becauseof differencesin the photoelastic an operation.fherefore s,,r, changessign for the tensorfor the two twin segments.Photoelasticity- two orientation states,but the other coefficientsdo the changein refractive indices with stress-is a not. Under ,An appropriate stress o, the difference fourth-rank tensor like elasticity.Orientation states in free energybetryeeil Dauphin6 states is proportional differingin elasticconstants will alsodifter in photo- to 51123qE.-As shqwn in Figure 5, a uniaxial stress elasticcoefficients. at 45" to X,e' and X" is effective in switching the ,B-quartzis hexagonal,crystal class622. On cool- ferrobielasticdomains. 914 ROBERT E. NEWNHAM

elastoelectricsince Dauphin6 twins differ in piezo- electricconstants as well as elasticconstants. Sal ammoniac (NHnCl) is another possible ferro- elastoelectric.Ammonium chloride undergoesa near second-ordertransition at - 30"C accompaniedby a ooll (t, X-anomalyin the specificheat. The crystal structure is cubic, both above and below the transition, but \/ the spacegrdup changesfrom Pm3m at room tem- perature to P43m at low temperatures.The NH4CI structureresembles CsCl with N at (0, 0, 0) and Cl at (l/2, l/2, l/2). Hydrogenslie along the body diago- nals forming N-H-CI hydrogen bonds. There are two possible orientations for the tetrahedral NHn group with hydrogensat x,x,x; x,*,*; *,x$; *,*,x (x : 0.153), or at fi,x,xi xfi,xl x,xfr; *,X&. Neutron Frc. 5. Ferrobielastic switching of Dauphin6 twins in quartz diffraction data recordedat roorn temperaturefavor produced by uniaxial stressapplied at 45" to Xz and Xr. As the Frenkel's model in which there is random disorder from mechanical stressis increasedslowly 4.9 to 5.0 newtons/ (Levy cm2, the striped twin pattern changes abruptly. Specimen between the two orientations and Peterson, dimensions are 5 X 5 X 3 mm. Orthogonal property axes 1952).Measurements below the transition at liquid (X$zXi correspond to the [10.0], ll2.Ol, [00.1] crystal- air temperaturehave establishedan ordered model Iographic axes, respectively. The crystal is viewed along Xr with only one set of positionsoccupied (Goldschmidt polarizer between crossed and analyzer. Domains are visible and Hurst, 1951). because of the photoelastic effect; the contrast in brightness disappears when the stress is removed (after Aizu, 1973). In the absenceof external forces the two orienta- tion states are equal in energy, so that domains Atomic movementsin the Dauphin6 twin opera- undoubtedly exist at low temperatures.Reflection tion are small and do not involve the breaking of across (100) brings the two statesinto coincidence. Si-O bonds. In shifting from one orientationstate This is a very subtletype of twinning sincethe physical to the other, silicon atoms are displacedby 0.3 A, properties of the two orientation states are nearly and oxygens by about twice that amount. Across identical. Only through third-rank tensor properties the composition plane there is a slight difference such as piezoelectricitydnd the electro-optic effect in bond angles.Dauphin6 twinning disappears at the can the two statesbe distinguished. a-B transformation. Crystal class43m has but one independentpiezo- electric modulus d,r, relating polarization along Ferroelastoelectricity [00] to a shearingstress about [100]:Pt : dnzozr. Ferroelastoelectricityis a new cross-coupledfer- For the two orientation states,4", is equal in mag- roic phenomenon;at presentthere are no known nitude but opposite in sign. Reflection across (100) - examples.lhe domain statesof a true ferroelasto- takes Xr to Xr, and leavesX, and X' unchanged. electric differ in the piezoelectrictensor, when re- Therefore4r, transformsto -dr* for two domains ferred to a common set of axes. The crystal can relatedby a mirror parallel to (100). be switched from one state to another when an Ammonium chloride is a potential ferroelasto- electric field and a mechanicalstress are applied electric becauseits two orientation states differ in simultaneously.A ferroelastoelectricis not simply piezoelectriccoefficients. Applying a uniaxial stresso a ferroelectricwhich is also ferroelastic.Such ma- along [0ll] together with an electric field E along : terials can be switched by either an electric or [00] leadsto a differencein free energyAG 2dn$E. mechanicalforce. Both forces are required to switch Domain switching will take place if the driving a true ferroelastoelectric,for it is neither ferro- potential AG is large enough to overcome the re- electricnor ferroelastic. sistanceto domain wall motion. Sinceall polar classesare potentially ferroelectric, Ferroelastoelectricswitching has yet to be demon- a likely source of ferroelastoelectricsare the ten strated for NHrCl, though domains should be dis- non-polarpiezoelectric classes: 222, 32. 4, 42m, 422, tinguishablethrough the electro-opticeffect. Indirect 6, 6m2, 622, 23, and 43m. Quartz is potentialferro- evidence comes from attempted measurementsof DOMAINS IN MINERALS 9r5 the piezoelectriccoefficient (Bahrs and Engl, 1937). Only a very small piezoelectriceffect (20 times smaller than quartz) was observed in the low-temperature phase,and measurementson different sampleswere inconsistent, suggestingthe possibility of domain effects. Confirmation of domain formation comes from optical experimentsin which laser light of frequency / generatessecond-harmonic light of frequency 22. Ftc. 5. Magnetic structure of siderite at low temperatures. The symmetry requirements for second harmonic Siderite has the calcite structure with Fe'* ions at (0, 0, 0) and, (Vz, r/2, t/z) in the unit cell. The magnetic moment of generation(SHG) are almost identical to those for the Fe atom at the origin is parallel to [111] and anti- piezoelectricity. Ammonium chloride crystals gen- parallel to the moment at the cell center. Both spin direc- erateharmonic signalsat low temperaturesin keeping tions are reversed in the other antiferromagnetic domain. with the acentric point group 432. Small-angle optical-harmonic-scatteringexperiments provide evi- (1962) have studied the piezomagneticeffect in iron denceof domainsin which the senseof the NHn* ion carbonate crystals at liquid hydrogen temperature orientation appears to alternate in a somewhat using a magnetic torsion balance in which a press regular manner(Freund and Kopf, 1970). containing the specimenis suspendedbetween the A similar transition from m3m to 43m symmetry pole pieces of the magnet. Q'r" was measured,but may occur in spinel compounds (Grimes, 1973). 'v,t?s Qzzz below the limit of observation.The mag- Re-evaluationof diffraction data has suggestedthat nitude of Qr"" is sensitiveto bias during annealing. the crystal structure is properly referred to acentric When cooled through the N6el point without stress space group F43m rather than the conventional bias, the effect was smaller, presumablybecause of Fd3m symmetry.The discrepancyis associatedwith antiferromagnetic domains. Domains in antiferro- a small displacement(- 0.03A1 of the octahedral magnetic siderite are of the 180"-typein which all cation along [Il] directions so that alternate tetra- spins are reversed(Fig. 6.). The magneticstructures hedral $oups expand and contract. This symmetry of neighboring domains are related by reflection provides the possibility of antiferroelectricdomains across (2i0) accompanied by time reversal. m' which may explain the high dielectric constants convertsQrr" to - Qrr" so that the piezomagnetic observedin Ni-Mn and Mg-Mn ferrites. coefficientis of opposite sign for the two domains. Siderite is therefore a potential ferromagnetoelastic Ferromagnetoelastic Effect crystal in which domainscan be switchedby applying The domains of a ferromagnetoelasticmaterial mechanicalstress and magneticfield simultaneously. differ in piezomagneticcoefficients. Siderite (FeCOr) The field should be directed along X' together with is antiferromagnetic below 30K. The magnetic a shearingstress about X'. structure (Fig. 6.) consistsof antiparallel Fe'* spins aligned along the hexagonalc axis (Pickart, 1960). Feromagnetoelectricity Sideritebelongs to crystal class32, and the magnetic The coupling between the electric and magnetic point group is also 32. Symmetry elements3' and variables of a material is called the magnetoelectric m', in which the spatial operation is accompanied effect. More specifically,it is an induced magnetiza- by time-reversal,are absent. Crystals with magnetic tion linearly proportional to an applied electric field. symmetry 3m arc potentially piezomagnetic(Birss, In analytic form the electrically-inducedmagneto- 1964). There are two independent piezomagnetic electric effect is M, : auE,, and the magnetically coefficients,Q"", and Qrr". Qr* relatesa tensilestress induced effect is P; : a;;Hr. The subscriptsrefer along X, to a magnetizationin the same direction; to the three directions of a right-handed Cartesian X, is the crystallographic[20] directionperpendicular coordinate system and the cqr coemcientsare the to both the 2-fold (X,) and 3-fold (X.) symmetry field-independentmagnetoelectric coefficients. axes. Piezomagneticcoefficient Q,r, gives the mag- Landau and Lifshitz (1958) employedsymmetry netization component along X, resulting from a argumentsto predict the existenceof the magneto- shearingstress about Xr. electric effect and demonstratedthat long-range Borovik-Ramanov, Aleksanjan, and Rudashevskij magneticorder is a necessary,although not sufficient, 9t6 ROBERT E. NEWNHAM requirement. Magnetoelectricity is permissible in 58 induced magnetizationfor one domain is opposite of the 90 magnetic point groups (Birss, 1964). to that of the other. If a magneticfield is then ap- Recent advances in experimental techniques and plied, the energiesdiffer for the two time-reversed theoretical understanding have been reviewed by structures,making one more probable than the Bertaut and Mercier (197 l). Magnetoelectric co- other. Domain wall motion is easiestjust below the efficients have been measured for about twenty N6el point where coercive fields are smallest. materials, including Cr2O3 (eskolaite), LiFePO+ (triphylite), and LiMnPOa (lithiophilite). Summary LiFePO+ undergoes a paramagnetic-antiferromag- A unified approachto the study of domains in netic phase transition at 50 K. Magnetic suscepti- crystalscan be derivedfrom the free energyfunction bility data collected in the paramagnetic region show by consideringthe various work terms. In the ab. typical Curie-Weissbehavior with an effective atomic sence of applied forces, the free energies of the moment of 5.45 ,ps and an extrapolated Curie con- differentorientation states are identical,but domains stant of 88 K (Santoro and Newnham, 1967). sometimesdifier in energyunder mechanicalstress Bozorth and Kramer (1959) obtained comparable or under electric or magneticfields. Under these results on a mineral specirnen of composition circumstances,domain wall motion may occur, as LiMnoTFee3POa:Tp = 42K, Pett: 6.1 ,,r8,O = low-energy twin domains glow at the expense of 80 K. high-energydomains. If the domains differ in spon- Triphylite is isostructuralwith olivine; lattice taneousmagnetization, magnetic fields will usually parameters for the orthorhombic unit cell are a = alter the domain configuration giving rise to mag- 10.31, b: 6.00, c = 4.69 A. The spacegroup is netic hysteresis,the characteristicfeature of ferro. Pnma with four molecules per unit cell. Divalent magnetism,one of the three primary phenomena. iron atoms occupy mirror plane positions (equi- Ferromagnetic,ferroelectric, and ferroelasticbehav- point 4c) with coordinates :t(0.28,0.25,0.98; ior have been reportedin severalminerals. 0.22,0.75,O.48). Low-temperature neutron diffrac- Secondaryferroic phenomenaare less well known. tion studies gave a magnetic structure in which Theseare inducedeffects caused by propertytensor two of the Fe2* spins are parallel to lb, the other differencesbetween orientation states. The best -b. two to All four spins are reversed for the known exampleis the ferrobielasticeffect in quartz antiferromagnetic 180o domain. where Dauphin6 twin-walls can be shifted by me- The magnetic structure of triphylite conforms to chanicalstress. The strrdyof ferroic phenomenais magnetic point group mmm', one of the magneto- still in its infancy.Many new examplesand applica- electric groups. The symbol m'means that the mirror tions will appearin the years ahead. operation perpendicular to c includes a time re- versal operator. Time reversal flips the spins by Acknowledgments 180o since magnetic moments are associated with It is a pleasure to acknowledge the support we have electric only non-zero moving charge. The magneto- received from the National Science Foundation (Grant No. electric coefficients for mmm' ?rs c12 : .o,2t.A value GH-34547), Air Force Cambridge Laboratories (Contract of 1Fn has been reported by Bertaut and Mercier F19628-73-C-108),and the OIfice of Naval Research(Con- (1971).In gaussianunits, o is dimensionless. tract N00014-67-A-385-0023).I also wish to thank Professor Faul Ribbe for their advice. Magnetoelectric measurementsprovide ample evi- L. E. Cross and Professor H. dencefor the existenceof antiferromagneticdomains. Appendix The magnetoelectric coefficient ,ar: is identical in magnitude for the two domains but opposite in sign. Summary of tensor properties (Nye, 1957; Birss, If the sample is raised above the N6el temperature 1964). Many of the physical properties of crystals and cooled through the transition, the sign of a can can be formulated in tensor notation. Tensors are be positive or negative. Rapid cooling produces both defined in terms of transformations from one orthog- kinds of domains. Powder specimens exhibit no onal axial system to another. Let x7, x2, .r3 be the magnetoelectric effect unless annealed in bias fields old axes and xy', x2', x3'be the new set. ai,i(i, i = 1, to remove the degeneracybetween the two domains 2, 3) is a set of nine numbers representing the (Shtrikman and Treves, 1963). The principle of the cosines of the angles between the new axes xr' and method is quite simple. In an electric field the the old axes. The axes transform according to the DOMAINS IN MINERALS 917

equationsxt' : Qilxi whereSummation is automati- Properties Defining Relation Tensor cally understoodfor repeatedsubscripts. Electric susceptibility x;i P; : x;1Ei Second-rank A zero-ranktensor is unchangedby the transforma- polar tion from old to new axes.Scalar properties such as Magnetic susceptibility Ml : x;iHt Second-rank density are zero-rank tensors. The transformation xij polar Magnetoelectric M; : aa1E1 law for a first-rank tensor is the sameas that for the effect Second-rank qii axial coordinates. Force, electric field, and other vector Magnetoelasticeffect Mt : Quxoix Third-rank axial quantities are first-rank tensors.The componentsof Qti* electric field for instance,transform as En' : q B . Piezoelectriceffect d;p P; : dri*oi* Third-rank polar The transformation law for a second-ranktensor is Elastic complianC€r;;7"1 cij : sijktdkt Fourth-rank polar the sameas that for the products of two coordinates. Electric susceptibilityis a secondrank tensor whose componentstransform as xi;t : e;r"eitx*t.A tensor References of rank n has n subscripts and transforms as the Annrueus, S. C. (1971) Ferroelasticity. Mat. Res. Bull. products of n coordinates. 6. 881-890. The tensorsjust describedare polar tensors.The Arzu, K. (1970) Possible species of ferromagnetic, ferro- transformation law for a polar tensor is unaffected electric, and ferroelastic crystals. Phys. Reo. B.2, 7 54-772. (1973) Second-prderferroic state shifts. .I. Pftys. by handedness.It is the same regardlessof whether Soc. Japan, 34, l2l-128. the old and new axesare both of the samehandedness Axs, J. D. (1971) Neutron studiesof displacive structural or not. For an axial tensor,a changein handedness phase transformations. Trans. Am. Crystallogr. Assoc. introduces a minus sign to the transformation law. 7, 89-109. Magnetic field is an axial first-rank tensor whose Benns, S., eNn J. ENcr (1937) Ztm piezoelektrischen Ef- fekt an Ammoniumchloridkristallen beim Umwadlung- componentstransform ds H1' - The positive tan,H,. spunkt -30.5". Z. Phys. lO5,474-477. sign appliesif the old and new axial systemsare both Brnreur, E. F.,.r.rvoM. Menclrn (1971) Magnetoelectricity right-handedor both left-handed.If the transforma- in theory and experiment. Mat. Res. BulI. 6, 907-922. tion is from right to left, or from left to right, the Brnss, R. S. (1964) Symmetry and Magnetisnr. North- negativesign is required. Holland Publishing Co., Amsterdam,252 p. (1969) The number of independent coefficients of a Bonc, I. Y., eNn H. C. HEeno Mechanical twinning and in experimentally deformed plagioclases. Con- tensor property slip increasesrapidly with rank. For a trib. Mineral Petrol. 23, 128-135. crystal belongingto triclinic point group 1, three Bonovn<-Rovrenov, A. S., G. G. Arrrslx.llN, eNo E. G. coefficientsare required to specifya first-rank tensor Ruoesnrvsxrr (1962) lnt. Conf. on Magnetism and property; a symmetricsecond-rank tensor requires Crystallography, Kyoto, Japan, Pap. 155. six, a third-rank tensornine, and a symmetricfourth- Bozonrn, R. M., exo V. Kuupn (1959) Some ferri- materials low tempera- rank tensor 21. Crystallographic magnetic and antiferromagnetic at symmetryreduces tures. .1.Phys. Rodium, 20, 393-441. the number of independentcoefficients in keeping Bucc, W. L. (1937) Atomic Structure of Minerols. Cornell with Neumann'sPrinciple: The symmetryelements University Press, Ithaca, New York, 292 p. of any physical property of a crystal must include FREUND,I., eNo L. Kopr (1970) Long range ordering in the symmetry elementsof the point group of the ammonium chloride. Phys. Reo. Lett. 24, lOlT-lO2l. (1948) New data on elpasoite and hagge- crystal. The number of FnoNoer-, C. independentcoefficients for manite. Am. Mineral. 32, 84-87. each point group are tabulated by Nye (1957). GoroscHr"rror,G. H., eNo D. G. Hunsr (1951) The struc- Magnetic symmetry and magneticproperties require ture of ammonium chloride by neutron diffraction. specialconsideration (Birss, 1964). Phys.Reo.83, 88-94. Defining relations for several important tensor Gorosurru, G. J. (1956) Ferroelectricity in colemanite. BulI. Am. Phys. Soc. l, 322. propertiesappearing in the free energyfunction are Gnruts, N. V. (1973) Antiferroelectricity among com- listed below. pounds with spinel structure? I. Phys. C: Solid Stote Phys. 6, L78-L79. Intensive HerNswonru, F. N., eNo H. E. Percn (1966) The struc- ExtensiveQuantities Tensor Quantities tural basis of ferroelectricity in colemanite.Can. l. Phys. Entropy S Zero-rank polar Temperature Z 44,3083-3107. Electric polarization P; First-rank polar Electric field Er Kresssw-Nerr-yuDovA,M. V. (1964) Mechanical Twinning Magnetization Ma First-rankaxial Magneticfieldrll; of Crystals. Consultants Bureau, New York, 213 p. Strain e;; Second-rankpolar Stresso;; Lrwoeu, L. D., eNo E. M. Lrnsnrrz (1960) Electrodynamics 918 ROBERT E, NEWNHAM

of Continuous Media. Addison-Wesley Publishing Co., SnEnwooo, R. C., I. P. Rnurrre, eNo H. J. Wrureus Reading, Massachusetts,P. ll9. (1959) Domain behavior in some transparent magnetic Llves, F. (1952) Mechanische Zwillungsbildung in Feld- oxides..I. Appl. Phys. 30, 217-225, spbten in Abhiingigkeit von Ordnung/Unordung der Si/ SnrurueN, S., eNo D. Tnpvrs (1963) Observation of the Al-Verteilung innerhalb des (Si,Al)nO"-Geriistes.Natzr- magnetoelectric effect in Cr,O" powders. Phys. Reu. 130, wissenshaften,39, 546-547. 986-988. Levv, H. A., rNp S. W. PnrnnsoN (1952) Neutron diffrac- Sr-rcr, G. A. (1960) Crystallography and domain walls in tion study of the crystal structure of ammonium chloride. antiferromagneticNiO crystals.J. Appl. Phys. 31, 1577- Phys.Reu.86,766-770. r582. MonrN, F. J. (1950) Magnetic suscEptibilityof oFe'O, and Sr'rrr, f, ^rNo H. P. J. WIrN (1959') Ferrites. John Wiley aFe,O, with added titanium. Phys. Rea. 78, 819-820. and Sons,Inc., New York, 369 p. Miicct, O., eNo F. Hrror, (1931) Einfache Schiebungenam Srenxrv, J. (1967) On the relationship of pericline and Anorthit. Neues lahrb. Mineral. Abt. A.64, 163-169. albite twinning to the composition and structural state of NvE, J. F. (1957) Physical Properties ol Ct'ystals.Oxford plagioclase feldspar. Schweiz. Mineral. Petrogr. Mitt. 47, University Press,London, 122 p. 257-268. Prcrenr, S. J. (1960) Antiferromagnetic ordering in FeCO'. Tsoues. L. A.. ,u.roW. A. Woosren (1951) Piezocrescene: Bull. Am. Phys.Soc. 5, 357. The growth of Dauphine twinning in quartz under stress. Rocrns, A. F., rNo P. K. Kenn (1952) Optical Mineralogy. Proc. Roy. Soc. (London), A.20E,43-62. McGraw-Hill Book Co., New York, 238 p. Wreoen, H. H. (1959) Ferroelectric propertiesof coleman- Rorn, W. L. ( 1960) Neutron and optical studiesof domains ite. I. Appl. Phys.3O' 1010-1018. in NiO. l . Appl. Phys. 31, 2000-2011. Wrrrrlrrs, H. J., R. C. Snsnwooo, eNo J. P. Rrvt'rrl Slrrr, M. A., exo L. E. Cnoss (1970 Dielectric properties (1958) Magnetic domains in aFe"Or. l. Appl. Phys. 29, of strontium titanate at low temperatures.Pftys. Reu. B.2, 1772-r773. 677-684. Sexrono, R. P., eNo R. E. NswxHtu (1967) Antiferro- Manuscript receioed, October 12, 1973; accepted magnetism in LiFePO+. Acta Crystallogr. 22, 344-347. for publication, April 16, 1974.