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Download the Scanned 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-
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