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, ferroelastoelectric, ferromagnetoelastic, and ferromagnetoelectric minerals are described.
Introduction
Twinning is widely used in mineral identification
and in elucidating the formation conditions of rocks.
The distribution of transformation twins in rockforming minerals enables one to establish the
thermal processes that have occurred in the rock.
Mechanical twinning is studied by petrologists in
the analysis of flow effects. In rock magnetism, it
is the arrangement of ferromagnetic domains which
determines remanent magnetization. These are but
a few examples of twin phenomena in minerals.
In the past twinned crystals have been classified
according to twin-laws and morphology, or according to their mode of origin, or on a structural basis,
but there is another classification scheme which
has not been widely used by mineralogists, one
which is based on the tensor properties of the twin
domains. An advantage of such a classification is
the straightforward relation between free energy and
twin structure, from which it becomes apparent
which forces and fields will be effective in moving
twin walls. In a ferroelectric, for instance, electric
fields are efiective in moving domain walls, giving
rise to hysteresis between electric polarization and
electric field. Analogous magnetic hysteresis loops
are caused by the reorientation of ferromagnetic
domains under applied magnetic fields. The domain
patterns in ferroelectric and ferromagnetic materials
are strongly affected by external fields, but there are
many other types of twinned crystals with hysteresis
and movable twin walls. Aizu (1973) has classified
such materials as ferroelastic, ferrobielastic, ferrobimagnetic, ferroelastoelectric, ferromagnetoelastic,
and ferromagnetoelectric.As explained later, each
type of domain reorientationarisesfrom a particular
term in the free energy function.
A ferroic crystal contains two or more possible
orientationstatesor domains;under a suitably chosen
driving force the domain walls move, switching the
crystal from one orientation stateto another. Switching may be accomplishedby mechanicalstress(a),
electricfield (E), magneticfield (I/), or somecombination of the three. Ferroelectric, ferroelastic, and
ferromagneticmaterials are well known examplesof
primary ferroic crystals in which the orientation
statesdiffer respectivelyin spontaneouspolarization
P,",, spontaneousstrain 6r"r and spontaneous
magnetizatiorrMr"t. It is not necessary,however,
that the orientation states differ in the primary
quantities(strain, polarization, or magnetization)for
the appropriate field to develop a driving force
between orientations. If, for example, the twinning
leadsto differentorientationsof the elasticcompliance
tensor,a suitably orientedstresscan producedifferent
strains in the two domains.The stressmay act upon
the difference in induced strain to produce wall
motion and domain reorientation. Aizu (1973)
suggestedthe term ferrobielastic to distinguish this
type of responsefrom other ferroics and illustrated
the effect with Dauphin6 twinning in quartz. Other
typesof secondaryferroic crystalsarelisted in Table l,
along with the differencesbetween domain states,
and the driving fields required to switch from one
stateto another.The derivation of the variousferroic
speciesfrom a free energy function is considered
next.
906
DOMAINS
IN MINERALS
907
Teer.p 1. Primary and SecondaryFerroics
Ferroic Class
Orientation StatesDiffer in
Switching Force
Example
Primary
Ferroelectric
Ferroelastic
Ferromagnetic
Secondary
Ferrobielectric
Ferrobimagnetic
Ferrobielastic
Ferroelastoelectric
SpontaneousPolarization
SpontaneousStrain
SpontaneousMagnetization
Electric Field
Mechanical Stress
Magnetic Field
BaTiOa
CaAlzSigOr
FesOr
Dielectric Susceptibility
Magnetic Susceptibility
Elastic Compliance
PiezoelectricCoefficients
SrTiOs(?)
Nio
SiOz
NH4C(?)
Ferromagnetoelastic
PiezomagneticCoefficients
Ferromagnetoelectric
MagnetoelectricCoefficients
Electric Field
Magnetic Field
Mechanical Stress
Electric Field & Mechanical
Stress
Magnetic Field &
Mechanical Stress
Magnetic Field & Electric
Field
FeCOs
LiMnPOr
Free Energy Formulation
elasticconstants.Spontaneousstrain is the change
The stability of an orientation state is governed in shapemeasuredrelativeto the prototype structure.
by the free energyG. In differential form dG is com- Some twinned crystalspossessspontaneousstrain.
Electric polarization can be expandedin a manner
prised of thermal energy and various work terms:
similar to strain, with contributions from a spondG : SdZ - e.;;do;;- PidEi - MidHi
taneouspolarizationP,", and severalinducedeffects.
S is entropy; 7 temperature; e;1 strain; oi, stress;
Pn electric polarization; Eo electricfield; Mo magnetization; and .EIomagnetic field. Rationalized Mrs units
are used to avoid awkward factors of 4zr. The directional subscripts refer to cartesian coordinates;
i, j : 1,2,3. Abrief summary of the tensorproperties
of crystals is given in the Appendix. The entropy
term is neglected in what follows since we assume the
experiments are performed under isothermal conditions.
Strain e is measured relative to the prototype
structure and can be written as a spontaneous strain
e1"y plus an induced strain. Induced strain may
arise from applied mechanical stress (elasticity),
from applied electric fields (piezoelectricity), or from
appiied magnetic fi elds (piezomagnetism).
eii :
e(')ii *
sti*to*t *
d n o ; E n*
QonrHn.
In this equation s,,i, is a component of the fourthrank elastic compliance tensor. The piezoelectric
coefficients dr,, constitute a third rank tensor, as
do the piezomagnetic coefficients Qooo.Compliance
and piezoelectricity are polar tensors whereas piezomagnetism is an axial tensor. Of the four terms in the
expressionfor e1;given above, only the second term
is always present. Piezoelectricity and piezomagnetism
are null properties which disappear for certain
symmetry groups, but all groups have non-zero
Pr. :
Prtr. *,toiE;
{
d ; 1 1 " o ; nI
aniHi
The second rank tensorS r;; &Ird di, represent the
electric susceptibility and magnetoelectric coemcients,
respectively. rrn is a polar tensor and an, is an axial
tensor. Only the ten polar crystal classes possess
spontaneous polarization P ( ). All materials have
"
finite electric susceptibility coemcients, so that
electrically-induced polarization is always present,
but magnetoelectricity is found only in solids with
certain types of magnetic symmetry.
Magnetization can be expanded in terms of the
spontaneous magnetization, and induced effects
arising from electric and magnetic fields, and mechanical stress.
Mr :
M r , > n* x n t H ; !
Q i i T " o i** a , t E ; .
Only ferromagnetic and ferrimagnetic crystals have
non-zero spontaneous magnetization. All materials
possess non-zero magnetic susceptibility coefncients,
(1n,), but the induced magnetization is often very
small.
Substituting the expressions for G,i, Pr, and M,
into the differential form for free energy, combining
terms, and integrating gives the thermodynamic
potential G, which applies to all orientation states.
'G
Let
represent the free energy for the first orienta2G
for the second. with the tensor
tion state and
ROBERT E. NEWNHAM
terms referredto a common axial system'The driving
'G - 'G.
potential for a state shift is the AG :
In the absenceof externalfieldsand forces,the energy
of all orientation states is equal, so that AG : 0.
Under external forces the differencein free energy
for the two orientation statesis
AG :
A e r " r ; i o n i*
AP(.riEi +
I
l A s ; ; e : o a ; o m*
!
Adi;lEp;*
w h e r e A e 1 " y ; ;i s
'e
*
1"1ni
LM.,;'iHi
*AxtiE;E; !
AQoioH$rt *
t.,",n,,
$AYa;H6H;
Aa;iH;Ei,
the differencein a
taneous polarization, spontaneousstrain, and all
piezoelectric coefficients are zero1,therefore ferroelectricity, ferroelasticity, and ferroelastoelectricity
are absent.Without magnetic order, the ferromagnetic, ferromagnetoelectric,and ferromagnetoelastic
effectsare also absent.Secondand fourth-rank polar
tensorsare identicalfor the two orientationstates,
eliminating any differencesin the susceptibility,permittivity, or elastic constantmatrices.D'omainsof
this type could not be switchedby any of the primary
or secondaryferroic mechanisms,and yet they could
be seen optically. The sign of the optical-activity
coefficientis different for the right- and left-handed
domains.
An even more subtle type of twinning could
occurin mineralssuchas elpasoite(Frondel, 1948).
The structureascribedto elpasoite(K2NaAlFo) is
a slight distortion of the (NHr).qFeFearrangement.
Ammonium ferrifluoride consistsof NHa ions packed
togetherwith octahedral(FeFe)3-groups;the structure belongs to cubic space group Fm3m. Metal
atoms occupy the same positions in elpasoite but
the fluorines are distributed less symmetrically in
space group Pa3 because of the K,Na ordering.
Coordinates difier only slightly for the two structures,but the point symmetryis loweredfrom m3m
to m3. With prototype symmetry m3m, the orientation statesin m3 are identical for all tensor prop
erties through rank four. All primary and secondary
ferroic phenomenaare therefore absentin elpasoite,
making the domainsimmovable.Optical properties
of the orientationstatesare also identical,making
them invisible as well as immovable. In casessuch
as these, domains could be difterentiatedby X-ray
diffraction or by etch-figuretechniques.
In addition to the primary and secondaryferroics
in Table l, tertiary and evenhigher-orderphenomena
are possible, although not likely. Effects such as
ferrotrielectricity(AG - E"\ and ferroelastomagnetoelectricity (AG - EoH) wil. appear as higher-order
terms in the free energyfunction. The coeffrcientsfor
such terms are generally small, increasingthe size
of the coercive fields. Furthermore, tertiary effects
will usually be obscuredby primary and secondary
ferroic behavior.
Mineralogical examplesof ferroic phenomenaare
discussedin the following sections.
certain componentof spontaneousstrain for orientation statesI and 2. AP1"y,and AMq,1nare the differencesin the fth componentof spontaneouspolarization and spontaneousmagnetization for the two
domains.Differencesin elasticcompliancecoefficients
by Ason*,.The remainingfour terms
are represented
in AG arise from differencesin electric and magnetic
susceptibility,and from differencesin piezomagnetic
and magnetoelectriccoefficients.
A wide variety of ferroic phenomenais possible,
dependingon which terms in LG are important. If
APr"r is non-zero, the material is ferroelectric,
provided the coercive field does not exceed the
electricbreakdownlimit. Materialswith Aer", # 0
are ferroelasticif the mechanicalstress required to
switch orientation statesdoes not result in rupture.
Ferromagneticdomains-the third type of primary
ferroic-possess finite differences in spontaneous
magnetization.Ferroic phenomenaare not mutually
exclusive. In barium titanate, for instance, 90o
domains are both ferroelectric and ferroelastic.
The transition betweendomain statesin a ferroelectric is said to be electrically first-order because
AG is proportionalto E. If APr"r : O and Ax I O,
then AG - E' and the material is potentially ferrobielectric.Other secondaryferroicsare listed in Table 1.
For a ferrobielastic AG - o', and for a ferrobimagnetic AG - .E['. Cross-coupledferroics include
the ferroelastoelectric(AG - Eo), ferromagnetoelastic (AG - Ho\ and ferromagnetoelectric(AG EH).
Almost all types of mimetic twins belong to one
or more of the ferroic classesin Table 1. There are,
however, a few casesin which the tensor properties
of the orientationstatesare nearly identical.ConFerroelectricity
sider a non-magnetic crystal belonging to crystal
A ferroelectric is a crystal possessingreversible
class432 with pseudosymmetry
m3m. The orientation statesare mirror imagesof one another.Spon- polarization, as shown by a dielectric hysteresis
DOMAINS
IN MINERALS
Ferromagnetism
loop. Domains in a ferroelectricdiffer in spontaneous
polarizationP,",, and can be switchedby an applied
The orientation states of a ferromagrretdiffer in
electricfield.
spontaneousmagnetization, and can be switched
A large number of ferroelectriccompoundsoccur by a magnetic field. This b,road definition encomin the perovskite and pyrochlore families, although passesferrimagnets(magnetite) and weak ferrothe minerals CaTiO. and CaNaNb2OuFare not magnets(hematite), as well as ordinary ferromagferroelectric.Barium titanate andPZil (leadzirconate- nets (iron). It does not include antiferromagnetic,
titanate) are of considerablecommercialimportance paramagnetic,and diamagneticsubstanceswhich
becauseof their high permittivities and large piezo- have no spontaneous
magnetization.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,FerOnis ferrimagneticwith
ferroelectric investigated extensively using mineral a magneticmoment of 4 Bohr magnetonsper molecule
is colemanite,CaBrOn(OH)'.HrO.
specimens
correspondingto the four unpaired electron spins
Ferroelectricity was discovered in the borate associatedwith Fe'* (Smit and Wijn, 1959).Tetramineral colemanite by Goldsmith (1956). At room hedral Fett spins are directed antiparallel to octatemperaturecolemaniteis 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 spontaneousmagnetizationequivaferroelectric phase belonging to point group 2. lent to one Fe'* momentper molecule.The direction
Spontaneouspolarization developsalong the mono- of easymagnetizationis (lll), giving rise to eight
clinic b axis, accompaniedby the formation of 180" orientation states for 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 increasingtemperatureP,", decreases domains 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 samemagneticstructure,but is transmost natural specimens
it rangesfrom OoCto -7oC. parent to red wavelengths.Sherwood,Remeika,and
Ferroelectric domains are sometimes visible in Williams (1959)observedsnake-likedomain patterns
polarizedlight, but the 18Oodomainsin colemanite in a (l I l) thin-sectionof magnesiumferrite.
cannot be distinguishedin this way since the optical
Domains in transparentferromagnetic and ferriindicatrix is identical for both orientation states. magnetic crystals are visible in polarized light beThere 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 orientationstatesin 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 anglebetweenthe magnetizationvector and the
when viewed along an optic axis.
direction of propagation.Faraday rotation coeffiThe structuralbasisof 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 antiferroone 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 centric phase.
the ferromagnetic component does not violate the
ROBERT E. NEWNHAM
910
symmetry elements of the antiferromagnetic spin
array (Birss, 1964).In hematite, weak spontaneous
magnetizationappearsalong the monoclinic two-fold
to one of the three diad axesin
axis,
-3m. corresponding
At 250K the spin direction changesto the rhombohedral axis [11], and the weak ferromagnetic
effect disappears.Below the spin-flop transition, the
magneticpoint group is 32.
Mugtr"iic domains in hematite have been observed using the Faraday effect (Williams, Sherwood, and Remeika, 1958). The white, gray, and
black regions in Figure 1 correspondto domains
with three different magnetic axes; magnetic fields
of only 10 oenteds produce significant differences
in the domain pattern. When cooled through the
spin-flop transition at -120"C, the domains disappear, and then reappear in a different pattern
on heating.
Ferroelasticity
A crystalis ferroelasticif ( 1) it has two or more
strain,and
orientationstatesdifferingin spontaneous
(2) can be transformedfrom one to another of
thesestatesby an external mechanicalstress.Spontaneousstrain is measuredrelativeto the prototype
structure. In the ferroelastic state the crystal symmetry is reducedto a subgroupof a higher symmetry
class by a small distortion, typically on the order
of parts per thousand,which is a measureof the
rmFrc" l. Magnetic domain pattern in a hematite crystal
observed by means of the Faraday effect. Three orientations
of the paiasitic ferromagnetism give rise to difterent light
intensities. The specimen is a thin platelet (-0.03 mm
thick) with major faces parallel to the rhombohedral (1ll)
plane. To observe the domains it was necessaryto tilt the
platelet with respect to the light beam; otherwise the Faraday effect is absent because the magnetization vectors are
perpendicular to the beam. (After Williams, Sberwood, and
Remeika. 1958).
spontaneousstrain e1"y.By analogywith the ferroelectric case,just as spontaneouspolarization can
be redirected by an electric field, so can the spontaneous strain of a ferroelastic be reoriented by
mechanicalstress.
Ferroelasticity differs from ferroelectricity and
ferromagnetism is one respect. Defining the zero
referencefor spontaneousstrain is more subtle than
that for spontaneouspolarization or for spontaneous
magnetization.If, in the absenceof external forces,
there is no electric or magnetic charge separation'
then P1", : M<"> : 0. To definee1"1'a reference
state of zero strain is required' The spontaneous
strain of various orientation statesmay differ in sign
or direction but must be equal in magnitude,otherwise equivalent free energiesare not obtained for
equivalent stresses.When measuredrelative to the
prototype structure,the e1"1valuesof all orientation
statesare equalin magnitude.Thereforethe prototype
state containing all the pseudosymmetryelementsis
the zero referencefor spontaneousstrain.
Ferroelasticity is a type of mechanical twinning
in which the lattice reorients rapidly in responseto
a mechanicalstress.Thereis no diffusionor breaking
of chemical bonds, only small rearrangementswith
of the order of 0.1 A. Abraatomic displacements
hams (1971) has discussedthe structuralbasis of
ferroelasticity, emphasizing the importance of
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 KlassenNeklyudova (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 triclinic feldspars show ferroelastic mechanical twinning. Twinning without change in form is exemplified by a-etrartz. As discussedlater, quartz is a
ferrobielastic material with no spontaneousstrain,
but with orientation states that differ in elastic
compliance.
Two viewsof the twin structurein triclinic feldspars
are shown in Figure 2. Twinning is common in all
the feldspars, and almost universal in microcline
(KAlSi,Os) and the plagioclases (NaAISLO'CaAlrSirO, series).The extinction anglesin twinned
crystalsconstituteone of the chief methodsof identifying feldspars. The two most important types of
DOMAINS IN MINERALS
t
a
.<
Ftc. 2. Plagioclase feldspars exhibit perfect cleavage parallel to (001), and less perfect parallel to (010). Albite
twin lamellae are usually present on (001) cleavage flakes
(left). Albite lamellae are parallel to the straight edge
formed by the intersecting (010) and (001) cleavage
planes. Flakes parallel to (010) sometimes show pericline
twins (right). Twin lamellae intersect the (001:OlO) edge
at an angle c, the so-called angle of tho rhombic section.
(After Rogers and Kerr, 1942).
twins in feldsparsare polysyntheticalbite and pericline twins. Albite twin lamellae are parallel to
(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-ranktensor 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
componentsof spontaneousstrain are €(6)28
ond e1g)12,
shearing strains about X, and X3, respectively.
The spontaneous strain associated with ferroelasticity in the plagioclase feldspars can be estimated from crystallographic data. For the varieties
exhibiting mechanical twinning the cell angles are
- + 0 . 5 o , = 1 1 6 . 0 o- +
a:93.5
0.5o,7 = 90.5
F
9tl
-+ 0.5o,comparedto 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: 0 and er"ler = G/360")
taneousstrain are e1s1r2
(a' - 90o) = 0.03. Compared to most ferroelastics, this is a rather large spontaneousstrain.
Basedon this analysis,it appearsthat the difference
in free energy for the two orientation states will
The most effective
take the form AG = 4 e*l21lo2s.
stress in moving domain walls should be o23, o
shearingstressabout X, or a*.
Mechanical twinning in anorthite was demonstrated by Mtigge and Heide (1931). Albite and
pericline twin lamellaespacedby about 0.03 mm
were introduced under uniaxial stress. Coercive
stresseswere not measured,but are known to be
considerablyless than 25 x loa N/cm'?. These
observations
were confirmedby Laves (1952). Untwinned cleavage flakes of disordered analbite
(Nao.rIG.rAlSiBOs)were pressedlightly with the
tip of a needlewhile viewed in a polarizing microscope. With changingpressurealbite twin lamellae
appearand disappearwith changingwidths.
Frc. 3. Projection of the triclinic albite along the c axis.
The tetrahedral (Al,Si) ions (solid circles) and oxygen
(open circles) form an aluminosilicate framework with Na.
ions in cavities. Sodium ions are not large enough to contact all corners of the cavity, so the framework is sheared
and the symmetry is lowered from monoclinic to triclinic.
Pseudo-mirror planes associatedwith the monoclinic prototype symmetry are shown as vertical dark lines (after
Brage, 1937).
9t2
ROBERT E. NEWNHAM
Borg and Heard (1969) have studied mechanical
twinning and slip in twelve types of plagioclase'
Mechanical albite and pericline twins were induced
by triaxial compression at 800'C and 8-10 kbar
confining pressure. The critical resolved shear stress
for albite and pericline twinning is approximately
1 kbar at strain rates of 10-4 to 10-5 sec-1.Twinning
was induced in seven specimens: disordered An1
and Anig; slightly disordered An32, An3e, and Ans.g;
and An76;and Ane5 with transitional'and primitive
anorthite structure, respectively. No microscopic
twinning was detectedin comparable tests on ordered
albite (Anr), ordered oligoclase (Anrn), peristeritic
feldspars(An,r, Anr.r), or in low-labradorite(Anut).
Pressure-twinning is easier in "high-temperature"
varieties, indicating that Al-Si ordering tends to
"freeze-in" the twin states,raising the coercive stress.
Ordered albite will not twin in response to prBssure
because strong chemical bonds have to be broken
during twinning. The Al-Si distribution in low albite
has triclinic symmetry, so that Al-Si interchange is
required in switching from one orientation state
to the other. No interchange is required in ordered
anorthite where Al and Si occupy alternate tetrahedra. Therefore glide twinning is easier in ordered
Ca-rich plagioclases.The relation between structural
state and ease of formation of pericline-albite twins
by gliding has been discussed by Starkey (1967).
at low temperatures.The TiOu octahedraof the ideal
perovskite structure rotate about a fourfold axis.
Alternate octahedra rotate clockwise and counterclockwise, causing the structure to crumple about
the Sr'* ions (Axe, 1971).On cooling through the
transition, the tetragonal c axis may develop along
any of thethreecubeedges,givingriseto 90" domains.
The domains are ferroelastic and optically-distinct.
For stress-free90o domains in SrTiO3,the difference
in free energywill be proportionalto (xsa- *rr) E'.
There is no apparent discontinuity in the dielectric
constantor its slopeat the cubic-tetragonaltransition,
but anisotropy in the permittivity developsat low
temperatures.Dielectric constantsas large as 25,000
have been reported (Saifi an{ Cross, 1970).Below
50K, electric double hysteresisloops are observed,
along with changesin weak-field permittivity under
DC bias. Such behavior may be associatedwith
ferrobielectricdomain wall movement.
Ferrobimagnetism
Anisotropic magneticsusceptibilitymay lead to
Magneticsusceptibilityis a polar
ferrobimagnetism.
second-ranktensor like strain and electric permittivity; therefore ferrobimagnetismhas the same
symmetry requirementsas ferroelasticity and ferrobielectricity. In materials with spontaneousmagnetization, 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 ferroNickel oxide (bunsenite)is both ferroelasticand
electric. Switching between orientation states occurs ferrobimagnetic.At temperaturesabove the N6e1
with the cubic
because of differences in the dielectric permittivity
point of 523 K, NiO is paramagnetic
tensor. Permittivity is a second-rank tensor, like
rocksalt structure. Below Tr, antiferromagneticorstrain or magnetic susceptibility. Any orientation dering of the Ni'. spinsresultsin a small rhombostates 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 ferrobiangle between cube axes changing from 90o to
electric and ferrobimagnetic.
90"4'. Crystallographic twinning occurs because
the contractionmay take place along any of the
Ferrobielectricity can be expected in non-polar
crystals with mimetic twinning and substantial four body diagonals.Each domain is optically unidielectric anisotropy. Antiferroelectric materials such axial rrith the optical axis parallelto the contraction
- tl" - 0.003 at
as NaNbO, and SrTiO' are promising candidates direction.The birefringence(n"
because the dielectric permittivities are large enough 5900 A) is large enqpghto make domains visible
in polarizedIight ($oth, 1960).
to make a sizeable contribution to the induced
crystal,domainwalls are easily
In a well-annealed
polarization term in the free energy function. Below
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
enersv is lowest for domainswith the contraction
change from cubic (class m3m) to tetragonal (4/mmm)
DOMAINS
IN MINERALS
axis parallel to the applied stress. The walls can
be moved distancesof severalmm and the movement observedwith a polarizingmicroscope.Only
small mechanicalstresses(< 10 N/cmr) are required to move domain walls. A multi-domain
specimencan be converted to an untwinned state
by pinching the crystal between thumb and index
finger (Slack, 1960).
Untwinned NiO crystals possessan anisotropic
magnetic susceptibility.For domains contracted
along [1111, the magneticsusceptibilityparallel to
[111] exceedsthose measuredin the perpendicular
directions.At low fiefds the anisotropyin susceptibility is 3.3 x 10-6emu/gram.In such a domain,
spins lie in the ( | 11) plane, perpendicularto the
[111] contractiondirection. As in most antiferromagnetic materiqfs,the mpgnetic susceptibilityis
largest perpendicularto the spins.
Moderate magnefic fields of 5000 oersteds are
suffrcient to move dgmain walls in well-annealed
crystals. Induced -agnelipj energy (and total free
energy) is minimized when the maximum magnetic
susceptibility is parallel to the applied magnetic
field. Antiferromagnetic domains with contraction
direction parallel to H arefavored over other orientations. The responseto pn applied field is highly
erratic becausethe walls are easily pinned by crystal
imperfections. Domain wall movement in ferrobimagneticNiO is illustratedin Figure 4.
Ferrobielasticity
Ferrobielasticcrystals are a class of secondary
ferroics in which orientation states differ in elastic
compliance, a fourth-rank polar tensor. Ferrobielastic switching in a-quartz has been reported by
Aizu ( 1973). Under appliedstress,the two twinned
regions strain differently. This creates a difference
in free energy favoring one domain over the other,
causingdomainwalls to move.Ferrobielasticityis a
second order effect in which the strain difierence
between orientation states is induced by applied
stress. When the stress is removed, the induced
strain and the differencein free energy disappear
also.Domain changesunder stresscan be observed
optically becauseof differencesin the photoelastic
tensorfor the two twin segments.Photoelasticitythe change in refractive indices with stress-is a
fourth-rank tensor like elasticity.Orientationstates
differingin elasticconstantswill alsodifter in photoelasticcoefficients.
,B-quartzis hexagonal,crystal class622. On cool-
913
Ftc. 4. Displacement of domain walls in antiferromagnetic
NiO by a magneticfield. The crystal is a thin plate approximately
1 mm on edge"lviththe major face parallel to (111). A magnetic
lq.Id qf 25000 oersteds was first applied along [ll2] and then
alodg [Il0] to produce ferrobimagnetic switching. Domains are
visible in polarized light because of the spontaneous strain
asgo$iated with antiferromagnetic ordering. Viewed between
crgssedpolarizor and,analyzer,the major portion of the crystal
is dark bec{psg the domains are contracted along [111]. Bright
stripeqsloping up to the left and up to the right correspond to
domains cgiitrqpted along [1-11]and [l1T], respectively(after
Roth, 19@),
ipg throggn the phase transition at 573"C, the
symmetry'is lowered to 32. Transformationtwins
d"evelopas p-guartz convertsto a-quartz.The transfgrmation twins, often called Dauphin6 twins or
electrical twins, consist of two orientation statesrelated by 180' rotation about [00.1], the trigonal
axis. Dauphin6twins combinetwo right-handed(or
two left-hqnded) individuals, often with irregular
composition planes. Such twinning renders the
crystalsuselessfor piezoelectricapplicationsbecause
it reversesthe directionof the X1 axis and the signs
of the piezoelectriccoefficients.Becauseof the importance of piezoelectrrcquartz in communications
applications,techniquesfor detwinningquartz were
developedduring World War II when quartz was
scarce.The mechanicaldetwinningof quartz demonstratedby Thomas and Wooster (1951), and
by others (spe Klassen-Neklyudova,
1964), is an
excellent example of ferrobielasticity.
When referredto the sameaxes,statesof Dauphin6
twin orientationdiffer in elasticconstants.Class32
has six independentcompliancecoefficients:s,,,,,
Srrzs,Srrea,Srr2g,Sgasa,
and srrrr. The twins are related
by 180" rotation about X, which reversesthe signs
of X, and Xr. Polar tensor coefficientswith an odd
number of I and 2 subscriptschangesign under such
an operation.fherefore s,,r, changessign for the
two orientation states,but the other coefficientsdo
not. Under ,An appropriate stress o, the difference
in freeenergybetryeeilDauphin6statesis proportional
to 51123
qE.-Asshqwn in Figure 5, a uniaxial stress
at 45" to X,e' and X" is effective in switching the
ferrobielasticdomains.
ROBERT E. NEWNHAM
914
ooll
(t,
\/
Frc. 5. Ferrobielastic switching of Dauphin6 twins in quartz
produced by uniaxial stressapplied at 45" to Xz and Xr. As the
mechanical stressis increasedslowly from 4.9 to 5.0 newtons/
cm2, the striped twin pattern changes abruptly. Specimen
dimensions are 5 X 5 X 3 mm. Orthogonal property axes
(X$zXi
correspond to the [10.0], ll2.Ol, [00.1] crystalIographic axes, respectively. The crystal is viewed along Xr
between crossed polarizer and analyzer. Domains are visible
because of the photoelastic effect; the contrast in brightness
disappears when the stress is removed (after Aizu, 1973).
Atomic movementsin the Dauphin6 twin operation are small and do not involve the breaking of
Si-O bonds. In shifting from one orientationstate
to the other, silicon atoms are displacedby 0.3 A,
and oxygens by about twice that amount. Across
the composition plane there is a slight difference
in bond angles.Dauphin6twinningdisappearsat the
a-B transformation.
Ferroelastoelectricity
Ferroelastoelectricityis a new cross-coupledferroic phenomenon;at presentthere are no known
examples.lhe domain statesof a true ferroelastoelectric differ in the piezoelectrictensor, when referred to a common set of axes. The crystal can
be switched from one state to another when an
electric field and a mechanicalstress are applied
simultaneously.A ferroelastoelectric
is not simply
a ferroelectricwhich is also ferroelastic.Such materials can be switched by either an electric or
mechanicalforce. Both forces are required to switch
a true ferroelastoelectric,
for it is neither ferroelectricnor ferroelastic.
Sinceall polar classesare potentially ferroelectric,
a likely source of ferroelastoelectricsare the ten
non-polarpiezoelectric
classes:222, 32. 4, 42m, 422,
6, 6m2, 622, 23, and 43m. Quartz is potentialferro-
elastoelectricsince Dauphin6 twins differ in piezoelectricconstantsas well as elasticconstants.
Sal ammoniac (NHnCl) is another possibleferroelastoelectric.Ammonium chloride undergoesa near
second-ordertransition at - 30"C accompaniedby a
X-anomalyin the specificheat. The crystal structure
is cubic, both above and below the transition, but
the spacegrdup changesfrom Pm3m at room temperature to P43m at low temperatures.The NH4CI
structureresemblesCsCl with N at (0, 0, 0) and Cl at
(l/2, l/2, l/2). Hydrogenslie along the body diagonals forming N-H-CI hydrogen bonds. There are
two possible orientations for the tetrahedral NHn
group with hydrogensat x,x,x; x,*,*; *,x$; *,*,x (x :
0.153), or at fi,x,xi xfi,xl x,xfr; *,X&. Neutron
diffraction data recordedat roorn temperaturefavor
Frenkel's model in which there is random disorder
between the two orientations (Levy and Peterson,
1952).Measurementsbelow the transition at liquid
air temperaturehave establishedan ordered model
with only one set of positionsoccupied(Goldschmidt
and Hurst, 1951).
In the absenceof external forces the two orientation states are equal in energy, so that domains
undoubtedly exist at low temperatures.Reflection
across (100) brings the two statesinto coincidence.
This is a very subtletype of twinning sincethe physical
properties of the two orientation states are nearly
identical. Only through third-rank tensor properties
such as piezoelectricitydnd the electro-optic effect
can the two statesbe distinguished.
Crystal class43m has but one independentpiezoelectric modulus d,r, relating polarization along
[00] to a shearingstressabout [100]:Pt : dnzozr.
For the two orientation states,4", is equal in magnitude but opposite in sign. Reflection across (100)
takes Xr to - Xr, and leavesX, and X' unchanged.
Therefore4r, transformsto -dr* for two domains
relatedby a mirror parallel to (100).
Ammonium chloride is a potential ferroelastoelectric becauseits two orientation states differ in
piezoelectriccoefficients.Applying a uniaxial stresso
along [0ll] together with an electric field E along
[00] leadsto a differencein freeenergyAG : 2dn$E.
Domain switching will take place if the driving
potential AG is large enough to overcomethe resistanceto domain wall motion.
Ferroelastoelectricswitching has yet to be demonstrated for NHrCl, though domains should be distinguishablethrough the electro-opticeffect. Indirect
evidence comes from attempted measurementsof
DOMAINS
IN MINERALS
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-harmoniclight of frequency 22.
The symmetry requirements for second harmonic
generation(SHG) are almost identical to those for
piezoelectricity. Ammonium chloride crystals generateharmonic signalsat low temperaturesin keeping
with the acentric point group 432. Small-angle
experimentsprovide evioptical-harmonic-scattering
denceof domainsin which the senseof the NHn* ion
orientation appears to alternate in a somewhat
regular manner(Freund and Kopf, 1970).
A similar transition from m3m to 43m symmetry
may occur in spinel compounds (Grimes, 1973).
Re-evaluationof diffraction data has suggestedthat
the crystal structure is properly referred to acentric
space group F43m rather than the conventional
Fd3m symmetry.The discrepancyis associatedwith
a small displacement(- 0.03A1 of the octahedral
cation along [Il] directions so that alternate tetrahedral $oups expand and contract. This symmetry
provides the possibility of antiferroelectricdomains
which may explain the high dielectric constants
observedin Ni-Mn and Mg-Mn ferrites.
9r5
Ftc. 5. Magnetic structure of siderite at low temperatures.
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
the Fe atom at the origin is parallel to [111] and antiparallel to the moment at the cell center. Both spin directions are reversed in the other antiferromagnetic domain.
(1962) have studied the piezomagneticeffect in iron
carbonate crystals at liquid hydrogen temperature
using a magnetic torsion balance in which a press
containing the specimenis suspendedbetween the
pole pieces of the magnet. Q'r" was measured,but
'v,t?sbelow the limit of observation.The magQzzz
nitude of Qr"" is sensitiveto bias during annealing.
When cooled through the N6el point without stress
bias, the effect was smaller, presumablybecauseof
antiferromagnetic domains. Domains in antiferromagnetic siderite are of the 180"-typein which all
spins are reversed(Fig. 6.). The magneticstructures
of neighboring domains are related by reflection
across (2i0) accompanied by time reversal. m'
converts Qrr" to - Qrr" so that the piezomagnetic
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 mechanicalstressand magneticfield simultaneously.
differ in piezomagneticcoefficients.Siderite(FeCOr) The field should be directed along X' together with
is antiferromagnetic below 30K. The magnetic a shearingstressabout X'.
structure (Fig. 6.) consistsof antiparallel Fe'* spins
Feromagnetoelectricity
aligned along the hexagonalc axis (Pickart, 1960).
Sideritebelongsto 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 magnetizaby time-reversal,are absent. Crystals with magnetic tion linearly proportional to an applied electricfield.
symmetry 3m arc potentially piezomagnetic(Birss, In analytic form the electrically-inducedmagneto1964). 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-independentmagnetoelectriccoefficients.
axes. Piezomagneticcoefficient Q,r, gives the magLandau and Lifshitz (1958) employedsymmetry
netization component along X, resulting from a argumentsto predict the existenceof the magnetoshearingstressabout Xr.
electric effect and demonstratedthat long-range
Borovik-Ramanov, Aleksanjan, and Rudashevskij magneticorder is a necessary,
althoughnot sufficient,
9t6
ROBERT E. NEWNHAM
requirement. Magnetoelectricity is permissible in 58
of the 90 magnetic point groups (Birss, 1964).
Recent advances in experimental techniques and
theoretical understanding have been reviewed by
Bertaut and Mercier (197 l). Magnetoelectric coefficients have been measured for about twenty
materials, including Cr2O3 (eskolaite), LiFePO+
(triphylite), and LiMnPOa (lithiophilite).
LiFePO+ undergoes a paramagnetic-antiferromagnetic phase transition at 50 K. Magnetic susceptibility data collected in the paramagnetic region show
typical Curie-Weiss behavior with an effective atomic
moment of 5.45 ,ps and an extrapolated Curie constant of 88 K (Santoro and Newnham, 1967).
Bozorth and Kramer (1959) obtained comparable
results on a mineral specirnen of composition
LiMnoTFee3POaT
: p = 4 2 K , P e t t : 6 . 1 , , r 8 ,O =
induced magnetizationfor one domain is opposite
to that of the other. If a magneticfield is then applied, the energiesdiffer for the two time-reversed
structures,making one more probable than the
other. Domain wall motion is easiestjust below the
N6el point where coercive fields are smallest.
Summary
A unified approachto the study of domains in
crystalscan be derivedfrom the free energyfunction
by consideringthe various work terms. In the ab.
sence of applied forces, the free energies of the
differentorientationstatesare identical,but domains
sometimesdifier in energyunder mechanicalstress
or under electric or magnetic fields. Under these
circumstances,
domain wall motion may occur, as
low-energy twin domains glow at the expense of
80 K.
high-energydomains.If the domains differ in sponTriphylite is isostructuralwith olivine; lattice taneousmagnetization,magneticfields will usually
parameters for the orthorhombic unit cell are a =
alter the domain configuration giving rise to mag10.31,b:
6 . 0 0 , c = 4 . 6 9 A . T h e s p a c eg r o u p i s 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 (equiFerromagnetic,ferroelectric, and ferroelasticbehavpoint 4c) with coordinates :t(0.28,0.25,0.98; ior have been reportedin severalminerals.
0.22,0.75,O.48). Low-temperature neutron diffracSecondaryferroic phenomenaarelesswell known.
tion studies gave a magnetic structure in which
Theseare inducedeffectscausedby propertytensor
two of the Fe2* spins are parallel to lb, the other
differencesbetween orientation states. The best
two to -b. All four spins are reversed for the known exampleis the ferrobielasticeffect in quartz
antiferromagnetic 180o domain.
where Dauphin6 twin-walls can be shifted by meThe 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 applicaelectric groups. The symbol m'means that the mirror
tions will appearin the years ahead.
operation perpendicular to c includes a time reversal operator. Time reversal flips the spins by
180o since magnetic moments are associated with
moving electric charge. The only non-zero magnetoelectric coefficients for mmm' ?rs c12 : .o,2t.A value
of 1Fn has been reported by Bertaut and Mercier
(1971).In gaussianunits, o is dimensionless.
Magnetoelectric measurementsprovide ample evidence for the existenceof antiferromagneticdomains.
The magnetoelectric coefficient ,ar: is identical in
magnitude for the two domains but opposite in sign.
If the sample is raised above the N6el temperature
and cooled through the transition, the sign of a can
be positive or negative. Rapid cooling produces both
kinds of domains. Powder specimens exhibit no
magnetoelectric effect unless annealed in bias fields
to remove the degeneracybetween the two domains
(Shtrikman and Treves, 1963). The principle of the
method is quite simple. In an electric field the
Acknowledgments
It is a pleasure to acknowledge the support we have
received from the National Science Foundation (Grant No.
GH-34547), Air Force Cambridge Laboratories (Contract
F19628-73-C-108),and the OIfice of Naval Research(Contract N00014-67-A-385-0023).I also wish to thank Professor
L. E. Cross and Professor Faul H. Ribbe for their advice.
Appendix
Summary of tensor properties (Nye, 1957; Birss,
1964). Many of the physical properties of crystals
can be formulated in tensor notation. Tensors are
defined in terms of transformations from one orthogonal axial system to another. Let x7, x2, .r3 be the
old axes and xy', x2', x3'be the new set. ai,i(i, i = 1,
2, 3) is a set of nine numbers representing the
cosines of the angles between the new axes xr' and
the old axes. The axes transform according to the
DOMAINS
IN MINERALS
equationsxt' : Qilxi where Summationis automatically understoodfor repeatedsubscripts.
A zero-ranktensoris unchangedby the transformation from old to new axes.Scalarpropertiessuch as
density are zero-rank tensors. The transformation
law for a first-rank tensor is the sameas that for the
coordinates. Force, electric field, and other vector
quantities are first-rank tensors.The componentsof
electric field for instance,transform as En' : q B .
The transformation law for a second-ranktensor is
the sameas that for the products of two coordinates.
Electric susceptibilityis a secondrank tensor whose
components
transformas xi;t : e;r"eitx*t.A tensor
of rank n has n subscripts and transforms as the
products of n coordinates.
The tensorsjust describedare polar tensors.The
transformation law for a polar tensor is unaffected
by handedness.It is the sameregardlessof whether
the old and new axesare both of the samehandedness
or not. For an axial tensor,a changein handedness
introduces a minus sign to the transformation law.
Magnetic field is an axial first-rank tensor whose
components
transformdsH1' - tan,H,. The positive
sign appliesif the old and new axial systemsare both
right-handedor both left-handed.If the transformation is from right to left, or from left to right, the
negativesign is required.
The number of independent coefficients of a
tensor property increasesrapidly with rank. For a
crystal belongingto triclinic point group 1, three
coefficientsare requiredto specifya first-rank tensor
property; a symmetricsecond-ranktensor requires
six, a third-rank tensornine, and a symmetricfourthrank tensor 21. Crystallographicsymmetryreduces
the number of independentcoefficientsin keeping
with Neumann'sPrinciple: The symmetryelements
of any physical property of a crystal must include
the symmetry elementsof the point group of the
crystal. The number of independentcoefficientsfor
each point group are tabulated by Nye (1957).
Magnetic symmetry and magneticpropertiesrequire
specialconsideration(Birss, 1964).
Defining relations for several important tensor
propertiesappearingin the free energyfunction are
listed below.
ExtensiveQuantities
Entropy S
Electric polarization P;
Magnetization Ma
Strain e;;
Tensor
Intensive
Quantities
Zero-rank polar
Temperature Z
First-rank polar
Electric field Er
First-rankaxial
Magneticfieldrll;
Second-rankpolar Stresso;;
Properties
917
Defining Relation
Electric susceptibility x;i P; : x;1Ei
Magnetic susceptibility Ml : x;iHt
xij
Magnetoelectriceffect
M; : aa1E1
qii
Magnetoelasticeffect
Mt : Quxoix
Qti*
Piezoelectriceffectd;p P; : dri*oi*
Elastic complianC€r;;7"1 cij : sijktdkt
Tensor
Second-rank
polar
Second-rank
polar
Second-rank
axial
Third-rank axial
Third-rank polar
Fourth-rank
polar
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