Piezoelectricity in Quartz

Mechanical stress creates electric displacement in a piezoelectric crystal. This is the direct piezoelectric effect. An electric field can also create strain, which is the converse effect.

The direct effect relates a symmetric stress tensor \(\sigma\) to electric displacement \(D\) through the rank three piezoelectric strain tensor \(d\):

\[ D_i = d_{ijk}\,\sigma_{jk}. \]

This page assumes the tensor rank and compact notation introduced in Defining Tensorial Properties. Read Importing Tensor Data first for units, crystal frames, and the doubleConvention used by the file below.

plottingConvention.default('y↑→x');

% define the quartz crystal symmetry and crystal frame
csQuartz = crystalSymmetry('32',[4.916 4.916 5.4054],...
  'X||a*','Z||c','mineral','Quartz');

% load the right-handed quartz piezoelectric strain tensor
quartzFile = fullfile(mtexDataPath,'tensor','Single_RH_quartz_poly.P');
P = tensor.load(quartzFile,csQuartz,...
  'propertyname','piezoelectric strain','unit','pC/N',...
  'doubleConvention');

setMTEXpref('defaultColorMap',blue2redColorMap);

Why crystal symmetry matters

A material-property tensor must be invariant under every operation of its crystal point group. Inversion changes the sign of a polar rank three tensor. A centrosymmetric crystal must therefore have \(d=0\).

Piezoelectricity is allowed in 20 of the 32 crystallographic point groups: every noncentrosymmetric group except 432. Quartz belongs to point group 32, so symmetry permits the effect and repeats it about the threefold crystal axis.

Handedness remains important even though the point-group symbol is the same. Inverting right-handed quartz produces left-handed quartz and reverses the piezoelectric tensor. The filename identifies P as the right-handed form.

The signed longitudinal response

directionalMagnitude contracts the same unit direction \(n\) into all three indices:

\[ q(n) = d_{ijk}\,n_i n_j n_k. \]

For a uniaxial stress along \(n\), \(q(n)\) is the electric-displacement component along that direction per unit stress. It is a signed longitudinal coefficient, not the magnitude of the displacement vector.

q = P.directionalMagnitude;

% plot one symmetry-reduced sector
plot(P);
mtexColorbar('title','longitudinal coefficient (pC/N)');

The sector contains both positive and negative response. Blue and red therefore mean opposite signs of the longitudinal component, not weak and strong polarization.

A rank three directional response is odd: \(q(-n)=-q(n)\). Plotting both hemispheres makes that sign reversal explicit.

close all;
plot(P,'complete','smooth','upper','lower');
mtexColorbar('title','longitudinal coefficient (pC/N)');

The same threefold pattern occurs on the two hemispheres with red and blue exchanged at antipodal directions. This is the sign reversal expected for an odd-rank tensor.

A radial surface

surf can use the absolute response as distance from the origin and the colour as its sign. The 'noScaling' option keeps the physical zero and the original pC/N values.

close all;
surf(q,'noScaling');
mtexColorbar('title','longitudinal coefficient (pC/N)');

The six lobes have the same radial magnitude in antipodal pairs, while their colours have opposite signs. The surface meets the origin in directions where the longitudinal response is zero.

Planar sections

plotSection draws the signed response as a polar radius in a chosen plane. A negative radius is placed in the opposite direction, so each curve is traced twice. Use the coloured hemisphere plots above, rather than these outlines, to read the sign.

The basal plane is normal to z.

close all;
plotSection(q,vector3d.Z);
xlabel('x');
ylabel('y');
drawNow(gcm);

The basal section is a three-petal rose. Its threefold repetition is the clearest planar expression of quartz point group 32.

A vertical section normal to x has a different outline.

close all;
plotSection(q,vector3d.X);
ylabel('y');
zlabel('z');
drawNow(gcm);

This section is a single oval rather than a three-petal rose because its plane contains the threefold z axis instead of cutting across it.

A polycrystal average needs handedness

The Tongue quartzite data contain one orientation for each of 382 grains. Mainprice, Lloyd, and Casey (1993) explain a decisive limitation of these measurements: routine electron-channelling patterns did not determine the handedness of each quartz grain. Every grain was indexed as right-handed. The paper therefore states that piezoelectricity cannot be calculated from this orientation set.

orientationFile = fullfile(mtexDataPath,'orientation',...
  'Tongue_Quartzite_Bunge_Euler');
ori = orientation.load(orientationFile,csQuartz,...
  'ColumnNames',{'Euler 1','Euler 2','Euler 3'});

orientationCount = length(ori)

orientationCount =
   382

The calculation below is still instructive as a counterexample. It uses calcTensor to rotate the right-handed tensor by every orientation and take their unweighted arithmetic mean. The result is an apparent aggregate in which all 382 grains have been assumed to be right-handed. It is not a prediction for the rock.

apparentMean = ori.calcTensor(P);
qApparent = apparentMean.directionalMagnitude;

close all;
plot(apparentMean,'complete','smooth','upper','lower');
mtexColorbar('title','apparent longitudinal coefficient (pC/N)');

responseRanges = array2table(...
  [min(q),max(q);min(qApparent),max(qApparent)],...
  'VariableNames',{'minimum','maximum'},...
  'RowNames',{'single crystal','all right-handed average'})

responseRanges =
  2×2 table
                                minimum     maximum
                                ________    _______
    single crystal                  -2.3        2.3
    all right-handed average    -0.59399    0.59399

The single-crystal range is -2.3000 to 2.3000 pC/N. Under the deliberately false all-right-handed assumption, the range is -0.5940 to 0.5940 pC/N, or 25.8 percent of the single-crystal extreme.

Differently oriented grains partly cancel, which explains the reduction. Unknown left-handed grains can reverse additional contributions. Their number and orientations are absent from this data set, so the apparent average cannot be corrected without new handedness information.

Next

Birefringence continues with a rank two optical property, whose even rank makes its directional response antipodal. Tensor Averages develops Voigt, Reuss, and Hill estimates for elastic stiffness and explains their mechanical assumptions.

Further reading

setMTEXpref('defaultColorMap',WhiteJetColorMap);