Quantitative characterization of plastic deformation of zircon and geological implications

Abstract

The deformation-related microstructure of an Indian Ocean zircon hosted in a gabbro deformed at amphibolite grade has been quantified by electron backscatter diffraction. Orientation mapping reveals progressive variations in intragrain crystallographic orientations that accommodate 20° of misorientation in the zircon crystal. These variations are manifested by discrete low-angle (<4°) boundaries that separate domains recording no resolvable orientation variation. The progressive nature of orientation change is documented by crystallographic pole figures which show systematic small circle distributions, and disorientation axes associated with 0.5–4° disorientation angles, which lie parallel to rational low index crystallographic axes. In the most distorted part of the grain (area A), this is the [100] crystal direction. A quaternion analysis of orientation correlations confirms the [100] rotation axis inferred by stereographic inspection, and reveals subtle orientation variations related to the local boundary structure. Microstructural characteristics and orientation data are consistent with the low-angle boundaries having a tilt boundary geometry with dislocation line [100]. This tilt boundary is most likely to have formed by accumulation of edge dislocations associated with a 〈001〉{100} slip system. Analysis of the energy associated with these dislocations suggest they are energetically more favorable than TEM verified 〈010〉{100} slip. Analysis of minor boundaries in area A indicates deformation by either ${{\left[ {0\bar{1}0} \right]}}$ (001) edge, or [100](100) and [001](100) screw dislocations. In other parts of the grain, ${{\left[ {1\bar{1}0} \right]}}$ cross slip on (111), ${{\left({11\bar{1}}\right)}}$ and (112) planes seems likely. These data provide the first detailed microstructural analysis of naturally deformed zircon and indicate ductile crystal-plastic deformation of zircon by the formation and migration of dislocations into low-angle boundaries. Minimum estimates of dislocation density in the low-angle boundaries are of the order of ∼3.10¹⁰ cm⁻². This value is sufficiently high to have a marked effect on the geochemical behavior of zircon, via enhanced bulk diffusion and increased dissolution rates. Therefore, crystal plasticity in zircon may have significant implications for the interpretation of radiometric ages, isotopic discordance and trace element mobility during high-grade metamorphism and melting of the crust.

Appendices

Appendix 1: Unit quaternions

Orientations can be described by a rotation with an angle ω around an axis ${\vec{r}}$ from a reference orientation. This information can be combined into a unit quaternion

$$q = {\left({\begin{array}{*{20}c}{{q_{0}}} \\ {{\vec{q}}} \\ \end{array}} \right)} = {\left( {\begin{array}{*{20}c}{{\cos {\left({\omega /2} \right)}}} \\ {{\vec{r}\sin {\left({\omega /2} \right)}}} \\ \end{array}} \right)}$$

(A1.1)

The elements of the unit quaternion in the convention used here (Morawiec 2003) can also be found from the commonly used Euler angles ${(\varphi _{1}, \Phi, \varphi _{2})}$

$$\begin{aligned} q_{0}&={\left| {{\hbox{cos}}{\left( {\Phi {{/2}}} \right)}} \right|}{\left| {{\hbox{ cos}}{\left({{\left({\varphi _{{{1}}} + \varphi _{2}} \right)}/2} \right)}} \right|} \\ q_{1} &= - \operatorname{sgn} {\left({{\hbox{cos}}{\left({\Phi {{/2}}} \right)}{\hbox{cos}}{\left({{\left( {\varphi _{{{1}}} + \varphi _{2}} \right)}/2} \right)}} \right)}{\hbox{sin}}{\left({\Phi {{/2}}} \right)}{\hbox{cos}}{\left({{\left( {\varphi _{{{1}}} - \varphi _{2}} \right)}/2} \right)} \\ q_{2} &= - \operatorname{sgn} {\left( {{\hbox{cos}}{\left({\Phi {{/2}}} \right)}{\hbox{cos}}{\left({{\left({\varphi _{{{1}}} + \varphi _{2}} \right)}/2} \right)}} \right)}{\hbox{sin}}{\left({\Phi {{/2}}} \right)}{\hbox{sin}}{\left({{\left({\varphi _{{{1}}} - \varphi _{2}} \right)}/2} \right)} \\ q_{3} &= - \operatorname{sgn} {\left( {{\hbox{cos}}{\left({\Phi {{/2}}} \right)}{\hbox{cos}}{\left({{\left({\varphi _{{{1}}} + \varphi _{2}} \right)}/2} \right)}} \right)}{\hbox{cos}}{\left({\Phi {{/2}}} \right)}{\hbox{sin}}{\left({{\left({\varphi _{{{1}}} + \varphi _{2}} \right)}/2} \right)} \\ \end{aligned}$$

(A1.2)

Appendix 2: Calculation of dislocation energy in zircon

The line energy of a dislocation in an anisotropic medium

$$E_{L} = \frac{{Kb^{2}}}{{4\pi}}\ln (b/R)$$

(A2.1)

is determined by its Burgers vector and an effective energy factor K which depends on the elastic constants C _ij . For zircon, Burgers vectors of a-type ${(\vec{b}_{a} = [010]a)}$ are longer then Burgers vector of c-type ${(\vec{b}_{c} = [001]c)}.$ The effect of the elastic anisotropy has previously been investigated in cubic and hexagonal crystals (Foreman 1955). Under particular constraints on the elastic constants, Foreman has obtained a general result for dislocation lines along the third axis and an arbitrary Burgers vector b =  [b ₁, b ₂, b ₃ ] (cf. Fig. 13)

$$\begin{aligned} Kb&= K_{1} b_{1} + K_{2} b_{2} + K_{3} b_{3} \\ K_{1}&= (\bar{c}_{{11}} + c_{{12}}){\sqrt {\frac{{c_{{66}} {\left( {\bar{c}_{{11}} - c_{{12}}} \right)}}}{{c_{{22}} {\left( {\bar{c}_{{11}} + c_{{12}} + 2c_{{66}}} \right)}}}}} \\ K_{2}&= {\sqrt {\frac{{c_{{22}}}}{{c_{{11}}}}}}K_{1} \\ K_{3}&= {\sqrt {c44c55}} \\ \bar{c}11&= {\sqrt {c11c22}} \\ \end{aligned}$$

(A2.2)

Fig. 13

Sketch of the geometries involved (left) Foreman and (right) our geometry with dislocation line along [100]

For analyzing dislocations along the crystallographic [100] direction, the coordinate system must be rotated. The resulting matrix of elastic constants in the new system (1 = [010], 2 = [002], 3 = [100])

$${\hbox{c =}}{\left({\begin{array}{*{20}c} {{\hbox{C}}_{11}}& {{{\hbox{C}}_{13}}}& {C_{12}}& {0}& {0}& {0} \\ {C_{13}}& {C_{33}}& {C_{13}}& {0}& {0}& {0} \\ {C_{12}}& {C_{13}}& {C_{11}}& {0}& {0}& {0} \\ {0}& {0}& {0}& {C_{44}}& {0}& {0} \\ {0}& {0}& {0}& {0}& {C_{66}}& {0} \\ {0}& {0}& {0}& {0}& {0}& {C_{44}} \\ \end{array}} \right)}$$

(A2.3)

satisfies Foreman’s necessary condition (his equation (26)) and the energy factors can be obtained

$$\begin{aligned} K_{1}& = ({\bar{c}}_{11} + C_{13}){\sqrt {\frac{{C_{{44}} {\left( {{\bar{c}}_{11} - C_{13}} \right)}}}{{C_{{33}} {\left( {{\bar{c}}_{11} + C_{13} + 2C_{{44}}} \right)}}}}} \\ K_{2}&= {\sqrt {\frac{{C_{{33}}}}{C_{11}}}}K_{1} \\ K_{3}&= {\sqrt {C_{{44}} C_{66}}} \\ \bar{c}_{{11}}& = {\sqrt {C_{{11}} C_{33}}} \\ \end{aligned}$$

(A2.4)

These expressions allow calculation of the line energy of all dislocation with line vector [100] independent of their Burgers vector. The energy factors for the two possible edge dislocations become

$$\begin{aligned} K_{c}&= K_{2} \quad{\hbox{for}} \, b_{c} = [001]c \\ K_{a}& = K_{1} = {\sqrt {\frac{{C_{{11}}}}{{C_{{33}} }}}}K_{2} \quad{\hbox{for}} \,b_{a} = [010]a \\ K_{a}& = {\sqrt {\frac{{C_{11}}}{{C_{33}}}}}K_{c} \\ \end{aligned}$$

(A2.5)

Both the values differ only by a factor given by the ratio between the main elastic constants C₁₁ and C₃₃. Elastic constant data for zircon shows considerable variation (Bhimasenachar and Venkataratnam 1955; Rhyzova et al. 1966; Ozkan et al. 1974)(see Table 4), even if the discredited values of Bhimsenachar and Venkataratnam (1955) are ignored (Sirdeshmukh and Subhadra 2005). However, the conclusions for the line energy are consistent.

Table 4 Comparison of published C ₁₁ and C ₃₃ elastic properties for zircon

With the ratio ${{b_{a}} \mathord{\left/ {\vphantom {{b_{a}} {b_{c}}}} \right. \kern-\nulldelimiterspace} {b_{c}} = 6.61/5.98 = 1.1}$ between the lengths of the two Burgers vectors, the ratio between the line energies of the dislocations becomes

$$\frac{{E_{{_{a}}}}}{{E_{c}}} = \frac{{K_{a} b^{2}_{a} }}{{K_{c} b^{2}_{c}}} = 0.93 \times (1.10)^{2} = 1.14$$

(A2.6)

and the dislocations of [010]a type have higher line energy.

Furthermore, the energy of a boundary of given disorientation angle (formed by edge dislocations in spacing h)

$$E_{B} = \frac{{E_{L}}}{h} = - \frac{{Kb}}{{4\pi}}\theta \ln {\left(\theta \right)}$$

(A2.7)

depends only linearly on the Burgers vector. The ratio between the boundary energies

$$\frac{{E_{{_{a}}}}}{{E_{c}}} = \frac{{K_{a} b_{a} }}{{K_{c} b_{c}}} = 0.93 \times (1.10) = 1.02$$

(A2.8)

decreases the energy advantage of the [001]c dislocations, but the boundary energy of the [010]a type still remains larger. For this reason [001]c dislocations should be energetically favorable.