Quantifying strain birefringence halos around inclusions in diamond

Abstract

The pressure and temperature conditions of formation of natural diamond can be estimated by measuring the residual stress that an inclusion remains under within a diamond. Raman spectroscopy has been the most commonly used technique for determining this stress by utilising pressure-sensitive peak shifts in the Raman spectrum of both the inclusion and the diamond host. Here, we present a new approach to measure the residual stress using quantitative analysis of the birefringence induced in the diamond. As the analysis of stress-induced birefringence is very different from that of normal birefringence, an analytical model is developed that relates the spherical inclusion size, R
i, host diamond thickness, L, and measured value of birefringence at the edge of the inclusion, \Updelta n(R_{\text{i}} )_{\text{av}} , to the peak value of birefringence that has been encountered; to first order \Updelta n_{\text{pk}} = (3/4)(L/R_{\text{i}} ) \, \Updelta n(R_{\text{i}} )_{\text{av}} . From this birefringence, the remnant pressure (P
i) can be calculated using the photoelastic relationship \Updelta n_{\text{pk}} = - (3/4)n^{3} q_{\text{iso}} P_{\text{i}} , where q
iso is a piezo-optical coefficient, which can be assumed to be independent of crystallographic orientation, and n is the refractive index of the diamond. This model has been used in combination with quantitative birefringence analysis with a MetriPol system and compared to the results from both Raman point and 2D mapping analysis for a garnet inclusion in a diamond from the Udachnaya mine (Russia) and coesite inclusions in a diamond from the Finsch mine (South Africa). The birefringence model and analysis gave a remnant pressure of 0.53 ± 0.01 GPa for the garnet inclusion, from which a source pressure was calculated as 5.7 GPa at 1,175°C (temperature obtained from IR analysis of the diamond host). The Raman techniques could not be applied quantitatively to this sample to support the birefringence model; they were, however, applied to the largest coesite inclusion in the Finsch sample. The remnant pressure values obtained were 2.5 ± 0.1 GPa (birefringence), 2.5 ± 0.3 GPa (2D Raman map), and 2.5–2.6 GPa (Raman point analysis from all four inclusions). However, although the remnant pressures from the three methods were self-consistent, they led to anomalously low source pressure of 2.9 GPa at 1,150°C (temperature obtained from IR analysis) raising serious concerns about the use of the coesite-in-diamond geobarometer.

Full text

Extended Abstract
1
9
th
International Kimberlite Conference Extended Abstract No. 9IKC-A-00282, 2008
Quantifying Strain Birefringence Haloes Around
Inclusions in Diamond
Dan Howell
1
, Adrian Jones
1
, David Dobson
1
, Ian Wood
1
,
Lutz Nasdala
2
& Jeff Harris
3
1
Earth Sciences Department, University College London, UK
2
Institute of Mineralogy & Crystallography, University of Vienna, Austria
3
Department of Geographical & Earth Sciences, University of Glasgow, UK
Birefringence in diamond has long been recognized as
an anomalous optical property for a cubic mineral
(Lang, 1967). Various causes have been postulated for
this phenomenon but they all identify birefringence as
being the result of the photoelastic effect (change of
refractive index with stress) (Poindexter, 1955). Until
now birefringence studies (a non-destructive technique)
have seldom been utilized because of the time
consuming nature of the analysis. The MetriPol system
is a new automated birefringence analysis system that
can produce very accurate data in a matter of seconds
(Glazer et al. 1996).
When analyzing stress-induced birefringence, it is
important to consider the difference from normal
birefringence. Whilst normal birefringence is the result
of constant anisotropy throughout the sample, stress-
induced birefringence is the result of anisotropy that is
governed by the stress field. This means that a beam of
light that passes through the radial stress field around
an inclusion under remnant pressure will not record the
peak value of anisotropy that it encounters but, in
effect, a value that is an average over the entire
thickness of the diamond. This means that it is
important to understand the relationship between the
inclusion size, diamond thickness, remnant pressure
and recorded value of birefringence before any analysis
can be interpreted with confidence. To quantify this
relationship a theoretical model has been developed.
All studies into quantifying the elastic effects around
inclusions in diamond that remain under remnant
pressure are based upon a number of assumptions. In
this study these assumptions are:
(i) The inclusion is spherical.
(ii) Both the host and inclusion have cubic
crystal symmetry.
(iii) When formed, the diamond fits the inclusion
perfectly with no interfacial region of
different composition or structure.
(iv) Deformation is linear elastic and no brittle or
plastic deformation has occurred.
(v) The remnant pressure of the inclusion is the
only cause of strain in the diamond lattice
that results in birefringence.
The model uses an inclusion (radius = R
i
) under
remnant pressure (P
i
) in the centre of a semi-infinite
slab of diamond of thickness L. The remnant pressure
creates a radial stress field in the host diamond that
decays as 1/r
3
. By splitting up a 2D section, that runs
through the centre of the inclusion, into s number of
slices (Fig. 1), an optical indicatrix (related to the radial
stress regime) can be calculated for a point in the
centre of each slice. For a beam of light passing
through the slab at a distance, d, from the centre of the
inclusion, the retardation is calculated by summing the
retardation from the indicatrix in each slice (1 to s).
The peak value of retardation will be obtained when
the light passes immediately next to the inclusion-
diamond interface (when d=R
i
), and will decrease as it
moves away from it (d>R
i
).
Fig. 1: schematic image explaining the parameters of
the theoretical model.
It is clear that the measured birefringence value
(through the radial stress regime) will be significantly
less than a theoretical maximum birefringence value
obtained from a stress that is equal to the remnant
pressure of the inclusion. Using several different values
for each of the 3 main variables (R
i
, L & P
i
), the model
has been used to calculate the peak measured
birefringence (when d=R
i
). This model has defined a
relationship that involves all of these 4 variables, so
that the remnant pressure on an inclusion can be
estimated from the birefringence measurements.

Extended Abstract
2
To test the validity of this relationship, a variety of
analyses have been performed on diamonds with
inclusions in. Techniques including X-ray analysis
(Harris et al., 1970), 2D Raman mapping of the host
(Nasdala et al., 2003; 2005), and Raman spot analysis
of the inclusion (Liu et al., 1990, Izraeli et al., 1999)
have already been documented for measuring remnant
pressure on inclusions in diamond. So the results from
these techniques have been used to compare with the
results from MetriPol birefringence analysis system
and the theoretical model.
There are a number of criteria that govern the use of
diamond samples in this study. These are not only
related to the assumptions already listed, but also due
to the limitations of the MetriPol system and sample
preparation capabilities. The birefringence analysis
requires 2 parallel faces to view through. This means
that natural diamonds with good octahedral faces can
be utilized. However, it is rare to find samples in which
the birefringence is only caused by an inclusion under
remnant pressure, and not by plastic deformation,
variations in nitrogen concentration, or dislocations
(Lang, 1967). The diamond should also only have 1
inclusion in (to prevent any overlapping stress fields
complicating the interpretation). Garnets are the
preferred mineral inclusion due to their cubic crystal
symmetry. This means that cubic minerals will apply
an equal stress to the host diamond along each of its 3
main crystallographic axes.
From many collections of diamonds sampled, 3 natural
octahedra containing garnet inclusions were chosen
from Udachnaya, Siberia (sample numbers UD2, 5 &
7). A prepared diamond plate containing 4 coesite
inclusions from Finsch, South Africa (F125), has also
been analyzed. Despite being monoclinic, coesite is
interesting due to its calibrated pressure sensitive
Raman peak (Hemley, 1987) and the resultant
extraordinarily high levels of birefringence observed.
The birefringence seen in these samples is shown in
Figures 2, 3 & 4.
Using the MetriPol birefringence analysis and the
model described above suggests that the garnet in UD7
is under 0.53 ± 0.06 GPa of remnant pressure. Without
liberating the garnet inclusions from the host diamond,
it is not possible to verify this result. The results from
the liberated inclusions will be presented at the
conference. This is not so simple to do for samples
UD2 & 5, as the birefringence patterns in them are far
more complex. Figure 4 shows that the birefringence is
not just focused around the inclusion but pervasive
throughout the entire sample. In UD2 the birefringence
pattern seems to be following the growth stratigraphy
of the stone, while in UD5 the pattern suggests that it
has been plastically deformed (near Type II diamond).
For quantitative purposes, this highlights the
importance of focusing on birefringence that is only
caused by the inclusion. This makes finding suitable
Fig. 2: Images of UD7 taken under crossed-polars (A
& C) and the MetriPol system (B & D).
Fig. 3: Images of F125 taken under crossed-polars (A)
and the MetriPol system (B).
Fig. 4: Images of UD2 (A & B) and UD5 (C & D),
show how growth zoning (UD2) and plastic
deformation (UD5) can overwhelm the birefringence
caused by remnant pressure on an inclusion.
samples very difficult, as the other causes of
birefringence are quite common. For example, the
entrapment of inclusions can result in dislocations in
the host diamond due to lattice closure errors, fractures
are commonly seen around inclusions, the diamond
may have a complex growth stratigraphy caused by
varying levels of nitrogen being incorporated into the
lattice, and the sample may undergo plastic
deformation due to non-hydrostatic stresses in the
mantle. All of these possibilities need to be considered
when interpreting the quantitative birefringence
analysis with confidence.

Extended Abstract
3
Applying the birefringence analysis and theory to the
largest coesite inclusion in F125 gives a remnant
pressure of 2.45 ± 0.44 GPa. This is in agreement with
the result from the Raman peak shift of 2.55 ± 0.35
GPa. This result suggests that as a first order
approximation, the model can be applied to non-cubic
mineral inclusions with good agreement. However, to
be able to place a high level of confidence on the
results of uniaxial and biaxial inclusions will require
understanding of the effect of anisotropy on the radial
stress field. Another important factor that may have an
effect on the stress field is the shape of the inclusion.
Finite element analysis calculations are currently being
performed to assess these.
In conclusion, these preliminary results suggest that
quantitative birefringence analysis with the MetriPol
system is a very useful technique for assessing remnant
pressure on inclusions in diamond. The rapid nature of
the technique and the near isotropic nature of the effect
suggest that this method may provide a comparable
alternative to 2D Raman mapping. However, we are
aware that many more analyses are required to support
there results. This quantitative birefringence analysis is
currently also being successfully used to investigate the
other causes of birefringence in diamond to assess the
stresses and strains that they produce.
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