In-situ diffraction measurements of lattice response due to shock
loading, including direct observation of the α−ε phase transition in
iron

D. H. Kalantar^1\*^, G. W. Collins^1^, J. D. Colvin^1^, J. H. Eggert^1^,
J. Hawreliak^2^,

H. E. Lorenzana^1^, M. A. Meyers^3^, R. W. Minich^1^, K. Rosolankova^2^,
M. S. Schneider^3^, J. S. Stölken^1^, J. S. Wark^2^

Lawrence Livermore National Laboratory, Livermore, CA 94550, USA

Department of Physics, Clarendon Laboratory, University of Oxford,
Oxford, OX1 3PU, UK

University of California -- San Diego, La Jolla, CA 92093, USA

Received Date Line (to be inserted by Production)

Abstract

In-situ diffraction is a technique to directly probe the lattice
response of materials during the shock loading process. It is used to
record diffraction patterns from multiple lattice planes simultaneously
during shock loading. The application of this technique is described for
laser-based shock experiments. The approach to analyze in-situ
wide-angle diffraction data is discussed. This is presented in the
context of single crystal \[001\] iron shock experiments where uniaxial
compression of the bcc lattice by up to 6% was observed. Above the α−ε
transition pressure, the lattice showed a collapse along the \[001\]
direction by 15-18%. Additional diffraction lines appear that confirm
the transformation of the iron crystal from the initial bcc phase to the
hcp phase.

*Keywords*: x-ray diffraction, laser shocks, phase transformation,
dynamic lattice measurement.

# Introduction {#introduction .HVIS-Heading-1}

When a material experiences shock loading at a very low pressure, it
typically responds by deforming elastically. At a pressure above the
Hugoniot Elastic Limit (HEL), stresses that occur in the lattice
initiate and propagate dislocations \[1, 2\]. The movement of the
dislocations results in the rearrangement of the structure, which is
plastic deformation. The number density of dislocations that are
generated and the speed with which they propagate depend on the rate of
the applied strain, and the energy threshold for the dislocations, which
in turn depends on the lattice and its orientation relative to the
applied stress.

A standard technique to study the shock response of materials is to
measure the rear surface particle speed as the shock releases into
either vacuum or an impedance-matched window using a velocity
interferometer \[3\]. The rear surface velocity profile is related to
the dynamics of the lattice response, providing information on the shock
propagation in the solid under dynamic high pressure loading. The
specific wave profile depends on properties such as the HEL, the
threshold and kinetics of dislocation motion, the magnitude of the
shock, and the existence of phase transformations in the pressure range
accessed.

Identification of features in the wave profiles with physical processes
is typically based on a correlation with static high pressure
measurements such as x-ray diffraction. Because high-pressure dynamic
loading can markedly affect the structural modifications of solids, such
correlations traversing static and dynamic regimes are intrinsically
limited in nature. To capture potential complications introduced by
lattice kinetics, real-time studies of the lattice are required using a
dynamic technique such as *in-situ* x-ray diffraction \[4-6\]. *In-situ*
x-ray diffraction provides direct information on the lattice response
during shock compression.

High intensity lasers provide a capability to access high-pressure
states of material, with diagnostic capabilities that are generally not
available on other facilities. Specifically, large lasers with multiple
beams provide both a high-pressure capability and simultaneous x-ray
diagnostic capability. We have performed a series of experiments using
lasers to study the deformation of single crystals by *in situ* dynamic
x-ray diffraction \[7-9\]. The technique of *in-situ* dynamic
diffraction and how it is applied in these laser experiments is
described. Key results from shocked single crystal iron experiments are
presented as examples \[10\].

# In-situ dynamic diffraction  {#in-situ-dynamic-diffraction .HVIS-Heading-1}

*In-situ* diffraction has been implemented on laser-based high pressure
platforms to provide simultaneous measurements of multiple lattice
planes \[8, 9\]. This is a large solid angle measurement of x-rays
diffracted from several different lattice planes using a static detector
that subtends a large solid angle surrounding the shocked sample.
Measurement of the change in lattice spacing of different planes
simultaneously provides the detailed information necessary to determine
the state of the material under compression, and its deformation.

The target and experimental configuration for *in-situ* diffraction are
shown in Figure 1. The target (Figure 1a) consists of a single crystal
sample, a thin foil used as the source of x-rays for the measurement,
and a cylindrical gold shield to protect the detector from direct
illumination by the x-ray source. The single crystal sample is coated
with a 15-20 micron thick ablation layer of parylene-N and a 0.1 micron
aluminum overcoat to prevent shine through of the focused laser beam
early in the pulse. A high intensity laser pulse is used to provide the
shock loading pressure by direct laser irradiation of the parylene
ablation layer. The laser focal spot size that is typically used is 2-3
mm in diameter in order to generate a large area of uniform pressure in
the single crystal sample.

Probe x-rays are created by direct laser irradiation of a separate thin
metal foil that is positioned 1-2 mm from the crystal sample. K-shell
x-rays are emitted from the hot plasma that is created. A 200 micron
diameter pinhole aperture defines the size of the x-ray source. X-rays
passing through this aperture diverge spherically from the source and
are incident on the single crystal sample at a range of angles,
diffracting according to the Bragg diffraction condition:

$$n\lambda = 2d´\text{sin}(q)$$

![](media/image1.wmf)

1.  a\) Example target showing the crystal and backlighter foil
    components. b) Geometry for in-situ wide-angle diffraction
    measurements showing the large angle coverage of the film detector.

where *d* is the lattice spacing, *λ* is the x-ray source wavelength,
and *θ* is the angle of x-rays relative to the lattice plane, as shown
in Figure 2.

When the lattice undergoes a deformation due to the shock loading, the
spacing of the lattice planes change. Provided the range of incidence
angles for the x-rays is large enough to meet the Bragg condition for
both the uncompressed and compressed lattice spacings, x-rays will
diffraction from both lattice configurations. Since the source of x-rays
is located close to the single crystal sample, the Bragg condition may
be met for several different lattice planes simultaneously.

A large area segmented film holder is positioned surrounding the target,
subtending up to a full π-steradian solid angle (Figure 1b). This
detector intercepts x-rays that are diffracted from the many different
lattice planes, resulting in an image of diffraction curves, such as
shown in Figure 3. Here, diffraction curves from different lattice
planes of shocked single crystal \[001\] iron recorded at a range of
loading pressures are shown. By varying the timing of the probe relative
to the pressure drive, diffraction curves from both the unshocked and
shocked lattice may be recorded on the same film. Several curves in the
images in Figure 3 show shifted features due to the shock loading of the
crystal. However, not all curves show evidence of compression since
these curves result from x-rays diffracting from different spatial
locations on the sample. When these regions fall outside the area on the
crystal that is shock loaded, the shifted curves do not appear.
Illustration of this is discussed below.

![](media/image2.wmf)

2.  Geometry for *in-situ* Bragg diffraction. X-rays are incident at a
    wide angle, satisfying the Bragg condition for different lattice
    spacings at different locations in the crystal. The diffracted
    signal shifts at the detector due to compression of the lattice.

![](media/image3.wmf){width="6.3125in" height="4.0in"}

3.  Example film images showing multiple plane diffraction from single
    crystal \[001\] iron recorded at laser intensities of a) 2.0x10^10^
    W/cm2^2^at 532 nm (1.3 GPa), b) 6.0x10^10^ W/cm2^2^at 532 nm (3.8
    GPa), and c) 3.1x10^10^ W/cm2^2^at 351 nm irradiation (26 GPa). The
    diffraction lines from the static lattice are identified with
    squares, lines from the uniaxial compressed lattice are identified
    with triangles, and lines corresponding to the hcp phase are
    identified with crosses.

# Approach for analysis {#approach-for-analysis .HVIS-Heading-1}

In the example of diffraction from a lattice plane that is parallel to
the surface of the crystal, such as the (002) plane of \[001\] Fe, the
Bragg condition is met on a circular region on the surface of the
crystal, and a cone of x-rays is diffracted, intercepting the film in a
conic section. However, more generally the lattice planes are not
parallel to the surface of the crystal. The effective probe depth for
the x-rays is typically only a few microns. As a result, the region on
the crystal that meets the Bragg condition cannot be approximated by a
single cone of x-rays. As a result, it is necessary to include the
detailed source and sample geometry in interpreting the film images.

Software routines have been developed to calculate the diffraction
pattern for the laser experiments where the monochromatic x-ray source
is located close to the crystal. One such approach is described here. In
this case, the crystal is represented by a grid of points. A bundle of
rays from the x-ray source to the grid of points on the crystal surface
(pixels) are calculated for a given lattice plane defined by *h, k, l*
indices. The direction of reflection off the lattice plane is calculated
for each ray. These rays are then propagated to the film plane. The
intersection of these rays with the film plane are calculated in
coordinates representing position in the plane of the film.

An image is created to represent the locations on the crystal where the
Bragg diffraction condition is met. This is defined with an intensity Y:

Y = exp( - (nλ - 2 d sin(θ))^2^/a^2^).

where *λ* is the wavelength of probe x-rays, *d* is the lattice spacing,
and *θ* is the angle of the x-rays relative to the lattice plane. A
representative linewidth *a* is applied to generate narrow curves on the
crystal corresponding to where the diffraction occurs (where *θ* is
equal to the Bragg angle). This image has a value of 1 at locations on
the crystal where the Bragg condition is met, and 0 where it is not.
This map of where diffraction occurs on the crystal is then warped to
the film plane coordinates as defined by the ray tracing from each point
on the crystal, resulting in a simulated film image of diffraction from
the given *h, k, l* lattice plane. This is repeated for the different
*h, k, l* indices that are allowed for the given crystal symmetry to
create a simulated diffraction film image.

An example is shown for single crystal iron in Figure 4. Here, we show
the calculated image due to 1.85 Å x-rays that are diffracted from the
(002) and (±1±12) and (±101) lattice planes of static (uncompressed)
\[001\] iron crystal. The image showing where diffraction occurs on the
crystal is shown in Figure 4a, and the corresponding simulated film
image is shown in Figure 4b.

Diffraction patterns are calculated for the static lattice, and compared
with the experimental film images in order to determine the specific
geometry of the target and film for each experiment. Patterns are then
calculated assuming different lattice deformations to compare with the
diffraction curves due to the compressed lattice. In order to make this
comparison, a range of different configurations may be considered by
varying the unit vector lengths and rotation of the crystal in the
simulation.

In the example of \[001\] Fe, the (002) planes are parallel to the shock
loading direction. The compression of the lattice in this direction is
determined directly from the shift of the diffraction curve from the
(002) planes. Determination of whether the lattice compression is purely
uniaxial or hydrostatic is based on comparison of the film data with
simulated images that assume compression of the lattice in the
perpendicular directions as well. The pattern from static (uncompressed)
iron is shown in Figure 5a, with additional features overlaid that were
calculated based on uniaxial compression of the initial bcc lattice by
0.95 along \[001\]. The same image calculated assuming that the crystal
is compressed by 0.95 along all three lattice directions is shown in
Figure 5b. This would be the case for a lattice that plastically
deformed due to the generation and propagation of dislocations.

![](media/image4.wmf)

4.  a\) Simulated image showing locations on the crystal where
    diffraction occurs from several different lattice planes. b)
    Simulated film image showing the locations of the diffraction
    curves. These sample images are simulated for a single crystal
    \[001\] iron lattice with iron K-shell x-rays at 1.85 Å.

The test cases that are shown in Figure 5 illustrate the sensitivity of
the shifted diffraction lines to the lattice deformation. The relative
shift of the diffraction curves from different lattice planes provides a
quantitative measure of the overall lattice deformation -- whether it
undergoes uniaxial compression, or there is a relaxation that results in
compression perpendicular to the shock direction. Comparison of the
relative shifts of the different diffraction curves may also indicate a
rotation of the lattice, and the appearance or disappearance of
diffraction curves may be an indication of a phase change in the
structure of the lattice.

This approach of simulating the diffraction pattern based on initial
assumptions of the lattice configuration provides insight into the
response of a material to pressure loading. A more general analysis
capability is being developed where the peak positions of each
diffraction curve are identified, and the overall lattice deformation is
calculated based on an optimum fit.

![](media/image5.wmf){width="6.125in" height="3.75in"}

5.  a\) Simulated film image showing diffraction from a) static and
    uniaxially compressed bcc lattice, and b) static and hydrostatically
    compressed bcc lattice.

# Shock experiments in single crystal iron {#shock-experiments-in-single-crystal-iron .HVIS-Heading-1}

The technique of *in-situ* dynamic x-ray diffraction has been used to
probe shocked single crystals with x-rays as they undergo deformation by
shock compression. Experiments have been done previously to study
shocked single crystals of silicon and copper \[7, 9\]. Recent
experiments were done to study the deformation of single crystal iron
\[10\]. We present analysis of the shocked iron experiments to
illustrate the analysis of the in-situ diffraction measurement. In these
experiments, features in the diffraction data are consistent with an
initial uniaxial compression and subsequent collapse and transformation
from the bcc phase to the hcp phase.

Previously, the presence of a phase transformation in shock-loaded iron
was identified at 13 GPa based on the observation of structure in wave
profile measurements \[11, 12\]. Separately, a solid-solid phase
transformation in iron under static loading conditions was observed at
the same pressure \[13\]. The correlation of the static measurement with
the dynamic loading result was taken as evidence that iron undergoes
this solid-solid transition on shock loading. This is supported by
microstructural analysis of the residual deformation in recovered
samples \[14, 15\], but direct confirmation of this transition during
shock loading requires *in-situ* measurements.

Single crystal iron samples were prepared with a 16 µm thick parylene
ablation layer, and a 0.1 micron overcoat of aluminum. A single beam of
OMEGA \[16\] or Janus was used to irradiate the single crystal samples
with a 2-6 ns flat-in-time laser pulse shape. Additional laser beams
were used to create a bright source of x-rays by direct laser
irradiation of a thin iron foil located 1.3 mm from the center of the
crystal. X-rays diffracted from the shocked single crystal were recorded
with static film. Additional experiments were conducted at the Vulcan
laser \[17\], with similar thick crystals, as well as with much thinner
10 micron single crystal iron foils. \[10\]

The single crystals were subjected to a high intensity pressure drive by
direct laser irradiation at intensities of approximately 2x10^10^
W/cm^2^ to 5x10^11^ W/cm^2^ at 351-1056 nm wavelength. At these low
intensities, LASNEX \[18\] simulations of the laser interaction with the
Al-coated CH suggested that the pressure scales nearly linearly with
intensity. In addition, LASNEX simulations with 351 nm and 532 nm laser
wavelength provided a scale factor of 0.76 to relate the 532 nm laser
intensities to the 351 nm intensities and a scale factor of 0.30 to
relate the 1054 nm laser intensities to the 351 nm intensities. The
absolute pressures achieved in these experiments were determined based
on a third order elasticity constants. \[10\]

The deformation of the iron lattice was recorded at a range of laser
intensities. Three examples of the diffraction data are shown in Figure
3. In each image, there are multiple diffraction curves present, each
resulting from x-rays diffracted from a single lattice plane. Several of
the lattice planes show both static and compressed features. Since the
crystals were cut with a \[001\] orientation, the shift of the (002)
diffraction provide a direct measurement of the lattice compression
along the shock propagation direction.

The precise geometry for the source, crystal, and film planes were
determined for each image to match the diffraction lines from the
uncompressed lattice planes. The relative shift of the diffraction from
the different lattice planes were then compared with calculations such
as those shown in Figure 5 under a range of lattice deformation
conditions. The compression of different lattice planes observed in
Figures 3a and 3b are consistent with an overall uniaxial compression of
the initially bcc lattice. The shift of lattice planes that are at an
angle with respect to the shock direction are consistent with an overall
compression of the lattice only along the \[001\] direction as indicated
by the shift of the (002) diffraction.

In the case of the film image recorded at higher pressure shown in
Figure 3c, multiple compressions are evident. Several lattice planes
show a shift that is consistent with a nearly uniaxial compression of
the lattice by 6% along the \[001\] direction (parallel to the shock).
Additional broader curves appear at a larger offset. Most of these
additional diffraction curves are consistent with 15-18% compression of
the lattice along the shock direction. However, one additional
diffraction curve appears that cannot be explained with the symmetry of
the compressed bcc lattice. This additional diffraction curve is
consistent with diffraction from the (1-210) plane of hcp iron. Other
film images were recorded showing other hcp planes, confirming that iron
undergoes a solid-solid transformation on the ns time-scale of these
laser shock experiments.

The relative positions of the different diffraction lines from the hcp
lattice constrain the lattice unit cell. In Figure 3c, the collapse of
the initially bcc lattice is evident by the broad shifted lines adjacent
to the (002) and (101) diffraction curves. These are consistent with a
planes in uniaxially compressed bcc and are also identified with the hcp
structure. The combination of these diffraction curves and the
additional appearance of the additional (1-210) hcp line are consistent
with the uniaxial collapse of the bcc structure and shuffle of alternate
\[110\] planes of atoms. \[10\]

The measured compression of the lattice along the \[001\] direction is
plotted as a function of laser intensity in Figure 6. In this figure,
the compression was determined from the shift of the (002) diffraction
curve. The lattice deformation is uniaxial along the shock direction at
low intensity and the plane spacing decreases with increasing laser
intensity (pressure) to a maximum relative compression of 0.94.

![](media/image6.wmf)

6.  Compression of the lattice as a function of laser irradiation
    intensity (pressure) determined by the shift of the (002)
    diffraction curves. For images that showed low and high compressions
    on the same film record, both compressions are indicated. The open
    squares represent the lower compression

At laser intensities above approximately 1.5x10^11^ W/cm^2^, two shifted
diffraction curves are evident in the film images. The uniaxial
compression feature maintains a fixed shift corresponding to
approximately 0.94 compression for a range of laser intensities. The
second shifted curve is consistent with 15-18% compression of the
initial bcc lattice. Both compressions are shown in the figure. The open
squares represent the low compression observed in images that also
showed a corresponding higher compression. This discontinuous behavior
is consistent with a volume collapse and transformation of the initially
bcc structure to the hcp structure. The presence of both compressions
simultaneously is assumed to be due to a 2-wave structure in the shock.
The x-ray measurements are time-integrated over 1-3 ns, during the shock
loading of the iron crystal.

# Pressure calibration {#pressure-calibration .HVIS-Heading-1}

The lattice compression measurements shown in Figure 6 are plotted both
as a function of intensity and pressure. The pressure scale that is
shown is based on third order elasticity constants. At low intensity
where only a single compression of the (002) planes was observed, the
shift of the diffraction curves from each lattice plane was consistent
with an overall uniaxial compression of the lattice. Third order
elasticity constants for iron \[19, 20\] were used to determine the
pressure corresponding to the uniaxial lattice deformation for each low
pressure image. The average scale factor from laser intensity to
pressure was applied to determine a pressure for each data point. This
mean scale factor had a 10% statistical error.

The pressures scale determined based on third order elasticity constants
is consistent with the shock Hugoniot for iron and the phase
transformation at 13 GPa. Free surface wave profile measurements of
these shocked crystals were made, such as shown in Figure 7. This shows
an example wave profile measurement from a 250-270 micron thick single
crystal \[001\] iron sample. This is a spatially averaged lineout from a
line imaging VISAR. The release shows a two-wave structure due to the
elastic and plastic waves. The elastic response is enhanced relative to
the HEL for iron, due to the low initial dislocation density and
kinetics of generating and propagating dislocations. Modeling of these
waveforms is ongoing using the LASNEX hydrodynamics code and also using
a kinetics model. \[21, 22\]

![](media/image7.wmf)

7.  VISAR free surface wave profile of the breakout from \[001\] single
    crystal iron.

# Summary {#summary .HVIS-Heading-1}

Shock experiments provide information on the mechanical properties of
materials, including the elastic response, plastic deformation, equation
of state, and phase transformations. Laser based shock experiments have
been developed to characterize the response of materials under high
pressure and high strain rate conditions.

One diagnostic technique that has been applied to laser shock
experiments is *in-situ* diffraction. This technique utilizes a laser
produced plasma as a source of monochromatic x-rays that are incident on
a sample crystal that is shock loaded by direct laser irradiation.
X-rays that diffract from the shocked sample are recorded during shock
loading. The amount of shift of the different lattice planes provides
information on the deformation of the lattice as it deforms. Time
resolution of these measurements provides information on the dynamics of
the lattice deformation.

The technique of *in-situ* diffraction has been used to study the
response of single crystal \[001\] iron to shock loading. At low
pressures, the iron responds with a nearly uniaxial deformation with a
compression of up to 6% along the shock direction. This maximum
compression corresponds to shock loading just below the α−ε
transformation pressure of 13 GPa. Above this pressure, both a uniaxial
compression by 6% and a collapse and transformation to hcp are observed.
Both states are observed due to the 2-wave response of the lattice.
These results provide direct confirmation of the hcp phase during the ns
time-scale of the laser shock experiments.

# Acknowledgements {#acknowledgements .HVIS-Reference-Heading}

This work was conducted This work was conducted under the auspices of
the US DOE by the University of California Lawrence Livermore National
Laboratory under Contract No. W-7405-Eng-48. Experiments were conducted
at the University of Rochester Laboratory for Laser Energetics under the
NLUF grants program. Additional support was provided by the Department
of Energy under Grants DEFG0398DP00212 and DEFG0300SF2202, by the UK
EPSRC under grant GR/R25699/01, and by LDRD 04-ERD-071 at LLNL.

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