CP620, Shock Compression of Condensed Matter - 2001
edited by M. D. Furnish, N. N. Thadhani, and Y. Horie
© 2002 American Institute of Physics 0-7354-0068-7/02/$ 19.00
PICOSECOND TIME-RESOLVED X-RAY DIFFRACTION :
ESTIMATION OF LOCAL PRESSURE
Yoichiro Hironaka, Fumikazu Saito, Akio Yazaki, Kazutaka G. Nakamura,
and Ken-ichi Kondo
Materials and Structures Laboratory, Tokyo Institute of Technology, 4259 Nagatsuta, Midori, Yokohama 2268503, Japan
Abstract We have performed time resolved X-ray diffraction experiments with picosecond time
resolution on Si single crystal compressed by laser irradiation. From the measured diffraction profiles,
temporal and spatial distribution of the strain in the sample have been estimated using the direct search
of optimization method based on the dynamical X-ray diffraction theory. The maximum compression
of 1.05% was measured at the irradiation power density of 4.7X109 W/cm2. We discussed pressure
distribution analyzing observed data.
we perform pump and probe X-ray diffraction
experiment for investigating the dynamics of a laser
irradiated Si (111) single crystal[5]. Strain profiles
and EOS are also obtained.
INTRODUCTION
The time resolved X-ray diffraction from shock
compressed samples can give important information
on the dynamics of transient phenomena such as
phase transition and shock induced plasticity.
Transient X-ray diffraction for shock compressed
material has been studied using plate impact and
flash X-ray technique [1-3]. However in order to
investigate mechanisms of atomic motion induced
by the shock wave front, more severe temporal and
spatial resolutions on measurement will be
required[4]. In particular, time-resolved recording
on experiment is needed for detail of the dynamics
in transient phenomena.
Recently, ultra short pulse X-rays can be
generated by high intense femtosecond laser
irradiation on metals. This enable to perform ultra
fast X-ray diffraction of shocked material by
combining with laser shock technique. In this paper,
EXPERIMENT AND RESULTS
The pulsed X-rays (Fe Kal and Ka2 ) were
generated by irradiation of femtosecond laser on the
Fe target. The pulse width of the X-rays were
measured to be 10 ps. Shock was generated by
picosecond pulse (300 ps, 780 nm) was focused on
the Si (111) wafer with the power density of
4.7X109 W/cm2. The Si wafer was slightly
translated to ignore the effect of the damage during
the accumulation of signal (600 shots). Figure 1
shows the results of diffraction patterns obtained at
every 60 ps. At the early time of laser irradiation
(0~300 ps) the new peak is grown up at the larger
angle. This corresponds to shock-compression of Si.
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theory for the analysis of Figure 1, by assuming
uniaxial strain perpendicular to the surface.
According to the dynamical X-ray diffraction
theory, the depth dependent scattering amplitude
from a strained crystal is expressed as formula
m
*m
I"
slY
—— = (1 + ik)X2 - 2(y + ig)X + l + ik
dA
fg
A ==
mi sis u ^i MI MJ an mi
f/Fsin0B,* = ^_,A = -f
fg
(1)
J
S
f
A is therefore a dimensionless measure of the depth
t relative to the crystal surface. re and A, are
classical electron radius and X-ray wave length,
respectively. V is the unit cell volume, and structure
factor is / = f^g + if^g, e(t) means strain at depth
t. The reflecting power on the surface of strained
crystal is calculated by analytical solution of
equation (1) and layered assumption [7]. We
separated 100 layers within lOjjm in depth.
Information of each layer which has constant strain
is expressed by the 4x4 real matrix (equation(2)).
(2)
AR =
Fsin<9
2sin(2AR)
FIGURE 1. The results of Time resolved X-ray diffraction
profile for laser perturbed Si(l 11) single crystal at every 60ps.
e2AI + e~2AI + 2cos(2AR)'
DISCUSSIONS
Strain Profile Analysis
e2AI
TI =
e
2AI
_e-2Al
+e~
2AI
+2cos(2AR)
where 8j means thickness of y'-th layer.
We used DSOM (Direct Search of Optimization
Method) based on the dynamical X-ray diffraction
SR and SI
2
are real and imaginary part of y(y + ig) - (1 + Ik)2 ,
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Figure 2 shows the time-evolution of the obtained
strain profile. The maximum compression of 1.05%
is obtained at the delay time of 180ps. The induced
wave has no steady state because the temporal
profile of irradiated laser is Gaussian. After the end
of laser irradiation (~300ps), expansion wave starts
to propagate from the surface and caches up the
wave.
respectively. The reflecting amplitude on the
surface of crystal is calculated by equation (3).
r=
(3)
EOS Determination
Here, we can translate strain distribution to the
density distribution under the uniaxial assumption
directory. Usually, we use three conservation laws
and equation of state for the calculation of flow.
The conservation of mass and momentum gives
new particle velocity and density. The energy
conservation law and EOS give P-V relation. Thus
we can calculate new pressure value using new
density value. Here, we note, the form of
conservation of mass and momentum are
independent from the ensemble (equation (4), the
form of energy conservation law depends on the
ensemble).
R(0) is the reflecting intensity at diffracted angle of
0. The vector % is the boundary condition at the
bottom of layers. The components of % are (0,0,0,1)
for the case the rear surface of crystal contact with
vacuums. For the case of infinite crystal, boundary
condition of (XI, XR, 0, 1) are used. XI and XR
mean imaginary and real part of reflecting
amplitude from the perfect crystal.
In the equation (3), we optimized value of each
matrix of ^ using DSOM to the observed data Thus,
we obtain strain distribution inside of the crystal.
— =0 (mass)
(4)
-6.0
^T +"hH +hH = ° (momentum)
-5.5
-5.0
Delay time=180 ps
Using finite differential method on first order
assumption, equation (4) are developed to forms of
equation (5) and equation (6).
-4.5
300 ps
-4.0
-3.5
-3.0
(5)
-2.5
-2.0
-1.5
-1.0
The notation of i and n means position (depth)
inside of the crystal and time, respectively. At the
infinite crystal, the particle velocity of Ui+1 will be
zero. In the equation of (5), we have already
estimated the value of spatial and temporal
distribution of density. Thus, the particle velocity
of Uj is calculated by equation (5). In this sense,
the particle velocity distribution will be calculated
by repetitively applying the formula (5) from deep
-0.5
0.0
4
6
Depth (urn)
FIGURE 2. Estimated strain distribution inside of the Si crystal
by DSOM computation based on the dynamical X-ray diffraction
experiment.
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inside of the crystal (infinite crystal) to the surface.
Then the pressure distribution will be calculated
using formula (6) in the same sense. Figure 3
shows the results of pressure distribution just after
the laser irradiation using analysis mentioned above,
and we plot analyzed value as the P-V relation in
Figure 4 with the line of known EOS of Si(lll).
Our solution shows good agreement with known
EOS of Si crystal.
CONCLUSION
We successfully obtained the signal of time
resolved X-ray diffraction for the laser perturbed Si
single crystal with 60ps time step. We performed
DSOM for the estimation of strain distribution
based on the dynamical diffraction theory and we
confirmed maximum compression of 1.05% at the
irradiation power density of 4.7X109W/cm2.
According to the analysis using conservation laws,
the estimated value shows good agreement with the
known value of Si single crystal.
ACKNOWLEDGEMENT
This work was supported by Core Research for
Evolutional Science and Technology (CREST)
program of Japan Science and Technology
Corporation (1ST).
2
Depth/um
REFERENCES
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2. Kondo, K., Sawaoka, A., and Saito, S., Proc. 4th
International Conference on High Pressure
Kyotol974, 845 (1974).
3. D'Almeida, T. and Gupta, Y. M., Phys. Rev. Lett.
85, 330 (2000).
4. Wark, J. S., Whitlock, R. R., Hauer, A, Swain, J.
E., and Solone, P. J., Phys. Rev. B, 35, 9391
(1987).
5. Hironaka, Y., Yazaki, A, Saito, R, Nakamura, K.
G., Kondo, K, Takenaka, H., Yoshida, M., Appl.
Phys. Lett. 77, 1967 (2000).
6. Klar, B. and Rustichelli, E, Nuovo Cimento B 13,
249 (1973).
7. Wie, C. R., TombreUo, T. A. and Vreeland, J.,
Appl. Phys., 59, 3743 (1986).
4
FIGURE 3. Estimated pressure distribution just after the laser
irradiation.
0.424
0.426
0.428
0,430
Specific Volume / cm* g*1
FIGURE 4. The relation between pressure and specific volume
estimated using analysis mentioned in the text.
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