Reconstruction and interpretation of photon Doppler velocimetry spectrum for ejecta particles from shock-loaded sample in vacuum
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Reconstruction and interpretation of photon Doppler velocimetry
spectrum for ejecta particles from shock-loaded sample in vacuum
Xiao-Feng Shi,1 Dong-Jun Ma,1, a) Song-lin Dang,2 Zong-Qiang Ma,1 Hai-Quan Sun,1 An-Min He,1 and Pei
Wang1, 3, b)
1) Institute of Applied Physical and Computational Mathematics, Beijing 100094, China
2) Jiangxi University of Applied Science, Nanchang 330103, China
3) Center for Applied Physics and Technology, Peking University, Beijing 100871, China
(Dated: 11 January 2021)
The photon Doppler velocimetry (PDV) spectrum is investigated in an attempt to reveal the particle parameters of ejecta
arXiv:2012.00963v2 [physics.app-ph] 8 Jan 2021
from shock-loaded samples in a vacuum. A GPU-accelerated Monte-Carlo algorithm, which considers the multiple-
scattering effects of light, is applied to reconstruct the light field of the ejecta and simulate the corresponding PDV
spectrum. The influence of the velocity profile, total area mass, and particle size of the ejecta on the simulated spectra
is discussed qualitatively. To facilitate a quantitative discussion, a novel theoretical optical model is proposed in which
the single-scattering assumption is applied. With this model, the relationships between the particle parameters of ejecta
and the peak information of the PDV spectrum are derived, enabling direct extraction of the particle parameters from the
PDV spectrum. The values of the ejecta parameters estimated from the experimental spectrum are in good agreement
with those measured by a piezoelectric probe.
I. INTRODUCTION A standard PDV setup is shown in Fig. 1. The photodetector
records a mixture of reference and backscattering light. The
The strong shock wave released from the metal– reference wave is in the carrier frequency, and the backscat-
vacuum/gas interface may eject a great number of metal tering wave from ejecta particles has a shifted frequency due
particles.1–5 Most of these particles are of micrometer-scale to the Doppler effect. The interference of the two light waves
in size. This phenomenon of ejecta, or microjetting, was in the photodetector leads to temporal beats of light intensity.
first observed by Kormer et al. in a plane impact experi- The beat signal consists of a large number of harmonics with
ment in the 1950s.5 And the earliest available technical re- different amplitudes and phases. The heterodyne signal may
port on ejecta is from research by the Atomic Weapons change according to variations in the particles’ position and
Research Establishment, Aldermaston(UK).6 The physics of velocity. A discrete Fourier transform is applied to sweep
ejecta are understood as a special limiting case of impulse the beats over time, giving a two-dimensional spectrogram on
driven Richtmyer–Meshkov.7,8 In recent decades, extensive the “frequency/velocity–time” plane. In the spectrogram, the
investigations on particle ejection have been performed be- brightness of each point represents the corresponding spectral
cause of its important role in many scientific and engi- amplitude. The spectrogram is composed of the integral of all
neering fields, including explosion damage,9 pyrotechnics,10 particles’ scattering effects. Hence, interpreting the spectro-
and inertial confinement fusion.11,12 Many experimental ap- gram in detail remains a challenging task.
proaches have attempted to measure the ejection produc- There have been several studies on the interpretation of
tion, such as the Asay foil,3,13 foam recovery,14 piezo- the PDV spectrum. Buttler27,28 used the spectrogram bound-
electric probes,4,15 Fraunhofer holography,16,17 X-ray/proton aries to determine the velocities of the spike and bubble of
radiography,2,18 Mie scattering,18,19 , and photon Doppler ve- Richtmyer–Meshkov instability in loaded metal surface. The
locimetry (PDV).20–26 The main quantities of interest are the evolution of the PDV spectrogram in a gas environment was
particles’ velocity, diameter, and total area mass. Most ap- discussed by Sun et al.,24 and the upper boundary of the spec-
proaches can only measure some of these ejecta parame- trogram was used to obtain the particle size by considering
ters. To reveal the full particle field of ejecta, multiple mea- aerodynamic deceleration effects. Fedorov et al.25 discussed
surement approaches must be equipped. However, in real- the influence of different particle sizes on the spectrogram
world conditions, these approaches are hard to apply simul- boundary in further detail. Recently, Franzkowiak et al.23
taneously because of limitations on the experimental space and Andriyash et al.22,26 reconstructed the light field of ejecta
or configuration. Recently, PDV has attracted considerable and obtained the simulated PDV spectrum using single- and
attention22,23,26 owing to its ability to recover the total area multiple-scattering theory, respectively. They varied the parti-
mass and the distributions of particle velocity and diameter at cles’ parameters and fitted the simulated PDV spectrum to the
the same time. In addition, the light path of PDV is rather experimental data. In this way, the particle velocity profile,
concise and its application is convenient. In some complex diameter, and total area mass were recovered. Andriyash et
experimental configurations, PDV may be the only approach al. considered the aerodynamic deceleration effects in a gas
that can measure the ejecta particles. environment, whereas Franzkowiak et al. only discussed the
case of a vacuum.
Franzkowaik et al.23 and Andriyash et al.22,26 proposed
a) Electronic mail: ma_dongjun@iapcm.ac.cn similar approaches for recovering the ejecta parameters from
b) Electronic mail: wangpei@iapcm.ac.cn the PDV spectrum through reconstruction and then fitting.2
II. RECONSTRUCTION OF PDV SPECTRUM
A. Theoretical background
The photodetector records reference and backscattering
light waves. The scattering process of incident light is illus-
trated in Fig. 2. The scattering light from the ejecta is gov-
erned by the superposition of waves propagating in the ejec-
tion particles along different light paths i:
Ebs (t) = ∑ Ei (t) (1)
i
where Ebs and Ei are the electric vectors of total and partial
scattering waves, respectively.
The light intensity measured by the detector can be repre-
sented as:
2
2
FIG. 1. Standard PDV setup. (1) Laser; (2) Reference light; (3)
I (t) = (Er (t) + Ebs (t)) = Er (t) + ∑ Ei (t)
i (2)
Incident light; (4) Ejecta; (5) Metal plate; (6) Shock; (7) Detonation; = Er2 (t) + ∑ Ei2 (t) + 2 ∑ Er (t) Ei (t) + ∑ Ei (t) E j (t)
(8) Backscattering light; (9) Optical circulator; (10) Photodetector; i i i6= j
(11) Photoelectric signal; (12) PDV spectrogram; (13) Instantaneous
spectrum. where Er2 and Ei2 denote the intensity of the reference and
scattering light signals, respectively. The third term repre-
sents the heterodyne beats between the reference and scat-
tering light, and the last term represents the heterodyne beats
However, some assumptions were introduced in the recon-
between the different scattering light paths. The Fourier trans-
struction of the light field. Franzkowaik et al. assumed that
form of I(t) is determined by the relation:
only backscattering light was present, while Andriyash et al.
set the light scattering direction to be uniform and random in R
I (ω ) = dt exp
(iω t)I (t)
space. These assumptions affect the accuracy of the spectrum R
reconstruction, and thus influence the recovery of the ejecta ≈ dt exp(iω t) 2 ∑ Er (t) Ei (t)
i (3)
parameters. The fitting model is another factor that affects the
interpretation of the PDV spectrum. Different convergence = 2|Er | · ∑ |Ei |
criteria may produce different results. The quantitative rela- i ω =ωi −ωr
tionship between the ejecta parameters and the PDV spectrum
remains unclear. Hence, it is difficult to obtain definite ejecta where |E| is the amplitude of the light wave field. In Eq. (3),
parameters from the PDV spectrum. These issues provide the only the third term appearing in Eq. (2) remains. This is be-
motivation for the present work. cause the frequencies of the first two terms are too high to be
measured by the detector and the value of the last term is much
In this study, we improve the reconstruction method of smaller than that of the third term. ωr is the carrier frequency,
the ejecta light field, and propose a novel model for extract- and ωi is the Doppler-shifted frequency, which corresponds to
ing the ejecta parameters directly from the PDV spectrum. a sequence of scattering events along path i:
Mie theory, which gives a rigorous mathematical solution to
Maxwell’s equations, is applied to calculate the light scatter- ωr
∑ nsk,i − nk,i vk,i
ωi = (4)
ing effects, and a Monte-Carlo (MC) algorithm is used to de- c k
scribe the light transport process realistically. This reconstruc-
tion method provides a high-fidelity simulation for the PDV where c is the speed of light, nk,i and nsk,i are the directions
spectrum. The procedure is discussed in detail in Section II. of wave propagation before and after scattering by particle k,
The influence of the ejecta parameters on the PDV parameters and vk,i is the velocity of this particle.
is then explored through MC simulations in Section III.A. In To reconstruct the PDV spectrogram [Eq. (3)], the key is to
Section III.B, we propose an optical model that reveals the re- obtain |Ei | and ωi , i.e., the detailed scattering process in the
lationships between the PDV spectrum characteristics and the particles. For multiple-particle systems, the scattering field
ejecta parameters. With this model, the ejecta parameters can can be described by the transport equation:29–31
be extracted directly from the PDV spectrum, instead of fitting
to experimental data. In Section III.C, the estimated ejecta ∂
n + σs + κ I (r, n,t)
parameters from an experimental PDV spectrum are verified ∂r
(5)
against those measured by a piezoelectric probe. Finally, the Z
σs p n, n′ exp ik0 n − n′ vt I r, n′ ,t dn′
conclusions to this study are presented in Section IV. =3
where m0 is the total area mass of ejection and ρ0 is the density
Incident
n1 s n1 of the particle material.
light
Combining Eqs. (8) and (9), we can rewrite Eq. (7) as:
v
Scattering
3m0 ∑ di2 Kext (di )
Particles light i
pr = exp − (10)
Free Ejection 2ρ0 ∑ di3
sureface i
The photons staying in the medium are scattered or ab-
FIG. 2. Multiple scattering of light waves in ejection particles.
sorbed by particles. The probabilities of scattering and ab-
sorption are calculated by the formula:
where I is the light intensity in the field, which depends on
ps = Ksca /Kext
both the detection position r and the direction n. σs and κ (11)
pa = Kabs /Kext
are the coefficients of scattering and absorption, respectively.
p (n, n′ ) is the scattering phase function. The right-hand side where the scattering coefficient Ksca and the absorption coef-
of this equation represents the contributions of scattering light ficient Kabs are calculated by Mie theory33 as:
from other positions. ∞
At the boundaries of the ejection, the light intensity has the
Kext = α22 ∑ (2n + 1) Re (an + bn )
form:
n=1
∞
( Ksca = α 2 ∑ (2n + 1) |an |2 + |bn |2
2 (12)
I (r = 0, n = n0 ,t) = I0
n=1
I (r = 0, n = −n0 ,t) = Ibs = (∑ |Ei |)2 (6)
Kabs = Kext − Ksca
i
where α is a dimensionless particle diameter parameter, α =
where n0 is the direction of incident light, which is usually π d/λ , λ is the light wavelength, and an , bn are Mie coeffi-
perpendicular to the free surface. I0 and Ibs are the intensities cients. It is clear that ps + pa = 1.
of incident light and backscattering light, respectively, which If the photon is absorbed by particles, it will completely
correspond to the input and output of the transport equation. disappear and be converted into the particle’s internal energy.
If the photon is scattered, its propagation direction and fre-
quency will change, as shown in Fig. 3. The phase function of
B. Monte-Carlo algorithm
the scattering angle θ is calculated by the formula:
Andriyash et al.22,26 used the discrete ordinate method to λ2
solve the transport equation [Eq. (5)]. In this paper, a more p(θ ) = (|S1 (θ )|2 + |S2 (θ )|2 ) (13)
2π Ksca
convenient and accurate method of the MC algorithm is ap-
plied to calculate the scattering effects. where S1 and S2 denote the scattering intensity components in
In the MC algorithm, the incident light is assumed to be the perpendicular and parallel directions:
a great number of photons. When passing through random
∞
(1)
(cos θ ) dPn (cos θ )
granular media, only part of the photons can penetrate. The 2n+1
S1 (θ ) = ∑ n(n+1) an dPdncos + b
n
θ dθ
n=1 (14)
proportion of permeable photons is approximated by Beer– ∞
(1)
θ) dPn (cos θ )
Lambert’s law:32 2n+1
S2 (θ ) = ∑ n(n+1) an dPn d(cos + b
n
θ d cos θ
n=1
pr = exp (−τ L) (7) (1)
where Pn and Pn are Legendre and first-order associative
where L is the thickness of the medium and τ denotes the in- Legendre functions, respectively.
verse extinction length, given by: The incident light is assumed to be non-polarized, so the
azimuth angle ϕ after scattering obeys the uniform random
∑ π4 di 2 · Kext (di ) distribution:
i
τ = NKext Ā = (8) 1
As L p(ϕ ) = (15)
2π
where N is the number of particles per unit volume, Kext is the
light extinction coefficient (determined by the particle diame- After scattering, the scattering angle and azimuth angle are
ter, light wavelength, and metal relative refraction index), Ā is added to the original light direction. The new direction cosine
the mean cross-section area of the particles, di is the diameter û = [ûx , ûy , ûz ] has the form:
of particle i, and As is the light exposure area.
û = √ 1 2 sin (θ ) [ux uy cos (ϕ ) − uy sin (ϕ )] + ux cos(θ )
For particles in the light exposure area, the total mass has x
1−uz
the form: ûy = √ 1 2 sin (θ ) [ux uz cos(ϕ ) − ux sin (ϕ )] + uy cos (θ )
1−uz
1
p
ûz = − sin (θ ) cos(ϕ ) 1 − u2z + uz cos (θ )
m0 As = ∑ π di3 ρ0 (9)
i 6 (16)4
(3) In one iteration, all photons take one step in the direc-
tion of their propagation. Some photons may penetrate the
current ejection layer, and the proportion pr is determined by
Eq. (10). For each photon, a random number is generated in
(0, 1). If the random number is less then pr , the corresponding
photon travels over the ejection layer boundary successfully.
Otherwise, the corresponding photon is absorbed or scattered
by particles in the ejection layer.
(4) For the photons that remain in the ejection layer, we use
Eq. (11) to determine whether they are scattered or absorbed.
If the photon is scattered, the direction change is calculated
by Eqs. (13)–(17). Because the phase function of the scatter-
ing angle is very complex, an acceptance–rejection method is
applied. The new frequency of the scattered photons is deter-
mined by Eq. (18).
(5) Overall, if the photon travels across the ejection layer
FIG. 3. Light scattering on a particle. boundary, its position is updated; if the photon is scattered, its
direction, frequency, and position are updated; if the photon is
absorbed, it is labeled as such and removed from subsequent
where u = [ux , uy , uz ] is the original direction cosine. When calculations.
|uz | ≈ 1, the direction cosine û is calculated by the formula: (6) After updating the state of the photons, we check which
of them have reached the free surface or left through the top of
ûx = sin (θ ) cos (ϕ ) the ejection. For all photons that have reached the free surface,
ûy = sin (θ ) sin (ϕ ) (17) the ideal diffuse reflection is applied. If any photons have left
û = cos (θ ) u / |u |
z z z the ejection, they are labeled accordingly and removed from
subsequent calculations.
After scattering, the frequency of the photon will have
(7) Steps (3)–(6) are repeated until all photons have been
changed. The new frequency of scattering light has the form:
absorbed or have left the ejection. The frequency shifts of
outgoing photons are summarized as the spectrogram.
v̂ − v
ω̂ = ω 1 + n (18)
c
where v is the velocity of the last particle that the photon left, C. Particle models
v̂ is the current particle velocity, n is the photon direction, and
ω is the photon frequency before scattering. The MC algorithm indicates that the PDV spectrum is re-
After passing through the entire ejection layer, few photons
lated to the particle size d, velocity v, position z, and num-
reach the free surface. An ideal diffuse reflection is assumed ber N (i.e., total area mass m0 ). This algorithm can be ap-
for these photons. After reflection, the space angles θ and ϕ
plied in cases where these parameters are completely ran-
are uniformly random in [π /2, π ] and [0, 2π ], respectively. dom. In real situations, however, the particles of the ejecta
Because the ejection is blocked by the free surface, even-
usually satisfy certain distributions in terms of velocity and
tually all the photons are either absorbed by particles or diameter.2,16,17,19,34–37 For the sake of discussion, these as-
backscattered out from the top of the ejection. The frequency
sumptions are applied in this paper. Previous studies2,34 in-
shifts of these “out” photons are summarized as the theoretical dicate that the initial velocities of particles in the ejecta can be
PDV spectrogram:
approximated by an exponential law:
√ m(v) β v
r
I0 f (v) = = exp −β −1 (20)
I (ω ) ∝ ∑ |Ei | = ∑ Ii = nout |ω =ω̂ −ωr · (19) m0 vfs vfs
i i n0
where n0 is the number of initial photons and nout is the dis- where v f s is the velocity of the free surface and β is the veloc-
tribution of out photons in terms of their frequency. ity distribution coefficient. Under this exponential law, most
The detailed steps of the calculation procedure are as fol- of the particles are located in the low-velocity region, which
lows: is near the free surface. β determines the non-uniformity of
(1) First, the initial conditions of the photons and particles this distribution.
are set, such as the number and frequency of photons, and the In this paper, we only consider the ejecta in a vacuum en-
diameter, velocity, and position of the particles. The photons vironment. After being ejected, the particles retain an almost
start at the top of the ejection and then move towards the free constant velocity and the ejecta expands in a self-similar man-
surface. ner over time. The particle position z is only related to its ini-
(2) The step sizes of all photons are set to the same and tial velocity v and ejection time te , z = vte . The corresponding
equal to one thousandth of the height of the ejecta. PDV spectrum exhibits slight changes over time.23,385
The particle size distribution is assumed to obey a log- 0.010
4
normal law:17,36 10 photons
5
10 photons
0.008
ln2 (d/dm )
1 6
) (arb. units)
10 photons
n (d) = √ exp − (21)
2πσ d 2σ 2 10
7
photons
0.006
where σ is the width of the distribution and dm is the me-
dian diameter. These parameters depend on the roughness of
0.004
the metal surface, shock-induced breakout pressure, and sur-
I(
rounding gas properties. When σ = 0, the function becomes a
Dirac equation and all of the particles have the same diameter. 0.002
Obviously, this is the ideal situation. The particle distribution
can also be described by a power law,16,35,37 but this descrip-
0.000
tion may be invalid in the range of small particle sizes (less 0.8 1.0 1.2 1.4 1.6 1.8 2.0
than 10 µ m).17 In this paper, the particle size is assumed to
fs
be independent of its velocity.
With these assumptions, the determining factors of the PDV
FIG. 4. Theoretical PDV spectra with different initial numbers of
spectrum change to the velocity profile coefficient β , total area photons. The calculation assumes the ejection of Sn particles in a
mass m0 , median diameter dm , and size distribution width σ . vacuum environment. The particle velocities obey an exponential
The aim of this paper is to discuss the influence of these pa- distribution (β = 10) and the particle sizes follow a log-normal dis-
rameters on the PDV spectrum, and to explore how they can tribution (dm = 1.5 µ m, σ = 0.5). The total area mass is 20 mg/cm2 .
be extracted from the PDV spectrum most accurately. ω f s is the Doppler frequency shift corresponding to the free surface
velocity, ω f s = 2ωr · v f s /c. The probing wavelength is λ = 1550 nm.
D. Convergence and comparison 21
Quadro P600
The accuracy of the MC algorithm mainly depends on the Nvidia GTX960
initial number of photons. Theoretical PDV spectra with 104 , 16
Acceleration ratio
105 , 106 , and 107 initial photons are shown in Fig. 4. The
calculation assumes a vacuum environment and the particles
distribution assumptions are applied. In this case, the PDV 11
spectra have a single peak. As the initial number of photons
increases, the spectrum curves tend to be smooth. The differ-
ence between the spectra with 106 and 107 initial photons is
very slight. Thus, 107 initial photons are applied in the fol- 6
lowing calculations.
The high initial number of photons leads to considerable
computational cost. For the case of 107 photons, a single-core 1
CPU requires approximately 3 h to determine the spectrum. 104 105 n0 106 107
GPUs can be applied to accelerate the calculation. Although
the frequency of GPU processors is much lower than that of FIG. 5. Acceleration ratio of GPU calculation compared with CPU
CPUs, GPUs contain hundreds or thousands of stream proces- for different initial numbers of photons. The CPU is an Intel Xeon
sors that can work simultaneously. The acceleration ratio of a W-2102 and its basic frequency is 2.9 GHz. The Quadro P600 GPU
GPU compared to a CPU is shown in Fig. 5. The Intel Xeon has 384 stream processors; the clock speed of each processor is about
W-2102 CPU (frequency 2.9 GHz) and two GPUs (Quadro 1.3 GHz. The Nvidia GTX960 GPU has 1024 stream processors; the
P600 and Nvidia GTX960) are applied. As the initial num- clock speed of each processor is about 1.1 GHz.
ber of photons increases, the acceleration ratio of the GPUs
is enhanced. For the case of 107 photons, the acceleration
ratio reaches a factor of 8 for the Quadro P600 and a factor mass and particle size instead of the transport optical thick-
of 20 for the Nvidia GTX960. Because there are many judg- ness used by Andriyash et al. With the uniform scattering as-
ment events in the procedure, and the GPUs have few logical sumption, our simulation (Case 2) is almost the same as that of
units, it is difficult to improve the acceleration ratio with these Andriyash et al. (Case 3), which validates the adequacy of the
GPUs. Thus, the Nvidia GTX960, which requires approxi- present numerical method. However, when the Mie scattering
mately 500 s to compute each case, is used in the following theory is applied, there is a remarkable difference between the
calculations. present procedure (Case 1) and the results of Andriyash et al.
The PDV spectra simulated by the present procedure are (Case 3) and Franzkowiak et al. (Case 4). The difference
compared with those reported by Andriyash et al. and with Andriyash et al. is mainly in the low-velocity part. This
Franzkowiak et al. in Fig. 6. We use the equivalent ejecta area is because the change in the scattering phase function has a6
Free surface 0.010
0.020
=4
Case 1
=5
Case 2 0.008
=6
) (arb. units)
Case 3
0.015
) (arb. units)
Case 4 =7
0.006
=8
0.010 =9
0.004
=10
I(
=15
I(
0.005 0.002
0.000
0.000 0.5 1.0 1.5 2.0 2.5 3.0
0.8 1.0 1.2 1.4 1.6 1.8 2.0
fs
fs
FIG. 7. Simulated PDV spectra with different velocity coefficients.
FIG. 6. Comparison of PDV spectra calculated by different recon- The calculation was carried out for Sn particles in a vacuum environ-
struction methods. Case 1: MC + Mie scattering theory (proposed ment. The total area mass was 20 mg/cm2 . The log-normal distribu-
procedure); Case 2: MC + uniform scattering assumption; Case 3: tion (dm = 1.5 µ m, σ = 0.5) was applied to the particle sizes.
Discrete coordinates + uniform scattering assumption (Andriyash et
al.); Case 4: Single scattering theory (Franzkowiak et al.) . The data
for Case 3 were extracted directly from the paper of Andriyash et al. comes invisible. The coefficient β determines the distribution
The calculations were carried out for a transport scattering thickness
of particles in the ejecta. With larger β , fewer particles are
of τtr = 10, which corresponds to m0 = 10.7 mg/cm2 , dm = 1.5 µ m,
and σ = 0.5. The material is Sn and the ejection velocity profile has
located at the top of the ejecta and the incident light can pen-
β = 8. etrate deeper. This results in the movement of the spectrum
peak and a decrease in the observability of high-velocity par-
ticles.
great influence on the multiple scattering, which is the main The simulated PDV spectra with different values of the total
form of scattering in the low-velocity dense part. The differ- area mass m0 are shown in Fig. 8. The changes in the spectra
ence with Franzkowiak et al. is in the location of the spectrum can be divided into two sections. When m0 ≥ 10 mg/cm2, the
peak. Franzkowiak et al. applied the single-scattering the- spectrum displays a single peak. With a decrease in m0 , this
ory and assumed that all of the light scattered backward. This peak moves to the left, and its magnitude and slope exhibit
implies that the optical thickness is overestimated, and so lit- slight changes. When m0 ≤ 5 mg/cm2, a new peak appears
tle light would reach the deep region of the ejecta. Thus, the around the free surface, and the spectrum exhibits a double-
spectrum moves toward high velocities. These differences in peak shape. A smaller area mass produces a more remarkable
spectra indicate that the scattering assumption may introduce new peak. For m0 = 2 mg/cm2, the original peak disappears
some reconstruction inaccuracy that cannot be neglected. and the spectrum again exhibits a single peak. The double-
peak spectrum has been observed in previous experiments20,39
and simulations,26,40 and is the result of the direct exposure of
incident light at the free surface.
III. INTERPRETATION OF PDV SPECTRUM There are two parameters that determine the particle size
distribution—the median diameter dm and the distribution
A. Influence of ejecta parameters width σ . Their influence on the PDV spectrum is illustrated in
Figs. 9 and 10, respectively. dm and σ exhibit similar effects:
The results of numerical calculations that demonstrate the as dm or σ increases, the original peak of the spectrum moves
sensitivity of the PDV spectrum to changes in the ejecta pa- towards the low velocities and the peak value decreases. A
rameters (β , m0 , dm , and σ ) are presented in Figs. 7–10. new peak then appears in the position of the free surface and
The PDV spectra were simulated using the MC algorithm de- the original peak gradually attenuates. This change in the
scribed in the previous section for Sn particles in a vacuum form of the spectrum peak is similar to that for the area mass.
environment. The initial ejecta parameters were set to β = 10,
m0 = 20 mg/cm2, dm = 1.5 µ m, and σ = 0.5. In each figure,
one of the parameters changes and the others remain constant. B. Theoretical optical model
The simulated PDV spectra with different values of the ve-
locity coefficient β are shown in Fig. 7. With an increase in The simulations described above using the MC algorithm
β , the spectrum peak moves towards the low velocities and provide a qualitative understanding of the influence of the
its magnitude decreases. Furthermore, the spectrum shape be- ejecta parameters on the PDV spectrum. However, how to
comes sharper and the high-velocity part of the spectrum be- solve the reverse problem, i.e., extracting the ejecta param-7
0.010 0.010
m0=2 =0
m0=5 =0.2
0.008 0.008
) (arb. units)
) (arb. units)
m0=8 =0.4
m0=10 =0.6
0.006 0.006
=0.8
m0=20
=1.0
m0=40
0.004 0.004
I(
m0=80
I(
0.002
0.002
0.000
0.000
0.5 1.0 1.5 2.0
0.5 1.0 1.5 2.0
fs fs
FIG. 8. Simulated PDV spectra with different total area mass. The FIG. 10. Simulated PDV spectra with different particle size coeffi-
area mass unit is mg/cm2 . The calculation was carried out for Sn cients σ . The calculation was carried out for Sn particles in a vacuum
particles in a vacuum environment. The velocity coefficient β = 10 environment. The total area mass was 20 mg/cm2 . The velocity co-
and the size coefficients dm = 1.5 µ m, σ = 0.5. efficient β = 10 and the median diameter dm = 1.5 µ m.
0.010
d =0.5
m
spectrum is calculated by the formula:
d =1.0
m
0.008
) (arb. units)
d =1.5
m
d =2.0
m ṽ8
1.0
v>ṽ v dv
π 2 ∗ f v / f v fs e v
∑ 4 di Kext
Z ∞ R∞ π 2 ∗ 0.8
i ṽ N (v)dv · 4 d Kext
τ (v)dv = =
ṽ As As (24)
R∞
3K ∗ m (v)dv 0.6
= ext ṽ
Ratio
2ρ0 d 3 /d 2
0.4
∗ only considers
where the equivalent extinction coefficient Kext
the backscattering and absorption effects:
0.2
Extiction ratio
∗
Kext = Kback + Kabs = gback Kext (25) Particle distribution
0.0
where the coefficient gback is 0.5–0.7 for particle diameters of 1.0 1.2 1.4 1.6 1.8 2.0
1–10 µ m. When gback = 1, the present model reduces to that v / v fs
of Franzkowiak et al.
Because the velocity profile is exponential, ṽ∞ m (v)dv =
R
FIG. 11. Extinction ratio and particle distribution with respect to
m0 f (ṽ)v f s /β . Equation (25) has the form: velocity.
Z ∞ ∗ 0.008
3Kext m0 v f s m0=20
τ (v)dv = f (v) (26) MC,
ṽ 2ρ0 d 3 /d 2 β 0.007
SS, m0=20
MC, m =5
) (arb. units)
0.006
When ṽ is equal to the free surface velocity, the integral 0
SS, m =5
represents the amount of light that is able to reach the free 0.005 0
surface. The optical thickness of the ejecta is defined as:
0.004
Z ∞ ∗ 0.003
3Kext m0
I(
τ0 = τ (v)dv = (27)
vfs 2ρ0 d 3 /d 2 0.002
0.001
Combining Eqs. (23), (26), and (27), we can write Eq. (22)
as: 0.000
0.8 1.0 1.2 1.4 1.6 1.8 2.0
√
Kback d vfs v / v fs (i.e. fs )
I (ṽ) ∝ ∗ · · τ0 f (ṽ) exp − τ0 f (ṽ) (28)
Kext d2 β
FIG. 12. Simulated PDV spectrum by MC algorithm and single-
This equation provides the theoretical form for determining scattering (SS) model. The area mass unit is mg/cm2 . The particle
the PDV spectrum from the ejecta parameters in a vacuum en- settings are β = 10, dm = 1.5 µ m, σ = 0.5. The backscattering co-
vironment. In this formula, the PDV spectrum is proportional efficient is gback = 0.67.
to the product of the velocity profile and the extinction term.
These two parts are illustrated in Fig. 11. As the velocity de-
creases, the corresponding ejecta position moves closer to the this defect has only a very slight influence on the accuracy of
free surface, and the incident light becomes weaker because the present model.
of particle extinction. However, the particles become dense We now analyze the spectrum function [Eq. (28)]. First, we
deeper within the ejecta, and this enlarges the cross-section take its derivative:
area of scattering. With the contribution of these two parts,
′ ′ vfs vfs
the PDV spectrum exhibits a single peak shape. I (ṽ) ∝ τ0 f (ṽ) exp − τ0 f (ṽ) 1− τ0 f (ṽ)
β β
The simulated PDV spectra of the present model are com- (29)
pared with those of the MC algorithm in Fig. 12. With the cor- β vfs
= − I (ṽ) 1 − τ0 f (ṽ)
rection of the backscattering coefficient, there is a good agree- vfs β
ment between the results, both in the main peak position and where f ′ (ṽ) = −β /v f s f (ṽ).
the curve shape. However, in the case of a small ejecta mass When I ′ (ṽ) = 0, the solution provides the position of the
(m0 = 5 mg/cm2), the present model cannot simulate the peak spectrum peak:
around the free surface. Although the spectrum is nonzero
at the position of the free surface in this model, neglecting vfs
1− τ0 f ṽ peak = 0 (30)
the multiple-scattering results prevents the second peak from β
appearing.26,40 In this paper, we mainly consider the informa-
tion supplied by the original peak of the PDV spectrum. Thus, ṽ peak /v f s = 1 + ln(τ0 ) /β (31)9
The peak value of the PDV spectrum is: 1.8
Monte-Carlo
√ 1.7
Kback d¯ β −1 β ln( )/
I ṽ peak ∝ ∗ · · e ∝ (32)
Kext d2 v f s d 2 /d 1.6
1.5
Finally, the relative curvature at the spectrum peak is given
vpeak /vfs
by the second derivative of the spectrum function: 1.4
′′ 1.3
′′
vfs vfs
I (ṽ) /I ṽ peak = e τ0 f (ṽ) exp − τ0 f (ṽ)
β β 1.2
(33)
1.1
1.0
2
β 4 5 6 7 8 9 10 11 12 13 14 15
I ′′ ṽ peak /I ṽ peak
=− (34)
vfs
In summary, the relationships between the ejecta param- FIG. 13. Peak positions of PDV spectrum calculated by MC al-
eters and the characteristics of the PDV spectrum have the gorithm and theoretical formula. Two optical thickness are con-
form: sidered, where the square points denote τ0 = 23.4 and the circular
points denote τ0 = 8.6. The corresponding ejecta parameters are
ṽ peak /v f s = 1 + ln(τ0 ) /β m0 = 20 mg/cm2 , dm = 1.5 µ m, σ = 0.5 and m0 = 10 mg/cm2 , dm =
2.0 µ m, σ = 0.5, respectively.
β
I ṽ peak /v f s ∝ (35)
2
d /d
′′ 0.012
I ṽ peak /v f s /I ṽ peak /v f s = −β 2
Monte-Carlo
k d1*
where the independent variable is normalized by the velocity 0.010
of the free surface, v f s .
The above relationships allow some of the ejecta parame- 0.008
I(vpeak )
ters to be determined. First, the velocity profile coefficient β
is given by the relative curvature at the spectrum peak. In ad-
2 2
dition, the surface mean diameter d ∗ = d /d = dm e0.5σ can 0.006
be derived from the value of the spectrum peak. Finally, the
optical thickness of the ejecta τ0 is obtained from the position 0.004
of the spectrum peak, where τ0 is related to the ejecta area
2
mass m0 and the Sauter mean diameter ds = d 3 /d 2 = dm e2.5σ .
0.002
∗
If ds is assumed to be approximately d , the area mass m0 can 4 5 6 7 8 9 10 11 12 13 14 15
be determined.
Figures 13–15 compare these theoretical relationships with
the MC simulation results. Two optical thickness and a large
FIG. 14. Peak values of PDV spectrum calculated by MC algo-
range of velocity profiles are considered. It can be observed rithm and theoretical formula. Two optical thickness are consid-
that the theoretical relationships largely conform to the MC ered, where the square points denote τ0 = 23.4 and the circular
simulations in these cases, which verifies the present model to points denote τ0 = 8.6. The corresponding ejecta parameters are
some extent. m0 = 20 mg/cm2 , dm = 1.5 µ m, σ = 0.5 and m0 = 10 mg/cm2 , dm =
2.0 µ m, σ = 0.5, respectively.
C. Experimental verification
A set of ejecta PDV experiments performed by
In the above relationships, the peak value of the spectrum is Franzkowiak et al.23 was used to verify the present the-
difficult to use in the analysis of PDV experiments. In the ex- oretical model. The experiment was carried out in a vacuum
periments, the PDV spectrum is scaled by the reference light environment using Sn material with the surface machined into
intensity, probe reception, photoelectric conversion efficiency, 60 × 8 µ m grooves. The shock-induced breakout pressure
and circuit amplification factor, among other factors. Addi- was PSB = 28 GPa. The velocity of the free surface was
tional PDV experiments are required to calibrate this scaled found to be approximately 2013 m/s. We extracted the PDV
factor. For a single PDV vacuum experiment, only the ve- spectrum from the experimental spectrogram over the period
locity profile β and optical thickness τ0 of the ejecta can be 0.2 − 0.8 µ s, as shown in Fig. 16(a). The PDV data were then
extracted. If there is an additional particle granularity mea- averaged and smoothed using the low-pass filtering of the fast
surement, the ejecta area mass m0 can also be determined. Fourier transform. We converted the spectrum units [dBm] to10
250 -25
Monte-Carlo Exp.
-35 Smooth
PSD (dBm)
200
-45
-I"(vpeak )/I(vpeak )
-55
150
-65
100 -75
2nd Relative Deriv.
50
50
0
0
-50
4 5 6 7 8 9 10 11 12 13 14 15
-100
-150
1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7
FIG. 15. Relative curvature of PDV spectrum at the peak calculated v/vfs
by MC algorithm and theoretical formula. Two optical thickness are
considered, where the square points denote τ0 = 23.4 and the circu-
FIG. 16. (a) PDV spectrogram extracted from ejecta experiment of
lar points denote τ0 = 8.6. The corresponding ejecta parameters are Franzkowiak et al.23 and (b) second derivative of the smooth data.
m0 = 20 mg/cm2 , dm = 1.5 µ m, σ = 0.5 and m0 = 10 mg/cm2 , dm =
2.0 µ m, σ = 0.5, respectively.
volts and then took the second derivative to give the smoothed
PDV spectrum shown in Fig. 16(b). The peak of this spectrum
is located at ṽ/v f s = 1.32 and the corresponding relative
curvature is approximately −110. Combined with Eq. (35),
this suggests a velocity profile coefficient of β = 10.5 and an
optical thickness of τ0 = 28.79.
Schauer et al.36 conducted a Mie-scattering experiment
with similar conditions, where the surface roughness was 50 ×
8 µ m and the breakout pressure was about 30 GPa. The parti-
cle size distribution was measured to be dm = 0.6 µ m, σ = 0.5.
Using this data, the total area mass was determined to be
m0 = 7.5 mg/cm2.
In their PDV experiment, Franzkowiak et al. simultane- FIG. 17. Comparison of the experimental and simulated PDV spec-
ously measured the area mass with respect to velocity us- tra.
ing a piezoelectric probe. The PDV spectrum and area mass
given by our estimations and their experiments are compared 8
in Figs. 17 and 18, respectively. These two results are in good Piezoelectric probe
agreement, which verifies the present theoretical model. Estimation
6
m (v) (mg/cm 2)
IV. CONCLUSION
4
This paper has discussed the PDV spectrum of ejecta par-
ticles from shock-loaded samples in a vacuum. A GPU-
accelerated MC algorithm that rebuilds the PDV spectrum for
2
the ejecta particles has been proposed, and Mie theory was
applied to describe the scattering process. Compared with the
reconstruction methods of Andriyash et al. and Franzkowiak
et al., a reasonable scattering model is the key to simulating 0
1.0 1.1 1.2 1.3 1.4
the PDV spectrum accurately. The simulations using the MC v/vfs
algorithm indicate that the particle velocity profile, particle
size, and ejecta area mass have a significant influence on the FIG. 18. Comparison of the area mass and velocity profile between
shape and values of the PDV spectrum. As the velocity profile the piezoelectric probe measurement23 and our estimation.
coefficient or particle size increases, the spectrum peak moves11
to lower velocities. However, this change in the spectrum peak 5 V. A. Ogorodnikov, A. G. Ivanov, A. L. Mikhailov, N. I. Kryukov,
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