Q. F. Chen
1,a
, J. Zheng, Y. J. Gu
1
, Y. L. Chen
1
, L. C. Cai
1
, and Z. J. Shen
2
1
National Key Laboratory of Shock Wave and Detonation Physics, Institute of Fluid
Physics, P.O. Box 919-102, Mianyang, Sichuan, People's Republic of China
2
Laboratory of computational Physics, Institute of Applied Physics and
Computational Mathematics, P O Box 8009-26, Bejing 10086
In order to address the issue on determining multiple shock compresion states, we show the shock impedance matching solution for the first- to fourth-shock states in experimental shot
GAr429, which can be analyzed with the aid of the pressure-particle velocity( P u p
) diagram shown in Supplemental Figure 1.
The gas argon is confined in a cell with a 304 steel baseplate at the impact end and a
LiF-1/LiF-2 /Al
2
O
3
composite window at the other end. The shocked state of the baseplate was determined using the known equation of state of the Ta flyer and the steel baseplate, and the measured flyer velocity. A graphical representation of the impedance matching method is shown inset in Supplemental Figure 1 . The initial shock state of the steel baseplate is described in the a Electronic mail: chenqf01@gmail.com; chen_qifeng@iapcm.ac.cn
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pressure-velocity P u p
plane by the point labeled 0, which corresponds to the intersection of the steel baseplate and the Ta flyer Hugoniots. The shock state of the gas Ar is constrained to lie on the straight line of slope

D
0 1, Ar
, where

0
is the initial density of the gas Ar sample. The u
1,Ar of the gas Ar is determined by the intersection of the baseplate release isentrope from state 0 and the line defined by the shock impedance of the gas Ar, indicated by point 1. The initial shock generates P u p
state 1 on the principal Hugoniot of the sample. The shock traverses the sample and reflects off the LiF-1 window, producing P u p
state 2, the intersection of the LiF Hugoniot with the first Ar reshock curve. The reflected shock then travels back through the sample and reflects off the steel baseplate, producing P u p
state 3, the intersection of the second Ar reshock curve with the reshock Hugoniot for the baseplate. The second reshock then traverses the sample a third time and re-reflects off the window, producing P u p
state 4, the intersection of the third Ar reshock curve with the reshock Hugoniot for the LiF window. As the shock reflects between the baseplate and the window, the multi-compressed sample becomes fairly thin, eventually reaching a constant state within a shorter time and forming a steady state, which can be seen from the second longer step of the particle velocity profile of the DPS output signals in Figure 2 . Calculations on the states of multi-shock compression dense gas argon were performed using the self-consistent fluid variational theory (SFVT) (red short dash in Supplemental Figure1 ) in the approximation recursion conditions,
 
1

2
and C
01
/ C
02

/
02
[ 2 ] , where

and C
0
are the slope and the sound speed of sample in the relation u s

C
0
  u p
, respectively. The model predictions over the experimental pressure range examined in these calculations indicate that being a good approximation for gas sample. As the above analysis, we can obtain a multiple
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measurements per shot from the present diagnostic technique, i.e
., the first-, second-, and fourth-shock states of argon plasmas and the shock Hugoniots of the LiF window.
200
160
120
80
40
0
0
300
250
200
150
100
50
0
S se
0 pl at e Sh o ck
Hu
Ise ntr opic hoc ot
of
ba k
H ugoni g o n io t o f T a fly er
Ar principal Hugoniot
re lea se
of
ba se pla te
1
0 1 2 3 4 5 6 7 8
H ugoni
4 ot
of
L iF
3rd Ar reshock
R es hock
3
H ugoni
R es hock 2nd Ar reshock
Is ent ot
of
bas epl ropi c rel eas at e
2 e pat ot of
L iF
1st Ar h of
bas
Hugoni reshock
GAr429 epl
Shock at e
Ar principal
Hugoniot
1
1 2 3 4 5
Particle velocity (km/s)
6 7 8
Supplemental Figure 1 .
Graphical solution to determine pressures and particle velocities for reverberation experiments. Inset shows a graphical representation of the impedance matching method.
Multi-shock compression states are analysed below. Supplemental Figure 2 represents the sketch of the shock and reshock pressure vs. volume. Firstly, according to the transmitting time
Δ t
12
= t
2
t
1
of Figure 2 and the thickness of the gas sample, the first shock velocity D
1,Ar
is accurately measured. The shock pressure P
1,Ar
and particle velocity u
1,Ar
can ben written as
P
1, Ar
= r
0, Ar
D
1, Ar u
1, Ar
, (1) where the initial argon density

0, Ar was measured by a method of draining and a special pressure vessel. The pressure and particle velocity can be deduced by impedance matching technique and
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the isentropic release path of the baseplate (see inset in Supplemental Figure 1 ), which is given by the formula,
P s
 exp
  i
  

P i
 
0
C
0
2

 i
 1

Sx
 
0 x

1

Sx

3 exp
 i
 x

 dx 

, (2) where

0
is the Grüneisen parameter and P i
(corresponding to the state 0 inset in Supplemental
Figure 1) is the shock pressure of the baseplate before the isentropic release. V V
0
and
V V
0
, where V
0
, V i,B
, and V s,B
are the specific volume at normal conditions, and the i initial volume of the isentropic release, and one in the end of the release process, respectively. In virtue of the formulas (1) and (2), the shock pressure P
1,Ar
and particle velocity u
1,Ar
(corresponding to state 1 inset in Supplemental Figure 1) can be obtained from the continuity condition of baseplate-sample interface. Also the first shock density,  is determined by
1, Ar formula below u
1, Ar
u
0, Ar
= ( P
1, Ar
P
0, Ar
)( V
0, Ar
V
1, Ar
)
(3) where the gas sample has the initial particle velocity u
0,Ar
=0 and the density
 
1, Ar
1/ V
1, Ar
.
Secondly, as the shock hits the front LiF-1 window, the shock would be reflected back into the argon specimen and then compress the sample again. Another shock wave was generated to transmit into the LiF-1 window. By the crossing time Δ t
23
= t
3
– t
2 of Figure 2 and combined with the thickness of the LiF-1 window, the shock velocity in the LiF-1, D
1,LiF
could be accurately measured. The shock pressure of the LiF-1, P
1, LiF
 
0, LiF
D
1, LiF u
1, LiF
, where the particle velocity u
1,LiF
was measured by DPS. Therefore, the reshock pressure and particle velocity of the argon sample, P
2,Ar and u
2,Ar
(corresponding to state 2 in Supplemental Figure 1) , are equal to the first shock pressure and particle velocity of the LiF window based on the continuity condition of sample-LiF interface, i.e
.,
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м
P
2, Ar
= п u
2, Ar
=
P
1, LiF
(4) u
1, LiF
The second shock density

2, Ar
can be acquired by the formula, u
2, Ar
u
1, Ar
= ( P
2, Ar
P
1, Ar
)( V
1, Ar
V
2, Ar
) (5) where V
2,Ar is the second shock specific volume, and the second shock density of the argon is
 
2, Ar
1/ V
2, Ar
.
Thirdly, the fourth-shock state depends on the combination of the DPS measurement and the reshock Hugoniot of the LiF window. Based on the continuity condition of the sample-window interface, the fourth particle velocity of the sample argon, u
4,Ar
, is equals to the reshock particle velocity of the LiF window, u
2,LiF
, which is determined from the second step of the DPS output recorded signals. Afterwords, the fourth shock pressure, P
4,Ar
, can be acuqired by the impedance-matching methods (refer to Supplemental Figure 1) , which relies on the reshock
Hugoniot of the LiF window (magenta dash in Supplemental Figure 1) . The reshock Hugoniot can be determined by the principal Hugoniot and the Grüneisen equation of state.
Referred to the Hugoniot relations of the LiF, the fomulas are listed as пп E
E
1 H
2 H
-
-
E
0
E
0
=
=
1
2
P
1 H
( V
0
V
1 H
)
1
2
P
2 H
( V
0
V
2 H
)
E
2 R
E
1 H
= P
2 R
+ P
1 H
)( V
1 H
V
2 H
)
. (6)
Meanwhile, the Grüneisen equation of state of LiF can be written as, пп по
P
2 H
P
P
2 R
P
1 H
1 H
=
= (
( g g
V
V )
)
1 H
1 H
(
( E
E
2 H
2 R
-
E
1 H
)
, (7)
E
1 H
) where the parameters’ definition can be found in Supplemental Figure 2 . The notation P , V , and E are the pressure, the volume, and the internal energy, respectively. The subscripts 0, 1H, 2H, and
2R represent the initial, the lower-shock, the higher-shock, and the re-shock states, respectively.
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The formulas (7) are incorporated as
P
2 R
P
2 H
= ( g V )
1 H
( E
2 R
E
2 H
) . (8)
According to the formulas (6) and (8), the reshock relation between the pressure and volume is
P
2 R
=
P
2 H
-
1
2
( g V ) (
1 H
P
2 H
P
1 H
)( V
0
V
2 H
1 -
1
2
( g V ) (
1 H
V
1 H
V
2 H
)
)
. (9)
The reshock relation of pressure versus particle velocity is, u
2 R
= u
1 H
+ ( P
2 R
P
1 H
)( V
1 H
V
2 H
)
. (10)
Then, the re-shock pressure of the LiF window, P
2,LiF
, can be obtained from the measured re-shock particle velocity u
2,LiF
by DPS. Based on the continuity condition, the fourth shock pressure of the dense argon plasma, P
4,Ar
, is equals to the reshock pressure of the LiF, P
4,Ar.
= P
2,LiF
Supplemental Figure 2.
The sketch of the shock and reshock pressure vs. volume.
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The sample holders of the target configuration have slight discrepancies for various shots. The thicknesses of the multi-layer holders and flyer used in the present experiments are listed in the
Supplemental Table I.
Supplemental Table I. The thicknesses of the flyer and the multi-layer holder used in the present experiments
Expt.No.
GAr707
GAr430
GAr408
GAr429
GAr518
Ta Flyer
(mm)
3.20
3.20
3.20
3.20
3.30
304SS Baseplate
(mm)
4.99
4.97
5.00
4.98
5.07
Ar Sample
(mm)
4.935
4.523
4.520
4.507
4.143
LiF-1
(mm)
1.50
1.48
1.52
1.48
1.48
LiF-2
(mm)
4.00
4.00
4.00
3.98
4.04
Sapphire
(mm)
2.08
2.02
2.00
2.00
1.99
The present diagnostic method can simultaneously perform multi-measurements for per shot by the Doppler pins system (DPS) combined with pyrometer and the skillfully designed for the composite window of the holder. We can determine not only the multi-shock states of the dense argon plasma, but also the shock Hugoniot of the LiF-1 window.
The Hugoniots of the LiF can be obtained by combining the measured shock velocity with the PMT and the measured particle velocity of the LiF with the DPS in our experiments (see Table
I ). The comparison between the present measured results of the Hugoniot for the LiF and others
[D=5.18+1.31
u km/s] 2 are given in Supplemental Figure 3. Two results are in good agreement.
- 7 -

10
Expt.(This work)
The linear fitting from our Expt. (D=5.301+1.285u)
LiF Hugoniot relation (D=5.18+1.31u)
[2]
9
8
7
1.5
2.0
2.5
3.0
Particle velocity (km/s)
3.5
Supplemental Figure 3
The comparison between the present experimental Hugoniots of the LiF and those in Ref. [2].
1 Q. F.Chen, J. Zheng, Y. J. Gu, Y. L. Chen, and L. C. Cai, Phys. Plasmas 18,112704 (2011).
2 Experimental Data on Shock compression and Adiabatic Expansion of Condensed Matter , edited by E.F. Trunin (Russian Federal Nuclear Center-VNIIEF, Sarov, 2001).
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