Radiation from explosive-driven shocks in the noble gases

1. Quont.Spmtrosc.Radial. Transfer.Vol. 4, pp. 839-845.Perpamon Press Ltd., 1964.Printed in Great Britain

RADIATION

FROM EXPLOSIVE-DRIVEN IN THE NOBLE GASES W. G.

SHOCKS

VULLIET

General Atomic Division of General Dynamics Corporation, John Jay Hopkins Laboratory for Pure and Applied Sciences, San Diego, California (Received 17 Jdy 1964) Abstract-Shock temperatures were calculated for xenon, krypton, and argon using the strong-shock approximation and Composition B explosive as the driver. Temperatures of 38,000 31,000 and 23,OOO”K were obtained for Xe, Kr and A, respectively. Further calculations were made on the emergent radiation flux from the cold preshocked gas by taking into account radiation absorption in the shocked gas itself, the radiation-heated shock precursor, and the cold gas ahead of the precursor. The frequency integrated power fluxes on a target 100 cm ahead of the shock are 4*l(lO)l*, 3*5(1O)la, and 1*4(10)1sergs per cm* per set for Xe, Kr and A, respectively.

I. INTRODUCTION

THE TEMPERATURES of strongly shocked noble gases can be calculated for a given postshock gas velocity and preshock gas density from the equation of state and the usual momentum, mass, and enthalpy conservation theorems by requiring that the postshock gas velocity be the expansion velocity of the high-explosive detonation products. The emergent radiation flux can then be calculated from a knowledge of the mean free path of the radiation in the hot shocked gas itself, in the radiation-heated shock precursor, and in the cold preshocked gas. II.

THE

SHOCK

TEMPERATURE

The state of ionization of the shocked gas is described by the Boltzmann-Saha equation, which is transcendental in the temperature. Hence, an iterative method was used to find the shock temperature. The procedure was as follows. First, the postshock temperature was selected and the density estimated, knowing that the density ratio across a strong shock in an ionizing gas is about 9. From the temperature and a knowledge of the energy levels and statistical weights of excited states for various stages of ionization(l) one can compute the partition function for each type of ion. Next, the average number of free electrons per ion can be guessed and the lowering of the ionization potential for each type of ion due to the electric field of adjacent ions and electrons can be computed.(s) Using the lowered ionization potential, the ionic partition functions, the selected temperature, and the above-mentioned guess for the average number of free electrons per ion, the Boltzmann-Saha equation can be used to calculate a better value for the average number of free electrons per ion. Such an iteration will converge very rapidly since the successive ionization potentials are usually widely spaced compared to kT, where T is the 839

Preshock density, P 0, (cm-a)

2qlo)r9 24(10)19

2*4(10)‘9 2qlo)so

2qlo)is

2*qlo)i@ 2*4(10)19

2*4(1o)is 2*4(10)‘9

Postshock temperature, T, (“IQ

40.500 38,200

45,000 38,200

38,200

31,000 36,000

23,200 24,400

6 6

15

6

:

15 15

2

14 14

15 15

15

6

6 6

III

II

1 1

I

Partition function for indicated ionization, B

0.45 0.55

1.12 1.22

1.85

1.75 1.31

157 1.68

9.0

50.7

14.6 15.9

Argon

27.5 32.8

9.0 9.0

8.9 8.9

9.0 9.0

49.5 30.4

Krypton

9.0

Density ratio, x

40.9 45-l

(eV/atom)

If1

Enthalpy,

Xenon

Free electrons_ per ion, 2

TABLE 1

l*qlo)g 1.2(10)9

2.0(10)9 2.3(10)s

3*6(10)8

4*1(10)9 3*o(lo)ro

3.3(10)9 3*6(10)9

(d y

n$ns)

Pressure,

0.90

7*7(10)5

7.5(10)5 7*8(10)5

0.046 0.044

0.081 0.068

0.17

8.0(10)5 5.2(1~)5

7*1(10)5 7*8(1O)s

0.13 0.14

r

69(1O)s 7*3(10)5

Velocity PI, (cm/=)

1*4(1o)‘s

3*5(10)‘3

4*1(10)15

Et (erg/cm%ec)

i r;

b

*

841

Radiation from explosive-driven shocks in the noble gases

shock temperature and k is Boltzmann’s constant. Having the degree of ionization, the postshock enthalpy can be calculated and hence the density ratio across the shock (see the Appendix), thus we can correct our original estimate of the postshock density and, if necessary, recompute the degree of ionization. However, such a recomputation only need be done for one temperature selection since the density ratio is not sensitive to changes in the postshock temperature. The postshock pressure can be calculated from the degree of ionization, the temperature, and the density. The postshock gas velocity can be calculated from the usual conservation theorems of mass, momentum, and enthalpy (see Appendix). Table 1 shows the results of these calculations.

9-

3 I-

I 2 %TEMPERATURE (I04“K) 0 XENON A KRYPTON 0 ARGON

i ‘5-

4 8

9 F IO LOGlo PRESSURE (DYNES/CM*1 Fm.

II

1

To determine the temperatures which are obtainable with high-explosive shocking, it is necessary to couple the data in Table 1 with data on the velocities and pressures obtainable with explosives. The solid curve in Fig. 1 is the velocity-pressure data on Composition B explosive. It was taken from the work of DEAL(~)which was corrected for measured shock velocities in air and argon by MAUTZ.(4) The intersection of the gas curves with that of the explosive corresponds to the shock conditions, and this gives temperatures of 38,000, 31,000 and 23,OOO”Kfor xenon, krypton and argon, respectively. These curves were computed for a preshock standard temperature and pressure (STP) density. The xenon points which lie to the extreme left and right in Fig. 1 are computed from preshock densities which are l/10 and 10 times the STP density, respectively, and from a postshock temperature of 38,OOO”K. Thus we see from the low-density xenon point that the increased shock velocity approximately compensates for the increased degree of ionization such that the postshock temperature remains about the same as it was at STP density. At this low density, however, the actual temperature will be less as a result of radiative energy loss; this will be discussed in the next section. For high-density xenon, the decreased shock velocity has a greater effect on the temperature than the decreased ionization, and the temperature falls. The

842

W. G. VULLIJi!T

points for STP density which lie above the explosive curve show the temperatures which are attainable if the velocity and pressure of the explosive products could be increased. III.

THE

EMERGENT

RADIATIVE

FLUX

We will take the shock front to be an infinite plane to avoid complicating geometrical effects. First we examine the shocked gas to see if it is optically thick to Planckian radiation at the temperature calculated in the previous section. Using the approximation of RAIZER'~) for the continuous radiation mean free path, A, at a photon energy vkT in a gas containing pr ions per cubic centimeter and on the average Z free electrons per ion, h-1 = 4~11(1())“3-

Z3p12 ev T,,2 -v3 cm-l (T in “K).

Equation (1) shows that the photon mean free path is generally much shorter than the thickness of the shocked gas layer but longer than the gas kinetic mean free path for the shock parameters considered in Table 1. We also can compute the fraction of the total thermal energy of an optically thin element of gas which is radiating out through the shock front during the time the element is in the space between the shock front and one radiation mean free path behind the front. It is C7T4 F+= (2) Pl(Us - Ul)El

where El is the total thermal energy per ion behind the shock, Q is the Stefan-Boltzmann constant, and us and ur are the velocity of the shock and gas behind the shock, respectively. If r is of order unity, one might expect the effective shock temperature to be lower than the calculated value. Values of r are indicated in Table 1, and we see that except for the low density xenon, radiative losses through the shock front should not perturb the calculated temperature. In summary, we may say that if the mean free path for continuous radiation as calculated from equation (1) is long compared to a gas kinetic mean free path but short compared to the thickness of the shocked gas layer, and if the fraction r as given by equation (1) is small compared to unity, then the effective continuum temperature, seen at the shock front, should be the same as the calculated temperature shown in Table 1. If we wish to know the total radiation flux which falls on a target which is immersed in the cold, preshocked gas, we must estimate the absorption characteristics the preshocked gas presents to the shockfront radiation. We proceed by assuming the preshock gas to be un-ionized, in the ground state, and compute the energy absorbed to see if the temperature change is sufficient to invalidate these assumptions. This is done by assuming that thermodynamic equilibrium exists in the shock precursor-heated region, and we examine the characteristic times for ionic recombination to see whether this assumption is also invalid. Since the shock front radiates as a blackbody at temperature T, the average energy per atom, E, of the precursor-heated gas at density po cm-a is 8=-,

yoT4 POKS

(3)

Radiation from explosive-driven shocks in the noble gases

843

where y is the fraction of the shock radiation which is absorbed by the preshocked gas, and was taken to be the fraction of the blackbody radiation which lies at photon energies larger than the first excitation energy. For the shock temperatures and velocities in Table 1, E is less than 3 eV per atom, which, for Xe, means a temperature about 12,00O”K, and about 5 per cent ionization. Hence there is essentially no change in the radiation absorption characteristics when the gas is heated by the shock radiation. (MODEL(~)shows for very high shock temperatures that this is not the case and the precursor temperature determines the radiation which emerges into the cold gas.) Since photoionization is the principal radiation absorption mechanism in the precursor, the validity of the local thermodynamic equilibrium (LTE) assumption can be estimated by comparing the characteristic time for 3-body electron recombination with the lifetime of the precursor heated region, i.e. the mean free path of shock radiation in the preshocked gas divided by the shock velocity. Using BATES’values (7) for the recombination coefficient, we find that radiation-heated preshocked gas at STP density is essentially in thermal equilibrium, whereas gas at l/10 of STP density does not have sufficient time to come into thermal equilibrium. The effectiveness of bound-bound transitions absorbing some of the continuous shock radiation in the cold preshocked gas is estimated by calculating the equivalent width of the absorption lines. Assuming that the principal,line-broadening mechanism in the preshocked gas is collisions of excited atoms by neutral atoms, ADIBARTSUMYAN~~) arrives at a formula for the equivalent width at large optical depths in the line center, which can be rearranged to give WV = lo-4fplJ J(

4 vo )

(4)

where WV is the equivalent width in eV, f is the transition oscillator strength, I is the gas thickness in cm, and p is the density in cm-s; vo is the photon energy at the line center in eV. Assuming an oscillator strength of 0.5 for the xenon strong resonance line and taking I to be 100 cm of gas at STP density, WV is found to be 1.3 eV. Thus, it would seem that the bound-bound transitions can be effective in absorbing the shock continuum radiation at photon energies between that of the fist excited state and thephotoionization threshold. Assuming that only radiation reaches the target which has photon energies below the wing of the first resonance line, we calculate the target power densities, given in Table 1 under column Et, when the target is 100 cm from the shock front.

REFERENCES 1.

2.

C. R. MOORE, Atomic Energy Levels, Circular 467, U.S. Bureau of

Standards (1952). J. C. STEWARTand K. D. PYATT,Theoretical Study of Optical Properties, Air Force Special Weapons Center, Report AFSWC-TR 61-71, Vol. I, September 1961 (unpublished). W. E. DEAL, Phys. Fluids 1,523 (1958).

:: C. W. MAUTZ, Private communication. 5. Yu. P. RAIZ&, Zh. Eksp. Teor. Fiz. 10,769(1960)). 6. I. SH. MODEL.Zh. Eksu. Teor. Fiz. 5.589 (1957). 7. BATES,KIN&ON and -MCWFXIRTER, kroc. ioy. koc. (London) 267A, 297 (1962). V. A., Theoretical Astrophysics, Pergamon Press, New York (1958). 8. AMEZARTSUMYAN,

844

W.G.

VULLIET

APPENDIX

The strong-shock

conservation

equations for mass, momentum,

and energy are

PO& = pl(us--ul),

(1)

mpous2 = A + mpl(u8 - ud2,

(2)

m/2us2 = HI +m/2(u8- UI)~,

(3) where the subscripts 0 and 1 refer to preshock and postshock conditions, respectively. The u’s are gas velocities with respect to a laboratory reference, the p’s are gas particle densities, m is the ion or atom mass, and P is the postshock gas pressure, which is given by Pl =

p1(1+ Z)kT,

(4) where 2 is the mean number of free electrons per ion, k is Boltzmann’s constant, and T is the temperature. His the mean enthalpy per ion and can be calculated from HI = 5/2(1+ Z)kT+

(5)

where NCis the number fraction of ions in the it” stage of ionization (i = 1 for the neutral atom) and x is the energy required to ionize an ion in the jth stage of ionization. In both the equations for P and H, 2 is defined by

The number fractions, Nt, can be found from the Boltzmann-Saha exp( -xt/k 0,

equation (7)

where the &‘s are the partition functions for the excited states of ions in the it” stage of ionization, me is the electron mass, h is Plan&s constant, and pe is the electron density, defined by pe =

Zp.

(8)

For a given temperature and postshock density, the author found it convenient to calculate the density ratio across the shock, x, the postshock gas velocity, U, and the shock velocity as follows. Subtracting equation (2) from equation (3) and then squaring equation (1) and subtracting (3) results in two equations which can be solved simultaneously for the density ratio, X, in terms of the temperature, enthalpy, and degree of ionization (a: -=

1

1+x

(l+%kT 2H1

x =

PI -. PO

(9)

Radiation from explosive-driven

shocks in the noble gases

845

By using equation (9), the square of the gas velocity and shock velocity are easily found in terms of the enthalpy and density ratio:

(10)