Radiation-induced shocks in oxygen

Radiation-induced shocks in oxygen

WILLIAM W. ZUZAK' AND BOYE AHLBORN Departnlent of Physics, University of British Co/umbia, Ya??couver, British Co/umbia

Received June 20, 1969

The formation of pressure waves, induced by the absorption of radiation in the Schumann-Runge continuum of oxygen, was studied experimentally and theoretically. The 10-ps radiation pulse, produced by a capillary discharge at a color temperature of 6 x lo4 OK, passed through an LiF window into a test chamber containing oxygen. Surprisingly, the propagation velocity of the pressure waves was found to be independent of the initial filling pressure p~ ( u = 364 k 8 mls); however, as expected, their point of formation behind the LiF window varied inversely as po. The evolution of the pressure waves was conlputed numerically on the basis of a simplified model and the method of characteristics modified to include the radiant-energy input. It appears that the wave is driven for some time after the decay of the radiation pulse by release of frozen dissociation energy.

Canadian Journal o f Physics, 47, 2667 (1969)

I. Introduction

Pressure waves and mass motion in a gas can be produced by the addition of energy. The energy may enter the gas through the surface by thermal conduction, it can be dissipated as joule heat, or it may be introduced by absorption of radiation. The last mechanism has been illustrated by many laser spark experiments and by the work of Elton (1964) on "radiation-induced shock waves external to quartz discharge tubes. " In order to understand this interaction between electromagnetic radiation and fluid motion, it is useful to investigate such waves in a plane geometry with simple boundary conditions.

Consider an experiment in which dissociating (or ionizing) radiation of intensity W passes through a window and enters a test chamber filled with gas of density p,. The gas in the vicinity of the window will be dissociated (or ionized) and a radiation front will propagate into the gas. In a recent paper (Ahlborn and Zuzak 1969, henceforth called I) we studied the steady- state solutions of this interaction analytically and found that three different types of radiation fronts can exist, depending on the magnitude of Wlp,. For extremely high values of Wlp,, a supersonic radiation front with little particle motion and a small increase of density is expected. At one particular value of Wlp,, the radiation front should act like a steady detonation and obey the Jouguet-point conditions. For relatively low values of Wlp,, which one can hope to obtain in the laboratory, the radiation front

'Present address: Institut fLir Experimentalphysik, Kiel, Germany.

behaves like a leaky piston: it accelerates and compresses the gas ahead and leaves dissociated (or ionized) gas of reduced density but increased pressure and temperature behind. The leading perturbation of the compressed region travels as a shock into the unperturbed gas, while the "piston" itself eats its way into the compressed gas with subsonic velocity. Hence, in I this type ofwave was called a subsonic radiation front with preceding shock. In this paper we describe an experiment in which it was hoped to produce such radiation-driven shocks.

In principle, it is not difficult to produce radiation fronts in a laboratory experiment of reasonable dimensions. One requires ( I ) a gas with a large photodissociation (or ionization) cross section in some frequency interval dv, (2) an extremely intense light source which radiates in this frequency interval for a long period of time, and (3) a window which is transparent to this radiation.

Oxygen satisfies the first requirement. It has a large photodissociation cross section of the order of lo-'' cm2 in the Schumann-Runge continuum, which we take to extend from 1280 A t o 1800 A. The width of the radiation front is of the order of the mean free path of the dissociating photons L - 1lctN. Thus, for a particle density N - 1 0 ~ ' c m - ~ , we have L - 1 cm and the width of the radiation front can be small compared with the dimensions of a reasonably sized oxygen container.

Lithium fluoride transmits radiation above 1050 A and therefore satisfies the third requirement.

The second condition is not as easy to fulfill.

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2668 CANADIAN JOURNAL OF PHYSICS. VOL. 47. 1969

FIG. 1. Capillary discharge light source. B = brass, E = epoxy, P = polyethylene, and S = mechanical shutter.

As calculated in I, a constant photon flux of computed numerically by the method of charac- 4.7 x lo2' photons cmP2 s-' is required to teristics, provided the reaction processes and produce a Mach 3 shock in oxygen at an initial rates within the radiation front are known. The pressure of 0.01 atm. Steady-state sources in the characteristic equations (modified to include the vacuum u.v. fall short of this value by several radiant-energy input) are presentedin Section IV. orders of magnitude. We finally chose a capillary The numerical results in Section V provide an discharge as light source, whichemitted an intense insight into the mechanisms driving the shocks. continuum with a color temperature of about 6 x lo4 OK for a period of about 10 ps (see Section 11).

Experiments performed with this light source did indeed produce weak shocks or pressure waves (see Section I l l ) . Unfortunately, the time-integrated intensity of this light source is

tion front. When the radiation is terminated, the insufficient to produce a fully-developed radia- 0

pressure wave which initially starts out as part of a developing radiation front separates from the region of high dissociation and travels ahead into the undisturbed gas Or less as a wave. FIG. 2. Light pulse of capillary discharge. Intensity The evolution of these pressure waves may be scale G in arbitrary units.

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ZUZAK AND AHLBORN: RADIATION-INDUCED SHOCKS IN OXYGEN

- radiation fron f

. .

0 d FIG. 3. Experimental arrangement to measure radiation-induced pressure waves.

11. The Light Source The radiation source used in this experiment

is similar to the capillary discharge described by Bogen et al. (1965). The vacuum-tight cylindri- cally symmetric apparatus, illustrated in Fig. 1, consists of a fiber-glass-strengthened epoxy mold in which two brass electrodes are imbedded and into which a threaded polyethylene rod of 1.9 cm diameter is inserted. This rod has a 0.2-cm diameter capillary along the axis. The electrodes are electrically connected to a 25 pF condenser bank, without a trigger gap, to avoid energy losses and electrical noise. To operate the light source, the gas pressure in the system is first reduced to a very low value, so that no breakdown can occur. The condenser is then charged to the desired voltage and the discharge is triggered by letting some helium gas into the vacuum system until a breakdown point on the Paschen curve is reached. As the current is constricted through the capillary, some polyethylene is vaporized from the capillary wall. This material forms a hydrogen -carbon-oxygen plasma of high temperature and density which emits a very intense continuum radiation in the axial direction. Unfortunately, the overpressure inside the capillary (Bogen quotes pressures of several hundred atmospheres) drives the plasma out of the ends and carbon deposits on any cold surface and the LiF window. Large dump chambers are therefore mounted adjacent to each electrode to disperse this spent plasma. However, in spite of this expansion volume, it is necessary to operate the source at only about 3 kV and clean the window after every second shot. A typical oscillogram of the light pulse is shown in Fig. 2. The discharge is

overdamped and the half width of the pulse depends slightly on the diameter of the capillary. The erosion of the spark channel makes it necessary to replace the polyethylene insert after some hundred shots.

The intensity of the light source was determined by comparison with a carbon-arc standard (Null and Louzier 1962). At a wavelength of 5000 A, the peak intensity of our source exceeds the carbon arc by a factor of (1.9 f 0.2) x lo3. In the visible wavelength region, the intensity distribution can best be described by a black- body Planck curve of 60 000 OK. The intensity seemed to be lower in the ultraviolet, however, the accuracy of the measurements is drastically reduced here due to the very large intensity difference of our source and the standard.

111. Experimental Results

The experimental setup to detect the radiation- induced pressure waves is illustrated in Fig. 3. The test chamber has a diameter of 5 cm and is filled with oxygen at the desired pressure. The radiation passes through the LiF window (aperture 1.7cm and thickness 0.6cm), is absorbed in the gas, and produces a pressure pulse which travels in the x direction. The dispersion of the pressure pulse in the radial direction has no detectable effect on the propagation of the pressure wave in the axial direction. This was checked by inserting a coaxial tube of 1.8 cm diameter directly behind the window. A piezo- electric pressure probe2 can be placed at any

'Pressure transducer LD-15/89 of the Atlantic Research Corp., Alexandria, Va.

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CANADIAN JOURNAL OF PHYSICS. VOL. 47, 1969

FIG. 4. Radiation-induced pressure waves S in oxygen of p, = 400Torr, monitored with uncalibrated piezo probe at several distances d. Arrival time of signal t,. Time scale 10 ps/division. Pressure scale 50 mV/division.

distance, d, opposite the LiF window and measures the time of arrival t, of the pulse. Typical oscillograms are shown in Fig. 4. The pressure signal S is superposed on a slowly- decaying background signal with a high-frequency fine structure. The background signal is probably due to thermal heating of the probe by the radiation pulse since it could be reduced by coating the probe surface with reflecting alu- minum paint. We made no special effort to eliminate the background signal since it did not interfere with the measurement of the arrival time, t,, of the shock.

The oscillograms show that the pressure wave does not form immediately behind the window. At a distance d = 0.5 cm, Fig. 4(a), the pressure pulse is not visible, however at distances d = 1.0 and d = 1.5 cm, the signal is clearly present. The point of formation, x,, of the pulse was determined from similar oscillograms for several filling pressures. Results are plotted in Fig. 5. At high pressures the shock forms very close to the window, while at low pressures the pressure pulse is only detected several centimeters away from the window. For a given filling pressure, the amplitude of the pressure signal increases with distance to a maximum and then decays gradually.

The arrival time, t,, was measured as a function of d for several initial pressures p,. Figure 6 shows the result for p , = 400 Torr. The slope of the measured points gives the propagation velocity, v, of the pressure wave. We found v =

u 0 250 Torr 500

FIG. 5. Point of formation of pressure wave xr as function of filling pressure po.

364 + 8 m/s for all initial pressures. This result is surprising. One would expect v to depend onp, and the velocity should approach the speed of sound of room-temperature oxygen (328 m/s) a t distances far from the window. Two explanations for this unexpected behavior come to mind. Firstly, the absorption cross section for radiation outside the Schumann-Runge continuum is small but not zero. Therefore, this radiation penetrates

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ZUZAK A N D AHLBORN: RADIATION-INDUCED SHOCKS IN OXYGEN 2671

FIG. 6. Propagation of pressure pulse. p, = 400 Torr.

much further into the gas and causes preheating at distances far ahead of the radiation front. If the temperature of the gas ahead of the pressure pulse were increased by only 69 OK by this effect, the speed of sound in the molecular oxygen would be as high as the measured propagation velocity. Secondly, there is little recombination during the lifetime of the radiation pulse. No complete equilibrium is expected amongst the internal degrees of freedom. An unproportionally large fraction of the energy is stored in the form of dissociation energy. As recombination sets in, via three-body collisions, this energy is released and appears partly as random kinetic energy which implies an increase of temperature and pressure of the gas. Therefore, the frozen dissociation energy may continue to drive the pressure wave long after the light pulse itself has decayed. We will return to these two ideas in a later section.

shock should form quickly in high absorber density, but it would have a long development time in a low absorber density, because the induced motion would start slowly in this case. The density dependence of the formation time (or conversely of x,) for the pressure pulse is, in fact, demonstrated by the experimental results of Fig. 5.

To calculate the evolution of the pressure wave, one must first of all know the rate of energy input per unit mass q(x , t), which is related to the photon flux F(v, x, t) and to the absorber density N(x, t). N is generally below the value p/M (where M is the mass of the molecule) since the photodissociation depletes the total number of absorbers and results in a high degree of nonequilibrium. Because of the relatively long equilibrium times between the atomic and

IV. Model Descri~tion of the Radiation Front 4 f' Let us now imagine how a pressure wave

develops. At the beginning, most of the dissociating radiation is absorbed over the distance L = l/Noct near the LiF window. If a sufficiently large number of molecules is dissociated, the temperature and pressure will be much higher in this region than further away from the window. This pressure gradient drives the particles in the x direction into the test tube. The gradient depends

= UJ + 5

P I I/ ) d t f n t h =u,

2

e x On L-' and is therefore P ~ ~ P ~ ~ ~ ~ ~ ~ ~ ~ to the FIG. 7. Characteristics ( 0 ) and path lines density No of the oxygen molecules. Hence a (0-- 3) in the X-t plane.

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2672 CANADIAN JOURNAL OF PHYSICS. VOL. 47, 1969

molecular species, chemical equilibrium can, in general, not be assumed. It is therefore necessary to use a conservation equation for the absorbing particles. This requires a knowledge of the rate coefficients of the kinetic reactions. The particle drift velocity and the state of the gas during the evolution of the pressure pulse is finally computed by the methods of characteristics.

To facilitate the calculation, a few simplifying assumptions are made: (1) Only the reactions

h v + O , - O + O 11 I

O , + P $ - O + O + P

are considered in the radiation front (P stands for either the atom 0 or the molecule 0, respectively). (2) The radiation field is assumed to be plane, so that the analysis is strictly one dimensional. (3) The absorption of oxygen atoms is completely neglected. (4) Radiation outside the Schumann- Runge continuum is not taken into account.

IV.l. The Rate of Energy Input The decay of a beam of photons F(v, 0 , ~ ) dv passing through the LiF window into oxygen with

absorption cross section u(v) and absorber density N(x, t) is governed by the differential equation

This follows from the radiative transfer equation, where emission has been neglected. The equation may be integrated with respect to x and v to give the total photon flux

~ ( v , 0, t) exp [- u(v) sox N ( x , t) dx] dv

where the frequency interval v, to v, corresponds to the wavelength interval 1800 A to 1280 A. If we assume that the light source radiates as a black body at a temperature Tthroughout the spectrum and that this distribution is not a function of time, the photon flux at x = 0 may be expressed as

[4 1 F(v, 0 , t ) dv = const. x ~ ( t ) v ~ ( e " ' ~ ~ ~ - 1)-'dv

where the time-dependent factor G(t) accounts for the observed time variation of the intensity (see Fig. 2).

The energy flux E(x, t) is obtained directly from [3] by multiplying the photon flux by the energy of the photon, hv. Then differentiating this equation with respect to x, we obtain the rate of energy input per unit volume, ijv:

a q x , t) V 2

P I g v ( ~ . t) / = ~ ( x , t) Jvl u(v)jlv F(v, 0, t) exp - u(v) N(X, t) dx dv [ s: I where F(v, 0, t) is given by [4]. It is convenient to introduce the degree of dissociation y by the equation

16 I N(x9 t) = (1 - y)(plM)

where p is the local mass density and M the mass of the oxygen molecule. Equation [5] then becomes

[7] p(x, t)q(x, t) E &(x, t) = (1 - y) % Jv: u(v)hvF(v, 0, t) exp

where q(x, t) is now the rate of energy input per unit mass.

IV.2. DzfSerential Equation for the Absorbing Particles For the model under consideration, there are only two types of particles. From conservation of

mass, a decrease in the number of molecules must be balanced by an increase in the number of atoms. The differential equations governing the concentrations of the two types of particles differ only by a factor of two;

C81 an, a m l - + - = - 2 - + - at ax ( a:xN)

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ZUZAK AND AHLBORN: RADIATION-INDUCED SHOCKS IN OXYGEN 2673

where u is the macroscopic flow velocity and n, is the particle density of the oxygen atoms. The equation for the conservation of molecules may be written

where k is the reaction rate coefficient and the subscripts d and r denote dissociation and three-body recombination, respectively. aF/ax is equal to the number of molecules which have been photo- dissociated per unit volume and unit time. This term can be obtained by differentiation of [3]. The number density N may be eliminated by use of [6] such that [9] becomes

In order to use this equation in our computations, we need numerical values of the four reaction rate constants in the temperature range from 300 OK to 7000 OK. Collisional dissociation and three- body recombination coefficients in oxygen have been measured by various workers behind shocks at approximately 4000 OK (Rink et al. 1961 ; Camac and Vaughan 1961 ; Mathews 1959). Statistical thermodynamics predicts that the ratio of the reaction rate constants of a gas in thermodynamic equilibrium depends only on the temperature. By the principle of detailed balancing, we have

[I11 k d , kd2 = K(T) = AT-'/' exp (- D / ~ T ) moles cm-' - kr, kr2

where D is the dissociation energy, K(T) is the chemical equilibrium constant, and A is an empirical constant, determined experimentally. We use the experimental value A = 1.15 x lo3 of Rink et al. and assume that [ l l 1, which was derived for equilibrium conditions, is applicable in our unsteady nonequilibrium problem as well (Hurle 1967).

Following Rink, the dissociation coefficient is of the form

[I21 kd = B(D/kT)" exp (-DlkT)

where B is some constant and the exponent n is believed to have a value between one and three. Zel'dovich and Raizer (1966) point out that oxygen atoms are about three times as effective as the molecules in recombination reactions. Thus, with Rink's numerical values, we have

kd2 = +kd, = 3.06 x 10-3~-3/2e-D1kT cm3 particle-' s-' [I31 kr2 = +kr, = 4.40 x. I O - ~ O T - ' cm6 particle-' s- ' Remember that the experimental values for k, of Rink et al. were fitted to the function T-' around 4000 OK and the values may be seriously in error when extrapolated down to room temperature.

IV.3. Characteristic Equations with Energy Input The calculation of fluid-flow problems by the method of characteristics is a standard technique.

See, for instance, Shapiro (1954), Chapters 23-25, or Oswatitsch (1957), Chapter 3. Haskin (1964) describes a method of computation at fixed time intervals which is particularly applicable to our case. However, it must be modified to include the time-varying energy input supplied by the radiation.

Let us consider the x-t plane shown in Fig. 7 and assume that the state of the gas at points 1 and 2 is known completely. Any pressure disturbance will travel with the velocity u f c where u is the particle velocity and c the local speed of sound. The plus and minus signs stand for right and left travelling waves, respectively. The loci of the disturbances

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are called Mach lines or v (plus sign) and 5 (minus sign) characteristics. The loci of the particles

are called path lines. The q and 5 characteristics from points 1 and 2 in Fig. 7 intersect at some point 3. The basis of the method of characteristics is that along the Mach lines and path lines the thermodynamic quantities vary according to certain specified equations (see below) such that the state and velocity of the gas at point 3 may be calculated.

It is convenient to introduce Lagrangian notation, i.e., describe the motion in coordinates which are attached to the center of mass of the local volume elements. In Lagrangian coordinates, z, [14] and [15] may be written as

1161

The energy equation has a simple form along the path lines, since no kinetic energy need be considered :

where h is the enthalpy per unit mass and the rate of energy input per unit mass, q, is given by [7]. Following Haskin, the equations of momentum and mass in Lagrangian coordinates take the form

We now multiply [20] by the speed of sound c and add a term (I/pc)/(ap/at), to each side to obtain

We define the effective adiabatic exponent g to be the ratio of the enthalpy to the internal energy (g = hlu), so that the equation of state takes the form (Ahlborn and Salvat 1967)

Using [22] and [IS], we may rewrite the right-hand side of [21] in terms of the energy input q. We obtain

We now add and subtract [I91 from [23] to obtain two equations in characteristic form

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Thus, along the Mach lines, ( d~ ld t ) , , ~ = + pc/po, we have

It is interesting to note that the speed of sound c in the above derivation was introduced ad hoc and has not yet been defined. Classically, the speed of sound is defined in terms of the rate of change of pressure with respect to density at constant entropy,

where y" is the isentropic exponent. For plasmas in thermodynamic equilibrium, replacing y" with the effective adiabatic exponent g may result in an error of the order of 10% (Ahlborn 1966). However, for nonequilibrium plasmas, the speed of sound depends on the frequency of the sound wave (see Zel'dovich and Raizer 1966), such that the validity of the approximation y" = g is questionable. Nevertheless, in the absence of better knowledge, we assume c2 = gplp in our calculations such that the last term in [24] vanishes.

We have made no attempt to prove the mathematical rigor of the characteristic equations [24]; further work along such general lines appears to be necessary.

V. Numerical Results

For the numerical calculations, the thermodynamic quantities and velocities are made dimensionless: jj = pip,, jS = p/po, 2 = clc,, ii = ulc,, etc., where the index 0 refers to the initial values and the bar to a dimensionless quantity. g and y are already dimensionless. The Lagrangian spatial coordinate, z, and the time t are made dimensionless by the transformations

where for convenience we choose a, to be the maximum value of the absorption cross section in the Schuinann-Runge continuum (a, = 14.94 x 10-I8 cm2).

The various free parameters were chosen to approximate the actual experimental conditions as closely as possible. The photoabsorption cross section a(v) between 1280a and 1800 was taken from the data of Metzger and Cook (1964) and approximated by 21 points. The time- dependent factor G(t) was approximated numerically by 29 points taken from a typical oscillogram such as Fig. 2. The frequency distribution of the

photon flux was assumed to be that of a black body at 6 x lo4 OK. The computations were carried out for two values of the peak photon flux, Fo = 1.16 x 10'' and 0.232 x lo2' photons cm- ' S- and for an initial pressure of 0.1 atm. Results for an initial pressure of 1.0 atm have been reported in a previous note (Zuzak and Ahlborn 1969).

The dimensionless equations, an outline of the method of solution, and some details of the computer program are given in the appendix. It is necessary to take small intervals in the spatial and temporal coordinates and thus the computation times are rather long. The computations were terminated after 80 iterations in the time interval At, which corresponded to 16 ps. The result is a set of graphs giving pressure, particle velocity, density, degree of dissociation, energy input, and temperature as a function of position x for various times after the initiation of the light pulse. The example in Fig. 8 is calculated for Fo = 1.16 x cm-' s-' and k,, = 4.4 x T-' cm6 particle-2 s-I. Although the radiation front is not fully developed, the concept of the "leaky piston" is quite evident at t = 16 ps. One recognizes the compressed region (shock wave)

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CANADIAN JOURNAL OF PHYSICS. VOL. 47, 1963

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ahead of the energy absorption zone (radiation 40

front). The temperature in the compression zone is only slightly increased, but rises sharply inside the radiation front. The density drops below the 3 5

initial value in the energy-absorption region, the missing mass being accumulated in the compres- ,, sion zone. The pressure wave travels at 564 m/s and will eventually coalesce into a sharp shock p, front.

To illustrate the effect of recombination, k, was set equal to zero and the calculation repeated. The pressure profiles for frozen dissociation are 20

shown in Fig. 9(a). Figure 9(b) compares the pressure profiles a t t = 16 ps for the two cases. Over this relatively short time interval, the pulse shape and velocity are apparently not much influenced by recombination. However, without recombination, the peak amplitude decays at a Lo- faster rate and the pressure (and temperature) in the rarefaction region are substantially lower. The maximum degree of dissociation next to the 3 5 -

LiF window is y = 0.30. With full recombination, the degree of dissociation decreases at a rate AylAt = 0.12% per ps. Consequently, recom- 30-

bination releases energy over several hundred E microseconds and may continue to drive the 25- pressure disturbance long after the primary light pulse has decayed.

Figure IO(a) shows the pressure profiles for a 20-

photon flux, Fo = 0.232 x 1OZ2 cm-2 s- l . The curve at 16 ps is compared with that for Fo = 1.16 x loz2 cm-2 s-' in Fig. 10(b). The velocity 15-

of the pulse is now about 400 m/s and it has decreased by 5% in the time interval from 6 to 16

(b)

-

I I I I I

ps. The peak pressure decays rapidly after loo 02 04 0 6 ~ ( c m ) 08 10 12

t = 4 ps so that the maximum pressure gradient FIG. 9. Recombination effects in radiation front.

decreases similarly. (a) Pressure profiles at various times, recombination The propagation velocity of the computed neglected. (b) Comparison of pressure profile at t = 16

pressure pulse in Figs, 8,9, and 10 is PS, recombination included (-0-*) and neglected (-). Initial conditions as in Fig. 8.

higher than the experimental value of 364 + 8 m/s. Also, the profiles do not support the strength and coalesce into a shock). The lack of experimental result that, as the distance between agreement between the computational and the LiF window and the piezo probe is increased, experimental results is likely due to inaccuracy in the amplitude of the signal first builds up to a the value of the assumed photon flux, Fo, and maximum and then slowly decays. This indicates due to neglect of the radiation outside the that the actual pressure gradient is not as sharp Schumann-Runge continuum region for which as the computations predict (i.e., more time is the absorption cross section is small but not required for the pressure pulse to build up in negligible. The presence of vibrationally-excited

FIG. 8. Structure of radiation front in oxygen a t various times after initiation of light pulse: (a) pressure, (b) particle velocity, (c) density, (d) degree of dissociation, (e) energy absorbed per unit mass per second, and (f) temperature. Recombination effects are included. po = 0.1 atm, co = speed of sound at room temperature To, Fo = 1.16 x 10" s-I, and x = distance measured from the LiF window.

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FIG. 10. Pressure profiles of radiation front in oxygen generated with low radiation intensity. Fo = 0.232 x loZZ cm-2 s-I , po = 0.1 atm, and To = room temperature. (a) Development of pressure wave. (b) Comparison of pressure profiles generated with high and low photon flux at t = 16 ps.

"windows" in which the minimum cross section

(a) is of the order of lo-" cm2. The radiation in these "windows" obviously penetrates deep into the test chamber. If the transmission of the LiF window in this spectral region were known (and it depends critically on the condition of the surface of the window) the radiation in this region could be included in the theoretical computer calculations in a straightforward manner. According to Blake et al. (1966) the absorption cross section of molecular oxygen from 1800 A to 2000 A decreases from cm2 to 4 x lo-', cm2. Obviously, most of the radiation in this spectral region passes right through the absorption cell.

It is possible that vibrationally-excited diatomic 2 molecules and ozone play an important role in a

dissociation front in oxygen. Because equilibra-

molecules and ozone may also have an effect on the steepness of the pressure gradient.

10-

3.5

30

P B 2.5

2.0

VI. Improvements of the Model and Experiment

The most obvious improvement in the theoretical model would be to consider the radiation outside the Schumann-Runge continuum. The absorption cross section of molecular oxygen between 1050 A and 1280 A is extremely ir- regular. In this region, Metzger and Cook (1964) list three prominent absorption bands and seven

Warneck 1967). Ozone is known to have a region of continuous absorption between 2300 A and 2900 A with an absorption coefficient of

l,; ; 12.7 x 10-l8 cm2 at 2537 A. One can conclude from these considerations

tion between vibrational and translational degrees of freedom requires about lo7 collisions (Black-

- (b) man 1956), one could expect a high degree of nonequilibrium between these two degrees of freedom within the radiation front. Also,

- according to Hudson et al. (1966), these excited molecules absorb in a different part of the spec-

- trum than ground-state molecules. The forma-

6 = 1 , , 6 x 1 0 ~ m - 5 - , tion of a triatomic ozone molecule is a more

that it would be more realistic to consider four types of particles separately: 0, O,, O,, and vibrationally-excited molecules. It would then be necessary to use four different reaction equations similar to [lo]. Although the incorporation of these equations into the numerical computations is relatively straightforward in principle, (Zuzak 1968) it requires a knowledge of 24 reaction rate constants as a function of temperature for the various collisional processes and four photoabsorption cross sections as a function of wavelength. Unfortunately, most of these quantities are as yet unknown and therefore it does not seem practicable to attempt such computations at the present time.

Experimentally, it is necessary to obtain absolute measurements of the radiation intensity

- likely occurrence than the formation of a diatomic molecule at low degrees of dissociation and at relatively low temperatures (Sullivan and

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ZUZAK AND AHLBORN: RADIATION-INDUCED SHOCKS IN OXYGEN 2679

passing through the LiF window in the wavelength region from 1050 A to 2000 A. Also, detailed measurements of the pressure profile would further the understanding of the development of the radiation-induced pressure waves. Attempts in both directions are presently under way.

W. Conclusion

We have generated pressure waves in oxygen by absorption of a pulse of continuous radiation, and we have measured their propagation velocity and position of formation. The development of these waves was also investigated in an analytical model, which includes photodissociation and collisional recombination and dissociation. The calculation shows that the wave approaches the structure of a steady subsonic radiation front with preceding shock within about 10 ps after initiation of the light pulse. The quantitative agreement of calculated and measured propagation velocity is only fair. This indicates that other processes with relatively small photoabsorption cross sections, such as absorption by

ozone and vibrationally-excited molecules, con- tribute significantly to the dynamics of the subsonic radiation front. These processes, however, could not be included in the model, since some of the required reaction rate constants are not known. In order to study in a similar experiment the other types of steady radiation fronts, namely detonation-like and supersonic radiation fronts, the intensity and duration of the light pulse would have to be substantially increased.

Acknowledgment We would like to thank Mr. Ricardo Ardila for

help with the experiment, and Mr. H. Baldis, Mr. R. Morris, and Dr. J. H. Williamson for assistance with the development of the computer program. The authors are indebted to Prof. W. Lochte-Holtgreven, Kiel, for making the computer facilities of the Institut fiir Experimental- physik, Kiel, available, and to Mr. H. Carls for his interest in the computation and the plotting of the result. The work was supported by a research grant of the Atomic Energy Control Board of Canada.

Appendix

An excellent treatment of the equations of unsteady flow with no energy input by the method of characteristics is given by Haskins (1964) in a form directly applicable for calculations on a computer. Here Haskin's treatment is extended to include energy input.

Figure 11 shows the typical mesh in Lagrangian space and time. In order to determine conditions at the point D on the t i+, baseline, we require values of the flow variable at A and B on the ti baseline which may be obtained by linear or quadratic interpolation between the known values at zj-,, zj, and zj+ ,. The dimensionless equation of Section IV may be written in the difference form:

LA61 (zD - z,) = 0

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2680 CANADIAN JOURNAL O F PHYSICS. VOL. 47, 1969

ay YD - YB [A81 - = ------ = G(~)AoNS(~ - y) A(i)c2(es - I)-' exp at ~t

For the calculations, we use the equations of state

where T is the kinetic temperature. The enthalpy per gram

P = Etrnns + Erot + - + &vib + Edis

P is used in the form

/I = ---- - hv'

[7/2 + 3 y/2 + (1 - y) - (ehV1lkT - 1) - ' + yD/k T] ~ + Y P k T

with D = 5.1 eV and hvt/k = 2228 OK. The following abbreviations and numerical values have been used in the above set of equations :

AONS = MFo/poc0 = (McO2/(hv))*CONS = 0.131

BONS = 3 . 0 6 ~ 10-3/(amco~0312) = 0.1 195 x 10'

CONS = F ~ / ~ ~ c ~ ~ = Fo(hv)'/poco3 = 30.0

DONS = po*4.4 x 10-30/(MamcoTo) = 0.08

po/M = 2.69 x lo-'' cm-3

c = hv/kT

A([) = a(C)lam

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ZUZAK AND AHLBORN: RADIATI ION-INDUCED SHOCKS IN OXYGEN 2681

The numerical values of the constants are evaluated for an initial pressure ofp, = 0.1 atm and Fo = 1.16 x loz2 photons cmP2 s-'. A(6) is the dimensionless absorption coefficient normalized such that A(6) = 1 at the maximum value ct, = 14.94 x lo-'' cm2 at 1420 A. The time factor G(r), see [4], is normalized such that G = 1 at the instant of maximum intensity, go = 1.4. The subscripts A, B, C, and D in the above equations refer to points in Fig. 11. The square bracket indicates a value~calculated at the midpoint of the Mach or path line; for example, point E of Mach line AD. This is done by averaging the value at A and the calculated value at D. It should be noted that the value (ag/at), in [24], which should be evaluated along a path line at E and F, was actually taken at the point G, i.e.,(Ag), = g, - g,.

The procedure for obtaining a solution at a point D is as follows:

1. Calculate approximate values of y, and qD from [A81 and [A9].

2. Calculate u, and p, from the sum and difference of [A1 ] and [A21 (in effect, using values of the thermodynamic quantities at points A and C which are located approximately from [A41 and [A5]).

3. Calculate / I , from [A3], and p,, TD, and g, from eqs. [A101 to [A12].

4. Repeat step 3 using calculated values of p, in [A31 and obtaining new values of p,, TD, and g,.

5. Calculate c, from [A13]. 6. Check for convergence of u, y, and c

(usually to an accuracy of 0.001). 7. If no convergence is obtained, go to step 1

and repeat. 8. If convergence is obtained, calculate x,

from [A7]. 9. Move to next point on spatial coordinate

and repeat steps 1 to 8. Special procedures are necessary for the first

point at the LiF window and for the final point at the end of the interval. The point of formation of a shock may be located by calculating the point where the right-flowing Mach lines first intersect. The shocks may be treated according to the procedure outlined by Haskin (1964).

For the case of no energy input, the choice of the difference intervals At and Az depends upon how well a straight line approximates the actual curved Mach lines. In our case, the photon flux imposes an additional constraint in that the

FIG. 11. The mesh in Lagrangian coordinates.

interval Az must be small enough such that the energy absorbed in the interval is small compared with the energy incident upon it (i.e., such that straight-line segments may be used to approximate the exponential-like decay curves), For the curves shown in Figs. 8-10, we used dimensionless intervals of At = 0.264 r (0.2 ps) and 2 = 1.056 E (0.0263 cm).

AHLBORN, B. 1966. Phys. Fluids, 9, 1873. AHLBORN, B. and SALVAT, M. 1967. Z. Naturforsch. 22a,

260. AHLBORN, B. and ZUZAK, W. W. 1969. Can. J. Phys.

47, 1709. BLACKMAN, V. 1956. J. Fluid Mech. 1, 61. BLAKE, A. J., CARVER, J. H., and HADDAD, G. N. 1966.

J. Quant. Spectry. Radiative Transfer, 6,451. BOGEN, P., CONRADS, H., and RUSBULDT, D. 1965. Z.

Physik, 186, 240. CAMAC, M. and VAUGHAN, A. 1961. J. Chem. Phys. 34,

460. ELTON, R. C. 1964. Plasma Phys. (J. Nucl. Energy, Pt.

C), 6, 401. HASKIN, N. E. 1964. Methods Computational Phys.

3. 265. -, HUDSON, R. D., CARTER, V. L., and STEIN, J. A. 1966.

J. Geophys. Res. 71, 2295. HURLE. I. R. 1967. Rept. Progr. Phys. 30, 149. MATHEWS. D. L. 1959. Phvs. Fluids. 2. 170. METZGER,' P. H. and C ~ O K , G. R.' 1964. J. Quant.

Spectry. Radiative Transfer, 4, 107. NULL, M. and LOUZIER, W. 1962. J. Opt. Soc. Am. 52,

1 1 56 O S W ~ ~ ~ ~ S C H , K. 1957. Gas dynamics (Academic Press,

New York). RINK, J . P., KNIGHT, H. T., and DUFF, R. E. 1961. J .

Chern. Phys. 34, 1942. SHAPIRO, A. H. 1954. The dynamics and thermodynamics

of conlpressible fluid flow (Ronald Press Co., New York).

SULLIVAN, J. 0. and WARNECK, P. 1967. J. Chern. Phys. 46, 953.

ZEL'DOVICI~, YA. B. and RAIZER, Yu, P. 1966. Physics of shock waves and high-temperature hydrodynamic phenon~ena (Academic Press, New York).

ZUZAK, W. W. 1968. PI1.D. Thesis, University of British Colun~bia, Vancouver, British Columbia.

ZUZAK, W. W. and AHLBORN, B. 1969. Physica, 41, 193.

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Radiation-induced shocks in oxygen

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