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BARC/1999/E/043

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REDISTRIBUTION OF THERMAL X-RAY RADIATION IN CAVITIES: VIEW-FACTOR METHOD AND COMPARISON WITH DSN CALCULATIONS

by M.K. Srivastava Theoretical Physics Division and Vinod Kumar and S.V.G. Menon Solid State & Spectroscopy Group

31/30 1999 BARC/1999/E/043

GOVERNMENT OF INDIA ATOMIC ENERGY COMMISSION

REDISTRIBUTION OF THERMAL X-RAY RADIATION IN CAVITIES:

VIEW-FACTOR METHOD AND COMPARISON WITH DSN CALCULATIONS

by M.K. Srivastava Theoretical Physics Division and Vinod Kumar and S.V.G. Menon Solid State & Spectroscopy Group

BHABHA ATOMIC RESEARCH CENTRE MUMBAI, INDIA 1999 BARC/199S/E/043

BIBLIOGRAPHIC DESCRIPTION SHEET FOR TECHNICAL REPORT (as per IS : 9400 -1980)

01 Security classification: Unclassified

02 Distribution: External

03 Report status: New

04 Series : BARC External

05 Report type: Technical Report

06 Report No.: BARC/1999/E/043

07 Part No. or Volume No.:

08 Contract No.:

10 Title and subtitle: Redistribution of thermal x-ray radiation in cavities: view factor method and comparison with DSN calculations

11 Collation: 39 p., 9 figs.

13 Project No. :

20 Personal authors): 1) M.K. Srivastava 2) Vinod Kumar, S.V.G. Menon

21 Affiliation of authors): 1) Theoretical Physics Division, Bhabha Atomic Research Centre, Mumbai 2) Solid State and Spectroscopy Group, Bhabha Atomic Research Centre, Mumbai

22 Corporate authors): Bhabha Atomic Research Centre, Mumbai - 400 085

23 Originating unit: Theoretical Physics Division, BARC, Mumbai

24 Sponsors) Name: Department of Atomic Energy

Type: Government

Contd... (ii) -l- 30 Date of submission: December 1999

31 Publication/Issuedate: January 2000

40 Publisher/Distributor: Head, Library and Information Services Division, Bhabha Atomic Research Centre, Mumbai

42 Form of distribution: Hard copy

50 Language of text: English

51 Language ofsummary: English

52 No. of references: 24 reft.

53 Gives data on:

Abstract :A view-factor method for studying distribution of tbermal x-ray radiation inside a hohlraum cavity is developed. This problem is of much relevance these days in an indirect-driven inertial confinement fusion (1CF) system where, one is supposed to optimize the irradiation pattern to derive maximum coupling with the fusion capsule. The x-ray reemission factor from the hohlraum wall is calculated by solving the instantaneous flux conservation equation, which is coupled with scaling laws derived from self-similar solutions of a one-dimensional planar radiation hydrodynamic equations. The method is applied to a gold capillary hohlraum with two typical primary sources forirradiation , viz., a disc source and a symmetric two-ring source. These two sources are of direct relevance to the actual implosion experiments. The relevant view factors are derived analytically in the present report. The earlier literatures used die same derived from a two-dimensional numerical integration in r-0 geometry. Our calculations show excellent agreement with those obtained from the exact numerical simulation. Apart from this, it also reproduces extremely well the actual experimental results obtained for a gold capillary hohlraum heated by a disc source. Further, to test our model against some standard technique, we also solve the same probtem using well known discrete ordinate (DSN) method. This method is applied to a steady-state radiation-transport problem in two-dimensional r-z geometry. The primary sources on the hohlraum surface are used as boundary conditions for the problem. Numerical results of this method (which can be generalized to solve even more complex radiation- transport problems) for the capillaries are compared with those of the view factor method. Excellent agreement is found between the two results.

70 Keywords/Descriptors : X RADIATION; ICF DEVICES; NUMERICAL SOLUTION; VALIDATION; IMPLOSIONS; SYMMETRY; ENERGY TRANSFER; IRRADIATION

71 INIS Subject Category: G5110;G3620

99 Supplementary elements:

-n- Redistribution of Thermal X-ray Radiation in Cavities:

View Factor Method and Comparison with DSN Calculations

M. K. Srivastava1, Vinod Kumar2 and S. V. G. Menon2

1 Theoretical Physics Division 2 Solid State & Spectroscopy Group Bhabha Atomic Research Centre, Mumbai-400 085, INDIA

Abstract A view-factor method for studying distribution of thermal X-ray radiation inside a hohlraum cavity is developed. This problem is of much relevance these days in an indirect-driven inertial confinement fusion (ICF) system where, one is supposed to optimize the irradiation pattern to derive maximum coupling with the fusion capsule. The X-ray reemission factor from the hohlraum wall is calculated by solving the instantaneous flux conservation equation, which is coupled with scaling laws derived from self-similar solutions of a one-dimensional planar radiation hydrodynamic equations. The method is applied to a gold capillary hohlraum with two typical primary sources for irradiation, viz., a disc source and a symmetric two-ring source. These two sources are of direct relevance to the actual implosion experiments. The relevant view factors are derived analytically in the present report. The earlier literatures used the same derived from a two-dimensional numerical integration in r-9 geometry. Our calculations show excellent agreement with those obtained from the exact numerical simulation. Apart from this, it also reproduces extremely well the actual experimental results obtained for a gold capillary hohlraum heated by a disc source. Further, to test our model against some standard technique, we also solve the same problem using well known discrete ordinate (DSN) method. This method is applied to a steady-state radiation-transport problem in two-dimensional r-z geometry. The primary sources on the hohlraum surface are used as boundary conditions for the problem. Numerical results of this method (which can be generalized to solve even more complex radiation-transport problems) for the capillaries are compared with those of the view factor method. Excellent agreement is found between the two results. 1. Introduction

Study of thermal radiation distribution inside a closed cavity is of much interest these days, particularly, in connection with inertial confinement fusion (ICF)'. A very important issue in these ICF schemes is the symmetric implosion of the ICF target. In the usual direct-driven scheme this symmetry is achieved (with some acceptable degree of imperfection) by irradiating a large number of incident laser or ion beams on to the fusion target. However, it is seen that any deviation from the perfect symmetry (caused either by imperfection in the fuel surface itself or due to nonuniform illumination of the ICF target) is known to magnify in the due course (the so-called Rayleigh Taylor instability), resulting finally into hampering of the fusion process.

Now, in order to achieve the perfect symmetric implosion, a novel scheme has been thought of, what is popularly known as indirect-driven fusion2'5. This scheme is based on the concept of a radiation cavity known as hohlraum6 .In this scheme, the driving laser or particle beams first generate thermal X-rays inside a high-Z (Z is charge number) cavity. These X-rays are then repeatedly absorbed and reemitted by the case walls till a desired equilibrium thermal radiation (known as Planckian radiation) is obtained. Finally, this thermal radiant energy is deposited on to the surface of a fusion capsule, placed at the center of the cavity (see Fig. 1), in a nearly perfectly symmetric manner.

Another advantage of the indirect-driven scheme is that, unlike in the direct-driven system, it does not require the laser beams to be of very high optical quality and that they be arranged very symmetrically around the fusion capsule. Moreover, in the direct-driven system additional problem arises due to excitation of the collective plasma instabilities by the laser light. Much energy is lost in the form of hot electrons generated by the plasma instability. In the indirect-driven system what we get is just an energy- source in the form of intense and isotropic radiation and not the oscillating electric and magnetic fields of the laser light that might excite the plasma waves.

However, the above advantages are not, without a cost. It is apparent that the indirect- driven scheme is energetically more demanding than the direct-driven one. Not only is a part of the laser energy converted into the X-rays, but also a certain amount of energy is required to maintain the radiating temperature of the cavity wall. Thus, less energy is left available for the actual implosion of the ICF target. Moreover, additional constraint comes from the geometric configuration. The coupling of the radiant energy to the fusion' capsule is characterized roughly by the ratio of the surface areas of the fusion capsule and the reemitting cavity wall. This ratio, obviously less than one, can not be fixed very large due to practical design limitation.

Apart from application in the ICF, there are other wide range of interesting phenomena that can be studied by using a high-Z cavity. For example, using the currently available laser pulse of intensity ~ 1014 W/cm2 and duration ~ 10 nsec, one can expect to heat up the cavity wall (of diameter ~ a few mm) to a temperature of about 5xl06 °K. This temperature is close to what we observe in the interior of sun (~14xlOs degree K). Thus, the man made cavity can be used to investigate matter in a state of high density and temperature in the laboratory, which generally prevails only in stars. Another application of the high-temperature cavity could be in the field of material reprocessing where a suitably designed substance is treated by the focussed heating of a radiant energy flux.

Hohlraum: a miniaturized sun

A high-temperature cavity produced in the laboratory simulates in many respects the radiation environment prevailing in the sun6. In fact, the very concept of hohlraum to generate a very intense, isotropic and incoherent radiation originates from the sun itself. The sun is well known to sustain itself since billions of years by the fusion energy that it has been producing since its origin. The same idea is adopted in a hohlraum system, but at a smaller scale of operation. There are, however, many obvious but important differences.

• Whereas the material is distributed throughout the radiation field in the sun, in the hohlraum the same is distributed on the wall with fusion capsule placed separately at the center. In the sun both the generation of radiant energy and confinement takes place in the entire volume region in an intermixed manner whereas in the cavity the two phenomena are more or less decoupled, at least over the time scale of fusion.

• In the sun the initial source of energy comes from gravitational contraction. Once a condition of fusion breakeven is realized, the further course of evolution is sustained by the fusion-chain itself. However, in the hohlraum, the gravitational force is obviously of no significance at all. Here, we derive the initial source of energy by the intense pulse of laser or particle beams.

• Whereas the sun is able to maintain the interior temperature (~14xlO6 degree K) for billions of years due to its giant size and materials, the hohlraum can not do the same owing to the small size and materials. Here the whole idea is to implode the small fusion capsule so fast that the total fusion energy yield over a period of say, 10 ns exceeds the energy that is initially invested to generate the thermal X-rays (called breakeven condition). In a realistic fusion reactor system, perhaps many such repetitive implosions might be required to derive a continuous power.

• The interior region of the sun acts as a naturally formed closed cavity, which confines the thermal radiation via multiple absorption and emission. It is this confinement effect which is responsible for the enormous temperature inside the sun. However, the hohlraum cavity can not be made perfectly closed one due to many practical considerations. Some openings have to be there to allow the laser pulse to pass through for heating the wall (Fig. 1) or for the purpose of diagnostics.

The entire phenomena inside a hohlraum depend on many factors like the nature of the wall material, the strength and distribution of the primary source and the geometric configuration of the cavity wall. All these are designed with an intention to obtain a maximum coupling of the radiant flux to the fusion capsule. 2. Radiation confinement in a cavity

Consider an arbitrarily shaped cavity (Fig.2) with walls made of high-Z material. Inside the cavity and on a small wall element there is a source of externally supplied energy. It is not necessaiy to specify the detailed nature of the source; the externally supplied energy can be of any form (e.g., laser light or ion beam) but the output is in the form of thermal radiation. The detailed physics of the conversion process of laser light and ion beam energy into thermal radiation has been extensively studied7'10 earlier and, will not be discussed over here. We simply assume the source to be a Lambertian emitter that isotropically radiates its energy, depositing it on the various wall elements of the cavity interior. The argument behind this assumption is as follows.

When the cavity wall is irradiated by an intense laser pulse (of intensity ~1014 W/cm2 and duration ~ 10 ns), a very high temperature and dense plasma is created at the surface within a small fraction of the laser pulse duration. The electrons at such a high temperature (~1O7 °K) generate soft X-rays by the bremsstrahlung process. The high- energy electrons in the plasma lose energy while traversing in the vicinity of nuclei which is gradually converted into radiation according to classical electrodynamics. Now this radiation is quickly thermalized to yield a Planckian spectrum with peak corresponding to the soft X-rays (~ 1 KeV=1.16xlO7 °K). The time scale of conversion into X-rays and subsequent thermalization is usually in the range of 0.1 to 0.5 psec (1 psec = 10~12 sec) which is much smaller than the laser pulse duration. Thus, over the time scale of hohlraum experiment, we can safely ignore any physics related to the generation and thermalization of the X-rays.

Next, depending primarily on the amount of energy received, the wall at a particular point acquires a temperature. As a consequence, each wall element reemits (isotropically) a fraction of the energy that it has received. The reemission coefficient R, defined as the ratio of the radiated flux Sr to the total incident flux S on a wall element depends on the nature of the wall material (e.g., opacity of the wall material, the equation of state (EOS) etc.). This problem has been studied extensively in past11"16 for the several wall materials both analytically and by computer simulation. A detailed report17 on this is recently brought about separately. Here we summarize some results, relevant to the present work.

As said earlier, when an intense flux of thermal radiation falls on a high-Z material, a hot but dense plasma is created. Even for a radiation temperature of 1 KeV, the plasma is" opaque (the so called optically thick plasma). Consequently, the energy transport into the wall interior follows by the process of diffusion, which leads to the formation of a heat wave front called the Marshak wave or ablative heat wave (see Fig.3). Here, the diffusion process arises from an evolving sequence of absorption and reemission of the radiant energy in the intervening layers of the wall interior. Simultaneously, a part of the absorbed energy is converted into kinetic energy of the ablating material. The ablation of the material launches a shock wave inside the target due to rocket action. This shock eventually overtakes the ablative heat wave and compresses the cold interior region of the wall material. Furthermore, apart from ablation the optically thick plasma also reemits radiant energy into the vacuum. Calculation of the reemission coefficient R is based on the partition of the incident energy between the kinetic, internal and the reemitted energy. In the case of complete thermodynamic equilibrium between matter and radiation, it is possible to obtain an analytical solution to this problem. It has been shown that the ablative heat wave propagation admits a self-similar solution", which leads to a scaling law for the wall temperature Tr (and therefore the reemitted flux Sr) as a function of time t and the net absorbed flux Sw into the wall, namely, a 4 a Tr(t) = C't 's£' => Sr=oT =Ct s£ .(2.1)

1 where Sr is reemitted flux and a is Stefan-Boltzmann constant. The constants C, a , P' or C = aC'4, a = 4a', p = 40' are related to the physical properties of the wall material. These are usually obtained from the numerical simulation of the problem. The complete simulation of this process is based on solving the coupled radiation transport and hydrodynamics equations for the medium. Such calculations, usually employ the pre- computed opacity and equation of state (EOS) data18 (as a function of density and temperature). However, with the power law approximations to the opacity and EOS functions- in appropriate regions of density and temperature- the same can also be determined from the self-similar solutions. For example, for a gold target the scaling constants are C = 4.87, ct=8/13 and p—16/13 so that Eq.(2.1) becomes A _i | 16 13 3 Tr(eV) = 262.21 S™ => Sr=4.S7t^Sl (2.2)

14 2 Here the reemitted flux Sr and the absorbed flux Sw are in unit of 10 W/cm and time t is in nsec. The opacity data used to obtain Eq.(2.2) are taken from Ref.18.

3. Exchange of radiant energy between different wall elements: A view factor approach

The radiant energy flux inside the cavity can be calculated using the flux conservation equation. The schematic diagram in Fig.2 shows the geometry of the cavity along with a coordinate system for calculating the radiant energy exchange. The primary source of radiant energy occupies a small section of the interior surface having an area A^ and it irradiates the cavity with a radiant flux S0(r, t), which, in general, is time dependent. The cavity is assumed to have holes of area Ah (required for the diagnostics or the laser entrance) that simulates the escape route for the radiation. In other words, it represents a sink of energy. The total cavity area A is the sum of the hole area Ah and the area occupied by the material wall A^,, i.e. A = Ah + A,,. If we neglect the small amount of energy stored in the void of the cavity and, also the effects due to transit time of the photons to travel across the cavity interior, we can write an instantaneous energy balance at the vacuum-material interface of a wall element located at the point r as (see Fig.3)

= ^(r,0 + S,(r,0 = Sr(r,0 + Sw(r,0 (3.1) . where Ss(i% t) is the total flux incident on the wall surface at r at time t which directly comes from the source and Sj(r, t) is the total incident flux on the wall surface at r at time t that consists of reemitted contributions from all the other wall elements. They are formally given by (see appendix-A for details)

Ss(r,t)=l SQ(r',t)V(r,r')dA', St(r,t)= I Sr(r',t)V(r,r')dA' .„ „ cos V cos V (3-2) F(r,r')=—• =— n\r-rf where V(r,r') is called view factor19 (see appendix-A for the basic definition) which is determined by the geometry of the cavity. Here, *P and *F are the angles between the normals to the surface at the points P(r) and P(r'), respectively, and the line of sight PP'. Eq.(3.1) expresses the fact that the sum of the total flux falling at a given wall element (both from the external source and the other wall elements) is balanced by the sum of the reemitted flux Sr(r,t) and the net heat flux Sw(r,t) absorbed into the wall material. The reemission factor R(r,t) is given by

Jtfr.Q (33) S(r,0 S,(r,/)+S,(iy)

Next, eliminating Sw(r,t) between the equations (2.1) and (3.1), we get the following implicit equation for solving the reemitted flux Sr(r,t).

a Sr(r,t) = Ct [Ss (r, 0 + Si(r,t) - Sr(r,t)f (3.4) where Ss(r,t) and Sj(r,t) are related to the flux source and the reemitted flux via Eq.(3.2).

Here it is important to note that in the above model any complication arising from the plasma expansion (the plasma may be generated by heating of the wall either by the primary or secondary sources) that fills the cavity, have been tacitly ignored. This is valid as long as the length of the plasma extent after expansion is much smaller than the typical cavity dimension. A justification to this will be discussed later.

3,1. A completely closed cavity

A completely closed cavity, the interior of which is uniformly irradiated, possesses a uniform wall temperature. The reemitted flux is independent of space coordinate but can be a function of time. It follows from Eq.(3.2) that S;(t) = Sr(t) (here we use the fact that the total integral of view factor over a closed surface is unity (see appendix-A)). a p Therefore, equations (2.1) and (3.1) gives Sw= Ssand Sr = C t Ss . Using this in Eq.(3.3), the reemission factor R(t) is given by tt (3-5) a sf+ss

Assuming a temporally constant source flux Ss, we see from Eq.(3.5) that the wall becomes more reemissive the longer the source flux is present and for higher source fluxes. At t = 0, R(t) = 0, implying that initially the wall is cold enough not to emit any significant radiation. However, as time t increases more and more radiation is reemitted from the wali into the cavity interior. Finally, in the limit t-»oo, R(t)-^1 (irrespective of the nature of the wall material), implying thereby, that the cavity wall behaves like a perfect emitter.

3.2. Cavity with opening or a sink

It is of great practical interest to calculate the flux distribution for a cavity with opening and, more importantly, the cavity with nonuniform wall temperature. In this case one has to solve Eq.(3.4) coupled with Eq.(3.2) in a .self-consistent manner for a given source distribution S0(r,t) and the material constants for the wall. Unlike the closed cavity, the equilibrium-state for an open cavity is not necessarily one where temperature is independent of the space coordinates. Here different sections of the cavity wall will, in general, possess different temperatures, thus yielding a spatial profile for the temperature. Furthermore, this spatial profile changes with time t too. We shall now calculate the temperature distribution (both as a function of space and time) inside a capillary tube.

3.3. Applications to capillaries

In a number of applications requiring high intensity X-ray flux, it is desirable that the source of the primary X-rays be separated from the object receiving it. For example, in opacity studies of dense matter it is important that a sample foil be heated by an intense X-ray flux, which is not accompanied by laser light. Similarly, in the ICF system the fusion capsule is heated by X-rays which is not accompanied by the laser or ion beams. All these necessitate an arrangement, which could transport the X-ray flux at a distance with high degree of efficiency. The mechanism of energy distribution offers such a possibility. A capillary system happens to be a very efficient and practical device to transfer the X-ray flux from one end to other.

A. Disc source at one end

A simple arrangement for the transport of thermal radiation is schematically shown in Fig.4. It consists of a capillary of radius r0 and length 1, the interior of which is heated by a source of thermal X-rays located at one end. Obviously, the interior surface is heated in a nonuniform manner. Different wall elements acquire different temperatures according to their axial location relative to the source. The farther wall elements will naturally have lower temperature. However, additional flux due to the wall reemission enhances the energy transfer to the other end, and therefore the temperature. In the ideal case of a perfect emitter (R(z,t) = 1, where z is axial distance from the disc source), all the flux supplied by the source will appear at the open end irrespective of the capillary length. However, in a realistic case when R(z,t) < 1, one is required to solve Eqs.(3.4) and (3.2) to obtain the various fluxes. For simplicity, let us assume

• The flux So emitted by the disc source is constant in space and time.

• The wall property possesses azimuthal symmetry so that all the quantities are functions of z and t only.

• Feedback of the capillary wall to the source is neglected. In other word the source disk behaves like a perfect emitter. This is quite justified, for the source is usually at a much higher temperature than the rest of wall.

Under these conditions, Eq.(3.4) can be written as (see Eqs.(B4) and (B9) in appendix-B)

Z ro

, S{(ZJ)= \dZ'Sr(Z',t)F<\Z-Z% L= — (3.6) a r0 Z2+2 ! 1- where G(Z) is axial view factor between a point on the disc source and another point on the wall surface at distance Z. Similarly, F(jZ-Z'j) represents axial view factor between two wall points at distances Z and Z\ Analytical expressions20"21 for these axial view factors, are derived in appendix-B. Eq.(3.6) can be easily solved by using standard numerical procedure (i.e., writing integral as a discrete sum and solving the resulting matrix equation by, say, Gauss elimination method, see Ref.l for the numerical algorithm). Numerical results using Eq.(3.6) and their comparison with DSN method for some typical problems (both simulation1 and experiment22) will be discussed later.

B. Two internal ring sources

Cylindrical hohlraums with ring sources, as shown in Fig.5, are most suited for an implosion experiment with a DT glass micro-balloon. In an experiment1, the two rings on the interior cylindrical surface were irradiated using ten laser beams in two bundles of five (GEKKO-XII glass-laser). The beams were injected through the open ends of the cylindrical cavity. The X-ray flux produced by these ring sources heated the rest of the interior. The pellet was positioned on the cylindrical axis midway between the two sources. One of the main experimental results was that by varying the distance between the two ring sources, an optimum distance was found for which the irradiation of the pellet and, therefore, its implosion was most symmetric. We assume for simplicity

An empty cylindrical cavity (no pellet present) • Ring sources radiate in one direction, i.e., only towards the cylindrical axis.

• The source flux So is constant and is same for both the rings sources.

• Wall property is axially symmetric so that all the fluxes are functions of Z and t only

With above assumptions, the axial temperature profile can be calculated as a function of time by solving Eqs.(3.4) and (3.2), implicitly. The source flux SS(Z) is given by

Ss(Z)=IS0(Z)F(\Z-Z'\dZ', 1- o (3.7) = S0 for ZX

= So for Z3

Next, for the constant So, using Eq.(BlO) of appendix-B in the above equation, we finally obtain the following.

for Q

for ZX

for Z2

= 1 + /(-£)-/(-£,) + /(-£) + I(-f4) for Z3Z4

=z,-z(/= 1,4),

Again numerical results for a typical hohlraum target', and comparison with DSN calculations will be discussed later.

4. Discrete ordinates method

23 The discrete ordinate method , also known as DSN method, is very often used for solving problems involving particle transport phenomena. For example, the transport of neutrons and y-rays in reactors, a-particles in fusion plasmas, thermal radiation in stellar atmosphere, etc., are all modeled using the DSN method. The main advantage of this method is its flexibility for extension to more general and complex problems. Multidimensional problems, time-dependent evolution of the particle distribution functions, and a variety of boundary value problems can be easily tackled by this method. For the present problem, we solve only the time-independent radiation transport equation24. Assuming frequency integrated spectrum for the radiation inside an empty cavity, the time-independent transport equation to be solved is

Q.VI(r,Q,t) = 0 with boundary condition at r = rs as

TV TC where I(r, Q, t) represents radiation energy flowing per unit area per sec at point r at time t due to photons moving along the direction Q. The explicit time dependence of I(r,Q,t) that arises due to the transit time of photons has been ignored in Eq.(4.1) like in the view factor approach. In other words the time t enters just as a parameter in the formulation of the whole problem. Further, the boundary flux I(rss Q,t) consists of two parts, one coming directly from the source and the other from the wall reemission. TE(rs) represents temperature of the direct emitting source located at the boundary point rs and Sr(rs,t) represents the reemitted flux from the boundary point rs at time t. This is calculated, at any stage, from the following scaling relation provided the incoming flux Sjn, which falls on the wall surface is known from the solution of Eq.(4.1).

(4.2)

Now for each value of t, the implicit equation (4.2) must be solved for all the wall elements to obtain new values of Sr for a given value of Sin. This new values of Sr is used in Eq.(4.1) to solve for a fresh value of Sin. In this way we solve Eqs.(4.1)-(4.2) self consistently using many iterations till we obtain the converged solution to a desired accuracy. It is to be noted that Eq.(4.1) solves for all the wall elements (at each time t) decoupled from each other unlike in the view factor approach where the fluxes from each wall element is coupled to those from all the other wall elements. In other words if the hohlraum walls are divided into, say, M elements, the view factor method requires solving M coupled non-linear equations, while the DSN method requires solution of M uncoupled non-linear equations for each iteration. Usually, the required number of iterations are much smaller than M.

In the DSN method, numerical solutions are obtained by discretization of the space variables z and r and the angle variable Q. In the Nth order approximation, a total of N(N+2)/2 angular directions are considered for a two-dimensional problem. The discrete version of Eq.(4.1) can be written as21

10 4m\ B. . ! / . j -B. . i /. . !• 1+ (4-3)

=0 {_ . i m2 l'J'm~2 where the indices (i, j, m) denote the mid-values of a 2-D space-angle mesh. The end values of the mesh are denoted by the indices (i ± 1/2, j ± 1/2, m ± 1/2). Further, the constants \im and £m are the direction cosines of the angle Qm and the term containing parameters a^,.,^ and am.1/2 represents the photons streaming in the angular space.

Once the radiating temperature of the external source is specified, the discrete set of Eq.(4.3) is solved using appropriate boundary conditions23. After the completion of one mesh-angle sweep, the reemitted fluxes are computed using the latest value of Sin(rs,t) from Eq.(4.2). In the next mesh-angle sweep, these re-emitted fluxes are used as the boundary conditions. The whole procedure is, then, repeated until the flux values are converged to a desired accuracy. Further, for a new value of time t, the reemitted fluxes are modified as per the scaling law (2.1).

5. Numerical results

A. Capillary with a disc source on one side

Fig.6 shows radiation fluxes Ss, Sr, Si5 and Sw and the reemission coefficient R inside ls a gold (for which C=4.87, a=8/13, p=16/13) capillary of length 25r0 after a 1 nsec of heating. The tube is heated with a source at one end which emits a constant flux of 2 magnitude So = 4 x 10'" W/cm . This flux corresponds to the temperature Tr = 250 eV IM 14 2 (use Tr(eV)=l76.67 S0 , where So is in unit of 10 W/cm ). The fluxes in Fig.6 are normalized to So. They are calculated by solving the implicit equation (3.6) as a function of the normalized axial distance Z. It is interesting to note that the fluxes calculated by using our analytical expressions for the view factors (derived in appendix-B) exactly simulate those obtained from the earlier numerical calculations'.

It is seen in Fig.6 that the incident and reemitted flux dominate along most of the capillary length and it is only near the opening that the source flux becomes significant due to the lower reemission coefficients.

Next, Fig.7 shows radiation temperature distribution along the capillary axis for the above mentioned parameters. Again this compares very well with those reported by Tsakiris1. We have chosen three cases of interest for our calculation:

11 • No reemission from the wall (Curve-A). Here the temperature distribution inside the cavity is determined only by the primary source So and the view factor at distance Z. l/4 The temperature is, thus, given by Tr(eV)=l76.67 Ss .

• Reemission after 1 nsec (Curve-B). Here the temperature is calculated from the reemitted flux Sr obtained by solving Eq.(3.6).

• Completely reemitting wall (Curves-C)=>R=1, Sw=0, Sr=Ss+S; This case corresponds 1 ct 1/p to the asymptotic limit t->oo for which Sw = (SrC" t' ) ->0. The fluxes become independent of wall parameters C, a and p.

The no-reemission case dominates in the limit t-»0 when the wall is cold. In this limit the wall absorbs more radiation than what it emits. However, as is evident from the curves A, B and C that as time t increases, more and more fraction of the incident energy is reemitted from the walls. Finally, when the wall becomes almost perfectly reemitting after a prolonged heating, the temperature distribution becomes same irrespective of the nature of the wall material. As far as the axial distribution is concerned, the temperature decreases monotonically away from the source. Further, this decrease becomes smaller and smaller with increasing emission from the wall. Finally, for the perfectly reemitting wall the temperature profile attains almost a flat shape .except near the opposite open end where the radiation leakage could be important (Curve-C). Thus, we find that high reemissivity of the hohlraum wall plays a crucial role in maintaining a uniform temperature distribution. ' "

The results of DSN calculation for N = 16 are also shown in Fig.7. The convergence of the solution requires about 10 to 20 iterations. The case of perfectly reemitting wall is simulated, by equating the incident and the reemitted fluxes and the no-reemission case is simulated by equating the reemitted flux to be zero. It is seen that the results of DSN method compare so well with those obtained by the view factor approach. Particularly, for the case of perfectly emitting walls the two results match exactly. Further, as DSN calculations consider only a finite number of photon flight directions, the radiation temperature at large distances from the source is somewhat lower for the case of no reemission from the walls. However, this discrepancy can be minimized, by further increasing the order of DSN calculations.

B. Experimental results for a disc source22

Another interesting example of the disc source is in an actual experiment22. The X-ray source is generated from a laser-heated gold foil (of 1.1 mm thickness and diameter 500 \xm) attached at one end of a cylindrical cavity (of about 40 mm length). The gold foil has the same diameter as that of the cylindrical cavity. A total energy of 200 Joules from a laser source of wavelength 0.44 \xm and pulse duration 0.3 nsec is deposited on to the gold foil. This is found to generate X-rays at a temperature of about 154 eV for a period of about 0.675 nsec. The experimentally measured radiation temperatures along the capillary axis -together with error bars- for the two cases: (i) closed capillary and (ii) open capillary are shown in Fig.8. The open-cavity experiment is simulated by removing about

12 75% of the cavity-wall. The corresponding axial temperature profile is expected to simulate the case of no-reemission from the wall as already discussed.

Fig.8 also exhibits results obtained by using the view factor as well as the DSN approaches. In these calculations, the open-geometry (Curve-A) is simulated by assuming the walls to be non-reemiting. The results of the closed-capillary experiment (Curves-B) using gold disc at one end and heated up to 0.675 nsec are compared with our view factor and DSN (N=8) calculations. It is seen that they compare very well. Thus, both our analytical view factor method and the DSN method are well justified even in light of actual experiments.

C. Capillary with two symmetric ring sources

Fig.9 shows axial temperature profile for the two ring sources placed symmetrically on either side of the center of the cylinder. The length of the cylinder 1 = 20r0, and width of the emitting sources is equal to the radius r0. Both the sources are placed at distance 5r0 from the center. These sources are generated by the two way laser-heating. The laser beams enter from both the open ends in an actual implosion experiment. The source 14 2 strength is taken to be S0=1.65xl0 W/cm which corresponds to a temperature of 200 eV. Curves A, B, C, and D correspond, respectively, to the case of no-reemission, the reemission after 1 nsec, the same after 10 nsec and a perfectly reemitting wall. It is seen that as time t increases the temperature profile between the two ring sources become more and more uniform Moreover, the resulting temperature due to superposition of the two fluxes on either side becomes higher. Thus, we obtain an intense source of uniform radiation (over an extended volume between the two rings), which is very crucial for a symmetric implosion.

However, the existence of the two open ends naturally introduces a perturbation to the irradiation uniformity. This perturbation becomes less significant the larger the separation between the two-ring sources. But for larger separation, the temperature dip at the middle also persists for longer periods. Another possibility to minimize the perturbation is to partially close the open ends.

It is again seen in Fig.9 that the results of both the view factor and the DSN methods match so excellently. Further, these results are also in well agreement with those obtained by Tsakiris1.

6. Power transfer to the cavity

In order to optimize energy distribution inside the cavity, it would be of interest to calculate the power transfer in different channels. The source power, Ps, incident on the entire wall will be given by (use Eqs.(3.6) and (B5)) / Ps (L) = \SS (Z) 2zr0 dz = Pofs (L), 0 (6.1)

where Po is power emitted by the disc source and fs(L) represents the fraction of the primary source power which is incident on the wall surface. The rest of the source power, say, Ph(L), shines through the opposite open end so that Ph =PO-PS Thus, the fraction of the source power that directly shines through the open end is fh(L) = Ph(L)/P0 = l-fs(L).

Further, the power PS(L) (which is independent of time t because the source flux So is assumed to be constant) consists of two parts, which are functions of both the capillary length L and time t. One is Pw(L,t), the power absorbed into the wall up to time t and the other is Pr(L,t), the power reemitted through the two ends, after undergoing multiple absorption and reemission up to time t. That is,

/ / Pw(L,t)=\2m-0dzSw(Z,t), Pr(L,t) = J2m-0dzSr(Z,t) 0 ° (6.2) P(Lj) P(Lt) L Z

Again, writing in terms of the power fractions fw(L,t) and fr(L,t) corresponding to the wall absorption and reemission respectively, we have

,/) = 2\dZZr(Z,t)G(Z), f2(L,t) = 2\ 0 0

Here f](L,t) and f2(L,t) represent, respectively, the reemitted fractions that appear towards the source and the open end. Table-I shows fw, f,, f2, fs, fh, and ft (= f2 + fh), the total fraction that is transported at the open end for various normalized lengths L when the disc source heats up to 1 nsec. It is interesting to note that for the larger lengths although the total power transfer to the other end decreases due to increasing wall absorption, there is relatively more contribution to the power transfer from the wall reemission than from the direct source (f2» fh). One can see that even for the moderate lengths (say, about three to

14 four times the capillary radius) the direct contribution is almost negligible as compared to that from the wall reemission.

As for variation with time is concerned it is seen1 that in the beginning, the wall is cold and nonreemitting. Thus, most of the power is absorbed by the wall and a small amount escapes through the opening. At later times (t >17 nsec, see Ref.l) exactly the opposite happens and for even longer times the power absorbed by the wall asymptotically approaches zero and the power escapes through the opening. This shows that longer a wall element is heated, the better a reemitter it becomes. We find similar conclusions for the two ring sources too.

Capillary efficiency

In the context of using the capillary as a device for transporting thermal radiation, it is useful to define its efficiency in the following way.

,t) = P0)f2(L,t')dt', (6.4)

Eo=Pot

Here r|e, the ratio of the total energy appearing at the open end of the capillary to the total energy delivered by the source. This turns out to be almost same as f, shown in Table-I and is not written explicitly.

Table-I: Power fractions in different channels for different capillary lengths

4 2 (disc source with So = 4x10' W/cm and t-1 nsec)

L fw f. f2 f, fh f,=fh+f2 1 0.1836 0.23 0.2045 0.6180 0.3820 0.5865 2 • 0.3243 0.31 0.2001 0.8284 0.1716 0.3717 3 0.4265 0.34 0.1517 0.9083 0.0917 0.2434 4 0.4988 0.35 0.1065 0.9443 0.0557 0.1632 5 0.5489 0.36 0.0725 0.9629 0.0371 0.1096 6 0.5834 0.37 0.0487 0.9737 0.0263 0.0750 7 0.6072 0.37 0.0327 0.9804 0.0196 0.0523 8 0.6240 0.37 0.0220 0.9848 0.0152 0.0372 9 0.6360 0.38 0.0149 0.9880 0.0120 0.0269 10 0.6448 0.38 0.0102 0.9902 0.0098 0.0200

15 7. Validity of the model

There are certain limitations of our model, which is worth discussing here. Many assumptions were made for simplicity. However, these must be justified in light of actual hohlraum experiments. Some important points are highlighted below.

1. Firstly, we assumed radiation and matter to be in equilibrium. This is well justified, for the equilibrium condition is achieved over a time scale of atomic transitions, which is much smaller (< 1 psec) than the time scale of laser heating (~ 5-10 nsec).

2. Secondly, the present model does not consider any complications arising from inevitable expansion of the wall plasma. Consequently, most of the calculations carried out so far may not be strictly valid in the limit of longer time. Further, the smaller radius of the capillary tube may also pose some limitations. A useful criterion for choosing the radius r0 is based on the plasma expansion velocity, which is of order of the sound velocity. Thus, for validity we must have ro>Csxs where Cs is the maximum velocity of sound (this is the case near Z«0) and xs is the duration of the 1 source pulse. In practice, for a source at a temperature Tr of 110 eV (^So^l.SxlO" 2 2 6 W/cm ), the sound velocity Cs = (ZT/M)" ~ 6.5x10 cm/sec for a gold target with charge number Z=79 and mass number M =197.2 amu. Now, if the pulse duration xs is chosen to be 5 nsec, then the plasma filling inside the cavity can be ignored if we 4 choose capillary radius r0 > 325 micron (1 micron = 10" cm). This criterion is quite justified in an actual experiment using r0 > 500 micron. However, for the same reason, the radius can not be increased just indefinitely, because then the total power load would also be accordingly very large.

3. Thirdly, we assumed that the conservation of fluxes expressed by Eq.(3.1) is true at every instant of time t. In other words, the transit time of photons between any two points inside the cavity was just ignored. Now, this may be justified if the transit time 10 ~ 1/c « xs where c is velocity of light in vacuum (=3xl0 cm/sec). For a typical case of 1 = 10r0 = 0 5cm (for r0~500 micron), the photon transit time ~ 1.67x10" sec which is much less than xs~5 ns. Thus, in practice the assumption of instantaneous balance of fluxes at any point inside the cavity is well justified.

4. It is known that the scaling relation (2.1) has been derived on the basis of self-similar solutions" of the governing radiation hydrodynamic equations. Now, this is strictly valid only in the asymptotic limit and for a power law dependence of the incident flux or the boundary temperature Tr (Tr ~ f where x is some exponent). Now it does take some finite time to reach the asymptotic limit, which depends on the incident flux level as well as the wall properties. Moreover, temporal profile of the flux falling on the hohlraum wall, is generally of a more complicated nature (not just a power-law profile). However, if the time needed to reach the asymptotic power level is much faster than the time scale of flux variation, then the use of Eq.(2.1) is quite justified.

In this regard it is worth noting that for a closed cavity when temperature is constant throughout (the exponent x=0), the scaling relation (2.1) is perfectly valid. This case

16 corresponds to Sr = S, (use Sr a constant in integral 3.2) so that Sw = Ss from Eq.3.1. Thus a p scaling law (2.1) reduces to Sr = Ct Ss which, shows that the reemiited flux follows a power law variation of the incident flux.

However, for the open geometry case, where Sj =0, only time-variation of the source flux is used to drive the heat wave into the wall while the rest is continuously reemitted from the wall element into the vacuum. For this general case, no self-similar solution is available. Still, there exists two asymptotic limits in which the solution turns out to be self-similar'. The first is in the limit t-»0 when the wall element is cold and R«I. In this limit, all the energy is absorbed by the wall, thus Sr = 0 and Sw = Ss.so that the wall element behaves like a part of a closed-geometry system In this case the scaling relation a p reduces to Sr(t->0) sCt Ss . The second one is in the limit t->oo when the wall becomes highly reemissive with R = 1 and, therefore, Sw = 0 and Sr s Ss. Consequently, the wall 1/4 element temperature approaches the asymptotic limit T(t-»°o) = (Ss/a) =constant and the 1/p 1/p wall flux approaches as Sw(t-»oo) = C" t^ Ss . Using constants for the gold target, we l/2 13 6 obtain Sw=0.28 t' Ss " . This is exactly the scaling law derived from the self-similar solution for a constant wall temperature18. A numerical simulation for the gold target for 14 16 2 Ss = 10 - 10 W/cm over a wide range of time t confirms the validity of the self-similar solutions in the two asymptotic regimes1. However, for the intermediate times some marked differences are found, particularly, for higher fluxes at later times. In general, for partially open cavities, the theoretical model reproduces the simulation results quite well within the accessible radiant flux range of 10l3-1015 W/cm2.

8. Summary

One of the most important problems in an ICF system is the symmetric implosion of the ICF target. From this point of view the conventional direct-driven system suffers from many limitations. A viable alternative is what is known as indirect-driven ICF system in which a laser or ion beam is first converted into an intense, isotropic thermal X-rays inside a high-Z cavity via multiple absorption and reemission. Subsequently these X-rays implode a fusion capsule placed well inside the cavity. These cavities have many more applications too, e.g., in studying material properties at exceptionally high density and temperature.

One of the key issues in the above problem is to calculate the irradiation pattern inside the cavity as a function of space and time. This is determined by many factors like the nature of the wall material, the geometric configuration of the hohlraum cavity, the strength and location of the thermal source, the structure and intrinsic properties of the fusion capsule etc. In general, this problem is very involved. Many physics related phenomena like, the conversion of laser into thermal X-rays, the radiation-matter interaction, the propagation of ablative heat wave inside the target, the multiple absorption and reemission of the fluxes inside the cavity and the implosion of the fusion capsule etc. are involved into the problem. Only a sophisticated computer simulation can reveal the full details. However, one can still get some physical insight by solving the

17 problem analytically under certain simplified but well justified assumptions. Although, much have been said earlier, we shall briefly outline some salient features of our study.

Firstly we calculated the flux and temperature profiles inside a gold capillary tube for a given disc source. The basic formalism assumes a view factor approach, which is discussed in detail in appendices A and B. We derived analytical expressions for the axial view factors, G(Z) and F(|Z-Z'j, in closed forms. Using these expressions in the flux conservation equation (3.1) coupled with self-similar scaling relation (2.1), we calculated various fluxes and radiation temperature inside the cavity as a function of space and time (Eq.3.6). Our analysis shows that for a closed cavity the temperature (or the fluxes) turns out be a function of time t alone. However, for the open cavity or cavity with a sink (which is more realistic in actual experiments) they are functions of both space and time. A general conclusion is that, initially, when the wall is cold, there is more absorption. As time increases, the temperature increases and more and reemission occurs from the wall. Finally, in the limit t->oo, the wall behaves like a perfect emitter and the temperature profile becomes independent of the nature of the wall material. Similarly, for a given time t, the temperature falls monotonically away from the source along the axis.

Further, we also solved the same problem using DSN method (in two-dimensional r-z geometry) discussed in details in section-4. It is seen tha't the results of the two methods match exactly. Moreover, they also agree well with those obtained by Tsakiris1 who calculated view factors by 2-D numerical integration in r-0 geometry. We also tested our calculations against an actual hohlraum experiment. Interestingly, they agree so well that our analytical view factor solutions can be taken as a standard for the benchmark problems related to some capillary hohlraum experiments.

We also derived an expression for the power transfer from a disc source through various channels. We found that the total power that appears at the open end, at a given time, depends crucially on the capillary length L. This total power is derived from two sources. One is primary source that directly comes from the source and the other is a secondary source, which is generated via multiple absorption and emission. It is seen that for longer capillaries (even of moderate size, say, l=5r0) the latter source plays a more important role. This can lead to a very efficient transport of the thermal radiation in a very short period. For example, we calculated explicitly the capillary efficiency (defined in section-6) and found that about 11% of the total energy can be transported by a gold capillary of length 1 = 5r0 just within 1 nsec (see Table-I).

Next, we also carried out a similar exercise for the two uniform ring sources. This type of arrangement is very much suited for implosion experiments. The goal in this type of experiment is to obtain symmetric pellet irradiation and, therefore, implosion. We found that for relatively large distance between the two ring sources (~10r0), it is possible to produce a fairly extended region with almost uniform wall temperature. However; this is a necessary but not sufficient condition. For example, for larger distance between the two ring sources, the required asymptotic state is approached slowly. The heating period is limited by the pulsed source energy available. Another way to achieve asymptotic state with shorter heating period is to increase the thermal flux delivered by the source since the reemission coefficient increases not only with time but also with wall flux. This again

18 implies a more powerful, pulsed energy source. A compromise is, thus, made with respect to uniform temperature zone depending on the laser power availability and the pellet characteristics. All these require further study of the problem.

At the end, we discussed the validity of our model in section-7, which will not be repeated here. The typical estimate suggests that most of the assumptions are well justified even in light of actual hohlraum experiments. At least, we hope that most of the physical conclusions would remain the same even in a more rigorous and refined treatment of the present problem.

Acknowledgement:

The authors are thankful to Dr. D. C. Sahni, Head, Theoretical Physics Division, B.A.R.C, and Dr. S. K. Sikka, Director, Solid State & Spectroscopy Group for their constant encouragement shown to the present work. They are also thankful to Dr. N. K. Gupta for some useful discussions.

19 References:

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2. J. Nuckolls, The Feasibility of Inertial-Confinement Fusion, Physics Today, 35, 24, (1982).

3. M. Nakamura, K. Kondo, H. Nishimura, T. Endo, H. Shiraga, S. Miyamoto, Y. Kato, and S. Nakai, Numerical Method for Finding Uniform Irradiation Conditions of a Fusion Capsule Driven by X-ray Radiation, Laser & Particle Beams. 10, 421, (1992).

4. J. Lindl, Development of the Indirect-Driven Approach to Inertial Confinement Fusion and the Target Physics Basis for Ignition and Gain, Physics of Plasmas. 2, 3933,(1995).

5. M.D. Rosen, The Physics Issues that Determine Inertial Confinement Fusion Target Gain and Driver Requirements: A Tutorial, Physics of Plasmas. 6, 1690, (1999).

6. R. Sigel, Laser Induced Hydrodynamics: An Introduction, in Laser-Plasma Interactions. Vol. 4, Ed. by H. B. Hooper, SUSSP Publications, pp-53-87, (1989).

7. R. Sigel, K. Eidmann, F. Lavarenne and R. F. Schmalz, Conversion of Laser Light into Soft X-rays, Part-I: Dimensional Analysis, Physics of Fluids. B2, 199, (1990).

8. K. Eidmann, R. F. Schmalz and R. Sigel, Conversion of Laser Light into Soft X-rays, Part-II: Numerical Results, Physics of Fluids. B2, 208, (1990).

9. F. Ze, Time Evolution of X-ray Conversion Efficiency, J. Appl. Physics. 66, 1935, (1989).

10. K. Eidmann and W. Schwanda, Conversion of Laser Light into Soft X-rays with 3 ns and 30 ps Laser Pulse, Laser & Particle Beams. 9, 551, (1991).

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20 15. R. Sigel, R. Pakula, S. Sakabe and G. D. Tsakiris, X-ray Generation in a Cavity Heated by 1.3 or 0.44 /an Laser Light, III: Comparison of the Experimental Results with Theoretical Predictions for X-ray Confinement, Phvs. Rev.. A38, 5779, (1988).

16. M. Murakami and J. Meyer-ter-Vehn, Indirectly Driven Targets for Inertial Confinement Fusion, Nuclear Fusion, 31,1315,(1991).

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21 Appendix-A View factor: exchange of radiant energy between two black surfaces

The view factor between any two arbitrary surfaces represents the exchange of radiant energy as viewed from one to the other19. Consider two differential area elements dA, at temperature T, and dA2 at temperature T2, arbitrarily oriented and have their normals at angles 4*, and T2 to the line of length r12 joining them (see Fig.Al). By definition, the total energy dE12 leaving the element dA, per sec. and incident upon dA2 is given by

* r\2 where dQ, is the solid angle subtended by dA2 when viewed from dA,, I, is specific intensity and S, is the emitted flux from the surface dA,. The above equation can be written in the following form.

dE12=SlVl2dAldA2, Fl2 = cosW (A2)

where Vl2 is called view factor. Similarly, one can define the view factor V21 for radiation leaving dA2 and arriving at dA,. Obviously, VI2 = V2, but dE12 * dE2l unless S, = S2 (the closed cavity condition where temperature is same throughout). Next, if we define S,_>2 as the total flux emitted from the surface A, (= JdE,2/dA2), which is incident at dA2, then

1^^ (A3)

Obviously, if S, = S,_>2, i.e., the flux emitted from the surface Ai is the same as that received by the surface dA2 and St is constant (the closed cavity condition) then from Eq.(A.3), the view factor V,2 is normalized to unity.

v iAl n Mi = 1 (A4)

22 Appendix-B

View factors for a cylindrical geometry

A. Exchange of radiant energy between a disc and a wall element

Fig.B 1 shows two points P and P1 respectively, on the curved surface and the circular base of a cylinder that exchanges radiant energy with each other. The circular base is emitting radiation flux S0(r',t) at P" whose Cartesian coordinate in terms of radius r' and the polar angle G' is r' = (r'cosG1, r'sinQ1, 0), (see Fig.Bl). Similarly, the flux received on the curved surface at P is Ss(r,t) whose coordinate is r s (r0cos9, r0cosG,z) where r0 is the radius of the cylinder. The two fluxes are related by (use Eq.A3)

C0S Ss(r,t)= jS0(r',t)V(r,r')dA', V(r,r')= ^^ (Bl) ;zjr-r' where *F and 4" are the angles between PP1 and the normals at P and P' respectively. The normal at P' is parallel to the cylindrical axis and cuts'the opposite circular base at Q1 whose coordinate is (r'cosQ1, r'sinG', 1) where 1 is the length of the cylinder. Similarly, the normal at P is parallel to the circular, base which cuts the cylindrical surface at a diametrically opposite point Q whose coordinate is (-r0cosG, -r0sin9, z) (obtained by replacing 6 by 6+TT in the coordinate of P). Now, the various length vectors in Fig.Bl can be written as

P'Q' = /k, 0

PP' = (-r'cosO' + r0 cos#)i + (-r'sinSf + r sin#)j+ zk, where i, j and k are unit vectors along the x, y and z directions. Using the above length vectors we evaluate cosT and cosT', and hence V(r,r') as follows

PPP'Q' - Z |PP'||PQ| 45 ' |PP|P'Q'|

r0 rQ rQ

Next, using the above equation in (Bl), we obtain the following expression for a given 1 1 2 uniform disc source So (use dA = r dG dr = r0 R dO dR for the area element).

23 It is interesting to note that Ss(Z,t) is independent of absolute value of r0. This is because the total energy emitted by the capillary disc source is proportional to the area 2 2 rtr0 whereas the view factor decreases as r0' (see Eq.B2) due to increasing beam divergence along the axis. Therefore, the flux received at any point on the curved surface, which is given by the product of the total energy emitted and the view factor becomes independent of r0.

Now, the evaluation of integral G(Z) in a closed analytical form is not so straight forward. An approximate analytical expression for G(Z) has already been obtained by Tsakiris1 which holds good only in the asymptotic limit (Z-»oo). However, it badly fails in regions close to the emitting source. For example, the approximate expression of Tsakiris1, which gives G(Z)->0 as Z->0, firstly increases with increasing Z up to some value and then decreases monotonically till it goes to zero in the asymptotic limit. However, contrary to this, the actual numerical values of G(Z) show that G(Z)-»0.5 as Z-»0 which then continues to decrease monotonically with increasing Z (without any intermediate rise) till it goes to zero in the asymptotic limit.

We evaluated the integral G(Z) in a closed analytical form20"21 which exactly simulates the numerical results. We shall briefly outline the procedure to evaluate the integral. First of all we change the angular integral in Eq.(B3) from 0 to 2TC to 0 to n by using the 2a a formula |0 f(x) dx=2 fo f(x) dx if f(x) = f(2a-x) (here we apply for a cosine function 2 2 noting that COSO=COS(2TI-0)). Next, we substitute cosO = (l-u )/(l+u ), where u = tan2O/2 in the resulting equation. We get the following result, after some algebraic manipulation

Further, we substitute u = B2tanv and use some standard integrals for the sine and cosine functions. We finally get the following.

At the end, we again substitute 1-R2 = Z2-2ZtanX (Z is fixed) and derive the following result after some algebra.

Z2+2 -z (B4) Z2+4

Obviously, as Z-»0, G(Z)-»0.5. Further, G(Z) decreases monotonically and tends to zero as Z->oo. This obviously implies that as one views the source from farther and farther distance, the less flux is available on the cylindrical surface. It is to be noted that the flux

24 is being emitted only along one side, i.e., towards the interior region of the cylindrical cavity. We can easily show that the integral of G(Z) is normalized to unity, i.e.,

1 p -, oo \dZ G(Z) = H(Z)= -Z VZ2 + 4 -Z U> \dZ G(Z) = //(oo) - //(-oo) = 1, 4 L J -oo (B5)

//(oo) -> - , //(-oo) -> -- and H(0) = 0

2 2 The exact expression for G(Z) and Tsakiris's approximate expression, Ga(Z) = Z(l+Z )~ , are compared in Fig.B3.

B. Exchange of radiant energy between two points on the wall

Fig.B2 shows two points P and P1 on the curved cylindrical surface that exchanges radiant energy with each other. Suppose, the radiant flux emitted from P' (having coordinate rocos0', r0sin6', z'), is Sc(r\t) which is received as dSj at P with coordinate (r0cos9, r0sin9, z). Therefore, the total flux Sj(r,t) incident at P (r0cos9, r0sin9,z) which consists of contributions from all the emitting wall elements can be written as

Sr(r>,t)V(r,r')dA', F(r,r') = — — (B6)

where *F and 4" are the angles between PP' and the normals at P and P1 respectively shown in Fig.B2. The normals at P and P' are parallel to the circular base which cut the cylindrical surface at the diametrically opposite points Q and Q1 whose coordinates are (- 1 r0cos9, -r0sin9, z) and (-r0cos9', -r0sin9', z') (obtained by replacing 0 and 0 by Q+n and G'+7t respectively, in the coordinates of P and P1). Now, the various length vectors in Fig.B2 can be written as

r PP' = r0(cos#- cos#') i + ro(sin#- sin#') j+ (z - z)k,

PQ = 2r0 (cos 9 i + sin 9 j), P'Q' = 2r0 (cos ff i + sin 6' j)

Using the above length vectors we evaluate cos^P and cos^F', and hence V(r,r') as follows

PP'.PQ 1-cosO „., PP'.P'Q' l-cos) +f, f=

It is to be noted that VF=XP' which is geometrically obvious (see Fig.B2). There are some interesting points to be noted in the above expression. Firstly, the view factor depends essentially on the difference of the two angles and the z-coordinates. Secondly, for Z=Z', 2 V(r,r) = l/47rr0 .which implies that the view factor between any two points at the same

25 axial distance is same for a fixed radius of the cylinder. Thirdly, if the two points are very far off along the axial direction then the view factor varies as ^*. Lastly, if the angles are same, i.e., 9 = 0' then view factor is zero irrespective of the axial location.

Next, in Eq.(B6) if Sr is only a function of Z and t and not of the angles (i.e., the wall property is axially symmetric) then we can further simplify the problem. Using dA'= 2 rod9'dz' = r0 dOd£ in Eq.(B6) we obtain the following.

(1 ,{Z,t) = \Sr{Z\t)F(\Z-Z'\)dZ\F{^) = - J o [2(l-cos

The integral F(§) can be obtained in a closed analytical form as follows* First, we write 1- cosO = 2sin2®/2 and put cotO/2 = u in the above integral, which gives the following.

00 2du 4 F{4) = f where g = -±

Next, resolving the integrand in partial fractions and using some standard integrals, we get

1 — • (B9)

It is obvious that F(£)=0.5 for 4=0. Further, it decreases monotonically and tends to zero as §->oo. The integral F(§) is normalized to unity, i.e.,

00 J dZ' = 2 -00 (BIO) = -i and r+4

The exact expression for F(^) (see Eq.B9) and Tsakiris's approximate expression, Fo(£) = (l/2)(l+7t2^2/16) "2, are compared in Fig.B4.

26 HOHLRAUM

Laser Laser Beams o Beams Fusion Capsule

Fig. 1: A schematic diagram of an indirect-driven ICF hohlraum. Laser beams from the two ends heat up the hohlraum walls to generate thermal X-ray radiation. These X-rays then undergo multiple absorption and reemission by the wall. Finally, an intense pulse source of isotropic X-rays is produced which implodes a fusion capsule placed at the centre.

P(r') P(r)

Fig.2: A schematic diagram showing coordinate system for calculating the view factors.

27 ABLATIVE HEAT WAVE sr

DENSITY S.+ S,

Fig.3: A schematic diagram of an ablative heat wave propagation inside a planar target. The flux conservation (Ss + S; = Sr + Sw) is maintained at the wall element. S8 is the source flux directly from the source, and Ss is the incident flux that consists of net reemitted contributions from the other wall elements. Sw is the absorbed flux that goes into the wall and Sr is the reemitted flux. The temperature and density profile of the propagating ablative heat wave is shown in the figure.

X-RAY RE-EMITTED SOURCE X-RAYS

Fig.4: A capillary where radiation is emitted by an uniform disc source at one end. The energy is transported to the other end predominantly by multiple absorption and re- emission. After prolonged heating, a steady temperature profile is established along the capillary axis.

28 RING SOURCES

Figure-5: A cylindrical hohlraum heated with two uniform ring sources of thermal X- rays. After sufficient heating, the portion between the two sources attains a uniform radiation density. The end temperature profile is, however, nonuniform due to leakage of radiation through the two open ends.

dA,

dA,

Fig.Al: The coordinate system for the view factor between two points P, and P2.

29 r

Fig.Bl: The coordinate system used for the calculation of view factor G(Z)

0'

P'

Fig.B2: The coordinate system used for the calculation of view factor F(|Z-Z'|)

30 0 12 3 4 5 normalized axial distance (Z) Fig.B3: Comparison of the exact and approximate1 expressions for the axial view factor G(Z).

31 Fig.B4: Comparison of the exact and approximate1 expressions for the axial view factor F(|Z-Z'|).

32 10-5 0 5 10 15 20 25 normalized axial distance Fig.6 Radiation fluxes vs. axial distance at 1 ns for S = 4x1014 W/cm2 o 1. Source flux 2. Heat wave flux 3. Re-emitted flux 4. Incident flux 5. Re-emission Factor

33 150 Q. I 100

0 5 10 15 20 25 normalised axial distance (Z) Fig.7 : Radiation temperature vs axial distance for a disc source at 250 eV. Solid lines - View Factor results. Dots - DS results. N A: No re-emission B : Gold parameters (at 1 ns) C : Perfect reemitter

34 4 8 12 16 normalised axial distance

Fig.8 : Radiation temperature for a disc source at 154 eV. Solid lines - View Factor results. Dots-DS results. N Dots with bars - experimental data. A: No re-emissipn B : Gold parameters (at 0.675 ns) C: Perfect reemitter

35 320

// \v/ v\ 40- 0 5 10 15 20 normalised axial distance

Fig.9: Radiation temperature for two ring sources at 200 eV. Solid Lines - View Factor results. Dots - DS results. N A: No re-emission B : Gold wall (at 1 ns) C: Gold wall (at 10 ns) D : Perfect reemitter

36 Published by: Dr. Vijai Kumar, Head Library & Information Services Division Bhabha Atomic Research Centre, Mumbai - 400 085, India.