Analytical Dependence of the Ignition Dynamics Parameters on the Low-Z Impurity Concentration

Analytical Dependence of the Ignition Dynamics Parameters on the Low-Z Impurity Concentration

Mohammad Mahdavi and Sayed Ebrahim Abedi

Physics Department, University of Mazandaran, P.O. Box 47415-416, Babolsar, Iran Reprint requests to M. M.; E-mail:m.mahdavi@umz.ac.ir

Z. Naturforsch.69a, 645 – 653 (2014) / DOI: 10.5560/ZNA.2014-0061

Received November 24, 2013 / revised July 1, 2014 / published online November 5, 2014

In this paper, thermonuclear burning of the deuterium-tritium (D/T) plasma of an inertial con- finement fusion (ICF) target is studied in the presence of low-Z impurities (lithium, beryllium, and carbon) with arbitrary concentrations. The effect of impurities produced due to the mixing of the ther- monuclear fuel with the material of the structural elements of the target during its compression on the process of target burning is studied. Also, the effect of impurity concentration on the plasma ignition parameters such as ignition temperature, confinement parameterρR, and ignition energy are dis- cussed. The models are constructed for an isobaric and an isochoric plasma for the case in which the burning is initiated in the central heated region of the target and then propagated into the surrounding relatively cold fuel. In ICF spherical implosions of the D/T fuel, the ignition parameters as igni- tion temperature and parameterρRin the hot spot are approximately 7 – 10 keV and 0.2 – 0.4 g cm⁻² respectively, and these values are increased by the presence of impurities.

Key words:Thermonuclear Plasma; Low-ZImpurities; Ignition Parameters; Deuterium-Tritium.

PACS numbers:28.52.Cx; 52.55.Pi; 52.40.Mj 1. Introduction

Laboratory-based ignition via inertial confinement fusion (ICF) [1] will be achieved by imploding a spher- ical capsule containing a frozen layer of deuterium and tritium (D/T) fuel on the MJ-class National Ig- nition Facility (NIF) [2] currently under construction at Lawrence Livermore National Laboratory. Virtually all ICF ignition target designs are based on a spheri- cal low-Z ablator containing a solid, cryogenic D/T- fuel shell surrounding a low-density D/T vapor, or slightly below the triple point [3]. The ablator is made of low-Z chemical elements in order to maximize en- ergy absorption of laser radiation and the resistance of the material to embrittlement in a radioactive (tri- tium) environment. At least three ablator designs are made of beryllium, hydrocarbon, and high density car- bon [4–7]. Also, hydrides are used as ablator ma- terials. For the most part, these are DT hydrides of light metals [8,9]: beryllium hydride (BeDT), lithium hydride (Li₂DT), and composite hydrides of beryl- lium and lithium (Li₂BeD₂T₂ and Li₂Be₂D₃T₃). In this paper, we study the ignition and burning of D/T plasma with an arbitrary concentration of impurity nu- clei such as lithium, beryllium, and carbon. The prob-

lem is solved under the assumption of a uniform distri- bution of impurities in the D/T plasma.

Compression of spherical plasma fuel occurs via a laser pulse in the time of approximate nanosec- ond (ns) for the implosions with a velocity of 3·10⁷cm s⁻¹. At this ablation materials expand out- ward; the remaining mass is firstly accelerated to- ward the center of D/T the fuel, then mixed and com- pressed [10–14]. Once the laser irradiation ceases, the fuel shell begins to decelerate, further compress- ing and heating the nuclei while the shells kinetic en- ergy is converted to thermal energy via PdV (P and dV are presented as pressure and volume element) work. At this time, the fuel consists of a highly com- pressed shell enclosing a hot region of igniting fuel in the center. Initial fusion reaction begins in a hot rel- atively small region of thermonuclear fuel and burn- ing will continue with wave spreading to other parts of the fuel. OMEGA Laser (The OMEGA Laser has been operational since 1995 and is one of LL’S primary research tools.) and NIF are the only facility world- wide that have been performing ignition-scaled cryo- genic target implosions that are required for most ICF target designs [15,16]. Cryogenic-D/T-target implo- sions using the triple-picket pulse shape have produced

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record areal densities of∼0.3 g cm⁻², corresponding to a peak density of∼250 g cm⁻³[15]. This areal den- sity is 50% higher than reported at the 2008 interna- tional atomic energy agency fusion energy conference (IAEA FEC) [17,18].

Rayleigh–Taylor instability (RTI) occurs whenever a fluid of densityρ2lies on the top of a fluid of less den- sityρ1(ρ₁<ρ2)in a gravitational field [19]. Equiva- lently, it happens when the lighter fluid pushes and ac- celerates the heavier one, such as it is the case in many experiments on high-energy density physics [20–22].

In particular, in inertial confinement fusion, RTI is the main factor that determines the minimum energy required for achieving ignition conditions. Therefore, any method for the suppression or mitigation of the instability growth during the implosion process is of great relevance to ICF [23]. The ablator material can penetrate not only into the peripheral regions of the D/T fuel near its boundary with the ablator. The im- purities can be distributed over the entire D/T plasma, first, due to diffusion and, second, due to the mixing of the inner and peripheral layers of the fuel [24].

In this research, dependence of ignition and burning of D/T plasma on the impurity concentration such as lithium (⁷₃Li), beryllium (⁹₄Be), and carbon (¹²₆ C) are presented analytically. For this purpose, in Section2, the effect of low-Z impurities on the ignition of ICF targets are presented (how the ignition conditions of isobaric and isochoric ICF targets depend on the con- centrations of impurities in an equimolar D/T fuel). In Section3, we calculate the fuel gain for these targets as functions of the impurity concentration. Finally, con- clusions are presented in Section4.

2. Effect of Low-ZImpurities on the Ignition of ICF Targets

For the ignitor being a fully ionized homogeneous equilibrium (T_e=T_i =T) spherical plasma contain- ing deuterium and tritium nuclei with equal atomic fractions x_fand different sorts of inactive nuclei with atomic fractionsx_j, we have

2x_f+

∑

j

x_j=1. (1)

For the ignition regime, the heat power density de- posited by the fusion products must be greater than the power densities lost by plasma thermal radiation(W_r),

the work of the pressure force(W_m), and electron heat conduction(W_e):

W_dep−W_loss=W_f−(W_r+W_m+W_e)≥0. (2) The heating rate of the plasma is determined by the rate of D/T reaction and the energy transferred to the plasma byα-particles(W_α)and neutrons(W_n):

W_f=fαWα+f_nW_n and

(Wα=nDnThσviε_α, Wn=nDnThσviεn,

(3)

whereεα =3.5 MeV andεn=14.1 MeV are the ini- tial energies ofα-particles and neutrons, respectively.

fα and fn are the energy fractions transferred to the plasma byα-particles and neutrons.hσviis the fusion reaction rate averaged over the Maxwellian distribu- tion.n_Dandn_Tare the number densities of deuterium and tritium nuclei,

n_f=n_D=n_T=x_fn=x_f ρ

m_nA^?, (4) wheren andρ are the plasma number and the mass densities, respectively.m_nis the mass of a neutron, and A^?is the average number of nucleons per plasma nu- cleus,

A^?= (A_D+A_T)x_f+

∑

j

A_jx_j

=2.5

"

1−

∑

j

x_j

# +

∑

j

A_jx_j.

(5)

In the D/T reaction, 80% of the D/T fusion power is associated with neutrons and the remaining 20% is as- sociated with theα-particles. In this case, the fusion power depositedW_f in the plasma given in (3) can be written as

W_f∼=2·10⁴²x²_fρ²

A^?2 hσvi(f_α+4f_n)(erg cm⁻³s⁻¹). (6) We now discuss separately the interaction of α- particles and neutrons with the hot plasma and suggest approximate expressions for fα and fn. Here, we use a simple approximate expression for quantities charac- terizing this process in a D/T plasma at temperatures below 30 keV. The energy ofα-particles is almost en- tirely transferred to the hot ignitor plasma via Coulomb

collisions with electrons. The fraction ofα-particle en- ergy deposited inside the considered hot homogeneous sphere of radiusRis given as [25]

f_α=









 3 2

R λα

−4 5

R λα

2

for R λα

≤1 2,

1− 1

4R/λ_α+ 1

160(R/λ_α)3 for R λα

≥1 2,

(7)

whereλα =8.4·10⁻³(A^?T^3/2/Z^?ρ) (cm)is the mean free path, andZ^?is the average number of electrons per plasma nucleus:

Z^?=2x_f+

∑

j

Z_jx_j. (8)

The neutrons released in D/T reactions interact pri- marily by elastic collisions with plasma nuclei because the elastic scattering cross section of neutrons by low- Z nuclei is substantially higher than the inelastic scat- tering [1]. The average neutron particles’ energy frac- tion deposited inside the considered hot homogeneous sphere of radiusRcan be expressed as

fn=

∑

i

2A_i

(1+A_i)²ψ_i. (9) The mean scattering probability ψ_i is the ratio of the average distanceL_i of the neutrons to path lengthλni

before an elastic collision with a nucleus ofith kind:

ψi= L_i

λ_ni= 2x_iσni

3A^∗m_nρR, λni= 1

n_iσ_ni, L=2R 3 ,

(10)

whereσniis the elastic cross section of neutrons with the ith nucleus. The cross sections for elastic scatter- ing of 14.1 MeV neutrons by light nuclei (including the hydrogen isotopes) and the impurity nuclei under study are close to one another and lie within a rela- tively narrow range of (0.8 – 1) barn [26]. Then (9) can be written as

fn=

"

0.14 1−

∑

j

x_j

! +0.7

∑

j

A_j (1+A_j)²x_j

#ρR A^∗ . (11) In the absence of impurities (x_j =0) f_n=0.056ρR, and the parameter ρR is valid up toρR>6 g cm⁻². In the D/T plasma,ρ λni'4.7 g cm⁻². This is much larger than theρRof a typical igniting hot spot while

Impurity concentration x Fraction of transmitted energy fn

Fig. 1 (colour online). Fraction of transmitted energy from neutrons to D/T plasma including impurities such as Li, Be, and C, in terms of impurities concentrations withρR= 7 g cm⁻².

it can be comparable to the total fuelρR. We there- fore neglect the neutron energy deposition for central ignition. In the presence of impurities, an increase of the neutron path length leads to a decrease in the en- ergy fraction fn(Fig.1). The termWr, corresponding to energy losses for plasma self radiation in criterion (2), is described by the well-known expression for the emissivity of a fully ionized plasma [27]:

W_r=32π 3

e⁶ mehc³

2πK_BT 3m_e

¹₂(Z²)^∗Z^∗ρ² (A^∗mn)²

∼=1.73·10²⁴

(Z²)^∗Z^∗ (A^∗)²

T^1/2ρ²

erg cm³s

(12) and

(Z^∗)²=

∑

i

Z_i²xi= 1−

∑

j

xj

! +

∑

j

Z²_jxj, (13) where K_B, h, c, and (Z²)^∗ are the Boltzmann con- stant, the Planck constant, speed of light, and mean squared charge number, respectively. The hot fuel sphere also exchanges energy with the plasma environ- ment through mechanical work. The mechanical work performed by a lump of matter at pressureP, the vol- ume of which changes by an amount dV, is dE=PdV.

The corresponding contribution to the power balance of a homogeneous sphere can be written as

W_m= 1

V dE

dt

=3P

Ru, (14)

where u =βvs is the velocity of the surface of the sphere expressed via the speed of soundv_s= (^P

ρ)^1/2. Now making use of the ideal-gas equation of state, P=ΓBρT, whereΓBis the gas constant:

Γ_B=3

2C_V'9.6·10¹⁴

1+Z^∗ A^∗

erg

keV g. (15) CVis the specific heat capacity withγ=⁵₃:

C_V= (Z^∗+1)K_B

A^∗(γ−1)m_n. (16) Then the mechanical work can be rewritten as

Wm∼=9·10²²

1+Z^∗ A^∗

³₂ βT³²ρ

R, erg

cm³s, (17) whereβ=0 [28,29] for a hot plasma with an isobaric distribution, andβ= [_γ+1² ]¹² [30] with an isochoric dis- tribution. The loss-power related to electron heat con- duction is described by [28]

W_e=−x∇T R

∼=1.3·10²⁰ T⁷² (Z^∗+4)R²

erg cm³s,

(18) wherexand∇T are the electron conductivity and the gradient of the electron temperature on the surface of the hot region, respectively. Substituting (6), (12), (17), and (18) in (2), the ignition condition for a D/T plasma containing impurities can be obtained as

h

8.07·10⁴⁰hσvi(f_α+4f_n)x_f−2.77·10²³T¹²x_ri (ρR)²

−6.4·10²²βT³²x_m(ρR)−2.61·10¹⁹x_eT⁷² ≥0, (19) where

x_f= 2.5

A^∗

1−

∑

j

x_j 2

, xr= 2.5

A^∗ 2

(Z²)^∗Z^∗

,

x_m= 2.5 A^∗

1+Z^∗ 2

3 2

, x_e= 5

Z^∗+4

. (20)

The factorsx_f,x_r,x_m, andx_edetermine the effect of im- purities on fusion reaction rate and the rates of energy

losses related to self-radiation, mechanical work, and heat conduction, respectively. In absence of impurities (x_j=0), the above factors are equal to one. A nega- tive effect of impurities on ignition process is related to a decrease in fusion reaction rate and an increase in the rate of energy losses for plasma self-radiation. This effect is substantially stronger than the weak positive effect related to a decrease in the energy lost for matter acceleration and electron heat conduction.

The minimum ignition temperature can be obtained from (19) for any type of target by equating the rates of thermonuclear heating and radiative energy losses and assuming thatfn=fα=1 as

T_min^ig =3.7

(Z²)^∗Z^∗ (1−∑jxj)²

²₅

keV. (21)

For a pure D/T plasma, the minimum ignition temper- ature is 3.7 keV. Lithium, beryllium, and carbon im- purities with an atomic fraction of 10% increase the value ofT_min^ig to 4.6, 5.1, and 6.3 keV, respectively (see Fig.2). The limiting concentration is determined by the equationWf=Wrat the plasma temperature of 60 keV at which the fusion reaction rate reaches its maximum value ofhσvi '9·10⁻¹⁶ (cm³s⁻¹) and fα =f_n=1.

These limiting concentrations of lithium, beryllium, and carbon atoms are 68%, 58%, and 42.5%, respec- tively.

Impurity concentration x Minimum ignition temperature Tig (min) (keV)

Fig. 2 (colour online). Minimum ignition temperature for D/T plasma including impurities Li, Be, and C, in terms of impurities concentration.

Ignition occurs when the alpha-particle heating of the hot spot exceeds all the energy losses [31]. Fig- ure 3 shows the ignition curves ρR−T for isobaric and isochoric D/T plasma with 10% impurity con- centrations of lithium, beryllium, and carbon atoms in the hot spot. It is shown that, for pure D/T plasma, (ρR)^min_ig '0.19 g cm⁻²atT_ig=7 keV for an isobaric target (β =0) and (ρR)^min_ig =0.4 g cm⁻² at T_ig =

ρ R (g/cm2) ρ R (g/cm2)

T (KeV) T (KeV)

Fig. 3 (colour online). Isobaric (left) and isochoric (right) ignition curve for D/T plasma containing 10% impurity of Li, Be, and C.

Impurity concentration x Impurity concentration x

Eig ρ2 100 KJ Eigρ100 KJ

Fig. 4 (colour online). Dependence of ignition energy on impurities concentrations in isobaric (left) and isochoric (right) plasma targets.

10 keV for an isochoric target (β =0.87). Also for the 10% impurity concentration of lithium, we have (ρR)^min_ig =0.34 g cm⁻²atT_ig=9.5 keV for an isobaric target and (ρR)^min_ig =0.564 g cm⁻² at T_ig=13 keV for an isochoric target. These figures show that mini- mum ignition temperature and(ρR)^min_ig increase with increasing impurity concentration for all types of tar- gets.

The ignition energy depends on the ignition param- eterρRand the ignition temperatureT_igas

E_ig=2πΓ_BT_ig(ρR)³_ig

ρ² (22)

=4.9·10⁸ 2.5 A^∗

1+Z^∗ 2

(ρR)³_ig ρ² T_ig (J).

Therefore, the minimum ignition energy corresponds to the minimum value of the parameter ρRin the ig- nition curve(ρR)_ig−T_ig. Forβ =0 andT_ig=10 keV (an isobaric target), the minimum ignition energy can be rewritten as

E_ig=7.12·10⁷ ρ²





A^∗ 2.5

Z^∗+1 Z^∗+4

(1−x)²−0.09(Z²)^∗Z^∗



 (J). (23) Also forβ =0.87 andT_ig=15 keV (an isochoric tar- get), we have

E_ig=3.15·10⁸ ρ²







 β ^2.5_A∗

¹₆

Z^∗+1 2

¹³₆

(1−x)²−0.06(Z²)^∗Z^∗









3 2

(J).

(24) Figure4shows the dependence of the ignition energy on impurity concentration in isobaric and isochoric tar- gets.

3. Effect of Low-ZImpurities on Fuel Gain

The fuel gain of a homogeneous plasma with inac- tive impurities is defined as the ratio of the energy gen- erated to total fuel energy (hot spot+cold fuel):

G=N_DTεrφ E_tot,f = εrφ

3T_ig

1−∑jx_j 1+Z^∗

, (25)

whereεr=17.6 MeV is the energy released in one fu- sion reaction,N_DTis the total number of coupled nuclei of D/T, and the burn efficiencyφis defined as the ra- tio between the number of D/T nuclei involved in the fusion reaction to their initial number:

φ=n⁽⁰⁾_f −n_f n⁽⁰⁾_f

. (26)

The time dependent number density can be obtained during fusion reaction as

n_f= n⁽⁰⁾_f 1+n⁽⁰⁾_f hσvit_c

t_c= R 4vs

, (27)

where the effective confinement time t_c is only one fourth oft_conf=_v^R

s. The burn efficiency can be shown as

φ= 1

1+ [n⁽⁰⁾_f hσvit_c]⁻¹

= 1

1+_ρRhσvi^4mfvs . (28) With the burn parameterH_B=^4m_hσvi^fvs, we can rewrite

φ= ρR

ρR+H_B. (29)

m_f=2.5m_nis the average mass of a D/T plasma nu- cleus with equal molar concentration. In the low-burn limitρRH_B, the burning efficiency isφ≈_H^ρR

B, and also in the limit of full burnρRHB, the burning effi- ciency isφ≈1. The burn parameter isH_B=6.4 g cm⁻² for the burning efficiency(20≤T ≤25 keV)of a pure D/T plasma. Then the burning efficiency will be

φ= ρR

ρR+6.4. (30)

Presenting impurities in the fuel:

H_B=1.65·10⁴ T⁵²

1+Z^∗ A^∗

¹₂ g cm²

. (31)

At temperatures between(20 – 25)keV:

φ= ρDTR ρDTR+7.1_1+Z^∗

A^∗

¹₂

. (32)

Assuming that the burning occurs at temperatures higher than 20 keV, then an increase in the impurity concentration leads to a decrease in the temperature of the burning plasma, and the critical value at which the mean temperature of the burning plasma decreases to 20 keV is determined. The critical concentration of impurities as lithium, beryllium, and carbon at 20 keV temperature are 43%, 37%, and 31%, respectively. For a typical impurity by a concentration lower than the critical, the burning efficiency obtained is

φ= ρDTR ρ_DTR+7.1h _2+(Z

j−1)x

2.5+(Aj−2.5)x

i¹₂ ,

0≤x≤x_cr.

(33)

For a concentration higher than the critical:

φ= ρDTR

ρDTR+7.1

[2+(Z_j−1)x]³ [5(1−x)]⁵2[2.5+(A_j−2.5)x]¹2

,

x≥xcr. (34)

Impurity concentration x

Burning efficiency

Fig. 5 (colour online). Burning efficiency of homogeneous D/T plasma in presence of impurities such as Li, Be, and C.

Fig. 6 (colour online). Fuel gain of a homogeneous D/T plasma in terms of impurities concentrations such as Li, Be, and C.

Figure5 shows the burning efficiency of a homoge- neous plasma with the parameter ρR=3 g cm⁻² as a function of the atomic fraction of low-Z impurities, calculated using formulas (33) and (34). Also, it is assumed that the burning of a homogeneous plasma can be performed with the parameterρDTR=3 g cm⁻² and the initial temperature 15 keV. The burning effi- ciency will slightly increase by increasing the impu- rity concentration at the range of 0≤x≤x_cr, which is related to an increase in the plasma confinement time. At the range of x>xcr, the burning efficiency rapidly decreases due to a decrease in burning temper- ature.

The fuel gain can be obtained by substituting (33) in (25) at the range ofx<x_cr:

G=17.6·10³ T_ig

1−x 2+ (Z_j−1)x

· 1

3+7.1

2+(Zj−1)x 2.5+(Aj−2.5)x

¹₂

, x<xcr. (35)

Also, for the range ofx>x_cr, the fuel gain can be ob- tained by substituting (34) in (25). The fuel gain of a homogeneous D/T plasma in terms of impurity con- centrations such as lithium, beryllium, and carbon are shown in Figure6. It is apparent that the fuel gain is dependent on the atomic fraction and the type of impu- rities. Also, at the critical impurity concentrations, the fuel gain is nearly half as compared to that in a pure D/T plasma.

4. Conclusion

According to above discussions, the presence of low-Zimpurities in the D/T plasma fuel of an ICF tar- get cause substantial changes in ignition parameters.

The negative effects of impurities in the D/T plasma burning are a decrease in fuel gain and an increase in target ignition energy due to a decrease in fusion re- action rate and an increase in radiation energy loss. In the presence of a 10% impurity of lithium, the fuel gain decreases about 25% and the ignition energy increases almost 2.1 times. By increasing the concentration of impurities, the values of the parameterρR and igni- tion temperature required to substantiate ignition in- creases. For 10% of lithium and beryllium impurities, the minimum value of the parameterρRand pure D/T

increases fuel about 1.5 and 1.8 times, respectively. If the concentration of impurities in the D/T plasma is

higher than a critical fraction, the burning efficiency and gain decreases suddenly.

[1] J. Nuckolls, L. Wood, A. Thiessen, and G. Zimmerman, Nature (London)239, 139 (1972).

[2] W. J. Hogan, E. I. Moses, B. E. Warner, M. S. Sorem, and J. M. Soures, Nucl. Fusion41, 567 (2001).

[3] T. C. Sangster, R. Betti, R. S. Craxton, J. A. Delettrez, D. H. Edgell, L. M. Elasky, V. Yu. Glebov, V. N. Gon- charov, D. R. Harding, D. Jacobs-Perkins, R. Janezic, R. L. Keck, J. P. Knauer, S. J. Loucks, L. D. Lund, F. J. Marshall, R. L. McCrory, P. W. McKenty, D. D.

Meyerhofer, P. B. Radha, S. P. Regan, W. Seka, W. T.

Shmayda, S. Skupsky, V. A. Smalyuk, J. M. Soures, C.

Stoeckl, B. Yaakobi, J. A. Frenje, C. K. Li, R. D. Pe- trasso, F. H. Séguin, J. D. Moody, J. A. Atherton, B. D.

MacGowan, J. D. Kilkenny, T. P. Bernat, and D. S.

Montgomery, Phys. Plasmas14, 058101 (2007).

[4] J. A. Frenje, C. K. Li, J. R. Rygg, F. H. Séguin, D. T.

Casey, R. D. Petrasso, J. Delettrez, V. Yu. Glebov, T. C.

Sangster, O. Landen, and S. Hatchett, Phys. Plasmas 16, 022702 (2009).

[5] S. W. Haan, M. C. Herrmann, T. R. Dittrich, A. J. Fet- terman, M. M. Marinak, D. H. Munro, S. M. Pollaine, J. D. Salmonson, G. L. Strobel, and L. J. Suter, Phys.

Plasmas12, 056316 (2005).

[6] S. W. Haan, P. A. Amendt, T. R. Dittrich, B. A. Ham- mel, S. P. Hatchett, M. C. Herrmann, O. A. Hurricane, O. S. Jones, J. D. Lindl, M. M. Marinak, D. Munro, S. M. Pollaine, J. D. Salmonson, G. L. Strobel, and L. J.

Suter, Nucl. Fusion44, 171 (2004).

[7] S. W. Haan, P. A. Amendt, D. A. Callahan, T. R Dit- trich, M. J. Edwards, B. A. Hammel, D. D. Ho, O. S.

Jones, J. D. Lindl, M. M. Marinak, D. H. Munro, S. M.

Pollaine, J. D. Salmonson, B. K. Spears, and L. J. Suter, Fusion Sci. Technol.51, 509 (2007).

[8] M. Tagagi, T. Norimatsu, T. Yamanaka, and S. Nakai, J. Vac. Sci. Technol. A10, 239 (1992).

[9] Yu. A. Merkul’ev and S. A. Startsev, in: Proceedings of the 26th European Conference on Laser Interaction with Matter, Prague 2000, p. 120.

[10] M. J. Edwards, J. D. Lindl, B. K. Spears, S. V. Weber, L. J. Atherton, D. L. Bleuel, D. K. Bradley, D. A. Calla- han, C. J. Cerjan, D Clark, G. W. Collins, J. E. Fair, R. J. Fortner, S. H. Glenzer, S. W. Haan, B. A. Hammel, A. V. Hamza, S. P. Hatchett, N. Izumi, B. Jacoby, O. S.

Jones, J. A. Koch, B. J. Kozioziemski, O. L. Landen, R. Lerche, B. J. MacGowan, A. J. MacKinnon, E. R.

Mapoles, M. M. Marinak, M. Moran, E. I. Moses, D. H.

Munro, D. H. Schneider, S. M. Sepke, D. A. Shaugh- nessy, P. T. Springer, R. Tommasini, L. Bernstein, W.

Stoeffl, R. Betti, T. R. Boehly, T. C. Sangster, V. Yu.

Glebov, P. W. McKenty, S. P. Regan, D. H. Edgell, J. P.

Knauer, C. Stoeckl, D. R. Harding, S. Batha, G. Grim, H. W. Herrmann, G. Kyrala, M. Wilke, D. C. Wilson, J.

Frenje, R. Petrasso, K. Moreno, H. Huang, K. C. Chen, E. Giraldez, J. D. Kilkenny, M. Mauldin, N. Hein, M.

Hoppe, A. Nikroo, and R. J. Leeper, Phys. Plasmas18, 051003 (2011).

[11] J. A. Marozas, F. J. Marshall, R. S. Craxton, I. V. Igu- menshchev, S. Skupsky, M. J. Bonino, T. J. B. Collins, R. Epstein, V. Yu. Glebov, D. Jacobs-Perkins, J. P.

Knauer, R. L. McCrory, P. W. McKenty, D. D. Mey- erhofer, S. G. Noyes, P. B. Radha, T. C. Sangster, W.

Seka, and V. A. Smalyuk, Phys. Plasmas13, 056311 (2006).

[12] R. L. McCrory, R. E. Bahr, R. Betti, T. R. Boehly, T. J. B. Collins, R. S. Craxton, J. A. Delettrez, W. R.

Donaldson, R. Epstein, J. Frenje, V. Yu. Glebov, V. N.

Goncharov, O. V. Gotchev, R. Q. Gram, D. R. Harding, D. G. Hicks, P. A. Jaanimagi, R. L. Keck, J. H. Kelly, J. P. Knauer, C. K. Li, S. J. Loucks, L. D. Lund, F. J.

Marshall, P. W. McKenty, D. D. Meyerhofer, S. F. B.

Morse, R. D. Petrasso, P. B. Radha, S. P. Regan, S.

Roberts, F. Séguin, W. Seka, S. Skupsky, V. A. Sma- lyuk, C. Sorce, J. M. Soures, C. Stoeckl, R. P. J. Town, M. D. Wittman, B. Yaakobi, and J. D. Zuegel, Nucl. Fu- sion41, 1413 (2001).

[13] H. Takabe, H. Nagatomo, A. Sunahara, N. Ohnishi, S.

Kato, A. I. Mahdy, H. Azechi, and K. Mima, Fusion Engin. Design44, 105 (1999).

[14] J. D. Lindl, P. Amendt, R. L. Berger, S. G. Glendinning, S. H. Glenzer, S. W. Haan, R. L. Kauffman, O. L. Lan- den, and L. J. Suter, Phys. Plasmas11, 339 (2004).

[15] T. C. Sangster, V. N. Goncharov, R. Betti, P. B. Radha, T. R. Boehly, D. T. Casey, T. J. B. Collins, R. S. Crax- ton, J. A. Delettrez, D. H. Edgell, R. Epstein, C. J.

Forrest, J. A. Frenje, D. H. Froula, M. Gatu-Johnson, Y. Yu. Glebov, D. R. Harding, M. Hohenberger, S. X.

Hu, I. V. Igumenshchev, R. Janezic, J. H. Kelly, T. J.

Kessler, C. Kingsley, T. Z. Kosc, J. P. Knauer, S. J.

Loucks, J. A. Marozas, F. J. Marshall, A. V. Maximov, R. L. McCrory, P. W. McKenty, D. D. Meyerhofer, D. T.

Michel, J. F. Myatt, R. D. Petrasso, S. P. Regan, W.

Seka, W. T. Shmayda, R. W. Short, A. Shvydky, S.

Skupsky, J. M. Soures, C. Stoeckl, W. Theobald, V. Ver- steeg, B. Yaakobi, and J. D. Zuegel, Phys. Plasmas20, 056317 (2013).

[16] J. Lindl, O. Landen, J. Edwards, E. Moses, and NIC Team, Phys. Plasmas21, 020501 (2004).

[17] V. N. Goncharov, T. C. Sangster, T. R. Boehly, S. X.

Hu, I. V. Igumenshchev, F. J. Marshall, R. L. McCrory, D. D. Meyerhofer, P. B. Radha, W. Seka, S. Skupsky,

C. Stoeckl, D. T. Casey, J. A. Frenje, and R. D. Pe- trasso, Phys. Rev. Lett.16, 165001 (2010).

[18] T. C. Sangster, V. N. Goncharov, R. Betti, T. R. Boehly, D. T. Casey, T. J. B. Collins, R. S. Craxton, J. A. Delet- trez, D. H. Edgell, R. Epstein, K. A. Fletcher, J. A.

Frenje, Y. Yu. Glebov, D. R. Harding, S. X. Hu, I. V.

Igumenschev, J. P. Knauer, S. J. Loucks, C. K. Li, J. A.

Marozas, F. J. Marshall, R. L. McCrory, P. W. McK- enty, D. D. Meyerhofer, P. M. Nilson, S. P. Padalino, R. D. Petrasso, P. B. Radha, S. P. Regan, F. H. Seguin, W. Seka, R. W. Short, D. Shvarts, S. Skupsky, V. A.

Smalyuk, J. M. Soures, C. Stoeckl, W. Theobald, and B. Yaakobi, Phys. Plasmas17, 056312 (2010).

[19] Lord Rayleigh, Scientific Papers, Vol. 2, Dover publi- cations, New York 1965.

[20] N. A. Tahir, D. H. H. Hoffmann, A. Kozyreva, A.

Tauschwitz, A. Shutov, J. A. Maruhn, P. Spiller, U. Ne- uner, J. Jacoby, M. Roth, R. Bock, H. Juranek, and R.

Redmer, Phys. Rev. E63, 016402 (2000).

[21] A. R. Piriz, J. J. Lopez Cela, O. D. Cortazar, N. A.

Tahir, and D. H. H. Hoffmann, Phys. Rev. E72, 056313 (2005).

[22] A. R. Piriz, J. J. Lopez Cela, and N. A. Tahir, Phys.

Rev. E80, 046305 (2009).

[23] A. R. Piriz, G. Rodriguez Prieto, I. Muñoz Díaz, and J. J. López Cela, Phys. Rev. E82, 026317 (2010).

[24] S. Y. Guskov, DEVIIin, and V. E. Sherman, Fizika Plazmy37, 1096 (2011).

[25] O. N. Krokhin and V. B. Rozanov, Soviet J. Quantum Electron.2, 393 (1973).

[26] L. P. Abagyan, N. O. Bazazyants, M. N. Nikolaev, and A. M. Tsibulya, in: Group Constants for Calculations of Reactors and Protection, M. N. Nikolaev (Ed.), En- ergoizdat, Moscow, 1981 [in Russian].

[27] Ya. B. Zel’dovich and Yu. P. Raizer, Physics of Shock Waves and High Temperature Hydrodynamic Phenom- ena, GIFML, Vols. 1, 2, Moscow 1963; Academic, New York 1966, 1967.

[28] S. Atzeni and J. Meyer-Ter-Vehn, Oxford Science Pub- lication, Oxford University Press, Oxford 2004.

[29] A. G. Aleksandrova, N. A. Popov, and V. A. Shcherba- kov, Vopr. At. Nauki Tekh., Ser. Yad. Fiz. Issl.,1, 95 (1993).

[30] N. A. Popov, V. A. Shcherbakov, V. N. Mineev, P. M.

Zaydel, and A. I. Funtikov Usp. Fiz. Nauk178, 1087 (2008) [Phys. Usp.51, 1047 (2008)].

[31] R. Betti, P. Y. Chang, B. K. Spears, K. S. Anderson, J. Edwards, M. Fatenejad, J. D. Lindl, R. L. McCrory, R. Nora, and D. Shvarts, Phys. Plasmas17, 058102 (2010).