Contents
C O N T E N T S
List of Figures iv
i n t r o d u c t i o n v
1 l a s e r-plasma interaction 1
1.1 Fourth state of matter: Plasma . . . 1
1.2 LASER produced plasmas . . . 2
1.3 Absorption mechanisms . . . 4
1.3.1 Inverse Bremsstrahlung absorption . . . 4
1.3.2 Resonance absorption . . . 6
1.3.3 Brunel effect . . . 7
1.4 X-Ray Emission from LASER produced plasmas . . . 7
1.4.1 Free to Free Emission: Bremsstrahlung . . . 7
1.4.2 Free to Bound Emission: Recombination . . . 9
1.4.3 Bound to Bound Emission: Atomic Lines . . . 9
1.4.4 Characteristics of LASER-plasma X-ray sources . . . 11
1.5 Radiation transport in plasmas . . . 12
1.6 Parametric instabilities . . . 13
1.6.1 Stimulated Brillouin Scattering . . . 13
1.6.2 Stimulated Raman Scattering . . . 14
1.6.3 Two Plasmon Decay . . . 15
1.6.4 Self-focusing and filamentation . . . 16
1.6.5 Hot Electrons . . . 17
2 p r i n c i p l e s o f i n e r t i a l c o n f i n e m e n t f u s i o n 19 2.1 Thermonuclear fusion . . . 19
2.2 Inertial Confinement Fusion . . . 22
2.2.1 Inertial confinement by spherical implosion . . . 24
2.3 Symmetry and stability . . . 26
2.3.1 Rayleigh Taylor Instability (RTI) . . . 27
2.3.2 Kelvin Helmoltz instability (KHI) . . . 28
2.3.3 Richtmeyer Meshkov instability (RMI) . . . 29
3 f oa m s 31 3.1 Interaction of foams with high intensity LASER . . . 32
3.1.1 The Homogenization Process . . . 32
3.1.2 Absorption mechanism in porous materials . . . 33
3.2 Applications of foams in Inertial Confinement Fusion . . . 35 i
Contents
4 t h e a b c l a s e r fa c i l i t y 39
4.1 ABC Laser . . . 39
4.2 Experimental chamber & Main diagnostics . . . 41
4.2.1 Plasma density profile: The Nomarski Interferometer . . 41
4.2.2 Optical Spectrometer . . . 44
4.2.3 Soft X-ray detection and Imaging . . . 44
4.2.4 Fast Particles: Diamonds detectors . . . 51
4.2.5 Time resolved optical diagnostic: The Streak Camera . . 53
4.3 Target Area . . . 55
4.3.1 Choosing Target . . . 55
4.3.2 Preparation of selected Foams on target Holder . . . 55
5 d ata a na ly s i s 59 5.1 Aluminum Targets . . . 59
5.1.1 Interferometer Analysis . . . 59
5.1.2 Transmission Grating Analysis . . . 62
5.1.3 Streak camera Analysis . . . 65
5.1.4 Diamond detectors Analysis . . . 66
5.2 Solid Polystyrene Targets . . . 69
5.2.1 Interferometer Analysis . . . 69
5.2.2 Streak camera Analysis . . . 71
5.2.3 Diamond detectors Analysis . . . 71
5.3 Polystyrene Foam Targets . . . 73
5.3.1 Interferometry Analysis . . . 73
5.3.2 Transmission Grating Analysis . . . 76
5.3.3 Streak camera Analysis . . . 77
5.3.4 Diamond detectors Analysis . . . 80
5.4 Comparison of behavior between the analyzed materials . . . . 82
5.4.1 Comparison of TOF measurements between different target materials . . . 82
5.4.2 X-Ray emission from different materials . . . 83
5.4.3 Diamond detectors . . . 84
5.4.4 Silicon Surface Barrier detectors . . . 84
6 c o n c l u s i o n s 87
Bibliography 89
List of Figures
L I S T O F F I G U R E S
Figure 1.2.1 Laser produced plasma characteristics . . . 4
Figure 1.4.1 Emission mechanisms in plasmas . . . 10
Figure 2.0.1 Cross section of the main fusion reactions . . . 20
Figure 2.2.1 Energy balance in an ICF reactor . . . 24
Figure 2.2.2 ICF by spherical implosion . . . 25
Figure 2.2.3 ICF target design for direct drive approach: a first scheme . . . 25
Figure 2.3.1 Concept of ignition margin . . . 26
Figure 2.3.2 Evolution of Rayleigh Taylor instability . . . 28
Figure 2.3.3 Evolution of Kelvin Helmoltz instability . . . 29
Figure 3.0.1 Miscroscope view of Foams . . . 31
Figure 3.2.1 Evidence of smoothing action of porous materials . . . 36
Figure 3.2.2 Laser ablated loading efficiency as a function of foam thickness . . . 37
Figure 3.2.3 LIGHT: the first ICF target involving foams . . . 38
Figure 4.1.1 The ABC Laser layout . . . 39
Figure 4.2.1 Nomarski interferometer scheme . . . 43
Figure 4.2.2 Nomarski Interferometer: image formation . . . 43
Figure 4.2.3 Spectrometer Ocean Optics HR4000CG-UV-NIR . . . . 44
Figure 4.2.4 Silicon Surface Barrier detectros . . . 46
Figure 4.2.5 Disposition of SSB and an example of their typical signal 47 Figure 4.2.6 Absorption Coefficient of Beryllium and Nichel Filters 48 Figure 4.2.7 Transmission grating characteristics . . . 50
Figure 4.2.8 Micro channel plates scheme . . . 50
Figure 4.2.9 Disposition of Diamond detectors in the experiemtal chamber . . . 52
Figure 4.2.10 Typical signal obtained from Diamond detectors . . . . 53
Figure 4.2.11 Streak Camera scheme . . . 54
Figure 4.3.1 The support of targets . . . 56
Figure 4.3.2 Example of cataloged Foam . . . 57
Figure 5.1.1 Interferometer analyis for Aluminum . . . 60
Figure 5.1.2 Aluminum electron density profile . . . 61
Figure 5.1.3 Transmission grating measurements on Aluminum . . 62
Figure 5.1.4 Streak Camera imaging for Aluminum . . . 65
Figure 5.1.5 Streak camera measurements for Aluminum . . . 66
Figure 5.1.6 Diamonds’ signal for Aluminum . . . 67
Figure 5.2.1 Interferometer analysis for solid Polystyrene . . . 69 iii
List of Figures
Figure 5.2.2 Electron density profile for Polystyrene . . . 70 Figure 5.2.3 Streak camera measurments for Polystyrene . . . 71 Figure 5.2.4 Diamond’s signal for Polystyrene . . . 72 Figure 5.3.1 Interferometer analysis for porous Polystyrene . . . . 73 Figure 5.3.2 Porous Polystyrene electron density profile . . . 75 Figure 5.3.3 Transmission grating measurements for porous Polystyrene 76 Figure 5.3.4 Streak camera measurement for "thick" porous Polystyrene 78 Figure 5.3.5 Streak camera analysis of "thin" porous polystyrene . . 79 Figure 5.3.6 Streak camera measurements for porous polystyrene
with thickness comparable to transparency length . . . 79 Figure 5.3.7 Diamonds’ signal for "thick" porous Polystyrene . . . . 80 Figure 5.3.8 Diamonds’ signal for porous Polystyrene with
thick-ness comparable to transparency length . . . 81 Figure 5.3.9 Diamonds’ signal for "thin" porous Polystyrene . . . . 81 Figure 5.4.1 Measured protons velocity for different target materials. 83 Figure 5.4.2 Velocity of plasma expansion in function of target
material. . . 83 Figure 5.4.3 Amplitude of the signal received from diamond . . . . 84 Figure 5.4.4 Amplitude of signal received from SSB . . . 85
i n t r o d u c t i o n
I N T R O D U C T I O N
The incessant need of energy has stimulated the development of new tech-nologies for its production. Nowadays the next step to take seems to be toward nuclear fusion.
In this context plasma physics plays a central role. Indeed fusion reactions can occur only if the relative energy of the nuclei involved is high enough to overcome Coulomb repulsion and can be shown that, in order to favor fusion reactions, it is convenient to work in a ionized gas with high density and temperature.
These requirements led thermonuclear fusion research to develop two differ-ent approaches: Inertial Confinemdiffer-ent Fusion (ICF) and Magnetic Confinemdiffer-ent Fusion. The main difference between these two techniques arises from the strategy adopted to reach ignition condition, i.e. to obtain self-sustained fu-sion reactions. Both methods work at high temperatures but, while magnetic confinement, to facilitate fusion, tries to increase the time that ions spend near each other by using intense pulsed magnetic fields, ICF relies on the inertia of the fuel mass to provide confinement in order to fuse nuclei before target explosion occurs. In fact at the high plasma density typical of ICF, the magnetic fields required to obtain confinement would be so high that are impossible to achieve.
In inertial Confinement fusion the necessary heating and fuel compression can be provided by high power laser beams. Two different schemes of irra-diation have been proposed: direct and indirect drive. In indirect drive, a pellet containing the fuel, typically a mixture of deuterium and tritium, is located inside an high Z case, an hohlraum. Laser energy is first absorbed by this enclosure which, once heated, emits X-rays which fill the hohlraum and drive the capsule implosion. On the other hand, in direct drive approach laser beams deliver their energy directly onto the outer layer of the pellet. The irradiation of the fuel target represents the crucial point of the process, in fact during laser-matter interaction, due to irradiation inhomogeneities, both parametric and hydrodynamic instabilities could be generated, lead-ing to target pre-heatlead-ing and compression inefficiency[1, 2]. Avoidlead-ing this phenomenon is one of the challenges we have to face to move a further step toward the access of a new source of energy. One of the key points to make this problem less severe lies into target design.
In this respect, since the late 90’s it was considered of interest to coat fusion pellet with light foams which would act as absorbers of the laser radiation. Made from plastic polymer, these foams present a porous structure with v
i n t r o d u c t i o n
cavities separated by solid filaments or membranes, with dimensions of the cavities much greater than those of the solid elements. This inhomogeneous structure induces peculiar absorption mechanisms when laser light impinges on its surface. In particular radiation reaches regions of the material deeper than in the corresponding solid and an homogenization process takes place in which the solid filaments evaporate and create a plasma which fills the surrounding cavities. In this conditions absorption doesn’t involve only the superficial layer of the material but actually concerns a considerable part of the volume [3, 4, 5, 6]. This distinctive property could be useful in reducing the unwanted effects of irradiation inhomogeneities and enhancing absorption in ICF scheme. Indeed it has been shown [7, 8] that covering the outer shell of the pellet with a layer of a porous plastic material, laser imprinting could be sensibly reduced with subsequent inhibition of undesired hydrodynamic instabilities. Moreover this materials seems to enhance the efficiency of conversion of laser energy into shock wave energy, as shown in [9].
The aim of this thesis is to study the behavior of such porous material inter-acting with high energy laser beams. In particular we are going to compare the response of different materials to irradiation, focusing our attention on the differences between an homogeneous solid material and the analogous inhomogeneous porous one.
The present work of thesis has been carried out in the frame of a collab-oration between the Physics Department of Pisa University and the ENEA research center in Frascati (Rome). The experimental campaign that I at-tended has been performed at ABC laser facility led by Dr. Riccardo De Angelis. The ABC laser is able to deliver two beams, each one up to 100 J energy and, thanks to the different diagnostics at our disposal, from a single shot we were able to collect a set of data pertinent with a wide range of various aspects of laser-matter interaction. This allows us to have a quite complete view of the subject investigated.
Nomarski interferometer allows us to perform the plasma electron density profile reconstruction; spectral analysis of the X-Ray radiation is performed by means of a transmission diffraction grating; visible streak camera provides information about temporal evolution, hence on expansion velocity, which can be related to plasma temperature. X-ray emission was also monitored with a system of semiconductor detectors, which, with an adequate filtering, provides information on different spectral regions. In order to persecute our aim, in addition to the diagnostic system set up and data collection, a consistent part of the work was that of preparing different targets of porous polystyrene, so as to have a wide collection of different irradiation situations.
i n t r o d u c t i o n The thesis is organized as follows;
Chapter 1 resumes basic concepts of laser-plasma interaction with particular attention to the absorption and emission mechanisms. The final part is dedi-cated to an overview of parametric instabilities which, as mentioned above, could compromise the ignition process.
Chapter 2 is dedicated to a brief description of thermonuclear fusion, fo-cusing our attention on the inertial confinement approach with direct drive scheme. In this context we describe the main hydrodynamic instabilities which could occur during laser irradiation.
Chapter 3resumes the acquired knowledge on the behavior of foams inter-acting with high energy laser beam and shows the possibility to use them as ablators.
Chapter 4offers an overview of the main diagnostics available at ABC facility, providing a brief description of their working principles and resuming the methods used to perform data analysis. Last section of the chapter illustrates the method adopted in order to select and prepare polystyrene foams targets. Chapter 5represents the heart of this work and shows the results obtained with different types of targets. First data to be presented are those of Alu-minum, which represents a well known situation and could be used as reference. Then we show results obtained for solid and porous polystyrene targets which, despite their identical chemical composition, will reveal re-markable differences in their absorption and emission properties.
l a s e r-plasma interaction
1
L A S E R - P L A S M A I N T E R A C T I O NIn this chapter we introduce the basic concepts of plasma physics, we de-scribe the formation process of a laser produced plasma and the subsequent interaction between the electromagnetic wave and this new-born plasma such as absorption and emission mechanisms as well as parametric instabilities formation.
1.1 f o u r t h s tat e o f m at t e r: plasma
Plasma is essentially a ionized gas in which charged particles outnumber neutral ones.
The net uncompensated charge is supposed to be small compared to the total amount of charged particles of either sign, therefore plasma can be considered globally neutral.
Another characteristic of this state of matter is that an electrostatic force tends to avoid a net charge separation and, due to this fact, electrons and ions move approximately together. In these terms we can refer to a plasma as an electrically quasi-neutral medium which shows a collective behavior. We use quasi-neutral term because this feature takes place only on quite large volumes and time intervals. The space and the time in which the charge compensation could fail are called characteristic distance and time of charges separation. The first one is represented by the Debye screening length, λD,
and the second one is τpe= ω1pe, where ωpe is the electron plasma frequency
given by the relation:
ωpe =
s 4πnee2
me
(1.1.1)
in which me, e, ne are respectively the electronic mass, charge and number
density.
This frequency is the one characteristic of the electron density oscillations, called Langmuir waves, generated by a perturbation to the initial electron density distribution.
The Debye screening length is the radius of a sphere in which is accumulated enough polarization charge to shield the electrostatic field on long distance and it is fundamental to establish in which situation plasma assume or not a
l a s e r-plasma interaction collective behavior. It can be expressed as λD = s kbTe 4πnee2 = vth ωpe (1.1.2)
where kb is the Boltzmann constant, T is electronic temperature and vth =
kbTe/me)1/2 is the thermal velocity of electrons.
The other characteristic parameter, τpe, is the time needed by the plasma to
restore neutrality after its perturbation.
For what concern equilibrium state is rather difficult to find laser produced plasmas in thermodynamic equilibrium, whilst we can more often talk of partial equilibrium in which the velocity distribution of ions and electrons are two Maxwellian characterized by different temperatures, electronic Te
and ionic one Ti.
Usually plasma studies involve the investigation of its behavior under small perturbation of this partial equilibrium.
1.2 l a s e r p r o d u c e d p l a s m a s
High intensity laser radiation impinging onto a solid target induces its surface to turn into plasma state. The subsequent expansion of material produces a plasma with peculiar temperature and density profile.
A laser pulse of sufficient duration will produce this density gradient before the end of the laser pulse itself, making possible the interaction with the expanding plasma. The longitudinal scale length of the density gradient is determined by the expansion velocity of plasma and the laser’s focusing con-dition. It is given by Ln≡ ∇nnee ' min[csτ, φ]where τ and φ are respectively
laser pulse duration and the focal spot diameter on the target, csis the sound
velocity at plasma temperature.
Plasma can be described as a fluid state with two components, electronic and ionic one. Though it can be considered neutral as a whole, its behavior when electromagnetic stimulation is performed remember us of the free charge composition.
The propagation of an electromagnetic wave, with frequency ωL and wave
vector kL, in a plasma is dominated by the presence of electrons, and looking
at the dispersion relation, given by
ω2L =ωP2+k2Lc2 (1.2.1)
we will find out that exists a special density, called critical density, for which the wave-vector of the propagating wave becomes imaginary, and the
1.2 laser produced plasmas
electromagnetic wave will be reflected out of the plasma region. From the equation (1.2.1) we can easily get the following expression for such critical density ncr= meω2 4πe2 = 1.1×1021 λ2L(µm) # cm3 (1.2.2)
where λLis the wavelength of the impinging laser.
This means that the interaction between laser and plasma develops in those regions called under-dense, ne < ncr, whereas in the over-dense regions
(ne >ncr) we will have an evanescent electromagnetic wave. The distance in
the over-dense region in which the light wave decays is called skin depth:
δsd ' 1 2 c q ω2p−ω2L (1.2.3)
For what concern the plasma temperature, we can get an estimation of it by equating the heat electron flux to absorbed laser energy,
Iabs =kBTene kBTe
me
12
f (1.2.4)
this leads to the following relation
Te = Iabs f ne 23 m13 kB '3×107 I abscmW2 f necm#3 23 eV (1.2.5)
where f <1 is the flux limiter parameter that takes into account the devia-tions from classical description of heat conduction.
Since we are not considering any energy loss mechanism, the equation (1.2.5) overestimates the real value.
Due to the high electron conduction in the under-dense region the temper-ature scale length LT ≡ ∇TTee will be very long, whilst, in the over-dense
region behind the critical density, a cooler plasma is in contact with the solid material, thus we will find a very short scale length.
The figure 1.2.1 shows the typical spatial distribution of temperature, density and expansion velocity of a plasma generated by irradiation of a solid target by a nanosecond laser pulse as a function of distance from the target surface. Of particular interest is the behavior of temperature, indeed it has a fast decrement in correspondence of the critical density. This fact confirms that laser radiation cannot penetrate beyond this point and consequently plasma heating is less efficient than in the underdense region.
l a s e r-plasma interaction
Figure 1.2.1:Density, Temperature and expansion velocity profile of a plasma gener-ated by irradiation of a solid target by a nanosecond laser pulse.[2]
1.3 a b s o r p t i o n m e c h a n i s m s
Laser light propagating in a plasma can interact with it in various ways. In this section we will describe the main absorption mechanisms such as Inverse Bremsstrahlung absorption, also known as collisional absorption, resonance absorption, ion turbulence absorption and Brunel effect.
Another contribution to energy absorption is due to the damping of electronic and ion waves produced by the parametric instabilities activation, whose description will be presented in detail in section 1.6.
A key parameter for these mechanism is the critical density ncr. Indeed most
of them takes place in a underdense plasma, therefore in order to have a bigger portion of plasma directly heated by impinging laser it is convenient to work with high values of ncr. We have already seen, from equation (1.2.2),
that ncr inversely depends on the square of the laser wavelength, hence
shorter values of λL allow to laser light to reach deeper regions of plasma.1
1.3.1 Inverse Bremsstrahlung absorption
An electromagnetic wave propagating trough the plasma interacts with the electrons, which, while oscillating, collide with ions providing to transfer energy from laser to the plasma.
This mechanism is known as collisional absorption or inverse Bremsstrahlung absorption.
In order to obtain the form of the absorption coefficient we have to calculate
1 This is one of the reasons why the experiments developed at the National Ignition Facility work with the third harmonic of a Neodymium laser.
1.3 absorption mechanisms
the dielectric function under the right conditions, namely we have to include in the Newton’s equation the electron-ion collisional term.
−iωme~ue= −e~E+ ~Fext (1.3.1)
where ue is the velocity of electrons, E the electric field and last term in the
equation is due to inclusion of collisions. The~Fextterm can be written trough
a friction term∝ νe, which is collisional frequency.
−iωmeu~e = −e~E−νemeu~e→ ~ue= − e
me(νe−iω)
~E (1.3.2)
with some algebraic passage
~j= − e2ne
ime(ω+iνe)
~E (1.3.3)
Therefore, using Ohm’s law and dielectric function definition: ~j= σ(ω)~E e(~k, ω) =1+4πi σ(ωω) (1.3.4) we have: e(~k, ω) =1− ω2p ω2(1+ iνωe) (1.3.5)
damping of the electromagnetic wave is due to the imaginary part of e(~k, ω). By direct substitution of the above relation in the one we find from waves equation assuming transversal propagation, namely k2= ω2
c2 e(~k, ω), a
com-plex quantity is obtained. The real part of it gives the usual plasma’s dis-persion relation, whilst its imaginary part gives the coefficient for collisional absorption kib = νe c ω2p ω2 1 q 1−ω2p ω2 (1.3.6)
Explicating νe for electron-ion collision
kib =3.1×10−7Zn2elnΛ 1 ω2 1− ωp ω 2−12 T−32 e (eV) (1.3.7)
where lnΛ is the Coulomb logarithm which, for Timmei <10Z2< Te, is given
by lnΛ'24−ln √ ne Te(eV) .
In an uniform plasma, the fraction of absorbed laser energy after a propaga-tion over a distance L is
l a s e r-plasma interaction
It is easy to see that collisional absorption decrease with temperature and increase with Z and density, then, when laser light interacts with inhomoge-neous plasmas, this kind of absorption takes place mostly near the critical surface ncr.
Due to the high dependence of Coulomb cross section on electron velocity,
σei∝ v−e4, we see that the increase of laser intensity can produce a reduction
in IB absorption, in fact, under the action of an electric field, electrons acquire a quiver velocity given by
vq =
eEo
mω (1.3.9)
and we can define effective electron velocity as ve f f =
q v2
e+v2q. Clearly at
higher intensities corresponds higher quiver velocities and consequently a reduction of efficiency of absorption mechanism.
Moreover the rate of energy gain from the laser field by plasma’s electrons ex-ceeds the rate at which they share it with each other. This fact leads to a lack of low energy electrons. As mentioned above, the cross section of electron-ion collisions inversely depends on electrons’ velocity, hence, a reduction in low energy electrons means a decrement of good candidates to perpetrate Inverse Bremsstrahlung absorption.
In the previous discussion we assumed a random ion motion. It has been demonstrated[10] that in the hypothesis of correlated motion of the ions we have an enhancement of absorption. This effect is also known as ion turbulence absorption.
In the case of short laser pulse (τL < 100ps) and under ideal conditions,
ion-acoustic fluctuations can cause an increment of the absorption up to 20%. For long pulse experiments, low Z targets and λL > 1µm, ion turbulence
absorption can exceed inverse Bremsstrahlung.
1.3.2 Resonance absorption
Resonance absorption takes place when a "p" polarized electromagnetic wave impinges on a plasma surface with oblique orientation. In fact, in this case, we have the electric field lying on the plane of incidence with a component in the direction of density gradient. This condition leads to charge separation and, at critical surface where ωL = ωp, to excitation of Langmuir waves.
Then, conversion of laser energy into thermal plasma energy occurs trough the damping of the plasma oscillations.
Despite oblique incidence is necessary to excite this kind of waves we have to maintain the angle of incidence θ not too large in order to be able to approach critical surface as near as possible. In fact, due to refraction, the plasma density at the turning point of laser light is given by netp =ncrcos2(θ).
1.4 x-ray emission from laser produced plasmas
Hence, a greater angle of incidence produces a bigger difference between the turning point density and the critical one. The point at which critical density is reached is the one where the electromagnetic waves can excite resonances. Therefore we have to find the perfect compromise between two conflicting requirements: we want to have the electric field component in the direction of density gradient as great as possible maintaining θ as small as possible. Previous studies [11] have shown that resonance absorption coefficient has its maximum for an angle of incidence satisfying the relation sin(θ) '
0.8(ωLn/c)−
1
3. In such condition resonance absorption can absorb up to
50% of the light reaching the critical surface.
1.3.3 Brunel effect
Brunel effect is another absorption mechanism that involve very intense Laser light obliquely impinging on a sharply bounded overdense plasma. This radiation pulls out in the vacuum electrons which will acquire kinetic energy K=1/2mev2q, where vqis the quiver velocity, then these electrons are
sent back into the plasma where they will loose a large part of the gained energy[12].
Brunel effect prevails on resonance absorption for vq/ω> Ln, where Lnis the
density gradient scale length. Moreover this effect depends on the ratio vq/c,
consequently it is of major importance in interaction with femtosecond laser, where shorter values of Lnand ultra high laser intensity can be achieved.
1.4 x-ray emission from laser produced plasmas
Plasma, once gained thermal energy, emits radiation trough three different ways: Free to Free, Free to Bound and Bound to Bound electron transitions. The first one is due to free electrons interacting with ions’ Coulomb potential. The experienced acceleration produces a continuum electromagnetic spec-trum, termed Bremsstrahlung emission.
Then we have the Recombination mechanism in which free electrons are cap-tured by ionized atoms, this will lead to emission of a continuum spectrum characterized by the so called absorption edges.
The third mechanism consists in electron transition between levels of ionized atoms and produces a line spectrum.
1.4.1 Free to Free Emission: Bremsstrahlung
Motion of free charges in plasma is influenced by the electric field generated by themselves. In particular electrons, due to their lower mass, will suffer an acceleration greater than the ions and consequently it is possible to consider
l a s e r-plasma interaction
only the electrons’ contribution to emission.
The process can be schematically described considering an electron moving toward an ion with velocity v and impact parameter b; the characteristic interaction time will be τ=2b/v and so the main emission frequency will be ν ' (2πτ)−1 = v/(4πb). In order to evaluate the energy, ∆E, radiated
during a single interaction we consider the maximum acceleration suffered by the electron, namely amax = Ze
2
b2m
e, therefore we will have
∆E= 4 3c3 Z2e6 b3m2 ev (1.4.1)
Taking in account that in the unit time the number of electron-ion collision with impact parameter ranging from b to b+db is 2πnivbdb, the power
radiated per electron will be
w= Z bmax bmin ∆E·2πnivbdb ' 8 3πni Z2e6 m2 ec3 Z bmax bmin db b2 (1.4.2)
where bmax =λd and bmin is given by the De Broglie length.
At this point, considering a Maxwellian distribution for electrons, we can evaluate the power radiated per unit volume. The exact numerical solution gives WB =1.6×10−27Z2n2eT 1 2 e(eV) erg seccm −3 (1.4.3)
The equation (1.4.3) shows a weaker dependence from temperature than in the case of blackbody radiation where emission scales as T4.
In order to have the spectral distribution the (1.4.2) has to be integrated over the frequency. To do this we can use the mentioned relation between the impact parameter b and ν.
The expression obtained is
Wν B = 32π 3 2π 3kBTeme 12 Z2e6n2e mec3 exp − hν kbTe '6.8×10−38Z2n2eT12 e exp − hν kbTe erg s cm −3Hz−1 (1.4.4)
where the integration over ν has been performed between νmin = v/4πλD
and νmax =mev2/2h.
Exploiting the relation between spectral intensity per unit frequency and that for unit of wavelength we obtain
Wλ B =WBν c λ2 '2×10−27Z2n2eT−12 e 1 λ2 exp hc λkBTe erg s cm −4 (1.4.5)
1.4 x-ray emission from laser produced plasmas
Even if the distribution shape is similar to the one for blackbody, at the same temperature the Bremsstrahlung maximum is situated at wavelength longer than in the case of black body, namely λBrem(Å) =6200/Te(eV)whilst for
the blackbody we have λBB(Å) =2500/Te(eV).
1.4.2 Free to Bound Emission: Recombination
In this process a free electron captured by a ionized atom produces a photon with energy
hν= 1 2mev
2
e+∆E (1.4.6)
where the first term in the right side is the kinetic energy of electron and ∆E = EnZ is the energy of the final atomic state. The kinetic energy of incident electron takes values over a continuum. The minimum energy of the emitted photon, corresponding to the capture of an electron at rest, is hν=EZn, therefore the spectrum will be a continuum characterized by abrupt interruption, called absorption edge.
Spectral intensity of recombination emission can be expressed in term of the bremsstrahlung one Wν r =WBν·2.2×10−32 Z3n2e T 3 2 e ∞
∑
n=1 1 n3exp Z2E H n2k BTe − hν kbTe erg seccm −3Hz−1 (1.4.7) where EH =13.6eV is the Hydrogen ionization energy.We report also the ratio between spectrally integrated recombination and Bremsstrahlung emission Wr WB '2.4Z 2E H kbTe (1.4.8)
from this relation it is evident that for high temperature and low Z, Bremsstrahlung process prevails on recombination one.
1.4.3 Bound to Bound Emission: Atomic Lines
Atomic Lines, generated by the transition of bounded electrons from an excited state to a lower one, give information about the species present in plasma.
Moreover studying the characteristic of those lines we can recover some other information about the plasma itself. In fact, looking at the emission coefficient, we see that it depends both on parameters strictly related to the kind of atoms involved and on the physical condition of plasma.
Three physical processes can cause the broadening of such lines. The first one is called natural broadening and it is due to the finite radiative lifetime of the
l a s e r-plasma interaction
bounded states involved in the transition. Then we have Doppler broadening which depends on thermal motion of radiating system
∆νD '6.65×10−9ν
r
Te(eV)
A (1.4.9)
where A is mass number.
Lastly we have Pressure broadening related to the interaction of radiating atom and the rest of plasma
∆νP '
σvrelnp
2π (1.4.10)
where σ is the total cross section of the collisional process, vrel is the relative
velocity of the interacting particles and npis the number density of the
per-turbers.
The most important pressure broadening in plasmas is the so called Stark broadening. It is due to the interaction of radiating atom with microscopic electric fields generated by the surroundings particles. This type of broaden-ing prevails in high density plasmas, for highly ionized species.
Figure 1.4.1 shows the spectrum of the radiation generated by a laser pro-duced plasma, where the contributions of the three described emission mechanisms are identified. The same picture includes also the emission spectrum of a blackbody. Despite the fact that two curves have similar shapes, the maximum of the spectral emission in the laser produced plasma occurs at higher wavelength compared to the blackbody spectrum as pointed out in 1.4.1.
Figure 1.4.1:Comparison between radiation spectrum from a laser produced plasma and the one of a blackbody at the same temperature.[2]
1.4 x-ray emission from laser produced plasmas
1.4.4 Characteristics of LASER-plasma X-ray sources
Focusing a high power pulsed laser onto the surface of a target it is possible to generate radiation in X-ray range. For each incident laser pulse an X-ray pulse is emitted into the 4π solid angle with the angular distribution weakly peaked along the target axis. More specifically, if solid massive target are used, the distribution of out coming radiation is limited to the solid angle not obstructed by the target.
Laser pulses duration can range between a few tens of femtoseconds(Ti:Sapphire-CPA Lasers) to tens of nanosecond (Nd Q-switched) with energies ranging from few mJ to tens of KJ, reaching intensities up to 1020W/cm2.
In the case of nanosecond interaction at moderate intensities, say below 1015W/cm2, the impinging laser pulse vaporizes and ionizes the shallow layer of the target creating an expanding plasma which will absorb the re-maining laser pulse. The resulting plasma temperature can range from 0.1 up to 1 KeV. Transport of heat will produce more plasma which ensures a steady flow from the ablation front that compensate the loss of matter due to the hydrodynamic expansion. This process creates an inhomogeneous plasma with increasing density approaching to the target, with a typical scale length of the order of several hundred of micrometers.
Just beyond the critical surface, where the plasma has both high temperature and high electron density, we find the peak of X-ray emission.
This X-ray source has some peculiarity in terms of spectrum, pulse duration, size and angular distribution. We will now briefly describe these characteris-tics.
As seen above, there are three main mechanisms of radiation emission from the plasma, each of which prevails over the others depending on the tar-get material: when we are working with medium-high atomic number (Z) materials, the resulting plasma is populated by ions with several bound electrons, and both recombination and line emission carry most of the out-going radiation energy. For medium Z values, plasma is mainly populated from Hydrogen like trough Be like ions and in these conditions a single line intensity can overcome continuum emission by several orders of magnitude. Finally, the radiation emitted by very low Z targets is almost free from lines and the continuum is mainly generated by Bremsstrahlung emission. Duration of X-ray pulse is generally the same as the one of the laser pulse2
. Indeed it directly depends on plasma density and temperature, which acquire higher values during laser irradiation and rapidly decrease when it ends. Therefore the rise time of X-ray emission is given by the time needed to heat the plasma, while the time scale of plasma cooling sets the fall time of the
2 exceptions are made by those pulse generated by ultrashort lasers, for example with a laser pulse of'1ps X-ray pulse up to 20ps could be generated.
l a s e r-plasma interaction emission.
The transverse size of the source is given by the focal spot dimensions φ or by the hydrodynamic expansion at the sound velocity in the medium dh= csτwhere τ is the laser pulse duration, namely d=max(φ, csτ). Since
the orthogonal dimension is given by the hydrodynamic expansion, if the focal spot dimensions are smaller than the contribution of the hydrodynamic expansion the X-ray source shows a shape with longitudinal and transversal dimension depending on heat transport and matter ablation mechanisms. Otherwise, plasma develops a shape with the diameter of the order of the focal spot size and length roughly equal to csτ.
For what concerns the angular distribution we see that this one shows a cylin-drical symmetry, whose axis is orthogonal to the target surface, regardless of the angle of incidence of the laser beam.
It has been found [13] that angular distribution fits the function Iλ(θ) = Iλcosn(θ)where θ is the angle of emission and n is a parameter that ranges from'0.3 to '3 accordingly to experimental condition.
1.5 r a d i at i o n t r a n s p o r t i n p l a s m a s
Emitting power of a body is related to its spectral absorptivity via the emitting power of the blackbody
e(ν, T) =α(ν, T)eBB(ν, T) (1.5.1)
where c is the speed of the light, eBB(ν, T) = 4cuBB(ν, T), and uBB is black
body spectral density.
Radiation propagating in a plasma will be increased in intensity by emission processes and decreased by absorption, over a distance dx
dI=ξdx−Ikdx (1.5.2)
where ξ is the emission coefficient and k the absorption one.
The quantity that describes the decrement of intensity suffered by radiation is called optical depth τ and is given by
τ=
Z x
0 kdx (1.5.3)
therefore the last relation could be written in the form dI
1.6 parametric instabilities
where S= ξ/k is a source function. In the case of thermodynamic equilibrium
the source function is related to the Planck function according to the relation
S= ξ k = c 4πu BB( ν, T) (1.5.5)
Since, in general, S depends on time and space co-ordinates, here we report the solution for a stationary and uniform plasma I(τ) =I0e−τ+S(1−e−τ),
with I0 the source term at x=0.
It is possible to classify different plasmas trough their optical depth. We refer to optically thin plasma when τ <<1, in this case emission prevails over absorption and the intensity is given by I(τ) =ξ x, where we consider
the case of pure plasma self-emission(I0 = 0). On the other hand, when
τ>>1 we speak about optically thick plasma: in this case plasma suffers of
strong self absorption which modifies the original spectrum and in the limit of thermal equilibrium it approximates that of a blackbody.
1.6 pa r a m e t r i c i n s ta b i l i t i e s
Studying plasma with a long density scale length compared to laser wave-length, i.e. Ln λL, we discover many other physical processes which
lead to scattering of incoming laser light with consequent reduction of laser absorption (Stimulated Brillouin Scattering) and to formation of ionic and elec-tronic plasma waves with consequent plasma heating due to their damping. These phenomena, called Parametric Instabilities, are characterized by an intensity threshold, typically ILλ2L '1015Wµm2/cm2. This condition could
be easier satisfied due to the presence of filamentation or self focusing, which produce the enhancement of laser intensity.
1.6.1 Stimulated Brillouin Scattering
In stimulated Brillouin scattering (SBS) an initial low-frequency ion density perturbation (ionic wave) is reinforced by generation of a reflected light wave which interfering with the incoming laser produces a variation in the wave pressure. We can read this phenomena also as a back-scattering process between photon and a phonon which absorb recoil momentum. This explains the name stimulated, in fact the phonon as to be generated parametrically. Assuming that matching conditions between impinging laser light, scattered radiation and ion plasma wave are satisfied, namely
l a s e r-plasma interaction
~
kL= ~kB+ ~ki (1.6.2)
this process feed the growth of an instability. In typical situations ωL>>ωi andk~Lk − ~kB, thus
~k i '2k~L '2 ωL c , ~ kB ' −~kL' − ωL c (1.6.3)
consequently backscattered wave angular frequency is given by
ωB =ωL−2kLcs= ωL 1−2cs c (1.6.4)
where csis the sound velocity in plasma cs =
q
ZkTe+3kTi
mi .
The reflected wave is red-shifted, this fact is due to the acquiring of momen-tum by ions and leads to a large rejection of incoming laser light significantly reducing plasma heating.
Activation threshold is given by
ISBS =7×1012 Te LλL ncr ne W cm2 (1.6.5)
therefore in order to avoid SBS formation and consequently promote absorp-tion it will be convenient to use short λLradiation.
1.6.2 Stimulated Raman Scattering
Production of Stimulated Raman Scattering (SRS) process is similar to SBS but this time are electron density fluctuations to be involved.
Matching conditions leads to the conclusion that SRS can only occur in region with ne< n4cr. We can obtain frequency shift of Raman scattered wave
combining matching condition with electron plasma wave dispersion relation ωL=ωR+ωe ωe2=ω2p+3k2evth2 =ω2p(1+3λ2Dk2e) (1.6.6) ωR =ωL 1−r ne ncr[1+3λ 2 Dk2e] (1.6.7)
Frequency shift of backscattered wave depends on ratio between electron and critical density, this means that measuring this quantity it is possible to get information about plasma density.
It has been shown from experiments [14] that for sufficiently long scale length '10% of laser energy can be converted by SRS into hot electrons giving rise
1.6 parametric instabilities
to intense emission of hard X-rays.
In the limit of ne<< n4cr activation threshold for SRS is given by:
ISRS =
4×1017
LλL
W
cm2 (1.6.8)
1.6.3 Two Plasmon Decay
In regions of plasma in which electron density reaches about a quarter of critical density Two Plasmon Decay (TPD) can occur. In this process laser light decays in two electron plasma waves called plasmons.
The incoming electromagnetic wave and the two electron waves have to satisfy energy and momentum conservation equations, ωL= ωB+ωRand
~
kL = ~kB+ ~kR where ωB,R and~kB,R are the frequency and wave vectors of
plasmons.
In the limit of weak Landau Damping of electron waves, namely k2B,Rλ2D <0.1,
the last two conditions together with the photons and the plasmons dispersion relations, namely ω2L= ω2p+k2Lc2c ω2B,R= ω2p+3k2B,Rv2e (1.6.9)
give an expression for electron density at which the instability occurs:
ne ncr ' 1 4 1−3 2(k 2 B+k2R)λ2D (1.6.10)
Therefore, taking extreme values for k2
B+k2R, we can see that exists a range
of allowable nein which TPD can develop
0.19ncr.ne. ncr 4 1−3 2k 2 Lλ2D (1.6.11)
From dispersion relations and conservation conditions we can also get the two expressions for plasmons’ frequency
ω2B ≈ ωL 2 1+3 4(k 2 B−k2R)λ2D , ω2R ≈ ωL 2 1− 3 4(k 2 B−k2R)λ2D (1.6.12)
with kB > kR. Since the two plasmons have frequency shifted in opposite
direction with respect to the exact ωL
2 we can refer to them as blue and red
plasmon. The first one is characterized by (ωB,~kB) and with a component of
wave vector in the direction of pump wave (~kL·~kB > 0); the second one is
characterized by (ωR,~kR) for which~kL·~kRcould be either positive or negative.
Non linear coupling of plasmons with incident electromagnetic wave light gives rise to radiation with frequency corresponding to half-integers harmon-ics of the incident laser light, in particular three-half harmonic. Detection of
l a s e r-plasma interaction
such a radiation is a typical signature of the occurrence of TPD instability and could be used to get information about plasma density.
TPD instability influences also X-ray emission, indeed, as in the case of SRS, it leads to intense emission of hard X-rays. In conclusion we report the threshold intensity for TPD instability:
ITPD =5×1012
Te
L1/4λL
W
cm2[15] (1.6.13)
where L1/4(µm)is the electron density scalelength at ne'ncr/4
1.6.4 Self-focusing and filamentation
Charged particles under the action of an inhomogeneous oscillating elec-tromagnetic field experience a non linear force called Ponderomotive force
~Fp= − e2 4mω2L∇E
2∝ ∇I
mω2L (1.6.14)
where E is the electric field of the incoming radiation and I its intensity. Since~Fpdepends on the inverse of particle’s mass it acts strongly on electrons
whilst is negligible on ions. In laser plasma interactions this will produce an electrons depletion channel along the direction of laser propagation. This condition gives rise to processes like Self Focusing and Filamentation.
Migration of electrons from the inner region of the laser beam generates a gradient in the refractive index which will decrease outwards. This gradient acts like a lens and focuses the impinging radiation providing to maintain high focusing and high level of intensity over a longer distance than typical laser values. If this phenomenon affects the whole laser beam we are in presence of self focusing. On the other hand if some points of the laser show a greater intensity than the surroundings, the so called "hot-spots", multiple filaments will be generated. In this case the phenomenon is called filamentation.
This effects have repercussions on electron temperature and consequently on emission, absorption and scattering of radiation. In particular Filamentation affects the spectral distribution and intensity of X-ray radiation emitted from the plasma region involved in this phenomenon.
In order to control this kind of instability, several smoothing techniques have been adopted, but though some of this methods results to be very effective in reducing small scale filaments, whole-beam focusing is found to be unaffected. In the picture of the other instability, self focusing and filamentation play a special role, indeed their presence sensibly changes plasma density profile and affects laser characteristics, making easier to achieve threshold condition and getting plasma to become a fruitful soil for instability growth.
1.6 parametric instabilities
1.6.5 Hot Electrons
During Laser plasma interaction, population of hot electrons could be pro-duced. The Presence of suprathermal electrons is suggested by observation of X-Ray spectrum. Indeed we find to have emission in the hard X-Ray region, well above the typical range of thermal emission of laser produced plasma. Many mechanisms can contribute to the formation of hot electrons, we are going to briefly describe them.
Inhibition of electron thermal conduction prevents laser energy absorbed at the critical density to propagate, this will lead to an overheating of plasma region just behind critical surface with consequent production of hot elec-trons and ions; in this case we find that hot elecelec-trons temperature scales as Th ∝(Iλ2)23[2]
In resonance absorption some electrons could travel trough the intense electric field region having an ideal velocity which make them feel a constant field for an adequate period of time; this combination of factors make electrons able of being accelerated. Electrons crossing this region in a fraction of period, ∆t6T/2, will gain as much energy as their initial kinetic energy. In this case Th ∝(Iλ2)13.[16]
Another way in which electrons can be accelerated is by the electric field of an intense laser radiation or by that of electron plasma waves. As we mentioned for acceleration deriving from resonance absorption, electrons have to interact with the electric field for an optimal period of time in order to maximize acceleration effect. In the case of transverse electromagnetic wave the maximum electron energy gain is:
∆emax = 1 2mv 2 q 1 1− vx vφ 2 (1.6.15)
where vqis the quiver velocity of the electrons and vx is the velocity
compo-nent in the propagation direction of the wave with phase velocity vφ. Since phase velocity of an electromagnetic wave in plasma is larger than the speed of light in vacuum, electrons remain under the action of electric field of the same sign only for a short time. Therefore this mechanism doesn’t produce fast electrons, unless we are not dealing with very high intensity laser. In fact, for electrons with a low initial speed vx/vφ 1 and taking λ= 1µm, I =1015W/cm2we obtain ∆emax'200eV.
Better conditions in terms of phase velocity are found in longitudinal plasma waves. They show a lower phase speed, hence electrons can spend more time
l a s e r-plasma interaction
in a region of accelerating field, for an ideal crossing time we have an energy gain of: ∆emax =4mv2q vq vphi 12 (1.6.16)
where vq is the electron quiver velocity due to the electric field of the plasma
wave.[17]
In order to compare the efficiency in hot electrons production of these two last phenomena, it is useful to write 1.6.16 as a function of plasma parameters:
∆emax' 1 2mv 2 th8η2 δne ne 12 (1.6.17)
with η =vφ/vth. Typical values of the mentioned parameters for laser plasma interaction with long scalelength plasmas are η = 10, δne/ne = 0.1 and
Te=0.5KeV. These values lead to a maximum energy gain∆emax =125KeV.
This greater value is reached for laser intensities lower than the one required in the case of transversal waves.
In Inertial Confinement Fusion (ICF) Experiments, generation of hot electrons obstacles achieving ignition condition, since they pre-heat the fusion pellet with subsequent expansion.
This issue was the drive in searching other schemes of interaction for ICF in which hot electron generation was inhibited.
p r i n c i p l e s o f i n e r t i a l c o n f i n e m e n t f u s i o n
2
P R I N C I P L E S O F I N E R T I A L C O N F I N E M E N T F U S I O NThermonuclear fusion research has developed two different approaches try-ing to reach suitable condition for ignition, i.e. the condition in which fu-sion reactions are self-sustained, Inertial Confinement Fufu-sion and Magnetic Confinement Fusion. As it is possible to guess from the names, the main difference between them concerns the confinement issue.
Throughout this chapter we resume the basic concepts of thermonuclear fusion, focusing our attention on ICF scheme, then we expose some of the problems one has to face in this context, in particular we spend some words describing hydrodynamic instabilities such as Rayleigh-Taylor, Richtmyer-Meshkov and Kelvin-Helmholtz instability.
In the following discussion we will take as reference the Deuterium Tritium fusion reaction D+T → α+n. Indeed it is of particular interest in fusion
studies because it has an high Q value1
of 17.59 MeV, and, as it possible to see in figure 2.0.1, among the main suitable fusion reactions, it presents the highest cross section at lower temperatures. However this reaction produces high energy neutrons which can activate the surrounding materials. Such activated media could decay toward a more stable situation with emission of ionizing radiation which could induce mutagen effects on biological system. For this reason are under investigation fusion reactions without neutrons emission. Multiple processes have no neutrons as product, but due to dif-ferent issues, such as the poor availability of the reagents, the more suitable reaction seems to be the proton-boron1p+11B→3α.
2.1 t h e r m o n u c l e a r f u s i o n
The main hindrance to fusion process is the Coulomb repulsion between two approaching nuclei. Indeed in order to have two nuclei fused together strong interaction has to be performed, this is a short range force but with much higher intensity than the Coulomb repulsion which, instead, is a long-range force. Hence the approaching nuclei have to own sufficient kinetic energy in order to increase their possibility to reach the distance at which nuclear forces starts to act. This fact lead to the requirement of high plasma temperatures. For conciseness we suppose to have a steady nucleus and the other moving to reach it, in this situation the radial behavior of the potential energy of the
1 Q=
∑imi−∑fmf
p r i n c i p l e s o f i n e r t i a l c o n f i n e m e n t f u s i o n
Figure 2.0.1:Cross sections of the main fusion reactions.[1].
two nucleon system puts in evidence that the approaching nucleus has to overcome a Coulomb potential barrier, Vc = Z1Z2e
2
r , in order to reach the deep
nuclear well of depth U0'30−40 MeV.
The distance at which strong interaction begins to act is about the sum of the radii of the two nuclei rn '1.44×10−13(A1/31 +A1/3), therefore the height
of the Coulomb barrier is given by
Vb'Vc(rn) =
Z1Z2
A1/31 +A1/32 MeV '1 MeV (2.1.1)
Classically two nuclei of relative energy e < Vb cannot reach each other,
quantum mechanics, otherwise, allows for tunneling a potential barrier of finite extension making possible to have fusion events for nuclei with energy smaller than the barrier height. Obviously this fact has repercussions on the cross section of the process, it could be parametrized as the product of three terms: σgeom ' (1/λDeBroglie)which is a geometrical cross section,R a factor
that contains both the probability that two nuclei coming to contact fuse and all the information concerning the specific nuclear reaction, and lastly the barrier transparencyT
2.1 thermonuclear fusion
the last term is the one taking into account the possibility of tunneling occurrence and it could be approximated by the Gamow factor:
T ≈ TG =e−(eGe )
1/2
eG= (παfZ1Z2)22mrc2 (2.1.3)
where eG is the Gamow energy with mr reduced mass and αf is the fine
structure constant. Due to the dependence of the fusion cross section on mass and atomic number, reactions with light elements will be favored over others with heavier ones.
Once reached suitable conditions to obtain fusion reactions, the following issue is to satisfy Ignition conditions. In fact in order to make possible using fusion reactions for energy production, released power has to exceed all possible power losses.
To impose this condition we start introducing specific fusion power PF(W/m3)
given by the product between the number of reactions in unit time and volume R12, that is reaction rate, and single reaction fusion energy eF; last one is
divided between charged particles and neutral ones therefore PF= PC+PN
where PC ≡R12eC =n1n2hσvi12eC PN ≡R12eN =n1n2hσvi12eN (2.1.4)
where n1,2are the densities of the nuclei of the two species.
In order to reach ignition condition PC2has to overcome PLwhich accounts
all possible power losses. Assuming n1 =n2 =n/2
PC = n 2 4 <σv> eC PL = τecon fth (2.1.5)
where τcon f is the energy confinement time. Imposing PC >PL we have:
PC PL > 1 → nτcon f > 4 neC eth hσvi ≡ F(T) (2.1.6)
that is known as Lawson Criteria, it puts condition over confinement time and plasma density and, since eth is proportional to T, gives an ideal work
temperature in order to reach ignition condition; it is evident that increasing the value of nτcon f it is possible to work at lower temperatures.
In magnetic confinement fusion the value nτcon f is increased acting on the
confinement time which in this approach is of the order of second. Plasma is confined trough a magnetic field, whose pressure has to be greater than the one exercised by the plasma, namely
B2 2µ0
>nKT (2.1.7)
p r i n c i p l e s o f i n e r t i a l c o n f i n e m e n t f u s i o n
since the maximum reachable magnetic field is of the order of 10T the highest allowed value of plasma density is'1014 #cm3.
On the other hand inertial confinement fusion acts on the density, which reaches values of'1025 #cm3. For such high densities the required magnetic
field in order to confine the plasma would be ' 106T which is well above the technical limit, hence the confinement time is entirely given by the pellet inertia and it is'10−9s.
2.2 i n e r t i a l c o n f i n e m e n t f u s i o n
The strategy adopted in Inertial Confinement Fusion is to give up in confin-ing plasma and tryconfin-ing to reach high temperatures and density before the disaggregation of target occurs, therefore the only confinement time is due to pellet inertia.
Suppose that target has a spherical shape of radius R, time necessary to its disaggregation is the same that the one taken by a sound wave to reach the center of the sphere starting from the surface
τcon f = R
cs (2.2.1)
where csis the sound speed of the medium.
In order to have the possibility to observe fusion reactions, confinement time has to be longer than characteristic fusion time τf us = (hσvin)−1, therefore
we need to compare them and to impose that
τcon f
τf us >
1 (2.2.2)
usually this condition is expressed in terms of the confinement parameter that is the product of mass density and sphere radius, from the above relation taking τf us and τcon f definitions is possible to obtain the following condition
ρR>
mfcs
hσvi (2.2.3)
Since the external part of target consists in a rarefying region with decreasing temperature and density not all the fuel initially present in the pellet is available for fusion, therefore another relevant point to take in account is the burn efficiency φ, which is given by the ratio between the total number of fusion reactions and the initial number of fuel’s pairs.
An approximate formula for burn efficiency taking account of burn depletion is given by
φ≈ ρR
2.2 inertial confinement fusion
where HB ≈8csmfhσviis the burn parameter [18]. From the expression of φit is possible to distinguish two different regimes: one of low-burn when ρR<<HB and that of full-burn when ρR>> HB ⇒φ'1.
Although reactors are necessary in order to use ICF process for energy production, they bring some more request on target feature. One of them concerns the energy released from a single micro explosion which has to be limited to a few GJ in order to avoid damages to the structure hosting the reaction. This will lead to limitation over allowable amount of fuel mass, considering that complete burn of 1 mg of DT releases 337MJ, assuming a burn up of 30%, targets’ masses could not exceed 10mg.
Due to this limitation over fuel masses, in order to reach sufficient burn efficiency is fundamental to reach high level of fuel compression, indeed from equation (2.2.4) and assuming HB '7 g/cm2it is possible to see that
burning 30% of DT fuel requires
Hf =ρR'3 g/cm2 (2.2.5)
hence, for a spherical fuel volume, the fuel density needed is
ρ= v u u t4π 3 H3f Mf ' 300 M1/3f where Mf = 4π 3 (ρR)3 ρ2 (2.2.6)
this means that 1mg solid DT mass with density ρ=0.225 g/cm3 has to be compressed by a factor
ρ ρDT
'1500 (2.2.7)
For this type of fusion approach being able to dispose of such high density is the main task as well as the most outstanding one.
Another issue arising from the need of reactors comes out just considering their energy balance. A scheme of the process is depicted in figure 2.2.1. Energy Edis delivered to the target by a driver pulse, here fusion reactions
release an amount of energy Ef us, therefore target gain is given by G =
Ef us/Ed. Then Ef usis converted into thermal energy and hence in electricity
by a thermal cycle with efficiency ηth. At this point a fraction f of this
electric power is used to feed the driver which converts it in pulse energy with efficiency ηd, whilst the remaining part is available for the grid3. The
energy balance for this cycle can be written as f ηdηthG = 1, assuming to
have an efficiency ηth =40% and imposing f <0.25, we obtain the condition
Gηd >10; this means that in order to take advantage of fusion reactions for power production, if it is possible to achieve driver efficiency in the range 3 This simple scheme neglects some effects as exothermic neutron reactions, that could enhance power production by a factor M=1.25, and plant auxiliary equipment that use some of the produced power.
p r i n c i p l e s o f i n e r t i a l c o n f i n e m e n t f u s i o n
Figure 2.2.1:Scheme of the Energy balance of an ICF reactor.
10−33%, target gains in the range G=30−100 are needed.
It could be seen that performing an uniform heating of fusion fuel to the ignition temperature is not possible to reach such gain values, what seems to be a better method consists on the ignition of a small part of the target, called the hot spot. Once ignited would be the central part of the target itself, thanks to a propagating burn wave, to ignite the remaining compressed cold fuel; proceeding in this way the amount of energy to spend on fuel will be significantly reduced. This technique requires that the hot spot has to be able to self-heat and it has to produce enough energy to succeed in producing the burning wave. In order to do this it has to satisfy some features [1] which are
ρhRh >0.2−0.5 g/cm2 and Th=5−12 KeV (2.2.8)
2.2.1 Inertial confinement by spherical implosion
From previous discussion it is clear which conditions have to be satisfied in order to reach ignition, to achieve them, in ICF context, two different schemes have been suggested, indirect-drive approach, where the capsule is radiated by X-Rays generated in a Hohlraum, and direct-drive approach in which the required plasma density is obtained driving the implosion of a spherical shell with high power laser radiation impinging on its surface.
The process of inertial confinement fusion by spherical implosion, resumed in figure 2.2.24
, consists of four different stages, the first one sees irradiation of the capsule with subsequent ablation of surface, then we have the implo-sion, in this stage material reaching the center converts its kinetic energy into internal one. At this point we have an highly compressed shell with a central hot spot of ignited fuel, a burn wave starting from it ignites the whole
2.2 inertial confinement fusion
material which then burns.
Target design plays an important role trying to reach ignition conditions. In direct drive approach the best choice seems to be a spherical hollow shell capsule consisting of an outer layer of a plastic ablator and an inner one of cryogenic solid DT, the central cavity of the capsule is filled with DT vapor which will form part of the hot spot. A simple scheme of this type of pellet is shown in figure 2.2.3. Laser beams heat the outer layer which passes to gaseous form increasing its volume, for moment conservation the non-ablating inner capsule material is pushed inward by ablation pressure. At this point a first shock wave, driven by the laser pulse, bursts inside the capsule. In order to enhance compression effect instead of using a single stronger pulse it’s preferable to use many weaker shocks, therefore, thanks to the modulation of laser pulse, a series of shocks follow the first one leading to capsule implosion.
Once turned off the laser, almost all the ablator material has been vaporized and the inner shell is moving inward with high velocity, once reached the center collides with it generating high pressure and temperature in the hot spot which reaches a confinement parameter value suitable for ignition. Soon after a burn wave is formed which propagating outwards causes the ignition of the remaining fuel, this, due to the fast expansion, explodes.
Figure 2.2.2:Main stages of inertial confinement fusion by a spherical implosion.
p r i n c i p l e s o f i n e r t i a l c o n f i n e m e n t f u s i o n
2.3 s y m m e t r y a n d s ta b i l i t y
The whole process is very sensitive and small system perturbation, such as irradiation non uniformities caused by defects on target surface and by disposing of finite number of laser beams, can cause large implosion asymmetries which could lead to ignition failure.
According to the ratio between perturbation wavelength and shell radius we could distinguish two types of perturbations: long wavelength perturbations and short one.
The first type leads to formation of a deformed hot spot which has the surface shaped with the same periodicity and amplitude of the driver non uniformity. Nevertheless the hot spot would release nearly the same fusion energy, because it will have about the same volume and average density of the spherical one, but increasing its surface area, power losses due to the fraction of escaping α particle and thermal conduction are incremented. Previous studies have shown that the level of allowable perturbations in order to obtain ignition depends on target’s features, indeed if the hot spot generated by the implosion was just of the dimension necessary for ignition also a small perturbation to the shape would be crucial. On the other hand if we have a larger hot spot than the ignition margin of the target, ignition could be performed with higher level of surface deformation, because more power losses due to asymmetry could be tolerated. The two different situations are shown in figure 2.3.1
(a) (b)
Figure 2.3.1: Concept of ignition margin. Non-uniform irradiation results in the generation of a deformed hot spot which is represented by the grey area. the hot spot (a) will not ignite, while (b), having a large margin, can ignite despite the large deformation[1]
Short wavelength perturbations are even more dangerous because they could work as seed for birth and subsequent growth of hydrodynamic instabilities; this means that the small amplitude perturbations of target’s surface grow exponentially in time destroying any possibility for ignition.
2.3 symmetry and stability
The prominent instability concerning ICF is the Rayleigh Taylor one (RTI), it could compromise ignition achievement by itself. Related to it we find also Kelvin Helmholtz instability, which plays a role in the non linear evolution of RTI, and Richtmeyer Meshkov which could facilitate RTI formation.
2.3.1 Rayleigh Taylor Instability (RTI)
When two non-miscible fluids with different densities ρ1 and ρ2 are posed
one over the other in a gravitational field they are in hydrostatic equilibrium for every ρ2/ρ1 ratio. Nevertheless when some perturbation is applied to
the plain separation surface, if the upper fluid has greater density than the lower one, the small perturbations would rapidly grow in time. As the perturbation’s amplitude becomes comparable to its wavelength the pace of growth rate slows down thanks to the linear growth saturation phenomenon and the initially sinusoidal perturbation evolves in an asymmetric one. At this point bubbles of the lighter fluid rise up and spikes of the denser one fall down. This phenomenon is easily explained by an energetic argument. Indeed the situation in which the heavier fluid is lower than the lighter one has minor potential energy, therefore once it goes down there is no reason why it should climb up again, in other words the heavier fluid moves towards a more favorable energetic situation. In figure 2.3.2 is shown the evolution of Rayleigh Taylor instability in time, the typical "mushroom" shape is produced by the action of the Kelvin Helmholtz instability described in 2.3.3.
As shown in [19] the same considerations could be applied at every bound-aries between two non miscible fluids of different densities in an accelerated frame, which is the situation in which ICF pellet stands.
This kind of instability takes place in two different stages of the implosion process,first on the surface, when the ablating material pushes onto the inner denser shell accelerating it, and then when the rising pressure of the central hot spot decelerates the incoming more denser shell.
As we mention above classical RTI refers to superimposed fluid without tangential velocity shear. Neglecting effects of ablation, density gradient and thermal conduction, linear theory provides an expression for growth rate of sinusoidal perturbations with wavelength λ of the unstable surface
σRT = r 2π λ Ata = p kAtas−1 (2.3.1)
where a is the acceleration acting on fluids and Atis the Atwood number of
the interface
At=
ρ2−ρ1
ρ2+ρ1
