Contents of talk

● Introduction to self-similar analysis● Self-similar solution with full account of charge separation● Ion energy spectrum & comparison with experiment ● Maximum ion energy● Application to Coulomb explosion

Plasma expansion into vacuum and ion accelerationproblem in terms of self-similar solution

M. MurakamiInstitute of Laser Engineering, Osaka University, Japan

Introduction to self-similar analysis

Similarity between different scale systems　(1) Karman’s Eddy

500km5cm

= 107

Satellite pictureLaboratory experiment

500 km5 cm

Similarity between different scale systems　(2) Blast Wave

Laboratory laser exp. Atomic explosion Nova explosion

1 cm 1 pc（3光年）100 m

100m1cm

= 104 3pc100m

= 1015

(1) Π -theorem

Even without knowing the system equations, one can introduce combined newvariables by dimensional analysis, with which the number of the independentparameters are reduced, and the system can be significantly simplified.

(2) Lie group analysisGiven the differential equations system, one can find combined new variables byappropriate variable transformation, with which the number of the independentparameters are reduced, and the system can be significantly simplified. One caneven find linearity indwelling in the nonlinear system.

Representative tools for finding self-similarity

The following two methods are systematic and powerful methods to find self-similarity of the system under consideration, which may work complementarily.

Dimensional analysis - spherical pig problem -

If a pig is assumed to be spherical,Then the only parameter is the radius.

M ∝ R3

V ∝ R 3

S ∝ R 2

Mass

Volume

Area

R≈

Try to simplify the system as much as possible, but be careful not to loose the essence.

Dimensional analysis (Π-theorem, Buckingham 1914)

In many cases, similarity solutions can be obtained by dimensional analysis.This is based on the fact that the equations of physics are homogeneous withrespect to dimensions, and writing them in dimensionless form may lead to areduction of the number of independent variables.

Suppose a physical quantity a as a function of other n quantities, a1, a2, …, an

have independent dimensionscan be expressed dimensionally in terms of the variables of the 1st group

a = F(a1, ⋅ ⋅⋅, ak , ak+1, ⋅ ⋅⋅, an )

a[ ] = a1[ ]q1 a2[ ]q2 ⋅ ⋅ ⋅ ak[ ]qkDimensions denoted by [ ] :

Dimensionless variables:

Π = aa1q1 a2

q2 ⋅ ⋅ ⋅akqk

Π j =ak+ j

a1q1j a2

q2 j ⋅ ⋅ ⋅akqkj, ( j= 1,2,⋅ ⋅ ⋅,n − k)

ak+ j⎡⎣ ⎤⎦ = a1[ ]q1j a2[ ]q2 j ⋅ ⋅ ⋅ ak[ ]qkj , ( j= 1,2,⋅ ⋅ ⋅, n − k)

Substitution for and for :a ak+ j

Here the right-hand-side must not depend onwith independent dimensions, because the left-hand-sidedoes not. Otherwise, invariance with respect to system ofunits would be violated. Therefore:

Π =F a1,⋅ ⋅ ⋅,ak , Π1a1

q11 ⋅ ⋅ ⋅akqk1 ,⋅ ⋅ ⋅,Πn+ka1

q1,n+k ⋅ ⋅ ⋅akqk,n+k( )

a1q1 ⋅ ⋅ ⋅ak

qk

Π Π j

Π = Φ(Π1,Π2,⋅ ⋅ ⋅,Πn−k ) (Π-theorem)

⇒ for k = n −1, Π = Φ(Π1) similarity solution⇒ for k = n, Π = Φ0

A typical example of Π-theorem

　　　　　 (cm) :　 (sec) :

(g/cm3) :

(erg) :

ρ0[ ] = ML−3

E[ ] = ML2T−2

r[ ] = Lt[ ] = T

The number of dimensions of the system：T, M, L ( k = 3)

The number of nondimensional parameters :

Π-Theorem

The number of parameters that characterize the system: n = 4

RadiusTimeDensityEnergy

n − k = 1

Π1 =ρ0E

⎛⎝⎜

⎞⎠⎟1/5 rt2/5

Hydrodynamic fields :ρ(r, t), u(r, t), p(r, t)

ρ = ρ0G(Π1)

u = rtU(Π1)

p = ρ0rt

⎛⎝⎜

⎞⎠⎟2P(Π1)

Similarity ansatz:

Inserting ansatz into hydro-equations (PDE’s)leads to a set of ordinary equations (ODE’s).

where G(Π1), U(Π1), P(Π1) are still unknown functions.

Taylor-Sedov (-Von Neumann) Blast Wave

rf (m)

t (sec)

Lie Group Analysis - symmetry of differential equations -

Marius Sophus LieNorwegian mathematicianEnd of 19th century

Discovery of the symmetry

Introduction of a new combined variable

Partial differential system 　　 Ordinary differential system

Higher order system　　　 Lower order system

Nonlinear system　　　 Linear system

⇩

Finding of self-similar solution

⇩

⇩

Symmetry in differential equation system

The system can be significantly simplified, if it is converted intoa symmetric system under a specific variable transformation.

symmetry

Application to the linear heat conduction problem (stretching groups)

∂T∂t

=∂2T∂x2

∂ ˆ T ∂ˆ t

=∂2 ˆ T ∂ˆ x 2

ˆ x = λxˆ t = λα tˆ T = λβT

invariant

∂ ˆ T ∂ˆ t

=∂(λβT)∂(λαt)

= λβ−α∂T∂t

∂2 ˆ T ∂ˆ x 2

=∂(λβT)

[∂(λx)]2= λβ−2 ∂2T

∂x2

⇒1st eigenvalue

α = 2

⇒ˆ x = λxˆ t = λ2t

New combined variable ˆ x ˆ t =

xt≡ ξ

Tt = Txx, T(x,0) = δ(x), −∞ < x < ∞, t ≥ 0

Application to the linear heat conduction problem (cont.)

0

0.2

0.4

0.6

0.8

1

0 20 40 60 80 100x

T

t = t0

t = 4t0

t = 2t0

ˆ x = λxˆ t = λα tˆ T = λβT

T(λx, λαt) = λβT(x,t) T(x, t) = tβ /αΦ x

t1/β⎛ ⎝

⎞ ⎠

β = −1 ⇒ T = t−1/ 2Φ(ξ)Determination of the structure⇒Tdx =

−∞

∞

∫ const

2 ′ ′ Φ + ξΦ + Φ = 02nd-order ODE

T(x, t) = exp(−x2 /4t)

4πtSolution

2nd eigenvalue

Simple planar (SP) self-similar solution of isothermal rarefaction wave

t0

t3t2t1

∂ρ∂t

+ ∂(ρu)∂x

= 0

∂u∂t

+ u ∂u∂x

= − cs2

ρ∂ρ∂x

ξ = x / cst

v / cs = ξ −1

Density profilesρ /ρ0 = exp(ξ −1)

(Grevich 1966)

Comparison between the present model and the semi-infinite model

1. Self-similar solution of finite-mass & quasi-neutral plasma

∂ni∂t

+∇⋅niv = 0

∂v∂t

+ v ⋅∇v = − Zemi

∇φ

ne(r, t) = Zni(r, t) = nec(t) expeφ(r)Te

⎛⎝⎜

⎞⎠⎟

cs ≡ZTemi

1. Fluid equations for ions:

2. Boltzmann statistics for electrons:

∂v∂t

+ v ∂v∂r

= − cs2

n∂n∂r

3. Reduced partially differential equation system:

where is the sound speed.

⎧⎨⎪

⎩⎪← One-fluid description!

ν =1:planar2 : cylinder3 :spherical

⎧⎨⎪

⎩⎪

∂n∂t

+ 1rν−1

∂∂rrν−1nv( ) = 0

Murakami et al., Phys. Plasmas 12 (2005) 062706

Self-similar solution of finite-mass & quasi-neutral plasma (cont’d)

4. Similarity ansatz:

ξ = r

R(t), v = Rξ, n = n0

R0R

⎛⎝⎜

⎞⎠⎟νN(ξ)

5. Reduced ordinary deferential equation :

RRcs2 = − ′N

ξN= 2 Variable separated!

6. Self-similar solution :

N = exp(−ξ2 ), eφT

= −ξ2

R = 2cs ln(R / R0 )

Self-similar solution of finite-mass quasi neutral plasma expansion

0

0.2

0.4

0.6

0.8

1

0

1

2

3

0 1 2 3

Density N

Velocity V

Similarity coordinate ξ

N = exp(−ξ2 )

V = ξ

（planar, cylinder, sphere）

100

101

102

103

104

105

106

10-1 100 101 102 103 104 105

ν = 1, 2, 3: 等温膨張 R(t)

R(t)/R0

cst/R0

Temporal evolution of the system characteristic scale

R ∝ t

Isothermal case

(a) Finite mass

(1) Isothermal expansion

(3) others　 (a) Maxwellian

Ion energy spectra obtained from quasi-neutral model

dNdε

∝ ε(ν−2)/2 exp(−ε) =exp(−ε) ε (ν = 1)exp(−ε) (ν = 2)exp(−ε) ε (ν = 3)

⎧

⎨⎪

⎩⎪

(a) Finite mass

(b)half-infinitely- stretched plane

dNdε

∝ exp(− ε ) ε

(2) Adiabatic expansion（planar／initially isentropic distribution）

PlaneCylinderSphere

dNdε

∝ 1− γ −12( ε −1)⎛

⎝⎜⎞⎠⎟2/(γ −1)

ε

dNdε

∝ (1− ε)1/(γ −1) ε

dNdε

∝ exp(−ε) ε

(b)half-infinitely- stretched plane

The analytical model excellently reproduces the experimental results on ion kinetic energy spectrum (planar geometry)

10-6

10-5

10-4

10-3

10-2

10-1

100

0.1 1 10Ion kinetic energy ε (keV)

Experiment

(α =1, ε0=1.7 keV)

Norm

alize

d sp

ectru

m d

N/dε

Present model

1Maximum ion energy predicted by the present analytical model

Murakami et al., Phys. Plasmas 12 (2005) 062706

Sn solid plane

Laser

The analytical model excellently reproduces the experimental results on ion kinetic energy spectrum (spherical geometry)

Murakami et al., Phys. Plasmas 12 (2005) 062706

10-4

10-3

10-2

10-1

100

0.1 1 10

Normalized spectrum dN/d

ε

Ion kinetic energy ε (keV)

Experiment

Present model

1

(α =3, ε0= 3.0 keV)

Maximum ion energy predicted by the present analytical model

Xe liquid jet

Laser

hot electrons

cold ions

At t = 0 the electron component of a finite plasma massis heated to a uniform temperature Te(r, 0) = Te0.Hot electrons expand and create an ambipolar electricfield E(r, t), which drags the cold ions.

・There are no collisions between electrons and ions.・At all times, the electron temperature is very quickly leveled off across the plasma volume: Te(r, t) = Te(t).

Assumptions

2. Self-similar solution of finite-mass & non-quasi-neutral plasmaMurakami & M.M.Basko, Phys. Plasmas 13 (2006) 012105

Governing equations

where ν stands for the geometrical index: ν = 1, planarν = 2, cylindricalν = 3, spherical

⎧⎨⎪

⎩⎪

In the framework of two fluid approximation, expansion of theconsidered plasma is governed by the following system of equations,

∂ni∂ t

+ 1∂ rν−1

∂∂ r

rν−1nivi( ) = 0∂ne∂ t

+ 1∂ rν−1

∂∂ r

rν−1neve( ) = 0⎧⎨⎪

⎩⎪massconservation

∂vi∂ t

+ vi∂vi∂ r

= − Zemi

∂Φ∂ r

∂ve∂ t

+ ve∂ve∂ r

= eme

∂Φ∂ r

− Temene

∂ne∂ r

⎧⎨⎪

⎩⎪

momentumconservation

1rν−1

∂∂ r

rν−1 ∂Φ∂r

⎛⎝⎜

⎞⎠⎟ = 4π e ne − Zni( )Poisson

equation

(1)

(2)

(3)

(4)

(5)

Similarity ansatzThe key assumption is that the velocity, v(r, t), is linear in radius. This isalways correct for the asymptotic stage of free expansion of a finite mass.

ve(r, t) = vi(r, t) = Rξ

ξ = r

R(t), R ≡ dR

dt

ne(r, t) = ne0R0R(t)

⎛⎝⎜

⎞⎠⎟ν

Ne(ξ), Ne(0) = 1

Zni(r, t) = ni0R0R(t)

⎛⎝⎜

⎞⎠⎟ν

Ni(ξ), Ni(0) ≠ 1

(6)

(7)

(8)

(9)

NiNe

ξf

Cold ions preserve a sharp edge at ξ = ξf(still unknown).

Functions, vi and Ni, are then definedonly for 0 ≤ ξ ≤ ξf

●

●

Coherency of the R(t) and λD(t) for self-similarity

The present system contains two characteristic scale lengths, R and λD

In general, a self-similar solution cannot be found, if the consideredsystem has two (or more) temporal characteristic scales.

●

●

This can be interpreted in turn that one can find a self-similar solution ifthe system scale lengths coherently evolve in time, i.e.,

●

Λ = R(t)λD(0, t)

= R(0)λD(0,0)

= R 4πnee2

Te

⎛

⎝⎜⎞

⎠⎟

1/2

= const

neRν = constrecalling mass conservation,

Te(t)∝R(t)2−ν ∝ ne

1−2/ν

Self-similar solution

Continuity equations (1) and (2) are automatically satisfied for any R(t), Ne(ξ) and Ni(ξ). The electron equation of motion (4) yields

This is the well-known Boltzmann relation generalized to the case of .

Ne = exp(φ − µeξ2 )

µe > 0

The ion equation of motion (3) yields

for 0 < ξ < ∞

φ(ξ) = −ξ2 for ξ < ξfSpatial profile

Temporal evolution

R(t) =

2cs02 t / R0, ν = 1

2cs0 ln R(t) / R0( ), ν = 22cs0 1−R0 / R(t), ν = 3

⎧

⎨⎪⎪

⎩⎪⎪

cs0 =ZTe0mi

⎛

⎝⎜⎞

⎠⎟

whereφ = eΦ / Te

µe = Zme /mi

⎧⎨⎩

The Poisson equation (5) is reduced to

Ni = Ne +2νΛ2

= exp −(1+ µe )ξ2( ) + 2ν

Λ2for 0 ≤ ξ ≤ ξf

and a boundary-value problem

1ξν−1

ddξ

ξν−1 dφdξ

⎛⎝⎜

⎞⎠⎟= Λ2 exp(φ − µeξ

2 ),

φ(ξf ) = −ξf2 , dφ(ξf )

dξ= −2ξf , lim

ξ→∞ξν−1 dφ

dξ= 0

which allows us to calculate the position of the ion front

for ξf < ξ < ∞

⎧⎨⎪

⎩⎪

ξ = ξf .

NiNe

ξf

Main results of the solution (1) Ion energy spectrum

dNi

dεi= Aε0

εi

ε0

⎛⎝⎜

⎞⎠⎟2νΛ2

+ exp −(1+ µe )εi

ε0

⎡

⎣⎢

⎤

⎦⎥

⎧⎨⎩

⎫⎬⎭

dNi

dεi∝ε1/2 , Λ <<1ε1/2 exp(− ε), Λ >>1

⎧⎨⎪

⎩⎪

Coulomb explosion⇐

10-4

10-3

10-2

10-1

100

101

102

10-2 10-1 100 101

0.5Λss

= 0.251

24

8

16

32

64

128

256

d Ni/dε i (

arb.

uni

t)

1

1

Normalized ion kinetic energy ε i / ε0

µe−1 = 1000

0 ≤ εi ≤ εi,max

ξf2 =

W π1/3Λ4/3 / 2µe( ) / 2, Λ <<1

W(Λ2 / 2), Λ >>1

⎧⎨⎪

⎩⎪

Main results of the solution (2) Maximum ion energy

εi,max = ε0ξf2

is the bulk fluid velocity of the expanding plasma (can be calculatedunder the condition of quasi-neutrality)v∞

ε0 =12miv∞

2 =2ZTe0 ln

R(t)R0

, ν = 2 (γ = 1, isothermal)

2ZTe0 , ν = 3 (γ = 4 / 3, adiabatic)

⎧⎨⎪

⎩⎪

W(x) is the Lambert W-function defined by

x =Wexp(W)⇒W =

x, x1ln(x / ln x), x1

⎧⎨⎩

ξf2 =

W Λ2

2⎛

⎝⎜⎞

⎠⎟, Λ1, ν = 1,2, 3

12W π1/3

2µeΛ4/3

⎛

⎝⎜⎞

⎠⎟, Λ 50, ν = 3

⎧

⎨

⎪⎪

⎩

⎪⎪

The maximum energy of accelerated ions is proportional to , where is theposition of the ion front; may be called the acceleration factor.

ξf2 ξf = ξf (Λ, µe ,ν)

ξf2

Solution of boundary-value problem

Generalization of the main resultWe believe our result for to be more general than the self-similar solution itself. εi,max

Poisson equation:

Boltzmann distibution:

Energy balance:Zni

ne

+

−

∂2φ∂x2

= 4πene

ne = ne0 expeφTe

⎛⎝⎜

⎞⎠⎟

⎫

⎬⎪⎪

⎭⎪⎪

⇒ ∂2φ∂x2

= 4πene0 expeφT

⎛⎝⎜

⎞⎠⎟

Argument: For µe<<1 and Λ>>1 one can integrate the Poisson equation over the electronsheath to obtain the electric field at the ion front and the maximum ion energy:

12Ef2 = 4πnefTe

Ef = 8πnefTe ∝ nef ∝ ref−γν/2 εi,max = Ze Efdrf

rf 0

∞

∫ ∝ rf−γν/2drf

rf 0

∞

∫⇒

The integral converges whenever γ ν > 2, which means that all the acceleration occursDuring the initial phase of expansion.

Time

等温膨張領域

Isothermal Expansion

断熱膨張領域

Adiabatic Expansion

Laser On

エネルギースペクトル

の構造はここで決まる！

Laser Off

エネルギースペクトル

の構造は殆ど変化しない

εi,max ≈ ε0W(Λ02 / 2)

where ε0 =12miv∞

2 =2ZTe0 ln

R(t)R0

, γ = 1 (isothermal)

2ZTe0 / ν(γ −1), γ >1 (adiabatic)

⎧⎨⎪

⎩⎪

where is measured at the onset of the asymptotic phase of expansion.

Λ0 = Λ(t0 )

Te0

Te(t)

Time t

ne(t, 0)

t0

Generalized form of the maximum ion energyFor any , the maximum ion energy in cylindrically and sphericallyexpanding plasmas is given by

γ ≥1

• Self-similar solution exists only in the framework ofthe quasi-neutral hydrodynamics.

P.Mora (PRL, 2003) obtained an accurate numericalsolution for µe = 0.

In the limit of the ions at the front are acceleratedto the infinite energy.

•

•

εi,max = 2ZTe0 ln cstλD

⎛⎝⎜

⎞⎠⎟

2

= 2ZTe0 lncstλD

⋅ ln cstλD

12mivbulk

2⇓

The simple planar solution is the plane-parallel isothermal expansion of a plasma, initially occupying a half-space.

Comparison with the SP solution (P. Mora, 2003)

t→∞

Conjecture

As a final conclusion, we may conjecture that in all situations, where ions are accelerated due to the free plasma expansion, the maximum ion energy is given by

εi,max = ε0Gpl where

and the factor, Gpl is a weak (logarithmic) function of the plasma parameters; it may becalled a plasma logarithm (by analogy with the Coulomb logarithm). For D>>λD we have

Gpl ≈ lnDλD

= 5 − 30 whereR0 = the plasma size, or the laser focal-spot size, orthe distance to the ion detector.

D =⎧⎨⎩

Our self-similar solution Gpl =W12R0λD

⎛⎝⎜

⎞⎠⎟

2⎡

⎣⎢⎢

⎤

⎦⎥⎥

The SP solution Gpl = lncstλD

ε0 =12mivbulk

2 2ZTe0 ln

R(t)R0

, isothermal

2ZTe0 , adiabatic

⎧⎨⎪

⎩⎪

Application to Coulomb explosion

10-4

10-3

10-2

10-1

100

101

0

2

4

6

8

10

0 1 2 3 4 5 6

Norm

alize

d de

nsiti

es N

e, Ni

Elec

tric

field

E

Self-similar coordinate ξ

Ni

Ne

E

1 µe−1 = 2000

ν = 3 Λss = 2

The self-similar solution can described plasma expansions of any size

Λ ~ 1 Λ ~ 106

0

0.2

0.4

0.6

0.8

1

1.2

0.1 1 10 100

Ener

gy tr

ansf

er ra

tio η

tra

Initial plasma size Λuni

Simulation

Theory

4.5

450

1400

4.5

45

450

4500Te0 = 4500eV

Rf0 = 10nm

Rf0 = 2.2 nm

ne0 = 1021cm−3

ne0 = 1023cm−3

45

(µe−1 = 100)

µe−1 = 100

µe−1 = 1000

Energy transfer efficiency from electrons to ions

Summary

• A new self-similar solution is found to describe the plasma expansion intovacuum with full account of the charge separation.

• The experimental results have turned out to be excellently reproduced by thepresent solution.

• The maximum ion energy has been also obtained in a simple analytical form,for example for spherical geometry, εi,max = 2ZTe0 ln(Λ

2 / 2)

Thermodynamic property in terms of the adiabatic index γ

T ~ n1-ν/2 defines the polytropic characteristic of the system in the form,●

T∝ neΓ−1 with Γ =

⎧⎨⎪

⎩⎪

0, ν = 1

4 / 3, ν = 31, ν = 2

Γ = γ ⇒ adiabaticΓ ≠ γ ⇒ non adiabatic

(A) ν = 1 (planar case)The system is needed to be kept heated such that T(t) ~ R(t) for any γ.

(B) ν = 2 (cylindrical case)The system is needed to be kept heated such that T(t) ~ const (isothermal) for any γ.

(C) ν = 3 (spherical case)

Since γ > 1,

⎧⎨⎪

⎩⎪

γ > 4 / 3 cooling (mono − atomic nonrelativistic plasmas)γ = 4 / 3 adiabatic (relativistic electrons)γ < 4 / 3 heating (multi− ionized nonrelativistic plasmas)