The plasma length and time scales

The plasma length and time scales
Prof. L. Conde
Departamento de Física Aplicada
Página personal: http://plasmalab.aero.upm.es/~lcl/
Curso2015-2016
Plasmalengthandtimescales,…
Thedampingand/orattenuationofsmallamplitudefluctuationsofthe
equilibriumtakesplaceovercharacteristiclengthandtimescales.
r·E =
e
(ni
✏o
ne )
E ' E1 small amplitude perturbations
ni 6= ne
•
•
•
Space fluctuations: Debye length
Time fluctuations: Plasma frequency
Number of charges: Plasma parameter
r
r
o kB T
o kB T
⇥
=
⇥De =
Di
e2 nio
e2 neo
k B Te
⇥pi =
s
e2
nio
o mi
⇥pe =
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s
e2 neo
o me
k B Ti
De
pi
pe
=
r
Di
me
mi
2
TheDebyelengthandshielding,…
The initial equilibrium:
Eo = 0
neo ' nio = no
k B Te ; k B Ti
where we introduce an small perturbation in the electric charge,
⇥sp = e [ni (r)
r'1 (r)
E(r) ' Eo + E1 (r) E1 (r) =
⇥ext = q (r)
ne (r)]
and E1(r) represents the perturbed electric field.
The field E1(r) is governed by the Poisson equation,
q
e
⇥ · E1 =
(r) + [ni (r)
⇥o
⇥o
ne (r)]) where,
⇢ext + ⇢sp
·E=
✏o
n (r) = n
o
exp
✓
e 1 (r)
±
kB T
◆
Smallamplitudeperturbationsofthecharge/electricfieldmeansthatthethermal
energy|eφ(r)|dominatesovertheelectrostaticenergykBT
= e, i
e 1 (r)
⌧1
kB T
we approximate
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n (r)
n
o
✓
1±
e 1 (r)
kB T
◆
3
Debyeshielding,…
r2 ⌅1 (r) =
1
⇥o

⇤ext +
✓
◆
2
e no
⌅1 (r) +
k B Te
that introduces the Debye lengths,
1
2
=
1
2
De
+
1
2
Di
✓
r
2
1
2
◆
✓
2
e no
⌅1 (r)
k B Ti
⇥D =
⇤1 (r) =
r
◆
o kB T
e 2 no
q
(r)
⇥o
= e, i
Setting,
q e r/
⇤1 (r) =
4⇥ o r
The perturbations of the electric charge and/or
plasma potential decay in space with an exponential
rate governed by λD
The Debye length accounts for thermal effects, the
size ∼λD of the perturbed region increases with kBT
This is a linearization only valid whereas,
e 1 (r)
⌧1
kB T
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We deal with a non linear
Poisson equation otherwise,
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Approximatemagnitudesintypicalplasmas,…
Plasma
ne (cm-3) KBTe (eV)
λDe (cm)
fpe (Hz)
neλDe3
Interstellar gas
1
1
700
6,0 ×104
4 ×108
Solar corona
109
100
0,2
2,0 ×109
8 ×106
Solar atmosphere
Gas discharge
1014
1
7,0 ×10-5
6,0 ×1011
40
Tokamak
1014
104
2,0 ×10-3
2,0 ×1012
6 ×106
Values from NRL Plasma Formulary 2007. This practical reference could be
downloaded for free at; http://wwwppd.nrl.navy.mil/nrlformulary/
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Timescale:theplasmafrequency,…
Again the initial equilibrium:
Eo = 0
neo ' nio = no
k B Te ; k B Ti
and we consider a one small perturbation of the
electric charge in one dimension as in the figures,
⇥=
Z
S
e no (A X)
E1 · ds = A E1x =
⇤
=
⇥o
e
no (A X)
⇥o
e
no X
The electric field,
⇥o
gives us an equation of motion for ions or electrons
 2
2
d
e no
(
X)
+
( X) = 0
dt2
m ⇥o
E1x =
that defines the electron/ion plasma frequencies,
s
e 2 no
⇥p =
= e, i
m o
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Theplasmaandcouplingparameters,…
Inordertoshieldouttheelectricfieldperturbationsaminimumnumberofcharges
arerequiredinsideasphereofradiusλD.ThisdefinestheplasmaparameterNDfor
ionsandelectrons,
NDe =
4⇥
no
3
3
De
k B Te
k B Ti
De
NDe
Di
NDi
The plasma parameter is closely related with the coupling parameter ΓC the ratio
between thermal and electrostatic energies.
C
rc
=
rd
rd ⇠ no 1/3
rc is the closest colliding approach as
in the figure
m v2
E(r, v ) =
2
e2
rc =
4⇥ o kB T
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average packing of charges
e2
4⇥ o r
E(rc , vth ) = 0
1/3
C
e 2 no
=
4⇥ o kB T
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Couplingparameter,…
For the coupling parameter we also have,
3
c
1
1
=
⇥
⇥
(4⇤)3
n2o
and we obtain,
c
✓
no e 2
o kB T
◆3
1
=
2
36 ND
1
1
1
=
=
(4⇤)3 n2o ⇥6D
(4⇤ ⇥ 9)
✓
3
4⇤ no ⇥3D
◆2
1/3
equivalent to,
Description
C
e 2 no
=
4⇥ o kB T
Plasma parameter
Limits
ΓC >> 1 (ND << 1)
ΓC >> 1 (ND >> 1)
Coupling
Debye sphere
Strongly coupled
Sparsely populated
Weakly coupled
Densely populated
Strong
Weak
Cold and dense
Hot and diffuse
Laser ablation plasmas
Inertial fusion
experiments
White dwarfs
Neutron stars
Ionospheric plasmas
Magnetic fusion experiments
Space plasmas
Electric discharge plasmas
Electrostatic influence
Characteristic
Examples
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Theplasmaskindepth,…
We consider the plasma electrons
B
=
t
B = Bz (x, t) uz
Bz (0, t) = Bo (t)
r^E
E(x, t) = Ey (x, t) uy
We also have,
r ^ B = µ o Je +
E
t
1
c2
'
e neo ve
and the equation of motion for electrons
dvey
=
dt
e
Ey (x, t)
me
Bz
= e µo neo ve
x
Bz
=
t
Ey
x
2
⇥ Ey
⇥x2
=
⇥ 2 Ey
⇥x2
=
Ey (x, t) =
2
e
me neo µo
pe
c
2
Ey =
Ey
p (⇥Bo /⇥t) e
Themagneticfieldexponentiallydecreasesintheplasmaalong
thesocalledskindepthofLondoncharacteristiclength
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1
c2
⇣
2
e neo
o me
⌘
Ey
Bz (x, t) = Bo (t) e
x/
p
p
=
q
x/
p
c
pe
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Magnetizedplasmas:Larmorradius,…
The force experienced by electric charges in this course (non-relativistic),
= e, i
F↵ = q↵ n↵ (E + v↵ ^ B)
this gives the cyclotron frequency,
↵ = q↵ B? / m↵ and using the
perpendicular speed to the magnetic field lines we obtain the Larmor radius
rL, =
m v?
|q | B
RL, =
RL,i =
r
a new length is defined using the particle thermal speed
Vth,
mi
RL,e
me
Magnetized electrons
and unmagnetized ions
RL,e ⌧ RL,i > L
Magnetized electrons
and ions
RL,e ⌧ RL,i < L
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Ordersofmagnitude,…
Low pressure discharges
KT≈1-3 eV, n≈108 cm-3
Solar corona
KT≈100 eV, n≈109 cm-3
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Fusion reactor
KT≈104 eV, n≈1015 cm-3
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TheEarthionosphericplasma,…
The black curves are
for neutral particles
and the red line is the
altitude dependent
electron density.
The sum of all different
ion densities ni equals
the electron density ne
The ions are produced by the absorption by the neutrals of parts of the solar
radiation spectrum. The maximum rate takes place at the F1 and F2 peaks.
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