6.#Particles#of#Ionized#Gases# andtheir#Interactions · Nel gas e nei plasma si definisce la della...

1

B

J

Department of Electrical, Electronic, and Information Engineering (DEI) - University of Bologna1

6. Particles of Ionized Gases and their Interactions

2

a. Photonsb. Electronsc. Atoms or molecules at the ground stated. Atoms or molecules at an exited state:

a - one electron excited atomsb - multiple electron excited atomsc - metastable state excited atomsd - vibrational excited moleculese - rotational excited molecules

e. Positive ions:a - singly ionized ionsb - multiply ionized ionsc - excited ions

f. Negative ions (atoms or molecules)

Department of Electrical, Electronic, and Information Engineering (DEI) - University of Bologna

6. Particles of Ionized Gasesand their Interactions

2

3

W=

=

chhnne

n

n

p

Fotone

EM) onde delle (velocit /10998.2Plank) di (costante 10626.6h

8

34

smcJs

=

= -

2eee

30e

19

vm21

elettrone)dell' (massa 10911.0melettrone)dell' (carica 101.622q

=

=

-=-=-

-

e

kgCe

Elettrone

0


Electron

Photon (Particle with null mass and null charge)

(EM wave speed)

(electron charge)

(electron mass)

(electron energy)

(photon energy - Plank constant: h = 6.62610-34 Js)

(photon momentum - EM wave speed in vacuum: c = 2.998108 m/s)

PLASMA PARTICLES

4

continuocontinuum

i12

23

1

23

i

g1

g2

g3

e

q An atom with its optical electron (electron in the outer shell or optical shell), can be found in the ground, non-excited state or in an excited state

q The energy levels (energy states) are a discrete series of infinite values of energies between 0 and i (12, 12, 13, 14, . i), corresponding to different quantum numbers. The electron remains in an excited level for a time of the order of 10-8 10-7.

q At energies below the i the optical electron is bound to the atom. When the optical electron reaches an energy higher than i, acquiring this energy through collisions or interaction with photons, it becomes free and the atom becomes positively charged, i.e. it becomes a positive ion. i is also called series limit of the shell. The electron immediately below this shell becomes the new optical electron.

q Each energy level corresponds to different electronic arrangements. The number of electronic arrangements (or more precisely, quantum states) with the same energy level is called the degeneracy or statistical weight of that level and is denoted by gl, g2, etc.

Atom

3

5

Metastable states: the excited atom permanence in such states is tc >> 10-7 s.

Energy states of ion: even in this case, the optical electron of an ion can be in an infinite number of energy levels up to escape the ion which becomes charged of an additional positive charge.

Negative ion: the electron affinity is the binding energy of an electron bounded to a neutral atom giving rise to a positive ion. For a given negative ion only one value of binding energy, which is the affinity, is known.

Energy of atomic levels in eV: the energy of atomic levels (atomic states) is measured in eV:

1 eV = e (1 V) = (1.602 10-19 C) (1 V) = 1.602 10-19 J

Temperature: high temperatures can be measured in eV: e TeV = k TK (both energies in Joule) TK = (e/k) TeV

1 eV = 11600 K


6

Interactions between particles:

Collisional interactions: interactions between particles with mass

Radiative interactions: interactions between particles and photons


4

7

A collision between two particles (or impact) causes a change of their motion and/or energy state when they approach,

If the total kinetic energy is conserved, the collisions are called elastic collisions. All other collisions are non-elastic collisions.


COLLISIONAL INTERACTIONS

8

Il flusso (densit di flusso) del fascio di particelle di tipo 1 per unit di superfice :

1 = n1 g 1m2s

dove n1 la densit delle particelle di tipo 1 (numero di particelle per unit di volume) e g la loro velocit supposta uguale per tutte le particelle del fascio.

The flux density of a beam of type 1 particle is the number of the particle of type 1 crossing the surface unit per unit of time:

where n1 is the number density of he particle of type 1 (number of particles per unit of volume [1/m3]), g is the velocity of the particles (assumed to be equal for all particles).


COLLISION CROSS SECTION

5

9

1 = n1g1 g1

Q12

Collision cross sectionThe total collision cross section presented by particle 2 (field particle) for collisions with particle 1 (test particle), is:

Q12 (g) =

number of particles of type 1 colliding with the particle of type 2 per unit of time

1 [m3]

The type p collision cross section presented by the field particle 2 to collisions with particle 1 resulting in reactions of type p, is:

Q12 (g) =

number of particles of type 1 colliding with the particle of type 2 per unit of time

1 [m3]


Q12p (g) =

number of test particles collidind with the field particle and resulting in a p reaction per unit of time

1 [m3]

2

2

10

Differential collision cross section

The differential collision cross section presented by the field particle for collisions with test particles is:

I12 (g,)d =

number of test particles scsttered by the field particle in a direction contained in d per unit of time

1 [m3]

Q12 (g) = I12 (g,)d4

is the differential solid angle around the direction defined by the scattering angle and the azimuthal angle . d

For azimuthal symmetry, as , it results:d = sin d d


Q12 (g) = 2 I12 (g, )sin d

2

6

11

d = dS/r2

d = r2 sin d d/r2

d

d

r d

r sin d

dS


Expression of as a function of the scattering angle and the azimuthal angle .

d

12

The collision frequency is the number of collisions that a test particle (particle 1) undergoes with field particles (particle 2) per unit of time:

The reaction rate is the number of collisions of the test particles (type 1 particles of density n1) colliding with the field particles (type 2 particles of density n2) per unit of volume and of time:

R12(g) = n1n2 g Q12(g) [m-3s-1]

n121= = -

Rn

n gQ 121

2 12 [ ]s

lg

n Q12 12 2= =n

1

12

[ ]m

The mean free path is the path covered by the test particle between two consecutive collisions with the field particle:

Quantities related to collision between particles


7

13

Every particle of type s of the plasma (or a gas particles) is characterized by velocity with a distribution characteristic of the s particles. The velocity distribution function of the s type species is defined by fs(vs) where vs is the velocity of the s particle. The number of the s particle with velocity vs between vs and vs+dvs is:

ns fs(vs) d3vswhere:

+ - = f ( )d vs

3svs 1

+ - = n f ( )d v ns s

3s svs

Therefore the velocity distribution function results to be normalized:

Characteristics velocity of the particles of a gas or a plasma


vz

d3v

vx

vy

=-=

=

=

=

-+

=

0 :risulta infatti

:specie della la definisce si s specie ogniPer

:da definita fluido del La

: fluido del La

nm

: specie della La

v)d(f

:come s specie della la definisce si plasma nei e gas Nel

s

s

s

sss

s3

s

s

s

s

sss

s

sss

UuuU

uu

vvu

r

r

r

rr

r

diffusione divelocit

massa divelocit

totalemassa didensit

massa didensit

media velocit

Plasma velocities

The average velocity of the type s particle species is:

The mass density of the s species is:

The total mass density of the plasma is:

The mean mass velocity of the plasma is:

The diffusion velocity of the s species is:

Indeed it is:


8

15

Current density of a plasma

totalecarica didensit caric

didensit convezione dicorrente didensit

conduzione dicorrente didensit

elettricarente -cor didensit

,ne = = :s specie della a

la e s, specie della la

ne

s. specie della la

ne= :dove

+ ne=

:ottiene si , Poich

ne= :

la particella sua ogni di e caricacon s carica specie unaPer

sss

s

s

ss

ss

ss

ss

s

=

CsC

C

Cs

s

rr

r

r uu

UJ

Juj

U+u=u

uj

s

ss

ss

ss

For a charged particle of type s with charge es, the current density of the type s particle species is:

As us = u + Us, it results:

where:

Js is the the conduction current density of the type s species.

rCsu is the the convection current density of species s, where rCs is the charge density of the species s. The total charge density of the plasma is:

total plasma charge density.


16

=

ssUJJ

u

Jujj

s

s

sss

C

sssC

Cs

ne= =

ne=

+ =

:conduzione ditotale corrente didensit

:convezione ditotale corrente didensit

:totalecarica didensit

:totalecorrente didensit

r

r

r

)n (ne = ;)n (ne =

);n (ne = :ha si

ionizzati, ntesingolarme ioni e elettroni da costituito plasmaun In

eei

eiC

eiC

UUJuu

i - -

- rr

total charge density:

total current density:

total convection current density:

total conduction current density:

Current density of a plasma

Plasma with two charged particle species: electrons and singly ionized ions. It is:


9

17

In a plasma the particles of type 1 have a velocity v1 and a velocity distribution f1(v1). The particles of type 2 of the plasma have velocity v2 and a velocity distribution f2(v2).

The differential flux density of the type1 particles with velocity v1 between v1and v1+dv1, (dv1=d3v1) with reference to the type 2 particle, considered non-moving, is

dG1 = n1 f1() d (v1-v2)with module:

dG1 = n1 f1() d |v1-v2|The number of collisions p, that the particles of type 1 with velocity v1 between v1 and v1+dv1, have with a particle 2 per unit of time, is:

n1 f1() d |v1-v2| (|v1-v2|)Q12p



18




The differential reaction rate for type p collisions of particle 1 with velocity v1between v1 and v1+dv1, and particle 2 with velocity v2 between v2 and v2+dv2, is

dR12 = n1 f1() d |v1-v2| (|v1-v2|) n2 f2() dHence the total reaction rate is given by:

R12 = dR12 =

= n1 n2 f1() d |v1-v2| (|v1-v2|) f2() d

For the collisions between electrons and heavy particles (atoms, ions, molecules), as the heavy velocity is much smaller than the electron velocity, the velocity of heavy particles may be considered to be null. Hence:

Reh = ne nh fe()|ve| (|ve|)de

Q12p

+

+

Q12p

Qehp

+

10

19

Maxwellian distribution function

The equilibrium velocity distribution function of the s type species is defined by fs(vs) = fsM(vs) where fsM(vs) is the Maxwellian distribution function. It is obtained when considering the energy statistically distributed.

fsM(vs) =

where k = 1.38110-23 [J/K] is the Boltzmann constant and T is the temperature of the s particle species.

The function fsM defines the temperature T of the s species. T is the parameter that determines the equilibrium distribution of the species. Hence the temperature can strictly be defined for an equilibrium plasma or gas only. A less rigorous definition of T is that the average energy of the particles of the gas is given by 3kT/2.

ms2kT

3/2

exp msvs2

2kT


20

At equilibrium the equilibrium average velocity of the species s is obtained by weighting the velocity of a particle by means of the the maxwelliandistribution function of the species:

us = |vs| fsM() ds = < |vs| > = 789:;? msvs2 fsM() ds = <

>? ms|vs|2 > =

@?

kT

+

Maxwellian distribution function- equilibrium average velocity- equilibrium average kinetic energy

+


11

21

Plasma in a Maxwellian condition(Equilibrium plasma)

1 >> mm

uu

:risulta

,m

8kT = u,m

8kT = u :poich

a,temperatur stessa alla atomo) stesso dello neutri e (ioni pesanti

particelle e elettroni di ionizzato teparzialmen gasun Per

e

h

h

e

hh

ee

=

ppAs it is:

At partially ionized plasma there are electrons, ions, and neutrals (ions and neutrals of the same atom). At equilibrium with equal temperature of electrons and heavy particles (ions and neutrals) the average velocity of electrons is much higher than that of heavy particles.

It results:


22

Elastic collisions

'

:risulta elastico urtoun per urto,l' dopo e prima relative velocitle sono e urto,l' dopo e prima massa di centro del velocitle sono e Se

mmmm

:massa di centro del velocitla e tiva-rela velocitla definisce si collidono che particelle due delle sistema ilPer

21

21

g = gG' =G

g'gG'G

v-v =G

v-v = g

21

21

+

In elastic collisions the internal energies and identities of the colliding particles are unchanged. In an elastic collision the particles change their directions of motion, but the total initial momentum and kinetic energy of the particles is conserved. All other types of collision are called non-elastic.

For two colliding particles the relative velocity and the velocity of the centre of mass are:

When G and g, and G and g are the the velocity of the centre of mass and the relative velocity before and after the collision respectively, for an elastic collision it is:


12

23

The loss of momentum in the z-direction along the direction of velocity g, of the test particle when scattered in c for any f, is:

m1g m1gcosc = m1g m1g cosc = m1g(1 - cosc)

The number of test particles of a beam scattered elastically into d per time unit is given by:

G1I>?(C)(g,c,f) d = G1I>?

(C)(g,c,f) sinc dc df.

Therefore the total loss of momentum of the beam of particle 1 due to collisions with a particle 2 per unit of time, is:

m1g(1 - cosc) G1I>?(C)(g,c,f) d =

= m1g G1 (1 - cosc) I>?(C) (g,c,f) sinc dc df

4

4

Elastic collisions


m1g

zc

2

1

24

Elastic collisionsThe following quantities are defined

- Flux density of momentum of the test particle beam: m1g G1- Cross section for momentum transfer for elastic collisions:

- Probability of elastic scattering into d:

Therefore it results:

Q12(1)(g)= I12

(e)(g,,)(1 cos )d4

p12 (g,,)d = I12

(e)(g,,)dQ12

(e)

p12 (g,,)(1 cos )d4 =

Q12(1)

Q12(e)


13

25

When comparing Q>?(>) with Q>?

(C), the factor (1 - cos c) gives Q>?

(>) more weight from large angle scattering events [ i.e.,f or c = , (1 - cos c) = 2; for c = 0, (1 -cos c) = 0].

For elastic electron atom scattering at sufficiently low energies (ee < 10 eV) generally Q>?

(>) Q>?(C).

At higher energies Q>?(>)can be less than Q>?

(C) by a factor as large as one half.

Elastic collisions


26

Elastic collisions

Total elastic collision cross sections of electrons in the alkali metals Na, K, Rb, and Cs.

Q12(e)(g) = 2 I12

(e)(g, )sin d

Total elastic collision cross sections for azimuthal symmetry is:

Total elastic collision cross sections of electrons in nobel gases Xe, Kr, A, and He.


14

27Department of Electrical, Electronic, and Information Engineering (DEI) - University of Bologna

Comparison of momentum

transfer cross sections (----) with total elastic cross sections ( )

for electron collisions with He,

Ne, and A

Comparison of low-energy momentum transfer ( --- and - - ) and total elastic ( ) cross sections for

electron collisions with argon.

Elastic collisions

28

The average fraction of the kinetic energy lost by a test particle of speed v1 incident on a stationary field particle 2 is:

12 (v1,0)m1v1

2 /2= 2m12(m1+m2 )

Q12(1)(v1)Q12(e)(v1)

This result is most frequently applied to collisions in which the test particle is an electron and the field particle is a heavy particle such as an atom, molecule, or ion. Since at low electron energies Q>?(>) Q>?

(C) we obtain:

eh (ve,0)meve

2 /2!2memh

where m12 = m1m2/(m1+m2) is the reduced mass.

where me T.

Elastic collisions


15

29

Elastic collisionsThe average energy lost by a test particle per collision, as a result of elastic collisions with a type 2 field particle is:

This is obtained by neglecting the second term in the expression of . For collision-dominated gases the ratio of the diffusion speed U to the thermal speed v for any species, is usually very small, so that the order of the second term of is much less than the first one.

< 12 > =2m12

(m1+m2 )32k(T1-T2 )

A good estimation of the average energy lost by electrons in elastic collisions with atoms, particularly at temperatures below the threshold energy of the first excited level, is given by:

The average momentum lost by a test particle per collision, as a result of elastic collisions with a type 2 field particle is:

< p12 > = m12Q12(1)(v1)Q12(e)(v1)

(U1-U2 ) m12(U1-U2 )


= 2memh

32

k(Te -Th )

As a matter of fact in most gases the presence of other charged particles will modify the nature of the interaction between any two given particles when their separation becomes large. The surrounding charged particles behave collectively, on the average, in such a manner as to effectively shield two particular particles from interacting with each other when their separation exceeds a distance of the order of the Debye length.

Charged particle collisionsThe collisions between charged particles are elastic collisions. They are due to interaction between these particles cased by the Coulomb potential.

The Coulomb potential varies inversely with the distance of separation between the particles. The infinite character of Q(1) would appear as the Coulomb interaction decreases with increasing distance so slowly that all test particles in the incident beam interact significantly with the field particle, no matter how distant he test particles are from the field particle. The geometrical interaction area which the field particle presents to the beam is therefore effectively infinite.


16


D = 0kTmee

2

(r) = q40r

exp 2Dr

The shielded Coulomb potential, in the presence of surrounding charged particles, is:

r flD 0,243117 f0

2lD 0,029553 f03lD 0,004790 f04lD 0,000873 f05lD 0,000169 f0

Charged particle collisions

The Debye length in a plasma is the distance from a charged particle that defines the shield effect against the Coulomb field induced by the charge of the particle of surrounding charged particles. The expression of it, that will be derived in the following chapter, is:


][Tln Zn 103.64 =

:secondo al Z,carica di ionicon collisioni subisce elettroneL'

nT

Z1 101.24 =

: e , il ln dove

][T

ln Z105.85 = Q

:liana)-Maxwel onedistribuzi sulla (pesato

della medio Valore

13/2i

6-ei

ei

e

37

22

10-ei

-Ln

n

L

L

L

s

m

Coulomb di logaritmo

Zcarica diione -elettrone urto per moto diquantit di ntotrasferime per urtod'sezione

The number of collision per second of the electrons with ions a singly ionized ion is:

Charged particle collisions

The Coulomb logarithm ln is of the order of 10 for a large range of ne and T. For Z = 1, T = 2000 K, ne = 1020 m-3 ln 5, Qei 7 10-16 m2.

where the Coulomb logarithm lnL is:

The collective behaviour of the charged particles thus results in making the momentum transfer cross section effectively finite. The collision between charged particles are named also Coulomb collisions. The energy-averaged momentum transfer cross section for electron-ion singly ionized (Z = 1) is:

17

Non-Elastic collisions

For these reactions due to particle collisions the kinetic energy of the colliding particles is utilized. In the case of electron impact, as theelectron velocity is much higher than the heavy particle velocity, the collision depends only on the electron velocity. In a plasma it depends on the electron thermal velocity of on the electron temperature Te when it is defined.

In non-elastic electrons-atom collisions, a free electron collides with an atom. The atom changes its energetic excitation state as the optical electron of the atom changes energy level. The excitation energy energy of this transition is exchanged with the kinetic energy of the free electron that collides with the atom. Reactions due to non-elastic electron-atom collisions are: a. excitationb. de-excitationc. ionizationd. recombination

1

l

ui

In order that the excitation occurs, the electron must have a kinetic energy equal or higher than the threshold energy lu(e lu).



Collisional excitation of an atom in the state the state u (u > l) due to electron impact is described by the following reaction:

A(l) + e(e) A(u) + e(e')

with e'= e - lu

34

1

l

ui

continuum

18



Excitation cross sections for the 2ps levels of A, Kr, and Xe by electron impact.

Excitation cross sections for the alkali metals Na, K, Rb, and Cs by

electron impact.

36

In the de-excitation by electron impact the free electron takes away the threshold energy lu as kinetic energy. Therefore the de-excitation cross section has no threshold energy.



Collisional de-excitation of an atom in the state the state u (u > l) due to electron impact is described by the following reaction:

A(u) + e(e) A(l) + e(e')

with e'= e + lu

1

l

ui

continuum

19

37

In this case also there is a threshold energy for the reaction. The threshold energy is li(e li).


Non-Elastic collisionsDue to collisional ionization a neutral atom in the state l by electron impact becomes ionized. In this case the optical electron uses part of the kinetic energy of the free electron to become a free electron:

A(l) + e(e) A+ + e(e') + e(e)

with e= e + e + li

1

l

ui

continuum

38

Radiative recombination of a free electron, which collides with a singly ionized ion, passes to the l level of the atom that becomes neutral, becoming the optical electron of the atom. The energy of ionization and the kinetic energy of the recombining free electron escapes as radiation.

A+ + e(e) A(l) + h

with h = e + li

1

l

ui

continuum

Three body recombinationis the consequence as three bodies collide: two electrons and an ion. An electron becomes the optical electron of the atom and the second electron takes away, as kinetic energy, the ionization energy and the kinetic energy of the recombined electron.

A+ + e(e) + e(e) A(l) + e(e)with e = e + e + li



20

39

Non-Elastic collisionsRadiative recombination radiation

Spectrum of the radiation of radiative recombination to the 6s level of Cs of a

plasma with a Maxwellian

distribution of free electrons

40

Evaluation of the relative importance of the radiation recombination and the three body recombination processes in a plasma:

Reh = nenh g Qeh [m-3s-1]

Rei(r) neni ne2, Ri2e(3) ne2 ni ne3

RR

1n

ei(r)

i2e(3)

e

ab


21

41

Radiative Processes Particle Theory and Wave Theory

Electromagnetic radiation are fluxes of energy due to vibration of the electromagnetic field caused by the deceleration of charges. When the optical electron of an atom passes from an higher level to a lower level, the energy difference between the two level can be emitted as the kinetic energy of a colliding body or as radiation characterized by a defined wave frequency n (or wave length l where l = n/c, with c = = 2.9979108 m/s is the speed of light). The energy and the momentum of the emitted radiation are:

where h = 6.626110-34 m2kg/s is the Plancks constant and n is the unitary vector in the direction of the propagation of radiation.

= h

p =hcn


42

Radiative Processes Particle Theory and Wave TheoryThe radiation energy emitted by the transition from an upper to a lower atomic level is characterized by the energy elu = hnlu and the frequency nlu. Hence a fraction of of the energy elu = hnlu at the frequency nlu does not exist. The energy of the flux of radiation at a given wave length (a beam of light of a given colour) is a multiple of elu = hnlu. Hence a radiation quantity of energy elu is considered a particle, a photon, with energy, momentum and without mass.The photon number density is defined by nn (number of photons per unit of volume). The distribution function f(pn) expresses the number of photons per unit of volume with momentum between pn and pn + dpnand its given by:

nnfn(pn)dpnwhere:

f K dLMNM K = 1


22

43

Radiative Processes Particle Theory and Wave TheoryThe differential flux density of photons passing through the unit of the surface component perpendicular to the direction of pn with momentum between pn and pn+dpn is:

c [nnfn(pn)dpn] >

OPQ

The specific intensity of radiation (radiation with frequency nbetween n and n + dn and direction W in the infinitesimal solid angle W and W + d W) is given by the energy transported by the differential flux density of photons with momentum within pn and pn+dpn:

I(n,W)dndW = c[nnfn(pn)dpn]hn = RS

TPnnfn(n,W)n3dWdn

UOPQ

where: dpn = pn2dpndW = (h/c)3 n2 dn dW

I(n,W) is an intensity of radiation and is a quantity of the wave theory.Department of Electrical, Electronic, and Information Engineering (DEI) - University of Bologna

Radiative transitions Bound-bound radiation: line radiation in a discrete spectrum resulting from electronic transition between an upper to a lower bounded state.

Free-bound radiation: radiation of a continuous spectrum resulting from radiative recombination: a recombining free electron follow into the optical shell of an atom (it is also possible the reverse process of radiative ionization with the absorption of radiation bound-free absorption).

Free-free radiation: radiation of a continuous spectrum: a. bremstralung radiation, b. cyclotron radiation.

Radiative Processes


23


Radiative Processes

Line and continuous absorption processes.

46

Line radiation from bound-bound processes emitted by a Cs seeded A plasma at low temperature (T ~ 1000 K, Te ~ 4000 K). The radiation are emitted by de-excitation processes of neutral Caesium.

1

lui


Radiative Processes

24

47

Recombination radiation from free-bound processes emitted by a Cs seeded A plasma at low temperature (T ~ 1000 K, Te ~ 4000 K). from an Ar-Cs plasma. The radiation are emitted by recombination processes of of free electrons with Caesium ions with transition to the first excited level of the neutral Caesium.

1

lui


Radiative Processes

48

Line radiation are emitted/absorbed when an electron bound to an atom, moves between two levels of the atom.

1

lui

Radiative Processes

a. Spontaneous emission:

A(u) = A(l) + hluwith u > l (u - uper and l lower exited state). The energy of the emitted photon corresponds to the energy difference of the two levels. The spontaneous de-excitation takes place in times of the order of 10-7-10-8 s.

b. Induced emission:A(u) + hlu = A(l) + 2hlu

the atom at level u collides with a photon of energy equal to the energy difference u-l and de-excites to level l. Two photons are produced by the reaction: one of them is the colliding photon, the second one is the photon carrying away the energy released by the de-excitation. The two photons are emitted in the same direction of the incident beam.

25

49

c. Absorption:A(l) + hlu = A(u)

An atom at level l collides with a photon of energy equal to the energy difference between level u and level l, and excites passing from the level l to level u.

Line radiation are emitted/absorbed when an electron bound to an atom, moves between two levels of the atom.

1

lui

Radiative Processes


50

Qlu() is the radiative absorption cross section with transition from l to u state.

The differential reaction rate for the process becomes:

dRlu(lu) = nlnf (p)cQlu()p2dpd == nlnf (p)cQlu() 2 (h/c)3dd

dRlu(lu) is the total number of transitions l-u per unit of time and volume (corresponding to the number of photons hnlu) caused by the absorption of radiation with frequency between and + d and with direction within and +d.

By integrating over all frequencies that can give a contribution to the transition l-u, the reaction rate for the u-l transition caused by radiation with direction within and +d is obtained:

Radiative absorption


Rlu = nlnuh3

c2d Qlu ( )f (p)

0

+

2d

=

D+

0 lulu dR = R

nnn

(pn = hn/c)

(lu) = nunl

26

51

The wave theory description of the reaction rate for the l-utransition caused by absorption of radiation with direction within and +d , as a function of the radiation intensity, is:

Rnlu(lu) = nlBluIlu()d

where:

Blu is the Einstein coefficient for absorption. Therefore from the two expression of the reaction rate for the process derived above, it results:

+

0

)(22

3

lulu d)(Q)(fchn =)(IB nnnnnnn

lupW

n nn+

0lu d)I( =)(I W,W

Wave theory description


Rlu = nlnuh3

c2d Qlu ( )f (p)

0

+

2d(lu)

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The reaction rate for the u-l transition (number of transition per unit of volume and of time) due to the induced emission of radiation in a direction within and +d is:

Rnlu(ul) = nuBluIlu()d

where Qul() is the radiative induced emission cross section with transition from u to l state.

Therefore the energy emitted per unit of volume and time in a direction within and +d, due to the l-u transition is:

hnluRnlu = hnlu nlBluIlu()d =

Wave theory description


Rlu = nlnuh3

c2d Qlu ( )f (p)

0

+

2d

Rlu = nlnuh3

c2d Qlu ( )f (p)

0

+

2dnh4nlu ul

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53

Einstein coefficientsBlu Einstein coefficient for absorptionBlu Inlu(W)dW probability per unit of time that an atom will

undergo an l to u transition with the absorption of radiation from the solid angle dW.

Bul Einstein coefficient for induced emissionBul Inlu(W)dW probability per unit of time of induced emission into

the solid angle dW corresponding to a de-excitation with an u to l transition.

Aul Einstein coefficient for spontaneous emission(Aul /4p)dW probability per unit of time that an atom will undergo

a spontaneous u to l transition with the emission of radiation into the solid angle dW. Spontaneous emission is isotropic with respect to direction. For strong lines the value of Aul is of order 108 s-1.


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Cross section for radiative bound-bound absorptionThe cross section for radiative bound-bound absorption and emission of a photon, corresponding to the atomic transition from the level u to the level l may be written (SI units):

Qlu ( ) = 140mec

flu( ) = 2.65 106 flu( ) = a flu( )

where in SI units a = 2.6510-6, ful is an atomic constant and is called the absorption oscillator strength for the l to u transition. Values of flurange between zero and one.The function f(n) is called the absorption line shape factor and satisfies the normalization condition:

( ) 0

+

d = 1Department of Electrical, Electronic, and Information Engineering (DEI) - University of Bologna

28

Line broadening1.Natural broadening When measuring the radiation caused by spontaneous emission, a finite time Dt is necessary for the measurement of the energy of it. It then follows, as a consequence of the Heisenberg uncertainty principle, that the energy of any excited level cannot be precisely specified, but it is uncertain by an amount DeHe h DnHelu (DeDt = De/Dn h/2, where h = h/2p is the reduced Planck constant). Therefore, as a consequence, the uncertainty in the energy is reflected on an uncertainty on the frequency (and on the wavelength) of the photon emitted. Hence the absorbed line shape is broadened.

The absorption line shape function for the l to u transition due to natural broadening is:

where DnHelu is the half broadening, broadening at half maximum of the line, due to natural broadening. The same broadening effect is also observed in line emission.

-- -

luHe ( ) = 1

lu

He / 2( lu )

2 + ( luHe / 2)2

55

56

Line broadeningThe natural broadening shape function is of the Lorentzian profile. The width of a naturally broadened line at half-maximum for visible radiation is of order DlHe ~ 10-4 . Natural broadening in partially ionized gases mostly is obscured by other broadening mechanisms.

2. Pressure-broadening due to interactions with other particlesThe Lorentzian broadening lione shape is described by:

where DnLlu is the half broadening, broadening at half maximum of the line, due to pressure broadening. Particular mechanisms which contribute to pressure broadening:a. Resonance broadening (also called Holtzmark or self-broadening)

interactions with like neutral particlesb. Van der Waals broadening (or Lorentz broadening) interactions

with unlike neutral particlesc. Stark broadening - interactions with charged particles

luL ( ) = 1

lu

L / 2( lu )

2 + ( luL / 2)2


29

57

3. Doppler broadening the frequency of radiation relative to a moving atom will differ from that measured by a stationary observer, because of the Doppler effect. Even if the atom were able to absorb only a single frequency nlu as measured in the atom's frame of reference, to a stationary observer it would appear capable of absorbing different frequencies, depending on its relative motion.For particles with a Maxwellian velocity distribution, the line-shape factor for Doppler broadening is:

where: , and DnDlu is the half broadening (broadening at half maximum) due to the Doppler effect for the transition l to u. The Voigt profile V(nlu, Alu, DnDlu) describes line-shape when the pressure and Doppler broadening mechanisms act simultaneously. The shape factor is:

a = 1/ 2 ln2

luD( ) = 1

a luD exp -

( lu )a lu

D

2

luV( ) = 1

a luD V( lu , Alu, lu

D )


Line broadening

Optically thin and optically thick plasmasThe photon mean free path in a plasma for the radiative process l-u, in which photons interact with field particles l, is given by:

When the value of luL() at the line centre for a Lorentzian broadening fluL(nlu) is used, the mean free path of the photons of the process l-u is approximately given by:

For a characteristic plasma length (plasma thickness) LC, with reference to the radiative process l-u, the plasma is called: - optically thick plasma: llu > LC (the l-u photons leave the

plasma)

llu = 1

nlQlu =

1nl a flulu

L ( )

llu =

2aflu lu

L

nl


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59

1

l

ui

Il inverse process is the radiative ionization. When an atom collides with a photon of energy at least equal to the energy of ionization, this process may take place.


Free-bound (bound-free) radiation

Free-bound radiation are the radiative recombination radiation that originates from the collision of electrons with a positive ions followed by recombination. By the electronic transition through the kinetic energy state to a bound energy state of the atom a photon is emitted.

60

Radiative recombination radiation

A+ + e(e) A(l) + h, with

h = meve2 + il

When defining as Qil(ve) the electron-ion collision cross section for radiative recombination, the differential reaction rate for the collisions of ions with electrons of speed between ve and ve+dve is given by:

dRil = neniveQil(ve)f(ve) d3ve

As dve = ve2dved, the radiative power density due to recombination with electrons of speed value between ve e ve+ dve and direction within d:

dPil = h neniveQil(ve)f(ve)ve2dved


31

Radiative recombination radiationThe energy hn of the photo coming from a recombination is:

Often the expression of Qil(ve) is given by Q0il/ve2 (where Q0il is a constant). The radiative power density due to recombination with electrons of speed value between ve e ve+ dve and direction within d:

Assuming Maxwellian distribution of free electron energy and isotropic radiation, the radiative power density of a wavelength between l and l+d l for unit of solid angle is:


62

In a thin band of wave length Dl in the neighbourhood Dl e Dl + d Dl, it is:

From the light intensity measurement at two different bands centred on the lk (k = 1, 2) wavelengths in a zone of the spectrum dominated by recombination radiation, the electron temperature is obtained:

From the value of Te from one of the two measured values of DPil(l1) or DPil(l2), assuming ne = ni, the electron density may be derive.

Radiative recombination radiation


32


64

Free-free radiationThe free-free radiation are due to the acceleration of a free electron or a charged particle due to the absorption of a photon, or the deceleration of it with the emission of a photon (bremstralungradiation).

From the centripetal acceleration of a charge, which moves around the lines of force of a magnetic field, cyclotron radiation is generated.

The radiative power emitted by a charge in a collision is given by the flux of the Pointing vector outgoing from a closed surface that contains the charge:

where Ze is the charge of the particle and is its acceleration.


33

65

Free-free radiation bremstralung radiation

For an electron which is decelerated by an ion of charge Ze, the acceleration is:

where n is the unit vector in the direction electron-ion. The force meeis the electrostatic force exerted by the ion on the electron. For electronic Maxwellian energy distribution in the plasma, the bremstralung power density emitted is:


66

Free-free radiation cyclotron radiation

The acceleration of a particle of charge Ze, due to an induction field B, in the plane perpendicular to B is

The power density emitted by the particles of charge Ze, mass mZe and density nZe is:

In a Maxwellian plasma on the temperature. If we assume that , it is:


34

To analyse a light beam, the properties of the transparent

solid, whose angle of refraction varies with the wavelength of the of the

refracted radiation, is used.


68

1

k

ui

e

ku ki

Through the statistical mechanics it is possible to determine the equilibrium relationship between various species of particles of a plasma and between the particles of the same species at different energies.

Thermodynamic equilibrium in plasmasEquilibrium relations


35

69

At equilibrium the energy of free electrons is given by the Maxwellianenergy distribution:

The relation between the population density nu and nl of two atomic states is given by the Boltzmann relation:

The total number density of the atom given by n = n1 + n2 + n3 + . . Hence:

where Z(T) is the partition function. Therefore it is:

Equilibrium relations

feM (ve ) =

me2kT

32exp - meve

2

2kT

nunl

= guglexp - lu

kT


70

Partition function

Part

ition

func

tion,

Z(T

)

T, 103 KDepartment of Electrical, Electronic, and Information Engineering (DEI) - University of Bologna

36


Equilibrium relationsAt equilibrium the relation between the population density nk of an atomic state, ions and electrons is given by Saha equation:

and also:

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Black-body radiationBlack-body radiation is the type of electromagnetic radiation within or surrounding a body in thermodynamic equilibrium with its environment, or emitted by a black body (an opaque and non-reflective body), assuming uniform temperature. The radiation has a specific spectrum and intensity that depends only on the temperature.A perfectly insulated enclosure that is in thermal equilibrium internally contains black-body radiation and will emit it through a hole in its wall, provided the hole is small enough with negligible effect upon the equilibrium.


Spectralradiance[kW/(srm2 nm)]

Wave length [m]Spectral Radiance (power of radiation per steradiant, m2 and wave length) as a function of the wave length

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73

Black-body radiation


When the electromagnetic radiation is in equilibrium with the matter, the specific intensity of radiation is described by Planck function ( or black-body radiation function):

74

Equilibrium regimes

TE Thermodynamic Equilibrium: Complete balance between matter and radiation. The following relations are satisfied: Maxwell, Boltzmann, Saha, Plank.

LTE Local Thermodynamic Equilibrium: The matter is in equilibrium (all plasma particles are in equilibrium with each other but not with radiations). The following relations are satisfied: Maxwell, Boltzmann, Saha.

PLTE Partial Local Thermodynamic Equilibrium: A reduced energy exchange between heavy particles and electrons causes deviations from equilibrium. The electron-electron collisions ensure Maxwellian electronic distribution with Te > Th. The following relations are satisfied: different Maxwelliandistributions for electrons and heavy particle distribution, Boltzmann, Saha.


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75

l

u

Detailed balancing principle

The inverse of any given process is defined as that process which has for its initial and final states the final and initial states, respectively, of the given process. Thus for the transition between two energetic states of an atomic shell at thermodynamic equilibrium (TE) is:

dRul = dRlu

The principle of detailed balancing states that under conditions of thermodynamic equilibrium the differential reaction rates for each microscopic process and for the corresponding inverse process are equal.


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A(l) + e(e) A(u) + e(e)e = e + lu

The differential reaction rates for this process and its inverse are:

dRlu = nenl ve Qlu(ve) fe(ve) ve2dve ddRul = nenu ve Qul(ve) fe(ve) ve2dve d

From de = de it follows vedve = vedve. At ET it is:

dRlu = dRul, fe= feM, and

gl ve2 Qlu(ve) = gu ve2 Qul(ve)

This relation between Qlu(ve) and Qul(ve) obtained at TE depends only from the difference between ve2 and ve2 given by elu. Therefore this relation is always valid.

Detailed balancing principle:collisional processes

nunl

= guglexp - lu

kT


39

77

At TE the probability of radiative excitation (for absorption of a photon) must be equal to the probability of radiative de-excitation (for spontaneous and forced emission emission):

( ) ( )W,W, n

pn lulul

ulululu I B n = 4

A + I B n

l

u

At TE nu/nl is given by Boltzmann relation and I (,) is given by Plank function Bn(T):

1 - kT

hexp

BB

gg

B4A

= 1 -

kTh

exp

c2h

ul

lu

u

l

ul

ul

2

3

lulu

lu

np

n

n

2

3ul

ul

ul

ululul

ch8

= BA

B g = B g

np

In order that this equality will be satisfied for every

T, it follows:

Detailed balancing principle:radiative processes


78

From the measurement of line radiation in

a plasma at TLE, it is

possible to obtain the plasma

temperature.


40

79

Line emission in a TLE (PLTE) plasma: Temperature determination

Eu1 = a Au1 nu u1where a is a coefficient which depends on the plasma volume observed. By means of this measure, it is possible to determine the population density of the state:

For a plasma at TLE, it is:

1

l

uiAssuming the plasma in TLE and transparent

to radiation, the light output measured per unit of solid angle in the spectral range , which containing the line emitted by the transition u l, is:

nu = Eu1

a Au1 u1

Eu1E l1

Al1 l1Au1 u1

= nunl

= guglexp lu

kT

log gl Eu1gu E l1

Al1 l1Au1 u1

= log glnu

gunl

= lu

kT


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