where V is the cathode fall potential, I the discharge current, t the decay time, k is Boltzmann constant and N is the total number of atoms
4.8.1 Estimation of the ground state atoms effect on the beat shift
As w:e have calculated in (§ 4.3) and demonstrated experimentally
in (§ 4.5.2) a diange in the helium pressure of 1 Torr causes a beat shift of 6 MHz at 0.6328 ym.- This is about one order of magnitude larger than that expected due to an electron number density of led® cm"®, which is the minimum we require for detection.
In the decaying plasma of interest here we have seen in (§ 4.6) that the change in the helium pressure is expected to be far less than 1 Torr and hence the beat shift due to ground state atoms is expected,
to be less than that due to 10^® cm”® electron at 0.6328 ym. However by comparing the beat shift on tivo transitions (0.6328 and 1,15 ym)
where the relative beat shifts due to botli effects are different, it is possible to deduce both the electron density and ground state helium
density..
The beat shift on two transitions wi or (at 0.6328 ym) and
0)2 or V2 (at 1.15 ym) can be wrritten respectively.
4ita AN - (w
S y . = VI S 2--- ( 4 . S .1.1)
(2Li/t) + [47ra^AN^ - (u)p/aji)2 ]
4ïra AN ~ {a J u2)^
S V ■ = V2 2--- ( 4 .8 .1 .2 )
[ 4itag^AN^ - (Wp/w2)2 ]
where AN^ and AN^ (implicit in w ) are the change in the atom and electron number densities respectively. The polarizability of He changes by less than 1% in going from 0.6328 to 1.15 ym and so the same value has been used in both equations ( 4 .8 .1 ,1 ) and. ( 4 . 8 , 1 , 2 ) .
■Evaluating the parameters in equations (4.8.1.1) and (4.8.1.2) taking ratios we obtain
^’sig, vi 0.0129 AN - 1.76 AN.,
(4.8.1.3) 6 V - sig2 V2 0.0129 AN a - 5.83 AN'e
This equation shows that for AN =0, (6 a. sigi /A sig2) = 0,55, while
for ANg = 0, (6 = 1.82.
Assuming that (6 v . _ /6 v . ) is A and with some algebraicSlgi Slg2 operations vie get
AN
^ = 452.62 X - A (4.8.1.4)
AN e 1.82 - A
From equation (4.8.1.4) we can see that the ratio of atom to electron number density change can be estimated by measuring A, and hence the ratio of atoms to electrons effects at 0.6328 ym are estimated (see table 4.5a).
The measurement of A was performed by comparing the beat shift at 0.6328 ym and 1.15 ym under similar disdiarge conditions. For helium pressure of 1 to 20 Torr and current of 10 to 150 mA, A varies betiveen
0.50 and 0.54 (see table 4.5b). Comparing the values of A in table (4.5^,b) we find that the atom effect does not exceed 12% of the electron effect at 0.6328 ym. Hence we can take the beat shift at 0.6328 ym to be as a result of the electron effect only.
At early decay time the minimum measured beat shift is about
200 KHz. This suggests that the atom effect does not exceed 24 KHz which is equivalent to a gas number density change of 4 m Torr (see § 4.5.2).
The fact that the beat shift depends predominantly on the electron number density was demonstrated again by comparing the decay of the discharge current wdth that of the deduced electron number density (see next paragraph). The decay of the electron nimiber density follows that of the discharge current. There is no evidence of an atom effect, which
4.31
would lead to the beat shift increasing as the current decays. 4.8.2 The electron number density
Neglecting the ground state atom effect, which does not exceed 12% of the beat shift at 0.6328’ ym, the electron number density N
is related to the measured beat shift 6 "^sig equation (4.2.4) which can be written as
6 V • Slg 6n - - V ^Lj (o) /w)^ p (Hz) (4,8.2,1) The active length of the plasma is 10 cm, while the cavity mirror
separation is 72 cm. Substitution into equation (4.8.2,J) gives, for X equal to 0.6328 ym
= 8.582 X 1Q7 6 (cm''^) (4.8.2.2)
Electron number density at the tube centre deduced from the measured beat shift for different pressure and current values can be seen in figures (4.9a,b). The measured electron number density at the tube centre by this teclmique is compared wdth that deduced from the Stark broadening teclmique. These can be seen in figures (4.10a,b). It is evident that the electron ntunber density shows the same characteristic behaviour as a function of current and pressure.
In figure (4,11) electron number density as a function of current during the decay is plotted with initial current as a parameter. This graph shows the same linear dependence of electron density on current as figure (4.10a). This suggests that the electron number density at any particular time during the decay is the steady state value
corresponding to the current at that time,'
We now consider the loss processes for He* and electrons in order to demonstrate that they are consistent wdth the electron number density following the discharge current on the time scale involved, l\e must first consider the cooling of the electrons in the decaying plasma.
The electrons can lose energy through elastic and inelastic
collisions, but can also gain energy as a result of superelastic collisions with metastable or other excited states. For electron energies in excess of about 20 eV (v > 10® cm seC^) inelastic collisions can occur with ground state helium atoms, for wliich the total cross-sections is around 10“ cm^. The time between inelastic collisions is therefore around 3 x 10"? sec at a pressure of 1 Torr. Even the high energy electrons (primary electrons with energies around
200 eV) will cool to around 20 eV in times less than 3 y sec. The energy gain due to superelastic collisions during this time will be
negligible since the metastable number density is 10"® of the neutral atom density and the cross-section for the two processes are comparable
(see Chapter VI).
The electrons can only cool below 20 eV by elastic collisions. In elastic collisions the electron looses its energy according to the
relation
^ [U(t) - 1 V , (4.3:8.1)
where m and M are electron and atom mass respectively, U(t) is the electron energy, is the atom thermal energy and v is the collision frequency. If w^e assume v to be constant^^, then the energy decay is given by
U(t) - Uy = AUg exp C-t/T), (4.8.3.2)
where
T = M/(2my). (4.8.3.3)
and AU is the initial energy difference between the electrons and the gas. Substituting for v in terms of tire electron mean free path at 1 Torr, & , o we obtain for the time taken for the electrons to cool to within 10% of the gas temperature T
4.35
A Aün
t = 1.4 X 10"® ----1 An (n~iTT—) (4.8.3.4)
(j p T ^ '' '
where A is tlie molecular iveiglit and p is tire pressure (Torr) of the
gas.
For electrons initially at 20 eV, for a gas temperature of
0
0.04 eV (300 K) and assuming an electron mean free path at 1 Torr of