VUV Spectroscopy of Atoms, Molecules and Surfaces

52 High-resolution VUV spectroscopy of H −
velocity spread. As the velocity distribution narrows, the effect of intrabeam
scattering, causing heating, becomes more and more pronounced, and the
cooling rate decreases. Eventually an equilibrium situation is reached with
equal cooling and intrabeam-scattering rates, as reflected by the slow decrease
in the velocity spread and an almost constant mean velocity (2–8 s). In this
regime the relative velocity spread is predicted to obey an ID 0.4 dependence
(equation 161 in [21]), with ID ∝ exp(−t/τ) beingtheD − beam current and
τ the lifetime of the ion beam. This dependence is indicated by the solid
curve in figure 3.4(b) and is seen to agree with the data. After 8 s of cooling
the relative velocity spread is 7×10 −5 and thus reduced by a factor of ∼6
with respect to that of an uncooled beam. The deviation from the ID 0.4 law
reflects the limit of equal D − ion- and electron-beam temperatures.
The evolution of the mean longitudinal velocity of the D − beam can be
predicted from a simple one-dimensional model, disregarding magnetic fields
and assuming monochromatic beams with no transverse velocity components.
In the binary collision approach of Poth [21] the cooling force is then, with
the present conditions, proportional to the difference between the ion- and
electron-beam velocities, F = K(vD(t) − ve(t)) (equation 43c in [21]). The
electron-beam velocity is given by ve(t) = � 2Ve(t)/m where Ve(t) isequal
to the acceleration energy of the electrons supplied by the electron- cooler
cathode voltage Ucath, corrected for a small but significant contribution from
space-charge. The effect of rest-gas ion trapping is taken into account through
this last contribution which, as explained qualitatively above, has a large
influence on the time dependence of vD(t). Denoting by ne and nrgi(t) the
density of electrons and of trapped rest-gas ions, respectively, Ve(t) isgiven
on axis, i.e. in the center of the beam, by
Ve(t) = eUcath + e2 r◦ 2
4ɛ◦
� � ��
r◦
(ne − nrgi(t)) −1+2ln
R
(3.1)
where r◦ and R are the electron- and vacuum tube radii, respectively, ɛ◦
the vacuum permittivity and e the elementary charge (equation 98 in [21]).
Assuming nrgi(t) to approach an equilibrium value n∞ exponentially with a
characteristic ”trapping” time τt, i.e. writing nrgi(t) =n∞(1 − exp(−t/τt)),
the time dependence of vi can now be obtained by solving the Newtonian
equation MdvD/dt = K(vD(t) − ve(t)). The result is an expression of the
form
vD(t) = A exp(−t/τc) − B exp(−t/τt)+C, (3.2)
where τc =1/K is the cooling time and A, B, andC are positive constants.
The data of figure 3.4(a) are fitted well with this expression as indicated by