Download Ionization of the Thomas–Fermi atom in intense laser fields

J. Phys. B: At. Mol. Opt. Phys. 33 (2000) 829–842. Printed in the UK
PII: S0953-4075(00)08381-4
Ionization of the Thomas–Fermi atom in intense laser fields:
the static limit revisited
D Pitrelli†, D Bauer†‡, A Macchi‡§ and F Cornolti†§
† Dipartimento di Fisica, Università di Pisa, Piazza Torricelli 2, 56100 Pisa, Italy
‡ Theoretische Quantenelektronik (TQE), Technische Universität Darmstadt, 64289 Darmstadt,
Germany
§ Istituto Nazionale Fisica della Materia (INFM), sezione A, Piazza Torricelli 2, 56100 Pisa, Italy
E-mail: [email protected]
Received 4 October 1999, in final form 24 November 1999
Abstract. The problem of the Thomas–Fermi atom in strong static homogeneous electric fields
is revisited. The static field situation is a reasonable approximation to the otherwise fully timedependent problem of an atom in a laser pulse, provided that the laser frequency is not too high.
Exact results for the ionization degree versus the applied electric field, obtained numerically,
are compared with those from an approximate model which is much easier to solve. Very
good agreement is found. Although the approximate model was published ten years ago, an
unexplained discrepancy with both experimental results and the exact numerical solution of about
one order of magnitude in the ionization degree had remained. In this paper it is shown that
this discrepancy does not exist. A comparison with the simpler ‘over the barrier ionization’
model and available experimental data for rare gases is made. A very appealing feature of the
Thomas–Fermi model is its universality thanks to the scaling behaviour with respect to the nuclear
charge Z.
1. Introduction
The rapid development of lasers delivering intensities >1014 W cm−2 strongly triggered interest
in non-perturbative methods for studying the interaction of such intense laser light with atoms,
solids and plasmas [1, 2]. Conventional perturbation theory, even when extended to very high
orders, is not applicable to intense laser–atom interaction, but semiclassical approximations
turned out to be quite reliable in this regime, at least to estimate ‘rough’ features. One example is
the so-called ‘simple man’s theory’ of ionization and rescattering [3–5] which is able to explain
the cut-offs of electron energy spectra and high harmonic generation. Another example is the
simple ‘over the barrier’ ionization picture [6, 7] which can be used to estimate the intensity
necessary to produce a certain charge state of a given element.
Applying Thomas–Fermi theory (TFT) to an atom in an electric field can be considered
as an extension or improvement to those simple semiclassical approaches. Although there
are well known deficiencies in standard TFT (see, e.g., [8] for a review of TFT) such as the
incorrect asymptotic behaviour of the electron density both for r → 0 and r → ∞, strong
points of TFT are its self-consistency, simplicity and universality with respect to the atomic
number Z, thanks to self-similar solutions.
Since with currently available computers the numerical solution of the full time-dependent
Schrödinger equation of an atom in an electric (laser) field is not possible for more than two
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© 2000 IOP Publishing Ltd
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D Pitrelli et al
active electrons, approximate models are the only way out of this problem. However, even
if the full solution was possible simple models are desirable for understanding the essential
physics.
In this paper we study the Thomas–Fermi (TF) atom in static homogeneous electric fields.
For optical laser frequencies the inner atomic dynamics is very fast compared to one laser
period. Therefore, static models can be expected to give reasonable results provided that
rescattering effects have no strong influence on the observables of interest. If rescattering
effects can be neglected, for an atom in a laser pulse of peak field strength Ê we expect the
same ionization degree as for the same atom in a static field of strength Ê. The problem
of an atom in a static electric field was tackled with TFT ten years ago by means of an exact
numerical solution in [9] and in [10] with some approximations which reduce the computational
effort significantly. Unfortunately the results disagreed strongly. In this paper we revisit the
approximate method and show that excellent agreement with the full solution for the ionization
degree can be obtained. We present the universal curve for the ionization degree versus the
(accordingly Z-scaled) electric field and compare with experimental data and the simpler ‘over
the barrier’ model.
2. Basic equations
The Thomas–Fermi energy functional for an atom with nuclear charge Z in an external electric
field, pointing in the z-direction E = E ez , reads in atomic units (e = me = h̄ = 1)
Z
Z
n(r 0 ) 3 0 Z
3
2 2/3
5/3
3
1
ETF [n(r )] = d r 10 3π
d r − + Ez
(1)
n(r ) + n(r ) 2
|r − r 0 |
r
where n(r ) is the electron density. The TF energy functional (1) can be considered as a
semiclassical limit of density functional theory with the kinetic energy taken in the local
density approximation (LDA) and with the exchange correlation term completely neglected.
The electrons interact through the electron gas pressure ∼ n(r )5/3 (due to the Pauli exclusion
principle) and the electron–electron interaction ∼ |r − r 0 |−1 only.
Variation of the energy functional (1) with respect to the electron density yields
Z
n(r 0 ) 3 0 Z
2 2/3
2/3
1
d r − + Ez + 80 = 0.
n(
r
)
+
(2)
3π
2
|r − r 0 |
r
The ionization degree is
Z
Z−N
q=
Z
With
Z
8(r ) = −
N=
n(r ) d3 r.
n(r 0 ) 3 0 Z
d r + − Ez
|r − r 0 |
r
(3)
(4)
equation (2) becomes
1
(3π 2 )2/3 n(r )2/3
2
− 8(r ) + 80 = 0
(5)
which is a relation between the electron density and potential.
In standard TFT ions have a finite radius r0 . The electric field breaks the spherical
symmetry. We therefore expect a deformed surface r0 (ϑ) where ϑ is the angle with respect
Ionization of the Thomas–Fermi atom in intense laser fields
831
to the electric field direction. Poisson’s equation for our problem reads like in ordinary
TFT
4
18(r ) = 4πn(r ) =
(2(8(r ) − 80 ))3/2
for 0 < r < r0 (ϑ).
(6)
3π
Switching to scaled entities,
2Z 1/3
r̄ = α r
α=4
(7)
9π 2
t¯ = α 3/2 Z 1/2 t
n̄ = α −3 Z −1 n
Ē = α −2 Z −1 E
(8)
Ē = Z −2 α −1 E
8̄ = α −1 Z −1 8
and introducing the screening function
χ(r̄ ) = r̄(8̄(r̄ ) − 8̄0 )
we obtain
(
(χ (ξ, w)/ξ )3/2
1 ∂2
1 ∂
2 ∂
(1
−
w
χ
(ξ,
w)
=
+
)
ξ ∂ξ 2 ξ 3 ∂w
∂w
0
(9)
for
0 < ξ < ξ0 (w)
for
ξ > ξ0 (w)
(10)
where ξ = r̄ and w = cos ϑ.
The asymptotic values for the screening function are
χ (0, w) = 1
lim χ(ξ, w) = −ξ 2 wĒ
ξ →∞
and the relation between the density and screening function reads

3/2

 1 χ(ξ, w)
for 0 < ξ < ξ0 (w)
n̄(ξ, w) = 4π
ξ


0
for ξ > ξ0 (w).
(11)
(12)
Solving the full equation (10) numerically is still a quite demanding task. To our knowledge it
was done for the first time by Susskind et al [9] using an alternating direction relaxation
approach, calculating the separatrix ξ0 (w) explicitly. We used our full time-dependent
Thomas–Fermi–Bloch hydrocode [11, 12] to calculate the charge state and found excellent
agreement with the results for the ionization degree in [9].
3. Approximate equations
Ten years ago Brewczyk and Gajda [10] proposed an approximation to equation (10) which
leads to a set of ordinary differential equations which is much easier to tackle than the partial
differential one (10). In fact, the numerical effort to solve this set of ordinary differential
equations is not much more than that for standard TFT. In the following we will review
Brewczyk and Gajda’s method in quite a lot of detail because the ionization degrees presented
by them in [10] differ considerably from the exact ones calculated by Susskind et al [9] and
ourselves. However, as we will show in section 4 we found very good agreement with the
exact results using the approximate model.
Considering the boundary conditions for the screening function (11) we may expand the
screening function for ξ > ξ0 (w), i.e. outside the charge cloud, as
"
#
∞
X
A`
χ (ξ, w) = ξ(8̄(ξ, w) − 8̄0 ) = ξ −Ēξ P1 (w) +
P` (w) − 8̄0 .
(13)
ξ `+1
`=0
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D Pitrelli et al
Here P` (w) is the Legendre polynomial of order `. By equating the scaled potential (4)
Z
Z
Z
n̄(ξ 0 , w0 ) 1
2
8̄(ξ, w) = − dξ 0 ξ 0
(14)
+ − Ēξ P1 (w)
dw0 dϕ 0
|r̄ − r̄ 0 | ξ
with the expression for 8̄(ξ, w) which can be obtained from (13) one finds the coefficients
Z
Z
N
χ(ξ 0 , w0 ) 3/2
0 02
0
1
= 1 − 2 dξ ξ
A0 = q = 1 −
(15)
dw
Z
ξ0
Z
Z
Z
χ(ξ 0 , w0 ) 3/2
3
dw0 P1 (w0 )
.
(16)
A1 = d3 r̄ z̄n̄(r̄ 0 ) = − 21 dξ 0 ξ 0
ξ0
Thus, A0 is the ionization degree and A1 is the dipole moment. Since we scaled all quantities
according to equations (7) and (8), A1 has to be divided by α to obtain the dipole in
atomic units. In the following, although all the mathematics is in scaled ‘TF units’, in the
plots we present all quantities in atomic units. So far there are no approximations in the
treatment.
We now expand χ (ξ, w) in a Taylor series with respect to 1w = w − w0 around a fixed
but arbitrary direction w0 ,
χ (ξ, w) = χ0 (ξ, w0 ) + χ1 (ξ, w0 )1w + 21 χ2 (ξ, w0 )1w2 + · · ·
(17)
where χ0 (ξ, w0 ) = χ(ξ, w0 ), χ1 (ξ, w0 ) = (∂χ /∂w)w=w0 , etc. Plugging expansion (17) into
(10) leads to a set of coupled ordinary differential equations. Here we give the result up to
χi (ξ, w0 ), i = 0 and 1 only (also keeping higher orders turned out to not be necessary),
∂ 2 χ0
3/2
= 2ξ −2 w0 χ1 + ξ −1/2 χ0
∂ξ 2
(18)
∂ 2 χ1
1/2
= 2ξ −2 χ1 + 23 ξ −1/2 χ0 χ1 .
(19)
∂ξ 2
To initialize the numerical integration of the set (18) and (19) we need information about
χ0 , χ1 and their derivatives near the origin ξ = 0. This can be achieved with the help of a
generalization of the well known Baker expansion [13] for small ξ ,
χ(ξ, w) = 1 +
∞
X
P` (w)
`=0
∞
X
ak` ξ k/2 .
(20)
k=2
Inserting this expansion into equation (10) we obtain relations for the coefficients ak` . The
coefficients ak0 are given, for example, in [8] or in the classic paper by Feynman et al [14],
a20 = to be fixed
a30 =
4
3
a40 = 0
a50 = 2a20 /5
(21)
....
The coefficient a20 is the derivative of the screening function at the origin and determines the
charge state in standard TFT. All other coefficients ak0 are either 0 or can be expressed in terms
of a20 . While in standard TFT all ak` with ` > 0 vanish this is different for the static field case
where
a21 = a31 = 0
a41 = to be fixed
a51 = a61 = 0
a71 = 2a41 /9
....
(22)
Here, a non-vanishing a41 corresponds to a certain electric field, causing a deviation from a
purely spherical screening function. Coefficients ak` for ` > 1 are of no interest in this paper
but the first non-vanishing ones for ` = 2 and 3 are listed in the original work by Brewczyk
and Gajda [10].
Ionization of the Thomas–Fermi atom in intense laser fields
833
The Baker expansion (20) together with the values for the first non-vanishing coefficients
(21) and (22) give
χ0 (ξ → 0, w0 ) ≈ 1 + a20 ξ + 43 ξ 3/2 + w0 a41 ξ 2
(23)
χ1 (ξ → 0, w0 ) ≈ a41 ξ
(24)
2
and expressions for the first derivatives accordingly.
We solved the system (18) and (19) subject to ‘initial values’ to be determined from (23)
and (24) sufficiently near the origin ξ = 0.
The electric field does not appear explicitly in equations (18) and (19) like in the standard
TFT differential equation the ionization degree does not appear but is connected to the boundary
condition at ξ = 0, i.e. the value of a20 . Therefore, what still remains to be clarified is how
one can obtain the electric field Ē from the screening function calculated for certain values of
a20 and a41 . Firstly, we note that we are interested in the lowest ionization degree possible for
a certain electric field. This implies that we have to look for the upper limit of all physically
meaningful a20 for a given a41 . A value above this upper limit would yield a diverging screening
function which is nowhere zero and therefore does not lead to a physical density (12). A value
below this upper limit corresponds to a TF ion in an electric field but the charge state of this ion
is higher than that the electric field would be able to create. This corresponds to the physical
situation where an already highly charged ion is placed in an electric field which is not strong
enough to ionize this ion further. Secondly, we know that we have to look for the upper limit
of a20 in the direction where the radius ξ0 (w) of the field-deformed TF ion is greatest. For
Ē < 0 this is the case for w = 1.
The electric field can be calculated through equation (13). Inserting the separatrix ξ0 (w)
into this expression gives, since χ (ξ0 (w), w) = 0,
8̄0 + Ēξ0 (w)P1 (w) −
A1
A0
−
P1 (w) − · · · = 0.
ξ0 (w) ξ0 (w)2
(25)
From the numerically calculated approximate screening function χ0 (ξ, w) for certain a20 and
a41 ionization degree A0 and dipole A1 can be calculated through equations (15) and (16).
Thus there remain the two unknowns 8̄0 and Ē for which we need two equations. For each
choice of two angles w1 and w2 6= w1 we obtain the two necessary equations such that finally
we can calculate the field by
Ē =
A0 (ξ1−1 − ξ2−1 ) + A1 (w1 ξ1−2 − w2 ξ2−2 )
ξ1 w1 − ξ2 w2
(26)
where ξi = ξ0 (wi ), i = 1, 2. An analogous expression can be obtained for the chemical
potential −8̄0 .
If the method was exact we would obtain the same electric field for each choice of w1 and
w2 . However, since we truncated the expansion (13) after the dipole term, the value we obtain
for the electric field depends on the pair of directions we choose. The variation in the field
with respect to the chosen directions gives us an estimate for the quality of our approximation.
4. Results
In figure 1 typical results for the screening function and separatrices ξ0 (w) are presented.
It is interesting to compare the shape of the TF density with the field-deformed Coulomb
potential. In figure 1(a) it is shown that the separatrix ξ0 (w) is much closer to the nucleus
than the Coulomb barrier, indicating that the TF ion is strongly confined and does not extend
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D Pitrelli et al
Figure 1. Results for the separatrix ξ0 (w) and for the screening function χ0 (z). All lengths are
rescaled to atomic units. (a) Shaded contour plot for the tilted Coulomb potential −1/r + Ez for
E = −0.004 1615 au. Superimposed are the contours corresponding to the ‘height’ of the Coulomb
barrier (full) and the separatrix ξ0 (w) for the same field strength (broken). In (b) the separatrix for
E = −0.000 7955 (full, obtained with a41 = 0.0001), −0.004 1615 (broken, a41 = 0.001) and
−0.024 0836 au (chain, a41 = 0.01) are shown. In (c) χ0 in the field direction is plotted for the
situation in (a).
up to the barrier. At first glance this seems counterintuitive, but the TF pressure acts in such
a way that a density distribution extending up to the barrier would be unstable. The contour
line corresponding to the value of the potential of the barrier is indicated in figure 1(a). In
figure 1(b) two other examples for the separatrix are shown as well. The stronger the field the
smaller the TF ion is (due to its higher ionization degree). In figure 1(c) the screening function
in the field direction is shown for the situation in figure 1(a).
4.1. Ionization degree versus electric field
With equation (26) we calculated the electric fields for a number of computer runs solving
equations (18) and (19) for various a41 , leading each time to a certain minimum ionization
degree A0 and dipole moment A1 . For each outcome we calculated for w1 , w2 ∈ [−1, 1],
w1 6= w2 a set of electric field values {E(w1 , w2 )}. Taking the minimum and the maximum
electric field value of this set we produced the two full curves shown in figure 2. There the
ionization degree q is plotted versus the electric field necessary to produce this charge state.
Results obtained by Susskind et al [9] for the full solution of (10) are also included. The
agreement is very good.
Brewczyk and Gajda [10] also produced a plot like our figure 2 (their figure 5). However,
there is a strong discrepancy with our results. This was also pointed out in the paper by
Susskind et al [9]. According to [10] one would need, for instance, a field E ≈ 0.023Z 5/3 au
Ionization of the Thomas–Fermi atom in intense laser fields
835
Figure 2. Ionization degree versus the electric field to produce it. The two full curves correspond
to the maximum and minimum values we obtained for the electric field for a given value a41 with
the approximate method. Results for the exact ionization degree are also shown (). The broken
curve is the fit with equation (27).
to ionize the TF atom by 3.5%. Instead from figure 2 we see that it is possible to ionize the TF
atom to 50% with this field strength.
In table 1 we give the numerical values corresponding to figure 2. Note that in table 1
all data are given in scaled TF units (see (7) and (8) for conversion to au). We notice that the
dipole |A1 | is always smaller than 0.078 TFu and that the radii max ξ0 and min ξ0 decrease
with increasing electric field, like TF ions in standard TFT do. The small polarizability
justifies the truncation of expansion (13). The fact that the dipole |A1 | decreases for increasing
field is due to ionization. It is a somehow spurious effect since the freed electron density
is not present in the static calculation, whereas in a fully time-dependent Bloch hydrocode
treatment it might be still in the simulation box, giving rise to a much greater dipole moment.
However, this unphysical aspect does not affect the self-consistent determination of the electric
field.
We would like to emphasize that, different from published work so far, the field strength
interval shown in figure 2 covers six orders of magnitude. The gap between the two curves in
figure 2 is an estimate for the accuracy of the approximate method. The difference between
the upper and lower limit for the electric field is acceptably small everywhere. For very low
and very high fields the two curves agree particularly well. This is connected to the behaviour
of the dipole moment |A1 | which is small for a low field and becomes small again for very
high fields because the remaining, non-ionized part of the electronic charge is tightly bound
and therefore difficult to polarize. Therefore, the approximate model does not suffer from a
restricted range of validity concerning the electric field strength. Of course, there remain the
well known deficiencies of TFT itself such as, for example, the poor treatment of the outermost
and the innermost electrons. In the context of this work the erroneous treatment of the inner
electrons is not important since realistic electric laser fields are not sufficiently strong to ionize
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D Pitrelli et al
Table 1. Input data and results obtained by solving the system (18) and (19) numerically.
a41
0
0.000 0001
0.000 001
0.000 005
0.000 010
0.000 050
0.000 1
0.000 5
0.001
0.005
0.01
0.05
0.1
0.2
0.5
1.0
2.0
3.0
10.0
a20
−1.588 071 023
−1.588 071 588
−1.588 076 160
−1.588 094 889
−1.588 117 174
−1.588 283 082
−1.588 478 647
−1.589 910 935
−1.591 572 396
−1.603 376 128
−1.616 635 945
−1.704 738 895
−1.796 478 806
−1.952 460 934
−2.321 310 750
−2.792 600 973
−3.512 395 703
−4.088 791 779
−6.795 758 691
max ξ0
min ξ0
q = A0
|A1 |
|Ē|
8̄0
∞
43.195
30.209
23.213
20.633
15.498
13.611
9.871
8.505
5.819
4.850
2.996
2.361
1.822
1.253
0.925
0.673
0.556
0.311
∞
23.757
16.316
12.337
10.879
7.997
6.949
4.896
4.159
2.741
2.245
1.330
1.031
0.784
0.532
0.389
0.282
0.232
0.129
0
0.0178
0.0386
0.0654
0.0816
0.1345
0.1654
0.2614
0.3143
0.4646
0.5388
0.7159
0.7854
0.8454
0.9065
0.9390
0.9614
0.9708
0.9876
0
0.039
0.055
0.066
0.070
0.077
0.078
0.075
0.070
0.053
0.043
0.021
0.014
0.008
0.004
0.002
0.001
0.001
0.000
0
5.879E−6
2.693E−5
7.931E−5
1.269E−4
3.856E−4
6.235E−4
1.966E−3
3.262E−3
1.097E−2
1.888E−2
7.060E−2
1.284E−1
2.380E−1
5.536E−1
1.068E00
2.084E00
3.094E00
1.013E01
0
6.670E−4
2.104E−3
4.694E−3
6.630E−3
1.477E−2
2.086E−2
4.640E−2
6.545E−2
1.453E−1
2.048E−1
4.540E−1
6.398E−1
9.020E−1
1.422E00
2.007E00
2.835E00
3.470E00
6.329E00
inner electrons for elements with Z > 15 anyway. The mistakes made for the outer electrons
could be accounted for with corrections to standard TFT, such as the Thomas–Fermi–Dirac–
Weizsäcker approximation (see, e.g., [15]). However, in this paper we will restrict ourselves
to standard TFT.
The field E(q) necessary to produce a certain ionization degree q (plotted in figure 2 as
q(E)) can be empirically fitted by
E(q) = 0.0238q 2 + 0.2q 4 + 0.3q 8 + 1.7q 16
(27)
−5/3
within the interval 0 6 EZ
< 0.2 (corresponding to 0 6 q < 0.85) where the fit lies
between the two curves of figure 2. The fit is included in figure 2 (broken curve).
4.2. Ionization degree versus Thomas–Fermi ionization potential
To compare with experimental results it is advantageous to plot the threshold intensity necessary
to create a certain ionization degree versus the ionization potential of the corresponding charge
state. Therefore, we need to calculate the TF ionization potential. Grout et al [16] gave precise
values for the total energy of positive TF ions over the entire 0 6 q 6 1 range. The unperturbed
(i.e. a41 = 0) total energy of a TF ion is (in au)
2 1/3
12
q2
ETF (Z, N) =
(28)
a20 +
Z 7/3
7 9π 2
ξ0
where a20 and ξ0 depend on q. It can be shown (see, e.g., [8]) that the energy (28) can be
written in the form ETF = Z 7/3 f (q) or, like in [16],
ETF (Z, N) = Z 2 N 1/3 F (q) = Z 7/3 (1 − q)1/3 F (q).
(29)
The function F (q) is tabulated in [16]. From the energy we can calculate the TF ionization
potential for a certain element with atomic number Z and charge state Z − N,
Ip (Z, N) = ETF (Z, N − 1) − ETF (Z, N ).
(30)
Ionization of the Thomas–Fermi atom in intense laser fields
837
Figure 3. Ionization degree versus the ionization potential (31). , 80 values for the TF atom in
the static field (corresponding to 8̄0 = α −1 Z −1 80 in table 1).
Unfortunately, Ip (Z, N) is not a universal curve, i.e. it cannot be written as Ip (Z, q) = Z γ g(q)
with γ fixed and g(q) a function of q alone. However, making an 1/Z expansion the ionization
potential can be expressed as
∂ ETF
= −µ
Ip (Z, N ) ≈ Ip (Z, q) = Z 4/3 (1 − q)1/3 F 0 (q) − 13 (1 − q)−2/3 F (q) = −
∂N
(31)
where F 0 = ∂F /∂q. Thus the negative chemical potential −µ is a good approximation to the
ionization potential for q Z −2 . Since the smallest q of interest is Z −1 , the approximation
can be expected to be good already from Z ≈ 10 on. Since for smaller Z TFT is not reliable
anyway the approximation (31) adds no additional limits. For a more detailed discussion
on the relation between chemical potential, electron affinity, electronegativity and ionization
potential see, for example, the textbook by Parr and Yang [15]. The ionization degree q
versus the ionization potential (31) is shown in figure 3. If one compares the positive TF
ions’ chemical potential, which can be calculated through µ = −Zαq/ξ0 from the data
given in [16], with 80 = αZ 8̄0 from table 1 (included as in the plot) one finds excellent
agreement for all q, indicating once again that the remaining TF ion in the field is not strongly
deformed.
4.3. Comparison with the ‘over the barrier ionization’ model
In [6] a simple semiclassical model was introduced to explain experimentally measured
appearance intensities. Equating the height of the Coulomb barrier to the ionization potential
Ip one finds the appearance intensity
2
(Z ∗ ) =
Iapp (Z ∗ ) = Eapp
Ip4
16Z ∗ 2
(32)
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D Pitrelli et al
Figure 4. Comparison of the TF result and the ‘over the barrier’ ionization model. In (a) the
necessary intensity to remove all electrons up to a certain ionization potential is plotted versus this
ionization potential (full curve, TF result; +, OBI result for Ne; ∗, OBI for Ar; 4, OBI for Kr; ,
OBI for Xe). In (b) the same is plotted versus the ionization degree.
where Z ∗ is the charge state under consideration, for example, for Ne one would take Z ∗ = 1
and Ip = 0.79 au for the first electron, Z ∗ = 2 and Ip = 1.51 for the second electron, etc.
Thus the ‘over the barrier ionization’ (OBI) model is ‘semiclassical’ since the correct ionization
potentials are input data.
In figure 4 for Ne, Ar, Kr and Xe the prediction of the OBI model for the appearance
intensity is compared with the intensity I necessary to ionize the TF atom up to the same
ionization potential (a) and to the same ionization degree (b). We observe that in figure 4(a)
the agreement becomes better with increasing charge state. This is due to the fact that the
outer electrons are not correctly treated in TFT; they are too easily removed. From the fifth
electron on the agreement becomes quite good. In figure 4(b) the same data are plotted
versus the ionization degree q. The difference is that in (a) we compared the OBI data with
the results for the TF atom with the same ionization potential, whereas in (b) we compare
with the TF atom of corresponding ionization degree q. The disagreement with the OBI
model is huge in the latter case since one does not compensate for the fact that the ionization
energies predicted by pure TFT are different from the exact ones for the elements under
consideration.
4.4. Comparison with experimental results and similar work
In figure 5 the TF prediction for the intensity necessary to ionize up to a certain charge state is
compared with measured appearance and saturation intensities. The experimental data were
taken from [17, 18] where results for the ionization of rare gas elements with 200 fs, 800 nm
Ti:sapphire laser pulses are presented. As in the previous figure 4 in (a) the comparison is made
with respect to the ionization potential, whereas in (b) it is made with respect to the ionization
Ionization of the Thomas–Fermi atom in intense laser fields
839
Figure 5. Comparison of the TF result and experimental results. Experimental results from
[17, 18] (800 nm, 200 fs pulse) for rare gas atoms are plotted. +, Ne; ∗, Ar; 4, Kr; , Xe. The
connected symbols at higher intensity are saturation intensities of the corresponding charge state,
whereas the same symbols at lower intensity indicate the ‘appearance intensity’, here defined as
the intensity where the measured ion yield is 10−3 times the saturation yield.
degree. Again, in (b) there is a strong disagreement due to the error in the TF ionization
energies. In (a) the TF result (full curve) in most cases lies between the appearance intensity
(here defined as 10−3 times the saturation intensity) and the saturation intensity. The agreement
becomes better for elements with higher Z and with increasing charge state. Unfortunately
there are no experimental results available for higher ionization degrees where we expect TFT
to become more reliable.
Note that there is almost one order of magnitude in intensity between the appearance of
a certain charge state and the saturation of the same. Moreover, the appearance intensity is
an ambiguous concept, depending on the experimental apparatus, while the saturation also
depends on the appearance of the next charge state. From a purely static viewpoint, i.e.
from the stationary TF solution or a simple OBI approach, it is not possible to determine
accurate ionization rates. On the other hand, to predict the entire ionization yield probability
distribution, as measured, for example, in [17, 18], knowledge of the ionization rate and the
pulse is necessary. Therefore, every static theory that predicts threshold intensities between
the measured appearance intensity and saturation can claim to be successful. This is the
case for both OBI and TFT. TFT has the advantage of delivering one universal curve, to
be scaled to the desired Z, but it fails for the outermost electrons and elements with too
low Z.
The static TF treatment can only be expected to be applicable to the time-dependent case
if the laser frequency is low compared to inner atomic frequencies of the order of 1 au and
rescattering effects do not play an important role. For optical frequencies this is the case. In
fact, we believe that the disagreements in figure 5 are not connected to the breakdown of the
840
D Pitrelli et al
Figure 6.
Comparison of the TF result and experimental results for lower frequencies.
Experimental appearance intensities from [19] (9.55 µm, 1.1 ns pulse) for Ar (∗), Kr (4) and
Xe (), the symbols for each rare gas element connected with a full curve. The Ar data connected
with dotted curves are results from [18] (1053 nm, 600 fs) for the appearance intensity and saturation
intensity, respectively. The broken curve is the TF result in [9].
adiabatic viewpoint but are due to the well known limitations already present in standard TFT.
In order to underpin this we present in figure 6(a) a comparison with experimental results for
lower frequencies from [19] (9.55 µm, 1.1 ns pulse) and [18] (1053 nm, 600 fs). There is
no significant difference to the 800 nm results in figure 5 as far as the accuracy of the static
field TF approach is concerned. This is in contrast to results presented in [20] where static
field TF results are compared with experimental data from [6] (higher frequency) and [19]
(low frequency). In [20] the authors came to the conclusion that the Nd:YAG measurements
presented in [6] are already in a regime where the adiabatic approximation is no longer valid,
at least for the first three charge states of the rare gases Ne, Ar, Kr and Xe. They justified
this statement by presenting a graph (their figure 1) where the measured data are compared
with the prediction of static field TFT, referring to their approximate method proposed in [10]
(also discussed extensively in the current paper). However, it seems that they forgot to rescale
the electric field values they determined from the curve in figure 5 of [10] according to the
element and charge state under consideration. If one does the rescaling correctly one finds a
huge disagreement for all the data points since the TF result for q versus the electric field in
figure 5 of [10] itself disagrees by one order of magnitude in q with our result in figure 2, as
discussed already in section 4.1.
The only other comparison of experimental data with the static field TF result we are
aware of is made in [9]. However, their figure 3 of the intensity versus the ionization potential
differs from ours. In figure 6(a) their result is included (broken curve). This results in the
fact that in their final comparison with experimental data from [6], presented in their figure 4,
static TFT agrees better with He and Ne than with Xe which is contrary to both the usual TF
behaviour and our results.
Ionization of the Thomas–Fermi atom in intense laser fields
841
5. Summary and conclusion
In summary, we have reviewed Thomas–Fermi theory in a static electric field of arbitrary
strength. In particular, we revisited an approximate method to solve this problem. This method
was proposed ten years ago but due to an apparent discrepancy between results obtained with
this method and both experimental data and the exact numerical results of the static field
Thomas–Fermi problem, it seemed to be have been dismissed or not recognized at all. In this
paper we demonstrated that the method actually works well and is valid for all field strengths. It
reduces the numerical effort necessary to solve the static field Thomas–Fermi problem almost
to that of standard Thomas–Fermi theory, i.e. to the solution of ordinary differential equations
instead of partial differential ones. Comparison of the TF predicted laser intensity, necessary
to ionize up to a certain ionization potential, with results from the simple ‘over the barrier
ionization’ model and experimental appearance and saturation intensities were presented. The
strong point of Thomas–Fermi theory is to produce universal results which can be simply
scaled to the element and charge state under consideration. Moreover, the self-consistent
deformation of the charge cloud is taken into account. Weak points are mainly due to the well
known deficiencies already inherent in standard Thomas–Fermi theory: the outer electrons
are not correctly treated which leads to an underestimated intensity necessary to remove the
first few electrons of an atom and the Thomas–Fermi energies and ionization potentials are
not sufficiently accurate. For higher charge states and elements which are not too light the
Thomas–Fermi predictions merge with the simpler ‘over the barrier ionization’ results. The
comparison with experimental appearance and saturation intensities indicates that the adiabatic
assumption in which the laser is considered slow with respect to the inner atomic dynamics is
appropriate, at least for wavelengths > 800 nm. Results for the full time-dependent study of a
Thomas–Fermi atom in a laser field through Bloch’s hydrodynamic model will be discussed
in a forthcoming paper.
Acknowledgments
Helpful discussions with Professor B U Felderhof are gratefully acknowledged. This work
was supported in part by the European Commission through the TMR Network SILASI
(Superintense Laser Pulse–Solid Interaction), no ERBFMRX-CT96-0043, and by the Deutsche
Forschungsgemeinschaft under contract no MU 682/3-1.
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