Strong oscillations in the photoionization of 5s electrons in Xe@C
60
endohedral atoms
M. Ya. Amusia,
1,2
A. S. Baltenkov,
1,3,4
and U. Becker
4
1
Hebrew University of Jerusalem, Racah Institute of Physics, 91904 Jerusalem, Israel
2
Ioffe Physical-Technical Institute, 194021 St. Peterburg, Russia
3
Arifov Institute of Electronics, Akademgorodok, 700143 Tashkent, Uzbekistan
4
Fritz-Haber-Institute der Max-Planck-Gesellschaft, 14195 Berlin, Germany
共Received 19 November 1999; published 5 June 2000兲
The photoionization cross sections of 5s electrons in Xe atom in the Xe@C
60
endohedral complex have been
calculated. It is shown that within the wide frequency region, from the threshold to the giant resonance, the
cross section is qualitatively modified, acquiring additional resonances due to inner reflection of the photo-
electron from the surrounding carbon atoms. The strong dependence of this effect upon the radius of the
fullerene shell has been predicted.
PACS number共s兲: 33.80.Eh, 36.40.Cg
At present there are many papers devoted to the study of
the properties of endohedral atoms 关1兴, i.e., atoms located
inside the C
60
fullerene cage. The number of different atoms
which may be introduced into C
60
continuously increases and
has already reached about one-third of the Periodic Table
including the rare gases 关2,3兴. The unusual geometrical struc-
ture of endohedrals makes them extremely attractive objects
to study. The analysis of elementary processes of these par-
ticles will provide information on their very specific struc-
ture, which is affected to a large extent by the mutual influ-
ence of the C
60
cage and the inner atom. To understand this
influence, and for the development of experimental photo-
electron spectroscopy of these giant molecules, the compari-
son between photoionization spectra of an isolated M atom
and the same atom, but inside the endohedral M@C
60
,isof
great interest. Such studies would make it possible to analyze
the influence of a fullerene shell on optical characteristics of
an atom located inside the endohedral. From the other side,
the analysis of the carbon electron spectra from M@C
60
would permit us to investigate the opposite action—that
from the atom M upon C
60
. This is a very complicated the-
oretical problem, more complex by orders of magnitude than
that of an isolated atom. Therefore, considerable simplifica-
tions are inevitable. They are suggested, for example, in pa-
pers 关4–6兴, where the photoabsorption spectrum of atoms
inside the C
60
molecule was calculated using a jelliumlike
model for the fullerene shell.
The aim of the present paper is to study of the photoion-
ization of the Xe 5s
2
electrons when the Xe atom is located
inside the C
60
fullerene. Recently we have suggested a
simple model 关7兴, which in spite of its simplicity predicted
an important qualitative feature of the electron scattering in
the field of the C
60
shell—the cross-section oscillation due to
electron wave reflection from the fullerene carbon atoms.
This model was applied in 关8,9兴 to consider the photoioniza-
tion of M atoms inside the M@C
60
endohedral complexes. It
was demonstrated that the reflection of an electron from a
fullerene shell leads to resonances in the M-atom photoion-
ization cross sections, which without the C
60
shell would be
more or less structureless and smooth. Therefore, it is par-
ticularly interesting to see how the frequency dependence of
the 5s
2
-subshell photoionization cross section of the Xe
atom is modified in the Xe@C
60
endohedral. For an isolated
Xe atom it is strongly affected by the outer 5p
6
electrons and
particularly by the giant resonance in the 4d
10
subshell
关10,11兴. In this paper we want to demonstrate a prominent
effect which comes from the C
60
action—a kind of a resona-
tor which surrounds Xe in Xe@C
60
.
The analysis of the influence of the surrounding atoms on
the endohedral photoionization may be simplified consider-
ably by replacing the real potential of C
60
, formed as a su-
perposition of the constituent atomic fields, by a model po-
tential V(r) of the fullerene shell. It is quite natural to
assume, as it is usually done, that this potential is equal to
zero in all space except for the spherical shell formed by the
partially delocalized valence electron of the carbon atoms.
Inside this shell the function V(r) is nonzero and can be
approximated by different functions 关12–14兴. The param-
eters of this potential can be found, for example, using the
experimental data for the negative C
60
⫺
ion 关7兴. It is known
that the electron affinity in this ion is I⬇2.65 eV, while the
radius of the fullerene shell is R⬇6.64a
0
关15兴, a
0
being the
Bohr radius. The thickness of the carbon atom shell is ap-
proximately 2a
0
. Using these data, we may estimate the
depth of the fullerene shell potential well using the following
simple model 关12,13兴:
V
共
r
兲
⫽
再
⫺ V
0
for r
i
⭐r⭐r
o
0 elswhere
共1兲
where r
i
and r
o
are the inner and outer radii of the shell, and
⌬r⫽ r
o
⫺ r
i
is its thickness. The wave function of the addi-
tional electron of a negative ion in this spherically symmetric
well has the form
(r
ជ
)⫽
关
l
(r)/r
兴
Y
lm
(
,
). Its radial part
in the case of the s ground state is given by the expression
关5兴
0
共
r
兲
⬀
再
exp
共
r
兲
⫺ exp
共
⫺
r
兲
, r⭐r
i
exp
共
⫺
r
兲
, r⭓r
o
,
共2兲
where
is the wave vector
⫽
冑
2I. Note that throughout
this paper we use atomic units: ប⫽ m⫽ e⫽ 1.
PHYSICAL REVIEW A, VOLUME 62, 012701
1050-2947/2000/62共1兲/012701共4兲/$15.00 ©2000 The American Physical Society62 012701-1
Inside the fullerene shell the wave function
0
(r) has the
form
0
共
r
兲
⬀sin qr⫹ C cos qr. 共3兲
The constant C and the wave vector q⫽
冑
2(V
0
⫺ I) are de-
termined by the equality of the logarithmic derivatives of the
wave functions on the inner L
i
⫽
coth
r
i
and outer L
o
⫽⫺
surfaces of the fullerene shell:
coth
r
i
⫽ q
1⫺ C tan qr
i
C⫹ tan qr
i
, ⫺
⫽ q
1⫺ C tan qr
o
C⫹ tan qr
o
. 共4兲
The system of equations 共4兲 is transformed into the following
transcendental equation:
q
共
L
o
⫺ L
i
兲
⫹
共
q
2
⫹ L
o
L
i
兲
tan q⌬r⫽ 0, 共5兲
with a constraint: tan qr
i
tan qr
o
⫹1⫽0. The solution of Eq.
共5兲 for r
i
⬇5.75 and r
o
⬇7.64 关12,13兴 and
⬇0.441 关7兴 leads
to the following results: q⬇0.641 and C⬇0.443. This gives
for the depth of the potential well V
0
the following value:
V
0
⬇8.24 eV. This is small compared to the ionization po-
tential of the inner subshells of the M atom, and the potential
V(r) is located well outside the localized inner shells.
Hence, we can neglect the influence of the fullerene shell on
the wave function of the ground state, and should only take
into account the action of the potential V(r) upon the elec-
tron continuum states.
For the considered parameters of the potential V(r), the
logarithmic derivatives are almost independent of the par-
ticular choice of r
i
and r
o
: L
i
⬇
and L
o
⫽⫺
. Therefore,
they can be considered as localized at the point R and the
potential well, which in fact has the nonzero thickness ⌬r,
and can be replaced by a
␦
potential: V(r
ជ
)⫽⫺A
␦
(r⫺ R), as
it was suggested earlier 关7,16兴. The strength of the
␦
poten-
tial A is connected to the jump of the logarithmic derivative
⌬L⫽ L
o
⫺ L
i
by the relation A⫽⫺⌬L 关7,8兴. In this paper we
will use the model potential V(r
ជ
)⫽⫺A
␦
(r⫺ R) to describe
the photoionization of the atoms located inside the fullerene
shell.
Suppose that we know the regular u
kl
(r) and irregular
v
kl
(r) solutions of the radial Schro
¨
dinger equation for the
optical electron of the free M atom
1
2
冉
u
kl
⬙
⫺
l
共
l⫹ 1
兲
r
2
u
kl
冊
⫹
关
E⫺ U
共
r
兲
兴
u
kl
⫽ 0. 共6兲
The functions
v
kl
(r) obey the same equation in all space
except for r⫽ 0. Here U(r) is the self-consistent field created
by the atomic nucleus together with all atomic electrons,
acting upon the optical electron. The functions u
kl
(r) and
v
kl
(r)atkrⰇ1 are of the following form:
u
kl
共
r
兲
⬀sin
冉
kr⫹
1
k
ln 2kr⫺
l
2
⫹ ⌬
l
共
k
兲
冊
,
共7兲
v
kl
共
r
兲
⬀⫺ cos
冉
kr⫹
1
k
ln 2kr⫺
l
2
⫹ ⌬
l
共
k
兲
冊
,
where ⌬
l
(k) are the phase shifts of the wave functions in the
potential field of the M
⫹
ion.
Let us represent the Xe@C
60
endohedral as a Xe atom
located in the center of the fullerene shell. The radial parts of
the electron wave functions of the endohedral atom are de-
termined by the equation
1
2
冉
kl
⬙
⫺
l
共
l⫹ 1
兲
r
2
kl
冊
⫹
关
A
␦
共
r⫺ R
兲
⫺ U
共
r
兲
⫹ E
兴
kl
⫽ 0.
共8兲
The wave function of the Xe 5s
2
electrons is located rela-
tively close to the coordinates origin, within a region of ap-
proximate size which can be estimated as ⬃(I
5s
/Ry)
⫺ 1/2
⬍ R, where I
5s
⬇1.712 Ry is the ionization potential of the
Xe 5s
2
subshell. The binding energy of the 5s level is nearly
three times greater than the depth of the potential well V
0
.
Therefore, we will consider the wave function of the 5s elec-
tron of the endohedral Xe atom as coinciding with that of the
isolated Xe atom
5s
(r)⬇u
5s
(r).
The situation is quite different for the wave functions of
the continuous spectrum. For r⬍ R they are proportional to
the atomic wave functions, with the proportionality coeffi-
cient D
l
(k) being only k-dependent:
kl
(r)⫽ D
l
(k)u
kl
(r).
Outside the shell, i.e., for r⭓R, they become linear combi-
nations of the regular and irregular solutions of the Schro
¨
-
dinger equation 共6兲. By matching the logarithmic derivatives
of the function
kl
(r)atr⫽ R, we obtain, for the wave-
function amplitudes D
l
(k) inside the fullerene shell and the
photoelectron phase shifts
␦
l
(k) on the V(r) potential, the
following formulas:
D
l
共
k
兲
⫽ cos
␦
l
共
k
兲
冉
1⫺ tan
␦
l
共
k
兲
v
kl
共
R
兲
u
kl
共
R
兲
冊
, 共9兲
tan
␦
l
共
k
兲
⫽
u
kl
2
共
R
兲
u
kl
共
R
兲
v
kl
共
R
兲
⫹ k/2A
. 共10兲
The photoionization amplitude of the Xe atom 5s
2
sub-
shell is defined by the matrix element
具
kp
兩
r
兩
5s
典
, calculated
using the wave function
5s
(r) of the ground state of the Xe
atom and the continuum wave function
kl
(r) of the electron
with momentum k. The main contribution to this matrix
comes from the inner region of the fullerene sphere. There-
fore, we represent it as
具
kp
兩
r
兩
5s
典
⬇D
1
共
k
兲
冕
0
R
u
kp
共
r
兲
ru
5s
共
r
兲
dr⫽ D
1
共
k
兲
R
1
.
共11兲
The integral in Eq. 共11兲, R
1
, is the photoionization ampli-
tude of the 5s
2
subshell of the isolated Xe atom. Hence, the
partial cross sections
l⫾1
(
) of the endohedral atom photo-
ionization, up to a k-dependent factor D
1
2
(k), coincide with
the partial cross sections
l⫾1
a
(
) of the isolated Xe atom
5s
共
兲
⫽ D
1
2
共
k
兲
5s
a
共
兲
, 共12兲
where
⫽ I
5s
⫹ k
2
/2 is the photon energy.
M. YA. AMUSIA, A. S. BALTENKOV, AND U. BECKER PHYSICAL REVIEW A 62 012701
012701-2
Equation 共12兲 connects the partial photoionization cross
section of the Xe@C
60
endohedral and the free Xe atom with
the parameters of the fullerene shell, namely the radius R and
the electron affinity I, of the C
60
⫺
. Due to the connection
between oscillations of the wave functions inside and outside
the fullerene shell, the amplitudes D
l
(k) have a resonance
character. This is why in the photoionization cross sections
of endohedrals the resonance behavior appears, resulting
from the inner reflections of photoelectrons from the
fullerene shell 关4,5,8,13兴.
Taking into account Eqs. 共9兲 and 共10兲, the squares of the
amplitudes D
l
(k) can be written in the form
D
l
2
共
k
兲
⫽
共
k/2A
兲
2
关
u
kl
共
R
兲
v
kl
共
R
兲
⫹ k/2A
兴
2
⫹ u
kl
4
共
R
兲
. 共13兲
According to Eq. 共13兲, for low strength of the
␦
potential
(A→ 0), the amplitude D
l
(k) behaves as D
l
2
(k)→ 1. For
high strength (A→ ⬁) the amplitudes D
l
2
(k) are different
from zero only for k, which obeys the condition u
kl
(R)⫽ 0.
Thus, the stronger the
␦
-sphere potential is, the more pro-
nounced the resonance effects are.
The calculation results for the 5s
2
subshell of the free Xe
atom and endohedral atom in the Xe@C
60
complex are pre-
sented in Fig. 1. The wave functions of the isolated atoms in
the ground state u
nl
(r) and in the continuum u
kl
(r) were
calculated in the one-electron Hartree-Fock approximation
using the package of computing codes 关17兴. The irregular
solutions
v
kl
(r) were determined by a known formula
v
kl
共
r
兲
⫽ u
kl
共
r
兲
k
冕
dr
u
kl
2
共
r
兲
. 共14兲
Using these functions, the coefficients D
1
2
(k) determined by
Eq. 共13兲 were calculated. The following parameters of the
fullerene shell corresponding to the empty C
60
were used:
R⫽ 6.64a
0
and
⫽
冑
2.65/13.6⬇0.441.
As it is well known, the subvalent 5s
2
subshell of the Xe
atom is under extremely strong influence from the neighbor-
ing 5p
6
and 4d
10
electrons 共see 关10兴 and references therein兲.
Indeed, it has been known since long ago that the calculation
of
5s
a
(
) in the one-electron approximation leads to incor-
rect results. To determine the cross section precisely enough,
one must take into account the intershell correlations in the
framework of random-phase approximation with exchange
共RPAE兲; it is the virtual excitations of 5p
6
and 4d
10
elec-
trons as it was described in 关10兴. The multielectron correla-
tions qualitatively alter the frequency dependence of the
photoionization cross section of the 5s
2
subshell, thus lead-
ing to the appearance of a new minimum at about
⬇2.5 Ry
and a maximum at
⬇6.5 Ry, the latter being tightly bound
to the giant resonance in photoabsorption by 4d
10
electrons.
The photoionization cross section
5s
a
(
) of a free Xe atom
关10兴 is presented in Fig. 1 by a dashed line. The solid line is
the photoionization cross section
5s
(
) of the endohedral
Xe atom calculated with Eq. 共12兲. As seen from this figure,
the photoelectron reflection from the fullerene shell
drastically modifies the frequency dependence of the
5s
2
-photoabsorption cross section in a very broad frequency
region, up to the giant resonance domain. Additional reso-
nances in the cross section appear, namely at least one new
minimum and maximum. At the resonance energies, the
cross section of the endohedral atom is up to two to three
times greater than that of the free Xe atom.
The calculations of the coefficients D
1
2
(
⑀
) show that the
locations of resonance peaks on the photoelectron energy
⑀
scale are very sensitive to the magnitude of the fullerene
shell radius R. The results of D
1
2
(
⑀
) calculations for three
values of R, which are different from the radius of the empty
C
60
by less than one Bohr radius, are presented in Fig. 2. The
differences between these curves in the amplitude values and
in the peak locations become more pronounced with increas-
ing photon energy. This result is very important and interest-
FIG. 1. Photoionization cross section of Xe 5s
2
electrons. Solid
line, endohedral Xe@C
60
complex; dashed line, free Xe atom.
FIG. 2. The dependencies of D
1
2
(
⑀
) upon fullerenes radius R.
STRONG OSCILLATIONS IN THE PHOTOIONIZATION . . . PHYSICAL REVIEW A 62 012701
012701-3
ing for the photoelectron spectroscopy of the fullerenes with
heavy atoms inside the carbon cage. It can be expected that
the insertion of these atoms into C
60
results in an increase of
the geometrical size of the fullerene shell. Experimental
studies investigating the location of the resonance peak can
indicate that the fullerene shell radius increased after the in-
sertion of an atom inside.
In summary, we have demonstrated that the photoioniza-
tion cross sections of 5s
2
electrons of the Xe atom inside C
60
have prominent resonance structure in a broad frequency re-
gion. The experimental study of this structure by photoelec-
tron spectroscopy methods can be currently performed, and
very useful valuable information on the endohedral system,
in particular on their radii, can be obtained.
One of us 共A.S.B.兲 acknowledges the financial support of
the Deutscher Akademischer Austauschdienst 共DAAD兲 and
useful comments by V. Pikhut.
关1兴 J. Fink, T. Picher, M. Knupfer, M. S. Golden, S. Haffuer, R.
Frudlein, U. Kirbach, P. Kuran, and L. Dunsch, Carbon 36,
625 共1998兲.
关2兴 M. Saunders, H. A. Jimenez-Vazquez, R. J. Cross, S. Mrocz-
kowski, M. L. Gross, D. E. Giblin, and R. J. Poreda, J. Am.
Chem. Soc. 116, 2193 共1994兲.
关3兴 T. Ohtsuki, K. Ohno, K. Shiga, Y. Kawazoe, Y. Maruyama,
and K. Masumoto, Phys. Rev. Lett. 81, 967 共1998兲.
关4兴 M. J. Puska and R. M. Nieminen, Phys. Rev. A 47, 1181
共1993兲.
关5兴 G. Wendin and B. Wastberg, Phys. Rev. B 48, 14 764 共1993兲.
关6兴 J. Luberek and G. Wendin, Chem. Phys. Lett. 248, 147 共1996兲.
关7兴 M. Ya. Amusia, A. S. Baltenkov, and B. G. Krakov, Phys.
Lett. A 243,99共1998兲.
关8兴 A. S. Baltenkov, Phys. Lett. A 254, 203 共1999兲.
关9兴 A. S. Baltenkov, J. Phys. B 32, 2745 共1999兲.
关10兴 M. Ya. Amusia, Atomic Photoeffect 共Plenum Press, New York,
1990兲.
关11兴 U. Becker and D. A. Shirley, in VUV-and Soft X-Ray Photo-
ionization, edited by U. Becker and D. A. Shirley 共Plenum,
New York, 1996兲,p.13.
关12兴 Y. B. Xu, M. Q. Tan, and U. Becker, Phys. Rev. Lett. 76, 3538
共1996兲.
关13兴 J.-P. Connerade, V. K. Dolmatov, P. A. Lakshmi, and S. T.
Manson, J. Phys. B 32, L239 共1999兲.
关14兴 O. Frank and J. Rost, Chem. Phys. Lett. 271, 367 共1997兲.
关15兴 E. Tosatti and N. Manini, Chem. Phys. Lett. 223,61共1994兲.
关16兴 L. L. Lohr and S. M. Blinder, Chem. Phys. Lett. 198, 100
共1992兲.
关17兴 M. Ya. Amusia and L. V. Chernysheva, Computation of
Atomic Processes 共IOP Publishing Ltd, Bristol, 1997兲.
M. YA. AMUSIA, A. S. BALTENKOV, AND U. BECKER PHYSICAL REVIEW A 62 012701
012701-4
