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Resonances in the photoionization cross section of [email protected]60 endohedrals
12 April 1999
PHYSICS
LETTERS
A
Physics Letters A 254 ( 1999) 203-209
ELSEVIER
Resonances in the photoionization cross section of M @C60 endohedrals A.S. Baltenkov Arifov Institute Received
I5 October
of Electronics, Akadetngorodok. 700143 Tashkent. Uzbekistan
1998; revised manuscript received 4 January 1999; accepted Communicated by B. Fricke
for publication
2 February
1999
Abstract A model for the calculation of photoionization cross sections of an atom introduced inside the fullerene cage has been proposed. A comparative investigation of the photoionization cross sections of an isolated M atom and the same one in the M@Ceo endohedral has been carried out. It has been shown that the resonance peaks in the photoionization cross sections of the endohedral appear near the thresholds of ionization of atomic subshells. The positions of these peaks on the scale of the photoelectron energy are defined by the fullerene shell radius and the affinity energy of the electron to GO. @ 1999 Elsevier Science B.V.
Among the variety of clusters, carbon clusters are of special interest. Clusters of this type form a perfect geometrical structure. The most well known is the Cen cluster. This multi-atomic system is a nearly spherical shell formed by carbon atoms. The closeness of the atomic structure of Cm fullerene results in high stability and non-reactivity of these particles, which makes it possible to use them as containers for atoms of different kinds. Compounds of this type were called endohedrals and labeled as M@Cm unlike to usual chemical compounds of MCm, where an attached M atom is located outside the fullerene cage. The unusual geometrical structure of fullerenes and endohedrals make them extremely attractive to study. The analysis of elementary processes with these particles will provide information on the unique nature of these formations and on their electronic structure. The results of investigations will not only broaden existing ideas about these giant molecules, but also be used for practical problems such as formation of controlled
beams of these particles and formation from them of surface or bulk crystals. Current experimental investigations of C60 and fullerene-like objects are carried out using various methods, including photoelectron and electron spectroscopy. By means of these methods, in particular, well-founded evidence was obtained that a M atom in M@C60 endohedral is located inside the fullerene sphere, rather than on the external surface of the fullerene skeleton [ 1,2]. For the development of photoelectron spectroscopy of these giant molecules, the comparison of photoionization spectra of an isolated M atom and the same atom introduced inside the M@C6n endohedral is of great interest. This study will make it possible to analyze the influence of a fullerene cage on optical characteristics of an atom located inside endohedral. In the present paper the comparative investigation of photoionization of isolated atoms and corresponding endohedrals have been carried out and resonance peculiarities due to fullerene shell in cross
0375-9601/99/$ - see front matter @ 1999 Elsevier Science B.V. All rights reserved. PII SO375-9601(99)00095-X
204
A.S.
Balrenkov/Physics
sections of these processes have been analyzed. Two approaches are mainly used to calculate the electronic structure of fullerene-like systems. The first is the molecular calculation of the wave functions from the first principles [ 31. The use of ab initio calculation methods becomes practically impossible when solving problems where it is necessary to know the continuous spectra wave functions. The second approach is the approximation of the potential of the fullerene cage by some model potential well. The maximal simplification can be reached if we replace the real potential of C60, formed by the superposition of the atomic fields, by the model potential of the fullerene shell V(r). It is usually supposed that this potential is equal to zero in all space except the spherical layer formed by the smeared carbon atoms. Inside this layer the function V(r) is different from zero and simulated by the different potential functions [4,5]. One of the potentials of this type was used in papers [6,7] to describe some processes with negative C, ions. In articles [6,7] it was supposed that electron interaction with the fullerene shell can be described by a &like potential which is different from zero within an infinitely thin spherical layer with the radius R equal to the fullerene radius. This model can be used when the lengths of electron waves are much greater than the “thickness” of the fullerene shell. In this case the detailed form of the function V(r) is not important, because the electron is subject to the averaged action of the potential. It is this situation in C, where the affinity energy of the electron to the neutral fullerene I is comparatively low on atomic scale (I = 2.65 eV [ 81) and, therefore, most time the extra electron is moving in regions where we can consider it as free. That allows us to determine the wave function of electron from the boundary conditions. The situation here is similar to one in nuclear physics, where the weakly bounded nuclear system - deuteron - can be described by a logarithmic derivative of the wave function at zero [9] and this derivative is defined by the binding energy of deuteron, which is known from experiment. In the case of C,, the boundary condition for the electron wave function is not given at zero, but on the sphere with R radius. As any idealization, the model [6,7] has, naturally, the limited domain of applicability. It can be used to calculate the physical values which are determined by the large distances from the fullerene shell. These can be the photoionization cross sections
Letter.7 A 254
(1999)
203-209
of the negative Cc0 ion near the threshold, polarizability and magnetic susceptibility of these ions, etc. Let us use this model potential to describe the photoionization of M atoms located in the fullerene cage near the threshold of this process. We shall represent the M@C6a endohedral as a corresponding isolated atom located in the center of the fullerene shell that is described by the &like potential. The wave functions of an optical electron of the M atom in this spherically symmetrical well have the form fl(r) = [~c(T)/Y]Y,~~(B,(D). The radial part of the wave function of this electron is defined by the equation (throughout this paper we use atomic units, h=??r=e=l) 1 2
xl” -
1tz+ I)
-XI r2
I
+
[A@r - R) - U(r) +
=o.
,qX,
(1)
Here, U(r) is the potential energy of an optical electron in an isolated atom, A is the power parameter of the &potential. This parameter is connected with the fullerene radius R and the affinity energy of the electron to Cm by the following expression [ 71, 2A = -AL = p( 1 + cothPR)
,
(2)
where AL is the jump of the logarithmic derivative of the wave function at a point r = R and p = a. Radius R = 6.639 [4] is much greater than the “sizes” of atomic electron shells for atoms with mean values of nuclear charges. Therefore, the wave functions of an optical electron in the ground state can be considered to coincide with the corresponding wave functions of an isolated M atom. The situation is quite different for the wave functions of a continuous spectrum. For r < R they coincide with the atomic wave functions up to a constant. Outside the shell for r > R they are linear combinations of the solutions, regular uk/ ( r) and irregular ukl( r) at r = 0, of the Schrodinger equation for the M atom,
uil + k2 - v
- ZU(r)]
ukl= 0,
the functions ukl( r) obey the same equation. Functions uk/ ( r) and u&((r) have the following form at kr >> 1, uk,(r)
M sin
kr+tln2kr-q+A,(k)
, I
AS.
E
L’kltr)
-
kr + ; In 2kr - ;
cos
Raltenkov/Phv.~ics
+ A/(k)
1
Letters
,
(4)
where z is the charge of the ion created in photoionization of the M atom, Al(k) are the phase shifts of the wave functions in the potential field of this ion. The solutions of Eq. ( 1) for E = k2/2 > 0 have the form
-ok/(r)
sin&(k),
r > R,
(5)
where 6, ( k) are the phase shifts due to the photoelectron scattering by the potential of the fullerene sphere V(r) = -A6( r - R). The wave functions (5) must be continuous at I = R and their derivatives at this point are discontinuous. By integrating Eq. ( 1) near the potential well surface we obtain Rit ? II
X{‘(r)dr=itXi(R+e)
-~{(R-e)l~+e
.
(6)
Substituting functions (5) in Eq. (6) we have the following relations for the amplitudes DI( k) and the phases 6,(k), = cosSr(k)
1 - &St(k)-
1
Q/(R) Q/(R)
=
m+,(w)
=
I=0
l)e-i(s’+A’)P,
@*,W~;‘*,W
( 10)
1
where w = I,, + k’/2 is the photon energy. The differential cross section of the endohedral toionization is defined by the formula ?[I
+ p(w)P?(coso)]
1
’
pho-
(II)
here, 0 is the angle between the photoelectron momentum and the polarization vector of the photon [ IO], a(w) is the total photoionization cross section equal to the sum of the partial cross sections, =Dl’_,(k)c+;l_,(w,
a(w)
+D;+,(kb;L,(w). t 12) in Eq. ( I 1) is
The parameter of the dipole anisotropy
u;,(R) w(R)u~/(R) - k/AL ’
2.1i (2Z+
205
(7)
(8) p(w)
When deriving Eq. (8) it is taken into account that the Wronskian of Bq. (3) is wk[ = r&r(r)&(r) ui,(r)uk,(r) = k. The amplitude of the optical transition of an electron from the bound state of the atom in M@C6c into continuum is defined by the matrix element ($; IrI&) calculated with the wave functions for the ground state of the atom I+!Q(r) = [ U,I (r) /r ] I$,,( 8,~) and for the continuous spectrum
$A:(r, =
203-209
Here, the argument of the Legendre polynomials is described by cos6 = (kr/kr), where 6 is the angle between the momentum of the photoelectron and its radius-vector, and functions xk/( r) are defined by Eq. (5). The atomic wave functions u,,(r) are localized in the origin of coordinates in the domain with the size N (21,l) -‘P < R (I,,/ is the ionization potential of the nl-subshell of the M atom). Therefore, the inner domain of the fullerene sphere, where the continuum wave functions are distinguished only by the amplitude D,(k) from the wave functions of the isolated atom, makes the main contribution into the amplitude of photoeffect. Hence, the partial cross sections PI*] (w) of the endohedral photoeffect, up to a constant, coincide with the partial cross sections u;; I ( w ) of the isolated atom
df2
= -Ax/(R)
tg&(k)
(1999)
dd w) -=
K’tc
D/(k)
A 254
k-r
( ) 7
xdr)
~
r
’ (9)
= [1(1-- l>Dl’_,R;_,
-61(1-t
+(I+
l)(/-t-2)D;,,R;+,
l)D/-IDI+IRI_,R/,,
x cos(&+,
+ A/+, - 61-1 - A/-I)
I)[lD;_,R;_,
x ((21f
+ (I+
1
l)D,“+,R~+,]}-‘. (13)
00 RI%,
w(r)rukt~l(~)
=
dr .
f
14)
0
Eqs. (lo)-( 13) connect the photoionization cross section of the M@C6o endohedral and the M atom with the parameters of the fullerene shell, namely its radius and the affinity energy of the electron to the
206
AS. Baknkm/Physics
neutral Gee. We underline again that the derivation of these relations is based on the assumption that atomic electrons can be considered as independent of the electrons of the fullerene shell. It is evident that this assumption is valid, at least for the inner subshells of the M atom. Because of the coupling between oscillations of the wave functions inside and outside the fullerene sphere, the amplitudes Q(k) have a resonance character. Therefore, there are resonances in the photoionization cross sections of endohedrals. According to Eqs. (7) and (8)) the squares of the amplitudes have the form
D?(k) =
(k/W2 [u/dR)w(W
- k/AL12 + n:,(R)
’ (15)
In case of low power of the S-potential (A = -A L/2 -+ 0)) the amplitudes Df (k) -+ 1. For high power (A -+ 00) the amplitudes Df (k) are different from zero only for k which obey the condition &i(R) = 0. Thus, the more effective the potential of the S-sphere is, the more well defined the resonance phenomena are [ 111, The partial cross sections a(w) and @(o) coincide when Df (k) = 1. This occurs for k being the solutions of one of the following equations, &r(R)
Letters A 254 (1999)
203-209
side the fullerene shell; the second one to the H@CGO endohedral. The clusters of these types hardly exist in nature, however, the consideration of these simplest model systems makes it possible, formally using the derived formulas, to describe qualitatively peculiarities in photoelectron spectra of endohedrals. Suppose in the first case that the ionization potential of the ion is Zi = ~~12. Then the wave function of the ground s-state of the ion is uio( r) = &exp( -KT), and the wave functions of a continuum for I= 0 are UkO(r) = sin[kr
+ Ao(k)l
&O(r) = -cos[kr+Ao(k)l and for orbital functions are &l(r)
=
moments
krjl(kr) ,
h/(r)
, , different
( 17) from zero, these
= krq(kr)
,
(18)
where jr (kr) and vr (kr) are the spherical Bessel functions of the first and second kind [ 121, respectively, with the asymptotics equation (4), where z = 0. The phase da(k) in Eq. (17) is connected with K by the formula tg Aa( k) = -k/K [9]. The photoionization cross section of the negative M- ion in the model of zero-range potentials is [ 131
(19)
= 0,
(16) Since the number of nodes of the regular solution Ukl( R) increases without limit at k + 00, the number of resonance maxima in the photoionization cross section of the atom, located in the spherical potential cavity V(r) = -AS(r - R), is also unrestrictedly large. The formal consideration of Eq. (15) for k -+ CXJis possible, of course, only for the infinitely thin spherical potential. The real “thickness” of the fullerene shell is N 1, so for endohedrals the derived formulas are valid for electron lengths l/k 2 1 that correspond to the near threshold energies of photoelectrons. Let us analyze how functions Df (k) are varied with types of the atomic potential U(r) . We shall consider two limiting cases, nameIy a short-range point potential and the long-range Coulomb potential. The first of them corresponds to a negative M- ion contained in-
here (Yis the fine structure constant. The photoionization cross section of the same ion in the M- @X60 system, according to Eq. (lo), is U(W) = D:(k)&(w). The results of calculations of the cross sections g(w) and &(w) for K = 0.5 (Z; = 3.4 eV) and AL = -0.885 [7] are given in Fig. 1. As seen from this figure, the amplitude function 0: (k) considerably affects the form of the frequency dependency of the cross section near the threshold of photoionization of the M- ion. Over the range of the photoelectron energy E = k’/2 = O-6 eV, the cross section (T(W) is 4-5 times less than U+‘(W), while in the interval 69 eV the resonance peak appears in the cross section and a(w) is 2-3 times greater than U’(W). The dependencies of the amplitude functions 0: (E) on photoelectron energies for the orbital moments I= O-3 are given in Fig. 2. The amplitudes Di( E) and Dz( E) are given in reduced scale of 10 and IO3 times,
AS. Baltenkov/Physics
Photoelectron Fig.
1.The
0
energy E, (eV)
photoionization cross sections of M-
2
4
6
Photoelectron
@Cm and H@Cm
model aggregates. Solid curves correspond to the free M-
207
L-men A 254 (1999) 203-209
anion
and H atom.
8
Fig. 2. The amplitude functions D:(E) potential U(r)
The curves 0:
( l) and
IO
12
14
energy C, (eV) for the short-range point D:(c)
are given in reduced
scale of 10 and 10” times, respectively.
respectively.
tude functions
At the photoeffect threshold ( 15) are equal to
the ampli-
D;(e=O)=[l+RAL-AL/K]-*, D~(~=0)=[l+RAL/(21+1)]-*, 1 # 0.
(20)
All the curves 0; (E) have resonance maximums that are specially well defined for continuum states with 1 = 3. This is connected with the existence of quasi discrete f-levels formed by a centrifugal potential barrier [7]. In the case of the atomic potential U(r) = -z/r (z = I), the solutions of Eq. (3) u&f(r) and uk[( r) are the Coulomb wave functions [ 123. The photoionization cross sections of the ground state of the H atom, 25~2 GH(W) = -cr 3w4
exp[ -(4/k)
arctgk]
1 - exp( -2r/k)
’
(21)
and the H@Cba endohedral, calculated with these functions, are given in Fig. 1. In this case also, the amplitude function D:(e) essentially changes the frequency dependence of the photoionization cross
section of the ground level of hydrogen atom near the photoeffect threshold. For l M O-8 eV the cross section a( W) is less than oH (o) and for E “N 11 eV there is a resonance maximum where a(w) is more than two times greater than gH (w) The amplitude functions 0: (E) for the Coulomb potential are given in Fig. 3. All the curves in this figure, as well as the curves in Fig. 2, have resonance peaks. Their location on the energy scale E depends on the electron orbital moment in continuum. Therefore, in the case of ionization, for example, of the npstates of H atom the resonances in the total cross section of the photoeffect equation ( 12) appear at photoelectron energies E FZ:0.5 eV and E x 5.0 eV, corresponding to the resonances in s- and d-electron waves. For these energies, the parameter of dipole anisotropy p(o) Eq. ( 13) will also have the resonance structure. The consideration of these limited cases shows that the amplitude function Df ( E) essentially depends on the shape of the potential U(r) in which the optical electron moves. In the case of the short-range potential, there are narrow bands of energy E for which the electron density inside the fullerene shell increases in
A.S. Baltenkov/Physics
Lettm
A 254 (1999) 203-209
H atom
; ,’
;:
\\
,i
\ \
I
\
0
0
~‘~‘~‘~“‘~‘c’~‘~’ 2
4
‘.
6
8
Photoelectron Fig. 3. The amplitude potential U( r )
functions
10
12
14
16
18
0
Coulomb
several tens times (resonance in f-wave). In the case of the long-range Coulomb potential, the quasi discrete levels are absent and for this reason the resonance effects are not so great, however, they are also well defined. Let us consider not model cluster systems, but really existing endohedrals Ne@C60. These particles are observed in the experiments [ 141 in the mass-spectra of fullerene-containing soot. The “sizes” of the outer 2s and 2p subshells of this atom we can estimate using the formulas for hydrogen-like atoms [ 151, r=
$[3?? -l([+
I>1 1
72=n2[5322+1-31(1+1)].
. ..
- - _ .‘~:.,,__.<.~
o’,“‘.“‘,“l.“‘.‘,
energy E, (eV)
0: (t-) for the long-range
,’
l.
(22)
According to these formulas, the Ne atom practically wholly fills the cavity of the fullerene shell. Consequently, it can be supposed that the nucleus of the Ne atom in the Ne@& endohedral is located near the center of the fullerene shell, which makes it possible to use derived formulas for the calculations. The results of calculation of the photoionization cross sections of 1s and 2s levels of Ne atom and Ne@CGa endohedral are given in Fig. 4. The wave functions of the isolated Ne atom in the ground state U,I( r)
2
4
6
, _.I’ ___.. ..-
8
Photoelectron
,,’
10
,I’
Ne (Is)
\,
12
14
16
18
energy E, (eV)
Fig. 4. The photoionization cross sections of the Ne@CMf endohedral. Dashed curves are the cross sections of the Is and 2s subshells of the free Ne atom.
and in continuum uk/ (r) were calculated in the oneelectron Hartree-Fock approximation using the code package [ 161. The irregular solutions ukl( Y) were determined by formula [ 171
(23) The ionization potentials of these atomic levels are IIS z 89 1.4 eV and lzs N 52.5 eV. Since the continuum wave functions in potential fields of the Ne atom with vacancies in 1s and 2s subshells are different, the conditions of the resonance also differ. For this reason the photoionization cross section of the 2s state, unlike the 1s state, has two resonance peaks. The first of them is at the ionization threshold of the 2s subshell (W N” 52.5 eV), the second one at the photon energy o z 67.5 eV. For the first peak the photoionization cross section of Ne@Qe is more than 6 times greater than the cross section of the free Ne atom, for the second maximum almost 3 times greater. The cross section of the Is photoeffect, in the energy region under consideration E, has only one well-defined maximum
A.S. Baltenkm/Physics
at 0 22 903.5 ev. Thus in the photoionization cross section of an atom located in the center of the fullerene cage, there are resonances. Their positions on the o scale are defined by the fullerene radius and affinity energy of electron to c60. The experimental detection and study of these peculiarities by methods of photoelectron spectroscopy can be used for obtaining additional information on endohedrals and fullerenes. The author is grateful to Professor M. Amusia and Professor U. Becker for useful discussions and to Dr. B. Krakov and V. Pikhut for performing some numerical calculations.
References I II T. Guo. M.D. Diener, Y. Chai, M.J. Alford, R.E. Haufler, S.M. McClure, T. Ohno, J.H. Weaver, G.E. Scuseria, R.E. Smalley, Science 257 ( 1992) 1661. I2 I D.S. Bethune, C.S. Yannoni, M. Hoinkis, M. Devries, J.R. Salem. MS. Crowder, R.D. Johnston, Z. Phys. D 26 (1993) 153.
Letters A 254 (1999) 203-209
209
[ 3 ( M. Halser. J. Almlof. G.E. Seuseria. Chem. Phys. Lett. I8 I (1991) 497. 141 E. Tossatti. N. Manini. Chem. Phys. Lett. 223 (1994) 61. 151 B. Xu, M.Q. Tan, U. Becker, Phys. Rev. Lett. 6 (1996) 3538. 16) L.L. Lohr. SM. Blinder. Chem. Phys. Lett. 198 ( 1992) 100. 171 M.Ya. Amusia, A.S. Baltenkov. B.G. Krakov. Phv\, Lett. A 243 ( 1998) 99. 181S.H. Yang. C.L. Pettiette. J. Conceicno. 0. Cheschnovsky. R.E. Smalley. Chem. Phys. Lett. 139 (1987) 233. 191 H.A. Bethe, P. Morrison, Elementary Nuclear Theory (Wiley. New York; Chapman & Hall, London. I956 J. 1101 J. Cooper. R.N. Zare. J. Chem. Phys. 48 ( 1068) 942. IIll S. Flugge, Practical Quantum Mechanics (Springer,Berlin, Heidelberg. New York. 1971). 1121 M. Abramowitz, I.A. Stegun. Handbook of Mathematical Functions, Applied Mathematics Series. Vol. 55 (National Bureau of Standards, 1964). B.H. Armstrong, Phys. Rev. I31 ( 1963) I 132. M. Saunders, H.A. Jimenezvazquez. R.J. Cross. R.J. Poreda. Science 259 (1993) 1428. L.D. Landau. E.M. Lifshitz. Quantum Mechanics. Nonrelativistic Theory (Pergamon Press, Oxford. 1965). M.Ya. Amusia. L.V. Chernysheva. Computation of Atomic Processes ( IOP Publishing Ltd. Bristol. Philadelphia, 1997). GA. Korn. T.M. Korn. Mathematical Handhook (McGrawHill, New York, I96 I )
PHYSICS
LETTERS
A
Physics Letters A 254 ( 1999) 203-209
ELSEVIER
Resonances in the photoionization cross section of M @C60 endohedrals A.S. Baltenkov Arifov Institute Received
I5 October
of Electronics, Akadetngorodok. 700143 Tashkent. Uzbekistan
1998; revised manuscript received 4 January 1999; accepted Communicated by B. Fricke
for publication
2 February
1999
Abstract A model for the calculation of photoionization cross sections of an atom introduced inside the fullerene cage has been proposed. A comparative investigation of the photoionization cross sections of an isolated M atom and the same one in the M@Ceo endohedral has been carried out. It has been shown that the resonance peaks in the photoionization cross sections of the endohedral appear near the thresholds of ionization of atomic subshells. The positions of these peaks on the scale of the photoelectron energy are defined by the fullerene shell radius and the affinity energy of the electron to GO. @ 1999 Elsevier Science B.V.
Among the variety of clusters, carbon clusters are of special interest. Clusters of this type form a perfect geometrical structure. The most well known is the Cen cluster. This multi-atomic system is a nearly spherical shell formed by carbon atoms. The closeness of the atomic structure of Cm fullerene results in high stability and non-reactivity of these particles, which makes it possible to use them as containers for atoms of different kinds. Compounds of this type were called endohedrals and labeled as M@Cm unlike to usual chemical compounds of MCm, where an attached M atom is located outside the fullerene cage. The unusual geometrical structure of fullerenes and endohedrals make them extremely attractive to study. The analysis of elementary processes with these particles will provide information on the unique nature of these formations and on their electronic structure. The results of investigations will not only broaden existing ideas about these giant molecules, but also be used for practical problems such as formation of controlled
beams of these particles and formation from them of surface or bulk crystals. Current experimental investigations of C60 and fullerene-like objects are carried out using various methods, including photoelectron and electron spectroscopy. By means of these methods, in particular, well-founded evidence was obtained that a M atom in M@C60 endohedral is located inside the fullerene sphere, rather than on the external surface of the fullerene skeleton [ 1,2]. For the development of photoelectron spectroscopy of these giant molecules, the comparison of photoionization spectra of an isolated M atom and the same atom introduced inside the M@C6n endohedral is of great interest. This study will make it possible to analyze the influence of a fullerene cage on optical characteristics of an atom located inside endohedral. In the present paper the comparative investigation of photoionization of isolated atoms and corresponding endohedrals have been carried out and resonance peculiarities due to fullerene shell in cross
0375-9601/99/$ - see front matter @ 1999 Elsevier Science B.V. All rights reserved. PII SO375-9601(99)00095-X
204
A.S.
Balrenkov/Physics
sections of these processes have been analyzed. Two approaches are mainly used to calculate the electronic structure of fullerene-like systems. The first is the molecular calculation of the wave functions from the first principles [ 31. The use of ab initio calculation methods becomes practically impossible when solving problems where it is necessary to know the continuous spectra wave functions. The second approach is the approximation of the potential of the fullerene cage by some model potential well. The maximal simplification can be reached if we replace the real potential of C60, formed by the superposition of the atomic fields, by the model potential of the fullerene shell V(r). It is usually supposed that this potential is equal to zero in all space except the spherical layer formed by the smeared carbon atoms. Inside this layer the function V(r) is different from zero and simulated by the different potential functions [4,5]. One of the potentials of this type was used in papers [6,7] to describe some processes with negative C, ions. In articles [6,7] it was supposed that electron interaction with the fullerene shell can be described by a &like potential which is different from zero within an infinitely thin spherical layer with the radius R equal to the fullerene radius. This model can be used when the lengths of electron waves are much greater than the “thickness” of the fullerene shell. In this case the detailed form of the function V(r) is not important, because the electron is subject to the averaged action of the potential. It is this situation in C, where the affinity energy of the electron to the neutral fullerene I is comparatively low on atomic scale (I = 2.65 eV [ 81) and, therefore, most time the extra electron is moving in regions where we can consider it as free. That allows us to determine the wave function of electron from the boundary conditions. The situation here is similar to one in nuclear physics, where the weakly bounded nuclear system - deuteron - can be described by a logarithmic derivative of the wave function at zero [9] and this derivative is defined by the binding energy of deuteron, which is known from experiment. In the case of C,, the boundary condition for the electron wave function is not given at zero, but on the sphere with R radius. As any idealization, the model [6,7] has, naturally, the limited domain of applicability. It can be used to calculate the physical values which are determined by the large distances from the fullerene shell. These can be the photoionization cross sections
Letter.7 A 254
(1999)
203-209
of the negative Cc0 ion near the threshold, polarizability and magnetic susceptibility of these ions, etc. Let us use this model potential to describe the photoionization of M atoms located in the fullerene cage near the threshold of this process. We shall represent the M@C6a endohedral as a corresponding isolated atom located in the center of the fullerene shell that is described by the &like potential. The wave functions of an optical electron of the M atom in this spherically symmetrical well have the form fl(r) = [~c(T)/Y]Y,~~(B,(D). The radial part of the wave function of this electron is defined by the equation (throughout this paper we use atomic units, h=??r=e=l) 1 2
xl” -
1tz+ I)
-XI r2
I
+
[A@r - R) - U(r) +
=o.
,qX,
(1)
Here, U(r) is the potential energy of an optical electron in an isolated atom, A is the power parameter of the &potential. This parameter is connected with the fullerene radius R and the affinity energy of the electron to Cm by the following expression [ 71, 2A = -AL = p( 1 + cothPR)
,
(2)
where AL is the jump of the logarithmic derivative of the wave function at a point r = R and p = a. Radius R = 6.639 [4] is much greater than the “sizes” of atomic electron shells for atoms with mean values of nuclear charges. Therefore, the wave functions of an optical electron in the ground state can be considered to coincide with the corresponding wave functions of an isolated M atom. The situation is quite different for the wave functions of a continuous spectrum. For r < R they coincide with the atomic wave functions up to a constant. Outside the shell for r > R they are linear combinations of the solutions, regular uk/ ( r) and irregular ukl( r) at r = 0, of the Schrodinger equation for the M atom,
uil + k2 - v
- ZU(r)]
ukl= 0,
the functions ukl( r) obey the same equation. Functions uk/ ( r) and u&((r) have the following form at kr >> 1, uk,(r)
M sin
kr+tln2kr-q+A,(k)
, I
AS.
E
L’kltr)
-
kr + ; In 2kr - ;
cos
Raltenkov/Phv.~ics
+ A/(k)
1
Letters
,
(4)
where z is the charge of the ion created in photoionization of the M atom, Al(k) are the phase shifts of the wave functions in the potential field of this ion. The solutions of Eq. ( 1) for E = k2/2 > 0 have the form
-ok/(r)
sin&(k),
r > R,
(5)
where 6, ( k) are the phase shifts due to the photoelectron scattering by the potential of the fullerene sphere V(r) = -A6( r - R). The wave functions (5) must be continuous at I = R and their derivatives at this point are discontinuous. By integrating Eq. ( 1) near the potential well surface we obtain Rit ? II
X{‘(r)dr=itXi(R+e)
-~{(R-e)l~+e
.
(6)
Substituting functions (5) in Eq. (6) we have the following relations for the amplitudes DI( k) and the phases 6,(k), = cosSr(k)
1 - &St(k)-
1
Q/(R) Q/(R)
=
m+,(w)
=
I=0
l)e-i(s’+A’)P,
@*,W~;‘*,W
( 10)
1
where w = I,, + k’/2 is the photon energy. The differential cross section of the endohedral toionization is defined by the formula ?[I
+ p(w)P?(coso)]
1
’
pho-
(II)
here, 0 is the angle between the photoelectron momentum and the polarization vector of the photon [ IO], a(w) is the total photoionization cross section equal to the sum of the partial cross sections, =Dl’_,(k)c+;l_,(w,
a(w)
+D;+,(kb;L,(w). t 12) in Eq. ( I 1) is
The parameter of the dipole anisotropy
u;,(R) w(R)u~/(R) - k/AL ’
2.1i (2Z+
205
(7)
(8) p(w)
When deriving Eq. (8) it is taken into account that the Wronskian of Bq. (3) is wk[ = r&r(r)&(r) ui,(r)uk,(r) = k. The amplitude of the optical transition of an electron from the bound state of the atom in M@C6c into continuum is defined by the matrix element ($; IrI&) calculated with the wave functions for the ground state of the atom I+!Q(r) = [ U,I (r) /r ] I$,,( 8,~) and for the continuous spectrum
$A:(r, =
203-209
Here, the argument of the Legendre polynomials is described by cos6 = (kr/kr), where 6 is the angle between the momentum of the photoelectron and its radius-vector, and functions xk/( r) are defined by Eq. (5). The atomic wave functions u,,(r) are localized in the origin of coordinates in the domain with the size N (21,l) -‘P < R (I,,/ is the ionization potential of the nl-subshell of the M atom). Therefore, the inner domain of the fullerene sphere, where the continuum wave functions are distinguished only by the amplitude D,(k) from the wave functions of the isolated atom, makes the main contribution into the amplitude of photoeffect. Hence, the partial cross sections PI*] (w) of the endohedral photoeffect, up to a constant, coincide with the partial cross sections u;; I ( w ) of the isolated atom
df2
= -Ax/(R)
tg&(k)
(1999)
dd w) -=
K’tc
D/(k)
A 254
k-r
( ) 7
xdr)
~
r
’ (9)
= [1(1-- l>Dl’_,R;_,
-61(1-t
+(I+
l)(/-t-2)D;,,R;+,
l)D/-IDI+IRI_,R/,,
x cos(&+,
+ A/+, - 61-1 - A/-I)
I)[lD;_,R;_,
x ((21f
+ (I+
1
l)D,“+,R~+,]}-‘. (13)
00 RI%,
w(r)rukt~l(~)
=
dr .
f
14)
0
Eqs. (lo)-( 13) connect the photoionization cross section of the M@C6o endohedral and the M atom with the parameters of the fullerene shell, namely its radius and the affinity energy of the electron to the
206
AS. Baknkm/Physics
neutral Gee. We underline again that the derivation of these relations is based on the assumption that atomic electrons can be considered as independent of the electrons of the fullerene shell. It is evident that this assumption is valid, at least for the inner subshells of the M atom. Because of the coupling between oscillations of the wave functions inside and outside the fullerene sphere, the amplitudes Q(k) have a resonance character. Therefore, there are resonances in the photoionization cross sections of endohedrals. According to Eqs. (7) and (8)) the squares of the amplitudes have the form
D?(k) =
(k/W2 [u/dR)w(W
- k/AL12 + n:,(R)
’ (15)
In case of low power of the S-potential (A = -A L/2 -+ 0)) the amplitudes Df (k) -+ 1. For high power (A -+ 00) the amplitudes Df (k) are different from zero only for k which obey the condition &i(R) = 0. Thus, the more effective the potential of the S-sphere is, the more well defined the resonance phenomena are [ 111, The partial cross sections a(w) and @(o) coincide when Df (k) = 1. This occurs for k being the solutions of one of the following equations, &r(R)
Letters A 254 (1999)
203-209
side the fullerene shell; the second one to the H@CGO endohedral. The clusters of these types hardly exist in nature, however, the consideration of these simplest model systems makes it possible, formally using the derived formulas, to describe qualitatively peculiarities in photoelectron spectra of endohedrals. Suppose in the first case that the ionization potential of the ion is Zi = ~~12. Then the wave function of the ground s-state of the ion is uio( r) = &exp( -KT), and the wave functions of a continuum for I= 0 are UkO(r) = sin[kr
+ Ao(k)l
&O(r) = -cos[kr+Ao(k)l and for orbital functions are &l(r)
=
moments
krjl(kr) ,
h/(r)
, , different
( 17) from zero, these
= krq(kr)
,
(18)
where jr (kr) and vr (kr) are the spherical Bessel functions of the first and second kind [ 121, respectively, with the asymptotics equation (4), where z = 0. The phase da(k) in Eq. (17) is connected with K by the formula tg Aa( k) = -k/K [9]. The photoionization cross section of the negative M- ion in the model of zero-range potentials is [ 131
(19)
= 0,
(16) Since the number of nodes of the regular solution Ukl( R) increases without limit at k + 00, the number of resonance maxima in the photoionization cross section of the atom, located in the spherical potential cavity V(r) = -AS(r - R), is also unrestrictedly large. The formal consideration of Eq. (15) for k -+ CXJis possible, of course, only for the infinitely thin spherical potential. The real “thickness” of the fullerene shell is N 1, so for endohedrals the derived formulas are valid for electron lengths l/k 2 1 that correspond to the near threshold energies of photoelectrons. Let us analyze how functions Df (k) are varied with types of the atomic potential U(r) . We shall consider two limiting cases, nameIy a short-range point potential and the long-range Coulomb potential. The first of them corresponds to a negative M- ion contained in-
here (Yis the fine structure constant. The photoionization cross section of the same ion in the M- @X60 system, according to Eq. (lo), is U(W) = D:(k)&(w). The results of calculations of the cross sections g(w) and &(w) for K = 0.5 (Z; = 3.4 eV) and AL = -0.885 [7] are given in Fig. 1. As seen from this figure, the amplitude function 0: (k) considerably affects the form of the frequency dependency of the cross section near the threshold of photoionization of the M- ion. Over the range of the photoelectron energy E = k’/2 = O-6 eV, the cross section (T(W) is 4-5 times less than U+‘(W), while in the interval 69 eV the resonance peak appears in the cross section and a(w) is 2-3 times greater than U’(W). The dependencies of the amplitude functions 0: (E) on photoelectron energies for the orbital moments I= O-3 are given in Fig. 2. The amplitudes Di( E) and Dz( E) are given in reduced scale of 10 and IO3 times,
AS. Baltenkov/Physics
Photoelectron Fig.
1.The
0
energy E, (eV)
photoionization cross sections of M-
2
4
6
Photoelectron
@Cm and H@Cm
model aggregates. Solid curves correspond to the free M-
207
L-men A 254 (1999) 203-209
anion
and H atom.
8
Fig. 2. The amplitude functions D:(E) potential U(r)
The curves 0:
( l) and
IO
12
14
energy C, (eV) for the short-range point D:(c)
are given in reduced
scale of 10 and 10” times, respectively.
respectively.
tude functions
At the photoeffect threshold ( 15) are equal to
the ampli-
D;(e=O)=[l+RAL-AL/K]-*, D~(~=0)=[l+RAL/(21+1)]-*, 1 # 0.
(20)
All the curves 0; (E) have resonance maximums that are specially well defined for continuum states with 1 = 3. This is connected with the existence of quasi discrete f-levels formed by a centrifugal potential barrier [7]. In the case of the atomic potential U(r) = -z/r (z = I), the solutions of Eq. (3) u&f(r) and uk[( r) are the Coulomb wave functions [ 123. The photoionization cross sections of the ground state of the H atom, 25~2 GH(W) = -cr 3w4
exp[ -(4/k)
arctgk]
1 - exp( -2r/k)
’
(21)
and the H@Cba endohedral, calculated with these functions, are given in Fig. 1. In this case also, the amplitude function D:(e) essentially changes the frequency dependence of the photoionization cross
section of the ground level of hydrogen atom near the photoeffect threshold. For l M O-8 eV the cross section a( W) is less than oH (o) and for E “N 11 eV there is a resonance maximum where a(w) is more than two times greater than gH (w) The amplitude functions 0: (E) for the Coulomb potential are given in Fig. 3. All the curves in this figure, as well as the curves in Fig. 2, have resonance peaks. Their location on the energy scale E depends on the electron orbital moment in continuum. Therefore, in the case of ionization, for example, of the npstates of H atom the resonances in the total cross section of the photoeffect equation ( 12) appear at photoelectron energies E FZ:0.5 eV and E x 5.0 eV, corresponding to the resonances in s- and d-electron waves. For these energies, the parameter of dipole anisotropy p(o) Eq. ( 13) will also have the resonance structure. The consideration of these limited cases shows that the amplitude function Df ( E) essentially depends on the shape of the potential U(r) in which the optical electron moves. In the case of the short-range potential, there are narrow bands of energy E for which the electron density inside the fullerene shell increases in
A.S. Baltenkov/Physics
Lettm
A 254 (1999) 203-209
H atom
; ,’
;:
\\
,i
\ \
I
\
0
0
~‘~‘~‘~“‘~‘c’~‘~’ 2
4
‘.
6
8
Photoelectron Fig. 3. The amplitude potential U( r )
functions
10
12
14
16
18
0
Coulomb
several tens times (resonance in f-wave). In the case of the long-range Coulomb potential, the quasi discrete levels are absent and for this reason the resonance effects are not so great, however, they are also well defined. Let us consider not model cluster systems, but really existing endohedrals Ne@C60. These particles are observed in the experiments [ 141 in the mass-spectra of fullerene-containing soot. The “sizes” of the outer 2s and 2p subshells of this atom we can estimate using the formulas for hydrogen-like atoms [ 151, r=
$[3?? -l([+
I>1 1
72=n2[5322+1-31(1+1)].
. ..
- - _ .‘~:.,,__.<.~
o’,“‘.“‘,“l.“‘.‘,
energy E, (eV)
0: (t-) for the long-range
,’
l.
(22)
According to these formulas, the Ne atom practically wholly fills the cavity of the fullerene shell. Consequently, it can be supposed that the nucleus of the Ne atom in the Ne@& endohedral is located near the center of the fullerene shell, which makes it possible to use derived formulas for the calculations. The results of calculation of the photoionization cross sections of 1s and 2s levels of Ne atom and Ne@CGa endohedral are given in Fig. 4. The wave functions of the isolated Ne atom in the ground state U,I( r)
2
4
6
, _.I’ ___.. ..-
8
Photoelectron
,,’
10
,I’
Ne (Is)
\,
12
14
16
18
energy E, (eV)
Fig. 4. The photoionization cross sections of the Ne@CMf endohedral. Dashed curves are the cross sections of the Is and 2s subshells of the free Ne atom.
and in continuum uk/ (r) were calculated in the oneelectron Hartree-Fock approximation using the code package [ 161. The irregular solutions ukl( Y) were determined by formula [ 171
(23) The ionization potentials of these atomic levels are IIS z 89 1.4 eV and lzs N 52.5 eV. Since the continuum wave functions in potential fields of the Ne atom with vacancies in 1s and 2s subshells are different, the conditions of the resonance also differ. For this reason the photoionization cross section of the 2s state, unlike the 1s state, has two resonance peaks. The first of them is at the ionization threshold of the 2s subshell (W N” 52.5 eV), the second one at the photon energy o z 67.5 eV. For the first peak the photoionization cross section of Ne@Qe is more than 6 times greater than the cross section of the free Ne atom, for the second maximum almost 3 times greater. The cross section of the Is photoeffect, in the energy region under consideration E, has only one well-defined maximum
A.S. Baltenkm/Physics
at 0 22 903.5 ev. Thus in the photoionization cross section of an atom located in the center of the fullerene cage, there are resonances. Their positions on the o scale are defined by the fullerene radius and affinity energy of electron to c60. The experimental detection and study of these peculiarities by methods of photoelectron spectroscopy can be used for obtaining additional information on endohedrals and fullerenes. The author is grateful to Professor M. Amusia and Professor U. Becker for useful discussions and to Dr. B. Krakov and V. Pikhut for performing some numerical calculations.
References I II T. Guo. M.D. Diener, Y. Chai, M.J. Alford, R.E. Haufler, S.M. McClure, T. Ohno, J.H. Weaver, G.E. Scuseria, R.E. Smalley, Science 257 ( 1992) 1661. I2 I D.S. Bethune, C.S. Yannoni, M. Hoinkis, M. Devries, J.R. Salem. MS. Crowder, R.D. Johnston, Z. Phys. D 26 (1993) 153.
Letters A 254 (1999) 203-209
209
[ 3 ( M. Halser. J. Almlof. G.E. Seuseria. Chem. Phys. Lett. I8 I (1991) 497. 141 E. Tossatti. N. Manini. Chem. Phys. Lett. 223 (1994) 61. 151 B. Xu, M.Q. Tan, U. Becker, Phys. Rev. Lett. 6 (1996) 3538. 16) L.L. Lohr. SM. Blinder. Chem. Phys. Lett. 198 ( 1992) 100. 171 M.Ya. Amusia, A.S. Baltenkov. B.G. Krakov. Phv\, Lett. A 243 ( 1998) 99. 181S.H. Yang. C.L. Pettiette. J. Conceicno. 0. Cheschnovsky. R.E. Smalley. Chem. Phys. Lett. 139 (1987) 233. 191 H.A. Bethe, P. Morrison, Elementary Nuclear Theory (Wiley. New York; Chapman & Hall, London. I956 J. 1101 J. Cooper. R.N. Zare. J. Chem. Phys. 48 ( 1068) 942. IIll S. Flugge, Practical Quantum Mechanics (Springer,Berlin, Heidelberg. New York. 1971). 1121 M. Abramowitz, I.A. Stegun. Handbook of Mathematical Functions, Applied Mathematics Series. Vol. 55 (National Bureau of Standards, 1964). B.H. Armstrong, Phys. Rev. I31 ( 1963) I 132. M. Saunders, H.A. Jimenezvazquez. R.J. Cross. R.J. Poreda. Science 259 (1993) 1428. L.D. Landau. E.M. Lifshitz. Quantum Mechanics. Nonrelativistic Theory (Pergamon Press, Oxford. 1965). M.Ya. Amusia. L.V. Chernysheva. Computation of Atomic Processes ( IOP Publishing Ltd. Bristol. Philadelphia, 1997). GA. Korn. T.M. Korn. Mathematical Handhook (McGrawHill, New York, I96 I )