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An alternative description of coherently excited states
Instruments and Methods in Physics Research B23 (1987) 173-176 North-Holland, Amsterdam
173
Nuclear
AN ALTERNATIVE 0. SCHGLLER”,
DESCRIPTION J.S. BRIGGS
OF COHERENTLY
Untoersitiit Freihurg
STATES
2, and R.M. DREIZLER”
‘) Insrirur ftir Theoretische Physik der Unroemtiit Frunkfurt/Muin, -‘I Fukultiit ftir Ph,sik,
EXCITED
FRG
FRG
The density matrix describing excitation of the low-lying energy levels of atomic hydrogen under proton impact is calculated. both in the first Born approximation extrapolation below threshold
and in a close-coupling of parameters describing
expansion in target eigenstates. An alternative characterisation, based on an the angular distribution of low-velocity ionised electrons is suggested.
1. Introduction Sets of energy levels which are effectively degenerate can be coherently excited during an atomic collision. In this paper we study the coherence of the n = 3 level of the hydrogen atom and consider only the “total” density matrix. That is, the density matrix elements are integrated over all scattering angles of the incident proton, with the result that the diagonal elements are just the cross-sections for excitation of a particular /m component of a given level. In the limit of high velocities excitation cross-sections should converge to their first Born values and the same is expected for off-diagonal elements of the density matrix. The first Born result therefore acts as a benchmark, particularly as the corresponding density matrix possesses certain symmetry properties arising from the simple form of the transition amplitudes in this approximation. The rate of convergence to the first Born result is explicitly investigated by performing close-coupling calculations within the II = 1, 2 and 3 manifolds. In particular, the coupling of different degenerate /-sublevels within a principal shell due to the long-range Stark mixing is of major importance. Thus, whilst summed cross-sections to a given principal level n may rapidly converge to the first Born result, the separate contributions depart from the first Born result even for collision velocities of the order of several times the Bohr velocity.
The resulting coupled equations have been solved in the following two levels of approximation: (a) The first Born approximation. (b) Close-coupling involving the n = 1, n = 2 and n = 3 manifolds (ten states of differing n/I m 1,approximation ClO). The density matrix elements were then calculated from the integral of bilinear products of asymptotic amplitudes ~1,(b, t ) over two-dimensional impact parameter vector b: P‘I = / dba,(r
= m)al”(t
= cc),
(2)
2. Presentation of the results As outlined in a recent transition amplitudes for the 1s ground state to a proton impact solving the equation in the impact straight-line trajectory:
paper [5] we calculated the excitation of hydrogen from particular final substate by time-dependent Schrodinger parameter method using a
R(r)=b+u.t. 0168-583X/87/$03.50 0 Elsevier Science Publishers (North-Holland Physics Publishing Division)
(I) B.V.
1
2
v Fig. 1. The angular cross-section to the approximation experimental
3
(Proton)
L
5
(a u i
distribution parameter PO (total excitation n = 3 level) calculated in the first Born The - - and approximation cl0 -. points are due to Park et al (1976). IV. HIGH
ENERGY
COLLISIONS
whcrc I or J stand for a set of quantum nun~bcrs nlnr. The density matrix elements can be combined to state multipoles [ 11, defined within an n-manifold by
multipoles with Q = 0 are nonzero. Up to now the beam axis was considered as the quantisation axis. In a Coulomb field one expects a smooth extrapolation of physhical quantities across the threshold for ionisation i.e. the transition from excitation to a state of high II to i[~nisation to a state of low momentum k in the continuum is continuous. The process of detecting an elcctron of momentum k described by a wavefunction
We restrict discussion to integrated cross-sections, which possess axial symmetry. so that only the elements with nr’ = nr are non-zero and hence only the state
*i-(r)
= x(21+
1) exp( -iS,)Rk,(r)PI(k.P),
can then bc viewed 3s a projection
(4)
onto Y?~(r) of the
0.16 0.2
0.12
0.10 0.08 O.C6 0.04
0.02 CLOD
i 1
I
v
i
-_
3 2 Proton)
L
4.32
-.__-_.L_-
5
2
ia b 1
V
:?rotCJnI
1
/
3
L
/4 5
(a ii )
D.010
0.02
O.Cl
0.035
N c7
z
0.00 0.000
n” : -3
-0.01
-0.305 -0.02
-0.010
-0.03
L v
Fig. 2. The angular
distribution
2
3
(Protoni
parameters
L
5
(a u :
fiK defined by eq. (9) for (a) K = 1. (h) EC= 2. (c) k’= 3 and (d) K = 4. The curve designation is as in fig. 1.
total coherent
wavefunction
produad by the collision, repr~se~~t~ng the “atom” u!here the electron has an energy +k2. Mere the R,,(r) are radial ~a~,efun~t~o~s and the particular phases 6, ensure incoming wave boundary conditions. Choosing the momentum k as quantisation axis we can rewrite eq. (4) as
8, = - g
~~(r)-4nCexp(-iS,)Y,~(k.u)p?,,(r), /,,I
& = #kc{
In this form it is easy to prove from the further symmetry properties of the state multipoles that in first Born approximation uN fiK for &I K vanish identically. For rr = 3 one finds explicitiy
(Rc(T(Ol),:,)+ fiRe( T(I?),:,))
(6)
where q,,,(r) = Rk,(r)Ylm(u. r) is the wavefunction of the Ith partial wave quantised along the beam axis (SchMer et al. 1986). The angular distribution of ionised electrons of a given momentum k can be written as (Briggs and Day 1980)
0) It can be shown by integration over the angle cos-1 (k * c) that the coefficients PK are given by
It has been explained by Burgdijrfer (1984) that the coefficients gEI which appear as a sum (8) over state multipoles of the 3-dimensional rotation group (.i-representation) can be written in terms of a single state multipole in a representation of the higher-symmetry 4-dimensional rotation group of the pure Coulomb problem. As a result of the continuity across the threshold in the Coulomb case, it is meaningful perhaps to consider the extrapolation of the coefficients (8) below threshold into the region of finite t7, even though the direct physical significance of a continuum electron angular distribution is lacking. Particularly interesting is the influence of the truncation of the I sum (for finite I?) on the Born approximation ‘*sum rule” that onfy & and & are finite in the Limit ~2--) c;c (Briggs and Day 15830). ‘The parameters fili represent a cross-section and are therefore real. A consideration of the symmetry of the state multipoles however, allows (8) to be written in the obviously real form
T(O2),:,)
- VTRe( T(ll);,,))
P, = $Re(W);J
The parameter &, for a given shell n, is trivially found to be proportional to the total excitation cross-section (fig. 1). inThe combination /!?] reflects the coilisionaliy duccd dipole moment and is shown in fig. (2a). The parameter & is shown in fig. (2b). This quantity is related to the quadrupole moment of the atom after the collision, The parameter & is shown in fig. (2~). The coefficient /$, measures essentially the 3d contribution to the alignment of the n = 3 shell as would be measured in a radiative decay (fig. 2d).
Experimentally the elements of the density matrix can be deduced by observation of the radiation emitted by the cohcrcntly excited atom, either alone or in the presence of applied electromagnetic fields. The density matrix can bc compared directly to the calculated one, but the number of independent elements increases rapidly as II increases. Then it is meaningful to look for ways to combine the elements into physically rclcvant parameters. In this paper a classification is proposed in which the particular coefficients (linear combinations of state multipoles) which describe the angular distribution of ionised electrons is calculated for excitation, i.e. the coefficients are extended below threshold. This shoutd be a meaningful way of representing the density matrix, particularly for high ?I, since it can be shown in the limit II + 00, only two of these coefficients are nonzero in the Born approximation.
[3] J. Burgdiirfer, Lecture Notes in Physics Vol. 213, eds., K.O. Groeneveld, W. Meckbach and LA. Sellin (Springer. Berlin, 1984). [4] J.T. Park, J.E. Alday. J.M. George and J.L. Peacher, Phys. Rev. A14 (1976) 608.
[S] 0. Schiillcr, J.S. Briggs (1986) 2505.
and
R.M.
Dreizlcr,
J. Phys.
BlY
173
Nuclear
AN ALTERNATIVE 0. SCHGLLER”,
DESCRIPTION J.S. BRIGGS
OF COHERENTLY
Untoersitiit Freihurg
STATES
2, and R.M. DREIZLER”
‘) Insrirur ftir Theoretische Physik der Unroemtiit Frunkfurt/Muin, -‘I Fukultiit ftir Ph,sik,
EXCITED
FRG
FRG
The density matrix describing excitation of the low-lying energy levels of atomic hydrogen under proton impact is calculated. both in the first Born approximation extrapolation below threshold
and in a close-coupling of parameters describing
expansion in target eigenstates. An alternative characterisation, based on an the angular distribution of low-velocity ionised electrons is suggested.
1. Introduction Sets of energy levels which are effectively degenerate can be coherently excited during an atomic collision. In this paper we study the coherence of the n = 3 level of the hydrogen atom and consider only the “total” density matrix. That is, the density matrix elements are integrated over all scattering angles of the incident proton, with the result that the diagonal elements are just the cross-sections for excitation of a particular /m component of a given level. In the limit of high velocities excitation cross-sections should converge to their first Born values and the same is expected for off-diagonal elements of the density matrix. The first Born result therefore acts as a benchmark, particularly as the corresponding density matrix possesses certain symmetry properties arising from the simple form of the transition amplitudes in this approximation. The rate of convergence to the first Born result is explicitly investigated by performing close-coupling calculations within the II = 1, 2 and 3 manifolds. In particular, the coupling of different degenerate /-sublevels within a principal shell due to the long-range Stark mixing is of major importance. Thus, whilst summed cross-sections to a given principal level n may rapidly converge to the first Born result, the separate contributions depart from the first Born result even for collision velocities of the order of several times the Bohr velocity.
The resulting coupled equations have been solved in the following two levels of approximation: (a) The first Born approximation. (b) Close-coupling involving the n = 1, n = 2 and n = 3 manifolds (ten states of differing n/I m 1,approximation ClO). The density matrix elements were then calculated from the integral of bilinear products of asymptotic amplitudes ~1,(b, t ) over two-dimensional impact parameter vector b: P‘I = / dba,(r
= m)al”(t
= cc),
(2)
2. Presentation of the results As outlined in a recent transition amplitudes for the 1s ground state to a proton impact solving the equation in the impact straight-line trajectory:
paper [5] we calculated the excitation of hydrogen from particular final substate by time-dependent Schrodinger parameter method using a
R(r)=b+u.t. 0168-583X/87/$03.50 0 Elsevier Science Publishers (North-Holland Physics Publishing Division)
(I) B.V.
1
2
v Fig. 1. The angular cross-section to the approximation experimental
3
(Proton)
L
5
(a u i
distribution parameter PO (total excitation n = 3 level) calculated in the first Born The - - and approximation cl0 -. points are due to Park et al (1976). IV. HIGH
ENERGY
COLLISIONS
whcrc I or J stand for a set of quantum nun~bcrs nlnr. The density matrix elements can be combined to state multipoles [ 11, defined within an n-manifold by
multipoles with Q = 0 are nonzero. Up to now the beam axis was considered as the quantisation axis. In a Coulomb field one expects a smooth extrapolation of physhical quantities across the threshold for ionisation i.e. the transition from excitation to a state of high II to i[~nisation to a state of low momentum k in the continuum is continuous. The process of detecting an elcctron of momentum k described by a wavefunction
We restrict discussion to integrated cross-sections, which possess axial symmetry. so that only the elements with nr’ = nr are non-zero and hence only the state
*i-(r)
= x(21+
1) exp( -iS,)Rk,(r)PI(k.P),
can then bc viewed 3s a projection
(4)
onto Y?~(r) of the
0.16 0.2
0.12
0.10 0.08 O.C6 0.04
0.02 CLOD
i 1
I
v
i
-_
3 2 Proton)
L
4.32
-.__-_.L_-
5
2
ia b 1
V
:?rotCJnI
1
/
3
L
/4 5
(a ii )
D.010
0.02
O.Cl
0.035
N c7
z
0.00 0.000
n” : -3
-0.01
-0.305 -0.02
-0.010
-0.03
L v
Fig. 2. The angular
distribution
2
3
(Protoni
parameters
L
5
(a u :
fiK defined by eq. (9) for (a) K = 1. (h) EC= 2. (c) k’= 3 and (d) K = 4. The curve designation is as in fig. 1.
total coherent
wavefunction
produad by the collision, repr~se~~t~ng the “atom” u!here the electron has an energy +k2. Mere the R,,(r) are radial ~a~,efun~t~o~s and the particular phases 6, ensure incoming wave boundary conditions. Choosing the momentum k as quantisation axis we can rewrite eq. (4) as
8, = - g
~~(r)-4nCexp(-iS,)Y,~(k.u)p?,,(r), /,,I
& = #kc{
In this form it is easy to prove from the further symmetry properties of the state multipoles that in first Born approximation uN fiK for &I K vanish identically. For rr = 3 one finds explicitiy
(Rc(T(Ol),:,)+ fiRe( T(I?),:,))
(6)
where q,,,(r) = Rk,(r)Ylm(u. r) is the wavefunction of the Ith partial wave quantised along the beam axis (SchMer et al. 1986). The angular distribution of ionised electrons of a given momentum k can be written as (Briggs and Day 1980)
0) It can be shown by integration over the angle cos-1 (k * c) that the coefficients PK are given by
It has been explained by Burgdijrfer (1984) that the coefficients gEI which appear as a sum (8) over state multipoles of the 3-dimensional rotation group (.i-representation) can be written in terms of a single state multipole in a representation of the higher-symmetry 4-dimensional rotation group of the pure Coulomb problem. As a result of the continuity across the threshold in the Coulomb case, it is meaningful perhaps to consider the extrapolation of the coefficients (8) below threshold into the region of finite t7, even though the direct physical significance of a continuum electron angular distribution is lacking. Particularly interesting is the influence of the truncation of the I sum (for finite I?) on the Born approximation ‘*sum rule” that onfy & and & are finite in the Limit ~2--) c;c (Briggs and Day 15830). ‘The parameters fili represent a cross-section and are therefore real. A consideration of the symmetry of the state multipoles however, allows (8) to be written in the obviously real form
T(O2),:,)
- VTRe( T(ll);,,))
P, = $Re(W);J
The parameter &, for a given shell n, is trivially found to be proportional to the total excitation cross-section (fig. 1). inThe combination /!?] reflects the coilisionaliy duccd dipole moment and is shown in fig. (2a). The parameter & is shown in fig. (2b). This quantity is related to the quadrupole moment of the atom after the collision, The parameter & is shown in fig. (2~). The coefficient /$, measures essentially the 3d contribution to the alignment of the n = 3 shell as would be measured in a radiative decay (fig. 2d).
Experimentally the elements of the density matrix can be deduced by observation of the radiation emitted by the cohcrcntly excited atom, either alone or in the presence of applied electromagnetic fields. The density matrix can bc compared directly to the calculated one, but the number of independent elements increases rapidly as II increases. Then it is meaningful to look for ways to combine the elements into physically rclcvant parameters. In this paper a classification is proposed in which the particular coefficients (linear combinations of state multipoles) which describe the angular distribution of ionised electrons is calculated for excitation, i.e. the coefficients are extended below threshold. This shoutd be a meaningful way of representing the density matrix, particularly for high ?I, since it can be shown in the limit II + 00, only two of these coefficients are nonzero in the Born approximation.
[3] J. Burgdiirfer, Lecture Notes in Physics Vol. 213, eds., K.O. Groeneveld, W. Meckbach and LA. Sellin (Springer. Berlin, 1984). [4] J.T. Park, J.E. Alday. J.M. George and J.L. Peacher, Phys. Rev. A14 (1976) 608.
[S] 0. Schiillcr, J.S. Briggs (1986) 2505.
and
R.M.
Dreizlcr,
J. Phys.
BlY