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12. The Dynamics of HeavyIon Collisions
To obtain information on the behaviour of electron states in strong fields using
heavyion collisions as a tool, a full understanding of the scattering dynamics is
essential. In this chapter a comprehensive discuss
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11.2 The TwoCentre Dirac Equation
ifill
(
0
)
r, , (O = ''
X
IfIxll=
''
1
X=
(
gAr)
'f ()
r
1 x
Il
(O)
(ll
X  x u, (O
)
)
.
307
(11.10)
Here
are the spinor spherical harmonics, Sect. 3.3. We now also introduce a
multi pole expansion of the twocentre
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12. The Dynamics of HeavyIon Collisions
If the collision proceeded infinitely slowly, the electrons would remain in
their quasimolecular state rl>k(r,R(t forever. The finite time dependence causes
excitations into other states rI>" 1,* k. Since we h
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12. The Dynamics of HeavyIon Collisions
various channels, because for R + 00 (V AB + 0), Xn(R) become eigenfunctions of
i.e. plane waves with good momentum P:
p2/2/l,
Xn(R)
R+oo
i
) .
exp ( ;P.R
(12.45)
Physically, however, the asymptotic relati
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338
12. The Dynamics of HeavyIon Collisions
More complicated processes, such as transfer of energy between nuclei and electrons (or positrons) during the reaction, are not covered by this expression.
For
and
(12.66) arises, which led to the singlepartic
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12. The Dynamics of HeavyIon Collisions
Substituting these expressions into (12.79) gives the timedependent wave
function at the final time tf in terms of the amplitudes Ck':
",i+)(r, tf) =
L
CPn (r,R (tfeiXn(tj) .
(12.84)
'mn
Here the amplitudes a
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12. The Dynamics of HeavyIon Collisions
Fig. 12.11. Definition of vectors in the quasi molecule

z
y
by numerical integration of the coupled differential equations (12.19,32). In the
following we shall discuss the approximations inherent in these nu
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318
12. The Dynamics of HeavyIon Collisions
Fig. 12.2. Schema of how the supercritical resonance
moves through the discretized continuum. Strong
and very much localized couplings occur between
neighbouring continuum states
E
me 2
E,s
t cr
+tcr
me 2
imp
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12. The Dynamics of HeavyIon Collisions
from T = 0 to T = 10  20 S are seen. When the delay time T is much longer than
the collision time on a Rutherford trajectory (rcoll1O 21 s), the spontaneously
produced positrons show up as a conspicuous line
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12. The Dynamics of HeavyIon Collisions
(12.32b)
akE= /
while the equations for n with En >  me remain formally unchanged. The additional terms in (12.32) describe the decay of a vacancy in the supercritical resonance state into the positron continu
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12.2 Expansion in the QuasiMolecular Basis
317
which obviously relates the change in the occupation amplitudes ak' to the time
variation of the basis of instantaneous eigenstates. Making use of the orthogonality of the wave functions CPt, by projection
O
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12. The Dynamics of HeavyIon Collisions
The next chapter discusses experimental results and compares then with theoretical calculations.
Bibliographical Notes
The theory of atomic collisions is the subject of several textbooks, including [Mo 33, Br 7
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12. The Dynamics of HeavyIon Collisions
Fig. 12.1. Relative motion of the nuclei in
Rutherford scattering and definition of the coordinate system
x
z
with the nuclei initially moving in the direction of the z axis. The angular momentum then points in
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11. Superheavy Quasimolecules
Bibliographical Notes
The formation of super heavy quasimolecules in heavyion collisions and their applications in the
physics of strong fields was discussed by MUller et al. [Mil 72b, c]. The solution of the twocentre
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12.6 Status of Numerical Calculations
341
b)
oj
"R'
/
/
R
z
/
.
/
/
z
.
Z'
yJt.
y
Fig. 12.12. (a) Rotation of internuclear axis by angle dO; (b) Rotation of electronic coordinate by
dO
tronic coordinate r, except for an overall rotation of the coordinate
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11. Superheavy Quasimolecules
5
'0 r'rR(fm]
30
50
3sIr
Fig. 11.4. The U + U correlation
diagram (Zt = Z2 = 92). (_. )
include the effect of the finite nuclear
radius, (  ) for point nuclei
[So 79a, Be 80b]
300
5(lj
1000
1500
E [keY]
energy f
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13. Experimental Test of Supercritical Fields in HeavyIon Collisions
tion like (13.6), where a = 1 and (Cj m e c2 )   55 keY. A systematic evaluation
of Kshell ionization data as a function of impact parameter was made by Bosch
et al. [Bo 80, 82]
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12.3 HeavyIon Collisions: A Quantal Description
e
323
Fig. 12.3. Definition of the coordinates
[He S1al
A
B
When the electron  electron interaction is neglected, the electronic Hamiltonian is simply a sum of the singleelectron twocentre Hamiltonians
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Summer 2016
12.6 Status of Numerical Calculations
Fig. 12.14. Radial coupling matrix elements in
the Pb + Pb quasimolecule between the 1 sa
state and the sa continuum. (  ) are for
point nuclei, (   ) take into account the
finite nuclear radius
<Esal oOR 11sa>
P
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13.1 Establishing Superheavy Quasimolecules
353
It is quite instructive to view the ionization process, somewhat idealized, as an
elastic collision between the Coulomb field of the projectile nucleus and the
ejected electron. The classical picture require
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13. Experimental Test of Supercritical Fields in HeavyIon Collisions
Fig. 13.1. Quasimolecular K xray spectrum of the
Br + Br system measured by Meyerho/ and collaborators
[Me 73b]. The spectrum extends up to the transition
energy of the united ato
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12. The Dynamics of HeavyIon Collisions
0=
=
L (4)mI H '  EI4>n> Xn
n
L [_l_(ihV+
2p
n
+ ( (d) 
I'
(X)')J
(12.58)
x,(R) .
Here (P>, (X"), (.1 > mean matrix elements taken with the electronic basis
functions. It is possible to show [He Sla] that fo
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12.2 Expansion in the QuasiMolecular Basis
319
region during the heavyion collision. In other words, coupling among the
modified continuum states remains weak.
The price paid is that lPR and tPe are not eigenstates of the twocentre Dirac
Hamiltonian, s
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11. Superheavy Quasimolecules
given numerical solutions for the twocentre Dirac continuum states for the first
time [Wi 84].
11.3 The Critical Distance R cr
The internuclear distance at which the binding energy of the lowest quasimolecular state (the
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12.5 Collisions with Nuclear Interaction
337
mechanics, where the relevant quantity is not the reaction time but the scattering
amplitude Sft which is a function of the bombarding energy.
For deep inelastic collisions between heavy, complex nuclei such a
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13. Experimental Test of Supercritical Fields in HeavyIon Collisions
15  20 MeV per nucleon). The first experiment to yield evidence of large
ionization probabilities at small impact parameters, which had been theoretically
predicted [Be 76b], was c
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12.2 Expansion in the QuasiMolecular Basis
315
The relative angle after the collision by integration [Or 65] is
8(00) =
00
00
b
= 2
a
=
voob
de
J dt8(t) = J d t  2 = b J 
00
00
J
.
00
d):
0
e cosh + 1
e
4 arctan (
00
R(t)
00
R
b
= 4_(e21)ll2arcta
Dr. AQ Khan Institute of Computer Science and Information Technology
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Summer 2016
13. Experimental Test of Supercritical Fields
in HeavyIon Collisions
This chapter describes the highlights of experimental research undertaken in
recent years to get information on the physics of superheavy quasimolecules and,
ultimately, to detect the d