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Pionic bremsstrahlung in heavy-ion collisions
291~
Nuclear Physics A428 (1984) 291~304~ North-Holland, Amsterdam
PIONIC BREMSSTRAHLUNG
IN HEAVY-ION
COLLISIONS+
David VASAK Gesellschaft fir Schwerionenforschung, 6100 Darmstadt 11, West Germany
PlanckstraRe
1, Postfach 11 05 41,
, Berndt MILLER, Thomas STAHL and Mark UHLIG,
Walter GREINER*
Institut fiir Theoretische Physik der Johann-Wolfgang-Goethe Robert-Mayer-StraRe 8-10, Postfach 11 19 32, 6000 Frankfurt am Main 11, West Germany
Universitat,
We apply the bremsstrahlung model for pion production to symmetric nuclear collisions at energies far below the single NN-threshold. All main features of recent neutral pion measurements can be explained by a single value of the deceleration parameter of the theory which is consistent with hydrodynamica1 calculations. Since the y-ray bremsstrahlunq also reflects the kinematics of its source, a simultaneous measurement-of pion and Y-ray bremsstrahlung should be used to further test the collision dynamics. Results of our calculations for both kinds of radiation are presented. An extension of this approach to asymmetric systems is proposed.
Subthreshold recent years,
pion production particularly
(or cooperative)
has become of considerable
effects. The idea to consider
anism for the production
of secondary
a bremsstrahlung-type
particles
not newl. If this model can be shown to describe collisions
In the following we apply the bremsstrahlung
2 do A=q dsldE
pion production
'
I =--(I 3 '
about the time develop.2 nuclei .
model to collisions
threshold
of equal
for pion production.
cross section for radiation of neutral pions is
to the Fourier-transformof
d2n 71 dodE
is
in nuclear
structure of the colliding
nuclei far below the single nucleon-nucleon
proportional
mech-
in hadronic collisions
it carld be a useful source of information
ment of the reaction and the spin-isospin
The inclusive differential
interest during
in connection with the search for cumulative
1 I; 3 II
P
the current density.
? ': sP,sT
' 13k(p)?
(1)
k
fsupported by the Bundesministerium fir Forschung und Technologie. "Invited Speaker at the Conference on Theoretical Approaches to Heavy Ion Reactions Mechanisms, Paris, 14-18 May 1984
0375-9474/84/$03.00 0 Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
292c
D. Vasak et al. / Pionic bremsstrahlung
in heavy-ion collisions
with n L
_C_ clj (PII*=; k
‘TSP
$
k
I*WT(A) *W,(A)]*
IK(p)I*
(2)
where E is the pion energy, p = (E*-m*)l'*, m = 135 MeV the mass of the neutral pion, M = 930 MeV that of a bound nucleon and o the pion solid angle. The pion 2 nucleon coupling constantgohas the usual value go /4~ = 14. Averaging over the spin direction
fi of both nuclei
(i = P projectile,
i = T target) is indicated
by i. The index k runs over the three isospin values of the pion; the factor l/3 in (1) and (2) accounts
for the fact that (1) describes
only one sort of
pions, in this case the neutral ones. For equal nuclei the factor cl0 has been set equal the geometrical
cross section 4nR2 with R being the half-density
radius of the gaussian density P(r) = '7~~ exp T-(r/*a)*l of the colliding
nuclei
n = P/p, the compression, condition).
(3)
(p, = 0.17 fm l/a = &
-3
is the normal nuclear density and
(o/A)~'~
is obtained
from the normalization
In
K(P) = z i=P,T
+I- [C(pi) ("i'p)/(ui'p)l' - 0)
exp[i(Et-c.iti(t))ldt
(4)
(prime denotes total time-derivative) all the information
about collision
are the (linear) trajectories system, parametrized multiple
dynamics
is contained.
$(t)
= +R(t)gz
of the nuclear centers in the center-of-mass
by the deceleration
of the "passing time" T
time T. We express T as a unique
= R/(2yin
vin), where
vin is the initial
c.m. velocity and yin = (l-v:,) -S1'2, i.e.,
T
=
with an
v*
(5)
S
adjustable
energy-independent
parameter
u.
0: the spin-vectors
boosted from the ' are the corresponding four-velocities, 'i particle's rest frame into the c.m. system, and z(p) the Fourier-transform of (3) with BT(t) = I$]* - (S.Gi(t)* taking time-dependent
Lorentz-contraction
(5) into account. The statistical
factors
D. Vasak et al. / Pionic bremsstrahlung in heavy-ion collisions
WS(A) and WT(A) result from single statistical spin-isospin-strength fluctuations
wS
estimation
293~
of the nuclei's
to which the pions couple and which is non-zero due to
during the collision
(see below).
In ref. 2 we have used
= WT ZJFi-. 7
(7)
The parametrization hydrodynamical
used here was strongly motivated
calculations
which show excellent
by the results of
agreement with the experimental
findings even at low energies.
In fig. 1 the calculated3proton
compared with the experimental
yield from a I2c + Ig7Au collision
spectrum
is
at 84 MeV/n.
A short remark to the role of the spin
z c
'2C+197A" 84MNn II'-
-
Experiment
----
Theory
2 E E 5
density is appropriate:
In the schema-
tic approach of ref. 2 it was assumed that the spin vectors of the projectile and target are not correlated This assumption
addition of the pion yields
II'-
at all.
leads to incoherent from both
nuclei. On the other hand, in the hydrodynamical collisions
picture of heavy ion
the nucleons
tion zone, consisting
l-
nuclear matter,
in the reac-
of hot compressed
do not "remember"
their origin. Thus there is only one spin direction,
lo-'-
I
0
and the pion yields
should be added coherently. I
I
100
50
[MeVI, 1 150
FIGURE 1 Comparison of the present theory and the ex erimental data for 12C(84 MeV/nucleon) + P g7A~+p + x. Full lines correspond to the experiment; dashed lines represent the calculations (from ref.3).
coherent addition pion production
at 90' in the c.m.
system, in case of coherence tive interference forward-backward
sideways
angular distribution
of pion radiation
appear preferably collisions
is a consequence
take place and lead to occupation addition
in the
and gives a In fact, a
of spin fluctuations4,
in the reaction zone, where the individual
light, the coherent
a nega-
leaves a
enhancement
smaller total pion yield. non-zerosource
While in-
leads to enhanced
which
nucleon-nucleon
of vacant states. Seen in this
is the natural one.
In order to avoid complications
caused by the Coulomb force when looking at
charged pion data5 we compare our calculations6
(cf. fig. 2) with recent ex-
periments on neutral pion production 7,8,9 . The measured pion angular distributions exhibit a forward-backward
behaviour
and thus favour the coherent addition
294~
D. Vasak et at. / Pionic ~rems~tra~l~ng in heavy-ion collisions
0
-1.0 -0.5
0 0.5 cos~~c‘m,l
50 100 150 200 pion cm. beg
0
1.0
FIGURE 2 (a) Theangulardistribution forpions produced at 84 MeVin in the energy region O-50 MeV (top), 50-100 MeV and loo-150 MeV, respectively. The data are from ref. 7. (b) The angle-integrated spectra at ion energies between 20 and 84 MeV/n. (c) The excitation function. The data are taken from refs.7 (solid dots),5 (open circles), 8 (open triangles), 9 (solid triangles), and 10 (solid squares). The dashed line exhibits the thermal pions. of the pion yields
from the projectile
and target nuclei: in fig, Z(a) the experimental
pion angular distribution
are compared with our results. The7 dashed curve in the upper energy cut is obtained by incoherent addition the deceleration adjusted
(here
has been
to v = 0.55). The solid lines show the results from coherent addition.
In this case the parameter negative
parameter
interference
v = 0.38 is somewhat smaller to c~pensate
mentioned
above.
clude that the spin distributions correlated
of the participating
or, in other words, that deceleration,
matter and pion production ent, the compression
are simultaneous
nuclei must be closely
creation of compressed nuclear
processes. Therefore,
to be consist-
has also to be taken into account by estimating
ence of the compression
on bombarding
for the
In the framework of our model we con-
energy from hydrodynamical
the depend-
calculations3,
295~
D. Vsak et al. / Pionic bremsstrahlung in heavy-ion collisions
although the validity of these results is questionable carbon and for energies
for nuclei as small as
as low as 20 MeV/n, where the mean free path of the
nucleus is large due to the action of the Pauli principle. Because the deceleration contribution
by the Coulomb force is very long range, its
to pion radiation
part of the available
is much less, but it takes away a considerable energy for heavy nuclei 2? We thus subtract the
scattering
energy Ecoul = ZpZT~Y/2R from the bombarding
In fig. 2(b) the experimental shown to coincide
energy Elab.
and theoretical
angle-integrated
in shape as well as in magnitude.
spectra are
The exponential
decay of
the cross sections with growing pion energy reflects the influence of the deceleration
as well as that of the gaussian form-factor,
the reaction zoneII. parameter
v is kept fixed at the value 0.38. At higher bombarding
experimental
is much
yield
other production isobarsI
mechanism
to contribute
too high. This is not surprising,
like the thermal production
Still, we find excellent
agreement with the pion production after these calculations
Ridge-GSI
collaboration"
GANIL8 (the yields are renormalized
and from Ar + Ca measured at 839 to be comparable
(dashed line) has been calculated
the upper bound for the thermal pion 12 with a hard in the shock-wave model
equation of state (Fermi gas) in which the temperatures higher than in the hydrodynamical 4T = ucl3
Rz
pound nucleus. Thermally
and the densities
d3p (2n)3
radius and T the temperature
created pions are obviously
of the compressed
negligible
low energy regime. The sum of the thermal and bremsstrahlung an overall quantitative An interesting
description
possibility
A classical
formula,
of the available
com-
in the very
pion yields gives
experimental
to test the time development
process would be a simultaneous
are
model:
7 [exp (E/T)-I]-' 0
where R, is the half-density
from a N + Ni
were already available
for the A-dependences
with the carbon data). For completeness,
otherm II
the deceleration
from the value v = 0.38 used at low energies.
collision at 35 MeV/n measured by the Stony Brook-Oak
energies the
since we expect
from highly excited
to the total pion yield. Moreover,
parameter may be different
yield
i.e., the shape of
In fig. 2(c) the pion excitation function is shown. The
data.
of the collision
measurement
of pion and y-ray bremsstrahlung8. 13 similar to (l), holds also for photons : z x f#) 1 exp [iw(t-z.Ei(t))ldt12 l-i;.;i(t)
(9)
296~
D. Vasak et al. / Pionic bremsstrahlung in heavy-ion collisions
where w is the photon frequency
and ?i the direction
of emission. Until now this 14 where also experimental
formula has been applied to low ion energy collisions
data15 exist. Our results, using the same nuclear trajectory duction, are presented
in fig. 3. If
move on a straight line the yield in the forward direction at 90' (c.m.) for symmetric differential
as for pion pro-
the sources of electromagnetic
radiation
is zero as it is
systems. These minima are clearly seen in the
cross section in the laboratory
intensity of high energy photons
system (fig. 3(a)). A significant
(25 MeV_'05 150 MeV) is predicted
(cf. fig.
3(b)).
*NJ3 s
15
c 5 E
10
3 B b
5 n 7.0
-0.5
0 0.5 cos kriaJ
10
u.
50 100 photon lab ekgy(MeV)
FIGURE 3 (a) The photon double-differential (b) The angle-integrated cross section at 84 MeV/n in the lab. in the lab.system. system. Recently, systems8.
photon bremsstrahlung
only. The reason is twofold:
the nuclear trajectories collisions
- the dependence
of the velocity
Firstly, our parametrization
of the deceleration
extrapolated parameter
T on the mass numbers
of refs.2and
Since, as shown above, 'the significant
the shape
of the total linear momentum
if AR = AT. To overcome these difficulties
trajectories
of
to asymmetric
nuclei has to be worked out. Secondly,
function must ensure conservation
during the collision the parametrized
model has been applied to syne-
cannot be unambiguously
AR and AT of the participating
y-ray spectra
spectra have been measured with asymmetric
But, up to now, the bremsstrahlung
tric collisions
150
6 by a classical
contribution
of the
we replace
dynamical model
16
.
29%
D Vasak et al. / Pionic bremsstrahlung in heavy-ion collisions
bremsstrahlung relativistic
pions occurs
concepts
in the low energy region only, we may apply non-
like friction etc.. We describe
nuclei by means of four collective
the collision
of two
degrees of freedom r, 8, eI and e2 (see
fig. 4), freezing all other collective
modes (e.g. surface vibrations). The corresponding
Lagrangian
L=${~;*+~r
2 ,$2
+ II 6: t I2 6;
is
1
(10)
- Vi') - V,(r) with the reduced mass p of the nucleus, moments of inertia II, I2 and the Coulomb and nuclear potentials
V,(r)
and V,,,(r)given by the folding integrals
Vi(?) = j d3rI' j d3r2' oI(?I')
x Q;-;,t;,) FIGURE 4 Definitions employed.
interaction
and a short range nuclear
iN
(r)
z-vo
(II)
i = 0,M
of the degrees of freedom
of a long range Coulomb
p,(Q
17
interaction
approximated
by a gaussian
18
,-(rlr0)*
(13)
V, = '2.36 MeV, r. = 2.3 fm with the overlap of the nuclear densities. The coupling to microscopic dependent
+ FI(r) = - K J rqP') &ere
v~,
degrees of freedom is accomplished
by a velocity
friction force
v2
- ;*(;')I p2(?-F)
are the velocity distributions
of the nuclei 1 and 2. The universal
pI(
d3r' = -;,(r)
(14)
and pI,p2 the density distributions
constant
K is the coefficient
of friction.
298~
LX Vasak et al. / Pionic bre~~~ahI~ng
in heavy-ion collisions
0.3 0.2 0.1 I
Oo
FIGURE 5 The c.m. velocity function$(t)]of a carbon nucleus in a symmetric collision at 50 MeVln as calculated in the dynamical model (solid lines) for zero impact parameter. With the value ~10~ ~eV.f~of the coefficient of friction the previously used form parametrized by T (dashed line) is well reproduced (up to a time-shift). Small friction (~=2000 MeV.fm2) gives rise to smaller decelerations. For b=1.6 fm the velocity component vz along the beam axis and vx perpendicular to it are also plotted (dotted lines).
In fig. 5 some results of the dynamical model are shown. For the impact parameter b=O the dynamical model indeed can reproduce ries. Furthermore,
the classical
also for impact parameters
collision
b (dotted lines).
we are now in the position to generalize
kinematics
I, The parameter
trajecto-
b # 0. The slope of the velocity function can be
seen to decrease with increasing Therefore,
the parametrized
dynamical model allows to find trajectories
the description
of the
in three respects:
T, which in principle
by a single parameters
depends on A p, AT and b, is replaced
, the coefficient
of friction.
2. The total cross section
is obtained
unambiguously
from
~1~ = 1 n* (b) b db rather than from u
*
= 0d.n
f (c.f. eq. (1);~~ is effectively
as used previously it has physically 3. The motion
reasonable
a parameter,
although
values).
of nuclei in collisions
at these energies
(down to, e.g. 20 MeV/n)
does not occur along straight lines, when b # 0. The contribution collisions
to the total cross section and their influence
distribution
could not be accounted
linear trajectories
for in the old modelzy6,
were allowed. Now even cases
nuclei stick together
nuclei we are forced to reconsider ations
(to firstorderboth,
statistical
can be included!
from collisions
the statistical
of unequal
model for isospin fluctu-
spin and isospin, can be described
model since it is based on the group structure
We propose the following
where only
where the participating
forming a rotating nuclear molecule
To apply these dynamics to pion bremsstrahlung
of such
on the angular
by the same
only).
picture:
the system of m participating nucleons in the collision gives rise to a non-zero mean isospin value T (m) which is completely determined by vector addition
rules. The computation
is quite easy if one benefits
[n/2, n/2 > x 11/2,1/2 > = (tl) I(ntl)/2, (n+1)/2 > of the Clebsch-Gordan
from the property
n = 0,1,2,...
(15)
coefficients.
A system of Z protons and N neutrons with total isospin T' according
(Z+N=m) can then be coupled to states
to
/Z/2, Z/2 ' x /N/2$4/2 > = 1 am(T')!T;(Z-N)/2 >
(16)
T' where
4T')
=
1(1/2,
can be interpreted
N/2,+T'I Z/2.-N/Z, as the probability
(z-~)/2)1' for the state
(17) IT',$%.
The mean isospin is defined simply by 7 (m) = 1 T'
a,(T') T'
(18)
3ooc
D. Vasak et al. 1 Pionic bremsstrahlung in heavy-ion collisions
In fig. 6 this expression is plotted against m (long-dashed line) for symmetric systems
(N = Z = m/Z), We see that the mean isospin may be approximated
quite
well by Y(m) = vm/3. Now we have to connect m, the number of participating numbers, Ap and AT , of the collision central symmetric
heavy-ion
partners.
collisions
nucleons with the mass
Under the assumption
all nucleons
participate
that in
we have
m = Al, + AT = 2A. The resulting
(20)
value WT = r(A) = V2A/3.
is somewhat modification
(21)
greater than the value in eq. (7) (this will lead to a slight of the parameter
v). For asymmetric
systems or/and non-zero
parameters we simply have to replace m by an effective
impact
number of participbnts,
i.e. m = AFff (b) + AFff (b)
(22)
with suitably chosen function Aieff (b). Because the final T3 component
is fixed by half the neutron excess
eq. 6) and T' cannot be smaller than 7. The dash-dotted
fine in fig. 6 displays r(m) for various realistic
when all involved nucleons excess this enhancement
participate.
for those
of
systems
It can be seen that for a large neutron
can be quite dramatic,
ment for pion bremsstrahlung
(c.f.
ITSI, N>Z gives rise to an enhancement
resulting
combinations
in a similar enhance-
of colliding
nuclei.
We conclude that 1. In particular
at low bombarding
can help to understand
energies
the bremsstrahlung
the (synnnetric) pion production
Within our bremsstrahlung
mechanism
cross section.
model the II'-data of refs. 7,8 and 9 can be
explained with the unique deceleration eration is similar to that obtained
parameter
v = 0.38. The decel-
from nuclear hydrodynamics3.
Other
pion production mechanisms 12,1g are suppressed in this enerqv reaime. 2. The forward-baclanardpeaking of the experimental angular distributions iS quantitatively
explained,
target are coherently threshold
if the pion radiation
added, supporting
pion production.
from the projectile
the cooperative
and
nature of sub-
301c
D. Vasak et al. / Pionic bretnsstrahlung in heavy-ion collisions
Number of h-ticipants
m
FIGURE 6 The mean isospin value T(m) is plotted as a function of the number of participants m for systems with N=Z. The dashed line gives the exact result according to eq. (18), which can be approximated by m (full line). To indicate the maximal possible isospin T = m/2 is drawn. With realistic target and projectile combinations all available nucleons coupled together (m = AP+AT) the dash-dotted curve is obtained. Significant deviations from the symmetrical case occur for m r 150. 3. Information contained
about the time-development
in electromagnetic
v-radiation
also processes
check the underlying
of the collision process is also
processes.
We therefore suggest that besides
such as electron
collision
emissions can be used to
dynamics.
4. To investigate
asynnietric collisions
and the A-dependence
bremsstrahlung
a slight modification
of the so far used parametrization
of the nuclear trajectories tical estimate
of the pion
is proposed together with an improved statis-
for the spin-isospin
strength.
Numerical work along these
lines is in progress. We thank G. Buchwald; G. Graebner and J. Maruhn for providing us with their results. We also acknowledge discussions with P. Braun-Munzinger, E. Grosse, J. Julien, Ch. Michel, H. Nell, F. Obenshain and P. Paul and are grateful that they made their data available to us prior to publication.We are grateful for a stimulating discussion with C.S. Warke.
D. Vasak et al. f Pionic ~remss~rahI~~~in heavy-ion cotIisions
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303c
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713.
G. Soff, J. Reinhardt, B. MUller and W. Greiner, Z. Phys. A 294 (1980) 137, and private communication.
21) Nevertheless we expect pion bremsstrahlung also from Rutherford trajectories as long as the total energy is above the pion threshold. It% should be observable in, e.g., U+U collisions at (or slightly below) the Coulomb barrier. There is also the process of fusion of nuclei by pion emission, allowing for the production of rather cold compound nuclei. This process was first predicted many years ago by M.G. Huber and his collaborators.(K. Klingenbeck, M. Dillig and M.G. Huber, Phys. Rev. Lett. 47 (1981) 1654).
Nuclear Physics A428 (1984) 291~304~ North-Holland, Amsterdam
PIONIC BREMSSTRAHLUNG
IN HEAVY-ION
COLLISIONS+
David VASAK Gesellschaft fir Schwerionenforschung, 6100 Darmstadt 11, West Germany
PlanckstraRe
1, Postfach 11 05 41,
, Berndt MILLER, Thomas STAHL and Mark UHLIG,
Walter GREINER*
Institut fiir Theoretische Physik der Johann-Wolfgang-Goethe Robert-Mayer-StraRe 8-10, Postfach 11 19 32, 6000 Frankfurt am Main 11, West Germany
Universitat,
We apply the bremsstrahlung model for pion production to symmetric nuclear collisions at energies far below the single NN-threshold. All main features of recent neutral pion measurements can be explained by a single value of the deceleration parameter of the theory which is consistent with hydrodynamica1 calculations. Since the y-ray bremsstrahlunq also reflects the kinematics of its source, a simultaneous measurement-of pion and Y-ray bremsstrahlung should be used to further test the collision dynamics. Results of our calculations for both kinds of radiation are presented. An extension of this approach to asymmetric systems is proposed.
Subthreshold recent years,
pion production particularly
(or cooperative)
has become of considerable
effects. The idea to consider
anism for the production
of secondary
a bremsstrahlung-type
particles
not newl. If this model can be shown to describe collisions
In the following we apply the bremsstrahlung
2 do A=q dsldE
pion production
'
I =--(I 3 '
about the time develop.2 nuclei .
model to collisions
threshold
of equal
for pion production.
cross section for radiation of neutral pions is
to the Fourier-transformof
d2n 71 dodE
is
in nuclear
structure of the colliding
nuclei far below the single nucleon-nucleon
proportional
mech-
in hadronic collisions
it carld be a useful source of information
ment of the reaction and the spin-isospin
The inclusive differential
interest during
in connection with the search for cumulative
1 I; 3 II
P
the current density.
? ': sP,sT
' 13k(p)?
(1)
k
fsupported by the Bundesministerium fir Forschung und Technologie. "Invited Speaker at the Conference on Theoretical Approaches to Heavy Ion Reactions Mechanisms, Paris, 14-18 May 1984
0375-9474/84/$03.00 0 Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
292c
D. Vasak et al. / Pionic bremsstrahlung
in heavy-ion collisions
with n L
_C_ clj (PII*=; k
‘TSP
$
k
I*WT(A) *W,(A)]*
IK(p)I*
(2)
where E is the pion energy, p = (E*-m*)l'*, m = 135 MeV the mass of the neutral pion, M = 930 MeV that of a bound nucleon and o the pion solid angle. The pion 2 nucleon coupling constantgohas the usual value go /4~ = 14. Averaging over the spin direction
fi of both nuclei
(i = P projectile,
i = T target) is indicated
by i. The index k runs over the three isospin values of the pion; the factor l/3 in (1) and (2) accounts
for the fact that (1) describes
only one sort of
pions, in this case the neutral ones. For equal nuclei the factor cl0 has been set equal the geometrical
cross section 4nR2 with R being the half-density
radius of the gaussian density P(r) = '7~~ exp T-(r/*a)*l of the colliding
nuclei
n = P/p, the compression, condition).
(3)
(p, = 0.17 fm l/a = &
-3
is the normal nuclear density and
(o/A)~'~
is obtained
from the normalization
In
K(P) = z i=P,T
+I- [C(pi) ("i'p)/(ui'p)l' - 0)
exp[i(Et-c.iti(t))ldt
(4)
(prime denotes total time-derivative) all the information
about collision
are the (linear) trajectories system, parametrized multiple
dynamics
is contained.
$(t)
= +R(t)gz
of the nuclear centers in the center-of-mass
by the deceleration
of the "passing time" T
time T. We express T as a unique
= R/(2yin
vin), where
vin is the initial
c.m. velocity and yin = (l-v:,) -S1'2, i.e.,
T
=
with an
v*
(5)
S
adjustable
energy-independent
parameter
u.
0: the spin-vectors
boosted from the ' are the corresponding four-velocities, 'i particle's rest frame into the c.m. system, and z(p) the Fourier-transform of (3) with BT(t) = I$]* - (S.Gi(t)* taking time-dependent
Lorentz-contraction
(5) into account. The statistical
factors
D. Vasak et al. / Pionic bremsstrahlung in heavy-ion collisions
WS(A) and WT(A) result from single statistical spin-isospin-strength fluctuations
wS
estimation
293~
of the nuclei's
to which the pions couple and which is non-zero due to
during the collision
(see below).
In ref. 2 we have used
= WT ZJFi-. 7
(7)
The parametrization hydrodynamical
used here was strongly motivated
calculations
which show excellent
by the results of
agreement with the experimental
findings even at low energies.
In fig. 1 the calculated3proton
compared with the experimental
yield from a I2c + Ig7Au collision
spectrum
is
at 84 MeV/n.
A short remark to the role of the spin
z c
'2C+197A" 84MNn II'-
-
Experiment
----
Theory
2 E E 5
density is appropriate:
In the schema-
tic approach of ref. 2 it was assumed that the spin vectors of the projectile and target are not correlated This assumption
addition of the pion yields
II'-
at all.
leads to incoherent from both
nuclei. On the other hand, in the hydrodynamical collisions
picture of heavy ion
the nucleons
tion zone, consisting
l-
nuclear matter,
in the reac-
of hot compressed
do not "remember"
their origin. Thus there is only one spin direction,
lo-'-
I
0
and the pion yields
should be added coherently. I
I
100
50
[MeVI, 1 150
FIGURE 1 Comparison of the present theory and the ex erimental data for 12C(84 MeV/nucleon) + P g7A~+p + x. Full lines correspond to the experiment; dashed lines represent the calculations (from ref.3).
coherent addition pion production
at 90' in the c.m.
system, in case of coherence tive interference forward-backward
sideways
angular distribution
of pion radiation
appear preferably collisions
is a consequence
take place and lead to occupation addition
in the
and gives a In fact, a
of spin fluctuations4,
in the reaction zone, where the individual
light, the coherent
a nega-
leaves a
enhancement
smaller total pion yield. non-zerosource
While in-
leads to enhanced
which
nucleon-nucleon
of vacant states. Seen in this
is the natural one.
In order to avoid complications
caused by the Coulomb force when looking at
charged pion data5 we compare our calculations6
(cf. fig. 2) with recent ex-
periments on neutral pion production 7,8,9 . The measured pion angular distributions exhibit a forward-backward
behaviour
and thus favour the coherent addition
294~
D. Vasak et at. / Pionic ~rems~tra~l~ng in heavy-ion collisions
0
-1.0 -0.5
0 0.5 cos~~c‘m,l
50 100 150 200 pion cm. beg
0
1.0
FIGURE 2 (a) Theangulardistribution forpions produced at 84 MeVin in the energy region O-50 MeV (top), 50-100 MeV and loo-150 MeV, respectively. The data are from ref. 7. (b) The angle-integrated spectra at ion energies between 20 and 84 MeV/n. (c) The excitation function. The data are taken from refs.7 (solid dots),5 (open circles), 8 (open triangles), 9 (solid triangles), and 10 (solid squares). The dashed line exhibits the thermal pions. of the pion yields
from the projectile
and target nuclei: in fig, Z(a) the experimental
pion angular distribution
are compared with our results. The7 dashed curve in the upper energy cut is obtained by incoherent addition the deceleration adjusted
(here
has been
to v = 0.55). The solid lines show the results from coherent addition.
In this case the parameter negative
parameter
interference
v = 0.38 is somewhat smaller to c~pensate
mentioned
above.
clude that the spin distributions correlated
of the participating
or, in other words, that deceleration,
matter and pion production ent, the compression
are simultaneous
nuclei must be closely
creation of compressed nuclear
processes. Therefore,
to be consist-
has also to be taken into account by estimating
ence of the compression
on bombarding
for the
In the framework of our model we con-
energy from hydrodynamical
the depend-
calculations3,
295~
D. Vsak et al. / Pionic bremsstrahlung in heavy-ion collisions
although the validity of these results is questionable carbon and for energies
for nuclei as small as
as low as 20 MeV/n, where the mean free path of the
nucleus is large due to the action of the Pauli principle. Because the deceleration contribution
by the Coulomb force is very long range, its
to pion radiation
part of the available
is much less, but it takes away a considerable energy for heavy nuclei 2? We thus subtract the
scattering
energy Ecoul = ZpZT~Y/2R from the bombarding
In fig. 2(b) the experimental shown to coincide
energy Elab.
and theoretical
angle-integrated
in shape as well as in magnitude.
spectra are
The exponential
decay of
the cross sections with growing pion energy reflects the influence of the deceleration
as well as that of the gaussian form-factor,
the reaction zoneII. parameter
v is kept fixed at the value 0.38. At higher bombarding
experimental
is much
yield
other production isobarsI
mechanism
to contribute
too high. This is not surprising,
like the thermal production
Still, we find excellent
agreement with the pion production after these calculations
Ridge-GSI
collaboration"
GANIL8 (the yields are renormalized
and from Ar + Ca measured at 839 to be comparable
(dashed line) has been calculated
the upper bound for the thermal pion 12 with a hard in the shock-wave model
equation of state (Fermi gas) in which the temperatures higher than in the hydrodynamical 4T = ucl3
Rz
pound nucleus. Thermally
and the densities
d3p (2n)3
radius and T the temperature
created pions are obviously
of the compressed
negligible
low energy regime. The sum of the thermal and bremsstrahlung an overall quantitative An interesting
description
possibility
A classical
formula,
of the available
com-
in the very
pion yields gives
experimental
to test the time development
process would be a simultaneous
are
model:
7 [exp (E/T)-I]-' 0
where R, is the half-density
from a N + Ni
were already available
for the A-dependences
with the carbon data). For completeness,
otherm II
the deceleration
from the value v = 0.38 used at low energies.
collision at 35 MeV/n measured by the Stony Brook-Oak
energies the
since we expect
from highly excited
to the total pion yield. Moreover,
parameter may be different
yield
i.e., the shape of
In fig. 2(c) the pion excitation function is shown. The
data.
of the collision
measurement
of pion and y-ray bremsstrahlung8. 13 similar to (l), holds also for photons : z x f#) 1 exp [iw(t-z.Ei(t))ldt12 l-i;.;i(t)
(9)
296~
D. Vasak et al. / Pionic bremsstrahlung in heavy-ion collisions
where w is the photon frequency
and ?i the direction
of emission. Until now this 14 where also experimental
formula has been applied to low ion energy collisions
data15 exist. Our results, using the same nuclear trajectory duction, are presented
in fig. 3. If
move on a straight line the yield in the forward direction at 90' (c.m.) for symmetric differential
as for pion pro-
the sources of electromagnetic
radiation
is zero as it is
systems. These minima are clearly seen in the
cross section in the laboratory
intensity of high energy photons
system (fig. 3(a)). A significant
(25 MeV_'05 150 MeV) is predicted
(cf. fig.
3(b)).
*NJ3 s
15
c 5 E
10
3 B b
5 n 7.0
-0.5
0 0.5 cos kriaJ
10
u.
50 100 photon lab ekgy(MeV)
FIGURE 3 (a) The photon double-differential (b) The angle-integrated cross section at 84 MeV/n in the lab. in the lab.system. system. Recently, systems8.
photon bremsstrahlung
only. The reason is twofold:
the nuclear trajectories collisions
- the dependence
of the velocity
Firstly, our parametrization
of the deceleration
extrapolated parameter
T on the mass numbers
of refs.2and
Since, as shown above, 'the significant
the shape
of the total linear momentum
if AR = AT. To overcome these difficulties
trajectories
of
to asymmetric
nuclei has to be worked out. Secondly,
function must ensure conservation
during the collision the parametrized
model has been applied to syne-
cannot be unambiguously
AR and AT of the participating
y-ray spectra
spectra have been measured with asymmetric
But, up to now, the bremsstrahlung
tric collisions
150
6 by a classical
contribution
of the
we replace
dynamical model
16
.
29%
D Vasak et al. / Pionic bremsstrahlung in heavy-ion collisions
bremsstrahlung relativistic
pions occurs
concepts
in the low energy region only, we may apply non-
like friction etc.. We describe
nuclei by means of four collective
the collision
of two
degrees of freedom r, 8, eI and e2 (see
fig. 4), freezing all other collective
modes (e.g. surface vibrations). The corresponding
Lagrangian
L=${~;*+~r
2 ,$2
+ II 6: t I2 6;
is
1
(10)
- Vi') - V,(r) with the reduced mass p of the nucleus, moments of inertia II, I2 and the Coulomb and nuclear potentials
V,(r)
and V,,,(r)given by the folding integrals
Vi(?) = j d3rI' j d3r2' oI(?I')
x Q;-;,t;,) FIGURE 4 Definitions employed.
interaction
and a short range nuclear
iN
(r)
z-vo
(II)
i = 0,M
of the degrees of freedom
of a long range Coulomb
p,(Q
17
interaction
approximated
by a gaussian
18
,-(rlr0)*
(13)
V, = '2.36 MeV, r. = 2.3 fm with the overlap of the nuclear densities. The coupling to microscopic dependent
+ FI(r) = - K J rqP') &ere
v~,
degrees of freedom is accomplished
by a velocity
friction force
v2
- ;*(;')I p2(?-F)
are the velocity distributions
of the nuclei 1 and 2. The universal
pI(
d3r' = -;,(r)
(14)
and pI,p2 the density distributions
constant
K is the coefficient
of friction.
298~
LX Vasak et al. / Pionic bre~~~ahI~ng
in heavy-ion collisions
0.3 0.2 0.1 I
Oo
FIGURE 5 The c.m. velocity function$(t)]of a carbon nucleus in a symmetric collision at 50 MeVln as calculated in the dynamical model (solid lines) for zero impact parameter. With the value ~10~ ~eV.f~of the coefficient of friction the previously used form parametrized by T (dashed line) is well reproduced (up to a time-shift). Small friction (~=2000 MeV.fm2) gives rise to smaller decelerations. For b=1.6 fm the velocity component vz along the beam axis and vx perpendicular to it are also plotted (dotted lines).
In fig. 5 some results of the dynamical model are shown. For the impact parameter b=O the dynamical model indeed can reproduce ries. Furthermore,
the classical
also for impact parameters
collision
b (dotted lines).
we are now in the position to generalize
kinematics
I, The parameter
trajecto-
b # 0. The slope of the velocity function can be
seen to decrease with increasing Therefore,
the parametrized
dynamical model allows to find trajectories
the description
of the
in three respects:
T, which in principle
by a single parameters
depends on A p, AT and b, is replaced
, the coefficient
of friction.
2. The total cross section
is obtained
unambiguously
from
~1~ = 1 n* (b) b db rather than from u
*
= 0d.n
f (c.f. eq. (1);~~ is effectively
as used previously it has physically 3. The motion
reasonable
a parameter,
although
values).
of nuclei in collisions
at these energies
(down to, e.g. 20 MeV/n)
does not occur along straight lines, when b # 0. The contribution collisions
to the total cross section and their influence
distribution
could not be accounted
linear trajectories
for in the old modelzy6,
were allowed. Now even cases
nuclei stick together
nuclei we are forced to reconsider ations
(to firstorderboth,
statistical
can be included!
from collisions
the statistical
of unequal
model for isospin fluctu-
spin and isospin, can be described
model since it is based on the group structure
We propose the following
where only
where the participating
forming a rotating nuclear molecule
To apply these dynamics to pion bremsstrahlung
of such
on the angular
by the same
only).
picture:
the system of m participating nucleons in the collision gives rise to a non-zero mean isospin value T (m) which is completely determined by vector addition
rules. The computation
is quite easy if one benefits
[n/2, n/2 > x 11/2,1/2 > = (tl) I(ntl)/2, (n+1)/2 > of the Clebsch-Gordan
from the property
n = 0,1,2,...
(15)
coefficients.
A system of Z protons and N neutrons with total isospin T' according
(Z+N=m) can then be coupled to states
to
/Z/2, Z/2 ' x /N/2$4/2 > = 1 am(T')!T;(Z-N)/2 >
(16)
T' where
4T')
=
1(1/2,
can be interpreted
N/2,+T'I Z/2.-N/Z, as the probability
(z-~)/2)1' for the state
(17) IT',$%.
The mean isospin is defined simply by 7 (m) = 1 T'
a,(T') T'
(18)
3ooc
D. Vasak et al. 1 Pionic bremsstrahlung in heavy-ion collisions
In fig. 6 this expression is plotted against m (long-dashed line) for symmetric systems
(N = Z = m/Z), We see that the mean isospin may be approximated
quite
well by Y(m) = vm/3. Now we have to connect m, the number of participating numbers, Ap and AT , of the collision central symmetric
heavy-ion
partners.
collisions
nucleons with the mass
Under the assumption
all nucleons
participate
that in
we have
m = Al, + AT = 2A. The resulting
(20)
value WT = r(A) = V2A/3.
is somewhat modification
(21)
greater than the value in eq. (7) (this will lead to a slight of the parameter
v). For asymmetric
systems or/and non-zero
parameters we simply have to replace m by an effective
impact
number of participbnts,
i.e. m = AFff (b) + AFff (b)
(22)
with suitably chosen function Aieff (b). Because the final T3 component
is fixed by half the neutron excess
eq. 6) and T' cannot be smaller than 7. The dash-dotted
fine in fig. 6 displays r(m) for various realistic
when all involved nucleons excess this enhancement
participate.
for those
of
systems
It can be seen that for a large neutron
can be quite dramatic,
ment for pion bremsstrahlung
(c.f.
ITSI, N>Z gives rise to an enhancement
resulting
combinations
in a similar enhance-
of colliding
nuclei.
We conclude that 1. In particular
at low bombarding
can help to understand
energies
the bremsstrahlung
the (synnnetric) pion production
Within our bremsstrahlung
mechanism
cross section.
model the II'-data of refs. 7,8 and 9 can be
explained with the unique deceleration eration is similar to that obtained
parameter
v = 0.38. The decel-
from nuclear hydrodynamics3.
Other
pion production mechanisms 12,1g are suppressed in this enerqv reaime. 2. The forward-baclanardpeaking of the experimental angular distributions iS quantitatively
explained,
target are coherently threshold
if the pion radiation
added, supporting
pion production.
from the projectile
the cooperative
and
nature of sub-
301c
D. Vasak et al. / Pionic bretnsstrahlung in heavy-ion collisions
Number of h-ticipants
m
FIGURE 6 The mean isospin value T(m) is plotted as a function of the number of participants m for systems with N=Z. The dashed line gives the exact result according to eq. (18), which can be approximated by m (full line). To indicate the maximal possible isospin T = m/2 is drawn. With realistic target and projectile combinations all available nucleons coupled together (m = AP+AT) the dash-dotted curve is obtained. Significant deviations from the symmetrical case occur for m r 150. 3. Information contained
about the time-development
in electromagnetic
v-radiation
also processes
check the underlying
of the collision process is also
processes.
We therefore suggest that besides
such as electron
collision
emissions can be used to
dynamics.
4. To investigate
asynnietric collisions
and the A-dependence
bremsstrahlung
a slight modification
of the so far used parametrization
of the nuclear trajectories tical estimate
of the pion
is proposed together with an improved statis-
for the spin-isospin
strength.
Numerical work along these
lines is in progress. We thank G. Buchwald; G. Graebner and J. Maruhn for providing us with their results. We also acknowledge discussions with P. Braun-Munzinger, E. Grosse, J. Julien, Ch. Michel, H. Nell, F. Obenshain and P. Paul and are grateful that they made their data available to us prior to publication.We are grateful for a stimulating discussion with C.S. Warke.
D. Vasak et al. f Pionic ~remss~rahI~~~in heavy-ion cotIisions
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21) Nevertheless we expect pion bremsstrahlung also from Rutherford trajectories as long as the total energy is above the pion threshold. It% should be observable in, e.g., U+U collisions at (or slightly below) the Coulomb barrier. There is also the process of fusion of nuclei by pion emission, allowing for the production of rather cold compound nuclei. This process was first predicted many years ago by M.G. Huber and his collaborators.(K. Klingenbeck, M. Dillig and M.G. Huber, Phys. Rev. Lett. 47 (1981) 1654).