Two kinds of number density distributions of the nucleus, harmonic well and Woods-Saxon models, are used with the t-matrix that is taken from the scattering experiments to find a simple optical potential. The parameterized two body inputs, which are kaon-nucleon total cross sections, elastic slope parameters, and the ratio of the real to imaginary part of the forward elastic scattering amplitude, are shown. The eikonal approximation was chosen as the solution method to estimate the total and absorptive cross sections for the kaon-nucleus scattering

Buck, Warren W.

Hong, Byungsik

Maung, Khin M.

English

NASA Technical Reports Server

N91-28086
KAON - NUCLEUS SCATrERING
J3z.r
j_ 6 of
Byungsik Hong, Warren W. Buck and Khin M. Maung
Department of Physics
Hampton University
Hampton, Virginia
Technical Monitors
John W. Wilson and L.W. Townsend
NASA Langley Research Center
Hampton, Virginia
ABSTRACT
Two kinds of number density distributions of the nucleus, harmonic well and Woods-Saxon models,
are used withib_ t-matrix that is taken from the scattering experiments to find a simple optical potential.
The parameterized two body inputs, which are kaon-nucleon total cross sections, elastic slope param-
eters, and the ratio of the real to imaginary part of the forward elastic scattering amplitude, are shown.
The eikonal approximation was chosen as our solution method to estimate the total and absorptive
cross sections for the kaon-nucleus scattering.
146
https://ntrs.nasa.gov/search.jsp?R=19910018772 2020-03-19T16:43:48+00:00Z
KAON - NUCLEUS SCAITERING
For the scattering of the high energy incoming particle, particulary a meson, by a nucleus, it is
expected that the situation can be considered in terms of its scattering by the constituent nucleons of
the nucleus individually. This means the many body problem may be reduced to a collection of the two
body interactions.
One of the successful scattering frameworks is the multiple scattering theory. A simple picture of
the scattering of elementary particles by a nucleus is to view the scattering in terms of the projectile
interactions with each single constituent of the nucleus (single scattering). There may be other terms
contributing to the scattering such as the projectile interacting with two consecutive constituents
(double scattering). Similarly there are contributions from three, four, and more successive scatterings.
The formalisms using this picture are called multiple scattering theories) '2'3 It is clear from this
description that the scattering from a nucleus is determined from the amplitude for the scattering of
the projectile from a single target constituent, i.e., a two body scattering amplitude. 4
A general multiple scattering theory for scattering between two nuclei (neglecting three body
interaction) has previously been developed by Wilson: "5The reaction for the hea T ion projectile which
is the previous application of Wilson's theory was well developed by Wilson et al. And the application
to the antiproton, antideuteron and antinuclei was also done by Buck et al. 7 giving good agreement with
the available experimental data.
A successful feature of multiple scattering theories manifests itself when combined with the optical
model. This allows the optical potential for elementary particle scattering from a nucleus to be
determined from more fundamental quantities such as the two body scattering amplitude and the target
number density function.
W(R) = A f d3r'pA(r ') t (e,r' - R) (1)
where R is the distance between the origin and projectile particle, r' is the separation between the origin
and the considered target constituent,A is the target mass number, e is the kaon-nucleon kinetic energy
in the center of the mass frame, p,4 is the target number density, and t is the energy dependent
kaon-proton transition amplitude obtained from scattering experiments. The origin was taken at the
center of the mass of the target nucleus. In the eikonal context, the scattering amplitude for the
kaon-nucleus scattering is given by
f (k,O) = k f . b_-[ d2be i q [eix (1,:,)_ 1] (2)
where
1 2/z f: W(b_)dz (3)x(bJ_) = 2k _2 ®
147
is theeikonalscatteringphaseshiftfunction,b is the impact parameter, and/z is the reduced mass of
the K-nucleus system. Using the above scattering amplitude and the optical theorem, we can obtain the
total and absorption cross section expressed as follows:
o,o , = 2f d2b [1 - e _ C*_)cos (Re x ObJ¢) ] (4)
and
/-
Oabsoq,aon = .J d2b [1 - e -ztm_ 0,_) ]. (5)
The number density pA of nuclear matter can be extracted from the corresponding charge density pc
by assuming 7
pc (r) = f pN (r')pa (r + r') d3r ' (6)
wherepc is the nuclear charge distribution, pN is the nucleon charge distribution, and pA is the nuclear
single particle density, pN was taken to be the usual Gaussian function
3 ) 3/z 3r_
pN (r) = ( -_ exp ( ---_ ) (7)
with the nucleon root-mean-square charge radius set equal to the proton value of 0.87fro 12.
For nuclei withA <20 we used an harmonic well form ofpc as
pc(r)=po[I+y(_)ZIe_p(-_) (8)
with the charge distribution parameters y and a listed in Table I. Substituting Eq.(7) and Eq. (8) into
Eq.(6), we get
P°a---_3[I + 37 - 3-3-Z-_-+ a_]exp(- _2-) (9)pA (r) = 8s 3 2 8s 2 16S4
s2 =a 2 r2N
4 6" (10)
where
For target nuclei with A _ 20 we choose a Woods-Saxon form of charge distribution
pc(r) =po [1 + exp (L-_ -R-) ]-1 (11)
where the parameters R and c are given by
148
and
R = to3 (12)
tc (13)
c = _-.-.-._
and r0.5 is the radius at half-density with tc representing the skin thickness. Values for R and tc are also
shown in table I. If we substitute Eqs. (7) and (11) into Eq. (6), we'll see the nuclear single particle
density that is also of Woods-Saxon form with the same R, but different normalization coefficient, p0,
and surface thickness. The latter is given (in fm ) by
IA = [5.08rNln (_)1-1 (14)
and
fl =exp (2.54_cN). (15)
We choose the kaon-nucleon scattering amplitude to be the function of momentum transfer q as
k oT(i + a)e -Bqz/2 (16)f (q) = -_
where k is the wave number of the kaon-nucleon system, or is the kaon-nucleon total cross section, a
is the ratio of the real to imaginary part of the forward elastic scattering, and B is the slope parameter.
Then the t-matrix in the coordinate space is
'e-eh2 .. e-r2/(2B)1
tCr) = --_]¥ ._"t eT(' + a)(2riB)a/2 (17)
where e is the total kinetic energy in the c.m. frame and/_' is the reduced mass of the colliding particles.
We compared our scattering amplitude, Eq. (16), with the experimental data of the K+p differential
cross section taken from Ref.13 and show the results in Figs. 1 and 2. In the case of the kaon-nucleus
scattering, the comparison to our calculation utilizing eikonal scattering amplitude, Eq. (2), with the
experimental data 9 for d 2 are displayed in Figs. 3 and 4. The optical potentials coming from the above
informations for K + and K- projectiles and C 12target are shown in Figs. 5 and 6.
The total cross sections of the K -x- proton system may be fitted by the following form:
or = ab (P_r b) + E gi (t_r b) (18)
i
where PLKabis the corresponding kaon laboratory momentum in GeV/c and the unit of a is mb. The
typical form ofgi (P_Kb) is that
149
gi (tdg_Kb) = mi
b 2]2 +1
(19)
where Pn_ is the resonance momentum of the ith peak and mi and ni are the constants which have to be
determined from the data.
Eq. (18) actually corresponds to the superpostion of Breit-Wigner formulae where Ob (p_,_b) is a
non-resonant background. Our fitting constants in the resonance energy range are displayed in table
II. The background functions are fitted by
ab (PLKaKb) = 23.2e -°'°°3e_x (20)
for K-p scattering and
as ( ab) = 16.8 -- 1
yea _ (21)
for K+ p scattering. When PtgaKb is less than 0.7 GeV/c in K-p scattering, we used the following function:
aT = 172.38e -2'°( xe_' + 0.1) (22)
In the more high energy region, the empirical formulae for Kp system are given by
24.63
ar = 20.18 + pLKa------T (23)
when PLxab > 20GeV/c in K-p system and
0.22
aT = 16.8 - pLKa-----g (24)
when/_K ab > 5GeV/c in K + p system,
The fitting formulae which we used for K _ - N systems are Eq. (25) through (28).
For K- n scattering,
aT =-1033.0 + 1060.0eLK ab-°'°3 + 28.0In_K ab
when ILKab >_2.5GeV/c and
aT = 23.91 + 17.0e -(e_Kb - l"°)z/°'a2
when/_K a_ < 2.5GeV/c. And for K + n scattering,
ar = 18.4 +175.0P_ ah-7_ + 0.2lnZP_K b
(25)
(26)
- 0.751nt_K ab (27)
150
whentCrab >- 2.5GeV/c and
aT = a (18.0 + 3.0e -(xP_ - 1"2)2/°°8) (28)
when pLKab < 2.5 GeV/c. The constant a is equal to 1 when P_Kb > 1.2GeV/c and 0.94 when
I_Kab < 1.2GeV/c.
We determined the fitting formulae of slope parameter, B, in the range of
0.1-< I t [ _< 0.4(GeV/c) 2. The resultant formulae are shown below.
B = 7.3 (29)
for K- p scattering and
B = -24.3 + 7.01n (s + 50.0) (30)
for K + p interaction. The square of the momentum transfer t (GeV/c)2,, and the invariant mass squared,
s(GeV/c2), are two convenient Mandelstam variables.
Our parameterized formulae, a, for K-p and K + p scattering, respectively, are as follows.
0.32 52_a = - sin{ PLrab -- 0.2)} (31)
1.0 + 0.9pLKabz
for K- p scattering and
a = - 1.86e -(_ab - °1)2/°41 --0.44 (32)
for K + p interaction where PLr'_bis the kaon laboratory monentum in GeV/c.
The total and absorptive cross sections for K +- --.4127 scattering using the harmonic well and
Woods-Saxon single particle densities are shown in Figs. 7 and 8. Although we cannot evaluate the exact
error due to the scaricty of experimental data, we expect that our calculation reasonably represents the
situation, because of the good agreement between our calculation and the experimental data of other
quantities such as the differential cross sections of kaon-proton and kaon-nucleus scattering.
In case ofK + p scattering our scattering amplitude was quite reasonable. Moreover, the calculation
of the diffcrential cross section gave us more accurate result as the energy of projectile went higher.
For K +- -C 12scattering the differential cross sections are estimated lower than the experimental results.
It may be natural because we used a very simplified, non-relativistic optical potential. At the very forward
angle the experiments show us the differential cross sections rise very rapidly, but the calculations
converge smaller values. This comes from the fact that we have neglected the contribution of Coulomb
interaction which is the long range interaction in the optical potential.
The total and absorptive cross sections for K- - nucleus scattering were bigger than those ofK +
- nucleus scattering. This, of course, indicates that K- meson interacts with nucleons stronger than
K + meson. According to the quark model, K + resonant peak requires the formation of five-quark
151
objectswhichhaveneverbeenfound,but theK-- N interactions are so strong that the cross sections
are roughly comparable to those of nucleon-nucleon scattering.
K + -nucleus total cross section shows almost perfect linear dependence of A and K- - nucleus
total cross section also represents linearity of A at high energies, but exhibits a little complexity at low
energies. Other things that we can give attention to are that the absorptive cross section ofK + - nucleus
scattering depends on A °82 linearly, but K- - nucleus absorptive cross section does not show fairly good
linear dependence of A °82 as in the case of total cross section at relatively low energies. The linear
dependence of A °'82 of absorptive cross section tells us that the nucleus is not perfectly black to the
kaon projectile.
The results of this work will be used in the hadron transport computer code at NASA Langley
Research Center (Ref. 18).
References
1. Watson, K.M., Phys. Rev. 1953, 89, 575
2. Frank, R.M.; Gammel, J.L.; Watson, ICM., Phys. Rev. 1956, 101, 891
3. Watson, K.M., Phys. Rev. 1957, 105, 1388
4. Wilson, J.W., A Dissertation for Ph.D, 1975
5. Wilson, J.W.; Phys. Lett. B 1974, 52, 149
6. Wilson, J.W.; Townsend, L.W., Can. J. Phys. 1981, 59, 1569
7. Buck, W.W.; Norbury, J.W.; Townsend, L.W.; Wilson, J.W., Phys. Rev. C 1986, 33, 234
8. Aguilar-Benitez, M. etal., "Particle Properties Data Booklet", North-Holland, Amsterdam,
April 1986
9. Marlow, D.; Varnes, P.D.; Colella, N.J.; Dytman, S.A.; Eisenstein, R.A.; Grace, R.;
Takeutchi, F.; Wharton, W.R.; Bart, S.; Hancock, D.; Hackenberg, R.; Hungerford, E.;
Mayes, W.; Pinsky, L.; Williams, T.; Chrien, R.; Palevsky, H.; Sutter, R., Phys. Rev. C
1982, 25, 2619
10. Siegel, P.B.; Kaufmann, W.B.; Gibbs, W.R., Phys. Rev. C 1984, 30, 1256
11. Goldberger, M.L.; Watson, K.M., "Collision Theory", John Wiley & Sons, New York,
1964
12. Borkowski, F.; Simon, G.G.; Walther, V.H.; Wendling, R.D., Z. Phys. A 1975, 275, 29
13. Price, LeRoy R.; Barash-Schmidt, Naomi; Benary, Odette: Bland, Roger W.; Rosenfeld,
Arther H.; Wohl, Charles G.," A Compliation of K+N Reactions' , Lawrence
Radiation Laboratory, University of California, Berkley, California, UCRL-20000
K+N, Sept. 1969
152
14. Giacomeli,G.,"TotalCrossSectionMeasurement",PergamonPress, Ltd, New York,
1970
15. Baldini, A.; Flaminio, V.; Moorehead, W.G.; Morrison, D.R.O., Landolt-BSrnstein
New Series 1/12a and 1/12b, "Total Cross-Sections for Reactions of High Energy
Particles", Springer Berlin Heidelberg, i988
16. Perl, Martin L., "High Energy Hadron Physics", John Wiley & Sons, Inc., 1974
17. Lusiguoli, M.; Restignoli, M.; Violini, G.; Snow, G.A., Nuovo Cimento 1966, 45,A792
18. Wilson, J.W.; Townsend, L.W.; Chun, S.Y.; Lamkin, S.L.; Ganapol, B.D.; Hong, B.S.;
Buck W.W.; Khan, F.; Cucinotta, F.A.; Nealy, J.E., "BRYNTRN; A baryon transport
model", NASA TP-2887, Feb. 1989.
Table I.
Nuclear charge distribution parameters from electron scattering data taken from Ref. 8.
Nucleus Distribution y or tc (fm) a or R (fm)
H 2 0 1.71
He 4 HW 0 1.33
Li 7 HW 0.327 1.77
Be 9 HW 0.611 1.791
Bll HWHW 0.811 1.69
C 12 _ 1.247 1.649
N14 HW 1.291 1.729
O 16 _ 1.544 1.833
Ne 2° WS 2.517 2.74
A127 WS 2.504 3.05
Ar 4° WS 2.693 3.47
Fe 56 WS 2.611 3.971
Cu64 WS 2.504 3.05
Br8O WS 2.306 4.604
AglO8 WSWS 2.354 5.139
Bat38 WS 2.621 5.618
Pb 2°8 2.416 6.624
153
Table II.
Kaon-nucleon total cross section parameters
i PI_ (GeV/c) mi (rob) ni (GeV/c)
ICp
1 0.8 11.
2 1.05 27.
3 1.60 9.
4 2.30 4.
5 3.50 4.30
0.21
0.2
0.35
1.
2.
K+p 1 0.80 -4. 0.40
2 1.30 2.50 1.
3
x7
2.0
1.5
1.0
0.5
0.0
I I I I
T
I
I
I I I I
1 0.8 0.6 0.4 0.2 0
COSSc.m.
Figure 1. Differential cross section of K+p scattering at 503 MeV
154
O,£1
.,..a
b
"t3
10 2
i0 1
i0 o
i0-i
10-2
10-3
E'' ''l''' ' I''''l' r, ,7.
I I I J
0 0.5 1 1.5 2
t (G_V/o) 2
Figure 2. Differential cross section +ofK p scattering at 6324.9 MeV
155
g]
E
3
b
10 2
101
10 o
10-1
10-2
I0-3
10-4
"FZ =:_
"_ ..,.
"Y" -r-
==zZ I __
0 10 20 3O 4O
8 ¢.F_..
50
Figure 3. Differential cross section of K +C 12scattering at 800 MeF/c
156
"Z"
E
3
b
10 3
10 2
tO t
tO 0
10-1
10-2
10-3
10-4
8 C. 1"_.
Figure 4. Differential cross section of K-C 12 scattering at 800 MeV/c
157
{D
g_
a=
g_
v
E
30
20
tO
0
-10
-20
-3O
0
I I |
l 2 3 4 5
r(fm)
Figure 5. Optical potential for K+C 12 at 800 MeV/c
158
IID
I..,
E
I,---I
4O
2O
0
-2O
-40
-60
-80
0
m
m
I I [lllllllll II]I] [ ] I_II I
1 2 3 4
r(f )
5
Figure 6. Optical potential for K C 12at 800 MeV/c
159
4O0
30O
2OO
100
0
-''"1 ' ' ' ' ''"t ' ' '" ''"1
_ \_\ "
102 10 3 10 4
k(+) Lab energy(MeV)
Figure 7. Total (solid) and absorptive (dashed) cross sections for K+A127
160
600
4OO
2OO
0
''"1 ' ' ' ' ''"1 '" ' ' ' ''"1
10 2 103 10 4
k(-) Lab energy(MeV)
Figure 8. Total (solid) and absorptive (dashed) cross sections for K tl/27
161


Kaon-nucleus scattering

Abstract

Similar works

Full text

Available Versions

NASA Technical Reports Server