Stability of a BUU Ground State

Stability of a BUU ground state

I.Pawelczak, J.Toke, W.U.Schroeder

Department of Chemistry, University of Rochester,

Rochester, NY14627

Abstract

Boltzmann-Uehling-Uhlenbeck (BUU) transport model was used to study the behavior of 197Au ground state nucleus. Different initial density and momentum distributions were considered.

It was found that nucleus is fairly stable in a certain time range.

I.Introduction

The Boltzmann-Uehling-Uhlenbeck equation (BUU) describes time evolution of the phase space distribution function of one particle, f(p,r):

(1.1)

Where, U is density dependent mean-field potential, V12 is relative velocity between particles 1 and 2. It includes mean filed and collision integral.

The phase – space distribution is represented by a collection of test-particles. BUU is modeling density distribution for each configuration and calculates average of all test-particles. Propagation is calculated on configuration based on average.

Equilibration of cold 197Au nucleus has been studied in this report. BUU model has been used to analyze how density and momentum density profiles of a ground state of 197Au nucleus evolve with time. Three types of initial density distribution were studied; Homogenously Charged Sphere, BUU Function (explained further in text) and the Complementary Error Function. Momentum distribution was calculated according to Fermi model approximation. Time range in which nucleus is fairly stable was found.

In first section of this report initial distributions are presented. Second one illustrates changes of density, momentum density profiles with time and emission rate for corresponding initial density distributions.

Initial Distributions

Here initial distributions are described for tested density functions. Spatial and momentum distributions, density and momentum density profiles, potential and total energy are included.

First homogenously charge sphere was studied as an initialization of phase-space. Test-particles are uniformly distributed with a radius parameter r0=1.124 fm. Each nucleon is represented by 400 test-particles. Since sharp sphere is only an approximation we studied another density functions that could possibly better describe a physical nucleus. Called BUU Function (described below) and the Complementary Error Function containing diffused surface were also used as initial density distributions.

Error function is defined to be:

,(1.2)

R is chosen randomly and distributed according error function from 0 to 9 fm and Radius is bulk radius of 197Au nucleus. Number of test-particles per nucleon is 450.

An explanation of how BUU function was obtained is essential. Simulation was performed for 207Fr up to 1000 fm/c using homogenously charged sphere distribution in initialization of phase space. Each nucleon was represented by 450 test-particles. To scale obtained density distribution to 197Au nucleus, cutoff was made at distance of 10 from the center of nucleus. At that point 10 nucleons were emitted beyond radius r=10 fm. Fermi fit was made to this density profile (below cutoff at r=10 fm) with following parameters: central density 0 = 0.1522 fm-3, half-density radius

C = 6.502 fm and surface diffuseness a = 0.4481. Figure 1 shows density profile of 207Fr below cutoff (r=10fm) at 1000 fm/c and fit (black line).

Figure 1

Density profile at 1000 fm/c, homogenously charged sphere was used in initialization of a phase- space. The abscissa is radius of nucleus, and the ordinate is nuclear density in units of fm-3. Fitting parameters: central density 0 = 0.1522 fm-3, half-density radius C = 6.502 fm and surface diffuseness a = 0.4481.

Initialization of spatial and momentum distributions of test-particles is shown in Figure 2 and 3, respectively. Each of them contains three columns representing different density distribution functions (from left: homogenously charged sphere, BUU function, Error Function).

Figure 2

Spatial distribution of test-particles graphed at time 0.5 fm/c for different density functions: a) homogenously charged sphere (400 test-particles per nucleon), b) BUU function(450 test-particles per nucleon distributed with radius r=10 fm), c) error function (450 test-particles per nucleon distributed with radius r=9 fm).

Momentum [GeV/c]Momentum [GeV/c]Momentum [GeV/c]

a) b) c)

Figure 3

Momentum distribution of test-particles graphed at time 0.5 fm/c for different density functions (columns): a) homogenously charged sphere (400 test-particles per nucleon), b) BUU function, c) error function (both- 450 test-particles per nucleon).The abscissa is the magnitude of the momentum vector of test-particles expressed in GeV/c and the ordinate is dN/dp.

Momentum distribution is determined by Fermi Gas approximation The Fermi Momentum for a 197Au nucleus was calculated according various nuclear density and is fixed in a bulk at ~0.27 GeV/c. However, it does vary on a surface.

Density profile at time 0.5 fm/c for each density distribution is presented in Figure 4. The first one from left with sharp surface – homogenously charged sphere, in the last two with diffused surface test-particles are distributed according BUU function and the complementary error function, respectively. Nuclear matter density of bulk calculated by BUU is consistent with experimental value, ~0.17 fm-3.

Radius [fm] Radius [fm] Radius [fm]

a) b) c)

Figure 4

Density profiles graphed at time 0.5 fm/c for different density functions:

a) homogenously charged sphere ,b) BUU function, c) error function .The abscissa is the radius of nucleus in fm and the ordinate is nuclear density in units of fm-3.

Initial momentum density profiles corresponding to specific density distributions are illustrated in Figure 5. In upper row occupation of minimum volume (h3) that can be associated with any state of a system is calculated over whole phase-space. Second row represents momentum density profiles determined in a distance of up to 4 fm from the center of nucleus. Fermi momentum is calculated as a function of density. It is constant in a bulk (~0.27 GeV) but varies on a surface of nucleus. As seen in first row of figure 5 the curve shapes reflect properties of corresponding density distributions. The occupation of low momentum states is equal one and decreases with decreasing density while momentum density of a bulk is constant. It means that the Pauli Principle is satisfied in initial conditions.

Momentum [GeV/c]Momentum [GeV/c]Momentum [GeV/c]

a) b) c)

Figure 5

Momentum density profiles; a) homogenously charged sphere, b) BUU function, c) error function. First row shows momentum density calculated over whole phase-space while second only in a range of 4fm from a center of nucleus. The abscissa is the momentum in GeV/c and the ordinate is momentum density corresponding to volume occupied by only one state (h3).

In our calculations simple density dependent, momentum independent mean field was used where one-body potential is given by equation:

(1.3)

with set of parameters: A = -0.109 GeV, B = 0.082 GeV,  = 1.333333, 0= 0.168 fm-3.

Density fluctuations in the center of nucleus produce repulsive potential as can be seen in Figure 6. Different power of density dependence of potential

(U ~ +1.333333) and Fermi energy (EF ~ 2/3) results in a depression of total energy on surface.

Radius [fm] Radius [fm] Radius [fm]

a)b)c)

Figure 6

Density dependent potential with medium parametrization (upper row) and total energy (lower row) as a function of radius for: a) homogenously charged sphere,

b) BUU function, c) error function.

Evolution of Density and Momentum Density Profiles

Changes of density and momentum density profiles of a cold 197Au nucleus with time are presented in Figures 7-12. Time evolution has been studied for three different initial density distribution function corresponding to: homogenously charged sphere, BUU function and the complementary error function. The simulations generating Figures 7-12 were run up to 300 fm/c. Density profiles plotted at time longer than 200 fm/c include only particles up to 20 fm from the center of nucleus. However, there are some free test-particles beyond that. The maximum distance test-particles can reach is about 40-50 from the center of mass at time 300fm/c.

As can be seen in Figures 7, 9 and 11, nucleus in its ground state has experienced ejection of nucleons. Density cut off is set to 1% of nuclear density, c=0.0017 fm-3. There are some differences between behavior of nucleons within nucleus with sharp and diffused surface at early stage. Diffusion of sharp surface and spontaneous emission of test-particles is observed in homogenously charged sphere. On the other hand ejection of nucleons in BUU and Error functions is preceded by compression of nuclear matter. It is significant in density profiles of BUU function at time ~20~60 fm/c. The most gradual changes in density profile among tested functions have been seen for error function.

Unexpected artificial behavior of nuclear matter in 197Au nucleus was noticed in simulation performed for sharp sphere. Changes in density profile look like wave propagation from surface toward the center of nucleus. It bounces back and forth. This anomaly is possible due to high nucleon compression in initial density distribution. The process can be easily seen in Figure 7 at 20 and 40 fm/c and is reduced for Error functions.

Momentum density profiles are calculated over whole phase-space. Enhanced ejection of nucleons with time increases volume occupied by all nucleons. That results in lower momentum density and change of a distribution shape. Increase of Fermi momentum was noticed from ~0.27 to ~0.32 GeV. It is due to violation of the Pauli Principle in simulation of a ground state BUU model.

Emission rate shown in Figure 13 and tables 1-3 was calculated for all initial density distribution functions. It contains two parts: short (from 0 to 300 fm/c) and a long time scale (up to 2000 fm/c). Rate is very similar for all initial density distribution analyzed in this report, generally increases with time. Majority of ejected nucleons in short time scale are protons. At that point emission of neutrons is close to zero, however it gradually increases. It is due to Coulomb repulsion. Simulation was also run for a homogenously charged sphere for infinite time (5000fm/c). At that stage almost half of nucleus disintegrated (80 nucleons were missing, including ~39 protons and ~41 neutrons).

Conclusion

BUU simulation of a ground state 197Au nucleus predicts ejection of nucleons increasing with time for all tested initial density distributions. Free test-particles have origin in whole nucleus and mostly high energy.

Emission rate is pretty similar for all density functions. At time 300 fm/c

1.1 nucleon is ejected for homogenously charged sphere, 1.7 for Error Function and 2.1 for BUU function. There are also some significant differences in behavior of nuclear matter depending on initial density distribution; strong expansion observed for compressed nucleus or compression of diffused one. Even though the lowest rate at time 300 fm/c is calculated for homogenously charged sphere, the most gradual changes in radial density among tested functions are observed for the error function.

Density distribution given by the complementary Error function is fairly stable in time range of 0-300 fm/c and eliminates spurious behavior seen in homogenously charged sphere. It can be used for study of equilibration of hot nucleus.

References

1. K. Morawetz, Phys. Rev. C62 (2000) 064602-1.

2. J. Aichelin, Phys. Rev. C31 (1985) 1730.

3. C. Gale, Phys. Rev. C41 (1990) 1545.

4. W. Bauer, Phys. Rev. C34 (1986) 2127.

5. P. Danielewicz Phys. Rev. C51 (1995) 716.

6. W. Bauer, BUU (isospin independent)

20 fm/c 40 fm/c 60 fm/c

Radius [fm] Radius [fm] Radius [fm]

80 fm/c100 fm/c 150 fm/c

Radius [fm] Radius [fm] Radius [fm]

200 fm/c 250 fm/c 300 fm/c

Radius [fm] Radius [fm] Radius [fm]

Figure 7

Density profiles at various times for 197Au; initial density distribution is given by a homogenously charged sphere.

20 fm/c 40 fm/c 60 fm/c

Momentum [GeV/c]Momentum [GeV/c]Momentum [GeV/c]

80 fm/c 100 fm/c150 fm/c

Momentum [GeV/c]Momentum [GeV/c]Momentum [GeV/c]

200 fm/c 250 fm/c300 fm/c

Momentum [GeV/c]Momentum [GeV/c]Momentum [GeV/c]

Figure 8

Momentum Density profiles at various times for 197Au, initial density distribution is given by homogenously charged sphere.

20 fm/c40 fm/c 60 fm/c

Radius [fm] Radius [fm] Radius [fm]

80 fm/c100 fm/c 150 fm/c

Radius [fm] Radius [fm] Radius [fm]

200 fm/c250 fm/c 300 fm/c

Radius [fm] Radius [fm] Radius [fm]

Figure 9

Density profiles at various times for 197Au; initial density distribution is given by the BUU function.

20 fm/c 40 fm/c60 fm/c

Momentum [GeV/c]Momentum [GeV/c]Momentum [GeV/c]

80 fm/c 100 fm/c150 fm/c

Momentum [GeV/c]Momentum [GeV/c]Momentum [GeV/c]

200 fm/c 250 fm/c 300 fm/c

Momentum [GeV/c]Momentum [GeV/c]Momentum [GeV/c]

Figure 10

Momentum Density profiles at various times for 197Au; initial density distribution is given by the BUU Function

20 fm/c 40 fm/c 60 fm/c

Radius [fm] Radius [fm] Radius [fm]

80 fm/c 100 fm/c 150 fm/c

Radius [fm] Radius [fm] Radius [fm]

200 fm/c 250 fm/c 300 fm/c

Radius [fm] Radius [fm] Radius [fm]

Figure 11

Density profiles at various times for 197Au; initial density distribution is given by the complementary error function.

20 fm/c 40 fm/c 60 fm/c

Momentum [GeV/c]Momentum [GeV/c]Momentum [GeV/c]

80 fm/c 100 fm/c 150 fm/c

Momentum [GeV/c]Momentum [GeV/c]Momentum [GeV/c]

200 fm/c250 fm/c 300 fm/c

Momentum [GeV/c]Momentum [GeV/c]Momentum [GeV/c]

Figure 12

Momentum Density profiles at various times for 197Au which initial density distribution is given by the complementary error function.

Figure 13

Emission rate for Homogenously charged sphere, BUU and Error function. Two columns show different time scale; first one up to 300 fm/c, second – 2000 fm/c.

Time [fm/c]

NUCLEON

PROTON

NEUTRON

0.5 / 0.000 / 0.000 / 0.000
20 / 0.000 / 0.000 / 0.000
40 / 0.025 / 0.025 / 0.000
60 / 0.048 / 0.048 / 0.000
80 / 0.050 / 0.050 / 0.000
100 / 0.070 / 0.070 / 0.000
150 / 0.320 / 0.305 / 0.015
200 / 0.448 / 0.440 / 0.008
250 / 0.895 / 0.875 / 0.020
300 / 1.128 / 1.088 / 0.040
1000 / 10.243 / 8.453 / 1.790
2000 / 28.153 / 18.740 / 9.413

Table 1

Emission rate for homogenously charged sphere.

Time [fm/c] /

NUCLEON

PROTON

NEUTRON

0.5 / 0.249 / 0.102 / 0.147
20 / 0.224 / 0.116 / 0.109
40 / 0.124 / 0.071 / 0.053
60 / 0.113 / 0.076 / 0.038
80 / 0.162 / 0.104 / 0.058
100 / 0.398 / 0.264 / 0.133
150 / 0.620 / 0.493 / 0.127
200 / 1.227 / 1.031 / 0.196
250 / 1.478 / 1.249 / 0.229
300 / 2.100 / 1.827 / 0.273
1000 / 10.331 / 8.384 / 1.947
2000 / 24.811 / 16.782 / 8.029

Table 2

Emission rate for the BUU function.

Time [fm/c] /

NUCLEON

PROTON

NEUTRON

0.5 / 0.031 / 0.013 / 0.018
20 / 0.004 / 0.002 / 0.002
40 / 0.042 / 0.042 / 0.000
60 / 0.100 / 0.100 / 0.000
80 / 0.109 / 0.098 / 0.011
100 / 0.162 / 0.147 / 0.016
150 / 0.440 / 0.422 / 0.018
200 / 0.784 / 0.747 / 0.038
250 / 1.216 / 1.131 / 0.084
300 / 1.700 / 1.607 / 0.093
1000 / 10.562 / 8.793 / 1.769
2000 / 25.967 / 17.853 / 8.113

Table 3

Emission rate for the Error function.