Evidence of self-affine target fragmentation process in 197Au-AgBr interactions at 10.7 A GeV

D.H. Zhang, F. Wang, J.X. Cheng, B. Cheng, Q. Wang, H.Q. Zhang, R. Xu

Institute of Modern Physics, Shanxi Normal University

Linfen 041004, China

Sept. 2, 2008

1、 Introduction

2、 Experimental details

3、Method of Study

4、 Experimental Results

5、 Conclusions

1. Introduction

In high energy interactions, the study of non-statistical fluctuations have entered int

o a new era since Bialas and Peschanski(NP B273(1986) 703) introduced an attractive

methodology to study non-statistical fluctuations in multiparticle production. They sug

gested that the scaled factorial moment Fq has a growth following a power law with dec

reasing phase space interval size and this feature signals the onset of intermittency in th

e context of high energy interactions. This scaled factorial moment method has the feat

ure that it can measure the non-statistical fluctuations avoiding the statistical noise. Up

to now, most of the analysis have been carried out in the relativistic produced particles

with the common belief that these particles are the most informative about the reaction

dynamics and thus could be effective in revealing the underlying physics of relativistic

nucleus-nucleus collisions. However, the physics of nucleus-nucleus collisions at high

energies is not yet conclusive and therefore all the available probes need to thoroughly

exploited towards meaningful analysis of experimental data.

1. Introduction

In relativistic heavy ion induced nuclear emulsion interactions, the target fragment

ation produces highly ionizing particles responsible for heavy tracks which are subdivi

ded into gray and black tracks. The gray tracks are the medium-energy (30-400 MeV) k

nocked-out target protons (or recoiled protons) with range 3 mm and velocity 0.3β

0.7. They are supposed to carry some information about the interaction dynamics bec

ause the time scale of the emission of these particles is of the same order (10-22s) as th

at of the produced particles. The general belief about these recoiled protons is that they

are the low energy part of the internuclear cascade formed in high energy interactions.

The black tracks with range <3 mm and velocity β<0.3 are attributed to evaporation fro

m highly excited nuclei in the thermodynamically equilibrium state. In the rest system

of the target nucleus, the emission direction of the evaporated particles is distributed iso

tropically.

1. Introduction

In the analysis of intermittency most of the studies are performed in the one-dimensio

nal space only, but the real process occurs in three dimensions. So one-dimensional ana

lysis is not sufficient enough to make any comment on the complete dynamical fluctuat

ions pattern. According to Ochs (PL B247(1990) 101), in a lower-dimensional projecti

on the fluctuations will be reduced by the averaging process. In two-dimensional analys

is generally the phase space are divided equally in both directions assuming that the ph

ase spaces are isotropic in nature. Consequently self-similar fluctuations are expected. I

t may happen that the fluctuations are anisotropic and the scaling behavior is different i

n different directions giving rise to self-affine scaling. So far only a few works have be

en reported where the evidence of self-affine multiparticle production is indicated by th

e data(Ghosh et al., EPJ A14(2002) 77, PR C66(2002) 047901, JP G29(2003) 983, IJM

P E13(2004) 1179, MPL A22(2007) 1759, Wang et al., PL B410(1997) 323, Wu and L

iu, PRL 70(1993) 3197).

1. Introduction

In most of the earlier works on intermittency, best linear fits were drawn in the total b

in range from some pre-conceived ideas. Actually, the plots are not perfectly linear in t

he whole bin range, rather nice linear behavior is apparent in selective bin ranges. So it

would be better to investigate intermittency in those bin ranges.

The intermittency pattern cannot only suggest the dynamical nature of fluctuation bu

t also reveals the inner fractal structure of the fluctuation codimensions dq (Ochs, PL B2

47 (1990) 101, Bialas and Gazdzicki, PL B252(1990) 483), which are related to the inte

rmittency indices aq as dq=aq/(q-1). Unique dq for a different order of moments suggests

monofractality whereas order dependence of dq signals the presence of multifractality.

Multifractality may be due to self-similar cascading, whereas monofractality is associat

ed with thermal transitions. Now, there is a feeling that self-similar cascading is not con

sistent with particle creation during one phase but instead requires a non-thermal phase

transition.

1. Introduction

According to Peschanski (PL B410(1991) 323) if the dynamics of intermittency is du

e to self-similar cascading, then there is a possibility of observing a non-thermal phase t

ransition. The signals of non-thermal phase transition can be studied with the help of th

e parameter λq=(aq+1)/q. The condition for non-thermal phase transition may occur whe

n the function λq has a minimum value at q=qc(Peschanski, NP B327(1989) 144, PL B4

10(1991) 323, Bialas and Zaeeswski, PL B238(1990) 413). Among the two different re

gions q<qc and q>qc, numerous small fluctuations dominate the region q<qc, but in the r

egion q>qc, dominance of small number of very large fluctuations occur. This situation

resembles a mixture of a "liquid" of many small fluctuations and a "dust“ consisting of

a few grains of very large density. The minimum of the function λq may be a manifestat

ion of the fact that the liquid and the dust phase coexist.

2. Experimental details

Stacks of NIKFI BR-2 nuclear emulsion plates were horizontally exposed to a 197Au beam at 10.7 A GeV at BNL AGS. BA2000 microscopes with a 100 oil immersion objective and 10 ocular lenses were used to scan the plates. The tracks are picked up at a distance of 5mm from the edge of the plates and are carefully followed until they either interacted with emulsion nuclei or escaped from the plates. Interactions which are within 30μm from the top or bottom surface of the emulsion plates are not considered for final analysis. All the primary tracks are followed back to ensure that the events chosen do not include interactions from the secondary tracks of other interactions. When they are observed to do so the corresponding events are removed from the sample.

  To ensure that the targets in the emulsion are silver or bromine nuclei, we have chosen only the events with at least eight heavy ionizing tracks of particles (Nh8).

3. Method of study

We adopted a procedure to study the self-affine scaling behavior of factorial moment

s, where the size of the elementary phase-space cells can vary continuously. In two dim

ension if the two phase space variables are x1 and x2, factorial moment of order q may b

e defined as (Bialas and Peschanski, NP B273(1986) 703)

M

mq

m

mmmq n

qnnn

MxxF

121

)1()1(1)(

Where δx1δx2 is the size of a two-dimensional cell, nm is the multiplicity in the mth cell,

<nm> is the average multiplicity of all events in the mth cell, M' is the number of two-di

mensional cells into which the considered phase-space has been divided.

To fix δx1, δx2 and M' we consider a two-dimensional region Δx1Δx2 and divide it int

o subcells with widths

3. Method of study

2

22

1

11 M

xx

M

xx

in the x1 and x2 directions where M1M2 and M'=M1·M2.

Here M1 and M2 are the scale factors that satisfy the equationHMM 21

Where the parameter H (0<H≤1) (called Hurst exponent) characterizes the self-affine p

roperty of dynamical fluctuations. The scaling behavior that we were looking into has t

he form

qaq xxxxF )()( 2121

3. Method of study

The power aq(>0) is a constant at any positive integer q and it is called intermittency ex

ponent which measures the strength of intermittency. If such a scaling behavior is found for H=1, the fluctuation pattern is called self-similar. If scaling behavior is found for H<1, the fluctuation is called self-affine.

It is clear that the scale factors M1 and M2 cannot be an integer simultaneously, so th

at the size of the elementary phase space cell would be continuously varying value.

The following method has been adopted for performing the analysis with non-integral value of scale factor (M'). For simplicity, we considered one-dimensional space (y) and let

M' = N + a

Where N is an integer and 0 ≤ a < 1. When the elementary bins of width δy=y/M' are used as the “scale” to “measure” the region y, N of them are obtained and a smaller bin of width ay/M' is left.

3. Method of study

Putting the smaller bin at the last place of the region and doing average with only the fi

rst N bins, <Fq(δy)> becomes

N

mq

m

mimimiN

ievq n

qnnn

NNyF

ev

1

)1()1(11)(

Where nmi is the multiplicity in mth cell of the ith event, and M' can be any positive real number and it can vary continuously.

Our work are performed in two-dimensional emission-azimuthal angle space. As the shape of the single particle distribution influences the scaling behavior of the factorial moments, the “cumulative” variables X(cosθ) and X(φ) are used instead of cosθ and φ. The cumulative variable X(x) is given by the relation as follows:

xdxxdxxXx

x

x

x

2

11

)()()(

3. Method of study

where x1 and x2 are two extreme points of the distribution ρ(x). The variable X(x) varie

s between 0.0 and 1.0 keeping ρ(X(x)) almost constant.

  To probe the anisotropic structure of phase space we have calculated factorial mome

nts for the qth order (q=3,4,5,6) with the varying values of Hurst exponent. The partitio

n numbers along Xcosθ and Xφ directions are chosen as Mφ= 3, 4, … , 30, and Mcosθ given

by

We have not considered the first two data points corresponding to Mφ= 1, 2 to reduce th

e effect of momentum conservation (Liu, et al., ZP C73(1997) 535) which tends to spre

ad the particles in opposite directions and thus reduce the value of the factorial moment

s. This effect becomes weaker as M increases.

HMM cos

4. Experimental results

We have plotted the natural logarithm of average value of factorial moments (ln<Fq>)

along Y axis and the natural logarithm of Xcosθ.X along X axis for 197Au-AgBr intera

ctions at 10.7 A GeV for different Hurst exponent values(0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, and 1.0). For each case linear behavior is observed in two or three regions. In order to find the partitioning condition at which the scaling behavior is best revealed, we have performed linear fit in first region, and have estimated the χ2 per degrees of freedom (DOF) for each linear fit. Interestingly best linear behavior is revealed at H=0.7 and not at

H=1 for each order of moment for the data set. The plots of ln<Fq> against Xcosθ.X at

H=0.7, and 1.0 for different order of moment are shown in figure 1, and 2, respectively. Table 1 represents the value of χ2 per DOF and the intermittency exponent for 197Au-AgBr interactions for different values of H and order of moment. From the table it is seen that χ2 per DOF is smaller at H=0.7 for different order of moment. So the dynamical fluctuation pattern in 197Au-AgBr interactions is not self-similar but self-affine.

4. Experimental results

Fig.1 Fig.2

4. Experimental results

The power-law behavior of the scaled factorial moments implies the existence of some kind of fractal pattern (Hua, 1990) in the dynamics of the particles produced in their final state. Therefore, it is natural to study the fractal nature of target fragments in 197Au-AgBr interactions under the self-affine scaling scenario.

4. Experimental results

The variation of anomalous fract

al dimension dq(dq=aq/(q-1)) with t

he order of moment q under the sel

f-affine scaling scenario (H=0.7) is

presented in right figure. From the

plot it is seen that dq increases linea

rly with the order q for the data set,

which suggests the presence of mul

tifractality of emission target fragm

ents in 197Au-AgBr interactions.

4. Experimental results

Recently Bershadskii (PR C59(1998) 364) showed that the constant specific heat a

pproximation is also applicable to the multifractal data of multiparticle production pro

cess. Starting from the definition of Gq-moment, he derived the following relation

for the multifractal Bernoulli fluctuations. In the above relation, Dq is the generalized

dimensions which is related intermittent indices aq as Dq=1-aq/(q-1), a is some other c

onstant and constant c can be interpreted as the multifractal specific heat of the syste

m. We have determined the multifractal specific heat for our data. Fig.4 shows the plo

t of Dq obtained from Fq-moment analysis as a function of lnq/(q-1) for 197Au-AgBr co

llisions at 10.7 A GeV in cosθ and φ two-dimensional phase space. Straight line is the

linear fit to the data, indicating good agreement between our data and multifractal Ber

noulli representation. The slope of the fitted line, which gives the value of the multifra

ctal specific heat c for our data, is 0.520.05.

1

ln)(

q

qccaDq

4. Experimental results

Fig.4, The generalized dimension Dq ver

sus lnq/(q-1) at H=0.7 in 197Au-AgBr int

eractions at 10.7 A GeV. Straight line is

the linear fit to the data, indicating good

agreement between our data and multifr

actal Bernoulli representation. The slope

of the fitted line, which gives the value

of the multifractal specific heat c for our

data, is 0.520.05.

Fig. 4

4. Experimental results

Finally we discuss the property of non-thermal phase transition in the emission of target fragments in 197Au-AgBr interactions.

Figure 5 presents the dependence of λq on the order q. From the plot it is seen that a slight minimum of λq is appeared at q=4, which may indicate the coexistence of two different phase, i.e. the "liquid" and "dust" phases.

 

Fig.5

Conclusions

From the present study of the 10.7 A GeV 197Au-AgBr interactions, it may be concluded that:

1). The effect of the intermittency is observed and the best power law behavior is exhibited at H=0.7 which suggested the dynamical fluctuation pattern in 197Au-AgBr interactions is not self-similar but self-affine.

2). The anomalous fractal dimensions of the intermittency is found to increase with the increase of the order of moments, which suggests the presence of multifractality of emission of target fragments in 197Au-AgBr interactions.

3). A slight minimum value of λq is observed at q=4, which suggested that there is a coexistence of the liquid and the dust phases.

Thank you !