FRACTAL BEHAVIOR OF NUCLEAR FRAGMENTS IN HIGH ENERGY INTERACTIONS
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Transcript of FRACTAL BEHAVIOR OF NUCLEAR FRAGMENTS IN HIGH ENERGY INTERACTIONS
November 19, 2003 16:51 00222
Fractals, Vol. 11, No. 4 (2003) 331–343c© World Scientific Publishing Company
FRACTAL BEHAVIOR OF NUCLEAR
FRAGMENTS IN HIGH
ENERGY INTERACTIONS
DIPAK GHOSH,∗ ARGHA DEB, MITALI MONDALSWARNAPRATIM BHATTACHARYYA and JAYITA GHOSH
Nuclear and Particle Physics Research Centre
Department of Physics, Jadavpur University
Kolkata – 700032, India∗[email protected]
dipakghosh [email protected]
Received May 22, 2002Accepted November 20, 2002
Abstract
The multifractal analysis of data on nuclear fragments obtained from 28Si-AgBr interactionsat 14.5 A GeV is performed using three different methods (the factorial moments, G-momentsand Takagi moments). The generalized fractal dimensions Dq is determined from all thesemethods. Data reflects multifractal geometry for the nuclear fragments. From the knowledge ofDq, the multifractal specific heat is calculated for this data and also for 16O-AgBr interactionsat 60 A GeV and 32S-AgBr interactions at 200 A GeV.
Keywords : Relativistic Heavy Ion Collisions; Target Fragmentation; Fractal Structure.
1. INTRODUCTION
B. B. Mandelbrot,1 the pioneer, opened a newwindow — Fractal Geometry — for looking intothe world of apparent irregularities. Once someproblem arose regarding the physical relevanceof non-differentiable curves and surfaces,2 it be-
came necessary to introduce a new class of
geometrical objectives called fractals. Althoughsome elementary ideas of fractality were known
by some physicists, the concept of fractal behav-
ior was explicitly formulated and made popular
by Mandelbrot. Fractal geometry allows one to
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mathematically describe systems that are intrinsi-cally irregular at all scales. A fractal structure hasthe property that if one magnifies a small portionof it, the same complexity is shown for the entiresystem. This property goes to suggest the absenceof analyticity or regularity in the system. In actu-ality, the analyticity requires that the structure befeatureless at very small scales, which can never bethe case for a self-similar structure.
Fractals fall into two categories: geometricallyself-similar or uniform fractals and non-uniformfractals which are called multifractals. The fun-damental characteristic of multifractality is thatthe scaling properties may be different for differ-ent regions of the system. The idea is thereforeto construct a formalism that is able to describesystems with local properties of self-similarity.
There is enough scope for studying the frac-tal properties of emitted particle spectra in highenergy nuclear collisions. Recent years have wit-nessed intense experimental and theoretical activityin search of scale invariance and fractality in nu-clear collisions at high energy, in short also called“intermittency.” In high energy interactions, inter-mittency is studied via the scaling properties of themoments of the relevant distributions as a functionof the bin size in phase space. Such studies3 suggestthat the mechanism for particle production has aself-similar property. The self-similar nature of thedynamics directly implies a connection between in-termittency and fractality. Here, we are dealing withnuclear emulsion detector4 which is itself the targetfor any high energy projectile beam. The particlesemitted after the impact are classified as “shower”,“grey” and “black” particles. The “shower” parti-cles are mainly pions, “grey” particles are fast tar-get recoil protons with energy up to 400 MeV and“black” particles are singly or multiply charged tar-get fragments of low energy (E < 30 MeV). Thecharacteristics about the dynamics of particle pro-duction in high energy nuclear collisions are notrevealed as yet in detail. One well known model,known as “evaporation” model tells that the par-ticles corresponding to “shower” and “grey” tracksare emitted from the nucleus very soon after theinstant of impact, leaving the hot residual nucleusin an excited state. Emission of particles from thisstate takes place relatively slowly. In order to es-cape from this residual nucleus, a particle mustawait a favorable statistical fluctuation, as a resultof random collisions between the nucleons withinthe nucleus, which takes the particle close to the
nuclear boundary, traveling in an outward direc-tion. After evaporation of this particle, a secondparticle is brought to the favorable condition forevaporation and so on, until the excitation energyof the residual nucleus is so small that the tran-sition to the ground state is likely to be affectedby the emission of rays. The evaporation model isbased on the assumption that statistical equilib-rium has been established in the decaying systemand the directions of emission of the evaporationparticles are distributed isotropically. However, theconcept of the evaporation model has not beenuniversally accepted. Barkas suggested5 that themechanism other than the evaporation process mustalso be considered to explain the emission of heavyfragments from excited nuclei, as the evaporationprocess cannot explain the emission of all the heavyfragments. Different experimental data also indicatethe existence of the non-equilibrium nature of pro-cesses to be responsible for the emission of slow,target-associated particles.6
Multifractality has become the focal point ofa number of theoretical and experimental inves-tigations in high energy nuclear collisions7–11 inorder to know the exact dynamics of multiparti-cle production. Most of the investigations on mul-tifractality have been carried out with the helpof produced pions with the common belief thatthey are the most informative about the colli-sional dynamics. However, a similar study of targetfragments can also provide useful and importantinformation in this regard. No such detailed studyhas been done with the target fragments so far.
Therefore in this work we deal with target frag-ments from nuclear interactions. It would be in-teresting to explore if the target-evaporated slowparticles also follow a multifractal structure. Thiswill not only provide a unified description ofthe whole production process but also provide anadditional parameter to understand the dynamicsof nuclear interactions.
The most notable property of fractals is theirdimensions.12 We have used three different meth-ods for extracting the fractal dimensions: factorialmoment method,13 G-moment method7–9,14 andTakagi moment method.5
The power law behavior of scaled factorialmoments reveals self-similarity and in general itindicates the existence of fractal properties.1
The G-moments are constructed to exhibit themultifractal properties of multiplicity distributionby Hwa and others.7,14 These G-moments are,
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Fractal Behavior of Nuclear Fragments 333
however, dominated by statistical fluctuations whenthe event multiplicity is low. Hwa and Pan modi-fied the old form of the G-moments by introducinga step function, which can act as a filter for the lowmultiplicity events.9–10,16
Recently, Takagi15 proposed a novel method forstudying the fractal structure where the difficultiesfaced in the conventional method were overcomed.Takagi pointed out that experimental data do notshow the expected linear behavior in a log-log plotand this is partially due to the fact that most meth-ods are unable to give the required mathematicallimit: the number of points tends to infinity.
These three methods have their own merits anddemerits, yet these are the most widely acceptedmethodologies for extracting the fractal dimensionsin particle physics.
In this paper, the work was done with the targetfragments of 28Si-AgBr interactions at 14.5 A GeVin emission angle space as well as in azimuthal anglespace. Previously, we worked on target-associatedparticles for 200 A GeV 32S-AgBr interactions and60 A GeV 16O-AgBr interactions.17–18 Here, wehave done one comparative study relating all thethree interactions for large range of energies from14.5 A GeV to 200 A GeV.
It is known in particular, that multifractalitycan be considered from a thermodynamic point ofview.3,19 Recently Bershadskii20 showed that theconstant specific heat approximation (CSH), whichis widely applicable in ordinary thermodynamicsis also applicable to multifractal data. FollowingBershadskii we have also calculated the multifrac-tal specific heat for all the interactions individu-ally from the knowledge of generalized dimensionDq obtained from factorial moment, G-moment andTakagi moment method.
2. EXPERIMENTAL DETAILS
The data were obtained from Illford G5 emulsionstacks exposed to 28Si beam of energy 14.5 AGeV from Alternating Gradient Synchrotron atBrookhaven National Laboratory (BNL AGS).21 ALeitz Metaloplan microscope with a 10X objectiveand 10X ocular lens provided with a semi-automaticscanning stage was used to scan the plates. Eachplate was scanned by two independent observers toincrease the scanning efficiency. The final measure-ments were done using an oil immersion 100X objec-tive. The measuring system fitted with it has 1 µmresolution along the X and Y axes and 0.5 µm res-
olution along the Z axis. After scanning, the eventswere chosen according to the following criteria:
(1) The incident beam track should not exceed 3
from the main beam direction in the pellicle.This is done to ensure that we have taken thereal projectile beam.
(2) Events showing interactions within 20 µm fromthe top and bottom surface of the pellicle wererejected. This is done to reduce the loss oftracks as well as to reduce the error in anglemeasurement.
(3) The incident particle tracks which induced in-teractions were followed in the backward direc-tion to ensure that they indeed were projectilebeam starting from the beginning of the pellicle.
According to the emulsion terminology,4 the par-ticles emitted from interactions are classified as:
(a) Black particles: They are target fragments withionization greater or equal to 10 I0, I0 being theminimum ionization of a singly charged par-ticle. The range of them is less than 3 mm,the velocity less than 0.3c and the energy lessthan 30 MeV, where c is the velocity of light invacuum.
(b) Grey particles: They are mainly fast target re-coil protons with energy up to 400 MeV. Theyhave ionization 1.4 I0 ≤ I < 10 I0. Their rangesare greater than 3 mm and they have velocities(v), 0.7c ≥ v ≥ 0.3c.
(c) Shower particles: The relativistic shower trackswith ionization I less than or equal to 1.4 I0 aremainly produced by pions and are not generallyconfined within the emulsion pellicle.
(d) The projectile fragments are a different class oftracks with constant ionization, long range andsmall emission angle.
To ensure that the targets in the emulsion are silveror bromine nuclei, we have chosen only the eventswith at least eight heavily ionizing tracks (black +grey).
According to the above selection procedure wehave chosen 350 events of 28Si-AgBr interactions at14.5 A GeV. The emission angle (θ) and azimuthalangle (φ) were measured for each tracks by tak-ing the coordinates of the interaction point (X0 ,Y0,Z0), coordinates (X1, Y1, Z1) at the end of the lin-ear portion of each secondary track and coordinate(Xi, Yi, Zi) of a point on the incident beam.
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It is worthwhile to mention that the emulsiontechnique possesses a very high spatial resolu-tion, which makes it a very effective detector forstudying the multifractal behavior of multiparticleproduction.
3. THE FACTORIAL MOMENTS
Consider x to be a phase space interval in which themultifractal analysis is carried out, and let n be thetotal multiplicity of target-associated slow particlesin the x interval. Subdivide x into M bins, each hav-ing phase space width δx = x/M . The normalizedfactorial moment Fq of order q is defined as,13
Fq(δx) = M q−1
M∑
m=1
nm(nm − 1) · · · (nm − q + 1)
〈nm〉q
(1)
where nm is the multiplicity in the mth bin. 〈. .〉indicates average over the whole sample of events.For given q and M values, Fq’s are calculated for allthe events and then averaged over the whole sam-ple of events to obtain 〈Fq〉. The unique feature of
this moment is that it can detect and characterizethe non-statistical density fluctuations in particlespectra, which are intimately connected with thedynamics of particle production.
If the non-statistical fluctuations are self-similarin nature, in the limit of small bin size, factorialmoment is given by
〈Fq〉 ∝ Mαq
i.e.
ln 〈Fq〉 = αq ln M + e . (2)
This property, in analogy with the turbulent fluiddynamics, is called “intermittency.” αq measuresthe strength of intermittency and is called theintermittency exponent and e is a constant. The in-termittency exponent αq is obtained by performingbest fits according to Eq. (2).
The authors9,16 have established a relationbetween the fractal co-dimensions (dq) and inter-mittency indices (αq)
dq = αq/(q − 1) . (3)
24
1.5 2.0 2.5 3.0 3.5 4.0
0
1
2
3
4
5
6
7
8
q=2q=3q=3
Fig. 1(a)
ln<
Fq>
lnM
1.5 2.0 2.5 3.0 3.5 4.0
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
q=2q=3q=4
Fig. 1(b)
ln<
Fq>
lnM
24
1.5 2.0 2.5 3.0 3.5 4.0
0
1
2
3
4
5
6
7
8
q=2q=3q=3
Fig. 1(a)
ln<
Fq>
lnM
1.5 2.0 2.5 3.0 3.5 4.0
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
q=2q=3q=4
Fig. 1(b)
ln<
Fq>
lnM(a) (b)
Fig. 1 The dependence of the logarithm of the factorial moments of order q = 2, 3, 4 on the logarithm of phase space partitionnumber M in (a) cos θ space and (b) φ space for 28Si-AgBr interactions at 14.5 A GeV.
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Fractal Behavior of Nuclear Fragments 335
Table 1 Parameters of factorial moment analysis in cos θ
space.
Inetractions q αq Dq
28Si-AgBr (14.5 A GeV) 2 0.192 ± 0.008 0.808 ± 0.008
3 1.41 ± 0.028 0.295 ± 0.014
4 2.738 ± 0.014 0.087 ± 0.005
16O-AgBr (60 A GeV) 2 0.021 ± 0.007 0.979 ± 0.007
3 0.338 ± 0.05 0.831 ± 0.03
4 1.065 ± 0.12 0.645 ± 0.04
32S-AgBr (200 A GeV) 2 0.041 ± 0.008 0.959 ± 0.008
3 0.227 ± 0.03 0.887 ± 0.015
4 0.593 ± 0.10 0.802 ± 0.033
Table 2 Parameters of factorial moment analysis in φ
space.
Interactions q αq Dq
28Si-AgBr (14.5 A GeV) 2 0.054 ± 0.006 0.946 ± 0.006
3 0.283 ± 0.019 0.858 ± 0.009
4 0.944 ± 0.076 0.685 ± 0.025
16O-AgBr (60 A GeV) 2 0.062 ± 0.007 0.938 ± 0.007
3 0.289 ± 0.03 0.855 ± 0.015
4 0.663 ± 0.10 0.779 ± 0.03
32S-AgBr (200 A GeV) 2 0.075 ± 0.010 0.925 ± 0.010
3 0.241 ± 0.040 0.879 ± 0.020
4 0.596 ± 0.11 0.801 ± 0.037
The monofractal structure of multiparticle spec-tra will show order-independent intermittency in-dices, whereas in the case of multifractality, dq > dq′
for q > q′. The generalized dimension Dq may beobtained from dq with the help of the following re-lation
Dq = 1 − dq . (4)
We have used the experimental data of the targetevaporated slow particles emitted in 28Si-AgBr in-teractions at 14.5 A GeV. We have divided the cos θspace and φ space into M bins and have calcu-lated the factorial moments Fq using Eq. (1) forM = 5, 6, . . . , 35, where q, the order for spatial fluc-tuation, is varied from 2 to 4 in steps of 1. We haveplotted ln〈Fq〉 against ln M in Fig. 1(a) and (b)respectively for cos θ space and φ space. It revealspower law behavior of factorial moments with binsize, signifying intermittent behavior. The best lin-ear fits for all the graphs have been performed. Ac-
cording to Eq. (2), the slopes of the plots give inter-mittency exponent αq. Using these values, we havecalculated the values of generalized dimension Dq
following Eqs. (3) and (4). The values of αq andDq are tabulated in Tables 1 and 2 respectivelyfor cos θ and φ space. The same analysis for targetfragments of 16O-AgBr interactions at 60 A GeVand 32S-AgBr interactions at 200 A GeV have al-ready been performed.17–18 The values of αq andDq have also been incorporated in Tables 1 and 2correspondingly.
4. THE G MOMENTS
The selected phase space interval of length x hasbeen divided into M bins of equal size, the widthof each bin being δx = x/M , and let n be the to-tal multiplicity of target-associated slow particles inthe x interval. Let nm be the multiplicity of the par-ticles distributed in the mth bin. When M is large,some bins may have no particles (i.e. “empty bins”).
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Let M ′ be the number of non-empty bins, whichconstitute a set of bins that have fractal properties.Hwa7 proposed a new set of fractal moments, Gq,defined as
Gq =M ′∑
m=1
pqm (5)
where Pm = nm/n with n =∑M ′
m=1nm, and q is the
order number.The summation is carried over the non-empty
bins only, so that q can cover the whole spectrumof real numbers.
In an attempt to circumvent the problem of sta-tistical noise, Hwa and Pan9 proposed a modifieddefinition of the G-moment as
Gq =
M ′∑
m=1
P qmΘ(nm − q) (6)
where Θ(nm − q) is the usual step function whichhas been added to the old definition Eq. (5) in orderto filter statistical noise:
Θ(nm − q) = 1 if nm ≥ q
= 0 if nm < q .
For very large multiplicity n/M q, the step func-tion is essentially unity and so the two definitionscoincide. But in the case of target fragmentation, nis a relatively small number and Θ function exertsa crucial influence on the G-moments. It imposesnon-analytical cut-off at positive integer values ofq. Thus in the real environment of high energycollision, however, the multiplicity is rather lowand the G-moments are dominated by statisticalfluctuations.
For an ensemble of events, the averaging is doneas,
〈Gq〉 =1
Ω
∑Gq (7)
where Ω is the total number of events in theensemble.
According to the theory of multifractals, a self-similar particle production process is characterizedby a power law behavior,7,14
〈Gq〉 ∝ M−τq (8)
where τq is the fractal index and can be obtained
25
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0-10
-8
-6
-4
-2
0
2
4
F ig.2(a)
ln
<G
q>
lnM
q=2 q=3 q=4
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0-10
-8
-6
-4
-2
0
2
4
Fig.2(b)
ln<
Gq>
lnM
q=2 q=3 q=4
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0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0-10
-8
-6
-4
-2
0
2
4
F ig.2(a)
ln
<G
q>
lnM
q=2 q=3 q=4
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0-10
-8
-6
-4
-2
0
2
4
Fig.2(b)
ln<
Gq>
lnM
q=2 q=3 q=4
(a) (b)
Fig. 2 The dependence of ln 〈Gq〉 on ln M for order q = 2, 3, 4 in (a) cos θ space and (b) φ space for 28Si-AgBr interactionsat 14.5 A GeV.
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Fractal Behavior of Nuclear Fragments 337
Table 3 Parameters of G-moment analysis in cos θ space.
Interactions q τ q τ stq D
dynq
28Si-AgBr (14.5 A GeV) 2 0.769 ± 0.007 0.854 ± 0.007 0.915 ± 0.014
3 1.08 ± 0.03 1.766 ± 0.028 0.657 ± 0.029
4 2.564 ± 0.061 2.564 ± 0.061 0.456 ± 0.044
16O-AgBr (60 A GeV) 2 0.78 ± 0.01 0.81 ± 0.03 0.98 ± 0.03
3 1.199 ± 0.04 1.682 ± 0.06 0.76 ± 0.03
4 1.12 ± 0.09 2.557 ± 0.27 0.52 ± 0.09
32S-AgBr (200 A GeV) 2 0.84 ± 0.01 0.89 ± 0.02 0.95 ± 0.02
3 1.47 ± 0.03 1.58 ± 0.05 0.94 ± 0.034 2.38 ± 0.11 2.77 ± 0.28 0.87 ± 0.10
Table 4 Parameters of G-moment analysis in φ space.
Interactions q τ q τ stq D
dynq
28Si-AgBr (14.5 A GeV) 2 0.776 ± 0.006 0.836 ± 0.01 0.940 ± 0.016
3 1.468 ± 0.019 1.747 ± 0.026 0.861 ± 0.022
4 2.071 ± 0.043 2.743 ± 0.069 0.776 ± 0.037
16O-AgBr (60 A GeV) 2 0.78 ± 0.01 0.85 ± 0.02 0.92 ± 0.02
3 1.42 ± 0.07 1.78 ± 0.06 0.82 ± 0.05
4 1.84 ± 0.10 2.96 ± 0.12 0.63 ± 0.05
32 S-AgBr (200 A GeV) 2 0.75 ± 0.01 0.82 ± 0.02 0.92 ± 0.023 1.52 ± 0.06 1.72 ± 0.06 0.90 ± 0.04
4 2.16 ± 0.18 2.77 ± 0.12 0.80 ± 0.07
from the slope of ln 〈Gq(M)〉 versus ln M plot.
τq = δ ln < Gq(M) > /δ ln(M) . (9)
We have divided the cos θ space and φ space intoM = 2, 3, 4, . . . , 20 bins. For each event, we havecalculated the G-moments of the order q = 2, 3, 4using Eq. (6). The logarithm of event-averaged G-moments of the order q = 2, 3, 4 have been plottedagainst ln M in Figs. 2(a) and (b) in emission an-gle space (cos θ as the phase space variable) andazimuthal angle space (φ as the phase space vari-able), respectively. A linear dependence of ln 〈Gq〉on ln M is observed, indicating self-similarity inparticle emission process. The exponent τq of thepower law behavior is obtained by least square fit-ting of the data points. The values are listed inTables 3 and 4 respectively for cos θ and φ space.
For the statistical contribution to 〈Gq〉, wedistribute n particles randomly in the specified xinterval, and using the same procedure we calculate〈Gst
q 〉 as in Eq. (7) and τ stq as in Eq. (9). The best-
fitted lines for ln 〈Gstq 〉 versus ln M plots are shown
by the dotted line in Figs. 2(a) and (b). The slopesτ stq are listed in Tables 3 and 4 correspondingly.The dynamical component of 〈Gq〉 can be esti-
mated from the formula given by Chiu,22
〈Gq〉dyn = [〈Gq〉/〈G
stq 〉]M1−q
which gives
τdynq = τq − τ st
q + q − 1 . (10)
If 〈Gq〉 is purely statistical, then 〈Gq〉dyn is M1−q
which is the result for trivial dynamics. Under such
a condition τ dynq = q − 1. Any deviation of τ dyn
q
from q − 1 indicates the presence of dynamicalinformation.
The generalized fractal (Renyi) dimensions,
Ddynq = τdyn
q /(q − 1) . (11)
For 28Si-AgBr interactions at 14.5 A GeV, the val-
ues of Ddynq have been calculated using Eqs. (10)
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and (11) and are listed in Tables 3 and 4 respec-tively for cos θ and φ space. We have also included
the values of τq, τ stq and Ddyn
q for 16O-AgBr inter-
actions at 60 A GeV and 32S-AgBr interactions at200 A GeV from our previous work18 in Tables 3and 4 correspondingly.
5. THE TAKAGI MOMENTS
In the multiparticle production process, the particledistribution is considered in a phase space x. Let asingle event contains n particles. The multiplicity nchanges from event to event according to the distri-bution Pn(x). The selected phase space interval oflength x has been divided into M bins of equal size,the width of each bin being δx = x/M . Then themultiplicity distribution for a single bin is denotedas Pn(δx) for n = 0, 1, 2, 3, . . ., where we assumethat the inclusive particle distribution dn/dx is con-stant and Pn(δx) is independent of the location ofthe bin. n hadrons, contained in a single event, isdistributed in the interval xmin < x < xmax. Themultiplicity n changes from event to event accordingto the distribution Pn(x), where x = xmax − xmin.
If the number of independent event is Ω, then theparticle produced in those events are distributed inΩM bins of size δx. Let N be the total number oftarget-associated slow particles produced in theseΩ events, and naj the multiplicity of black parti-cles in the fth bin of the ath event. The theoryof multifractals23,24 has been motivated to considerthe normalized density Paj defined by
Paj = naj/N .
This is of course also true when N →∝. Then onehas to consider the Takagi moment of order q as
Tq(δx) = ln
Ω∑
a=1
M∑
j=1
P qaj for q > 0
which behaves like a linear function of the logarithmof the “resolution” R(δx)
Tq(δx) = Aq + Bq ln R(δx)
where Aq and Bq are constants independent of δx.If such a behavior is observed for a considerablerange of R(δx), a generalized dimension may be
26
0 .0 0 .5 1 .0 1 .5 2 .0 2 .5 3 .0 3 .50
2
4
6
8
1 0
1 2
F ig . 3 ( a )
ln
<nq
>
q = 4
q = 3
q = 2
< n l n n > / < n >
l n < n >
- 0 . 5 0 .0 0 .5 1 .0 1 .5 2 .0 2 .5 3 .0 3 .50
2
4
6
8
1 0
1 2
F ig . 3 ( b )
< n ln n > / < n >
q = 2
q = 3
q = 4
ln
<nq
>
ln < n >
26
0 .0 0 .5 1 .0 1 .5 2 .0 2 .5 3 .0 3 .50
2
4
6
8
1 0
1 2
F ig . 3 ( a )
ln
<nq
>
q = 4
q = 3
q = 2
< n l n n > / < n >
l n < n >
- 0 . 5 0 .0 0 .5 1 .0 1 .5 2 .0 2 .5 3 .0 3 .50
2
4
6
8
1 0
1 2
F ig . 3 ( b )
< n ln n > / < n >
q = 2
q = 3
q = 4
ln
<nq
>
ln < n >
(a) (b)
Fig. 3 The dependence of ln〈nq〉 and 〈n ln n〉/〈n〉 on ln M for order q = 2, 3, 4 in (a) cos θ space and (b) φ space for28Si-AgBr interactions at 14.5 A GeV.
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Fractal Behavior of Nuclear Fragments 339
determined as
Dq = Bq/(q − 1) . (12)
Now evaluating the double sum of P qij for sufficiently
large Ω, Takagi15 expects a linear relation
ln 〈nq〉 = Aq + (Bq + 1) ln R(δx) .
While analyzing real data,25 it was observed26
that plot of ln 〈nq〉 against δx saturates for largex region. This deviation may be due to the non-flat behavior of dn/dx in the large x region. Takagisuggested that 〈n〉 would be a better choice ofthe “resolution” R(δx) because dn/d〈n〉 is flat bydefinition.23,26 Choosing R(δx) = 〈n〉, one has
ln 〈nq〉 = Aq + (Bq + 1) ln 〈n〉 (13)
a simple linear relation between ln 〈nq〉 and ln 〈n〉.The generalized dimension Dq can be obtained fromthe slope values using Eq. (12).
The case with q = 1 can be obtained by tak-ing an appropriate limit.24 The value of informationdimension D1 can also be determined from a newand simple relation suggested by Takagi15
〈n ln n〉/〈n〉 = C1 + D1 ln 〈n〉 (14)
where C1 is a constant.So far the methodology is developed for non-
overlapping bins but it is also be applicable foroverlapping bins.27
In the present case, the cosine of emission angleinterval is divided into overlapping bins, whose sizeis increased symmetrically in steps of 0.2 around thecentral value 0 (zero) and the azimuthal angle in-terval is divided into overlapping bins, whose size isincreased symmetrically in steps of 200 around thecentral value 1800. For each bin, we have calculated〈nq〉 with q = 2, 3, 4 and 〈n ln n〉/n for both thespace of target-evaporated slow particles emitted in28Si-AgBr interactions at 14.5 A GeV. Figures 3(a)and (b) represent the nature of variation of ln 〈nq〉with ln 〈n〉 for q = 2, 3, 4 and 〈n ln n〉/〈n〉 withln〈n〉, respectively for cos θ as the phase space vari-able and φ as the phase space variable. All theplots show excellent linear behavior. We have per-formed best linear fits to the data-sets and havecalculated the values of generalized dimension Dq
using Eqs. (12) and (13) and the values of informa-tion dimension D1 using Eq. (14). The values arelisted in Tables 5 and 6 respectively for cos θ asthe phase space variable and φ as the phase spacevariable. For comparison, we have also included thevalues of generalized dimensions for 16O-AgBr in-teractions at 60 A GeV and 32S-AgBr interactions
Table 5 Parameters of Takagi momentanalysis in cos θ space.
Interactions q Dq
28Si-AgBr (14.5 A GeV) 1 0.850 ± 0.011
2 0.794 ± 0.010
3 0.749 ± 0.026
4 0.712 ± 0.048
16O-AgBr (60 A GeV) 2 0.87 ± 0.046
3 0.83 ± 0.047
4 0.81 ± 0.051
32S-AgBr (200 A GeV) 2 0.839
3 0.793
4 0.753
Table 6 Parameters of Takagi momentanalysis in φ space.
Interactions q Dq
28Si-AgBr (14.5 A GeV) 1 0.780 ± 0.032
2 0.747 ± 0.029
3 0.711 ± 0.056
4 0.686 ± 0.083
16O-AgBr (60 A GeV) 2 0.87 ± 0.025
3 0.85 ± 0.018
4 0.84 ± 0.017
32S-AgBr (200 A GeV) 2 0.888
3 0.862
4 0.838
at 200 A GeV in Tables 5 and 6 correspondinglyfrom one of our previous work.18
6. THE MULTIFRACTAL
SPECIFIC HEAT
It is well known that in usual thermodynam-ics, constant specific heat approximation (CSH) iswidely applicable in many important cases, e.g. thespecific heat of gases and solids is constant, inde-pendent of temperature over a greater or smallertemperature interval.28 This approximation is alsoapplicable to multifractal data of multiparticle pro-duction process, which has already been shown byBershadskii.20
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340 D. Ghosh et at.
27
2 3 4
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
(f)(e)
(d)(c)
(b)(a)
2 3 40.60
0.65
0.70
0.75
0.80
0.85
0.90
0.95
1.00
1.05
Dq
28Si-AgBr
16O-AgBr
32S-AgBr
28Si-AgBr
16O-AgBr
32S-AgBr
2 3 40.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
1.1
1.2
1.3
1.4
28Si-AgBr
16O-AgBr
32S-AgBr
2 3 40.5
0.6
0.7
0.8
0.9
1.0
1.1
1.2
28Si-AgBr
16O-AgBr
32S-AgBr
1 2 3 40.60
0.65
0.70
0.75
0.80
0.85
0.90
0.95
1.00
1.05
28Si-AgBr
16O-AgBr
32S-AgBr
1 2 3 40.60
0.65
0.70
0.75
0.80
0.85
0.90
0.95
1.00
1.05
28Si-AgBr
16O-AgBr
32S-AgBr
Dq D
q
Dq
Fig. 4
Dq
Dq
Fig. 4 The q dependence of generalized dimension Dq in cos θ space obtained from (a) factorial moment, (c) G-moment,(e) Takagi moment, and in φ space obtained from (b) factorial moment, (d) G-moment, (f) Takagi moment.
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Fractal Behavior of Nuclear Fragments 341
Fig. 5 The plot of generalized dimension Dq versus ln q/(q − 1) in cos θ space obtained from (a) factorial moment, (c) G-moment, (e) Takagi moment, and in φ space obtained from (b) factorial moment, (d) G-moment, (f) Takagi moment.
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342 D. Ghosh et at.
Table 7 The multifractal specific heat in cos θ
space.
Interactions Method Specific Heat
28Si-AgBr (14.5 A GeV) Fq-moment 3.17 ± 0.31
Gq-moment 1.966 ± 0.148
Tq-moment 0.351 ± 0.03
16O-AgBr (60 A GeV) Fq-moment 1.404 ± 0.295
Gq-moment 1.945 ± 0.327
Tq-moment 0.262 ± 0.013
32S-AgBr (200 A GeV) Fq-moment 0.662 ± 0.127
Gq-moment 0.318 ± 0.195
Tq-moment 0.367 ± 0.037
Table 8 The multifractal specific heat in φ space.
Interactions Method Specific Heat
28Si-AgBr (14.5 A GeV) Fq-moment 1.08 ± 0.37
Gq-moment 0.694 ± 0.114
Tq-moment 0.263 ± 0.01
16O-AgBr (60 A GeV) Fq-moment 0.677 ± 0.079
Gq-moment 1.199 ± 0.396
Tq-moment 0.131 ± 0.006
32S-AgBr (200 A GeV) Fq-moment 0.515 ± 0.153
Gq-moment 0.481 ± 0.269
Tq-moment 0.213 ± 0.025
If the q dependence of Dq satisfies the conditionDq > Dq′ for q < q′, then the spectra is said toexhibit multifractality and the multifractal specificheat can be obtained from generalized dimensionDq by the relation20
Dq = (a − c) + cln q
q − 1. (15)
Here c is the specific heat and a is some other con-stant and q can be interpreted as the inverse oftemperature, q = T−1.28
The values of generalized dimensions Dq obtainedfrom three methods mentioned above for 28Si-AgBrinteractions at 14.5 A GeV , 16O-AgBr interactionsat 60 A GeV and 32S-AgBr interactions at 200 AGeV have been plotted against q in Figs. 4(a) and(b) respectively for cos θ space and φ space. FromFigs. 4(a) and (b), it is observed that in all thethree methods the values of the generalized dimen-sion Dq decrease with the increase of order num-
ber q. This reflects the multifractal geometry in thecase of target-evaporated particles. To obtain mul-tifractal specific heat, we have plotted Dq againstln q/(q−1) in Fig. 5 and performed linear best fits.The scatter points of Fig. 5 represent the data andthe lines represent the best fits. The slopes of thebest linear fits of the data-sets of Fig. 5 give the val-ues of specific heat (c). These values are tabulatedin Tables 7 and 8 respectively for cos θ space and φspace.
7. DISCUSSION
We have observed the bin size dependence of all thethree multiplicity moments, e.g. factorial moment,G-moment and Takagi moment both in cos θ and φspace. For each interaction, the values of generalizeddimension Dq extracted from these three differentmethods are different. However, for each case thevalues of generalized dimension Dq decrease withthe increase of order q, showing multifractal be-havior. One should note that all the three methodshave their own merits and demerits. The specialityof the factorial moment method is that, it enablesone to measure the non-statistical fluctuation, dis-entangling the statistical noise, which always con-taminates the dynamical fluctuation. In the study ofG-moments, special attention has been paid to thestatistical and dynamical components, and the dy-namical component of Dq has been extracted veryconveniently from the experimental data. But thesetwo methods suffer from a common problem thatexperimental data sets do not show the linearity ina log-log plot of moment against bin size as expectedfrom the mathematical formulations. This may bedue to the fact that the assumed mathematical limit(number of points tends to infinity) is not valid forthe real experimental data where number of parti-cles in each event is always finite. This difficulty hasbeen overcomed in the Takagi moment method.
The values of multifractal specific heat for28Si-AgBr interactions at 14.5 A GeV, 16O-AgBr in-teractions at 60 A GeV, and 32S-AgBr interactionsat 200 A GeV depend on the method of extraction(Tables 6 and 7). However, it is very interesting toobserve that in Takagi’s method the values of spe-cific heat for all three interactions are more or lessthe same for cos θ space as well as for φ space.
ACKNOWLEDGMENTS
The authors are grateful to Professor P. L. Jain,Buffalo State University, USA for providing us
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Fractal Behavior of Nuclear Fragments 343
with the exposed and developed emulsion platesused for this analysis. We also gratefully acknowl-edge the financial help from the University GrantsCommission (India) under the COSIST programme.
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