Unfolding of Crumpled Thin Sheets
TThis figure "foto_SA.png" is available in "png"(cid:10) format from:http://arxiv.org/ps/2102.09995v1 r X i v : . [ c ond - m a t . s o f t ] F e b Unfolding of Crumpled Thin Sheets
Leal F. C. B. ∗ and Gomes M. A. F. † Departamento de F´ısica, Universidade Federal de Pernambuco,Recife, PE 50670-901, Brazil (Dated: February 22, 2021)Crumpled thin sheets are complex fractal structures whose physical properties are influenced bya hierarchy of ridges. In this Letter, we report experiments that measure the stress-strain relationand show the coexistence of phases in the stretching of crumpled surfaces. The pull stress showed achange from a linear Hookean regime to a sublinear scaling with an exponent of 0 . ± .
03, whichis identified with the Hurst exponent of the crumpled sheets. The stress fluctuations are studied,the statistical distribution of force peaks is analyzed and it is shown how the unpacking of crumpledsheets is guided by long distance interactions.
The unpacking of structures with many folds is a pro-cess of immense importance, as seen in the unpacking ofnucleic acids [1, 2], proteins [3] and other biological struc-tures [4], such as the flowering of a flower bud, to namejust a few cases that occur with an absurdly high fre-quency in nature. Here, processes of this type are studiedin detail for the first time when considering the unpackingof crumpled surfaces. In addition to the intrinsic interestin the physical and mathematical aspects of unpackingprocesses, these are of great importance as they are thereverse operations of the crumpling mechanisms which,in turn, have been considered in many high-tech devicesand structures in the field of electronics and materialsscience [5–7].This Letter reports fully automated experiments ofunidirectional unfolding of near spherical crumpled pa-per balls made of square sheets of area L [8], undercontrolled ambient conditions, in order to obtain thecurve unpacking force versus deformation. The stretch-ing speed, | ~v | , was constant and the deformation δ , mea-sured along the pull axis x , is the full opening AB minusthe initial distance x as shown in Fig. 1-a.The unpacking of crumpled surfaces was performed forballs made with sheets of density µ = 75 g/m and sizes L = { , , , , , , ,
264 and 305 mm } . Wereproduced the experiment 10 times for each value of L ,with the stretching speed v = 0 . ± . mm/s . Ad-ditional tests, with speeds between 0 .
83 and 6 . mm/s showed no significant variation in stress-strain curves.A typical curve of force versus the stretching deforma-tion, F ( δ ), of a paper ball is shown in Fig. 1-b. Fluctu-ations around the force curve occur due the unfolding ofvarious types of local tangled folds, a phenomenon thathas been described in detail for the strain-strain curveassociated with the unfolding of crumpled thin sheet [9].Each peak signals that the system has moved from onelocal equilibrium state to another. This is a character-istic found in many types of physical systems exhibitingtransition between metastable states [10–13]. A simi- ∗ [email protected] † [email protected] FIG. 1. (a) Scheme of equipment used to stretch the paperballs. As the dynamometer is displaced with constant velocity ~v upwards the deformation “ δ ” increases. (b) The stretchingforce F versus δ for a typical paper ball. See second to fourthparagraphs in the text for details. lar phenomenon has been described in the unpacking ofcrumpled wires in two-dimensional cavities [14]. A rela-tionship between the extraction force and the length ofextracted wire can also be found in that work.In Fig. 2-a, we have a graph with 20 equivalent stretch-ing experiments with the continuous curve represent-ing the average force curve generated with the non-parametric regression method of Nadaraya and Watson[15, 16] whose purpose is to reduce fluctuations. In thissame graph, we can clearly see that the continuous F - δ curve has three distinct trends. Due to their importancewe have placed symbols to represent the two points oftransition between them. The cross (+) represents thepoint of change from the first region to the second ( δ (+) is the abscissa value of the point “+”) and the “ × ” thepoint at which the second region ends ( δ f is the abscissa -1 FIG. 2. (a) The sepia curves are 20 sequences of measurements of force, F , versus stretching, δ , equivalent to Fig. 1-b. To reducefluctuations in the 20 sequences we use the Nadaraya-Watson method, which can be seen in the continuous averaged curve.(b) Log-log graph of F versus δ Nadaraya-Watson curve for a sheet with L = 305 mm. The dashed line shows a F ∼ δ . ± . power law scaling. The continuous line shows a change in the F - δ curve in the second region, to the power law F ∼ δ . ± . .(c) Nine Nadaraya-Watson curves averaged on 10 equivalent sequences F versus δ , for L = { , , , , , , ,
264 and305 mm } , (for more details see the text). In the inset we have the schematic behavior of the average curves of F - δ for different L . value of the point “ × ”). δ f indicates the beginning of theregion, at the end of the F ( δ ) curve, where F starts togrow very quickly, that is, it indicates the region wherethe internal cohesion forces internal to the sheet startto control the physics of the process. Thus, δ f is asso-ciated with the transition from crumpled surface (CS)physics in three dimensions to physics controlled by thetwo-dimensional topology of the sheet. In Supplemen-tary Material A (Figure S1) the reader can see images ofthe CS unfolding and their respective δ .The trend of the average curves, described in Fig. 2-a,in the first and second regions obey two different scalinglaws. We see this in the log-log graph of Fig. 2-b whichshows a Nadaraya-Watson curve of F - δ for a sheet with L = 305 mm. The average curve obeys a double powerlaw dependence F ∼ δ n , (1)where n is 1 . ± .
01 for δ < δ (+) , that is a linearHookean behavior in the first region, and 0 . ± .
03 for δ f > δ > δ (+) . This sublinear behavior found here, typi-cal of auxetic materials, makes such materials exhibit theimportant feature of mitigating impacts [17].Typical stress-strain curves in usual non-fragile materi-als such as metals and polymers, initially exhibit a linearbehavior, characteristic of a Hookean regime, followed bya nonlinear, reversible regime, and then an irreversibleplastic regime. In all materials, the behavior of the curvein the plastic regime varies a lot depending on the compo-sition of the material, however, it usually has a “convexshape” and generally the curve ends at the breaking point[18, 19]. Differently, the stress-strain curve of CS stretch-ing has a “concave shape”, between the second and thirdregions in Fig. 2-a, due to the underlying structure thatis controlled by the 2D topology of the sheet. The sameaspect can be seen in stress-strain curves of stretching ex-periments with individual molecules of DNA or proteins until they are fully unfolded (or unpacked) [1, 2], whichare also influenced by an underlying topology of reduceddimension (1D) as compared to that of the embeddingthree-dimensional space.By varying the scale L of the system we were able toobserve other characteristics related to the slopes of the F - δ curve. The graph in Fig. 2-c shows nine unfold-ing curves of crumpled sheets that reflect the Nadaraya-Watson method for 10 equivalent experiments. To facil-itate the reader’s understanding, in the Figure 2-c insetwe have a scheme simplifying the behavior of the averagescurves of F - δ . The slope of the curves in the first regionis invariant by scale, that is, it does not change with L .The average curves shown in Fig. 2-c are associated withtwo unpacking behaviors. In the first behavior, the meancurves have only two regions, the first and the third, dif-ferently from the mean curve in Fig. 2-a (black contin-uous curve), which have three different regions. Thesecurves have L ≤ L c , where L c is the size that marks thetransition between the two behaviors (the dashed curvein the inset corresponds to the sheet with size L c ). Thesecurves lack the sublinear intermediate region, this meansthat they pass continuously from the first to the third re-gion. Because of this, their corresponding values of δ f ’sare less than δ (+) . Note that the point of change fromthe first region to the second (+) is the same for all thesecurves. The sheet with size L c does not have the secondregion and its δ f is equal to δ (+) . The experiment repre-sented in the graph of Fig. 2-c that comes closest to L c corresponds to L = 66 ± δ f = 58 ± F - δ represent CSs with L > L c , and have three differentregions similar to Fig. 2-a. The same behavior is exempli-fied in the L and L curves in the inset of Fig. 2-c; therethe stress-strain plots follow the same path in the graph F - δ and separate at the point with abscissa δ f . Thismeans that the work to unpack the ball L , W ( δ f ), isequal to the work to stretch the ball L , W ( δ f ), up tothe point of abscissa δ f . So W i ( δ fi ) = W j ( δ fi ), where i < j . Consequently W i ( δ ) = W j ( δ ) , (2)since δ ≤ δ fi . From detailed measurements of the work W ( δ f ) done to fully unpack the sheets in Fig. 2-c [SeeSupplementary Material B] it can be shown that botheq. 2 and the sublinear power law found in eq. 1, for δ f >δ > δ (+) , are consistent equations of the same dynamicalsystem.Now, we examine how the packed and the unpackedphases evolve by measuring a characteristic length u ofthe unpacked phase. In the Fig. 3-a we have a sequenceof images that shows the evolution of the stretching pro-cess of a CS of initial radius R = 18 ± L = 212 mm. The four images reveal thatthe unfolding process propagate from the clampled ex-tremities to the middle of the sheet with an up-downsymmetry along the strain direction. We measure thetotal length characterizing the degree of unfolding of theunpacked phase, u i = u ′ i + u ′′ i , where u ′ i and u ′′ i are thedisplacements shown in Fig. 3-a. The graph in Fig. 3-b shows the dependence of u versus δ for ten crumpledsheets (on each such sheets was performed between 16 to20 measurements), that takes us to the equation, u = δ. (3)The unpacked part of the sheet corresponds to an area A u that obeys a relationship similar to the eq. 2 (SeeSupplementary Material C for more detail), that is, A ui ( δ ) = A uj ( δ ) (4)where the A ui and A uj indexes indicate the equal un-packed areas of two CS with different L sizes. Equations2 and 4 mean that the mechanisms responsible for thecohesive forces of the paper ball are independent of theparameter L . The results presented in eqs. 3 and 4 arenot obvious, because the CS has a heterogeneous struc-ture with long-range correlations that make unpackinga complex process (more on this ahead). As shown inFig. 3-a the extension of the packaged part of the CS inthe direction x does not change with δ , so the packagedphase has a frozen structure that is maintained as far as“ δ f ” is not reached. Fig. 3-c shows the graph of δ f /ζ vs. L , where ζ is the maximum possible stretch of a crum-pled sheet, ζ = √ L − x . Curves with L > L c have δ f /ζ ≈ . ± .
02, that is, δ f is proportional to L , since ζ is proportional to L . CS are fractal structures withcomplex interactions, so we must expect that the propor-tionality between δ f and L has a non-trivial reason. InSupplementary Material D we show that the CS stretchhas a symmetry of unfolding of the packaged phase thatis related to the proportionality relationship between δ f and L . On the other hand, for sheets with L ≤ L c , δ f /ζ grows as δ f /ζ = (0 . ± .
06) + (0 . ± . L , that is δ f depends quadratically on L , meaning a different un-folding mechanism. The value found for L c by adjustingthe curve δ f /ζ versus L in Fig. 3-c is 66 ± δ + = 54 ± F - δ curve are the fluctuation peaks,which occur during sudden buckling changes in the facetsthat define the rough conformation, similar to what wasdescribed in the study of the noise emitted by a crumpledelastic sheet [20]. To find the relationship between thepeaks of the F - δ curve and the length of the facets (Λ)we analyze the magnitude distribution of the peaks forces F p . The magnitudes of F p correspond to the values of theforce measured at each peak, as highlighted in Fig. 1-b.The distribution of F p for 10 equivalent curves F - δ isshown in Fig. 4-a. The continuous line is a fit with thelognormal distribution which seems to capture quite wellthe characteristics of the data. Then, the magnitude ofthe peaks of force observed here and of the size of facets(Λ) reported in [21] obey a similar hierarchical relation-ship controlled by a same statistical distribution. Thedata in Fig. 4-a also presents a good fit for the Gammadistribution (dashed curve). For more details on the dis-tribution of F p see Supplementary Material E.It is important to observe how the length of the facetsis distributed over the surface: the topography of thecrumpled sheet as revealed by laser scanning [21] showsthat the average ridge roughness (or average elevation ofthe roughness) ¯∆, of a unfolded crumpled sheet of paper,within a box with range w (see these quantities in Fig. 4-b), scale as ¯∆ ∼ w H . This power law is a Hurst analysis[21] of the topography of the crumpled sheet, with anexponent of Hurst H = 0 .
71, for w > w c , and H = 1, for w < w c , where w c = 25 ± ∝ ¯Λ, thus¯Λ ∼ w H . (5)The parallel between F p and Λ pointed at the end of theprevious paragraph, and the matching between deforma-tion ( δ ) with the range ( w ) as the unfolding proceedsphysically means that ¯ F p ∼ δ H , (6)where H ≈ .
70 for δ (+) < δ < δ f and H ≈ . δ < δ (+) . Equation 6 is similar to eq. 1 and this leads usto identify the exponent of the first and second regionsof the F - δ curve with the Hurst exponent H .Two works that examine thin sheet folding when a ra-dial compression force f is applied showed that the sheetsgo through two regimes which are characterized by cur-vature patterns [23, 24]. In the first regime, the compres-sion radius Φ of the confinement volume is greater than FIG. 3. (a) Sequence of images showing the evolution of the stretching process, in order to find the total length u of theunpacked phase, u i = u ′ i + u ′′ i as a function of δ . The diameter of the packaged phase in the direction x , ϕ x , is constant. (b)The graph of u versus δ shows a linear fit u = (1 . ± . δ . (c) Graph δ f /ζ versus L , where the maximum stretch ( ζ ) is equalto √ L − x .FIG. 4. (a) Averaged peak size distribution P ( F p ) for tenequivalent samples of crumpled sheets with L = 264 mm.(b) Diagram of the cross section of a small local portion of acrumpled sheet, showing the roughness ∆ for a range w andthe length Λ of the facets. Φ c and f < f c , where Φ c and f c are the transition radiusand the force of transition to the crumpled regime. Thisinitial regime of smaller curvatures (less accentuated thanin a ridge), is characterized by having conics developablein elastic sheets, this means that the work of the force f is stored in the form of elastic energy. The fractal dimen-sion in this regime is similar to that valid for the uncrum-pled sheet, D = 2. For Φ < Φ c and f > f c , the sheet isin the crumpled regime, it has a hierarchy of ridges and f obeys the scaling f ∼ ( κ/L )( L/h ) β (Φ /L ) − ( α ( β +1) − ,where κ is the bending modulus and h is the thickness ofthe sheet. The exponent α is found through simulationfrom [24]: ¯Λ ≈ L. (Φ /L ) α , (7)with α = 1 .
65, and with β = 1 / D = 2 . c /L = 0 . L of an elasto-plasticsheet. After replacing Φ c /L with 0 . f ( h, Φ , L ) we get f c ( h, L ) = C ( h ) .L ( β − , with C ( h ) being a function h .In the graph of figure 2-c we have a record of the stretch-ing of sheets of different sizes that were compressed withthe hand grip force f m . Through the inverse functionof f c ( h, L ) we find the sheet with size L = L ′ satisfy-ing f c = f m , that means that all sheets with L > L ′ are in the crumpled regime and have a fractal dimension D = 2 .
5. The sheets with L ≤ L ′ are in the regimeof developable conics and consequently fractal dimensionequal to that of surfaces, D = 2. Assuming that L c = L ′ ,the change of regime found in the stretching of CS, be-tween L ≤ L c and L > L c , is caused by the unfoldingof the crumpled sheet from different structural regimes(regime of developable conics or the crumpling regime).The characteristic length L c is a function of the force f m and h . Using the relationship between mass ( M ) and theradius of the paper ball ( R ) [8] as a boundary condition, R = Φ ∼ L /D , from eq. 7 we have¯Λ ∼ L [(2 /D − .α +1] → ¯Λ ∼ L H , (8)where H = (2 /D − α + 1. For sheets with L < L c → D = 2 and H = 1. For sheets with L > L c → D = 2 . H = 0 . w . Therefore, on a sheetwith L > L c and scale w < w c we find the applicable con-ical regime. Since δ ∼ w then δ (+) is associated with w c ,therefore the linearity found at the beginning of the F - δ curve ( δ < δ (+) ) is related to the small curvature foldsof the applicable conical regime. In a scale w > w c , wefind the crumpled regime and for this reason the curve F - δ has a sublinear regime for δ (+) < δ < δ f . The expo-nent H = 0 .
65 indicates that CS roughness is not purelyrandom, but has memory for distances greater than theaverage ridge length [26].Each stretched piece of the originally crumpled sur-face is a continuous variety formed by the union of facetswhose average length is give by eq. 7. Thus, the num-ber of facets N ≈ ( L/ Λ) = ( L/ Φ) α . The hierarchy ofridges and F p are associated with the facet partitioningof the sheet [27], so the eq. 8 connects two properties ofCS, partitioning and long distance memory, due to theexponent H > .
5. In a three-dimensional perspective,the evolution of partitioning is a process of overlappingridges that occurs in a hierarchical order. The unpack-ing of a CS must obey the reverse process of that over-lap. The overlap of ridges creates a cohesive structurethat frustrates the unfolding when it does not follow theorder established by the hierarchy of ridges. For this rea-son, the diameter of the packaged phase in the direction x , ϕ x , in Fig. 3-a, is constant, as the energy required tounfold the ridges that are outside hierarchical order di-verges. The equality present in eq. 3 is a consequenceof this immobile state of the packaged phase and showsthat the work of the stretching force, W ( δ ), is used com-pletely to unfold the ridges that are in the correct order.If ϕ x changes with δ , then part of the work W ( δ ) wouldbe used for this change and the stretching tension wouldinvolve unfolding rigdes from different hierarchies at thesame time.In this Letter, we studied in detail for the first time the unfolding of crumpled systems, observing the force nec-essary to unpack and stretch these complex structures.The corresponding force curve presents different regimesas the sheet is unfolded: a linear Hookean regime at thebeginning of the stretching, as a result of the crumpledsurfaces elasticity, and a sublinear regime that has anexponent equal to 0 . ± .
03, identified with the Hurstexponent of the crumpled sheet which in turn is associ-ated with the surface roughness [21, 28]. We show thatthe behavior of the stress curve is guided by long dis-tance correlations and that the dynamics of the unfold-ing of the ridges is coordinated by a hierarchical order.In the unfolding process reported here there is the co-existence of two structural phases: a crumpled frozenfractal phase (packaged phase) and an open solid two-dimensional rough phase of a surface with fixed connec-tivity (unpacked phase).This work was supported by Coordena¸c˜ao de Aper-fei¸coamento de Pessoal de N´ıvel Superior (CAPES), Pro-grama PROEX 534/2018, . / Author contributions
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F. C. B. Leal ∗ and M. A. F. Gomes † Departamento de F´ısica, Universidade Federal de Pernambuco,Recife, PE 50670-901, Brazil (Dated: February 22, 2021)
A. Images of the unfolding of crumpled surfaces
The evolution of the unpacking of a crumpled paperball is shown in Fig. S1 in eight images taken from ap-proximately equal time intervals. The uniaxial unfoldingstress is directed along a diagonal of the sheet. Each im-age points to the region of the average curve where therespective deformation occurred. Although the imagesdo not belong to any of the 20 crumpled surfaces (CS)sampled in the graph, they describe a typical unpackingof a crumpled sheet of 210x210 mm . Photo (1) refers tothe axial pull strain δ = 0 and it represents the state ofmaximum packing. Images (1) and (2) show two config-urations of a CS in the first region. FIG. S1. Experimental stress-strain F - δ curves obtained from20 equivalent samples. The continuous black line refers to theNadaraya-Watson method used to obtain reliable averages.The eight photographs show the evolution of CS unpacking. The deformations recorded in the photos from (3) to(7) refer to the second region. These images show thattwo parts of the paper sheet progressively loosen them-selves from a packaged phase of the sheet as δ grows,forming an unpacked rough phase that coexist with amore compact truly crumpled phase. These two phasescoexist in the first and second regions of the F - δ curveshown in Fig. S1. However, at the point ( × ) the pack-aged phase is completely dismantled as we can see from ∗ [email protected] † [email protected] the photo (8). After this point, we have the third re-gion where the surface is entirely in the unpacked phase,although the sheet is still rough. So, the steeper slopefound in the third region is the result of a decrease inthe roughness and, eventually, a stretching of the paperfibers. That is, in the first and second regions, the forcesinvolved are locally transversal to domains of the sheet;they usually connect separate domains on the sheet. Dif-ferently, in the last region, the forces involved are basi-cally internal to the sheet. B. The work for unpacking
Through the numerical integration of the nine F - δ curves of Fig. 2-c of the manuscript, we obtain the corre-sponding work W ( δ f ) to fully unpack the sheets as shownas by the graph in Fig. S2. The work obeys the powerlaw W ( δ ) ∼ δ (1 , ± , . (S1)Deriving equation S1 in relation to δ we get F ∼ δ (0 . ± . , which is a reasonable approximation for thepower law fit found in the second region of the F - δ curveof Fig. 2-b which presents an exponent n = 0 . ± . -2 -1 -2 -1 FIG. S2. Work done, W , in SI units to unpack crumpledpaper balls for different L versus δ f . The data were obtainedthrough the numerical integration of the curves F - δ of Fig.2-c for nine different sizes L . The power law setting is W =(6 . ± . δ (1 . ± . f . C. Detailing the coexistence of phases - I
In Fig. S3 we have two sheets with sizes L and L equally stretched up to δ = δ f , that is, to the point atwhich the sheet L is completely unpacked and A u is theunpacked area of the sheet. They have been mapped toshow the parts of the sheets that are unpacked, A u and A u , and the part that is still packed, A p . The sheet L has the unpacked areas, A ′ u and A ′′ u , and the packedarea, A p . Through eq. 3 of main text we arrive at theexpressions u = δ f e u ′ + u ′′ = u = δ f (since the sheetwith size L has been stretched to δ f ) so the unpackedareas of the two sheets are the same. Therefore, if twopaper balls i = j with different L ’s are stretched to thesame δ , the average unpacked areas, A u , of both will bethe same, A ui ( δ ) = A uj ( δ ) . (S2) FIG. S3. Map showing the areas of the sheets that are un-packed, A u and A u , and the part that is still packed, A p .They have been stretched to δ f , at this point the sheet L isfully unpacked and the sheet L is partially unpacked. Theunpacked area of the two sheets are the same, A u = A u because δ f = u = u ′ + u ′′ = u . D. Detailing the coexistence of phases - II
The unfolding symmetry referred to in the main textis unidirectional, that is, the unfolding of the packagedphase grows along a single direction ( x axis), as can beseen in Figure 3-a. When stretching a crumpled sheetof an element of length dx , it is unfolded from a corre-sponding element of area dA , as shown in Fig. S4: thearea element dA is a rectangle with height du = dx andlength λ , whose area is equal to λdu = λdx . The two un-packed parts have equal average lengths, so u ′ = u ′′ , andtheir union forms an unpacked square sheet. This meansthat λ = δ is the length of the diagonal of a square, andthen dA = δdx . Integrating dA between, 0 and δ f , weobtain A f = Z δ f dA = Z δ f δ.dx = 12 δ f . When a sheet with size L is unfolded up to its δ f , A f = L , that is, the area of the unpacked phase, A f , is equalto the total area of the sheet, L : A f = 12 δ f → L = 12 δ f → L ∝ δ f . (S3)The result of eq. S3 shows that all crumpled sheets with L > L c have a dismantling dynamics described by eqs. 3and 4. FIG. S4. Scheme showing the infinitesimal unfolding of apackaged sheet. As the sheet is stretched of an element oflength dx , the area element dA (striped area) is unfolded.Note that the area element is equal to λdu . E. Extraction of peaks of stretching force
To find a hierarchical behavior in the statistical dis-tribution of peaks of force F p of the F - δ curves, weneed define the regions of interest. In the first and sec-ond regions, the forces involved are locally transverseto the sheet domains; they generally connect separatedomains on the sheet and are influenced by the hier-archy of the network of ridges. In the third region,the forces involved are internal to the sheet, this meansthat the corresponding slope for the curve F - δ is not re-lated to the hierarchical interactions of the network ofridges and therefore, the peaks found in the third re-gion were omitted from the analysis. The distribution ofmagnitude of peaks in the unpacking of crumpled sheetswas reasonably well described by a lognormal distribu-tion P ( x ) ∼ exp[ − (ln( x ) − µ ) / (2 σ )] / ( xσ ) (In Fig. 4the reader can find the distribution for sheets with size L = 264 mm).We fitted the distributions of magnitude of peak inthe unpacking of CS of sizes L = {
30, 55, 66, 77, 88,147, 206, 264 and 305 mm } , however, only the sheetswith the three highest L had a reasonable number ofpeaks to make a histogram. Probably the dynamome-ter used is not sensitive enough to record all fluctua-tions in the F - δ curve for sheets with small values of L . The standard deviation found for the unfolding ofCS with sizes L = 305 ,
264 and 206 mm were respec-tively σ ≈ . , .
45 (Fig. 4) and 0 .
42. Another dis-tribution that gives a good fit is the Gamma distribu-tion, P ( x ) ∼ x a − / [ b a Γ( a )] exp( − x/b ) with the shape parameter a = 6 . , . .
8. However, the χ2