Retrieving Nuclear Information from Protons Propagating through A Thick Target
aa r X i v : . [ nu c l - t h ] M a y Retrieving Nuclear Information from ProtonsPropagating through a Thick Target
B.G. Giraud [email protected], Service de Physique Theorique,DSM, CEA Saclay, F-91191 Gif-sur-Yvette, France
Lon-chang Liu [email protected], Theoretical Division, Group T16, MS B243,Los Alamos National Laboratory, Los Alamos NM 87545 USA
Abstract
The multiple scattering of high-energy particles in a thick target is formulated in an im-pact parameter representation. A formalism similar but not identical to that of Moli`ere isobtained. We show that calculations of particle beam broadening due to multiple Coulombscattering alone can be given in closed form. The focus of this study is on whether or not thebroadening of the Coulomb angular distribution prevents the retrieval of nuclear-interactioninformation from measuring the angular distributions of charged particles scattered froma thick target. For this purpose, we study multiple scatterings with both the nuclear andCoulomb interactions included and we do not make a small-angle expansion. Conditionsfor retrieving nuclear information from high-energy protons propagating through a block ofmaterial are obtained.Keywords: Multiple scattering; Broadening of particle beam profile.1
Introduction and basic formalism
Understanding and calculation of the broadening of a particle beam when it prop-agates through a block of materials are important not only to make multiple-scatteringcorrections for cross-section measurements in physics experiments but also to many appli-cations such as radiography by means of high-energy protons. Many different theories ofthe multiple scattering of electrons by thick targets have been formulated in the past sixtyyears [1]-[7]. The theory of Moli`ere [1][7] has received extensive attention because it givesthe best agreement with data concerning the broadening of Coulomb angular distribution,arising from the multiple scattering of charged particles from the atoms in thick targets. Anexcellent and succinct derivation of the Moli`ere theory was given by Mott and Massey [8].As one deals with a large (almost astronomical) number of scatterers in a thick tar-get, the numerical aspect of the calculation becomes extremely demanding. One can obtaingood results provided that careful approximations are carried out. Over the years, successfulparametrizations of the broadening of angular distributions due to Coulomb multiple scatter-ing have been established[9]. For hadronic projectiles, such as protons, nuclear interactionsalso contribute to multiple scattering. However, this latter aspect has not yet received suffi-cient attention in the literature. In this work, our focus is, therefore, on effects of multiplescattering on angular distributions in the Coulomb-nuclear interference region and in theregion where the nuclear interaction dominates. We formulate the multiple-scattering prob-lem in such a way that not only in the case with Coulomb multiple scattering alone cananalytical evaluations of the beam broadening become possible, but also the calculations aregreatly facilitated when both nuclear and Coulomb interactions are taken into consideration.For a very high energy (e.g., ≥
20 GeV) proton scattering from a single nucleus,the Coulomb cross section decreases rapidly with the increase of scattering angles in sucha way that the Coulomb cross section is negligible with respect to the nuclear cross sectionalready at scattering angles as small as several milliradians. However, the Coulomb peak israpidly broadened by proton-atomic nucleus multiple scatterings. Clearly, for the purposeof extracting the forward amplitude of the basic hadron-nucleus strong interaction, one2hould use targets as thin as possible and then employ Moli`ere-type theory to correct forthe Coulomb contribution from the measurement[10]. On the other hand, in many practicalapplications the thickness of the “target” is often fixed by specific needs, which is by no meansthin. It becomes, therefore, interesting to know how much nuclear information can still belearned from hadrons scattered from a thick target. Certainly, the feasibility of learningnuclear information can bring added value to probing materials with hadronic beams. Inother words, will the broadening of the Coulomb angular distribution render impossibleany study of the nuclear cross sections? One naturally expects that the survival of nuclearinformation, if any, depends on the target thickness, i.e., on the number of nuclear scatterersthat a proton encounters in a block of material. We use our formalism to examine thisquestion.In this section, after deriving the basic multiple-scattering formalism, we discuss theimportant bearing of an ability to retrieve nuclear information on many applications. InSection II, we show the broadening of angular distributions by Coulomb multiple scatteringin closed form. The broadening of angular distributions by combined Coulomb and nuclearinteractions are studied by means of semi-analytical models in regions of small momentumtransfers (Section III) as well as large momentum transfers (Section IV). We find that it ispossible to retrieve nuclear information from protons scattering from a thick target. Conclu-sions and suggestions are presented in Section V.It is well known that high-energy elastic scattering is basically forward peaked,which allows to a very good approximation to neglect the longitudinal momentum trans-fer. Glauber [11] has shown that, in an impact-parameter representation, every function O ( q ) of the tranverse momentum transfer q, whether O is an amplitude or a cross-section,can be parametrized in terms of a profile function φ ( b ) defined in the impact plane, O ( q ) = (2 π ) − Z d~b exp( i~q · ~b ) φ ( b ) = (2 π ) − ∞ Z b db J ( qb ) φ ( b ) , (1)where azimuthal symmetry is assumed, J is the Bessel function of the first kind, and q and b are the moduli of the transverse momentum transfer ~q and impact parameter ~b, re-3pectively. Nonessential complications ( e.g. spins, etc) are here understood. Conversely, theprofile function results from the inverse Fourier transform and is given by φ ( b ) = Z d~q exp( − i~q · ~b ) O ( q ) = 2 π ∞ Z q dq J ( qb ) O ( q ) . (2)Without loss of generality, we consider a target which consists of one kind of nuclei.For a thin target of thickness t and atomic density ρ the probability that a beam particleundergoes a scattering is p tot = t ρ σ tot , where σ tot is the total cross section [12]. (The subscript1 denotes the single scattering.) The transmission probability is, therefore, given by p trans =1 − p tot . From the definition of the differential cross section σ ( ~q ) , one obtains the sum rule σ tot = R d~q σ ( ~q ) . The scattering probability density p ( ~q ) is related to the differential crosssection by [12],[13] p ( ~q ) = t ρ σ ( q ). It is the probability that a particle experiences scatteringin the direction ~q. The sum rule of σ ( ~q ) leads to the sum rule p tot = R d~q p ( ~q ). Most often,only the modulus q counts, σ ( ~q ) = σ ( q ). Hence, σ tot = π R d ( q ) σ ( q ) . In thick targets the beam can bounce forward from many nuclei and/or electronicclouds and/or different atoms. These multiple scatterings are incoherent because the scatter-ers are separated far apart with respect to the ranges of the screened Coulomb and nuclearinteractions so that the scattering waves are already in the asymptotic region before thenext collision occurs. Furthermore, the target is not crystalline on a macroscopic scale, andthus the distance between scatterers are largely random. One must, therefore, add proba-bilities (not amplitudes) coming from individual scatterings. If one splits the thick targetwith thickness T into a large number N of thin targets each with thickness T /N, thenthe total multistep probability for a particle to be transmitted without any scattering is P transM ≡ P = (1 − σ tot T ρ/N ) N . The differential probability density for just one scattering in this situation with manythin targets, each with thickness
T /N, is P ( ~q ) = N σ ( ~q ) ( T /N ) ρ (cid:16) − σ tot T ρ/N (cid:17) N − . (3)This represents a combination of scattering from any one single layer and transmission4hrough the remaining ( N −
1) layers without scattering. The factor N in front of the righthand side accounts for the N layers, obviously.For double scattering one must count pairs of layers and fold two single-scatteringprobability densities p . Hence, P ( ~q ) = N ( N − (cid:18) T ρN (cid:19) σ ( ~q ) " − σ tot T ρN N − , (4)where σ ( ~q ) = Z d~q ′ σ ( ~q ′ ) σ ( ~q − ~q ′ ) . (5)For triple scattering, an identical argument induces the result P ( ~q ) = N (cid:18) T ρN (cid:19) σ ( ~q ) " − σ tot T ρN N − , (6)where σ ( ~q ) = Z d~q ′ Z d~q ′′ σ ( ~q ′ ) σ ( ~q ′′ − ~q ′ ) σ ( ~q − ~q ′′ ) (7)is a double convolution. Again, the first factor is the counting of all triplets of layers.It is useful at this stage to introduce the profiles φ ( b ) = Z d~q exp( − i~q · ~b ) σ ( ~q ) = 2 π ∞ Z q dq J ( qb ) σ ( q ) , (8)with φ (0) = σ tot , and, ∀ n,φ n ( b ) = Z d~q exp( − i~q · ~b ) σ n ( ~q ) = [ φ ( b )] n . (9)If we define Φ as the profile of P , then we see that,Φ ( b ) = N (cid:18) T ρN (cid:19) φ ( b ) (cid:16) − σ tot T ρ/N (cid:17) N − , (10)5nd, more generally, the profile of P n reads,Φ n = Nn T ρ φ N ! n " − σ tot T ρN N − n . (11)The “total multistep profile” due to P M = P Nn =1 P n can be written as,Φ M ( b ) = − σ tot T ρN + T ρ φ ( b ) N ! N − P = ( ν [ φ ( b ) /σ tot − N ) N − (1 − ν/N ) N , (12)where we have introduced the parameter ν = σ tot T ρ.
Like p tot , ν is dimensionless. However,it is not a probability for a thick target. Indeed, because the mean free path, Λ, of a beamparticle is 1 / ( ρσ tot ) , therefore, ν = T /
Λ represents the average number of collisions of abeam particle when it passes through a target of thickness T . Hence ν can be a very largenumber.In the limit, N → ∞ , we obtain,Φ M ( b ) = exp " ν φ ( b ) σ tot − ! − exp( − ν ) . (13)Here, we emphasize the nonlinear action of ν on Φ M ( b ). The second term in Eq.(13) comesfrom the limit of P , it also shows that ν has the meaning of a beam decay rate in a target ofthickness T . Note that this latter term is neglected in the Moli` e re theory [1] as well as in itsreformulation by Bethe [7] and by Mott and Massey [8]. Hence, we find a form similar butnot identical to that of the Moli` e re theory for the final, multistep probability density P M : P M ( q ) = (2 π ) − exp( − ν ) Z d~b exp( i~q · ~b ) ( exp " ν φ ( b ) σ tot − ) . (14)Again, we note that P M ( q ) depends nonlinearly on ν . It is appropriate to underline theimportance of the above-mentioned extra term in the present formulation. If T is smallrather than large, T = t, then ν is also small. The presence of this extra term reduces P M to p = t ρ σ , as should be for thin targets. On the other hand, φ ( b ) → b → ∞ .6onsequently, the first term in the curly bracket has a limit, exp[ ] → , that exactlycompensates the second term in the curly bracket, ensuring the convergence of the integrationfor P M ( q ).It is interesting to note from Eq. (14) that P totM ≡ Z d~q P M ( ~q ) = 1 − exp( − ν ) = 1 − P Mt , (15)where P Mt is the total transmission probability. This last equation is a sum rule for themultiple-scattering probability. Multiple-scattering differential cross sections can be relatedto probability densities by the general relation P M ( q ) = T ρ σ M ( q ) , (16)in the same way σ ( ~q ) is to p ( ~q ). In summary, three steps thus occur in this formalism: (i) Fourier transform thin target data σ into their profile φ , see Eq. (8); (ii) Find ν = T ρ σ tot and exponentiate T ρ φ , see Eq. (13); (iii) Fourier transform Φ M back into a probabilitydistribution P M , see Eq. (14).It is reasonable to parametrize the single-scattering distribution as σ = σ c + σ n with σ c being the Coulomb cross sections and σ n the sum of cross sections of nuclear scatteringand nuclear-Coulomb interference. The separation of σ into σ c and σ n induces the samefor the profile function: φ = φ c + φ n . The relationexp( βφ c ) exp( βφ n ) − βφ c ) −
1] + exp( βφ c ) [exp( βφ n ) −
1] (17)with β = ν/σ tot then leads to a split of P M as the sum of two probability densities, P Mc = exp( − ν )(2 π ) Z d~b exp( i~q · ~b ) ( exp " ν φ c ( b ) σ tot − ) (18)and P Mn = exp( − ν )(2 π ) Z d~b exp( i~q · ~b ) exp " ν φ c ( b ) σ tot exp " ν φ n ( b ) σ tot − ) . (19)7his allows a perturbative consideration of nuclear effects at those angles where Coulombscattering dominates. Note that the exponent in Eq. (18) contains a denominator σ tot andnot σ tot c . Hence P Mc is proportional to a pure Coulomb process with an effective value of ν, namely ν eff = ( σ tot c /σ tot ) ν. Alternately, at angles where nuclear scattering might dominate, the roles of φ c and φ n can be interchanged to generate similar formulae, namely, P ′ Mn = exp( − ν )(2 π ) Z d~b exp( i~q · ~b ) ( exp " ν φ n ( b ) σ tot − ) (20)and P ′ Mc = exp( − ν )(2 π ) Z d~b exp( i~q · ~b ) exp " ν φ n ( b ) σ tot exp " ν φ c ( b ) σ tot − ) . (21)One can parametrize the screened Coulomb interactions as σ c ( q ) = P α C α ( q + κ α ) − α . Thepowers α, screening momenta κ α , and normalizations C α are mainly functions of the charge Z of each individual nucleus. But σ n will depend on both Z and the mass number A. Theglobal normalization of σ M will also depend on the target thickness or the parameter ν .Hence, the theory is essentially driven by three parameters of a thick target, namely, Z, A and ν. Experimental measurements of σ M might, conversely, permit a determination of suchthree parameters when the nuclear nature of the target is unknown a priori , as is most oftenthe case for radiographic studies where a recovery of A, besides Z, would be precious. Successwill occur, however, only if multiple scattering does not spoil the information carried by A. This question is the main concern of the following sections.
The mean-square width h q i of the distribution σ represents a useful observable forthe broadening of the cross-section distribution σ and can be defined by an integral h q i = ( σ tot ) − Z d~q q σ ( ~q ) = 2 π ( σ tot ) − ∞ Z dq q σ ( q ) , (22)8f it converges. The use of Eq.(8) and elementary properties of the Fourier transform allowsus to write h q i = − lim b → b ddb b ddb ! φ ( b ) φ (0) , (23)where the operator between brackets ( ) comes from a two-dimensional Laplacian in cylin-drical coordinate space. The same procedure gives the mean-square width of σ M ( ~q ) as h q i M = − lim b → b ddb b ddb ! Φ M ( b )Φ M (0) , (24)where Φ M (0) = (1 − e − ν ). Assume, for the sake of the argument, that σ ( q ) is a Gaussian, σ ( q ) = C exp[ − q / (2 κ )] , (25)where C is a suitable normalization and the parameter 1 /κ is the interaction range. Forinstance, if one discusses screened-Coulomb interactions, then 1 /κ is of the scale of an atomicradius. Then one obtains σ tot = 2 π C κ , h q i = 2 κ , and φ ( b ) = 2 π C κ exp( − b κ / . (26)From Eqs. (13) and (24) one further obtainsΦ M ( b ) = exp h ν (cid:16) e − b κ / − (cid:17)i − exp[ − ν ] (27)and h q i M = ν h q i − exp( − ν ) . (28)Since ν is large in general, the denominator is ≃
1. Consequently, the multiple scatteringhas broadened the mean-square width by a factor ν , as might be expected from a Brownianmotion in the transverse-momentum space. The multiplication of h q i by ν also occurs ifwe start from a “polynomial Gaussian distribution” ( q/κ ) n exp[ − q / (2 κ )] . This growth9ate is very general and can be viewed as one more version of the central limit theorem. Asadditional evidence, one finds that if σ ( q ) is of the functional form σ ( q ) = C ( q + κ ) − n withan exponent n >
2, then h q i = κ / ( n −
2) and again h q i M = ν h q i / (1 − e − ν ) ≃ ν h q i . In the following, we illustrate the broadening of the cross-section distribution in thecase of a screened Coulomb scattering. We fit σ at small angles by a few terms of the sum σ c ( q ) = X m> C m ( q + κ m ) m , (29)where m can be half-integers as well as integers, and C m and κ m are fitting parameters. Itfollows that σ tot c = π X m C m ( m − κ m − . (30)For definiteness, we take two terms with m = 5 / σ c ( q ) = C / ( q + κ / ) / + C ( q + κ ) . (31)Hence, σ tot c = 2 πC / κ / + πC κ . (32)Dividing both sides by σ tot c , we obtain1 = 2 πC / κ / σ tot c + πC κ σ tot c ≡ a / + a . (33)Eq. (33) shows that both a / and a are dimensionless numbers between 0 and 1 . Anadvantage of using Eq. (29) is that its Fourier transform gives the profile function φ c interms of analytical functions which can be easily analyzed, i.e., φ c ( b ) = 2 πC / (1 + κ / b ) exp( − κ / b )3 κ / + πC b K ( κ b )24 κ , (34)10here K is the modified Bessel function of the second kind. One verifies easily that φ c (0) =2 πC / / (3 κ / ) + πC / (3 κ ) = σ tot c . The “Coulomb” multistep profile then readsΦ Mc = exp ( ν " a / (1 + κ / b ) e − κ / b + a b κ K ( κ b ) − − e − ν . (35)When b → ∞ , it is easy to verify that Eq. (34) induces exponential decreases with ranges κ − / and κ − . Numerical integrals with such integrands converge well. The final integral forthe Coulomb cross section then reads σ Mc ( q ) = 12 π T ρ ∞ Z db b J ( qb ) Φ Mc ( b ) . (36)Let κ be an average between the two momenta κ / and κ , which are both atomicscales. It is now convenient to scale momenta and lengths as q = κQ and b = B/κ.
Thedimensionless Q will be a few units or a few tens, if one wants to describe scattering anglesmoderately larger than the Coulomb peak. We also expect that the values of B contributingto the integral σ Mc ( Q ) = exp( − ν )2 πκ T ρ ∞ Z dB B J ( QB ) × ( exp " νa / (cid:18) κ / κ B (cid:19) e − κ / B/κ + νC B κ K (cid:18) κ κ B (cid:19) − ) (37)should be mainly between 0 and several units. In atomic units, all parameters κ, κ / /κ and κ /κ are of order 1 . It remains to estimate the dimensionless magnitudes of ν a / and ν a / . From Eq.(33), a / and a are moderate fractions of 1 . It is thus the large number ν thatdrives the integrand.It is also convenient to write σ ( q ) = κ − σ ( Q ) , φ ( b ) = κ − φ ( B ) (38)with σ ( Q ) and φ ( B ) being dimensionless. Because we work with systems of atomic scale,we further set κ to be 1 , meaning that our primary scale is “atomic”. In this scale, all lengths11nd momenta will, respectively, be given in units of atomic radius and its inverse.To show the shrinking of profiles by multiple scatterings we plot φ c and Φ Mc in Fig. 1as the “crosses” and solid curves when σ c = ( Q + 1) − / , and, respectively, as the “circles”and dashed curves when σ c = ( Q + 1) − . As one can see, the dashed and solid curves dodecay faster than their respective single scattering partners. The Fourier transform of Φ Mc B φ c ( B ) , Φ M c ( B ) Fig. 1. Crosses: φ c ( B ) for σ c ( Q ) = ( Q + 1) − / . Solid curve: the corresponding multistep Φ Mc ( B )if ν = 4 . Circles and dashed curve: φ c ( B ) and Φ Mc ( B ) (with ν = 4) for σ c = ( Q + 1) − . Allprofiles normalized to 1 at B = 0. then leads to the expected broadening of σ Mc , as shown in Fig. 2. (For graphical conveniencewe used ν = 4 in Figs. 1 and 2, which is much smaller than physical ν but is demonstrativeenough.) 12 Q σ c ( Q ) , σ M c ( Q ) Fig. 2. Crosses: σ c ( Q ) = ( Q + 1) − / . Solid curve: the corresponding σ Mc ( Q ) for ν = 4 . Circles: σ c ( Q ) = ( Q + 1) − . Dashed curve: the corresponding σ Mc ( Q ) for ν = 4 . All cross sectionsnormalized to 1 at Q = 0 . In this section, we introduce a semi-realistic model for σ ( Q ) which contains “nu-clear” information. We will investigate (a) changes of normalizations and not just shrinkingor dilation of shapes of σ ( Q ), and (b) how nuclear information may become lost. We will,therefore, illustrate the blurring of signal through the study of various relevant quantities,such as P M , P Mc , P Mn , P ′ Mc , P ′ Mn . We also use analytical models to ensure that the blur-ring comes from physics and is not a result of numerical imprecision. A good analyticalmodel must satisfy the following constraints: (i) positivity of the sum of σ c and σ n ; (ii)big contrasts between maxima and minima; (iii) analyticity in both the momentum and the13mpact parameter representations; and (iv) significant differences between the atomic andthe nuclear scales for profiles.Let σ c ( Q ) = (1 + Q ) − (1 + Q / − . (39)This is qualitatively realistic, because the factor, (1 + Q ) − , represents a screened Coulombscattering. The additional, artificial factor, (1 + Q / − , is here just for the convergenceof h Q i c . Then we further use, with σ n (0) = 0 , the following semi-realistic σ n : σ n ( Q ) = 11 Q / × exp(1 / − Q / h e (3600 − Q + Q ) − Q − (491485925 − Q + 215573 Q )+40919125 e ( Q − (3973881778272 − Q + 463199137 Q ) i (40)The quality of the model with respect to the requirements (ii) and (iv) are evidenced by Fig.3. As it is allowed by the split of σ into a “Coulomb” part ( σ c ) and a “nuclear” part( σ n ), our σ n can be positive or negative, as long as σ remains positive. Our σ n was finetuned to create four clear “nuclear” signals, namely two maxima of σ near Q = 8 and18 , and, as signatures of interferences, two sharp minima at Q = 5 and 12 . Furthermore,we adjusted its parameters so that the maxima do not exceed ∼
1% of the forward peakof σ c . Note also that our model σ n has only two maxima and, thus, carries no “nuclear”information for Q >
40. This is designed to track whether or not the maxima, if they survivethe blurring of angular distribution by multiple scatterings, would migrate towards largervalues of Q. The log σ and log σ c of our toy model are shown as functions σ ( Q ) in Fig.3. It is trivial to deduce σ n visually.The corresponding profiles read, in closed forms, φ c ( B ) = 200 π − K ( B )9801 + K (10 B )9801 + B K ( B )198 ! , (41)14 Q l og σ ( Q ) -8-6-4-20 σ = σ σ = σ Fig. 3. log σ (solid curve) and log σ c (dashed curve) as functions of Q . and, with u = − e / and v = − e / ,φ n ( B ) = 11 / π exp(1 / − B / × h B (37882968282999233748 + 24684880296681586800 u +18953703386795125 v ) − B (1727605574138750181696 +1184874254240716166400 u + 973536146177791625 v ) + 2734375 B × (1318896762458059001549772 + 935772074343960364976400 u +815387774668679551625 v ) − B (987420216355162578690908328 +716741544586342715769420000 u + 649720700222675824315625 v ) +2800 (12734093638656544401340038 + 9458232286390484639445000 u +8603026609006638128125 v ) B − u + 34558380311245783728125 v ) i . (42)These profile functions are shown in Fig. 4. The width of φ n is significantly smallerthan that of φ c , as one should expect when comparing a “nuclear” profile to an “atomic”15ne. A geometrical ratio of widths might be ∼ − or even ∼ − , but the model ratiowe choose, between ∼ / ∼ / , is sufficient for a pedagogical study and much moreconvenient numerically. B l og φ ( B ) -10123456 φ = φ φ = φ φ = φ Fig. 4. log φ (solid curve), log φ n (dotted curve), and log φ c (dashed curve) as functions of B . This choice of “data” gives, after a numerical implementation of Eq. (14), the totalmultistep probability distributions shown in Fig. 5. The solid curve is the same as thatin Fig. 3, namely, log σ . The dashed, linked-crosses, and linked-circles curves representlog P M for ν = 4 , , respectively. The result is striking, on two counts: (i) theforward peak is more and more damped, the distributions extending more and more towardslarger momenta, and (ii) the “nuclear information”, whether minima or maxima, becomesrapidly blurred beyond recognition. Furthermore, the broadening of distributions does notseem to push much residual information towards larger momenta. The broadening process16 Q l og Y ( Q ) -6-5-4-3-2-10 Y = σ Y = P M , ν = 4 Y = P M , ν = 9 Y = P M , ν = 16 Fig. 5. The dependences on Q of log σ of single scattering (solid curve) and log P M of mul-tiple scattering when ν = 4 (dashed curve), 9 (linked-crosses curve), and 16 (linked-circles curve),respectively. is also confirmed by the behavior of the component P Mc of P M , shown in Fig. 6. In ourmodel σ tot ≃ . σ tot c ≃ . . We chose a large nuclear contribution, σ tot n ≃ . , in orderto emphasize nuclear effects. However, at Q < , even this exaggerated nuclear informationdid not survive multiple scatterings.In Fig. 7, we show the various probability distributions P Mn ( Q ). We note again thatmultiple scatterings wash away nuclear information. A similar feature is also seen in the P ′ Mn given in Fig. 8. Besides the damping and information loss which are evident from Figs. 7 and8, we may stress a feature of Fig. 8, namely the transformation of “negative cross sections”into positive ones after multiple scattering. In order to create interferences, it was necessary,at the stage of making a model for σ n , to create negative values interfering with σ c . As has17 Q l og Y c ( Q ) -10-8-6-4-20 Y c = σ Y c = P Mc , ν = 4 Y c = P Mc , ν = 9 Y c = P Mc , ν = 16 Fig. 6. Solid curve: log σ c ( Q ) . Dashed, linked-crosses and linked-circles curves: log P Mc ( Q )for ν = 4 , , respectively. already been pointed out, this is allowed as long as σ remains positive; there is a degree offreedom in modeling σ n . The solid curve in Fig. 8 shows log | σ n | . One sees four arches, thefirst and the tiny third ones meaning negative values. Such “negative” arches disappear inthe dashed curves representing P ′ Mn . This disappearance justifies the use of models where σ n can be not everywhere positive as long as σ n + σ c is everywhere positive, as was discussedafter Eq. (40).An advantage of our use of special analytical forms for the cross sections, Eqs. (39,40),is that such forms induce analytical profiles, Eqs. (41,42), which in turn allow analytical formsfor the multistep profiles, Eqs. (13) and (18-21). Values of h Q i can then be easily obtainedfrom the use of Eq. (24). The rates of broadening as functions of ν can also be readilycalculated. Fig. 9 shows how, at values of ν smaller by several orders of magnitudes than18 Q l og P M n ( Q ) -4.6-4.4-4.2-4.0-3.8-3.6-3.4-3.2 ν = 4 ν = 9 ν = 16 Fig. 7. Dashed, linked-crosses and linked-circles curves: log P Mn ( Q ) for ν = 4 , , respec-tively. those estimated from geometric cross sections, the square-widths h Q i of P Mn , P Mc alreadyincrease linearly with ν. We have also noted a similar behavior of the widths of P ′ Mn and P ′ Mc . In summary, the signature of nuclear information (diffractive oscillations in the dif-ferential cross section) in the region of small momentum transfers is washed away by thebroadening of the angular distribution. This happens even with our model that has exagger-ated nuclear cross sections. In the next section, we examine if there exist momentum-transferregions where the multiple scattering of the proton does not completely blurr nuclear signals.19 Q l og Y n ( Q ) -10-8-6-4-20 Y n = | σ | Y n = P’ Mn , ν = 4 Y n = P’ Mn , ν = 9 Y n = P’ Mn , ν = 16 Fig. 8. Solid curve: log | σ n ( Q ) | for a nuclear signal. Dashed, linked-crosses and linked-circlescurves: probability distributions log P ′ Mn ( Q ) when ν = 4 , , respectively. Let r ≡ q / /q min be the ratio of the half-width of the Coulomb peak to the mo-mentum transfer at which the first minimum due to nuclear diffraction is observed for athin target. At high energies, both q / and q min occur at very small angles. Consequently, r = θ / /θ min with the θ ’s being the respective scattering angles corresponding to q / and q min . From Eq. (28), it is reasonable to expect a rule, h q i M ≃ ν h q i ≃ ν q / (= νr q min ) , hence that there is a critical value ν crit ≃ r − , above which nuclear signals will be obliteratedby the broadening of the Coulomb peak. In other words, nuclear signals can only be observedat q ≫ q min for ν > ν crit .In Fig. 10, we show the elastic scattering differential cross sections of protons scattered20 log ν l og < Q > Fig. 9. The dependences on log ν of log h Q i for P M ( Q ) (solid curve), P Mc ( Q ) (dashed curve)and P Mn ( Q ) (dotted curve). Note that the slopes ≃ ν >> . from a thin Pb target at ∼
20 GeV, which we have calculated by using the method ofoptical model of Ref.[14] with a screened Coulomb interaction. The calculated cross sectionsexhibit the main characteristics of high-energy proton-nucleus scattering, namely, a narrowforward Coulomb peak and the diffractive oscillations at larger angles. Here, the diffractivepattern constitutes the nuclear signal. One notes that the first diffractive minimum lies atabout ∼ θ / for the Coulomb peakis of order ∼ .
003 milliradians. It is therefore reasonable to assume that, for high-energyproton scattering from nuclei, r is of order ∼ − or less in general. At most one mightconsider r of order ∼ − . Accordingly, although the geometric size of a nucleus is typically ∼ − smaller than that of its atom, the range of “nuclear information profiles” athigh scattering energies may be taken ∼ to ∼ smaller than the range of the screenedatomic profile, and possibly much smaller. While the model used in the previous section21here ∼ . < r < ∼ . r is in order. θ c.m. [mrad] ( d σ / d Ω ) c . m . [ m b / s r] Fig. 10. Differential cross sections of p − Pb elastic scattering at 20 GeV.
We first introduce a model in which r = 10 − ; the profile function φ ( B ) is the sumof a “Coulomb” term, φ c ( B ) = B K ( B ) (43)and a “nuclear” term, φ n ( B ) = 4 × − B − / . (44)The profile φ c gives a bare Coulomb cross section of the form σ c ( Q ) ∝ (1 + Q ) − and theWoods-Saxon profile φ n makes, in practice, a window with range r = 1 /
100 indeed. Thecoefficient 800 in its exponent creates a “smoothed” Heaviside function. Both profiles are22ormalized so that σ tot c = 1 , and σ tot n /σ tot c = 4 × − except for a negligible factor 1 + e − . This cross-section ratio is quite compatible with the r suggested by Fig. 10. Hence, the set ofparameters given in Eqs. (43) and (44) is more realistic than that used in the previous section.The result for various angular cross sections σ M ( Q ; ν ) , compared with the single scattering σ ( Q ) , is shown in Fig. 11. An inspection of the figure shows that the Coulomb peak dampsand spreads and the nuclear signal fades when ν increases. The solid curve, representinglog σ ( Q ) , and the dashed curve, representing log σ M ( Q ) for ν = 2000 , exhibit somewhatsimilar oscillations. The dotted curve, corresponding to log σ M for ν = 10 , hardly oscillatesany more, i.e., nuclear signals are completely washed out. This confirms a loss of nuclearsignal at low and moderate momentum transfers when ν approaches ν crit ∼ r − = 10 . Q [10 ] l og σ ( Q ) -14-12-10-8-6-4-20 σ = σ σ = σ M , ν =2000 σ = σ M , ν =10000 Fig. 11. The dependences on Q of log σ (solid curve) and of log σ M for ν = 2000 (dashedcurve) and ν = 10000 (dotted curve). All with r = 10 − . The ν crit ∼ r − rule is also seen in the previous section, where the use of ∼ . < < ∼ . ν > ∼
10. To further verify this rule,we use the same φ c but use instead r = 0 . × − in φ n , namely, φ n ( B ) = 4 × − B − / . (45)The results are shown in Fig. 12. As we can see, the observation of nuclear signals is muchimproved; in agreement with the rule that r = 1 /
200 elevates ν crit to a higher value, ∼ × . Q [10 ] l og σ ( Q ) -14-12-10-8-6-4-20 σ = σ σ = σ M , ν =2000 σ = σ M , ν =10000 Fig. 12. The dependences on Q of log σ (solid curve) and of Y = log σ M for ν = 2000 (dashedcurve) and ν = 10000 (dotted curve). All with r = 0 . × − . As a last test of the ν crit ∼ r − rule, we keep φ c ( B ) = BK ( B ) and let φ n ( B ) = 2 × − B − / , (46)24hich is an obvious r = 1 / ν crit occurs between 10 − and 10 − . Q [10 ] l og σ ( Q ) -18-16-14-12-10-8 σ = σ σ = σ M , ν =10 σ = σ M , ν =10 Fig. 13. The dependences on Q of log σ (solid curve) and of log σ M for ν = 10 (dashed curve)and ν = 10 (dotted curve). All with r = 10 − . Because ν = T ρ σ tot ≃ T ρ σ tot c , the existence of a ν crit induces a critical target thick-ness T crit such that the retrieval of nuclear signal is possible for target thickness T sufficientlyless than T crit ; namely, T < T crit ≃ ν crit ρ σ tot c . (47)For p − Pb elastic scattering at 20 GeV, σ tot c ≃ . × mb = 6 . × − cm . Thedensity d and the atomic mass number A of lead are 11.3 g/cm and 208, respectively.Hence, ρ = ( d/A ) N Avog = 3 . × cm − , with the Avogadro number N Avog = 6 . × [1/mole]. Because from Fig. 10 it is likely that r ∼ − and our analysis indicates that25 able 1Target thickness T corresponding to ν = 10 for 20-GeV protons.Atom d [g/cm ] Z λ A ρ [10 cm − ] T [cm]Pb 11.3 82 1.000 208 3.27 4.6Cu 8.9 29 0.250 64 8.39 7Al 2.7 13 0.086 27 6.02 29Mg 1.74 12 0.077 24 4.37 44Be 1.85 4 0.018 9 12.38 67 ν crit ∼ , then Eq. (47) gives T crit ≃
46 cm. Hence, a nuclear signal can be retrieved at ν ≤ , which corresponds to T ≤ . . T crit [cm]. In so far as Q = 1 corresponds to θ ∼ .
003 milliradians, the survivor oscillation seen for ν = 10 in Fig. 13 between Q ∼ Q ∼ , compatible with the expected period ∼ π/r, would demand experimentalmeasurements at angles of order a few dozens of milliradians at most.The proton-nucleus Coulomb cross section σ tot c is ∝ Z R e ∼ Z / , where Z is thetarget charge and R e is the root-mean-square radius of electric charge distribution in an atomwith R e ∼ a /Z / and a being the first Bohr radius[15]. Hence, one can estimate σ tot c ; pA for proton scattering from a given nucleus A at 20 GeV by using σ tot c ; pA ∼ λ σ tot c ; p − P b withthe scaling factor λ = (82 /Z ) − / . Hence, for a same ν one has T ( pA ) = T ( pP b ) ρ P b / ( ρ A λ ).Results for a sample of atomic nuclei at ν = 10 are given in Table I. Of course, the priceone pays in studying the nuclear signals that survive the multiple-scattering broadeningis that one has to measure the angular distribution with good energy resolution at largeproton scattering angles. In the case of 20 GeV incoming protons, the angles are about tensof milliradians, where the magnitudes of the cross sections are quite small. However, suchmeasurements should be feasible with the currently available technology.26 Conclusion
The main mathematical and physical statement proposed by this work about multiplescattering consists in folding probabilities rather than scattering amplitudes. This is justifiedby the incoherence which is expected between the different scatterers of a thick, non crys-talline target. Simultaneously, an eikonal approximation, justified by the very high energyof the beam, allows a familiar impact parameter representation with profiles. Furthermore,small-momentum expansions[1] are not employed in the formulation. As a consequence ofsuch initial statements, a Poisson process is found, leading to an elementary formalism ofconvolutions and exponentiations in a context of Fourier-Bessel transforms.This Poisson process is nothing but a random walk in transfer momentum space.The central limit theorem is at work and the details of nuclear oscillations and interferencesbetween Coulomb and nuclear scattering are blurred very fast as soon as the parameter ν = T ρ σ tot , a measure of the number of collisions, exceeds a critical value of order r − . Here r is the ratio of the range of the nuclear profile to that of the atomic profile.Below this critical value of ν, and at moderate and large momentum transfers (at thecost of very small elastic cross sections in the latter case) our conclusion is that some nuclearinformation remains observable. Such information is contained in oscillations of the multistepangular cross section σ M ( Q ) with periods ∼ π/r, oscillations that are similar to those ofthe Bessel function, J ( r Q ) , which typically represents pure nuclear diffractive scattering.Our model analysis shows that the characteristic distances between successive cross-sectionmaxima and minima in the angular distribution remain essentially unchanged while each ofthese oscillations dampens as ν increases.From the point of view of retrieving nuclear signals from protons traveling through athick target, for which no sufficient attention was given in the literature, our work is moreof a general feasibility study rather than a specific numerical evaluation. We have made useof analytical and semi-analytical models to bring out the basic features of the underlyingphysics. We believe that the positive feasibility concluded from this study will sustain tests27n detailed numerical applications.One problem which has not been solved in the present work, however, is to find anestimate of the ν dependence of such periods ∼ π/r. Our numerical evidence suggests thatthe dependence is not very strong, despite all the causes for a broadening of the signal, butour models and calculations lack the precision needed to tabulate such periods into functionsof ν. This effort is under consideration for an extension of the present work.In summary, below ν crit , which is of order 1 /r with r being the ratio of the rangeof nuclear profile to the range of atomic profile, the nuclear signals can be retrieved fromproton scattering from a thick target of thickness T < T crit . We suggest a conservative upperbound, namely, T ≤ . T crit for practical considerations. References [1] G. Moli`ere,
Z. Naturforsch , 78 (1948).[2] E.J. Williams, Proc. Roy. Soc. , 531 (1938);
Phys. Rev. , 292 (1940); Rev. Mod. Phys. , 217 (1945).[3] S.A. Goudsmit and J.L. Saunderson, Phys. Rev. , 24 (1940); ibid. , 36 (1940).[4] H. Snyder and W.T. Scott, Phys. Rev. , 220 (1949).[5] W.T. Scott, Phys. Rev. , 245 (1952). (1940) and ibid.
36 (1940).[6] H.W. Lewis, Phys. Rev. , 526 (1950).[7] H.A. Bethe, Phys. Rev. , 1256 (1953).[8] N.F. Mott and H.S.W. Massey, The Theory of Atomic Collisions , Oxford at the ClarendonPress, third edition, 1965, pp.467-476.[9]
The European Journal of Physics , 166 (2000).[10] See, for example, G. Shen, C. Ankenbrandt, M. Atac, et al., Phys. Rev. , 1584 (1979).
11] R.J. Glauber, in
High-Enery Physics and Nuclear Structure , Proc. of the 2nd InternationalConference, Rehovoth, 1967, ed. G. Alexander, North-Holland, Amsterdam, 1967, p.311, andthe references mentioned therein.[12] Charles J. Joachain,
Quantum Collision Theory , North-Holland Publishing Co., Amsterdam-New York-Tokyo-Oxford (1983), pp.7-9.[13] L.S. Rodberg and R.M. Thaler,
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Nucl. Phys. B , 135 (1970).[15] K. Gottfried, Quantum Mechanics, Vol. I , p458, Addison-Wesley, N.Y. (1980): The root-mean-square of electron charge distribution is of the order of a /Z / with a = 0 . × fm.fm.