Correction to the Moliere's formula for multiple scattering
aa r X i v : . [ phy s i c s . a t o m - ph ] D ec Corre tion to the Molière's formula for multiple s attering.R.N. Lee ∗ and A.I. Milstein † Budker Institute of Nu lear Physi sandNovosibirsk State University,630090 Novosibirsk, Russia(Dated: November 8, 2018)Abstra tThe quasi lassi al orre tion to the Molière's formula for multiple s attering is derived. The onsideration is based on the s attering amplitude, obtained with the (cid:28)rst quasi lassi al orre tiontaken into a ount for arbitrary lo alized but not spheri ally symmetri potential. Unlike theleading term, the orre tion to the Molière's formula ontains the target density n and thi kness L not only in the ombination nL (areal density). Therefore, this orre tion an be re(cid:27)ered to as thebulk density orre tion. It turns out that the bulk density orre tion is small even for high density.This result explains the wide region of appli ability of the Molière's formula. ∗ Email:R.N.Leeinp.nsk.su † Email:A.I.Milsteininp.nsk.su 1. INTRODUCTIONMultiple s attering of high-energy parti les in matter is a pro ess whi h plays an impor-tant role in the experimental physi s. The basis of the theoreti al des ription of this pro essgoes ba k to Refs. [1, 2, 3, 4℄. Further development of the theory of multiple s attering wasperformed in numerous publi ations, see, e.g., Refs. [5, 6℄ and referen es therein. Detailedexperimental investigation of multiple s attering has also been performed, see Refs.[7, 8℄.The elebrated Molière's formula des ribes the angular distribution dWd Ω for small-angles attering. It was shown by Bethe in Ref. [4℄ that the most simple way to derive thisformula is to use the transport equation. As a onsequen e of this equation, the angulardistribution dWd Ω depends on the thi kness L and the density n only in the ombination nL ,whi h is the areal density of a target. One an expe t that the appli ability of the Molière'sformula is restri ted by the low density. However, the experimental results obtained forsmall s attering angles show that the deviations from the Molière's formula are small forall data available. In the present paper we explain su h surprising behavior al ulating theleading bulk-density orre tion to the Molière's formula.We start with the expression for the small-angle s attering amplitude. This expressionhas been obtained in Ref. [9℄ in the quasi lassi al approximation with the (cid:28)rst orre tiontaken into a ount. The appli ability of this approximation is provided by small s atteringangles and high energy ε of the parti le, ε ≫ m ( m is the parti le mass, the system of unitswith ~ = c = 1 is used). This amplitude has been obtained for arbitrary lo alized potentialwithout requirement of its spheri al symmetry. As known, the quasi lassi al wave fun tionhas a mu h wider region of appli ability than the eikonal wave fun tion. However, as itwas shown in Ref. [9℄, the s attering amplitude obtained with the use of the quasi lassi alwave fun tion oin ides with that obtained in the eikonal approximation, see also Ref. [10℄.Using the quasi lassi al s attering amplitude with the (cid:28)rst orre tion taken into a ount,we al ulate the orresponding ross se tion and average it over the positions of atoms inthe target. Dividing this ross se tion by the area of the target, we arrive at the angulardistribution dWd Ω . The leading term of this distribution oin ides with the Molière's formula.The orre tion depends not only on the areal density nL of the target but also on the bulkdensity n alone. We dis uss the magnitude of the orre tion for di(cid:27)erent target parametersand s attering angles. 2I. DIFFERENTIAL PROBABILITYLet us dire t the z axis along the initial momentum p of the parti le so that r = z p /p + ξ .The small-angle high-energy s attering amplitude in the lo alized potential V ( z, ξ ) has theform [9℄ f = − iε π Z d ξ e − i q · ξ ( e − i K ( ξ ) − e − i K ( ξ ) " ε ∞ Z −∞ dx x △ ξ V ( x, ξ ) − iε ∞ Z −∞ dx x Z −∞ dy y ( ∇ ξ V ( x, ξ )) · ( ∇ ξ V ( y, ξ )) , (1)where q = p ′ − p , p ′ is the (cid:28)nal momentum, K ( ξ ) = R ∞−∞ dxV ( x, ξ ) , ∇ ξ = ∂/∂ ξ , and △ ξ = ∇ ξ . The se ond term in bra es in Eq. (1) orresponds to the orre tion. For q = 0 ,the unity in the leading term an be omitted. The di(cid:27)erential ross se tion, orrespondingto the amplitude f and having the same a ura y as Eq. (1), reads dσd Ω = ε π Re Z d ξ d ξ e − i q · ( ξ − ξ ) e − i [ K ( ξ ) −K ( ξ )] × ε ∞ Z −∞ dx x △ ξ V ( x, ξ ) − iε ∞ Z −∞ dx x Z −∞ dyy ( ∇ ξ V ( x, ξ )) · ( ∇ ξ V ( y, ξ )) . (2)The total potential of atoms in the target has the form V ( r ) = X i u ( r − r i ) , (3)where u ( r ) is the potential of individual atom, whi h we assume to be spheri ally symmetri .We perform the averaging over the atomi positions using the pres ription h f i = Z Y i dx i d ρ i LS f, (4) orresponding to the dilute gas approximation. As a result, we obtain dWd
Ω = (cid:28) dσSd Ω (cid:29) = ε π Re Z d ρ e − i q · ρ ( F N − iNε F N − F − iN ( N − ε F N − F ) , (5)3here F = Z d ρ S exp {− i [ χ ( ρ ) − χ ( ρ − ρ )] } ,F = L Z d ρ S exp {− i [ χ ( ρ ) − χ ( ρ − ρ )] } [( ∇ ρ χ ( ρ )) · ( ∇ ρ χ ( ρ )) + i △ ρ χ ( ρ )]+ ∞ Z −∞ dx x Z −∞ dy y Z d ρ S exp {− i [ χ ( ρ ) − χ ( ρ − ρ )] } ( ∇ ρ u ( x, ρ )) · ( ∇ ρ u ( y, ρ )) ,F = ∞ Z −∞ dx ∞ Z −∞ dy Z Z d ρ S d ρ S exp {− i [ χ ( ρ ) − χ ( ρ − ρ ) + χ ( ρ ) − χ ( ρ − ρ )] }× " L − ( x − y ) L ( ∇ ρ u ( x, ρ )) · ( ∇ ρ u ( y, ρ )) . (6)Here χ ( ρ ) = R ∞−∞ dx u ( x, ρ ) , so that K ( ρ ) = P i χ ( ρ − ρ i ) . Then we pass to the limit N, S → ∞ and
N/S = nL = const . In this limit, F N = (cid:18) Z d ρ S [exp {− i [ χ ( ρ ) − χ ( ρ − ρ )] } − (cid:19) N → exp (cid:20) − nL Z d ρ (cid:0) − e − i [ χ ( ρ ) − χ ( ρ − ρ )] (cid:1)(cid:21) . (7)Substituting Eqs. (6) and (7) into Eq. (5), we (cid:28)nally obtain dWd Ω = ε (2 π ) Z d ρ e − i q · ρ exp (cid:26) − nL Z d ρ [1 − cos ( χ ( ρ ) − χ ( ρ − ρ ))] (cid:27) × (cid:26) − nLε ∞ Z −∞ dx Z d ρ sin ( χ ( ρ ) − χ ( ρ − ρ )) ρ · ∇ ρ u ( x, ρ ) − n Lε Z d ρ cos ( χ ( ρ ) − χ ( ρ − ρ )) ρ χ ( ρ ) × Z d ρ sin ( χ ( ρ ) − χ ( ρ − ρ )) ∇ ρ χ ( ρ ) (cid:27) . (8)At the derivation of this formula we have used the integration by parts. As a result, allterms proportional to L in F and F , Eq. (6), vanished. It is onvenient to rewrite Eq.(8) in another form. The di(cid:27)erential in the momentum transfer Q ross se tion dσd Q of high-energy s attering o(cid:27) one atom, al ulated in the quasi lassi al approximation with the (cid:28)rst orre tion taken into a ount, satis(cid:28)es the relation [9, 10℄ Z d Q (cid:0) − e i Q · ρ (cid:1) dσd Q = Z d ρ [1 − cos ( χ ( ρ ) − χ ( ρ − ρ ))+ 1 ε ∞ Z −∞ dx sin ( χ ( ρ ) − χ ( ρ − ρ )) ρ · ∇ ρ u ( x, ρ ) . (9)4sing this relation, we obtain with the same a ura y as Eq. (8) the following form dWd Ω = ε (2 π ) Z d ρ e − i q · ρ exp (cid:20) − nL Z d Q (cid:0) − e i Q · ρ (cid:1) dσd Q (cid:21) × (cid:26) − n Lε Z d ρ cos ( χ ( ρ ) − χ ( ρ − ρ )) ρ χ ( ρ ) × Z d ρ sin ( χ ( ρ ) − χ ( ρ − ρ )) ∇ ρ χ ( ρ ) (cid:27) . (10)The leading term dW M d Ω in Eq. (10), orresponding to unity in bra es, oin ides with theMolière's formula. The orre tion dW C d Ω des ribes the e(cid:27)e t of the bulk density of the targetand has not been known so far.III. DISCUSSIONLet us dis uss the magnitude and the stru ture of the orre tion obtained. At (cid:28)xed arealdensity nL (the number of target atoms per unit area), the orre tion behaves as n (or L − ), and in reases when L de reases. Estimations show that the relative magnitude of the orre tion is the largest when the main ontribution to the integral over ρ in Eq. (10) omesfrom the region ρ ∼ a , where a is the s reening radius of atom, a ∼ a B Z − / , a B is the Bohrradius, Z is the nu lear harge number. This ondition is ful(cid:28)lled when q ∼ nLa , where q is the momentum transfer. In this ase, the orre tion has the relative order δ = (cid:18) dW M d Ω (cid:19) − dW C d Ω ∼ Zαna εa R, R = ( Zα ) nLa , (11)where α = 1 / is the (cid:28)ne-stru ture onstant. Using the estimates ( εa ) − ≪ ( ma ) − ∼ αZ / ≪ ,Zαna . Zαa − B (cid:0) a B Z − / (cid:1) = α ≪ , we obtain δ . − R mε .
The upper bound for δ grows with R . However, when R ≫ , both the leading term and the orre tion are suppressed by the fa tor exp [ − b R ] , where b ∼ is some numeri al onstant.Therefore, in the whole region interesting from the experimental point of view R is not toobig so that δδ