The form factor of the Fermi model spatial distribution

Abstract

A u seful analyti c expression for the form fa ctor F (q) = f p (r)e iq . ·r(P.· of the F ermi di stribution p(r) = Po[l + e(r-c)/"l -l is de riv ed. This expre ss ion con s is ts of a sim p le te rm with elementary fun ction s plu s a rapidl y conve rge nt infinit e seri es with term s of alt e rn at.in g sign. Tables of th e form fa ctor a s a fim c tio n of q for several valu es of th e pa ra met.e rs c and a, as we ll a s t.h e numeric al valu es of the normalization co ns ta nt po a nd th e rm s radiu s co rres pondin g to th ese va lu es of th e paramete rs, are al so giv en. Key W ord s: An alyti c, fe rmi di s tributi on , form fac tor , mom e ntum tra n sfe r, no rm ali za ti o n co ns ta nt , roo t-m e a n-squ a re radiu s.In th e calc ulati on of a cross sectio n in th e Born approxim ati on th e s pati al di s tributi on of th e interac ti on may b e re prese nte d by th e inclu sio n of a form fac tor whi c h is its F ouri er tra nsform.The r e action ampJitud e for a di stribute d interacti on is th e n th e fo rm factor F tim es th e reacti o n amplitude for a p oint inter ac ti o n. Th e form facto r s qu ar ed thu s e nters ex pressions for th e cross section. Th e co n ce pt of th e form fa c tor has been utilize d in a to mi c ph ysics fo r so me tim e a nd no w is used wid ely in nucle ar a nd particle physics to c hara cterize th e e ffec ts of th e s pati al di s tributi on of th e inter ac ti on.Th e form factor F is ge nerally give n as a fun cti on of th e mo me ntum q tra nsfe rre d by th e incide nt particle (r e prese nte d by a pl ane wave) to th e interac tin g parti cle. If the s ph eri cally symmetri c s pati al di s tributi o n of the inter actin g parti cle is giv en by p(r), the n th e form factor as sociate d with it is gi ve n by th e Fouri er trans form of p(r) : F( q)= f p(r)e;q "cflr = 4; fox p (r)r sin qr d r .(1)The de nsit y is normalize d s_ o thatfrom whichThere are se veral collecti ons of form factors for vari ous models of th e s patial di stribution (for example , de nsity di s tributio ns in nu clei) in th e lite rature [1 , 2, 3, 4].One of the mos t widely used models is not include d in most collections of form tactors. It is de fine d by the spatial di stributio n:wher e th e normalization c onstant po is chosen so that (2) is satisfied. The parameters a and c d e termin e the s hape of the spatial distribution. 85

Appendiciario -- A hands-on manual on the theory of direct Dark Matter detection

“…The form factor of the Woods-Saxon distribution, however, must be integrated numerically, which is why the Helm parametrization is often preferred (see however Refs. [216,217]). Other less model-dependent and more accurate form factors can be found e.g.…”

Upcgen: a Monte Carlo simulation program for dilepton pair production in ultra-peripheral collisions of heavy ions

“…is the charge form factor of the ion emitting the photon, k ⊥ is the photon transverse momentum and J 1 is the Bessel function of the first kind. In our calculations, we use an analytic expression for the Woods-Saxon form factor derived in [21]:…”

Constraining nonlinear corrections to Maxwell electrodynamics using γγ scattering

“…There are several parameterization for the charge distribution, and they introduce an important theoretical uncertainty [21]. More realistic, [42] Gaussian Gaussian distribution parameters, Λ = 0.088GeV and Q 0 = 0.060GeV, are such as to obtain the lead root-mean-square radius [26,43]. Fermi 2P distribution parameters a = 0.549fm and c = 6.642fm describe the diffuseness and the radius of the lead ion respectively [44].…”

“…This model is considered much more realistic but has no closed form for its corresponding form factor. An expression, however, can be obtained in terms of a series [42]. Two other widely used distributions in the literature, Gaussian and of a homogeneously charged sphere, have simple form factors and are included for comparison [43].…”