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M1b. Scattering Theory 1 A

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M1b. Scattering Theory 1 A Scattering Theory 1 (last revised: July 8, 2013) M1b–1 M1b. Scattering Theory 1 a. To be covered in Scattering Theory 2 (T1b) This is a non-exhaustive list of what will be covered in T1b. Electromagnetic part of the NN interaction • Bound states and NN scattering • – basic deuteron properties (but not mixing) – absence of other two-body bound states Phenomenology of NN phase shifts (e.g., `ala Machleidt) • – Both S-waves and large scattering lengths. E↵ective range e↵ects. Sign changes – what parts of forces contribute to di↵erent phases – low vs. high L partial waves How high in energy and partial wave and how accurately do phase shifts need to be reproduced • for various nuclear applications? b. Reminder of basic quantum mechanical scattering Our first and dominant source of information about the two-body force between nucleons (protons and neutrons) is nucleon-nucleon scattering. All of you are familiar with the basic quantum me- chanical formulation of non-relativistic scattering. We will review and build on that knowledge in this lecture and the exercises. Nucleon-nucleon scattering means n–n, n–p, and p–p. While all involve electromagnetic inter- actions as well as the strong interaction, p–p has to be treated specially because of the Coulomb interaction (what other electromagnetic interactions contribute to the other scattering processes?). We’ll start here with neglecting the electromagnetic potential Vem and the di↵erence between the proton and neutron masses (this is a good approximation, because (m m )/m 10 3 with n − p ⇡ − m (m + m )/2). So we consider a generic case of scattering two particles of mass m in a ⌘ n p short-ranged potential V . While we have in mind that V = VNN, we won’t consider those details until tomorrow’s lectures. Let’s set our notation with a quick review of scattering (without derivation). The Hamiltonian (in operator form) is p2 p2 H = 1 + 2 + V, (1) 2m 2m where we label the momentum operators for particles 1 and 2. We switch to relative (“rel”) and Scattering Theory 1 (last revised: July 8, 2013) M1b–2 p2 p20 P/2 + k P/2 + k! = p ) 1 p0 P/2 k P/2 k! 1 − − Figure 1: Kinematics for scattering in lab and relative coordinates. If these are external (and 2 2 therefore “on-shell”) lines for elastic scattering, then when P =0wehavek = k0 = mEk =2µEk (assuming equal masses m). center-of-mass (“cm”) coordinates: r + r r = r r , R = 1 2 (2) 1 − 2 2 p p k = 1 − 2 , P = p + p , (3) 2 1 2 so that the Hamiltonian becomes (still operators, but the P, k, and k0 in Fig. 1 are numbers) P2 k2 H = T + H = + + V, (4) cm rel 2M 2µ with total mass M = m1 + m2 =2m and reduced mass µ = m1m2/M = m/2. (Note: Hrel is sometimes called the intrinsic Hamiltonian.) Now we note that V depends only on the cm variables (more on this tomorrow!), which means that we can separate the total wave function into a plane wave for the center-of-mass motion (only kinetic energy) and the wave function for relative motion: = P . (5) | i | i| reli As a result, the two-body problem has become an e↵ective one-body problem. From now on, we are only concerned with rel , so we will drop the “rel” most of the time. 42 | i Scattering theory Incoming plane wave exp(ikz) R = 0 Beam direction +z Outgoing spherical waves exp(ikR)/R Fig. 3.1. A plane wave in the +z direction incident on a spherical target, giving Figure 2: Scatteringrise problem to spherically-outgoing for an incident scattering plane waves wave in the +z direction on a spherical target (in the text r is used instead of R). [From F. Nunes notes.] 3.1.1 Partial wave scattering from a finite spherical potential We start our development of scattering theory by finding the elastic scat- tering from a potential V (R) that is spherically symmetric and so can be written as V (R). Finite potentials will be dealt with first: those for which V (R)=0forR R , where R is the finite range of the potential. This ≥ n n excludes Coulomb potentials, which will be dealt with later. We will examine the solutions at positive energy of the time-independent Schr¨odinger equation with this potential, and show how to find the scattering amplitude f(✓, φ) and hence the di↵erential cross section σ(✓, φ)= f(✓, φ) 2 | | for elastic scattering. We will use a decomposition in partial waves L=0, 1, , and the spherical nature of the potential will mean that each partial ··· wave function can be found separately. The time-independent Schr¨odinger equation for the relative motion with c.m. energy E, from Eq. (2.3.18), is [Tˆ + V E] (R, ✓, φ)=0, (3.1.1) − using polar coordinates (✓, φ) such that z = R cos ✓, x = R sin ✓ cos φ and y = R sin ✓ sin φ. In Eq. (3.1.1), the kinetic energy operator Tˆ uses the reduced mass µ, and is ~2 Tˆ = 2 −2µrR 1 ~2 @ @ Lˆ2 = R2 + , (3.1.2) 2µ −R2 @R @R R2 ✓ ◆ Scattering Theory 1 (last revised: July 8, 2013) M1b–3 Thus we can consider relative motion with total P = 0 (see Fig. 1) and describe elastic scattering 2 from a potential from incoming k to outgoing k0 with k = k0 and E = k /(2µ)(with~ = 1). We | | | | describe this quantum mechanically in terms of an incoming plane wave and an outgoing scattered (“sc”) wave: ik r (+) e · (r)= in(r)+ sc(r)= + sc(r) , (6) E (2⇡)3/2 which (assuming V falls o↵with r faster than 1/r), means that far way from the potential the wave function has the asymptotic form ikr (+) 3/2 ik r e (r) (2⇡)− e · + f(k, ✓,φ) , (7) E r! r !1 ✓ ◆ where the scattering angle ✓ is given by cos ✓ = kˆ kˆ . A schematic picture is shown in Fig. 2 with · 0 k = kzˆ. The scattering amplitude f modulates the outgoing spherical wave according to direction and carries all the physics information. The di↵erential cross section is calculated from dσ number of particles scattering into d⌦perunittime S r2 (k, ✓,φ)= = sc,r , (8) d⌦ number of incident particles per unit area and time Sin where Ssc,r and Sin are the scattered and incoming probability current densities, respectively. Taking k z without loss of generality (and with ~ = 1 and suppressing common (2⇡)3 factors), / 1 d ikz 1 ikz k S =Reb ⇤ =Re e− ike , (9) in in iµ dz in iµ / µ ✓ ◆ ✓ ◆ ikr ikr 1 d e− 1 e 1 k 2 1 S =Re ⇤ =Re f ⇤ ikf + f + , (10) sc,r sc iµ dr sc r iµ r O r3 / µr2 | | O r3 ✓ ◆ ✓ ◆ ✓ ◆ ✓ ◆ so the di↵erential cross section is dσ (k, ✓,φ)= f(k, ✓,φ) 2 . (11) d⌦ | | If we do not consider spin observables (where the spin orientation of at least one of the particles is known—polarized), then f(k, ✓,φ) is independent of φ; we consider only this case for now and drop the φ dependence. We expand the wave function in spherical coordinates as 1 u (r) 1 u (r) (r, ✓)= c l Y (✓,φ)= c˜ l P (cos ✓) , (12) l r l0 l r l Xl=0 Xl=0 where because there is no φ dependence, only the m = 0 spherical harmonic Y (✓,φ)= ✓,φ l, m = l l0 h | l 0 appears and we can use Y = 2l+1 P (cos ✓). The radial function u (r) satisfies the radial i l0 4⇡ l l Schr¨odinger equation (if there is noq mixing of di↵erent l values), d2u l(l + 1) l +2µV k2 u (r)=0. (13) dr2 − r2 − l ✓ ◆ Scattering Theory 1 (last revised: July 8, 2013) M1b–4 We have freedom in choosing the normalization of ul, which we will exploit below. For a central potential V (we’ll come back to non-central potentials tomorrow!), we resolve the scattering amplitude f into decoupled partial waves (the definition of fl sometimes has a extra k factor): 1 f(k, ✓)= (2l + 1)fl(k)Pl(cos ✓) . (14) Xl=0 [We can do a partial wave expansion even if the potential is not central, which means the potential does not commute with the orbital angular momentum (as in the nuclear case!); it merely means that di↵erent l’s will mix to some degree (in the nuclear case, this includes triplet S and D waves). The more important question is how many total l’s do we need to include to ensure convergence of observables of interest.] Using the plane wave expansion of the incoming wave, l+1 ikr ikr ik r ikr cos ✓ 1 l 1 ( 1) e− + e e · = e = (2l + 1)i jl(kr)Pl(cos ✓) (2l + 1)Pl(cos ✓) − , (15) r !!1 2ikr Xl=0 Xl=0 where we have used the asymptotic form of the spherical Bessel function jl: ⇡ l ikr l ikr sin(kr l 2 ) 1 ( i) e i e− jl(kr) − = − − , (16) r !!1 kr kr 2i ikr ikr we can resolve the asymptotic wave function into incoming (e− ) and outgoing (e ) spherical waves (so now e↵ectively a one-dimensional interference problem): l+1 ikr ikr (+) 3/2 1 ( 1) e− + Sl(k)e (r) (2⇡)− (2l + 1)Pl(cos ✓) − , (17) E r !!1 2ikr Xl=0 which identifies the partial wave S-matrix Sl(k)=1+2ik fl(k) .
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