Nuclear Direct Reactions to Continuum 4 How to get Nuclear Structure Information Munetake ICHIMURA (RNC) VII. Response Function 1. Response Function and Polarization Propagator 2. Mean Field Approximation 3. Tamm-Dancoff Approximation 4. Random Phase Approximation 5. Fermi Gas Model 6. Relations to familiar quantities 7. Discussions 1
VIII. Inclusive Breakup Reactions 1. Breakup Processes 2. Formalism 3. Decomposition of elastic and non-elastic breakup 4. Applications 2
VII. Response Function Here I sketch how to calculate the response functions (Recall V. DWBA ) 1. Polarization Propagator 1.1. One-body Density Operator For unified expression, we write ρ F (r) = k F k δ(r r k ) where F k = σ (α) a,k, (α = 0, 1, a = 0, x, y, z) 3
1.2 Polarization Propagator Introduce Polarization propagators Π F F (r, r ; ω) Φ A ρ 1 F(r) ω H A + i δ ρ F (r ) Φ A H A : Internal Hamiltonian of the nucleus A. Response Functions We can write R F F (r, r ; ω) = 1 π Im Π F F (r, r ; ω) 4
2. Mean Field Approximation 2.1 Hamiltonian This is the 0-th order approximation. Approximate H A by Mean Field Hamiltonian, H 0 H A H 0 = k ĥ k T c.m. ĥ k : Single-particle Hamiltonian for the k-th nucleon in A ĥ k = T k + U m.f k U m.f k : Mean field (Hartree-Fock field) 5
Single particle states h : occupied single particle state p : unoccupied single particle state They obey ĥ h = ϵ h h, ĥ p = ϵ p p, p2 p1 h1 h2 h3 6
2.2. Free Polarization Propagator The polarization propagator in the mean field approximation is called Free polarization propagator Π (0) F F (r, r ; ω) = Φ (0) A ρ 1 F(r) ω (H 0 E (0) 0 ) + i δ ρ F (r ) Φ (0) A Φ (0) A : Ground state of A in the mean field approximation H 0 Φ (0) A = E (0) 0 Φ (0) A 7
Note the operation of the density operator ρ F (r ) Φ (0) A = h 1 p h 1 p ρ F (r ) Φ (0) A h,p the sum of 1-particle-1-hole states. p2 p1 h1 h2 h3 8
We can write Π (0) F F (r, r ; ω) = Φ (0) A ρ F(r) h 1 p p,h 1 ω (ϵ p ϵ h ) + i δ Here we introduced ph Green s function h 1 p ρ F (r ) Φ (0) A = Φ (0) A ρ F(r)G ph (ω)ρ F (r ) Φ (0) A G ph (ω) = h 1 1 p h,p ω (ϵ p ϵ h ) + i δ h 1 p How to cope with the infinite sum p unocc? p runs continuously! cf. h run only finite number of states, and thus can be handled. 9
Further manipulation G ph (ω) = h h 1 g(ω + ϵ h ) h 1 with where g(ϵ) = p 1 p unocc ϵ ϵ p + i δ p = 1 p p full ϵ ϵ p + i δ p 1 h ϵ ϵ p + i δ h h = g sp (ϵ) h g sp (ϵ) = = p full 1 h ϵ ϵ h + i δ h 1 p ϵ ϵ p + i δ p 1 ϵ ĥ + i δ 10
The single particle Green s function g sp (ϵ) in r representation g sp (r, r ; ϵ) = r g sp (ϵ) r 1 = r ϵ ĥ + i δ r is known to be calculable. Calculation of g sp (r, r ; ϵ) (Ignore spins) Angular momentum representation g sp (r, r ; ϵ) = lm Y lm (Ω r ) g l(r, r ; ϵ) rr Y lm(ω r ) 11
The radial parts g l (r, r ; ϵ) = 2m N W (f l, h l ) f l(r < ; ϵ)h l (r > ; ϵ) where r < = min(r, r ), r > = max(r, r ), f l (r; ϵ) and h l (r; ϵ) : regular and singular solutions of the equation 1 d 2 2m N dr + 1 l(l + 1) 2 2m N r 2 = ϵ u l (r; ϵ) + U m.f (r) u l (r; ϵ) W (f, h) : Wronskian W (f, h) = f h f h 12
Thus g(r, r ; ϵ) = r g(ϵ) r = g sp (r, r ; ϵ) 1 ϕ h (r) h ϵ ϵ h + i δ ϕ h(r ) is calculable ϕ h (r) : Bound state wave function of the state h Now we can calculate Free Polarization Propagator = h Π (0) F F (r, r ; ω) Φ (0) A ρ F(r) h g(r, r ; ω + ϵ h ) h ρ F (r ) Φ (0) A and get Free Response Function R (0) F F (r, r ; ω) = 1 Im Π(0) F F π (r, r ; ω) 13
[Comment] In actual calculations, various refinements should be taken into account. Spins, Isospins isobar, Complex mean field (representing particle spreading width) Energy-dependent mean field (radial dependent) effective mass m N m (r) Perey factor Spreading widths of holes Orthogonality condition etc. For details, see Manual of the program RESPQ in http://www.nishina.riken.jp/researcher/ archive/program e.html 14
3. Tamm-Dancoff Approximation (Usually abbreviated TDA) Consider the nuclear correlations induced by the ph interaction V ph ph interaction V ph h2 p2 p1 h1 V ph = F F d 3 rd 3 r ρ F (r)w F F (r, r )ρ F (r ) 15
Polarization propagators in TDA Take account of the correlation x x = x x x x + + + x x The polarization propagator with this correlation is given by the solution of the equation Π TDA F F (r, r ) = Π (0) + F F (r, r ) F F d 3 r d 3 r Π (0) F F (r, r ) W F F (r, r )Π TDA F F (r, r ) 16
4. Random Phase Approximation Ring approximation ( Commonly abbreviated as RPA ) More elaborated approximation. Generalize Free polarization propagator Π (0) F F as Π (0) F F (r, r ; ω) = Φ 0 ρ 1 F(r) ω (H 0 E 0 ) + i δ ρ F (r ) Φ 0 + Φ 0 ρ F (r 1 ) ω (H 0 E 0 ) + i δ ρ F(r) Φ 0 17
Polarization propagators in RPA Given by solving the RPA equation Π RPA F F (r, r ) = Π (0) + F F (r, r ) F F d 3 r d 3 r Π (0) F F (r, r ) W F F (r, r )Π RPA F F (r, r ) x x = x x x x x + + + x Once this approximation had been called New Tamm-Dancoff Approximation 18
5. Fermi Gas Model Recall PWBA formula d 2 σ dω dω = K s m A Ṽ (q ) 2 R ρ (ω, q ) R ρ (ω, q) = 1 π Im Φ A ρ 1 (q) ω H A + iη ρ(q) Φ A ρ((p) = A k=1 e ip r k Let us calculate the free response function R ρ (ω, q) in a simple model. Fermi gas model provides the analytic form, from which we can learn some characteristic of the response functions. 19
Fermi gas model Π (0) (q, ω) = Φ (0) A ρ (q) h 1 p p,h 1 ω (ϵ p ϵ h ) + i δ h 1 p ρ((q) Φ (0) A = d 3 p θ(p F p)θ( p q p F ) (2π) 3 ( ) ω (p q) 2 p2 + iδ 2m N 2m N = d 3 p θ(p F p)θ( p q p F ) (2π) 3 ω ( q 2 ) + iδ 2m N q p m N h = p, p = p q 20
The free response function R (0) (q, ω) = 1 π ImΠ(0) (q, ω) Π (0) (q, ω) can analytically be calculated. It is known as the Lindhart function. A.L. Fetter and J.D. Walecka, Quantum Theory of Many-particle Systems, McGraw-Hill, Inc. (1971) 21
[Just for fun] Analytical form of R (0) (q, ω) p F : Fermi momentum ϵ F = p2 F 2m N : Fermi energy Set x = q 2p F, y = ω ϵ F For 0 x 1 R (0) (q, ω) = m Np F (2π) 2 y 4x 1 (x 4x y )2 y for 4x < 1 x for 1 x < y 4x < 1 + x For x > 1 R (0) (q, ω) = m Np F (2π) 2 1 (x 4x y 4x )2 for x 1 < y 4x < 1 + x 22
What spectrum R (0) (q, ω) has? 40 Ca q=1.75 fm -1 g NN =g ND = g DD = 0.6 M. Ichimura et al, PR 39 (1989) 1446 Peak at ω peak = q2 2m N 23
Whereis R (0) (q, ω) finite? 24
What region can we study? A reaction can reach very limited region. 1 90 Zr(p, n), T p = 300 MeV, θ = 0deg, 12 C(p, n), T p = 350 MeV, θ = 22deg M. Ichimura, H. Sakai and T. Wakasa, Prog. Part. Mucl. Phys. 56, 446 (2006) 25
6. Relation to familiar quantities Relation between R(q, ω) and familiar quantities. (1) GT strength B GT ±(ω) = X Φ X k t ± k σ k Φ A 2 δ(ω ω X ) R GT ±(q, ω) = X Thus Φ X k t ± k σ k e iq r i Φ A 2 δ(ω ω X ) B GT ±(ω) = R GT ±(q = 0, ω) (2) Fermi transition strength B F ±(ω) = X R F ±(q, ω) = X Φ X k Φ X k t ± k Φ A 2 δ(ω ω X ) t ± k e iq r i Φ A 2 δ(ω ω X ) B F ±(ω) = R F ±(q = 0, ω) 26
(3) E1 transition strength (> GDR) For the case J A = 0 B E1 (ω) = X 1 Φ X k t z,k r k Φ A 2 δ(ω ω X ) Response Function to the 1 states R IV1 (q, ω) = X 1 Φ X k = Φ X X 1 k δ(ω ω X ) = q 2 Φ X X 1 k Thus t z,k e iq r i Φ A 2 δ(ω ω X ) t z,k ( 1 iq ri + O(q 2 ) ) Φ A 2 t z,k r i Φ A 2 δ(ω ω X ) + O(q 4 ) 1 B E1 (ω) = lim q 0 q 2R IV1 (q, ω) 27
7. Discussion 7.1 Comments on the Fermi gas model (1) Fermi gas model is heuristic, but not necessarily realistic. (2) It may reasonably work for large q region (3) But for small q region, it is useless and even misleading. Spectrum at q = 0, 90 Zr to 90 Nb 28
(4) If you want to have R(q, ω), First calculate R(r, r ; ω), by the methods described in subsec. 2-4. Then take its Fourier transform R(q, ω) = R(q, q; ω) 7.2 Comments on calculation of R(r, r ; ω) (1) Choices of the mean field is crucial. (2) Choice of effective ph interaction is crucial. (3) Calculations are carried out in the angular momentum representation. Namely, calculate RSL,S J L (r, r ) (4) Taking suitable linear combinations of RSL,S J L (r, r ), we can calculate the response functions we want, such as R S, R L, R T, etc. 29
(5) How to include nuclear correlations beyond TDA or RPA in the framework of the present formalism is a longstanding subject There are lots of matters to be discussed about response functions. But they are out of scope in this lecture. For details about comments (3) and (4), see Manual of the program RESPQ in http://www.nishina.riken.jp/researcher/ archive/program e.html 30
7.3 Comments on PWIA (1) Factorized form d 2 σ dω dω = K V i(q) 2 R(q, ω) is very attractive nature to extract nuclear information R i (q, ω) (2) This doesn t hold in DWIA or more elaborate reaction theories. (3) PWIA is heuristic, but not realistic in general. (4) It may work for some cases, if one allows to use normalization factor as d 2 σ dω dω = N eff [ K Vi (q) 2 R(q, ω) ] N eff : Effective nucleon number 31
2 208 Pb(p, n) at 296 MeV. T. Wakasa, Private communication Looks OK, but to extract R i (q, ω) we need to know N eff from other independent data or by theoretical calculation. 32
Taddeucci s Prescription A kind od the N eff method. Applied to GT transitions, etc., very often. Set a semi-empirical ansatz d 2 σ(q, ω) dωdω = ˆσF (q, ω)r(q = 0, ω) ˆσ : unit cross section F (q, ω) : Normalized angular distribution F (q = 0, ω) = 1 e.g. R(q = 0, ω) = R F (ω) or R GT(ω) In N eff method, ˆσ = N eff K(q = 0) V (q = 0) 2 33
Calculate F (q, ω) by DWBA with simple nuclear structure model. * ω dependence is not care for. From observed database of d 2 σ(q, ω), and R(q = 0, ω) dωdω Evaluate ˆσ. Apply the formula to the newly observed data, and obtain R(q = 0, ω). Careful calibration is needed! 34
7.4 For more general cases My opinion is Structure models Response functions Reaction theories Feed back Modify the structure model (parameters in the model, etc.) Experimental Data Refine the reaction theory 35
VIII. Inclusive Breakup Reactions 1. Breakup Processes Consider the inclusive breakup reactions a + A b + anything a = b + x Assume b and x : structureless a b x x b A A X 36
The process is decomposed (1) Elastic breakup (2) Inelastic breakup (3) Transfer reaction (4) Breakup fusion (Incomplete fusion) Elastic breakup a b x b x A A We will consider the decomposition Elastic Breakup + Non-elastic Breakup d 2 σ inc de b dω b = d2 σ EBU de b dω b + d2 σ NEB de b dω b 37
2. Formalsm Hamiltonian H = T b + T x + H A + V xb + V xa + V ba = (T b + U ba ) + (T x + V xa ) + H A + (V xb + V ba U ba ) = (T a + U aa ) + (T bx + V bx ) + H A + (V xa + V ba U aa ) Wave functions H A Φ A = E A Φ A H X Φ X = (T x + V xa + H A )Φ X = E X Φ X (T bx + V bx )ϕ a = ϵ a ϕ a Distorted waves (T a + U aa )χ (+) a = E a χ (+) a (T b + U ba )χ ( ) b = E b χ ( ) b 38
Total energy of the initial state DWBA E i = E A + ϵ a + E a T fi = Φ X χ ( ) b V xb + V ba U ba Φ A ϕ a χ (+) a = Φ X χ ( ) b V post Φ A ϕ a χ (+) a Inclusive cross section d 2 σ inc = K de b dω Φ Xχ ( ) b V post Φ A ϕ a χ (+) b X δ(e i E b E X ) Using the completeness, we get a d 2 σ inc = K Φ A ϕ a χ (+) a V post, χ ( ) b de b dω b δ(e i E b H X ) χ ( ) b V post Φ A ϕ a χ (+) a 2 39
Assuming the excitation of A by V post is very small, we can write χ ( ) b V post Φ A ϕ a χ (+) a = Φ A χ ( ) b Φ A V post Φ A ϕ a χ (+) a Then we get d 2 σ inc de b dω b = K Φ A ϕ a χ (+) a V post, Φ A χ ( ) b Φ A δ(e i E b (T x + H A + V xa )) Φ A χ ( ) b Φ A V post Φ A ϕ a χ (+) a = K Φ A ϕ a χ (+) a V post, Φ A χ ( ) b Φ A δ(ω T x V xa ) Φ A χ ( ) b Φ A V post Φ A ϕ a χ (+) a where is the energy transfer ω = E a + ϵ a E b 40
Introducing the Green s function of x 1 G x (ω) = Φ A ω (T x + V xa ) + iδ Φ A 1 = ω T x U x + iδ with Optical potential of x on A U x = V x + iw x All excitations of A are included through U. Inclusive breakup cross section where d 2 σ inc de b dω b = K π Im d 3 r x d 3 r x S (r x)g x (r x, r x )S(r x ) S(r x ) = r x χ ( ) b Φ A V post Φ A ϕ a χ (+) a G x (r x, r x ; ω) = r x G x (ω) r x 41
[Comment] About the relation 1 Φ A ω (T x + V xa ) + iδ Φ A = Note Set 1 ω T x U x + iδ 1 Φ A ω (T x + V xa ) + iδ Φ A 1 Φ A ω (T x + V xa ) + iδ Φ A P = Φ A Φ A, Q = 1 P, ω + = ω + iδ By short manupulation = = 1 P ω + (T x + V xa ) P P ω + 1 T x P V xa P P V xa Q 1 ω + T x U x ω + T x QV xa Q QV xap [Excersize] When AB = 1, express P BP by P AP, P AQ, QAP, QAQ 42
3. Decomposition of elastic and non-elastic breakup An identity of the Green s function ImG x = (1 + G xu x)im [ G (0) ] x (1 + Ux G x ) + G xw x G x where Use G (0) x = 1 ω T x + iδ we get ImG (0) x = k k δ(ω k2 2m x ) k = k (1 + G xu x)im [ G (0) ] (1 + Ux G x ) χ ( ) k δ x ω k2 2m x χ ( ) k A. Kasano and M. Ichimura, PL 115B, 81(1982) 43
Now the first term gives Elastic Breakup Cross Section d 2 σ EBU de b dω b = K k χ ( ) k χ ( ) b Φ A V post Φ A ϕ a χ (+) a 2 δ(ω k2 2m x ) Consequently the seond term gives Non-elastic Breakup Cross Section where d 2 σ NEB de b dω b = K π ψ x W x ψ x ψ x (r) = G x χ ( ) b Φ A V post Φ A ϕ a χ (+) a = G x (r, r ; ω)s(r )d 3 r This formalism is called IAV model M. Ichimura, N. Austern and C.M. Vincent, Phys. Rev. C32, 431(1985) 44
4. Applications Jin Lei and A.M. Moro, PR C92, 044616(2015) d 2 σ/de p dω p (mb/sr MeV) d 2 σ/de p dω p (mb/sr MeV) 1000 100 10 1 0.1 0.01 0 20 40 60 80 θ lab (deg) 100 (b) θ 80 p =8 ο 60 40 20 58 Ni(d,pX) @ Ed =100 MeV EBU (CDCC) NEB (FR-DWBA) Total D. Ridikas et al. EBU (CDCC) NEB (FR-DWBA) PE+CN Total (a) E p =50 MeV 0 0 20 40 60 80 100 E p (MeV) FIG. 4. (Color online) Double differential cross section of protons emitted in the 58 Ni(d,pX) reaction at E d = 100 MeV in the laboratory frame. (a) Proton angular distribution for a fixed proton energy of E p = 50 MeV. (b) Energy distribution for protons emitted at a laboratory angle of 8 (arrow in top figure). The meaning of the lines is the same as in Fig. 3, and are also indicated by the labels. Experimental data are from Ref. [44]. 45
209 Bi ( 6 Li, αx) 46