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1 N=Z E α =400 MeV α

2 2 α + 2 C 0 + (RCNP) 2 C 6 O 24 Mg 28 Si 40 Ca E α = 400 MeV α folding model DWBA

3 3 Contents Chapter Introduction 5 Chapter 2 Experimental setup 9 2. Beam line Spectrometer Matching between beamline and spctrometer Focal plane detector system Target Data Acqisition System Chapter 3 Data Reduction 7 3. Particle identification Track reconstruction Energy calibrations Background substraction Differntial crosssection Efficiencies Chapter 4 Data Analysis DWBA Folding model density distributions elastic scattering Transition density Results of calculations Ratio Transition potential Uncertainty from α- nucleon interaction

4 4 Contents 4.0 Transition density by microscopic model Comparison between DWBA and CC conclusion Chapter 5 summary 45 Bibliography 57

5 5 Chapter Introduction α α α 4 α (RCNP) α α α γ Z = N B(IS, λ) B(Eλ) B(IS, λ) = 4B(Eλ)/e 2 L = 2 α 2 C Ex = 7.65 MeV (Hoyle ;J π = 0 + ) α α 2008 D.T.Khoa E α = MeV α+ 2 C. 2 C Holye [] resonating group method(rgm)

6 6 Chapter Introduction M3Y interaction folding model RGM 22.8 % 6.9 % Coupled Channel.2 Ex = 4.44 MeV(J π = 2 + ) Hoyle α Khoa Hoyle. Khoa α+ 2 C DWBA CC Ex = 7.65MeV RGM []

7 7.2 Ex = 4.44MeV DWBA CC [] α RCNP Grand Raiden α 2 C, 6 O, 24 Mg, 28 Si, 40 Ca RCNP 58 Ni, 90 Zr, 208 Pb α DWBA Single folding model 2 C RGM DWBA Coupled Channel

9 9 Chapter 2 Experimental setup (RCNP) C 24 Mg 28 Si SiO 40 Ca θ lab = RCNP 2. Beam line RCNP K40 AVF (Azimuthally Varying Field) K400 4 He ++ AVF 86 MeV 400 MeV Grand Raiden 3 5 na 94 kev 2.2 Spectrometer δp/p = Grand Raiden 2.2 Grand Raiden 2. Grand Raiden Grand Raiden 3 (D) 2 (Q) (SX) (MP) Q-SX-Q2-D-MP-D2-DSR D,D2 ( ) MP

10 0 Chapter 2 Experimental setup LAS WS Grand Raiden BLP-2 BLP- Ring Cyclotron N-BLP 0 50m superconducting solenoid magnets N AVF Cyclotron 2. RCNP

11 2.2 Spectrometer 2.2 Grand Raiden Mean orbit radius 3 m Total deflection angle 62 Angular range Focalcplane length 20 cm Tilting angle of focal line 45.0 Maximam magnetic field strength 8 kg Magnification-Vertical 5.98 Magnification-holizontal Momentum dispersion 5.45 m Momentum range 5 % Momentum resolution Acceptance angle - vertical ±70 mr Acceptance angle - horizontal ±20 mr Solid angle 5.6 msr 2. Specifications of the Grand Raiden Spectrometer

12 2 Chapter 2 Experimental setup 2.3 Matching between beamline and spctrometer WS 2.3. ion optics x 0 θ 0 δ 0 x = (x, θ, δ) 3 3 B = (b µν ), T = (t µν ), S = (s µν ), (µ, ν =, 2, 6) x = Bx 0 x fp = STBx 0 δ δ 0 δ = δ 0 B b 6 = b 62 = 0 b 66 = s 6 = s 62 = 0 s 66 = (x x 2 ) x 2 = T x, T = cos α ϕ T cos ϕ T (2.) θ 2 = θ + β α θ + Θ (2.2)

13 2.3 Matching between beamline and spctrometer 3 δ 2 = KΘ + Cδ (2.3) [2] β α + θ 2 θ K (/p out )( p out / α) α = 0 0 C (p in /p out )( p out / p in ) x fp = x 0 (s b T + s 2 b 2 ) + θ 0 (s b 2 T + s 2 b 22 ) + δ 0 (sb 6 T + s 2 b 26 + s 6 C) + Θ(s 2 + s 6 K) (2.4) θ fp = x 0 (s 2 b T + s 22 b 2 ) + θ 0 (s 2 b 2 T + s 22 b 22 ) + δ 0 (s 2 b 6 T + s 22 b 26 + s 26 C) + Θ(s 22 + s 26 K) (2.5) δ fp = δ 2 = KΘ + Cδ 0 (2.6) B = (b µν ), T = (t µν ), S = (s µν )(µ, ν = 3, 4, 6) first order matching (2.4) θ 0 δ δ 0 K = 0 b 2 = 0 (2.4) θ 0 focus matching (2.4),(2.5) δ 0 b 6 = s 6 s ( + s s 26 K s 2 s 6 K) C T lat disper (2.7) b 26 = (s 2 s 6 s s 26 )C (2.8) lateral dispersion matching angular dispersion matching 2.4 (b 6 = b 26 = 0) δ 0 lateral

14 4 Chapter 2 Experimental setup + P 0 P Focal plane Magnetic Spectrometer Target Achromatic Transport P 0 + P P 0 + P Momentum Ang./Mom. Dispersion Dispersion Matched Matched 2.4 lateral dispersion matchng mode lateral and angulardispersion matchng mode dispersion matching lateral + angular dispersion matchng Grand Raiden s µν (??) (2.8) lateral and angular dispersion matching WS 2.2 [2] Matrix elements Design values b = (x x) b 33 = (y y) 0.89 b 6 = (x δ) 37. m b 26 = (θ δ) rad b 2 = (x θ) 0 b 34 = (y φ) 0 Total bending angle 270 Total length m 2.2 WS

15 2.4 Focal plane detector system Focal plane detector system Grand Raiden 2 VDC (Vertical Drift Chamber) central ray 46.5 VDC X,U (X ) 48.9 (U ) wire wire X 6 mm U 4 mm VDC 3 single particle event TDC VDC 2 X U 2 3 VDC 2 2 coincidence VDC Target 2 C 24 Mg 28 Si SiO 2 40 Ca ZnS

16 6 Chapter 2 Experimental setup 2 C 2.2mg/cm 2 24 Mg 2.5mg/cm 2 28 Si 2.6mg/cm 2 40 Ca.63mg/cm 2 SiO 2 2.2mg/cm Data Acqisition System RCNP data aquisition system (DAQ) 2.6 VDC LeCroy3377 FERA/FERET ECL VME high speed memory module (HSM) flow controlling evebt module (FCET) HSM CAMAC function dead time 30 µs/event HSM VME CPU Gigabit Ethernet 2.6 DAQ

17 7 Chapter 3 Data Reduction 3. Particle identification RF (PMT) I x I 0 l ( I(x) = I 0 exp x ) l (3.) PMT x L x R Ī ( Ī = I 0 exp = I 0 exp x L l ) ( x L + x R 2l ( I 0 exp x r l ) = I 0 exp ) ( L ) 2l (3.2) L = x L + x R Ī Bethe-Broch Ī 3. Ī α RF Time of Flight 3.2 RF 3.2 Track reconstruction VDC

18 8 Chapter 3 Data Reduction 3. α 3.2 RF 3.3 VDC X d i d i d i+ α 3.4 VDC mm (FWHM) 8.7 kev 3.3 Energy calibrations E x E x = (E E 3 + M) 2 p 2 4 M (3.3)

19 3.3 Energy calibrations 9 Scttered Particle Cathod Plane d i Potential Wire Sense Wire 2mm d i d i 0mm Anode Wires Cathod Plane 3.3 VDC counts/channel close to cathode the foil around wires (TDC channel) TDC (X) arbitrary unit conversion (mm) drift length 3.4

20 20 Chapter 3 Data Reduction E E 3 E i = p 2 i + m2 α, (i =, 3) p 4 M VDC C 3.4 Background substraction α α 3.5 true

21 3.5 Differntial crosssection C 3.5 Differntial crosssection dσ dω (θ av) = Y (θ av ) N tgt N beam Ω(θ av ) ϵ (3.4) θ av Y (θ av ) N tgt N beam ϵ Efficiencies DAQ 2 VDC. (N total ) 3 2. VDC (N hit ) 4

22 22 Chapter 3 Data Reduction 3. X i ϵ Xi = N hit(x i X j U i U j ) N total (X j U i U j ) (3.5) 4. VDC 4 ϵ V DC = ϵ X ϵ X2 ϵ U ϵ U2 (3.6) VDC 75 % DAQ requested trigger accepted trigger 95%

23 3.6 Efficiencies C Ex = 4.44 MeV Ex = 7.65 MeV Ex = 9.64 MeV O Ex = 6.3 MeV Ex = 6.92 MeV Ex =.52 MeV Ex = 2.05 MeV

24 24 Chapter 3 Data Reduction Mg Ex =.36 MeV Ex = 6.43 MeV Mg Ex =.78 MeV Ex = 4.97 MeV Ca Ex = 3.74 MeV Ex = 3.90 MeV

25 25 Chapter 4 Data Analysis DWBA ECIS95?? 4. DWBA α a + A β b + B ˆV α a A V α U α ˆV α = V α U α (4.) a b A B a A A U α (DWBA) DW BA Tβα =< χ ( ) β (r β)ψ β (ξ β ) ˆV α χ (+) α (r α )ψ α (ξ α ) > (4.2) ψ χ (K α + U α )χ (+) α = E α χ (+) α, (K β + Uβ)χ ( ) β = E β χ ( ) β (4.3) E c K c = h 2 k 2 c/2µ c (c = α, β) χ ± c ) χ ± c (r) (e ikc r + f (2π) 2/3 c (Ω) e±ik cr r χ (±) c (4.4) DWBA ˆV α V α U α DWBA

26 26 Chapter 4 Data Analysis 4.2 Folding model DWBA single folding model Folding model U(r) = ρ A (r )ρ a (r 2 )v(r 2 )dr dr 2 (4.5) ρ A (r ) ρ a (r 2 ) r 2 = r + r 2 r double folding δ δ(r 2 ) U(r) = ρ A (r )v( r r )dr (4.6) α single folding α V fi (r) V fi (r) = v( r r ) = l,m ψ f (r )v( r r )ψ i (r )dr (4.7) v l (r, r )Y lm(r)y lm (r ) (4.8) ρ fi (r) ρ fi (r) = = l,m ψ f (r )δ(r r )ψ i (r )dr (4.9) ρ l fi(r, r )Y lm(r)y lm (r ) (4.0) V fi (r) = ρ l fi(r )v( r r )dr (4.) 4.3 density distributions 4.6 ρ(r) ρ ch (r) unfold

27 4.4 elastic scattering 27 ρ p ρ n ρ proton ch ρ neutron ch ρ ch (r) = ρ p (r )ρ proton ch (r r )dr + ρ n (r )ρ neutron ch (r r )dr (4.2) Foulier F ch (q) = G p ch (q) F p(q) + G n ch(q) F n (q) (4.3) F ch (q) F p (q) F n (q) G p ch (q) Gn ch (q) Foulier G p ch (q) Gn ch (q) e + d 3 He + 4 He [5] G p ch (q) = a 0.24τ τ τ τ 3 (4.4) G n ch(q) =.70τ τ ( τ) 2 (4.5) τ = q 2 /4M 2 nucleon G n ch (q) Gp ch (q) Gn ch (q) = F p (q) = F ch(q) G ch (q) (4.6) form factor form factor 4. ρ n (r) = N Z ρ p(r) (4.7) 4.4 elastic scattering single folding model -α v( r r, ρ 0 (r )) = V ( + βρ 0 (r ) 2/3 ) exp( r r 2 /α) iw ( + βρ 0 (r ) 2/3 ) exp( r r 2 /α) (4.8) ρ 0 β β = -.9 (β = -.9) (β = 0)2 V W α E α = 400 MeV α

28 28 Chapter 4 Data Analysis ρ(r) (fm -3 ) r (fm) 4. 2 C ( ) ( ) RCNP 2 C 6 O 24 Mg E α 400 MeV α 28 Si 40 Ca 58 Ni 90 Zr 6 Sn 44 Sm 208 Pb global V (A, Z) = a 0 + a ZA /3 (4.9) W (A) = b 0 + b A /3 (4.20) α(a) = c 0 + c A /3 (4.2) E α = 400 MeV 2 C DWBA

29 4.4 elastic scattering V (MeV) ZA -/ W (MeV) A /3 6 5 a (fm 2 ) A /3 4.3 α-n global interaction β =.9 β = 0

30 30 Chapter 4 Data Analysis Target β = -.9 β = 0 V(MeV) W(MeV) α(fm 2 ) V(MeV) W(MeV) α(fm 2 ) 2 C O Mg Ni Zr Sn Sm Pb Si Ca α-n Si Ca Transition density ( L = 0 O (0) = 2 r 2 i Y 00 ) [6] ( ρ L=0 (r, E x ) = β L=0 (E x ) 3 + r d ) ρ 0 (r) (4.22) dr ρ L= (r, E x ) = β (E x ) R 3 [3r 2 ddr + 0r 53 < r2 > ddr (r + ϵ d2 dr d )] ρ 0 (r)(4.23) dr ρ L 2 (r, E x ) = β L 2(E x )R 2L + ( r R ) L d dr ρ 0(r) (4.24) 4.24 Tassie β E x 00% β 2 0(E x ) = β 2 (E x ) = β 2 L 2(E x ) = 2π h 2 ma < r 2 > E x (4.25) 6π h2 R 2 /( < r 4 > 25 mae x 3 < r2 > 2 0ϵ < r 2 >) (4.26) 2πLR 2L 4 ma < r 2L 2, > E x (4.27)

31 4.6 Results of calculations 3 m, A, < r N >, R N half density radius ϵ = (4/E 2 + 5/E 0 ) h 2 /3mA(E 0, E 2 GMR GQR ) M(IS, λ) B(IS, λ) M(IS, λ) = ρl (r) 4π rk dr (4.28) B(IS, λ) = M(IS, λ) 2 (4.29) k λ = 0 k = 4 λ 0 k = λ + 2 (4.2) (4.7) ρ L 2 B(IS, λ) 4.6 Results of calculations M(Eλ) B(Eλ) γ N = Z B(IS, λ) = 4B(Eλ) e 2 (4.30) β L DWBA DWBA χ 2 2 C (Ex = 7.65 MeV) 6 O (Ex = 2.05 MeV)

32 32 Chapter 4 Data Analysis α+ 2 C (β = -.9) (β = 0)

33 4.6 Results of calculations O (Ex = 2.05 MeV) (β = 0) (θ cm 4.2 ) (θ cm 4.2 ) target state(e x ) J π B(Eλ) (e 2 fm k ) B(ISλ) (fm k ) β = -.9 β = 0 C 4.44 MeV 2 + 4± ± ± MeV ± ± 8.5 ± MeV 3 60 ± ± ± 8 O 6.29 MeV 3 (.55 ±.2) 0 3 (4.2 ± 0.9) 0 3 (6.2 ± 2.) MeV ± 4 95 ± 4 5± 4.52 MeV ± ± ± MeV ± ± ± 37 Mg.368 MeV ± 2 (.69 ± 0.4) 0 3 (2.00 ± 0.54) MeV ± 57 ± 9 80 ± 72 Si.779 MeV ± 2 07 ± 8 (.22 ± 0.2) MeV ± 84 ± ± 5 Ca MeV 3 (2.04 ± 0.7) 0 4 (5.5 ± 0.46) 0 4 (4.77 ± 0.47) MeV ± ± ± B(Eλ) B(IS, λ) B(Eλ) B(Eλ) k = 2 3 k = 3

34 34 Chapter 4 Data Analysis 4.7 Ratio 4.30 B(IS, λ) ref B(IS, λ) exp R = B(IS, λ) exp B(IS, λ) ref (4.3) RCNP 58 Ni Zr Pb 3 Z = N (4.30) R 2 + R 3 C Ca 0 + J π target state(e x ) R (β =.9) R (β = 0) 0 + C 7.65 MeV 0.40 ± ± ± 0.34 ± O 2.05 MeV 0.8 ± 0.0 ± ± 0.06 ± Mg MeV 0.39 ± ± ± 0.29 ± Si MeV ± 0.07 ± ± ± C 4.44 MeV ± 0.0 ± ± 0.26 ± O 6.97 MeV ± ± ± ± O.52 MeV ± ± ± ± 0.08 Mg.368 MeV ± ± ± ± 0.33 Si.779 MeV ± ± ± ± Ca MeV ± 0.40 ± ± 0.6 ± 0.22 Ni.45 MeV ± ± ± 0.03 ± C 9.64 MeV ± ± ± ± O 6.29 MeV ± 0.05 ± ± ± Ca MeV ± ± ± ± Ni 4.47 MeV.040 ± 0.22 ± ± 0.6 ± Zr 2.75 MeV ± ± ± ± Pb 2.6 MeV.46 ± 0.07 ± ± ± R 2

35 4.7 Ratio R Z 4.6 λ = 0 R Z 2.5 R Z 4.7 λ = 2 R 6 O 2 + Ex = 6.97 MeV Z = 8 (Ex =.52 MeV) Z =

36 36 Chapter 4 Data Analysis 2.5 R Z 4.8 λ = 3 R 4.8 Transition potential C Uncertainty from α- nucleon interaction 28 Si 40 Ca global interaction 2 folding model 24 Mg 58 Ni DWBA Ni Mg DWBA Si β =.9 β = 0 0%

37 4.0 Transition density by microscopic model Transition density by microscopic model C THSR Kamimura 3α-RGM [7] 2 3 RGM 4. 2 C DWBA % Comparison between DWBA and CC DWBA (coupled channel;cc) CC 4.. coupled channels method Ψ CC = ψ α (ξ α )χ α (r α ) + γ ψ γ (ξ γ )χ γ (r γ ) + ψ β (ξ β )χ β (r β ) (4.32) Ψ CC c = α, β, γ < ψ c E H Ψ CC >= 0, (4.33) (E c K c U c )χ c (r c ) = < ψ c (ξ c ) (H E) ψ c (ξ c )χ c (r c ) > (4.34) c c

38 38 Chapter 4 Data Analysis (C c ) χ c (r c ) χ (+) α (k α, r α )δ cα + eik cr c f cα (Ω cα r c (4.35) e κ cr c 2µ c ( E c ) χ c (r c ) C c, κ c = r c h 2 (4.36) 4..2 result of α + 2 C CC < ψ j (ξ) ψ i (ξ) > , 0+ 2, 3, 2+ 2 (Ex = 0.4 MeV), 4+ (Ex = 4.08MeV) CC Kamimura DWBA CC β = -.9 Khoa CC DWBA Khoa C CC conclusion folding model DWBA 0 + folding model 2 C THSR DWBA CC

39 4.2 conclusion 39 α

40 40 Chapter 4 Data Analysis transition potential (MeV) r (fm) transition potential (MeV) transition potential (MeV) r (fm) r (fm) ( ) ( )

41 4.2 conclusion Si (Ex=4.97 MeV) Mg Ni DWBA β =.9 β = 0 Ni Mg

42 42 Chapter 4 Data Analysis 6 5 r 2 ρ(r) (fm - ) r (fm) r 2 ρ(r) (fm - ) r (fm) r 2 ρ(r) (fm - ) r (fm) RGM ( ) ( ) DWBA

43 4.2 conclusion CC α+ 2 C (β = -.9) DWBA CC β = 0 DWBA CC

44 44 Chapter 4 Data Analysis CC ( ) β =.9 β = 0 DWBA

45 45 Chapter 5 summary 2 C, 6 O, 24 Mg, 28 Si, 40 Ca E α = 400 MeV α folding model DWBA C 40 Ca 3 consistent 0 + DWBA 2 C RGM DWBA CC RGM α Khoa double folding model J = 0

47 47 Acknoledgements RCNP

49 49 Appendix table of cross sections 5. α + 2 C θ (deg) Ex = 4.4 MeV 7.65 MeV 9.64 MeV ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± 0.22

50 50 Appendix 5.2 α + 6 O θ (deg) Ex = 6.2 MeV 6.9 MeV.52 MeV 2.05 MeV ± ± ± ± ± ± ±0.25.8± ± ± ± ± ± ± ±0.32 -± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ±0.2.02±0.08

51 5 5.3 α + 24 Mg θ c.m (deg) Ex =.37 MeV 6.43 MeV ± ± ± ± ±..62 ± ±.28.4 ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± 0.002

52 52 Appendix 5.4 α + 24 Si θ c.m (deg) Ex =.78 MeV 4.98 MeV ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± α + 28 Ca θ c.m (deg) Ex = 3.73 MeV 3.90 MeV ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± 0.25

53 53 Angular distributions of DWBA calculation α+ 2 C Ex = 4.44, 7.65, 9.64 MeV 4.4

54 α+ 6 O Ex = 6.3, 6.9,.52, 2.05 MeV 4.4

55 α+ 24 Mg Ex =.37, 6.43 MeV α+ 28 Si Ex =.78, 4.98 MeV 4.4

56 α+ 40 Ca Ex = 3.73, 3.90 MeV 4.4

57 57 Bibliography [] D.T.Khoa, Phys. Lett. B 660 (2008) 33 [2] Y. Fujita et al., Nucl. Inst. and Meth. B 26 (997) 274. [3] T. Wakasa et al., Nucl. Inst. and Meth. A 482 (2002) 79. [4] J. Raynal. computer code,ecis95, NEA [5] J.J. Kelly Phys. Rev. C 70 (2004) [6] M.N. Harakeh and A.van der Wounde. Giant Resonances. Clarendon Press, Oxford,200. [7] M. Kamimura, Nucl. Phys. A 35 (98) 456. [8] P.M. ENDT, At. Data Nucl. Data Tables, 23 (979) 3. [9] S. Raman et al., At. Data Nucl. Data Tables, 36 (989). [0] R.H. Spear, AT. Data Nucl. Data Tables, 42 (989) 55. [] G.R. Satchler and D.T. Khoa, Phys. Rev. C 55 (997) 285. [2] H. Sakaguchi et al., Phys. Rev. C 57 (998) 749. [3] T.N. Buti et al,. Phys. Rev. C 33 (986) 755. [4] R.Neu et al,. Phys.Rev. C 39 (989) 245.

α

α 2013 2 4 α α D. T. Choa DWBA coupled channel α 12 C 2 + 1 4.44 MeV Hoyle state 0 + 2 7.65 MeV Hoyle state DWBA coupled channel α Hoyle state 3α channel Hoyle state α Hoyle state RCNP Grand Raiden 12