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PHYS 6610: Graduate Nuclear and Particle Physics I PHYS 6610: Graduate Nuclear and Particle Physics I H. W. Grießhammer INS Institute for Nuclear Studies The George Washington University Institute for Nuclear Studies Spring 2021 II. Phenomena 1. Shapes and Masses of Nuclei Or: Nuclear Phenomeology References: [PRSZR 5.4, 2.3, 3.1/3; HG 6.3/4, (14.5), 16.1; cursorily PRSZR 18, 19] PHYS 6610: Graduate Nuclear and Particle Physics I, Spring 2021 H. W. Grießhammer, INS, George Washington University II.1.0 (a) Getting Experimental Information 0 heavy, spinless, composite target =) M E ≈ E me ! 0 !2 0 2 ds Za 2 q E 2 = cos F(~q ) (I.7.3) dW 2 q 2 E lab 2E sin 2 | {z } 1 Mott: spin- 2 on spin-0, me = 0, MA 6= 0 when nuclear recoil negligible: M E ≈ E0 E0 1 A recoil: = 2 0 2 2 2 E E q = (k − k ) ! −~q = −2E (1 − cosq) 1 + M (1 − cosq) A translate q2 () q () E max. mom. transfer: q ! 180◦ min. mom. transfer: q ! 0◦ For E = 800 MeV (MAMI-B), 12C(A = 12): p 1 q −q2 = j~qj Dx = j~qj 180◦ 1500 MeV0 :15 fm 90◦ 1000 MeV0 :2 fm 30◦ 400 MeV0 :5 fm ◦ [PRSZR] 10 130 MeV1 :5 fm PHYS 6610: Graduate Nuclear and Particle Physics I, Spring 2021 H. W. Grießhammer, INS, George Washington University II.1.1 “Typical” Example: 40;48Ca measured over 12 orders 12 orders of magnitude. q -dependence, multiplied by 10, 0:1!! [PRSZR] −1 -dependence [HG 6.3] 40Ca has less slope =) smaller size j~qj fm PHYS 6610: Graduate Nuclear and Particle Physics I, Spring 2021 H. W. Grießhammer, INS, George Washington University II.1.2 Characterising (Spherically Symmetric) Charge Densities 1 4p Z r F(~q2) := dr sin(qr) r(r); normalisation F(~q2 = 0) = 1 Ze q 0 In principle: Fourier transformation =) need F(~q2 ! 1): impossible =) Ways out: (i) Characterise just object size: sinqr 1 z }| { 4p Z r (qr)3 F(~q2 ! 0) = dr qr − + O((qr)5) r(r) Ze q 3! 0 1 1 4p Z q2 4p Z F(~q2 ! 0)= dr r2 r(r) − dr r2 × r2 r(r)+O((qr)5) Ze 3! Ze 0 0 | {z } | {z } = 1 mean of r2 operator 2 2 dF(~q ) (square of) root-mean-square (rms) radius =)h r i = −3! 2 d~q ~q2=0 PHYS 6610: Graduate Nuclear and Particle Physics I, Spring 2021 H. W. Grießhammer, INS, George Washington University II.1.3 Ways Out: (ii) Assume Charge Density and Compare FF To It Better parameterise as sum of a few Gauß’ians: 2 (r − Ri) rcharge(r) = ∑bi exp− 2 i d =) “line thickness” = parametrisation uncertainty [Mar] [PRSZR] A r (r = 0) decreases with A, but matter density r = r (r = 0) constant for large A: charge N Z charge 0:17 nucleons 11 kg tons 3 space shuttles 1 rN ≈ = 3 × 10 = 300 = ≈ fm3 b litre mm3 mm3 (1:4 × (2 × 0:7fm))3 rN plateaus in heavy nuclei =) Saturation of Interaction: attraction long-range, repulsion up-close!?! Nucleons “separate but close”: Distance between nucleons in nucleus ≈ 1:4× nucleon rms diameter. PHYS 6610: Graduate Nuclear and Particle Physics I, Spring 2021 H. W. Grießhammer, INS, George Washington University II.1.4 Matter Densities in Nuclei (ii) Assume a certain rN(r), calculate its FF, fit. [Tho 7.5] 1st min at qR ≈ 4:5 Very simple form for heavy nuclei: Fermi Distribution r0 A rN(r) = r−c = rcharge 1 + exp a Z experiments: radius at half-density c ≈ A1=3 ×1:07 fm: r 5 translate hard-sphere: R = hr2i ≈ A1=3 × 1:21 fm 3 experiments: a ≈ 0:5 fm: largely independent of A!; [PRSZR p. 68] related to surface/skin thickness t = a × 4ln3 ≈ 2:40 fm (drop from 90% to 10%) PHYS 6610: Graduate Nuclear and Particle Physics I, Spring 2021 H. W. Grießhammer, INS, George Washington University II.1.5 [HG 14.5; [nucl-th/0608036]; (b) Spin & Deformation Example: Deuteron[nucl-th/0102049]] So far: no spin, no deformation =) F(~q2) angle-independent is the exception, not the rule. Example Deuteron d(np) with JPC = 1+−: ~L =~J −~S, S = 1 L = 1J ⊗ 1S = 0 ⊕1 ⊕ 2 =) L = 0 (s-wave) or L = 2 d-wave – Parity forbids L = 1. angular momentum =) mag. moment =) spin-spin interaction/spin transfer( q ! 180◦, helicity) e 1 Charge FF: G (q2) = hm jJ0jm i avg. of hadron density, G (0) = 1 C 3 ∑ J J C mJ=−1 2 Md Magnetic FF: GM(q ) GM(0) =: kd, mag. moment kd = 0:857 MN 2 Quadrupole FF: GQ(q ) Z 3 2 2 2 quadrupole moment Qd := GQ(0) = Ze d r (3z − r ) rcharge(~r) = 0:286 fm 2 3 2 ! 0 2 4 ds Za 2 q E 6 2 2t 2 8t Md 2 4t(1 + t) 2 2 q 7 = cos × 6 GC + GM + GQ + GM tan 7 dW 2E sin2 q 2 E 6 3 9 3 2 7 lab 2 4 | {z } 5 | {z } helicity =) spin transfer Mott q2 with as before t = − 2 4Md =) Dis-entangle by q & t dependence. PHYS 6610: Graduate Nuclear and Particle Physics I, Spring 2021 H. W. Grießhammer, INS, George Washington University II.1.6 Deuteron Form Factors [Garçon/van Orden: [nucl-th/0102049]] does not include most recent data (JLab), but pedagogical plot 2 2 high-accuracy data up to Q & (1:4 GeV) Theory quite well understood: embed nucleon FFs into deuteron, add: deuteron bound by short-distance + tensor force: one-pion exchange Ny~s ·~qp(q)N =) photon couples to charged meson-exchanges and many p± p± p± more!! −1 open issues: zero of GQ, form for Q & 3 fm ,. PHYS 6610: Graduate Nuclear and Particle Physics I, Spring 2021 H. W. Grießhammer, INS, George Washington University II.1.7 (c) Nuclear Binding Energies (per Nucleon) 100 Mean Field Models Ò×ØÝ Shell Model(s) Effective 10 Microscopic Interactions Ab Initio Ì´ µ χ̸ π/ 3 4 Àe QCD Àe d Ô 3 1 À Proton Number Proton QCD Vacuum QCD Quark-Gluon Vacuum Ò Interaction 1 5 10 50 100 Neutron Number B=A rapidly increases from deuteron (A = 2): 1:1 MeV=A to about 12C: 7:5 MeV=A for A & 16 (oxygen) remains around 7:5:::8:5 MeV=A maximal for 56Fe–60Co–62Ni: 8:5 MeV=A small decrease to A ≈ 250 (U): 7:5 MeV=A =) “typically”, fusion gains (much) energy up to Fe; fission gains (some) energy after Fe. =) Fe has relatively large abundance: product of both fusion and fission. PHYS 6610: Graduate Nuclear and Particle Physics I, Spring 2021 H. W. Grießhammer, INS, George Washington University II.1.8 Bethe-Weizsäcker Mass Formula & Interpretation: Liquid Drop Model Know < 3000 nuclei =) roughly parametrise ground-state binding energies, not only for stable nuclei 2 2 Total binding energy: SEMF 2=3 Z (N − Z) d B = aV A − as A − aC − aa − (1935/36) Semi-Empirical Mass Formula A1=3 4A A1=2 −3 aV = 15:67 MeV volume term cf. rN ≈ 0:17 fm =) saturation =) Well-separated, quasi-free nucleons, next-neighbour interactions like in liquid. as = 17:23 MeV surface tension fewer neighbours on surface =) less B aC = 0:714 MeV Coulomb repulsion of protons =) tilt to N > Z aa = 93:15 MeV (a)symmetry/Pauli term Pauli principle =) tilt to N ∼ Z 8 < −11:2 MeV Z & N even opposite spins have net attraction d = 0 Z or N odd pairing term wf overlap decreases with A : +11:2 MeV Z & N odd =) A-dependence PHYS 6610: Graduate Nuclear and Particle Physics I, Spring 2021 H. W. Grießhammer, INS, George Washington University II.1.9 Valley of Stability around N & Z 2 2 2=3 Z (A − 2Z) d B = aV A − as A − aC − aa − A1=3 4A A1=2 aaA A =) Parabola in Z at fixed A with Z = min 2=3 . 2(aa + aCA ) 2 [HG 16.2/3] PHYS 6610: Graduate Nuclear and Particle Physics I, Spring 2021 H. W. Grießhammer, INS, George Washington University II.1.10 Valley of Stability Probe nuclear interactions A = 56 by pushing to ”drip-lines”: Facility for Rare Isotope Beams FRIB at MSU A = 150 PHYS 6610: Graduate Nuclear and Particle Physics I, Spring 2021 H. W. Grießhammer, INS, George Washington University II.1.11 (d) Application: Nuclear Fission [PRSZR 3.3] For A > 60, fission can release energy, but must overcome fission barrier. Let’s assume fission into 2 equal fragments. Estimate: infinitesimal deformation into ellipsoid (egg) with excentricity e at constant volume =) surface tension %, Coulomb &: E(sphere) − E(ellipsoid) e2 ∼ 2a A2=3 − a Z2A−1=3 5 s C Fission barrier classically overcome when ≤ 0: 2 Z 2as ∼ ≈ 48 e.g. Z > 114, A > 270 A a C Z p below: QM tunnel prob. / exp−2 2M(E − V) between points with r(E = V) Induced Fission: importance of pairing energy 238 239 n + 92 U ! 92 U: even-even ! even-odd =) invest pairing E = d=11p:2MeV = 0:7MeV d 239 235 236 n + 92 U ! 92 U: even-odd ! even-even d =) gain pairing energy p = −0:7MeV =) can use thermal neutrons (higher s!) 236 PHYS 6610: Graduate Nuclear and Particle Physics I, Spring 2021 H. W. Grießhammer, INS, George Washington University II.1.12 (e) First Dash into Nuclear Matter 0:17 nucleons Nuclear Interactions Saturate: rN ≈ ! const. in heavy nuclei fm3 Nucleons “separate but close”: Distance between nucleons in nucleus ≈ 1:4× nucleon rms diameter.
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