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 dσ Zα 2 θ E 2 = cos F(~q ) (I.7.3) dΩ 2 θ 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 − cosθ) 1 + M (1 − cosθ) A translate q2 ⇐⇒ θ ⇐⇒ E max. mom. transfer: θ → 180◦ min. mom. transfer: θ → 0◦

For E = 800 MeV (MAMI-B), 12C(A = 12): p 1 θ −q2 = |~q| ∆x = |~q| 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. . .

θ -dependence, multiplied by 10, 0.1!! [PRSZR] −1 -dependence [HG 6.3] 40Ca has less slope =⇒ smaller size |~q| 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

∞ 4π Z r F(~q2) := dr sin(qr) ρ(r), normalisation F(~q2 = 0) = 1 Ze q 0

In principle: Fourier transformation =⇒ need F(~q2 → ∞): impossible

=⇒ Ways out:

(i) Characterise just object size: sinqr ∞ z }| { 4π Z r (qr)3 F(~q2 → 0) = dr qr − + O((qr)5) ρ(r) Ze q 3! 0 ∞ ∞ 4π Z q2 4π Z F(~q2 → 0)= dr r2 ρ(r) − dr r2 × r2 ρ(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) ρcharge(r) = ∑bi exp− 2 i δ =⇒ “line thickness” = parametrisation uncertainty

[Mar]

[PRSZR] A ρ (r = 0) decreases with A, but matter density ρ = ρ (r = 0) constant for large A: charge N Z charge 0.17 nucleons 11 kg tons 3 space shuttles 1 ρN ≈ = 3 × 10 = 300 = ≈ fm3 b litre mm3 mm3 (1.4 × (2 × 0.7fm))3

ρN 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 ρN(r), calculate its FF, ﬁt.

[Tho 7.5] 1st min at qR ≈ 4.5

Very simple form for heavy nuclei: Fermi Distribution

ρ0 A ρN(r) = r−c = ρcharge 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( θ → 180◦, helicity) e 1 Charge FF: G (q2) = hm |J0|m i avg. of hadron density, G (0) = 1 C 3 ∑ J J C mJ=−1 2 Md Magnetic FF: GM(q ) GM(0) =: κd, mag. moment κd = 0.857 MN 2 Quadrupole FF: GQ(q ) Z 3 2 2 2 quadrupole moment Qd := GQ(0) = Ze d r (3z − r ) ρcharge(~r) = 0.286 fm   2 ! 0 2 4 dσ Zα 2 θ E  2 2τ 2 8τ Md 2 4τ(1 + τ) 2 2 θ  = cos ×  GC + GM + GQ + GM tan  dΩ 2E sin2 θ 2 E  3 9 3 2  lab 2  | {z }  | {z } helicity =⇒ spin transfer Mott q2 with as before τ = − 2 4Md =⇒ Dis-entangle by θ & τ 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 N†~σ ·~qπ(q)N

=⇒ photon couples to charged meson-exchanges

and many π± π± π± 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

Shell Model(s) Effective 10 Microscopic Interactions

Ab Initio

χ 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) δ 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. ρN ≈ 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   −11.2 MeV Z & N even opposite spins have net attraction δ = 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) δ 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 ε at constant volume =⇒ surface tension %, Coulomb &: E(sphere) − E(ellipsoid) ε2 ∼ 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 = δ=11√.2MeV = 0.7MeV δ 239 235 236 n + 92 U → 92 U: even-odd → even-even δ =⇒ gain pairing energy √ = −0.7MeV =⇒ can use thermal neutrons (higher σ!) 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: ρN ≈ → const. in heavy nuclei fm3 Nucleons “separate but close”: Distance between nucleons in nucleus ≈ 1.4× nucleon rms diameter. Fermi distribution at temperature T = 0 for N = Z: occupy all levels, 2 spins, proton & neutron, N = Z Z 3 d k 2 3 ρN = ρp + ρn = 2 [np(~k) + nn(~k)] = k (2π)3 3π2 F

|~k|≤kF r

2 3 3π ρN =⇒ Fermi momentum (max. nucleon momentum) k = ≈ 1.3fm−1 ≈ 260MeV ≈ 2m .

F 2 π

Liquid-gas transition for temperature T %

Liquid-Drop Model: heat =⇒ evaporation compare to water T via E-distrib. of collision fragments:

√

N(E) ∝ E exp[−E/T] (Maxwell)

need many fragments, angle-indep. exp: T ≈ 5MeV in ﬁnite symmetric nuclei. Neutron-E-distrib. in 235U ﬁssion: T = 1.29MeV excitation energy E/A [MeV] per nucleon [NuPECC LRP] PHYS 6610: Graduate Nuclear and Particle Physics I, Spring 2021 H. W. Grießhammer, INS, George Washington University II.1.13 Nuclei Are Not “Nuclear Matter”

2 2 B as Z (N − Z) δ SEMF of ﬁnite nuclei: = aV − − aC − aa − ≈ 8.5MeV A A1/3 A4/3 4A2 A3/2 B Inﬁnite nuclear matter: no surface, Coulomb negligible, no pairing =⇒ ≈ a = 15.6MeV. A V 1 grand canonical ensemble: Z = tr exp− [H − µ N − µ N ] with µ : chem. potentials T p p n n N −T lnZ = −PV = E − TS − µpNp − µnNn

=⇒ pressure −P = E − Ts − µpρp − µnρn with E,s: energy, entropy densities

Need to extrapolate or solve nuclear many-body problem: specify interactions!

−3 Descriptions agree at ρ0 ≈ 0.16fm — here χEFT (π, N, ∆(1232)) [Fiorilla/. . . [arXiv:1111.3688 [nucl-th]]] 4 symmetric 3 nuclear matter N=Z Empirical ﬁrst-order liquid-gas phase transition ] 3 − 2 for inﬁnite, symmetric (N = Z) nuclear matter at T=25MeV 20 critical temperature Tc = [16...18]MeV.

[MeV fm 1 15.1

P 10

0 Chemical potential at temperature T = 0: T=0 B -1 µN(T = 0) = MN − A = [939 − 16]MeV ≈ 923MeV 00.050.10.150.2 3 ρ [fm− ]

PHYS 6610: Graduate Nuclear and Particle Physics I, Spring 2021 H. W. Grießhammer, INS, George Washington University II.1.14 A First Phase Diagram of Nuclear Matter: N = Z

Tc ≈ 15MeV

Tc mπ ,MN =⇒ symmetric nuclear matter close to liquid-gas transition: just inside liquid phase

density ρ(T, µN) of stable nuclear matter depends on T, µ.

PHYS 6610: Graduate Nuclear and Particle Physics I, Spring 2021 H. W. Grießhammer, INS, George Washington University II.1.15 Nature is Not Symmetric: Dependence on Proton-Neutron Mix

Again relatively good agreement between descriptions – here χEFT [Fiorilla/. . . [arXiv:1111.3688 [nucl-th]]].

15 Critical line 14 Z 10 =0 A Z/A = 0.5 0.1 12 5 10 0.4 0 0.2 8 [MeV] 0.3 T [MeV] ¯ E -5 6 0.2 0.3 -10 4 0.4 0.1 2 -15 T =0 0.5 0.053 0 00.050.10.150.2 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 3 -3 ρ [fm− ] ρ [fm ]

=⇒ Nuclear-matter density ρ(N − Z) decreases as Z/A decreases.

– Nuclear Matter unbound for Z/A . 0.1. – Why is pure-neutron matter unbound (a gas)?? (Pauli-principle??)

– In early 1900’s, neutron “invented” to mitigate Coulomb repulsion between protons.

So why no binding when I take all protons away?

– LatticeQCD: Neutron matter might actually be bound for larger mπ > 600 MeV (controversial).

PHYS 6610: Graduate Nuclear and Particle Physics I, Spring 2021 H. W. Grießhammer, INS, George Washington University II.1.16 So Why Are There Neutron Stars??

−3 Gravity compacts interior =⇒ saturation point shifts: ρ0 ≈ 0.16fm → 3ρ0, holds neutrons together.

How to extrapolate to there – and how to extrapolate from Z ≈ 0.4A to Z . 0.1A (“neutron” star!)?

N − Z N − Z N − Z ha ( ) d(a /4) i N − Z 2 a ρ0 a Taylor in : E(ρ, ) = E0(ρ0, = 0) + + + ... A A A 4 dρ ρ0 A

Nuclei (SEMF): “(a)symmetry energy” aa(ρ0)/4 ≈ 22 MeV; nucl. matter: [29...33] MeV d(a /4) a slope L = 3 = [40...62] MeV. Method: compare different Z/A nuclei & extrapolate. dlnρ ρ0 2 N − Z N − Z d E 2 Taylor in ( − ): E( , ) = E ( , = 0) + ( − ) + ... ρ ρ0 ρ 0 ρ0 2 ρ ρ0 A A dρ ρ0 2 ρ = ρ0 + K(ρ0)(ρ − ρ0) + ... justiﬁed for ρ(neutron star) = 3ρ0?? d2E Compressibility of nuclear matter K(ρ) = 9ρ > 0 for stable nuclear matter at density ρ. dρ2 Test dependence on (ρ,N − Z) in neutron skin of heavy nuclei, collective excitations & extrapolate!

2 At ρ , N = Z: compressibility K = k2 (ρ ) d E = [210 ± 10]MeV. Wide agreement. 0 F 0 dρ2 ρ0

At ρ0, pure neutron matter: K ≈ 600MeV, error ±100MeV or more. People disagree! Number here from [Vretenar/. . . PRC68 (2003) 024310]

PHYS 6610: Graduate Nuclear and Particle Physics I, Spring 2021 H. W. Grießhammer, INS, George Washington University II.1.17 Preview: Nuclear Matter Phase Diagram for N = Z

When you compress nucleons, additional energy can be converted into new particles: baryons (Λ(1440),. . . ) mesons (kaon,. . . ), resonances/excitations (∆(1232,. . . ), exotics. . . =⇒ Inﬂuence on neutron-star radius,. . . ! =⇒ Later.

PHYS 6610: Graduate Nuclear and Particle Physics I, Spring 2021 H. W. Grießhammer, INS, George Washington University II.1.18 Preview: Nuclear Matter Phase Diagram for N 6= Z

Need third axis with chemical potential µI = µp − µn for Z − N to place neutron stars.

[NuPECC Long-Range Plan 2017 p. 89]

PHYS 6610: Graduate Nuclear and Particle Physics I, Spring 2021 H. W. Grießhammer, INS, George Washington University II.1.19 (f) Inelasticities: Excitations, Breakup, Knockout

SEMF does not explain nuclear level spectrum.

elastic peak

[PRSZR]

PHYS 6610: Graduate Nuclear and Particle Physics I, Spring 2021 H. W. Grießhammer, INS, George Washington University II.1.20 (g) Beyond the SEMF/Liquid Drop cursory look at [PRSZR 18, 19]

Difference Semi-Empirical Mass Formula SEMF – Experiment

Bethe-Weizsäcker: Semi-Empirical Mass Formula, good for qualitative arguments.

Magic numbers 2,8,20,28,50,82,126 for Z or N more stable than SEMF =⇒ Shell-like structure?

PHYS 6610: Graduate Nuclear and Particle Physics I, Spring 2021 H. W. Grießhammer, INS, George Washington University II.1.21 Example of Single-Particle Models: 3 Minutes on the Shell Model

Single-Particle Models: Individual nucleon moves in average potential created by all other nucleons. =⇒ Neglect feedback of motion onto potential. Saturation, short-range forces =⇒ V(r) ∝ ρ(r) −V0 Light Nuclei: Gaußian proﬁle; Heavy nuclei: Fermi/Woods-Saxon potential V(r) = r−c 1 + exp a Full QM: Solve Schrödinger Equation Analytically solvable models provide insight:

– Fermi Gas/Liquid Model: 3-dim. potential square-well with depth V0. 3 – 3-dim Harm. Oscillator E = (N + N + N + )h¯ω; ang. mom. l = N − 2(=# wf nodes − 1) h.o. x y z 2

Reﬁnement Spin-Orbit Coupling Vls(r)~l ·~s: ﬁne structure.

Experiment: Vls ≈− 20MeV < 0 huge (heavy & close constituents), sign opposite to atom. p n Reﬁnement Coulomb: proton sees charges, neutron not =⇒ V0 > V0 .

[PRSZR 18.1]

And, of course, many more reﬁnements. . .

PHYS 6610: Graduate Nuclear and Particle Physics I, Spring 2021 H. W. Grießhammer, INS, George Washington University II.1.22 Example of Single-Particle Models: 3 Minutes on the Shell Model

Each state with 2 protons & 2 neutrons (spin!); pairing =⇒ closed shells do not contribute.

=⇒ Spin-orbit indeed produces gaps at magic numbers 2,8,20,28,50,82,126. [Goeppert Mayer/Wigner/Jensen 1949 + developments: “Periodic table” of nuclei]

[PRSZR 18.7] PHYS 6610: Graduate Nuclear and Particle Physics I, Spring 2021 H. W. Grießhammer, INS, George Washington University II.1.23 Liquid Drop Is Example of A Collective Model

Collective Model: Nucleons loose individuality, form continuous ﬂuid/gas.

Example Collective Vibrations/Shape Oscillations: shape of nucleus deformed.

Example Compressibility of Nuclear Matter: “monopole mode” JPC = 0+−: radial oscillations.

Experiment: excitation energy ≈ 80A−1/3MeV any other mode

=⇒ Nuclear matter pretty incompressible. (but for interior of Neutron stars!)

Example Giant Electromagnetic Dipole Resonance: p & n oscillate against each other. =⇒ Coherent elmag. excitation ∝ Z2; Huge.

[PRSZR 19.4]

PHYS 6610: Graduate Nuclear and Particle Physics I, Spring 2021 H. W. Grießhammer, INS, George Washington University II.1.24 Example Collective Rotations

Non-spherical nucleus rotates around non-symmetry axis, inertia I:

~J2 J(J + 1) E = = “rotation bands” rot 2I 2I [PRSZR 18.14] =⇒ characteristic spacing ∆E ∝ (2J + 1).

Experiment: Inertia I < rigid ellipsoid, but > irrotational ﬂow (superﬂuid)

=⇒ Nucleus like raw egg.

PHYS 6610: Graduate Nuclear and Particle Physics I, Spring 2021 H. W. Grießhammer, INS, George Washington University II.1.25 Next: 2. Hadron Form Factors & Radii

Familiarise yourself with: [HM 8.2 (th); HG 6.5/6; Tho 7.5; Ann. Rev. Nucl. Part. Sci. 54 (2004) 217]

PHYS 6610: Graduate Nuclear and Particle Physics I, Spring 2021 H. W. Grießhammer, INS, George Washington University II.1.26