Nuclear Structure Ingredients for reaction models Lecture 1 - - PowerPoint PPT Presentation
Nuclear Structure Ingredients for reaction models Lecture 1 Nuclear ingredients for reaction models Models available Masses and their importance Masses of nuclei Experimental masses Mass models Liquid-drop models
An atomic nucleus composed of A nucleons (Z protons+N neutrons) is denoted by (Z,A) or ASym where Sym is the chemical symbol of the element (H,He,Li,Be,B,C,O,F,Ne,Na,Mg,Al,Si,….)
N Z
t1/2<10m 10m <t1/2<30d t1/2>30d stable unknown
Some specific features:
20 40 60 80 100 20 40 60 80 100 120 140 160 180 200
Fissioning nuclei a-unstable nuclei Nuclei with experimentally known masses Neutron Star matter
Stability and decay modes of existing nuclei
There are 82 stable elements, 285 stable nuclei (with a half-life larger than the age of the universe ~ 1010yr) The other nuclei (~8000) 0≤Z≤110 are unstable against either the weak interaction (b–,b+ decay or electron capture), or the strong interaction (a-emission or fission). Away from the neutron or proton drip lines, the nuclei become unstable against n- or p-emissions, respectively
a-unstable nuclei Proton emitters Spontaneous fission
EC/b+-unstable nuclei b--unstable nuclei Nuclei produced in the laboratory
5
Ground-state properties
(Masses, b2, matter densities, spl, pairing…)
Nuclear Level Densities
(E-, J-, p-dep., collective enh., …)
Fission properties
(barriers, paths, mass, yields, …)
Optical potential
(n-, p-, a-potential, def-dep)
g-ray strength function
(E1, M1, def-dep, T-dep, PC)
b-decay
(GT, FF, def-dep., PC)
STRONG ELECROMAGNETIC WEAK
Ground-state properties
(Masses, b2, matter densities, spl, pairing…)
Nuclear Level Densities
(E-, J-, p-dep., collective enh., …)
Fission properties
(barriers, paths, mass, yields, …)
Optical potential
(n-, p-, a-potential, def-dep)
g-ray strength function
(E1, M1, def-dep, T-dep, PC)
b-decay
(GT, FF, def-dep., PC)
STRONG ELECTROMAGNETIC WEAK
Coordinated by the IAEA Nuclear Data Section
MASSES - (ftp)
LEVELS - (ftp)
RESONANCES - (ftp) OPTICAL - (ftp)
DENSITIES - (ftp)
GAMMA - (ftp)
FISSION - (ftp)
Ground-state properties
densities Fission parameters
Nuclear Level Densities (formulas, tables and codes)
Optical Model Potentials (533) from neutron to 4He
Average Neutron Resonance Parameters
g-strength function (E1)
Discrete Level Scheme including J, p, g-transition and branching
GLOBAL MICROSCOPIC DESCRIPTIONS
ACCURACY (reproduce exp.data)
Concern of applied physics Concern of fundamental physics
RELIABILITY (Sound physics)
Phenomenological models
(Parametrized formulas, Empirical Fits)
Classical models
(e.g Liquid drop, Droplet)
Semi-classical models
(e.g Thomas - Fermi)
mic-mac models
(e.g Classical with micro corrections)
semi-microscopic
(e.g microscopic models with phenomenological corrections)
fully microscopic
(e.g mean field, shell model, QRPA)
PHENOMENOLOGICAL DESCRIPTIONS New concern of some applications Different possible approaches depending on the nuclear applications
EB = aV A − aSA2/3 − aC Z2 A1/3 − aA (N − Z)2 A + ∆(Z, N)
Ground-state properties
(Masses, b2, matter densities, spl, pairing…)
Nuclear Level Densities
(E-, J-, p-dep., collective enh., …)
Fission properties
(barriers, paths, mass, yields, …)
Optical potential
(n-, p-, a-potential, def-dep)
g-ray strength function
(E1, M1, def-dep, T-dep, PC)
b-decay
(GT, FF, def-dep., PC)
STRONG ELECROMAGNETIC WEAK
Strong nuclear force Electrostatic repulsion
generated by an effective n-n interaction ! Still phenomenological, but at the level of the effective n-n interaction Obviously more complex, but models have now reached stability and accuracy !
Ground-state properties
(Masses, b2, matter densities, spl, pairing…)
Nuclear Level Densities
(E-, J-, p-dep., collective enh., …)
Fission properties
(barriers, paths, mass, yields, …)
Optical potential
(n-, p-, a-potential, def-dep)
g-ray strength function
(E1, M1, def-dep, T-dep, PC)
b-decay
(GT, FF, def-dep., PC)
STRONG ELECROMAGNETIC WEAK
Nuclear masses, or equivalently binding energies, enter all chapters of applied nuclear physics. Their knowledge is indispensable in order to evaluate the rate and the energetics of any nuclear transformation. The nuclear mass of a nucleus (Z,A=Z+N) is defined as Mnucc 2 = N Mnc 2 + Z M pc 2 − B The atomic mass includes in addition the mass and binding of the Z electrons Matc 2 = Mnucc 2 + Z Mec 2 − Be where Mn is the neutron mass, Mp the proton mass and B the nuclear binding energy (B>0) where Me is the electron mass, and Be the atomic binding energy of all the electrons
Mp= 938.272 MeV/c2 Mn= 939.565 MeV/c2 ΔmZA = Mat − Amu
( )c 2 = Mat(amu) − A [ ]muc 2
where mu is the atomic mass unit (amu) defined as 1/12 of the atomic mass of the neutral 12C atom The number of nucleons (A=Z+N) is also conserved by a nuclear reaction. For this reason, the atomic mass Mat is usually replaced by the mass excess Dm defined by The mass excess is generally expressed in MeV through ΔmZA = 931.494 Mat(amu) − A
[ ] MeV
mu=1.66 1027 kg = 931.494 MeV/c2 To determine the atomic mass, the nuclear binding energy must be estimated from the nuclear force.
Z,N-1
Z,N Z-2,N-2 Z-1,N Z N
= Mat(Z,N) - Mat(Z-1,N+1) – 2Me = Mat(Z,N) - Mat(Z-1,N+1) = Mat(Z,N) - Mat(Z+1,N-1)
b decay: p
n conversion within a nucleus via the weak interaction Modes (for a proton/neutron in a nucleus):
On earth, only these 3 modes can occur. In particular, electron capture (EC) involves orbital electrons. Q-values for decay of nucleus (Z,N):
Note: QEC = Qb+ + 2Mec2 = Qb+ + 1.022 MeV
p n + e+ + ne e- + p n + ne n p + e- + ne
Favourable for n-deficient nuclei Favourable for n-rich nuclei
Qb+/c2 = Mnuc(Z,N) - Mnuc(Z-1,N+1) - Me QEC/c2 = Mnuc(Z,N) - Mnuc(Z-1,N+1) + Me Qb-/c2 = Mnuc(Z,N) - Mnuc(Z+1,N-1) - Me
b-unstable nuclei
a-unstable nuclei Proton emitters Spontaneous fission
EC/b+-unstable nuclei b--unstable nuclei Nuclei produced in the laboratory
In AME 2012 (wrt 2003): 225 new masses with 96 new p-rich and 129 new n-rich
About 2498 nuclear masses available experimentally (2016). Nuclear (astrophysics) applications require the knowledge of about 8000 0 ≤ Z ≤ 110 masses
In AME 2003 (wrt 1995): 289 new masses with 242 new p-rich and 47 new n-rich
(AME: Atomic Mass Evaluation)
Neutron drip line Sn(Z,A)= M(Z,A-1)+Mn- M(Z,A) < 0
In AME 2016 (wrt 2012): 60 new masses with 25 new p-rich and 35 new n-rich
In AME 2016
a smooth mass surface in the vicinity of known masses
recommended
Smooth trend in experimental nuclear masses away from shell closures, shape transitions and Wigner cusps along the N=Z line; in particular in the systematics of S2n, S2p, Qa S2n(Z,N)= M(Z,N-2)+2Mn- M(Z,N)
But the mass of the additional ~ 6000 masses needed for applications à to be determined from theory
recommended
B/A [MeV] The nuclear force is not known from first principles, but deduced from
Short range: strongly attractive component on a short range Repulsive core: repulsive component at very short distances (<0.5 fm) average separation between nucleons leading to a saturation of the nuclear force Charge symmetric: the nuclear force is isospin independent The binding energy per nucleon is a smooth curve, almost A-independent for A>12: B/A ~ 8 – 8.5 MeV/nucleon This implies that the interaction between nucleons is
Volume term: B/A ~ cst à roughly constant density of nucleons inside the nucleus with a relatively sharp surface à radius of the nucleus R ~ A1/3 Characteristic of the nuclear force (one nucleon in the nucleus interacts with
The saturation has its origin in the short-range nuclear force and the combined effect of the Pauli and uncertainty principles: The total binding energy is a subtle difference effect between the total kinetic energy and the total potential energy. The potential and kinetic energies of the nucleon almost cancel out totally leading to a shallow minimum at around 2.4 fm Nucleons do not interact with all the other nucleons, but approximately
neutron proton V r R V r R
Coulomb Barrier Vc
R e Z Z Vc
2 2 1
=
… …
Nucleons in a box: Discrete energy levels in nucleus
R ~ 1.3 x A1/3 fm
à Nucleons are bound by attractive force. Therefore, the mass of the nucleus is smaller than the total mass of the nucleons by the binding energy Dm=B/c2 Nuclei: nucleons attract each other via the strong force (range ~ 1 fm) à a bunch of nucleons bound together create a mean potential for an additional:
Liquid drop model (Myers & Swiateki 1966)
Droplet model (Hilf et al. 1976)
FRDM model (Moller et al. 1995, 2012)
KUTY model (Koura et al. 2000)
Weizsäcker-Skyrme model (Wang et al. 2011)
Shell model (Duflo & Zuker 1995)
ETFSI model (Aboussir et al. 1995)
Hartree-Fock-BCS model (2000)
Hartree-Fock-Bogolyubov model (2010)
Relativistic Hartree-Bogolyubov
2 4 20 40 60 80 100 120 140 160
ΔM [MeV] N M(Exp)–M(HFB-14)
Building blocks for the prediction of ingredients of relevance in the determination of nuclear reaction cross sections, b-decay rates, … such as
Nuclear mass models provide all basic nuclear ingredients: Mass excess (Q-values), deformation, GS spin and parity but also single-particle levels, pairing strength, density distributions, … in the GS as well as non-equilibrium (e.g fission path) configuration as well as for the nuclear/neutron matter Equation of State (NEUTRON STARS) The criteria to qualify a mass model should NOT be restricted to the rms deviation wrt to exp. masses, but also include
10 20 30 0.04 0.08 0.12 0.16 0.2
Baldo et al. (2004) Friedman & Pandharipande (1981)
energy/nucleon [MeV] density [fm
Challenge for modern mass models: to reproduce as many observables as possible
n(b) > m* p(b)
ST > –(2l+1)
100 200 300 400 500 0.2 0.4 0.6 0.8 1
Friedman & Phandaripande (1981) Wiringa et al. (1998) Akmal et al. (1998) Li & Schulze (2008)
energy/nucleon [MeV] density [fm
Neutron matter Symmetric matter
EB = aV A − aSA2/3 − aC Z2 A1/3 − aA (N − Z)2 A + ∆(Z, N)
The semi-empirical liquid drop mass model: (Bethe-Weizsäcker Formula, 1935): The nucleus is described as a collection of neutrons and protons forming a liquid drop of an incompressible fluid
A a A Z B
V
= ) , (
−asA2/3 −acoul Z 2 A1/ 3
−asym (N − Z)2 A
Volume Term: each nucleon gets bound by about the same energy Surface Term: ~ surface area (surface nucleons are less bound) Coulomb term: Coulomb repulsion leads to a reduction of the binding: uniformly charged sphere has E=3/5 Q2/R Asymmetry term: Pauli principle applied to nucleons: symmetric filling
protons neutrons neutrons protons lower total energy
Pairing correlation effect due to the attractive character of the nucleon force: each orbit can be occupied by 2 nucleons Pairing term: +D~12/A1/2[MeV] even number of like-nucleons are favoured (e=even, o=odd referring to Z, N respectively)
+D ee 0 oe/eo –D
In summary, the binding energy can be written as
B(Z,A) = aV A − aSA2/ 3 − acoulZ 2A−1/ 3 − asym N − Z A # $ % & ' (
2
A + δ
Or equivalently, the internal energy per nucleon e=–B/A
e(Z,A) = −aV + aSA−1/ 3 + acoulZ 2A−4 / 3 + asym N − Z A # $ % & ' (
2
−δ /A ⇒ e = e0 + f Z − Z0
( )
2
mass parabola B/A
[MeV]
A Binding energy per nucleon Experimental data versus liquid drop
A fit to experimental masses lead to aV ~ 15.85MeV; aS ~ 18.34 MeV; acoul ~ 0.71 MeV; asym ~ 92.86 MeV
aV ~ 15.7MeV; aS ~ 17.2 MeV; acoul ~ 0.70 MeV; asym ~ 23.3 MeV
Binding energy per nucleon along an isobar due to asymmetry term in mass formula
Mass parabola along an isobar:
decay decay decay decay
2D
As an example
125Te: only 1 stable isobar
Mass Parabola 3 stable isobars
124Sn 124Te 124Xe
Z Z
N-number of neutrons
Z=82 (Lead) Z=50 (Tin) Z=28 (Nickel) Z=20 (Calcium) Z=8 (Oxygen) Z=4 (Helium)
Magic numbers Valley of b-stability (location of stable nuclei) N=Z isobar
5 10 20 40 60 80 100 120 140 160
N
Some missiong energy : dW=Eexp – ELD à Shell correction energy
For nuclei with
B(Z,A) = aV A − aSA2/ 3 − acoulZ 2A−1/ 3 − asym N − Z A # $ % & ' (
2
A + δ −δW
Shell model: single-particle energy levels are not equally spaced
Magic numbers
shell gaps more bound than average. less bound than average need to add shell correction term dW(Z,N)
The shell effect
5 10 20 40 60 80 100 120 140 160
N
Shell correction energy: dW=Eexp - ELD
For nuclei with
But it remains difficult to predict reliably and accurately shell correction energies on the basis of simple analytical formula (e.g Myers & Swiatecki 1966) for experimentally unknown nuclei. Need more microscopic approaches like mean field theories, shell model, … to put the extrapolation on a safe footing. In particular, it is not clear if the N=28, 50, 82, 126 magic numbers remain in the neutron-rich region
B(Z,A) = aV A − aSA2/ 3 − acoulZ 2A−1/ 3 − asym N − Z A # $ % & ' (
2
A + δ −δW
Skyrme EDF
Liquid drop Deformation corr. Shell corr. +… Other corr. I=(N-Z)/A
Improve the accuracy by ~10% - 40% Revised masses
Ning Wang, Min Liu, PRC 84, 051303(R) (2011); leave-one-out cross-validation
Liu, Wang, Deng, Wu, PRC 84, 014333 (2011) keV # 2149 1988 46 2149
~ 7500 nuclei with 8 ≤ Z ≤ 124
M(Hilf et al.) - M(von Groote et al.) 20 ≤ Z ≤ 100
Experimentally known Exotic nuclei
Uncertainties in the prediction of masses far away from the experimentally known region
Two identical “droplet models” but with two different parametrizations Hilf et al. (1976) versus von Groote et al. (1976)
rms deviation on exp masses ~ 670 keV (1976) – 950 keV (2003) – 1020 keV (2012) – 1060 keV (2016)
But extrapolation to n-rich nuclei far away from the experimentally know region remains uncertain
10 8 6 4 2
50 60 70 80 90 100 110 120 130 140
Known Masses
Strong nuclear force Electrostatic repulsion
generated by an effective n-n interaction ! Still phenomenological, but at the level of the effective n-n interaction Obviously more complex, but models have now reached stability and accuracy !
written as a sum of the single-particle kinetic energies (T) and two-body interactions (potential energy)
form can be chosen for the potential V (e.g Skyrme or Gogny interaction)
The many body Schrödinger equation Hy=Ey is difficult to solve. To simplify the resolution of the Schrödinger equation, the mean-field approximation is used: each nucleon moves independently of other nucleons in a central potential U representing the interaction of a nucleon with all the other nucleons
Lawrence Livermore National Laboratory
LLNL-PRES-570332
54
§
Starting points:
for the nucleus (to be determined)
§
Goal: find the Slater determinant,
§
Method: Minimize the energy defined as the expectation value of the Hamiltonian
principle)
§
Resulting equations are non-linear: V depends on the (one-body) density r which depends on the fi(r) which depend
§
Produces magic numbers, reasonable values for binding energies, radii, etc. The self-consistent loop: the mean-field is constructed from the effective interaction, instead of being parameterized
vij = t0(1 + x0Pσ)δ(r r rij) + 1 2t1(1 + x1Pσ) 1 ~2 ⇥ p2
ij δ(r
r rij) + δ(r r rij) p2
ij
⇤ + t2(1 + x2Pσ) 1 ~2p p pij.δ(r r rij)p p pij + 1 6t3(1 + x3Pσ) n(r r r)α δ(r r rij) + i ~2 W0(σi + σj) · p p pij × δ(r r rij)p p pij ,
Dobaczewski et al., 2004
Skyrme and pairing interactions adjusted on all available masses à rms ~ 500-700 keV
srms (2353 AME’12)
M(exp)-M(HFB)
s(2353M)=500keV
s(HFB27) s(HFB24) s(FRDM) 2353 M (AME 2012): 512 keV 549 keV 654 keV 2353 M (AME 2012): model error 500 keV 542 keV 648 keV 257 M from AME’12 with Sn<5MeV: 645 keV 702 keV 857 keV 128 M (28≤Z≤46, n-rich) at JYFLTRAP (2012): 508 keV 546 keV 698 keV
2.5 3 3.5 4 4.5 5 5.5 6 6.5 2.5 3 3.5 4 4.5 5 5.5 6 6.5
R
exp [fm]
R
th [fm]
Charge radii for 782 nuclei rms deviation = 0.027fm
Charge distribution of 208Pb
2 4 6 8
Exp BSk20
0.02 0.04 0.06 0.08 0.1
r [fm]
208Pb
ρch [fm -3]
BSk21
0.1 1 10 1 10
exp [b]
288 experimental data with Q > 0.1b
Ms
* / M = 0.80
Mv
* / M ~ 0.70
&
Ms
* > Mv *
Gandolfi et al. (2012) Akmal et al. (1998) Li & Schulze (2008) Danielewicz et al. (2002) Lynch et al. (2008)
n [fm-3] n / n0
(J=32 MeV) (J=31 MeV) (J=30 MeV) (J=29 MeV) (J=30 MeV)
Energy per nucleon in neutron matter Pressure in symmetric nuclear matter From model-dependent HIC
Experimentally known
(0.51 < sexp <0.79 MeV)
M(Hilf et al.) – M(von Groote et al.) M(HFB-2) – M(HFB-24) 20 ≤ Z ≤ 100
0.5 0.55 0.6 0.65 0.7 0.75 0.8
1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 35 41
HFB model σ
rms [MeV]
D1M 95 12 FRDM
Girod, Berger, Libert, Delaroche
2 4 20 40 60 80 100 120 140 160
ΔM [MeV] N M(Exp)-M(D1M) M(exp)-M(D1M)
srms=0.50 MeV M(exp)-M(HFB27) srms=0.797 MeV
707 Radii: srms=0.031 fm (with Quadrupole corrections)
rms deviation = 0.031fm Including the quadrupole correction:
All Q-values in reaction codes must be estimated within the same model !
2 4 6 8
Exp BSk20
0.02 0.04 0.06 0.08 0.1
r [fm]
208Pb
ρch [fm -3]
BSk21
Experimentally known
Experimentally known
Experimentally known
HFB-24: Skyrme HFB mass model s(2408 exp masses)=551keV HFB-D1M: Gogny HFB mass model s(2408 exp masses)=797keV FRDM: Finite Range Droplet mass model s(2408 exp masses)=592keV M(D1M)-M(HFB-24) M(FRDM)-M(HFB-24)
20 40 60 80 100 50 100 150 200 250
But still major local differences impacting the determination of Q-values
1. To include the state-of-the-art theoretical framework
à GCM
l-r symmetry, …)
3. To consider different frameworks
2. To reproduce as many “observables” as possible (“exp.” & “realistic”)
(cf Lectures of Stephane Hilaire)