P461 - Nuclei I 1
Properties of Nuclei• Z protons and N neutrons held together with a
short-ranged force gives binding energy
• P and n made from quarks. Most of the mass due to the strong interactions binding them together. Recent JLAB results show masses inside nucleus might be slightly smaller than free particles
• P and n are about 1 Fermi in size and the strong force doesn’t compress. Size ~ range of strong force all nuclei have the same density and higher A nuclei are bigger (unlike atoms)
m M eV c m M eV c
M Zm Nm E Am
w ith A Z N m u M eV c
p n
nucleus p n b ind N
N
9 3 8 3 9 3 9 6
9 3 1 5
2 2
2
. / . /
. /
P461 - Nuclei I 2
Protons vs Neutron• neutron slightly heavier than proton and so it
decays. No reason “why” just observation
• quark content: n = udd and p = uud (plus g, qqbar) Mass up and down quarks 5-10 Mev
• three generations of quarks. Only top quark ever observed as “bare” quark. Somehow up quark seems to be slightly lighter than down quark
00
3331
35
22
,
)(10
886)mod(10
/56.939/27.938
ep
epnspecificyrs
sesallyrslifetime
cMeVcMeVmass
neutronproton
????31
32
udscbt mmmmmm
b
t
s
c
d
u
P461 - Nuclei I 3
Nuclei Force
• Strong force binds together nucleons
• Strong force nominally carried by gluons. But internucleons carried by pions (quark-antiquark bound states) as effective range too large for gluons
• Each p/n surrounded by virtual pions. Strong force identical p-p, p-n, n-n (except for symmetry/Pauli exclusion effects)
• Range of 1 F due to pion mass
p n n p
p p n n
0 0p
n
np
E t tm
s
xc
m
M eVF
M eVF
1 0
1 9 7
1 3 51
2 5
P461 - Nuclei I 4
Nuclear Sizes and Densities• Use e + A e + A scattering completely EM
• pe = 1000 MeV/c wavelength = 1.2 F now JLAB, in 60s/70s SLAC up to 20 GeV( mapped out quarks)
• Measurement of angular dependence of cross section gives charge distribution (Fourier transform)
• Can also scatter neutral particles (n, KL) in strong interactions to give n,p distributions
• Find density ~same for all but the lowest A nucleii
( )( )
/( ) /re
kg mr a b
0
11 0 1 8 3
a A F vo lum e A
b F surface area A
skin dep th rad ius A
1 0 7
0 5 5
1 3
2 3
1 3
.
.
/
/
/
P461 - Nuclei I 5
Nuclear Densities• can write density as an energy density
• Note Quark-Gluon Plasma occurs if
3
2273/1
/14.0/
/938107.12.1
FGeVvolE
cMeVkgmFAR p
3/1/ FGeVvolE
mF
F
kgcmg
m
kg
15
3
28314
3
17
101
10/10
10
P461 - Nuclei I 6
Nuclear Densities
)1979(9,42
,,,,
)(
PRL
PbSnCuAlCA
anythingAKL
P461 - Nuclei I 7
Nuclear Densities
)1979(9,42
,,,,
)(
PRL
PbSnCuAlCA
anythingAKL
84.A
P461 - Nuclei I 8
P461 Model of Nuclei
• “billiard ball” or “liquid drop”
• Adjacent nucleons have force between them but not “permanent” (like a liquid). Gives total attractive energy proportional to A (the volume) – a surface term (liquid drop)
• Repulsive electromagnetic force between protons grows as Z2
• Gives semi-empirical mass formula whose terms can be found by fitting observed masses
• Pauli exclusion as spin ½ two (interacting) Fermi gases which can be used to model energy and momentum density of states
• Potential well is mostly spherically symmetric so quantum states with J/L/S have good quantum numbers. The radial part is different than H but partially solvable shell model of valence states and nuclear spins
P461 - Nuclei I 9
Semiempirical Mass Formula
• M(Z,A)=f0 + f1 +f2 + f3 + f4 + f5 • f0 = mp Z + mn (A-Z) mass of constituents• f1 = -a1A A ~ volume binding energy/nucleon• f2 = +a2A2/3 surface area. If on surface, fewer
neighbors and less binding energy• f3 = +a3Z2/A1/3 Coulomb repulsion ~ 1/r• f4 = +a4(Z-A/2)2/2 ad hoc term. Fermi gas gives
equal filling of n, p levels• f5 = -f(A) Z, N both even
= 0 Z even, N odd or Z odd, N even
= +f(A) Z., N both odd f(A) = a5A-.5 want to pair terms (up+down) so nuclear spin = 0
• Binding energy from term f1-f5. Find the constants (ai’s) by fitting the measured nuclei masses
P461 - Nuclei I 10
Semiempirical Mass Formula
• the larger the binding energy Eb, the greater the stability. Iron is the most stable
• can fit for terms• good for making quick calculations; understanding
a small region of the nuclides.
A
Eb= E/A
volume
surfaceCoulomb
N/Z asymmetry
Total
P461 - Nuclei I 11
www.meta-synthesis.com/webbook/33_segre/segre.html
Number of neutrons
number of protons
most stable (valley)
P461 - Nuclei I 12
Semiempirical Mass• the “f5” term is a paring term. For nuclei near U
there is about a 0.7 MeV difference between having both n and p paired up (even A), odd A (and so one unpaired), and another 0.7 MeV for neither n or p being paired spin (even A)
• so ~5.9 MeV from binding of extra n plus 0.7 MeV from magnetic coupling
• easier for neutron capture to cause a fission in U235. U236 likelier to be in an excited state.
MeVf
MeVMMMnE
UnU
b
7.0236
11
6.6)(
5
236,921,0235,92
236235
MeVMeVnE
UnU
b 2.57.26.6)(
239238
P461 - Nuclei I 13
Fermi Gas Model• p,n spin ½ form two Fermi gases of
indistinguishable particles p n through beta decays (like neutron stars) and p/n ratio due to matching Fermi energy
• In finite 3D well with radius of nucleus. Familiar:
• Fermi energy from density and N/A=0.6
• Slightly lower proton density but shifted due to electromagnetic repulsion
N p p D EV
hm E( ) ( ) ( ) / / 2
33 1 2 1 28
2
Eh
m
N
a Aa F
E M eV p mE M eV
F
F F F
2 2 3
43
38
31 1
4 3 2 2 7 0
/
.
P461 - Nuclei I 14
Fermi Gas Model II
• V = depth of well = F(A) ~ 50 MeV
• Fermi energy same for all nuclei as density = constant
• Binding energy B = energy to remove p/n from top of well ~ 7-10 MeV V = EF + B
• Start filling up states in Fermi sea (separate for p/n)
• Scattering inhibited 1 + 2 1’ + 2’ as states 1’ and 2’ must be in unfilled states nucleons are quasifree
vs(ignore Coulomb)
n p n p
V
B
epn
P461 - Nuclei I 15
Nuclei • If ignore Coulomb repulsion, as n<->p through beta
decay, lowest energy will have N=Z (gives (N-Z) term in mass formula)
• proton shifted higher due to Coulomb repulsion. Both p,n fill to top with p<->n coupled by Weak interactions so both at ~same level (Fermi energy for p impacted by n)
n p
nep
enp
epn
npU 146,92238
P461 - Nuclei I 16
Nuclei: Fermi motion • if p,n were motionless, then the energy thresholds
for some neutrino interactions are:
• but Fermi momentum allows reactions to occur at lower neutrino energy.
MeVEpn
MeVEenpe
120
8.1
MeVpE
mMeVE
enp
pp
pthrsh 4.18.1
MeVmEp
MeVE
FF
F
2802
40
dN/dp
p
P461 - Nuclei I 17
Nuclei:Fermi motion •
),(
)1971(445,26
PbNi
XpeCe
pfree
pepe
PRL
solid lines are modified Fermi gas
calculation (tails due to interactions)
electron energy loss
P461 - Nuclei I 18
•
n in C nucleuspn
P461 - Nuclei I 19
Nuclei:Pauli Suppression • But also have filled energy levels and need to give
enough energy to p/n so that there is an unfilled state available. Simplest to say “above” Fermi Energy
• similar effect in solids. Superconductivity mostly involves electrons at the “top” of the Fermi well
• at low energy transfers (<40 MeV) only some p/n will be able to change states. Those at “top” of well.
• Gives different cross section off free protons than off of bound protons. Suppression at low energy transfers if target is Carbon, Oxygen, Iron...
• In SN1987, most observed events were from antineutrinos (or off electrons) even though (I think) 1000 times more neutrinos. Detectors were water…..
ee
pn
enpe
P461 - Nuclei I 20
C
Fe
Physics Reports 1972 C.H.
Llewellen-Smith
Fermi gas
“shell” model includes spin
effects
energy transfer
1-Suppression factor
P461 - Nuclei I 21
Nuclei: Fermi Suppression and Pauli Exclusion
• important for neutrino energies less than 1 GeV. prevents accurate measurement of nuetrino energy in detector