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