Atoms and Nuclei PA 322 Lecture 13

Unit 3: Nuclear Models

(Reminder: http://www.star.le.ac.uk/~nrt3/322)

PA322 Lecture 13 2

Topics

•  Nuclear forces –  basic features –  virtual particle exchange (Yukawa theory)

•  Liquid drop model –  semi-empirical mass formula –  binding energies

PA322 Lecture 13 3

Motivation for nuclear models

•  Motivation –  understand/compute properties of nuclei –  need framework analogous to that for atomic physics: QM description

•  Approach adopted –  nuclear models –  model: not application of physics from first principles, instead model is

approximation to full physical description that encapsulates most important physics (i.e. dominant effects)

–  several simplified nuclear models have been developed: each has predictive power and describes range of nuclear properties

•  Reason –  using full Standard Model (for particle interactions) for nuclear properties

not feasible: too complex, computationally impossible (approximations, e.g. perturbation theory doesn’t work as no “small” terms)

PA322 Lecture 13 4

Basic features of nuclear force

•  The nuclear force must exist to explain the fact that nuclei are bound. Otherwise nuclei would fly apart through effects of Coulomb repulsion

•  Basic features of nuclear force required: 1)  must be attractive & stronger than

Coulomb force at short range 2)  must be very weak at long range

–  otherwise would affect atoms & molecules etc. (not observed)

3)  must be repulsive at very short range –  otherwise nuclei would collapse

0

nucl

ear f

orce

F

distance R

very schematic!

attractive

repulsive

PA322 Lecture 13 5

Basic features of nuclear force (continued)

4)  must be charge independent, ie same for neutrons & protons (nn, np, pp) –  this follows from the observation that mirror nuclei – those with

N=X, Z=A-X and N=A-X, Z=X (e.g. ) - have same binding energy once Coulomb interaction of protons is taken into account

–  confirmed by nucleon scattering experiments 5)  must be spin dependent, i.e. whether spins of interacting nucleons

are parallel or anti-parallel –  (this is shown in nucleon scattering experiments) –  and is hinted at by stability of even-N/even-Z nuclides.

6)  must be non-central and non-spherically symmetric -  again shown by experiment

⇒ Nuclear force is very different from Coulomb force!

€

713N & 6

13C

PA322 Lecture 13 6

Origin of nuclear force

•  exchange of particles results in momentum transfer –  equivalent to a force

•  forces in particle physics –  virtual particle exchange –  Coulomb (electrostatic) force

involves exchange of virtual photons

•  Yukawa theory –  strong force involves

exchange of virtual mesons

p

n

PA322 Lecture 13 7

Origin of nuclear force: virtual particle exchange cartoon

p

n

n π -

not observable!

π -

not observable!

n

p

t1 t2

observed observed

n

p p

PA322 Lecture 13 8

Origin of nuclear force

•  Yukawa theory –  nuclear force involves exchange of virtual mesons (pions) –  energy to create the pions (ΔE) is “borrowed” from the vacuum

for a time Δt such that ΔE Δt ≤ ℏ

thus satisfying Heisenberg Uncertainty Principle –  range of strong nuclear force R can be used to estimate pion

mass mπ : ΔE ≈ mπ c2

Δt ≈ R/c (maximum speed is c) thus R ≈ cℏ / ΔE ⇒ mπ c2 ≈ ℏc/R

–  for R ≈ 2 fm (nucleus size ~ range of nuclear forces) ⇒ mπ c2 ≈ 100 MeV (~200 me)

PA322 Lecture 13 9

Origin of nuclear force •  Mesons predicted in Yukawa theory were originally believed to be

muons (which have mass ~100 MeV). We now know this is wrong – indeed muons are leptons not mesons.

•  Pions (π+, π-, π0) discovered in 1947 interact strongly with nucleons and have masses

mπ c2 ≈ 140 MeV (neutral pion is slightly lighter) –  Similar to what we just calculated

–  In the context of the “Standard Model” pions and nucleons are made of quarks held together by nuclear “colour” force mediated by gluons: very complicated.

–  But virtual pion exchange can be shown to be the largest contributor.

PA322 Lecture 13 10

Origin of nuclear force

Note, in the Standard Model the nuclear force is not exactly the same as the strong force which acts on hadrons, more correctly it is the residual strong force

The strong nuclear force between quarks is the result of interaction of virtual gluons. The residual of this force outside of quark triplets (neutron and proton) holds neutrons and protons together in nuclei.

PA322 Lecture 13 11 (http://en.wikipedia.org/wiki/Standard_Model)

PA322 Lecture 13 12

Origin of nuclear force

•  Key point for forces interpreted in virtual particle exchange framework

–  zero mass particles associated with forces with infinite range (e.g. EM forces)

–  finite mass particles associated with finite range forces (e.g. nuclear force)

PA322 Lecture 13 13

Liquid drop model

•  Model for nucleus used as framework for discussion of binding energies and nuclear stability

•  Terminology arises because of analogy with physics relating to properties of a drop of liquid (i.e. interactions between molecules) –  if we wanted to work out the potential energy of a drop of liquid

we would sum the contributions arising from each possible pair of molecules, i.e.

•  The liquid drop model for the nucleus takes essentially the same approach for working out the binding energy

pair moleculeeach for menergy ter potential theis where)(

ijijj i

ij uuU ∑ ∑≠

=

PA322 Lecture 13 14

Liquid drop model and the semi-empirical mass formula (SEMF)

•  The liquid drop model provides an approximate analytic formula for the binding energy B(Z,A) for a nucleus with atomic number Z and mass number A –  semi-empirical mass formula –  terms in equation come from theory, coefficients come from experiment –  recall that the mass of a nucleus mA is directly related to the binding energy

via the formula B = [ Z mp + N mn – { mA – Zme } ] c2

•  Semi-empirical mass formula gives the binding energy of a nucleus in the liquid drop model of the nucleus

•  SEMF is the sum of 5 terms: name, origin and meaning of each discussed on subsequent slides

43

31

32

5

2

4321 ),()2()1(),( −− +−

−−−−= AaAZAZAaAZZaAaAaAZB δ

€

(coefficients a1…5 are constants)

PA322 Lecture 13 15

Empirical facts that SEMF must explain: stability of nuclides

Overall pattern of stable nuclides –  for small Z, N~Z and A~2Z –  for large Z, N>Z –  most stable nuclei have even Z and

even N –  significant number with even-odd and

odd-even configurations –  only a very few odd-odd nuclei are

stable (eg. )

⇒  Empirical facts that semi-empirical mass formula must explain (or at least incorporate!)

A N Z number stable

even even odd

even odd

166 8

odd even odd

odd even

57 53

H21 N147

PA322 Lecture 13 16

Semi-empirical mass formula

Volume term:

•  Explanation: nuclear force has short range, each nucleon only experiences effect of its nearest neighbours ⇒ contribution to binding energy thus scales directly with number of nucleons A and is ~same for each nucleon

•  Equivalently we can say that the effect is proportional to the volume V of nucleus, but as V ∝ R3 and we know nuclei have R ∝ A1/3 we derive the same dependence on A

•  Term is +ve as nuclear force is attractive producing +ve contribution to binding energy of nucleus

Aa1 constants are tscoefficien 51…a

PA322 Lecture 13 17

nuclear force is short range ⇒  nucleons only interact with nearest neighbours ⇒ contribution to binding energy ∝ A

( ~same for each nucleon)

PA322 Lecture 13 18

nuclear force is short range ⇒  nucleons only interact with nearest neighbours ⇒ contribution to binding energy ∝ A

( ~same for each nucleon)

PA322 Lecture 13 19

nuclear force is short range ⇒  nucleons only interact with nearest neighbours ⇒ contribution to binding energy ∝ A

( ~same for each nucleon)

PA322 Lecture 13 20

nuclear force is short range ⇒  nucleons only interact with nearest neighbours ⇒ contribution to binding energy ∝ A

( ~same for each nucleon)

PA322 Lecture 13 21

nuclear force is short range ⇒  nucleons only interact with nearest neighbours ⇒ contribution to binding energy ∝ A

( ~same for each nucleon)

PA322 Lecture 13 22

Semi-empirical mass formula

Surface term:

•  Explanation: nucleons at the “surface” of the nucleus have fewer neighbours, this term thus effectively corrects the assumptions made for the volume term. The surface area of a sphere ∝ R2, but as we know nuclei have radii R ∝ A1/3, the surface term ∝ A2/3

•  Term is -ve as it is correcting for the reduction in binding energy due to this surface effect

32

2Aa−

PA322 Lecture 13 23

nucleons at “surface” have fewer neighbours

PA322 Lecture 13 24

nucleons at “surface” have fewer neighbours

PA322 Lecture 13 25

Semi-empirical mass formula

Coulomb term:

•  Explanation: the Coulomb term accounts for the electrostatic repulsion between the Z protons in the nucleus, each of which interacts with all the other protons or equivalently experiences the electrostatic field of the whole nucleus. Each proton interacts with (Z -1) other protons (it doesn’t interact with itself)

•  The potential energy for a sphere of charge Z and radius R can be shown to be ∝ Z(Z-1)/R, giving the form of the term quoted and of course because R ∝ A1/3

•  Term is -ve as the electrostatic effect is a repulsive force which lowers the binding energy

31)1(3

−−− AZZa

PA322 Lecture 13 26

Semi-empirical mass formula

Symmetry term:

•  Explanation: This term reflects the observed fact that light nuclei have similar numbers of neutrons and protons, i.e. N~Z. The term in brackets = (N-Z)2 so that equal numbers of neutrons and protons gives minimum value for this term. Factor 1/A reflects fact that this effect becomes less important as A increases.

•  This term has its origin in the Pauli Exclusion Principle applied to nucleons (which are fermions) and the way that it effects the filling of nuclear shells in the shell structure of the nucleus – covered in later lecture.

•  Term is –ve as effect lessens binding energy.

AZAa

2

4)2( −

−

PA322 Lecture 13 27

Semi-empirical mass formula

Pairing term:

•  Explanation: term reflects pattern of Z, N values found in stable nuclei: even-even combination strongly preferred, followed by even-odd or odd-even combinations and very few odd-odd cases. Encapsulates spin-coupling effects in the nucleus. The factor is empirically determined

•  Term is +ve so that it gives higher binding energy for even-even combinations.

43

5),(−AaAZδ

odd , 1odd 0

even , 1

NZANZ

−=

=

+=

δ

δ

δ

43−A

PA322 Lecture 13 28

Semi-empirical mass formula

43

31

32

5

2

4321 ),()2()1(),( −− +−

−−−−= AaAZAZAaAZZaAaAaAZB δ

binding energy B(Z,A) for a nucleus with atomic number Z and mass number A

volume term

surface term

symmetry term

Coulomb term

pairing term

B = [ Z mp + N mn – { m(AX) – Zme } ] c2

atomic mass energy

PA322 Lecture 13 29

Application of the Semi-empirical mass formula

•  Values of coefficients

a1=15.3 MeV a2=16.8 MeV a3=0.72 MeV a4=24 MeV a5=34 MeV

•  Provides a good fit to accurate binding energy values for large A, but poorer fit at low A (due to effects not included, e.g. shell structure of nucleus)

•  Semi-empirical mass formula can be used to predict – via binding energies –  stability of isotopes –  energy changes in nuclear processes and reactions

PA322 Lecture 13 30

Application of the Semi-empirical mass formula

•  Energy changes in nuclear processes and reactions –  simple example: beta-decay of radioactive

is this energetically possible? •  Evaluate binding energies from SEMF

BCu = 566.45 MeV BZn = 568.37 MeV [thus Zn isotope is more tightly bound than Cu isotope]

•  Mass energies of initial and final states:

ν++→ -6630

6629 e ZnCu

€

2966Cu

€

minit = m 66 Cu= 29mp + 37mn −B 66 Cu mfinal =m 66 Zn+me = 30mp + 36mn + me −B 66 Zn

minit −mfinal( ) = mn −mp −me + B 66 Zn−B 66Cu( )= 939.57 − 938.28 − 0.511+ 568.37 - 566.45( ) MeV= 2.669 MeV> 0

BNmZmm −+= np

€

(mass energies, if using atomic masses need to account for electron masses)