Chapter 4

The Masses of Nuclei

◎ The naturally occurring nuclei

● The nuclear binding energy

◎ The liquid drop model

● Mass parabolas and the stability line

◎ The implications of the semi-empirical mass formula

4-1 The naturally occurring nuclei

1. Apart from three or four exceptions, the naturally occurring elements up to lead are stable and lie on or near a line (the line of stability) in the Z, N plane.

2. These elements have a neutron number N which is equal to Z, Z+1,or Z+2 in nuclei up to A = 35 (except for 1H and 3 He, for which N =Z –1) but which thereafter increases faster than Z until in lead N ≈ 1.5 Z.

Line of Stability

1. We would like to understand why the stable nuclei have thisproperty and what happens if nuclei are produced in which N is greater or less than the stable optimum.

What we would like to know.

2. We would also like to understand why it is that for A > 209, there are no stable nuclei.

4-2 The nuclear binding energy

For the study of nuclei the nuclear mass (M) is a very important quantity. It is related to measurements of both binding energy (B) and separation energy (S).

The convenient unit for measuring the nuclear mass

: is called the atomic mass unit or for short amu.u

The mass of a 12C atom (including all six electrons) is defined as 12 amu (or 12 u) exact.

2

27

MeV/c )28( 49432.931

kg 10)10( 6605402.1 1 1

amuu

The mass of a proton 2MeV/c 27231.938 )12( 007276470.1 uM p

The mass of a neutron 2MeV/c 56563.939 )12( 008664898.1 uM n

(1)

Total binding energy B(A,Z)

2c)],([),( ZAMNMZMZAB np Definition:

The total binding energy B(A,Z) is defined as the total minimum work that an external agent must do to disintegrate the whole nucleus completely. By doing so the nucleus would no longer

be existent but disintegrated into separated nucleons.

This can also be considered as the total amount of energy released when nucleons, with zero kinetic energy initially, come close enough together to form a stable nucleus.

A

ZABZABave

),(),(.

An interesting measured quantity is the averaged binding energy per nucleon

(2)

(3)

Separation energy (S) (1). The separation energy of a neutron Sn

nXX NA

ZNAZ

1

1

2c)],(),1([ ZAMMZAMS nn (4)

),(c)(c),( 22 ZABNMZMZAM np

),1(c])1([c),1( 22 ZABMNZMZAM np

),1(),(

)},(c

c),1(c])1({[2

22

ZABZAB

ZAB)NMZM - (

MZABMNZMS

np

nnpn

),1(),( ZABZABSn (5)

Separation energy (S) (2). The separation energy of a proton Sp

)1,1(),( ZABZABS p (6)

pYX NAZN

AZ

11

(3). The separation energy of a α-particle Sα

2422

42 He

N

AZN

AZ YX

)2,4()2,4(),( BZABZABS (7)

The average binding energy per nucleon versus mass number A

Bave = B/A

nucleus boundtightly

most theis andenergy binding

nucleonper MeV 8.8 has Fe5626

1. The saturation property observed in the figure is the manifestation of short range characteristics of nuclear force.

2. The short range nuclear interaction can be studied by examining data collected from the (p,p) and (n,p) scatterings as well as from the binding energy of deuteron.

4He

8Be

12C

16O

24Mg

Bave = B/A

3. Overall description of the figure

The binding energy per nucleon Bave increases as the mass number A increases. At some particular values of A the value of Bave is apparently larger than neighboring A’s.

30 0 A(a).

The value of Bave varies slowly and is around 7.5 ~ 8.5 MeV.

240 30 A(b).

56 A(c).

The value of Bave reaches its maximum.

60 A(d).

The value of Bave decreases as the mass number A increases. This is due to the Coulomb repulsion.

4He

8Be

12C

16O

24Mg

A ≈ 56

(56Fe)

4He

8Be

12C

16O

24Mg

For the case when A = 4n (n = 1,2,3,4,…) the value of Bave is relatively larger when compared to neighboring nuclei.

Nucleons in a nucleus like to form α-particle clusters since aα-particle is a relatively stable 4-nucleon

system.

If we model A = 4n (for 2 < A < 25) nuclei as comprised of α-particle clusters and consider the energy difference ΔE defined as

)(),( nBZABE

We will soon discover that ΔE(8Be) = - 0.09 MeV. This shows that the nucleus 8Be is not naturally stable. Indeed the nucleus 8Be is not stable and has a measured half-life τ1/2 = 7.0 × 10-17 s.

The 9Be nucleus is stable and its natural enrichment is 100%.

(8)

ΔE(MeV) n n(n-1)/2 ΔE/[ n(n-1)/2] (MeV/pair)

12C 7.27 3 3 2.42

16O 14.44 4 6 2.41

20Ne 19.17 5 10 1.92

24Mg 28.48 6 15 1.90

1. In fact, if we divide ΔE by the number of α-particle pairs, given by n(n-1)/2, the result is roughly constant, with a value around 2 MeV.

2. This gives us a picture that, at least for light nuclei, a large part of the binding energy lies in forming α-particle clusters, around 7 MeV per nucleon, as can be seen from the binding energy of 4He.

3. The much smaller remainder, around 1 MeV per nucleon or 2 MeV between a pair of α-clusters, goes to the binding between clusters.

4. This phenomenon is usually referred to as the “saturation of nuclear force”. That is, nuclear force is strongest between the members of a group of two protons and two neutrons, and as a result, nucleons prefer to form α-particle clusters in nuclei.

4-3 The liquid drop model

1. A detailed theory of nuclear binding, based on highly sophisticated mathematical techniques and physical concepts, has been developed by Brueckner and coworkers (1954-1961).

2. A much cruder model exists in which the finer features of nuclear forces are ignored, but the strong inter-nucleon attraction is stressed. It was derived by von Weizsäcker

(1935) on the basis of the liquid-drop analogy for nuclear matter, suggested by Bohr.

Bave = B/A

Over a large part of the periodic table the binding energy per nucleon is roughly constant.

The mass density of nuclear matter is approximately constant throughout most of the periodic table.

These two properties of nuclear matter are very similar to the properties of a drop of liquid, namely constant binding energy per molecule, apart from surface tension effect, and constant density for incompressible liquids.

1. A spherical nucleus consists of incompressible matter so that R ~ A1/3.

2. The nuclear force is identical for every nucleon and in particular does not depend on whether it is a neutron or a proton.

Vpn = Vpp= Vnn (V denotes the nuclear potential)

3. The nuclear force saturates.

The essential assumptions of the liquid drop model:

The binding energy of a nucleus

2c)],([),( ZAMNMZMZAB np (9)

Definition:

From the liquid drop model ̶ Weizsäcker’s formula

Carl Friedrich von Weizsäcker, 1993 A German physicist (1912-2007)

A

ZNa

A

ZaAaAaZAB ACSV

2

3/1

23/2 )(

),( (10)

is the “volume term” which accounts for the binding energy of all the nucleons as if every one were entirely surrounded by other nucleons.

AaV

is the “surface term” which corrects the volume energy term for the fact that not all the nucleons are surrounded by other nucleons but lie in or near the surface.

3/2AaS

Nucleons in the surface region are not attracted as much as those in the interior of a nucleus. A term proportional to the number of nucleons in the surface region must be subtracted from the volume term.

A

ZNa

A

ZaAaAaZAB ACSV

2

3/1

23/2 )(

),( (10)

A

ZNa

A

ZaAaAaZAB ACSV

2

3/1

23/2 )(

),( (10)

3/1

2

A

ZaC

is the “Coulomb term” which gives the contribution to the energy of the nucleus due to the mechanical potential energy of the nucleus charge.

Assume a charged sphere of radius r has been built up, as shown in the figure (a). The additional work required to add a layer of thickness dr to the sphere can be calculated by assuming the charge (4/3)πr3ρ of the original sphere is concentrated at the center of the shell [see figure (b)]. The electrical potential energy of the nucleus is therefore

R

eZR

rdrrrV

R

Coulomb

22522

2

0

3

5

3

15

16

1)4(

3

4

3

34

R

Ze

where

3/1~ ARand

A

ZNa

A

ZaAaAaZAB ACSV

2

3/1

23/2 )(

),( (10)

A

ZNaA

2)(

is the “asymmetry term” which accounts for the fact that if all other factors were equal, the most strongly bound nucleus of a given A is that closest to having Z = N.

Three terms that were discussed previously are in a sense classical. The following terms that are to be discussed are quantum mechanical.

These include

(1) the asymmetry term

(2) the paring term

(3) the shell effect correction term

A

ZNa

A

ZaAaAaZAB ACSV

2

3/1

23/2 )(

),( (10)

The Pauli exclusion principle states that no two fermions can occupy exactly the same quantum state. At a given energy level, there are only finitely many quantum states available for particles.

Different system energies due to asymmetric configurations

1. If Z = N, then both wells are filled to the same

level (the Fermi level).

2. If we move one step up away from that situation, say in the direction of N > Z (or Z > N), then one proton must be changed into a neutron. All other things being equal (including equal proton and neutron mass), this state has energy ΔE greater than the initial state, where ΔE is the level spacing at the Fermi level.

3. A second step in the same direction causes the energy excess to become 2ΔE.

4. A next step means moving a proton up three rungs as it changes from proton to neutron and the excess becomes 5ΔE.

E ofunit 32,.... 25, 18, 13, 8, 5, , 2 1,

16,... 14, 12, 10, 8, 6, 4, 2, ZN

Cumulative effect

4. Therefore to change from N – Z = 0 to N > Z, with A = N + Z held constant, requires an energy of ~ (N – Z)2ΔE/8.

5. This is independent of whether it is N or Z that becomes larger and it means that, if all other things are equal, nuclei with Z = N have less energy and are therefore more

strongly bound than a nucleus with Z ≠ N.

6. The energy levels of a particle in a potential well have a spacing inversely proportional to the well volume, thus we put ΔE ~ A-1.

This is the asymmetry term.A

ZNaA

2)( (11)

A

ZNa

A

ZaAaAaZAB ACSV

2

3/1

23/2 )(

),( (10)

is the “pairing term” which accounts for the fact that a pair of like nucleons is more strongly bound than is a pair of unlike nucleons.

1. For odd A nuclei (Z even, N odd or Z odd, N even) →δ = 0.

2. For A even there are two cases;

(a). Z odd, N odd (oo) → – δ

(b). Z even, N even (ee) → + δ

MeV 12 ,),(2/1

PP a

A

aAZ (12)

A

ZNa

A

ZaAaAaZAB ACSV

2

3/1

23/2 )(

),( (10)

is the term accounts for the nuclear shell effect when Z or N is some magic number. This term is much less important than other terms. Therefore this term is not included in most of the applications.

A favorable set of values for the coefficients:

aV = 15.560 MeV aS = 17.230MeV

aC = 0.6970 MeV aA = 23.385 MeV

aP = 12.000 MeV

(13)

A

ZNa

A

ZaAaAaZAB ACSV

2

3/1

23/2 )(

),( (10)

If we make some rearrangement the Weizsäcker’s formula can be written as

22c),( ZZAZAM

3/12c

A

aaaM S

AVn

ApnA aMMa 4 ≈c)(4 2

3/1

4

A

a

A

aγ CA +=

For a fixed mass number A, this is the equation of parabola with respect to the variable Z.

We may differentiate the equation (14) and find the root (Z0, usually not an integer) of the following equation (15). Z0 is the optimum nuclear proton number for a fixed mass number A. The nuclear system with a specified mass number A is the most stable with proton number Z0.

(14)

0)c( 2

MZ

(15) 02 0 =+ Zγβ 3/20

)/(41

1

2/

2

Aaa

AZ

AC

(16)

4-4 Mass parabolas and the stability line

From the equation (15) the most stable nuclear systems of various mass numbers A are determined by the value of Z0. By using the relation A = Z0 + N we are able to plot stability lines on the N-Z plot. This follows exactly the shape of the empirical stability line in the figure.

From expression (16), we can recognize that the deviation of the stability line from N = Z or Z = A/2 is caused by the competition between the Coulomb energy, which favors Z0 < A/2, and the asymmetry energy which favors Z0 = A/2.

3/20

)/(41

1

2/

2

Aaa

AZ

AC

(16)

22c),( ZZAZAM (14)

For odd-A isobars, δ = 0, and equation (14) gives a single parabola, which is shown in the figure (a) for a typical case. We will see later that if

M(A,Z) > M(A, Z+1)

beta (electron) decay takes place from Z to Z+1

M(A,Z) > M(A, Z-1)

electron capture and perhaps positron decay takes place from Z to Z - 1

It is clear from the figure (a) that for odd-A nuclides there can be only one stable isobar.

(15)

22c),( ZZAZAM (14)

For even-A isobars, two parabolas are generated by the equation (14), differing in mass by 2δ. A typical case is given in the figure (b). Depending on the curvature of the parabolas and the separation 2δ, there can be several stable even-even isobars. Figure (b) shows that for certain odd-odd nuclides both conditions (15) are met so that electron and positron decay from the identical nuclide are possible and do indeed occur.

eeAZAZ ),1(),( decay

eeAZAZ ),1(),( decay

eAZAZe ),1(),( capture)electron ( EC

Three types of β-decay

4-5 The implications of the semi-empirical mass formula

1. The volume binding term aVA in the mass formula means that each nucleon interacts only its nearest neighbors and that the constant density is equivalent to a separation from nearest neighbors does not change with A. All this means that nuclear force saturates and is short range.

2. The nuclear density of nucleons is approximately 1 in every 7 cubic fermis so that the average separation is about 1.9 fm. Thus the range must be 1-2 fm.

3. From a study of nucleon-nucleon scattering at energies in the range 100-400 MeV. It was found that there is a strong repulsive force between nucleons at separations of less than about 0.5 fm.

0.5

Hard core

4. The liquid-drop model does accommodate the existence of excited states corresponding to the mode of vibration (vibrational states) which are one kind of collective motion that nuclei may have, the collective implying the motion of the constituent nucleons is correlated in some way to give the shape vibration. But this phenomenon can not be described by the model.

Water drop vibrations

5. There is another type of collective mode which is in the form of rotation for non-spherical nuclei. The quantum states associated

with nuclear rotational motion is called rotational states. These states can be identified from energy spectra of non-spherical nuclei.

A prolate deformed nucleus

~ The End ~