The nuclear models for pedestrian

1

Index

Introduction..........................................................................................................................................3

1. The compound nucleus.....................................................................................................................4

2. The residual interaction....................................................................................................................6

3. Rotational model..............................................................................................................................9

3.1. Hamiltonian of the rotational model.......................................................................................11

4. Wave Functions..............................................................................................................................13

4.1 first symmetry property of the wave functions........................................................................14

4.2. second symmetry property......................................................................................................15

5. The problem of the moment of inertia............................................................................................15

6. Electromagnetic transitions in Rotational Model...........................................................................17

7. Electric Quadruple Moments..........................................................................................................18

8. Nuclear Deformation......................................................................................................................21

9. Vibrational model...........................................................................................................................22

9.1. Appendix vibration model......................................................................................................23

10. Electric quadrupole transitions.....................................................................................................27

11. Particle-vibration coupling...........................................................................................................28

Conclusions........................................................................................................................................34

References..........................................................................................................................................35

2

V.R. Manfredi 1

Introduction

For ten years we gave a course in Nuclear Spectroscopy to students attending the

Science Faculty at Padua University and also to Erasmus students.

At the request of the students we collected the lessons in the present article.

We are grateful to Physics Italian Society for the editorial aid.

For many years the theories of atomic nucleus were essentially based on nuclear

models (macroscopic and/or microscopic) [Ref 1,2,3,4]. Only recently a complete

“abinitio” theory has been developed, including relativistic effects and mesons degree

of freedom [Ref 5].

The atomic nucleus

n

zX n

is mainly based on nucleons in interaction, Z protons and N

neutrons, with A=Z+N.

The main idea of theoretical nuclear physics is that the description of all properties of

atomic nuclei (the binding energy of the ground state, the spectrum of excitation, the

transition probabilities, the electric and magnetic moments, …) are obtained by the

interaction of free nucleons.

The complexity of the nucleon-nucleon interaction and the difficulties in solving

Schrödinger equation for many body problem ( A 10 ) lead to the introduction of a

lot of phenomenological nuclear models each one stressed on one aspect of nuclear

properties.

1Padua University Physics Department

Member of the New York Academy of Science

3

In nuclear reactions and nuclear spectroscopy fields, two different theories have been

developed.

The first one is models as independent particles, where each nucleon is in the average

mean field and the interaction between other nucleons is weak. In the nuclear reaction

field the compound nucleus model was proposed, otherwise in the nuclear

spectroscopy field the shell one.

1. The compound nucleus

Following N. B. [Ref 6], the so-called compound nucleus model of nuclear reactions

is briefly discussed.

It is assumed that when some target nucleus A is bombarded by an incident nuclear

particle a, both may coalesce to form a compound nucleus (A+a ).

In the compound nucleus (A+a), there are assumed to be strong interactions among

all the nucleus. The incident nuclear particle a loses its independent identity , and the

total energy of the excited compound nucleus is shared in a complicated manner by

all nucleons involved.

The compound nucleus ( A+a ) is taught of as being in a quasi-stationary quantum

state, whose mean life is long (~ 10�16�3 sec) compared with the time used by a

proton to cross nucleus (~ 10�22 sec).

Identically in the same level of excitation the same compound nucleus can be

produced by the collision (usually at a different bombarding energy) of other nuclei,

say, B and b, so that it is possible to have

4

( A+a) = [B+b].

In the N. Bohr postulates, the properties of the compound nucleus ( A+a ) are

independent of its mode of formation, i.e. the compound nucleus “forgets” how it was

formed.

The second step in the nuclear reaction is the dissociation of the compound nucleus.

This dissociation can generally take place in a large number of ways, sometimes

called “exit channels,” subject to the conservation laws for mass-energy, charge

angular momentum, etc.

The competition among various alternative modes of disassociation does not depend

on the manner in which the compound nucleus was formed, i.e., on the “entrance

channel”. Schematically, any nuclear reaction in which a compound nucleus is

formed can be represented as

Entrance Compound Exit

channel nucleus channels

A+a (elastic scattering)

A*+a (inelastic scattering)

B+b (nuclear transformation)

A+ a → (A+a) C+b+c (nuclear transformation)

D+d (nuclear transformation)

Etc. (nuclear transformation)

5

The asterisk, as in A*, denotes an excited level of a nucleus. As an explicit example,

the reaction on F19 in which a proton is captured and an α particle is emitted would

be written

F19

9 H1

1 Ne20

10 He4

2 O16

8

The same compound nucleus might instead emit a neutron, leaving Ne19

10 as the

residual nucleus, according to

F19

9 H1

1 Ne20

10 n1

0 Ne19

10

In the more compact notation which is usually used for nuclear reactions, these two

competing reactions would be written F19

p , and F19

p , n Ne19

without explicit

designation of the compound nucleus.

2. The residual interaction

In order to take into account also the residual interaction between the nucleus, the

total nuclear Hamiltonian H may be written:

R.I.1 H H 0 V R ,

6

Figure 1: Bohr's model of a neutron-nucleus collision

where

R.I.2 H 01

A

i H 0

� i�

1

A

i T i +V i1

A

i

pi

2

2m i

V i

and

(R.I.3) V ri j

A

V ij1

A

i V 1 .

H 0�i �

is the unperturbed single particle and V R is the nucleon-nucleon residual

interaction.

The residual interaction V R may be obtained by self consistent Hatree – Fock

equation, starting from the free nucleon-nucleon interaction.

Very often a phenomenological approach was followed, extract the single- particle

energies from closed shell nuclei plus one particle O17

, Ca41

, Pb209

, Bi209

, see fig.

But in order to determine the single particle spectrum i and the wave functions u i ,

the Schrödinger equation

R.I.4 H 0

� i�ui +i ui

is solved.

The antisymmetric wave function � , in order to take into account the Pauli

principle, is the solution of the equation.:

H 0 � E �

�0� ,

where

(R.I.5) E

0

1

A

i i .

7

In order to obtained the eigenstates of H, we diagonalize H on the basis of all

configurations � The Schrödinger equation is:

(R.I. 6) H�

E�

,

where

(R.I.7) n�

C n� � .

The eigenvalues E n and the eigenvectors C n are the solutions of the secular

problem:

R.I.8�

E�

�0�

�� � V R � Cn� En Cn� ,

where �V R � are the matrix elements of the operator V R , function of the two

body matrix elements V ij Ref .

In order to avoid the difficulty of infinite number of terms in (R.I.8), a truncation of

model space/see can be used and we take into account just “active orbits” (see fig ).

With the truncation of the model space, the structure of (R.I. 8 ) is:

(R.I:9)

E1

�0 �En 1 V R 1 1 V R 2 . 1 V R M

. E1

�0�En E2

0En 2 V R 2 .

. . .

. . E M

�0 �E N M V R M

The energies En are the theoretical spectrum of the system to compare with the

experimental data. The eigenvectors C n are determined by the condition

R.I.10�

Cn1�

Cn� nn1

8

By the knowledge of the wave function n is possible to calculate all observable

quantities such as the quadruple moment Q1 , the magnetic moment, 0 ...

As an example, we report in ( R.I.11 ) the quadruple moment Q of the lever n:

R.I.11 Q n Q n� �

Cn� C n� �Q

� ,

where the matrix elements �Q

� are functions of the structure of the operator Q

and of basis state i .

3. Rotational model

If there are many nucleons (neutrons and protons) outside the closed shells, there is a

permanent deformation of the atomic nuclei.

This deformation is in fact, the competition of the closed shells (spherical nuclei) and

the nucleons outside the closed shells. In zero order approximation, it is possible to

disentangle the motion of nucleons by the collective (adiabatic approximation).

9

Figure 2: Truncation of model space

m ( j1

n1, j2

n2, ..., jm

nm)

The validity of the adiabatic approximation is confirmed experimentally by the

existence of rotational motions. If the deformation is axially symmetric, the energies

of the rotational model are:

(RM1) E I � K

2

2JI I 1 K

2 2

2J3

K2

,

where I is the total angular momentum of the nucleus and K is the projection of I .

on the symmetric axe Z .

J J 1 J 2 and J 3 are the moments inertia, referred to principal axes. In fig the

spectrum of 168Er is reported for different values of K . The so called “ground state

band” correspond to K 0 and the (RM1) becomes :

(RM2) E I

2

2JI I 1 .

In the adiabatic approximation the moment of inertia J is constant in the band and it

is very easy to calculate the ratio between the energies of the different levels:

(RM3) E 4

E 2

10

3

E 6

E 2

7E 8

E 2

12 .

The adiabatic approximation allows one that the nucleus wave function may be

factorized as:

(RM4) vib Drot ,

where

vib describes the vibrations of the nucleus around the spherical scape,

D rot describes collective rotational motion,

describes the motion of the nucleons in the deformed field.

10

In the rotational motion we are only interested to and D .

The basic hypotheses are:

*) Nuclei with non spherical shape.

*) Factorization of the nuclear wave function.

We call OXYZ (S) is a fixed system (a lab system) and O X1Y

1Z

1S

1 a system in the

nucleus.

The orientation of S1 is determined by three Euler angles i

3.1. Hamiltonian of the rotational model

The Hamiltonian of the rotational model may be written in the form:

11

Figure 3: Spectrum of 168Er

7 1950.81

6 1820.14

5 1708.01

4 1615.36

3 1541.58

I E [keV]

k π = 3

–7 1432.97

6 1233.92

5 1117.60

4 994.77

2 821.19

3 895.82

I E [keV]

8 928.26

6 548.73

4 264.081

2 79.800

0 0

I E [keV]

k π = 2

+

k π = 0

+

8 1605.85

7 1448.97

6 1311.48

5 1193.04

4 1541.58

I E [keV]

k π = 4

–

Er100

16868

(RM5) HT T ro t H i n t x ' H compl ,

where

T rot is the kinetic energy of rotation,

H i n t X ' is the Hamiltonian of intrinsic motion,

H compl is the Hamilton of the coupling between the intrinsic motion and the rotational

motion.

The adiabatic approximation allows one to write the function Ψ as:

E' D i

If R is the angular momentum of the collective motion, J is the angular momentum

of the intrinsic motion and I is the total angular of the nucleus:

(RM6) I = R + J

for even-even nuclei (Z even N even) , J =0 and (RM6) I = R ,

the analysis of the experimental spectra suggests two different hypothesis:

*) the nuclear deformation is axially symmetric.

*) The nuclear deformation has a symmetric plane.

If the rotational motion is similar to a rigid body:

T rot

2

2

R1

2

J 1

R2

2

J 2

R3

2

J 3

1 → x' 2 → y' 3→z',

where R1 R2 and R3 are the components of the angular momentum of the nucleus.

If OZ' is the symmetry axes, we have:

(RM7) J 1 = J 2 =J,

12

the nuclear Hamiltonian may be written:

H= T rot

2

2JR

2R3

2

2

2J3

R3

2,

where

R2

R1

2R2

2R3

2Rx

2Ry

2Rz

2.

For even-even nuclei the rotational Hamiltonian may be written:

(RM7) H2

2II

2I

32

2J3

I 3

2,

where I is the total angular momentum and I 3 is the component along the Z' axe.

The energy spectrum for the rotational motion is:

(RM8) E I , K

2

2JI I 1 K

22

2J3

K2

,

4. Wave Functions

The eigenfunctions of the rotational Hamiltonian are:

DMK

I

i

I2D

MK

II I 1 D

MK

I

(RM9) IzD

MK

IM

D MK

I

I z ' DMK

IKDMK

I.

Where M is the projection of the total angular momentum I On the 0Z axe and K is

the projection on the OZ' axe.

13

The normalized wave functions are:

(RM10) |IMK>=2I 1

82

DMK

I

0

4.1 first symmetry property of the wave functions

*) symmetry property around the axe OZ'

The wave function is invariant for the rotation of the Euler angles (α, β,

γ+Φ) then :

(RM11) MK

I, , MK

I, ,

0 is invariant for the rotation and

(RM12) DMK

I, , e

iK �DMK

I, ,

As the consequence of the symmetry properties K=0 and J is perpendicular to the

OZ' axes.

14

Figure 4: Projection of I on Z axis and Z' axis

Z

Z’I

k

4.2. second symmetry property

This kind of symmetry is suggested by even values of I 0 2 4 for the

transformation

(RM13) (α, β, γ) → (α, β+π, -γ),

wee have:

(RM14)R

y 'D

MK

ID

MK

I,

I�KD

M ,�K

I, ,

,

for K=0

DM �

, ,I�K

DM ,�K

I, , .

As a consequence of symmetry properties of the transformation (α, β, γ) → (α, β+π,

-γ) I must be even.

The wave function is:

(RM15) | IMO › =2I 1

82

DMO

I, ,

0 ,

with I=0,2,4...,

in term of spherical harmonic we have:

(RM16) | IMO › = 1

2 M

I, 0 .

A rotational band is a set of states with the same intrinsic function, but different

values of I.

5. The problem of the moment of inertia

In order to calculate the moment of inertia a detailed motion of nucleons must be

calculated .

15

Historically, two different approaches were followed. The first one was similar to a

rigid body (J rig), the second one was similar to an irrotational liquid (J irrot).

In Fig we compare J rig , J irrot and the so called “ J sp ” (taken from 2�

state). The

agreement is not good at all.

In 1961 S.G. Nilsson and O. Prior obtained a good agreement with the

“experimental” data, by the introduction of a pairing interaction 2 between the

nucleons.

The result of Nilsson and O.Prior is :

J 22

'I 7

2

,

'E'

E'

'

U'V

'' V'U

''

2,

where E� �

x2

, U �

2V �

21

2 the matrix element of the pairing interaction is B j 1 j2 V p j3 j 4 X G j1 j2 j3 j4, G > 0

16

Figure 5: Comparison between the moment of inertia "experimental" (solid

line), rigid (dashed-dotted line) and irrotational (dashed line)

152 160 168 176 1840

50

100

150

200

62Sm 64Gd 66Dy 68Er 70Yb 72Hf 74W

A

(2 / h

2)J

[(M

eV

)–1]

6. Electromagnetic transitions in Rotational Model

The transition probability T between an initial state

I i M i and a final state J f M f is given by:

ET1 T1

,

Where is the wide of the nuclear level

For electromagnetic transition, the selection rule is:

I i I f I i I f ,

where is multi-polarity of the emitted transition.

The consequence of the parity conservation is:

ET2 i f 1,

for electric transitions

i f 1,�1 for magnetic transitions.

The structure of the multiple electric operator me is:

ET3 me i ei ri

,J �

,

i i

The transition probability T is given by:

ET4 T8 1

2 1 ! !� 2

1

2

E

c2 1 ,

where E is the difference between the energy of the final and the initial state and

is the reduced transition probability.

ET5 i I i I fMf

J f M f M x J i M i

2

17

By the use of the Winger-Eckert theorem and the properties of C.G. coefficients, we

have:

ET6 I f M f m I i M i I i M i I f M f

I f m I i

2I f 1

ET7Mf �

I i M i I f M f

2 1

2 I i 1 M i�

M f

I i M i I f M f

2 2 I f 1

2 I i 1

ET8 i I i I fM f

I i M i I f M f

2 I f m I i

2

2I f 1

2I f 1

2 I i 1

I f m I i

2

2I f 1

=I f m I i

2

2Ii 1.

Very often the reduced transition probabilities B are expressed as a function of

single particle transition. For the transition I i 2 I f 0 the single value is

( Weisskopf unit):

ET9 B E2 s.p.

1

4

3

5R0

22

G4 03

1052

cm4

.

7. Electric Quadruple Moments

The electric quadruple moment is given by:

( EQ1 ) Q I M I Q0�2�

I M I

where

EQ2 Q0

�2 �

i1

Z

Z i

2r i

2 16

5i1

Zr i

2

0

2

i i

Q0�2�

is a particular case 2 0 of the electric multiple operator me given

by:

EQ3 mei1

Z

r i

���

i i .

18

In fact

EQ4 me 25

16e Q�

�2�

where Q��2�

is the electric quadruple moment in the lab system.

In the system S' . , taken into account is the relation:

EQ5 �

�

�'

D��

'

�

�'

' '

we obtain

Q�

�2�

vD� v

2Q v

' �2�

where

EQ6 Q v

�2� 16

5

Z

i1 r i

12

i

1

i

1

r i'

i'

i' are the proton coordinate in the system S

' , Q v1�2�

, is the intrinsic electric

quadruple moment.

If

( EQ7 ) I M K2I 1

82

DMK

I

0

is normalized wave function,

we obtain:

EQ8 I M1K

1Q

�

�2 �I M K

�

2I 1

82 0 Q

�1

�2�

0 DM

1K

1

I

�

2I� M I M1

2I� K I1K

12 0 Q�

�2 �

0 M1���M K

1� K�� ,

( EQ5) if

( EQ6) M M1

I K K1

19

(EQ7 ) 0 andv 0

(EQ8 ) as the consequence

(EQ9 ) I I K Q0�2

I I K 2I K1

I K 0 Q01 �2

0 .

If Q0 is the intrinsic quadruple moment:

( EQ10) Q0 0 Q 01�2

0

and the C . G. coefficient is:

EQ11 2I D I II 2I 1

I I 1 2I 1 2I 3

the relation between the spectroscopic quadruple moment Q and the intrinsic

quadruple moment Q0 is:

( EQ12 ) Q I I K Q 0

1�2�I I K

3 K2

I I 1

I 1 2I 3Q0 .

Very often Q0 is called the “projection factor” and is the connection between the

spectroscopic factor Q and the intrinsic quadruple moment Q0 .

For the ground state band I K . The ( EQ12 ) becomes:

EQ13 QI 2I 1

I 1 2I 3Q0 .

For K 0 .

EQ14 QI

2I 3Q0

In the framework of rotational model it is very easy to establish the connection

between Q0 and the deformation .

( EQ15 ) Q0 e r1

3Z12

r12 1 16

5e r

1r

12J 0

�2�J 0

�2� 1y

1 1

20

Where e r

' is the electric charge density and 1r

12r

1 1

If the charge is uniform, we have:

( EQ16 ) e3 e Z

4 R03 ,

and the angular dependence of R is:

EQ17 R1

R0 1 0

�2� 1.

Finally

Q0 e

16

5 �4%� 0

�2 � 1

0

R �-1�

r14

r1

.

By integration we obtain:

EQ18 Q0

3

5Z R0

21 016 .

The ( EQ18 ) allows one to determine the deformation parameter by the

knowledge of Q0 .

8. Nuclear Deformation

The basic hypothesis is that the nuclear matter is uniformly distributed in a volume V

limited by the surface R=R .

As a function of spherical harmonics may be written:

R R0 1���

�� �, .

For spherical nuclei 2 in the system S1 only 1 is able to characterize the nuclear

shape.

By the use of Legendre polynomials, we have:

( ND1) R1

R0 1 0

2 1,

21

Where a20 and R0 is the mean radius of the nucleons. The deformation parameter

is also connected with the ratio

(ND2 ) R

R

a b

R,

where a is the major axe and b is the minor axe of the ellipsoid

(ND3 ) 0

�2 � 1 5

163cos

2 11 ,

we have:

ND4 a R0 15

4

10

( ND5 ) b R0 15

16

1

2.

As the consequence of (ND4 ) and ( ND5 ), we have:

( ND6 ) 4

3 5

R

R0

1,06R

R0

.

If 0 we have a prolate spheroid, if 0 we have an oblate spheroid.

9. Vibrational model

The vibrational model, similar to rotational model, is a collective description of the

even-even nuclei ( 1, 2, 3, …) . The main difference between the two models is that

in the rotational model the deformation parameter is a static quantity, in the

vibrational model the deformation parameter � ,� is a dynamic quantity.

To start with, we consider, for even-even nuclei, only harmonic vibrations, around

the spherical shape.

The nuclear surface, as a function of spheric harmonies, is :

22

(VM1) R , R0 1�0

7

���

�

�

�

�

�, R0 R ,

where:

R0 is the radius of the spheric nucleus,

�

�are the coefficients describing the deformation, in general time depend.

Because the radius R is a real quantity.

(VM2) ,

��1

�

,

��

.

In the case of harmonic vibrations, the Hamiltonian operator is::

(VM3) H�

�

2 ���

� ,

�

� 2

���

� ,

�

� 2,

where � and C� are connected to the frequency � of the vibration by the

relation:

(VM4) �

C�

�

.

In the Hamiltonian (VM3)( harmonic oscillations) there are only quadratic terms

�

� 2. In order to take into account also unharmonic terms, different contributions

can be written :

(VM5) �1 � 2 � 3 0 ,

,

� 1

,

� 2 � 2 0

9.1. Appendix vibration model

For the sake of completeness we briefly discuss two different modes: the so called “

breathing mode” and the Goldhaber and Teller [4] mode.

The “ breathing mode” is a spherical density oscillation, with the quantum number

23

O+

, T 0 . Its transition density is not only concentrated on the nuclear surface, but

involves changes of the nuclear density over the whole volume. Its excitation energy

lies at:

(VM6) E0

+ 80 A�1 03 M e V .

This energy is a direct measure of nuclear incompressibility. In Pb208 the

experimental breathing mode lies at 13. 8 MeV, in good agreement with the

theoretical predictions.

In the model proposed by Goldhaber and Teller (VM4) the proton sphere vibrates the

neutron sphere.

The parameters � i are coupled at zero angular momentum and the numbers of

time derivative ,

� iis even, in order to take into account the rotational and the time

-reversal invariance.

24

Figure 6: The model of goldhaber and Teller

–

–

––

–

– ++

++

++

z

By a canonical transformation, we introduce the operators b�

�:

(VM7) �

� �

2C�

b�

�+1

�b�

��,

(VM8) ,

�

�i �

2�

b�

�+1

�b�

��.

The operators b� have the following commutation rules:

(VM9) b�

� , b� '

� ' +

─ �� ' � � '

(VM10) b�

� , b� '

� ' b�

�+ , b� '

� ' +

─ 0 .

As function of b� operators, the Hamiltonian H

� can be written:

(VM11) H�

���

��

2b�

�b�

�+b�

�+b�

�

���

�

�b�+

b� 1

2

if

(VM12) N�

�b�

�+b�

�, N

����

�

N�

�

,

we have

(VM13) H�

���

�

�N

�

� 1

2 �N

�

2 1

2.

As a consequence of (VM13), we obtain the energy spectrum of even-even

vibrational nucleus.

25

In particular, the electric quadrupole moment of 2+ state Q

2+ is zero.

The wave function of ground state is denoted by | 0> , with the condition :

(VM14) b,�

0 › 0 , 0 0 1 .

In order to obtain the wave function of excited states, we apply the creation operator

on the ground state:

(VM15) 0 › , b�

�+0 › , b

�

�+, b

�

�+b�

�'+

0 › , b�

�+b�

�'+

b�

�' '

+0 › .

The harmonic description of even-even nuclei isn't in good agreement with the

experimental data.

In order to obtain a good agreement with the experimental data, we introduce in the

Hamiltonian “unharmonic terms” such as :

(VM16) 30b�+ ,

where 30 31 40 are constant, fitted from the experimental data.

26

Figure 7: The theoretical spectrum of the even-even vibrational nucleus. Where I is the total

angular momentum

EI π

3 hω2

2 hω2

hω2

0

6+

4+

3+

2+

0+

4+

2+

2+

0+

0+

As a simple example of unharmonicity , we report the work of A.K.K. Kerman and

C. M. Shakin [Ref 7] in the case of the spectrum N62

i .

A cause of the unharmonic term b2

�+b2

�'+b2

� , the number of phonons is mixed.

The structure of the wave function is:

|01 › |00 ›

01 a 12 › 1 a2

22 ›

22 a 22 › 1 a212 ›

02 › 20 ›

|41 › 24 ›

10. Electric quadrupole transitions

If the nucleus is similar to a fluid of uniform charge, we have:

(VM18) me x , r r�

�

�, d r =

(VM19) e�

�, d r

0

R�- ,<�

r��2

d r =

27

Figure 8: The spectrum of 62Ni

2.336 4+

2.302 2+

2.048 0+

1.172 2+

0

1.991.961.75

1.00

0

1.921.901.73

1.00

00+

E(MeV) Iπ

Ex. K.S.

E1/E2+

(VM20) e

3d r

,

�, R

,�3, .

If

(VM21) R R0 1,�

,� ,�, + e

3 Z e

4 R0

3 ,

at first order we have:

(VM22) me ,3 Z e

4 R0

3R0

,�3d r

,

�, 1 |

,'�

',

'�

',

'�

' , ,

(VM23) me ,3 Z e

4R0

,

,�.

For quadrupole transition we have:

(VM24) me 2,3 Z e

4R0

2

2b�

+─

�bn� .

From the structure of me operator it is very easy to calculate its matrix elements.

All the diagonals matrix elements are zero and the selection rules are :

(VM25) I 2, n 1 .

11. Particle-vibration coupling

In 1967 B.R. Mottelson proposed a simple mechanism to handstand the

unharmonicity in the atomic nuclei the particle vibration coupling.

The Hamiltonian of the model is:

(VM26) H T H S H p H i n t ,

where

H S describes the vacillations of even-even nucleus,

28

H p is the Hamiltonian of single-particle

(VM27) H p T V r ,

where T is the kinetic energy of the single-particle and H i n t is the coupling between

the surface oscillations of the even-even nucleus and single particle.

If we take into account only linear term in H coup and quadrupole 2 and

octupole 3 oscillations for the even-even nucleus, we have:

(VM28) H coupl K r�2

3

�

�

� ,

where

K r Vordf

dr, f r

1

1 er R .

Eigenvalues and eigenvectors of the coupled system are obtained by the standard

diagonalization procedure on the basis n1

2e , N2 R2 , N3 R3 R ; › , where j and R

are the angular momentum of the single-particle and of the states (N2,N3) of the

even-even nucleus. I is the total angular momentum of the add nucleus.

The matrix elements of H coupl are:

(VM29) j N2 R2 , N3 R3 R ; H coupl j ' N ' 2R ' 2, N ' 3 R ' 3 R' ; =

=�2

3

n l j k r n ' l ' j ' 1R2�R'3���J�10 2

where

2 if 3

3 if 2

29

In this model there are different parameters:

1) j :single particle energies

2) , :energies of the oscillations of even-even nucleus

In general also the matrix elements n l j k r n ' l ' j ' are fitted on the

experimental data.

In order to avoid to use free parameters, the following procedure has been adopted:

1) the single particle j are obtained by direct reactions,

2) the energies � are extracted by the even-even nucleus,

3) the radial wave function R nlj is obtained by the potential

(VM30) V r V 0 f r V S.O %

2 1

r

d f

d rσ~

e~

V coupl .

The parameters of the potential V (r)are taken from K. Bleuler et al [ Ref 8 ].

As example we apply the model to the nuclei:

(VM31) N28 i p Co , S50 n p I n , P82 b p Tl .

This choice is followed for different reasons:

1) the even-even nuclei have protons closed shells (28-50-82) and a few neutrons in

the single particles orbits. The simplest microscopic description of states 2+ ,4+

is based on quasi-neutron description [ Ref 9 ]. In the coupling of these states with a

proton , the antisymmetrization procedure is not very important. We assume also that

the role of the partial level occupation is not basic at all.

To start with, we consider the case of C57 o and I115 n .

1) N58 i p C57 o

30

The collective low-lying states of C57 o are: J%

3 2,5 2,7 , 2,9 2,1 1 2 . The

splitting of the multiplet (-0.7 MeV) is not small at all and then the particle-vibration

coupling is not weak.

The gravity centre of the multiplet is 1.65 MeV and E 2+

58 =1.45 MeV. This imply

a mixing of different phonons number. In ref [ ref 10 ] the model space is

1 f 7 2 , 2 S 1 2, and 2 d 3 2 (for proton hole) and tree quadrupole fonons and one

octupole.

The single particle energies are taken from A.G. Blair and A.D. Armstrong [ Ref 11 ]

and 2 1.45 M e V , 3 4.40 M e V See table

Nucleus Core 2 2 3 3 7 2� 3 2� 1 2�

Co57 N i58 1,45 0,19 4,4 0,13 0 3,31 2,36

Co59 N i60 1,.33. 0,21 4 0,11 0 2,98 2,12

�

�

2 1

j : A.G. Blair and D.D. Armstrong: t , He4 Reaction on the Even Ni Isotopes,

Phys. Rev.151 930 (1966).

We compare the experimental spectrum of Co57 with the theoretical one.

Nucleus if 7 0 2 K r if 7 02 if 7 0 2 K r ld 30 2 if 7 0 2 K r 2s1 02

Co57

Co59

58,40

58,80

48,27

48,23

-49,96

-50,25

Nucleus ld 3 02 K r ld 30 2 ld 3 02 K r 2s10 2 2s1 0 2 K r 2s1 02

Co57

Co59

41,63

41,30

-40,09

-40,05

44,24

44,38

31

In table we compare the theoretical transitions probability with the experimental

ones.

Single- hole strength

Co57

Co59

Ref�1� Present calc.

7 2─

I%

0,69 0,52

Ref�1� Present calc.

0,83 0,43

(1) L.Satpathy and S.C. Gujrathi, Nucl. Phys., 110A, 400 (1968).

the experimental values are obtained

K.S. Burton and L.C. Mc Intyre , from the reaction Fe54

, ,57

Co . As can be

seen, for the transition B E2 ;11 2─

9 the agreement between the experimental and

the theoretical one is not good at all.

2)We discuss here the case Sn116

─ I n115 .the comparison between the theoretical

calculation and the experimental data is quite good.

As it is well known, the collective properties of 2+ states of even-even Sn isotopes

are a smooth function of A.

As a consequence, also the collective states of odd In isotopes are quasi similar

( I n113 and I n117 ).

The model space for I n115 is to one hole state 1g9 0 2 ,2 1 02 , 2 3 02 and 1 f 50 2

and

= 2 and = 3 for the even-even nuclei. [ Ref 10 ].

32

Because the experimental data are not able to determine in an unique way the single

particle levels, we have adopted a fitting procedure of spacing j j ─ 1 g 90 2

j = 2B 20 3,2

B3 02 ,1 f 5 0 2 .

The comparison of the experimental spectrum of I n115 with the theoretical one is

quite good.

Four states, with I%=5 2+ , 7 2+ ,9 2+ ,11 2+ ,13 2+ , strangely excited an elastic

scattering of deutons and coulomb excitation, are interpreted as the member of a

quinted, obtained by the coupling of the proton hole 1g9 0 2 with the quadrupole

oscillations of Sn116 .

33

In fig. 9 it is very easy to note that the member of the multiplet are well reproduced

by the model.

Conclusions

As discussed in the previous sections, even if the particle vibration coupling is a very

simple model, it is also able to explain the energy spectrum, the transition

probabilities, the spectroscopic functions and so on.

But there are some limitations and difficulties in the use of this model.

34

Figure 9: Experimental and theoretical values of B(E2;I->I').

experimental values are from F.S. Dietrich et al. obtained by

anelastic scattering of deuterons.

0.0

0.5

1.0

1.5

2.0

E(MeV)

Exp. Calc.

9/2+

9/2+

13/2+

11/2+

5/2+

7/2+

1/2+

3/2+

9/2+

3/2–

1/2–

9/2+

3/2–

11/2+

5/2+13/2+

5/2–

7/2–9/2+7/2+3/2–5/2–

1/2–

7/2+

1) the first one is the determination of single particle energy j and the

deformation parameters � .

2) the second one is the coupling of the even-even nucleus | with a hole A-2

nucleus .

If we take into account both the contributions we have:

|

Very often the second contribution is very important.

References

[1] R. Nataf: “Les modelles en spectroscopie nucleaire” Parigi (1965).

[2] V.R. Manfredi: “modelli nucleari fenomenologici” Istituto della enciclopedia

italiana, 6, 570, A. Covello “Modelli nucleari microscopici” Istituto della

enciclopedia italiana, 6, 56.

[3] A. Bohr and B:R: Motelson: “Nuclear structure” Bejamin, New York, volume 1,

1969, volume 2, 1965.

[4] A. Bohr: “rotational states of atomic nuclei” Kohpenagen, (1954).

35

Figure 10: Coupling between different exitations

[5] Mesons in nuclei, ed by M. Roher and D. Wilkinson, 1,2,3,North Holland,

Amsterdam, New York, 1979.

[6] N. Bhor: “nature 137,144, (1936)

[7] A.K. Kerman and C. M. Shakin: “Physical letters” 1, 1951, (1961).

[8] K.Bleuler etal: “Nuovo Cimento” 52B, 45 (1967).

[9] E. U. Baranger: “Advanced in nuclear physics” 41, (1971).

[10] A. Covello, V.R. Manfredi and N. Aziz: “Nuclear Physics” A201, 215 (1973).

[11] A.G.Blair and A. .D. Armostrong: “Phys Rev” 151, 930 (1966)

The author is very indebted to the professor R.A. Ricci for his kind help in writing

this article.

36