ISSN 1062�8738, Bulletin of the Russian Academy of Sciences. Physics, 2012, Vol. 76, No. 10, pp. 1066–1069. © Allerton Press, Inc., 2012.Original Russian Text © N.A. Burkova, M.A. Zhusupov, R.S. Kabatayeva, 2012, published in Izvestiya Rossiiskoi Akademii Nauk. Seriya Fizicheskaya, 2012, Vol. 76, No. 10,pp. 1191–1194.

1066

INTRODUCTION

Studies of radiative capture of α�particles by 6Liand 7Li isotopes are of considerable importance for

theory and practical applications. The reactions have a resonance structure and are accompa�nied by the emission of monochromatic γ quanta withcertain energies, that is, are exothermic with Q = 4.459and 8.665 MeV, respectively, in the decay of a series of

quasi�stationary and isotopes states, whichis very convenient for nuclear physical (namely,γ�radial) diagnostics of the ionic hot plasma compo�nent [1]. Studies of the spectrum of γ quanta require atheoretical investigation of reaction cross sections,therefore, knowledge of interaction potentials in theα

6Li and α7Li systems in the subbarrier energy region.Note that nuclei�targets exhibit well�defined αd and αtcluster structures, as follows from anomalous smallnessof the disintegration thresholds of these nuclei in thecorresponding channels ( = 1.475 MeV, =2.467 MeV), whereas the next disintegration thresh�

olds into the and channels are 5.39 and7.25 MeV, respectively [2]. This makes it necessary totake into account the predominant cluster configura�tion in analyzing the mechanisms of α particle scatter�ing from lithium isotopes.

1. FOLDING POTENTIAL CALCULATIONS IN THE CLUSTER MODEL

1.1. The Selection of Jacobi Coordinates

Since we treat the 6Li nucleus target as an aggregateof α and d clusters, the selection of the relative Jacobi

coordinates for the α6Li system is as follows:

where is the coordinate of the relative

motion of clusters α and d in the 6Li nucleus and isthe coordinate of the relative motion of an incidentα particle and the center of mass of the 6Li nucleus tar�get. Recall that the 6Li nucleus has the following totalspin, parity, and isospin quantum numbers in the

ground state [2]:

1.2. The General Form of Folding Potential

The general form of the real part of the opticalpotential for α6Li interaction is calculated as the meanvalue with respect to the product of ground state wavefunctions of the nucleus target and incident particlefrom the sum of αα and αd potentials,

(1)

where the potential is the sum of partialαα and αd potentials,

(2)

Note that the orbital momentum of the relativemotion of clusters α and d in the 6Li nucleus is l = 0.

1.3. Wave Functions

Let us write the wave function of the 6Li nucleustarget in the ground state,

(3)

( )6,7 10,11Li B,α γ

10B* 11B*

dαε tαε

5Li n−

6Li n−

{ }d 6, ,R Rα α

� �

dRα

�

6Rα

�

( ) ( )πLi6. . , 1 ,0 .g s J T +

=

( )

( ) ( ) ( )

6

6 6

Li

Li Lid d d

6

, ,j jjm jm

R

R V R R R

αα

α α α αα α α= Ψ Ψ Ψ Ψ

�

� � � �

V

( )d,V R Rα αα

� �

( ) ( ) ( )d d d, .V R R V R V Rα αα α α αα αα

= +

� � � �

( ) ( ) ( )( )

( )( )( )

( )( ) ( )

( )( )

( )( )

( )( )

σ τ σ,τ σ τ σ,τ6

d

d d d

d

Lid d d d d

2

1 000 00 00 00,00 101 1 ,101 001 1 ,4

j

j s s

s

Rm s j jm m

m j

R m m C e α−α

α

Ψ = Φ α χ α χ α χ α Φ χ χ χπ∑ ∑

�

�

A Potential Description of α�Particle Elastic Scattering by the 6Li and 7Li Nuclei

N. A. Burkova, M. A. Zhusupov, and R. S. KabatayevaAl�Farabi Kazakh National University, al�Farabi avenue, 71, Almaty, 050012 Kazakhstan

e�mail: raushan19860722@mail.ru

Abstract—The results of analytic and numerical calculations of the real part of the α6Li interaction potentialin the cluster folding model are reported. Elastic scattering phases in the α + 6, 7Li systems are calculated andconstructed, a classification of multipole transitions for back photodisintegration reactions is given.

DOI: 10.3103/S1062873812040107

BULLETIN OF THE RUSSIAN ACADEMY OF SCIENCES. PHYSICS Vol. 76 No. 10 2012

A POTENTIAL DESCRIPTION OF α�PARTICLE ELASTIC SCATTERING 1067

where the expansion coefficients Cj and αj are takenfrom [3].

1.4. The Selection of Partial Interaction Potentials

Partial potentials are taken in the Gauss form. Thepotential for αd interactions is written as

(4)

where potential parameters are taken from [3] andhave the values V1 = –75.8469155 MeV and α1 =0.2 fm–2. The potential for αα interactions has theform (the so�called Buck potential) [4]

(5)

where potential parameters, V2 = –129 MeV and α2 =0.225 fm–2, were taken from [5].

( ) d

d dd d

21

1 63 3, ,2 2

RV R V e R R Rα−α

α α α α α

= = +

�

� � � �

( ) d

22

2 61, ,3

RV R V e R R Rαα−α

αα αα αα α α

= = +

�

� � � �

After certain transformations, we obtain the fol�lowing analytic equation for the real part of the foldingpotential for α6Li interactions:

(6)

where (αj + αm) ≡ f,

1.5. Numerical Calculation Results

The real parts of the optical interaction potentialsin the α6Li system calculated in this work by the con�volution method are compared in Fig. 1 with theresults obtained by other authors. In [6], a three�body6Li{αnp} nucleus model was considered. In [7], thepotential was obtained by the inversion method. Thedifferential cross sections of α6Li scattering calculatedwith the potentials obtained in this work and [7] areshown in Fig. 2. The theoretical cross sections havecomparable values for both models and substantiallydepend on energy. Moreover, at Ec.m. = 7.8 and11.1 MeV, correlation is observed for the positions ofmaxima in the angular distributions of α particles forboth models, and, at Ec.m. = 15 MeV, angular distribu�tion extrema calculated for different models are inantiphase at angles of θ = 68° and 98°. With the fold�ing model potential, theoretical cross sectionsincrease at large α�particle emission angles at all theenergies considered.

( )6d d' 'Li

2 2

1 6 1 6

6

1 23 2 3 2

11

4

,

s j s jm m m m

q R p Rj m

jm

R

V VC C e e

a bα α

αα

− −

π= δ δ

⎧ ⎫× +⎨ ⎬

⎩ ⎭∑

� �

�

V

1 19 ,4

a f≡ + α 2 19 ,2

a ≡ α 3 19 ,4

a ≡ α

=22

1 31

,4

aq a

a− 1 2

1 ,9

b f≡ + α 2 22 ,3

b ≡ α 3 2,b ≡ α

22

1 31

.4

bp b

b≡ −

0

–100

–120

6420

–80

–60

–40

–20

V, MeV

8R6α, fm

Fig. 1. Real parts of α6Li interaction potentials; the solidcurve corresponds to the results obtained in this work, thedashed line, to the results from [6], the dash�and�dot line,to the results from [7], and the dotted line, to the resultsobtained in [8].

102

180600

dσ/dΩ, mb sr–1

θ, deg120

104(a)

102

180600

dσ/dΩ, mb sr–1

θ, deg120

104(b)

102

180600

dσ/dΩ, mb sr–1

θ, deg120

104

(c)

Fig. 2. Differential α6Li scattering cross sections for the folding potential calculated in this work (solid curve) and inversionpotential from [7] (dashed curve) at Ec.m.(

6Li) = (a) 7.8, (b) 11.1, and (c) 15 MeV.

1068

BULLETIN OF THE RUSSIAN ACADEMY OF SCIENCES. PHYSICS Vol. 76 No. 10 2012

BURKOVA et al.

2. A CLASSIFICATION OF MULTIPOLE TRANSITIONS IN PHOTOPROCESSES

ON 10B AND 11B NUCLEI WITH EMISSION OF α�PARTICLES

Radiative capture and photodisintegration reac�tions are related with each other by the detailed equi�librium theorem, which allows these processes to beclassified using a unique formalism. Let us considerthe photodisintegration reaction reverse to α6Li scat�tering,

(7)

where are the spin, parity, and isospin of the cor�responding particles. Since the isospins of theentrance and exit channels are zero, isoscalar electro�

magnetic multipoles are only nonzero. The

nucleus in the ground state has the shell config�

uration. The nucleus has the config�uration. The number of quanta along the relativemotion coordinate of the 6Li and α nuclei is 4, and thetransferred orbital momentum takes on the values L =0, 2, 4. Clearly, only the L = 2 and 4 values give thetotal momentum J = 3, which corresponds to theground state of the 10B nucleus. The wave function of

the 10B nucleus, , in the 6Li + α fragmenta�tion channel can be represented as the superpositionof the D and G components,

(8)

At comparatively low energies considered in thiswork, the most probable electromagnetic transition isthe electric dipole transition of the isovector type.Transitions with the highest multiplicity are an addi�tion to the Е1 transition. However, if the dipole transi�tion is forbidden, in the first place because of isospinselection rules, the Е2 quadrupole transition becomesthe main transition. Reaction (7) relates exactly tosuch processes. Earlier, we thoroughly studied the

process (see review [9], the retrospective andmodern state of the problem), in which the isovectorЕ1 transition is also subject to forbidding by virtue ofisotopic spin selection rules. A variant of the construc�tion of the isoscalar dipole operator was suggested.This variant was based on the noncoincidence of thecenter of mass of the composite system and the centerof mass of charge distribution.

To preliminarily estimate the intensity of the Е1dipole isoscalar transition, we use the following equa�tions. If nucleus A consists of clusters a and b: A = {ab},the dipole moment of this nucleus is

(9)

( ) ( ) ( )10 6B Li3 ,0 1 ,0 0 ,0 ,+ + +

+ γ → + α

,J Tπ

10 B4 6s p

{ }6Li d= α

4 2s p

10 B3 0+

Ψ

10 B3 0 .D Ga D a G+

Ψ = +

6Li , d( )γ α

( )ˆ , ,Z

i cm a b

i

d e r R d d d dρ= − = + +∑� � � � � �

�

where is the relative coordinate of particles

a and b and is the dipole moment corresponding tothis coordinate. Then,

(10)

The intensity of nucleus fragmenta�tion is estimated as follows:

(11)

For the photodisintegrationchannel, a similar operator has the form

(12)

A comparison of Eqs. (11) and (12) leads us to con�clude that the contributions of isoscalar Е1 transitionsto the processes under consideration are of the saneorder of magnitude. The special features of these reac�tions can manifest themselves in the dynamic part ofthe corresponding matrix elements.

Next, let us consider possible continuous spectrumstates in terms of the predominant transitions deter�mined by us, (a) , (b) ,(c) , and (d) .

The photodisintegration reaction 11B(3/2–, 1/2) + γ →

is considered similarly. Here,

the isospins in the entrance and exit channels are non�

zero. The 11B nucleus has the configuration in

the ground state. The 7Li = {α]t} nucleus has the configuration. In this reaction too, the number ofquanta transferred along the coordinate of the relativemotion of the 7Li and α nuclei is 4, and the transferredorbital momentum takes on the values L = 0, 2, 4.According to the classification of the shell structure ofthe 11B nucleus, the D component predominates inthis channel. It follows that, for the Е1 and Е2 transi�tions, the (a) and (b) classification is valid.

3. A PHASE ANALYSIS OF THE ELASTIC SCATTERING OF α PARTICLES FROM THE 6Li AND 7Li NUCLEI

To preliminarily estimate the special features of theenergy dependence of direct photoemission of α par�ticles in the processes under consideration, informa�tion about the phases of elastic scattering from the 6Liand 7Li nuclei is necessary. There is no experimentalphase analysis data for these channels. We thereforeused the parameters of optical potentials obtained forthe folding model (convolution potentials) [6, 10].

ρ a br r= −

� � �

dρ

�

.a b a b

a b a b

m m z zd e

m m m mρ

⎡ ⎤= ρ −⎢ ⎥+ ⎣ ⎦

�

�

6Li d→ α +

d d

d d

d

d

324.256 10 .

m m z zd e

m m m m

m me e

m m

α α

ρ

α α

−α

α

⎡ ⎤= ρ −⎢ ⎥+ ⎣ ⎦

−= ρ ≈ × ρ+

10 6B Li+ γ → + α

6

6

Li

Li

33 22,24 10 .

m md e e

m mα −

ρ

α

−= ρ ≈ − × ρ

+

1ED p f⎯⎯⎯→ +

1EG f h⎯⎯⎯→ +

2ED s d g⎯⎯⎯→ + +

2EG d g i⎯⎯⎯→ + +

( ) ( )7Li 3 2 ,1 2 0 ,0− +

+ α

4 7s p4 3s p

BULLETIN OF THE RUSSIAN ACADEMY OF SCIENCES. PHYSICS Vol. 76 No. 10 2012

A POTENTIAL DESCRIPTION OF α�PARTICLE ELASTIC SCATTERING 1069

α�Particle scattering phases from the 6Li and7Li nuclei were calculated for the processes under con�sideration over the energy range 0–20 MeV withouttaking into account spin�orbit coupling splitting for apotential with the central part in the Woods–Saxonform,

We used the following parameters taken from [6]for calculations of α6Li scattering phases: R0= 2.1 fm,V0 = 116.86 MeV, and a = 0.815 fm. Figure 3 showsthat f phases experience a sharp change in passingthrough π/2 at the energy E = 2.34–2.35 MeV; that is,there is a well�defined resonance in the α6Li system atthis energy. It follows that the E1 amplitude can appearnontrivially just because the isoscalar E1 transitiongoverned by selection rules results in a resonating statein the continuous spectrum. Note that other phaseshave smooth energy dependences and, according toour estimates, do not contribute to resonance struc�ture formation.

For α7Li scattering, we used the following potentialparameters taken from [10]: R0 = 1.983 fm, V0 =119.205 MeV, and a = 0.878 fm. It follows from Fig. 4that the f phase also resonates at E = 1.50–1.51 MeV.The continuous spectrum can therefore appear as anintense resonance in the f scattering wave in the case ofisoscalar E1 transitions in processes of the type

CONCLUSIONS

To summarize, the α6Li potential obtained usingthe folding model quite satisfactorily describes elastic

scattering in this channel. The folding potential for thetwo�body nucleus model 6Li = {αd} on average givesthe result of the same order of magnitude for differen�tial cross sections as the potential obtained by theinversion method [7]. For the folding potential, we,however, observe the emission of particles at largerangles. The results obtained for elastic scatteringphases substantiate the resonance structure of reac�tions with the capture of α particles by lithium iso�topes.

REFERENCES

1. Cecil, F.E., Zweben, S.J., and Medley, S.S.,Nucl. Instrum. Methods A, 1986, vol. 245, p. 547.

2. Tilley, D.R., et al., Nucl. Phys. A, 2002, vol. 708, p. 18.

3. Dubovichenko, S.B., Phys. At. Nucl., 2010, vol. 73,no. 9, p. 1526.

4. Buck, B., Friedrich, H., and Wheatley, C., Nucl. Phys.A, 1975, vol. 275, p. 246.

5. Voronchev, V.T., et al., Yad. Fiz., 1994, vol. 57, p. 1964.

6. Kamal, M., et al., J. Phys. G: Nucl. Part. Phys., 1992,vol. 18, p. 379.

7. Mackintosh, R.S., Cooper, S.G., and Kukulin, V.I.,Nucl. Phys. A, 1999, vol. 645, p. 399.

8. Bacheller, D., et al., Nucl. Phys. A, 1972, vol. 195,p. 361.

9. Burkova, N.A., Zhaksybekova, K.A., and Zhusupov, M.A.,Fiz. Elem. Chastits At. Yadra, 2005, vol. 36, p. 801.

10. Sakhiev, S.K., Vestn.ENU im. L.N. Gumileva, 2005,vol. 44, p. 117.

( )( )( )0 01 exp .cV V r R a= + −

11,10 B1 .ED p fΨ ⎯⎯⎯→ +

–30840

60

120

180

240

20Ec.m., MeV

0

δ, deg

12 16

(a)

13

2

840

60

120

180

20Ec.m., MeV

δ, deg

12 16

(b)

1

2

Fig. 3. Elastic α6Li scattering phases in the folding model:(a) even phases, (1) s�phase, (2) d�phase, and (3) g�phaseand (b) odd phases, (1) p�phase and (2) f�phase.

–30840

30

90

150

210

20Ec.m., MeV

0

δ, deg

12 16

(a)

13

2

840

60

120

180

20Ec.m., MeV

δ, deg

12 16

(b)

1

2

270

Fig. 4. Elastic α6Li scattering phases in the folding model;curve denotations are the same as in Fig. 3.