Pertanika 4(2), 170-184(1981)

Shell-Model Calculations on 59Ni and 59Cu

M. YUSOF SULAIMAN and AMINUDDIN MUSTAFFAJabatan Fizik, Fakulti Sains and Pengajian Alam Sekitar, Universiti Pertanian Malaysia,

Serdang, Selangor, Malaysia.

Key words: State operator; matrix elements; binding energies.

RINGKASAN

Kertas ini melaporkan pengiraan model-petala keatas nukleus 59Nidan s9 Cu. Nukleus-nukleus inidianggap sebagai sistem tiga-zarah di luar teras tertutup se>Ni. Ketiga-tiga zarah ini bebas menduduki obit-obit 2p3ft, Ifs/i dan 2p\^. Untuk saling tindakan di antara dua zarah, kami telah menggunakan salingtindakan delta-permukaan terubahsuai. Elemen matriks Hamiltonian telah dikira dan kemudiannya di-pepejurukan dengan menggunakan program komputer. Dari kiraan ini, tenaga ikatan kepada tiga keadaanterendah bersama tiga keadaan lain yang mempunyai spin dan isospin yang sama dengan keadaan terendahitu telah diperolehi. Juga dikira adalah fungsigelombang keadaan-keadaan ini.

SUMMARY

This paper reports on a shell-model calculation of 59Ni and 59 Cu. The nuclei are considered as three-particle system outside a closed-core of s6Ni. The three particles are free to occupy the three orbits 2p$fi,if5/2 and 2p\/2- For the two-body force, we have used the modified surface-delta interaction. Matrixelements of the Hamiltonian were computed and later diagonalized with the aid of a computer program.From these calculations, the binding energies of the three lowest states and those of the next higher stateswith the same spins and isospins were obtained. Also calculated, were the wavefunctions of these states.

INTRODUCTION

When the shell-model was first proposed in 1949 (Mayer, 1949), because of its simplicity, it wasthen not quite sure whether it could be extended to account for nuclear properties other than those that wereoriginally intended i.e. magic numbers, angular momenta and parities of nuclei near closed-shells. However,with the many calculations that have been performed since then, it is now clear that nuclear shell model,not only could be used to explain other features of near magic nuclei but could also accountmany properties of nuclei whose nucleon numbers are far from close. The success of the shell-mode lieslargely on the model interaction used and its effectiveness in correlating the empirical two-body matrixelements. Also important, is the model space considered as previous calculations have shown that a limitedconfiguration space did not normally lead to good agreement with experiments.

In this paper, we will report on the model calculations of S9Ni and 59Cu using a modified surface-delta interaction (Plastino et ah (1966)). The calculations were performed for the active orbits lfs/2, 2p3/2

and 2pi/2 with a 56Ni closed core. We have calculated the binding energies of the three lowest states andthose of the next higher states with the same Jn, T values and the corresponding wavefunctions. The under-lying theory of these calculations will be given in the next section.

2. Theory

The essential step in any model calculation is the evaluation of the matrix elements of the physicaloperators of interest, such as the nuclear Hamiltonian, the residual two-body interaction etc. in the modelspace used. Before we give the general expression for the many particle multishell (multiorbit) matrix

Index to authors' names: MY. Sulaiman and A. Mustaffa.

170

M.Y. SULAIMAN AND A. MUSTAFFA

elements, we will first introduce state operators and general operators describing physical processes in thesecond quantization formalism. In the prescription for simplifying the many-particle multishell matrixelements, we will lean heavily on the approach proposed by French et al (1969).

2.1 Many-particle single-shell state operator

Consider a normalized and antisymmetrized two-particle state given by

| j a j b ; J M , T T Z > (2.1.1)

where

J - Ja + Jb is t n e t o t a l s P i n o f Particles a and b,T = ta 4- tb is the total isospin of particles a and b,M = ma + mb is the projection of J on the z-axis,Tz = taz + tbz is the projection of T on the z-axis.

This state can be created by operating a vacuum state with a state operator defined as,

1JT

MT.OaJb) = ~ 1

V +

1

V

Ja i

6 a b ma mb

*az tbz

Jb iMTZ

ma j b mb | JM . (2.L2)

+a JJb m b

where a+ is the single-particle creation operator, and < j a ma j b mb | JM >, <4-t a z4tbz I TTZ> are vector-coupling or Clebsch-Gordan coefficients for spin and isospin respectively.

Thus,

| j a j b ; J M , T T z > = Z J J m ( j a J b ) l > (2.1.3)MX 7

where | > represents the vacuum state.

For the construction of a state with more than two-particles, one must add one particle at a timewith the aid of the vector coupling coefficients and normalize the wavefunction afterwards. For a generaln-particle single-shell state operator, we then have,

j n ; a JM, T T Z > = z £ Gn) (2.1.4)

Here, a denotes any set of additional quantum numbers eg. seniority, reduced isospin etc, that may benecessary for a comple description of the state j n .

171

SHELL-MODEL CALCULATIONS ON 59Ni AND S9Cu

Likewise, the adjoint two-particle state operator is defined as,

MT7

JT

(2.1.5)

where a is the single-particle adjoint annihilation operator, and is related to the ordinary single-particleannihilation operator a by

a and a+ obey the well-known anticommutation relation,

(2.1.6)

= 5 ,

However, only a+ and a are spherical tensor operators and from (2.1.6), a+ and a are related according to

- ( U^

Taking into account the definition of the adjoint two-particle state operator (equation (2.1.5)) and thetransformation property of a and a+ (equation (2.1.7)), the adjoint n-particle single-shell state operatoris seen to obey the following property,

(-1 ) J + M + T + T Z < jn ; f l J _ M T _ T z , =' JT

(2.1.8)

2.2 Many-particle multishell state operator

From the single-shell state operators we will now discuss the construction of multishell state opera-tors. Suppose we wish to set up a shell-model calculation with the active nucleons occupying the orbitsPi,P2, • • • -, P -Let us, for each orbitp^take together all single-particle creation operators to form onenormalized state operator Z^i (p.ni). For the multishell state operator, a particular coupling scheme andphase will be defined in the standard form by,

where

i = 2N

i - ]n

i

i - 1

(2.2.1)

The set of quantum numbers specified by n = [ n i , n 2 nfc] denotes the number of particles in each ofthe orbit p l 5 p 2 , • • • >Pk- T n e s e t 7 = t7i» 72 > • • - • > 7 k l represents the spin and isospin of each single-shellcomponent. The diagram in equation (2.2.1) indicates how the 7. are coupled to intermediate spins andisospins denoted by the set F = [Fu F 2 , . . . , F k ] with Fj = 7 ^

172

M.Y. SULAIMAN AND A. MUSTAFFA

The set x = [xx, x2 , . . . x ] representing all further labels (eg, radial quantum numbers, seniority, reducedisospin etc.), that may be necessary to specify the multishell, many-particle state operator uniquely. It is under-

y y \r i y

stood that the state operators act to the right in the order Z k , Z , . . . Z 1.

Correspondingly, the adjoint many-particle multishell state operator Z is given by the coupling

scheme

(2.2.2)

The absence of a phase factor in the adjoint equation, is consistent with our definition of Z k

in equation (2.2.1).

2.3 General physical operator

A general physical operator O k can be expressed in a standard-form expansion of operators insecond quantization. This standard-form expansion is given by

q, y, w, SI K-(q, y, w, $2) F"« (q, y, w, 12) (2.3.1)

with the definition

(q,y,w,n,) =

The operator F k consists of coupled products of creation and annihilation operators and nothingelse. It thus depends only on the number of particles that are to be created and/or annihilated in each orbitand on the way the spins and isospins of these particles are coupled. The physical aspects of the operator

O k are combined in the expansion coefficients K. The operator F k , defined in the space spanned by

the orbits p 1 } p 2 , . . , ., p is decomposed into coupled products of single-shell operators fWi - fWl (q.).

Each fwi (q.) consists of operators a* and a for one and the same orbit p. only.The rank w. denotes

the spin and isospin of the coupled product of the operators a and a . The w. are coupled subsequently

to intermediate values £2.. The symbols q. and y. represent additional labels that may be necessary to

specify the coupled products of the operators a and a . The arguments in the expansion coefficients

K (q,y,w,fi) denote sets of labels, i.e. q - [qi,q2, . . ., qfc], y = (y i ,y 2 ) . . ., y ] , w= [ w b w2, . . ., w ] ,£2= [fii,$22 , . • MJ withS2! - w i

173

SHELL-MODEL CALCULATIONS ON 59Ni AND 59Cu

2.4 Many-particle multishell matrix elements

As has been mentioned earlier, the important step in a model calculation is the evaluation of many-particle mutishell matrix elements of the required operators.

Let us consider a general k-shell matrix element represented by

K K

2 K

N n v \ir 'i"1 { Zr^(n,x,7,r) I On« K-l )i = 2 *'** i Zr'*< (n\x\7 ' ,r') > (2.4.1)

where the states are represented by their state operators only and the symbol for the vacuum state l> issuppresed. The geometrical dependence of the matrix elements on the projection quantum numbers can befactored out with the aid of the Wigner-Eckart theorem.

Let us denote a matrix element reduced in space and isospace by,

>= K(q,y,w,Jl)(2.4.2)

The approach for the further evaluation of the reduced matrix element (2.4.2) is that successivelythe dependence on each subshell p. is peeled off. This requires that the operators a+ and a^ present in

each of the three parts of the reduced matrix elements be separated off and each subshell taken together.This necessitates the recoupling of angular momenta and isospins, which can be done with the use of

Racah algebra. Following that the necessary reshuffling of the operators a+ and a can be performedwhen their anticommutation relations are taken into account. * *

Using the above prescription, the reduced many-particle multishell matrix elements can be written in termsof the single-shell matrix elements, thus

<ZF* (n,x,7,r) P/< (n\x')7\r)>

q,y,w,£

| | | f * I ( q ) I H n ' x ' 7 ' ' > 7 r(qi)IHn'1x'17''1>7r V ( 2 r +y% i = 2

Ti r l

174

M.Y. SULAIMAN AND A. MUSTAFFA

w h e r e \ i a r e t h e 9-j s y m b o l a n d < n . x. 7 . Ill f * i III n. x.' 7 > a r e t h e s i n g l e - s h e l l m a t r i x - e l e m e n t s .

2.5 Single-shell matrix elements

For the present calculation, we will be interested only in one and two-body operators. We willtherefore be dealing with products of one creation or two annihilation operators acting in one or moreorbits. The expressions for specific operators in term of the creation and annihilation operators will begiven in section 3.

According to equation 2.4.3, the matrix elements of the one-or two-body operators acting in oneor more orbits can be evaluated from single-shell matrix elements. Any possible single-shell one-or two-body

constructed trom operators a anc1

q given in Table

operator fw (q) constructed from operators a+ and 'a' must belong to one of the nine categories labelled by

The calculation of the single-shell matrix element is rather lengthy. Therefore, if a computer is usedin the model calculation, the single-shell matrix element is usually precalculated and saved. Tables ofcalculated single-shell matrix elements are useful each time many-body, multishell matrix elements mustbe computed from equation 2.4.3.

3. The nuclear Hamiltonian

The nuclear Hamiltonian is given as

A AH = E0 + 2 T(k)+ 2 W(k,l) (3 01)

v — 1 v < 1

where Eo represents the binding energy of the core |0> i.e Eo = <0|H|0>. T(k) is the kinetic energy termand W(k, 1) is the two-body interaction.

For a shell model calculation, a shell model potential 2 U(k) is introduced and the Hamiltonian ink

(3.0.1) is rewritten,

A (H = E 0 + S \T(k) + U(k)

k = l ] [ I i(3.0.2)

= E ^o)

The basis wavef unctions are eigenfunctions of the unpertubed Hamiltonian H ( o ) and the residualinteraction is diagonalized subsequently. In second quantization, (3.0.2) is written as,

*****

where the Greek indices refer to the particular shell-model basis states that diagonalize the unpertubedHamiltonian. Since H (o) and V are scalars, i.e. they do not depend on orientation in the coordinate space,evaluation of equation (2.4.3) requires Hk = 0.

175

SHELL-MODEL CALCULATIONS ON 59Ni AND 59Cu

TABLE 1Single-shell one-and two-body operators and matrix elements

Category Operators <pT| | | f W (q) | | | p n ' V*>

q = 1 1 V 2 r + 1 5pp' 5nn '

q = 2 a* V n(2r+ I) < pn T1 pn~l V'> 8 ' .

-Vn(n- l ) (2r+ 1) £ U (r'p Tp ;Aw>A

< Pn r i ] p

< Pn + 2 r ' i ] P

n+16n, n + 2

p n r ' i

( - l ) ^ A - r ' / ( n + l ) n , n -£ j 2 A + 1

U ( r 'p [V ; Aw) < pn r l] pn~2 &(p2v)>

2A+ 1

n ( n - l ) \ / ( 2 r + l ) ( 2 r f +

< p n r | ] pn"2A(p2e)>

<Pnr ' i} p n - 2 A^ 2 , )>

Note: 1. U(abcd; eO is the U-coefficient which vanishes unless A (abe), A (edc), A (bdO and A (afc), where A (pqr) denotes that thevectors p, q, r can form a triangle.

2. < p " F i ] p e > denotes coefficients of fractional parentage

3. <p n r l ] p n - 2 n p 2 5 ) > ^ < p n r i ! p n ' 1 e x p n " 1 e l ] f>n~2v>€

x U(i>pFp ;e6j

denotes double-parentage coefficients.

176

MY. SULAIMAN AND A. MUSTAFFA

We will now look at the contribution of H ( o ) and V separately.

3.1 Single-particle energy contribution

The single-particle contributions to the Hamiltonian for k-orbit configuration with nf particles inorbit Pj coupled to spin and isospin yi and total spin and isospin Fk can be shown to be (French et al1969),'

<z r * II (0) p

i= 1*i*Pi (3.1.1)

where e denote the single particle energies.

3.2 T}\e residual two-body interaction

The two-body term in the Hamiltonian can be written as

- 1Pi <P2

Ps

- 2 <PiP2\\\ v ] i l p 3 p 4 > r

Pi <Pi

P3 < P4

r

Pi Pi

The expansion (3.2.1) does not yet give the correct expansions K, since for the desired standard formexpansion (2.4.2), a recoupling and/or reordering may be necessary to combine the operators aV and'a'p,that refer to one and the same orbit p^

For the possibly necessary ordering, the anticommutators of the operators have to be taken into account.The recoupling is achieved with the aid of Racah techniques.

In our calculation, we have used three orbits lf5/2 < 2p3/2 < 2p1/2. The three orbits restricted theordering in (3.2.1) to twenty one possible combinations as shown in Table 2.

3.3 The modified surface-delta interaction

The surface-delta interaction (SDI) is a mathematically simple interaction. Due to its simplicityand success in accounting for many nuclear properties, this interaction finds frequent use in shell-modelcalculation.

The assumptions and derivation of the surface-delta interaction can be found in Plastino et al,(1966) and its matrix elements are given as follows,

177

SHELL-MODEL CALCULATIONS ON 59Ni AND 59CuSDI / i 0 A i • • \ _ / M

n a + n b + n c + n d A

<JaJbl V U,2)| JcJd> J T - ( 1) AT

2(2J+1)

(2ja + ) (2jb . a. l f a + 1(J

( l + 8 . b ) ( l + 5 c d )

i U 0 X j d - i j c i | J O > [ l - ( - l

where the single-particle states are taken as [n a , l a , j a ] , a = a,b,c,d. The coefficients AT possessing thedimension of energy, are quoted as the strengths of the SDI, and depend on the values of the isospinT = 0 or T = 1. They are obtained empirically. The evaluation of the above matrix elements was done forthe coupling order 1 + s = j and for the radial wavefunctions that are positive near the origin.

TABLK 2All possible orderings of orbits satisfying the conditions

[ [a+ <£> a+ 1 ® [a (

Note : 1 . 5 = If;

Pi

3

3

3

33

3

5

5

5

5

1

5

5

5

5

3

3

3

3

5

5

5/2

Pi

3

3

33

335

5

5

5

1

33

33

1

1

1

1

1

1

If5/2 <

P3

3

5

15

5

3

5

i5

1

1

5

5

35

1

5

3

1

5

2p3/2 < 2p\/i f°r the operators

P4

3

1i

3

1

5

1

1

1

1

3

1

5

1

1i

ii

3 = 2p3/21 = 2p!/2

2. Because of the hermicity of the two-body matrix element, < px p2 I V I p 3 p4 > ^ = < p 3 p 4 I V I px p2 >

178

M.Y. SULAIMAN AND A. MUSTAFFA

It turns out that the spectra obtained with the SDI show some systematic discrepancies with respectto the experimentally observed level energies. The deviations occur for the spacing of the T = 0 and T = 1centroids of level energies. To remedy this, two additional J - independent terms are added, viz

(3-3.2)

The contribution of the additional terms is given as- 3 B + C for T = 0

(3.3.3)+ C for T = 1

where r is the isospin operator.

The extra terms leave off-diagonal matrix elements unaffected, while all diagonal matrix elements ofmatrices for a given mass number and isospin are changed by the same amount.

Thus the modified surface-delta interaction becomes,

< J a J b l V M S D I ( l , 2 ) | j c j d > J T = < J a J b l V S D I ( l , 2 ) | J c j d > J T

+ j [2T(T+1)-3J B + c l 5ab5cd6ac (3.3.4)

The coefficients Ao, Ai, B and C are obtained from fits to experimental data.

Table 3 gives the calculated values of the MSDI for the orbits 2p3/2, If5/2 and 2Pi/2 with Ao =A! = 1 MeVand B = C = 0.

4. Results and conclusions

The calculation was performed on | jNi3 i and ^Cu 3 0 for the active orbits 2p3/2, lf&/2 and 2p1/2

taking 2gNi28 a s t n e closed core.

The MSDI parameters used to calculate the two-body matrix elements and the single-particleenergies are given in Table 4. The parameters were obtained fron. a least-squares fit to the energies of 95states in A = 57-68 isotopes of Ni and Cu as obtained by Koops and Glaudemans (1977). The details ofthe calculations of the diagonal matrix elements of the Hamiltonian using first quantization can be found inAminuddin (1981). The off-diagonal elements were calculated using equation (2.4.3). The maximumdimension of the matrix is 20 x 20. For each Jn and T, separate diagonalization was attempted using theZILOG microcomputer of the Mathematics Department, Universiti Pertanian Malaysia, employing Jacobialgorithms. The coefficients of the fractional parentage were obtained from Bayman and Lande (1966).

The results for the binding energy and wavefunction are given in Table 5 and 6 respectively. For thebinding energy, experimental results are also given wherever available. It is pleasing to note that, althougha simple two-body interaction was used, the agreement in the binding energies between those calculatedand experiments was quite good.

Further tests of the wavefunctions on the spectroscopic factors and electromagnetic transition ratesare currently being undertaken. The results will be reported in a later issue. We are also attempting a similarproblem, using the more realistic Kuo-Brown interaction (Kuo-Brown, 1966) that is derived from the freenucleon-nucleon interaction.

179

SHELL-MODEL CALCULATIONS ON 59Ni AND 59Cu

TABLE 3Values of the two-body matrix elements (in MeV) calculated with the modified surface-delta interaction

for the orbits j - ,5, 3 and L The values listed are derived with the strength parameters Ao = Ai = 1 MeV,2 2 2

B = C = 0. The numbers 5, 3, 1 denote the orbits 2p3/2 and respectively.

a

55

'VI

555555

On

'vi

5555

55555555

i/iO

nLA

l

555

LAl

5

b

5

On

555

555

L/i

55

L/i

(SI

5555

L/i

5333333333333

c

LAl

1/1

55555

i/i

55

'v«

533333311555

(Si

55555555

d

LAl

5

LA.

Vt

5

(SI

33331133331111333333331111

J

01234512342301231201112233442233

T

10101001011010100110010101010101

<ab|V|cd>

-3.0000-1.6286-0.6857-0.9143-0.2857-1.4286-1.8142

0.4849-0.5938

0.5714-0.9071-1.3279-2.4495

1.1759-0.5237

0.3429-0.6761-0.7407-1.7321-1.1832-3.6000

0.0000-1.4286-1.3429-0.7429

0.0000-1.4286-1.1429

1.06900.6414

-1.02220.0000

a

5

On

555

(SI

onon

5555

(/i

55333333333333311

TABLE 4Strength parameters of the modified surface-delta

Parameters

A0

A iBC

P3/2f5/2

Pl/2

b

3333333311111111333333331111111

interaction £

c

33333331

On

5

On

53333333333113333111

nd(From Koops and Glaudemans, 1977)

(

;

]

;

;

;

;

;

]

]

]

]

]

]

i J

3 15 23 3I 1I 1I 2I 2I 1I 2L 2I 3I 3J 23 31 2I 23 0J 1I 2J 3

1I 2

L . 0L 1I 1L 1I 2I 2I 1L 0I 1

T

0100101001011001101001100101010

single-particle energies.

Strengths (MeV)

+ 0.2818+ 0.5293+0.3754+ 0.0711-10.2549

-9.4356

-9.1562

<ab|V|cd>

0.56570.3703

-0.39590.89440.0000

-1.30930.5237

-1.7889-0.8000-1.2000-2.0000

0.0000-0.6928

0.22130.9798

-0.9798-2.0000-1.2000-0.4000-1.2000

1.2649-0.5657-1.4142

0.6325-2.0000

0.0000-1.2000-0.8000

0.0000-1.0000-1.0000

180

M.Y. SULAIMAN AND A. MUSTAFFA

TABLE 5Binding energies with respect to 5 6Ni core in MeV

J*

3 -2

5 -2

1 -2

3 -2

1 -2

5 -2

1 References

59 Ni

59 Ni

(0000)°(5111)1

(5111)1

(0200)2

(4222)2

(0000)°(0000)°(0000)°(5111)1

(5111)1

(5111)1

(4222)2

(0200)2

(4222)2

(3333)3

T

3232

32

32

32

32

to

Theory

-31.54

-31.03

-30.97

-30.46

-30.34

-30.03

Expt1

-31.47

-31.13

-31.00

-30.59

-30.16

-30.28

the experimental binding energy can be

Basis states

(0200)2

(3111)1

(4222)2

(0000)°(3111)1

Basis states

(3111)1

(4222)2

(3311)3

(3111)1

(3111)1

(4222)3

(0000)°(3111)1

(3111)1

(0000)°

r

3 -2

1 -2

5 -2

3 -2

1 -2

5 -2

found in Koops and

TABLE 6Wave functions for 59 Ni and 5SP Cu

J*

(02)(22)(13)(02)(13)

J*

(31)(42)(33)(22)(42)(33)(42)(33)(33)(33)

= 1 - T= 32 2

(HID1

( l l l l ) 1

(0000)°( l l l l ) 1

(0000)°

- 3 - T = 32 2

(0200)2

( l l l l ) 1

(0000)°(Hll)1

(HID1

(0000)°( l l l l ) 1

(0000)°(0000)°(0000)°

T

12

12

12

1212

12

Theory

-35.31

-34.93

-34.72

-32.89

—32.77

-32.89

Glaudemans, 1977.

Expt1

-35.37

-34.88

-34.46

-

-33.41

Amplitudes

0.78950.32040.33170.3075

-0.2634

k = 2

0.2894-0.4387-0.5067

0.57590.3679

Amplitudes

k = 1

-0.25370.1056

-0.7715-0.0601-0.1460-0.0882-0.0152-0.5350

0.0954-0.0313

k = 2

-0.2416-0.2926

0.19950.23300.46000.39650.1732

-0.5090-0.2467

0.2079

181

SHELL-MODEL CALCULATIONS ON 59 Ni AND 59 Cu

TABLE6(Con ' t )59 Ni

(0000)°(5111)1

(51 I I ) 1

(5111)1

(5111)1

( 5 1 I I ) 1

(4222)2(4222)2(8222)2(5311)3

59 Cu

(0000)°(0000)°(0000)°(0000)°(0000)°(5111)1

(5111)1

(5111)1

(5111)1

(0200)2(2020)2(2020)2

(4222)2

( H 3 1 ) 3

Rasis statesLJ CLJIO J LCI L^O

(4222)2(0000)°(3111)1

(3111)1

(0200)2(4222)2

(0000)°(31II)1

(3111)1

(0000)°

iiasis states

(0000)°(31II)1

(0200)2(2020)2(1131)3

(3111)1

(3111)1

(4222)2

(6020)2

(0000)°(0000)°(31II)1

(3111)1

(0000)°

(42)(51)(42)(62)(53)(53)(42)(53)(53)(53)

(00)(31)(02)(20)(11)(20)(22)(ID(ID(02)(20)(11)(ID(ID

2

(HID1

(0200)2(0000)1

( l l l l ) 1

(0000)°(0000)°( l l l l ) 1

(0000)°(0000)°(0000)°

y = i - T =2

( l l l l ) 3

(2020)2

( l l l l ) 1

( l l l l ) 1

(0000)°( l l l l ) 1

( l l l l ) 1

(0000)°(0000)°( l l l l ) 1

( l l l l ) 1

(0000)°(0000)°(0000)°

32

= 12

Amplitude

k = 1

0.10950.25890.02540.07300.85440.06040.0005

-0.0616-0.0501

0.4184

k = 2

0.7512-0.1657

0.06260.3761

-0.0164-0.0373

0.0751-0.4651-0.0262-0.1962

Amplitudes

-0.2466-0.0010

0.53200.1937

-0.2794-0.3988

0.0115-0.2785-0.0651

0.4075-0.2000-0.0938

0.2367-0.1695

k = 2

- a 13250.09120.1937

-0.28610.79800.0760

-0.1728-0.2113-0.0981

0.2882-0.0689

0.1337-0.1100-0.1013

182