Calculation of Coulomb displacement energies in light nuclei

CALCWLATIQN OF ~~~U~E ~I~P~A~E~E~ IN LIGHT NUCLEI

ENERGZES

R. J. DE ~~IJ~~,

H. F. J. VAN ROYEN and I’. J. BRUSSAARD Fysisch finboratorism der Rijkswtir?ersiteit, Utreckt, The ~e~ker~~~d~ Received 28 Decem&r

1970

Abstract: Coulomb displacement energies are calculated in terms of first-order ~erturbatioR theory and the results are compared with the experimental data. In the model the Coulomb displacement energy is described as a sum of sin&+-particle Coulomb displacement energies, which result from changing a neutron outside a core into a proton with the same quantum numbers. In the calculations only two parameters were used, which were kept Constant for A = 13-28 and A =li29-40. From B comparison between theoretical and ~~~erirn~~tal Coulomb displacement energies one finds that for about 70 y$,of ail levels the deviation between the theoretical and experimental energies is smaher than 64 keV. A relatively large deviation of 200 keV is found for the lowest .P = Q+, 7’ = 1 states in doubly odd self-conjugated nuciei.

1. Introduction Coulomb d~spla~rn~~t energies can be determined from the measured excitation energies of ana~ogue states. If the excitation energy of one of the corresponding anafogue states is not known, one can - vice versa - estimate its energy by Gal~ulating the Coulomb displacement energy. In fact, the interest in analogue states has led to the calculation of Coulomb displacement energies presented in this paper. In the absence of charge-dependent interactions and of the proton-neutron mass difference, two corresponding analogue states in isobaric nuclei would have identical binding energies; mirror nuclei would have identical level schemes. Deviations from this simple situation are due to the approximate character of the above assumptions* There is (i) a small difference between the proton and neutron mass, (ii) an electrostatic interaction between the charge of the last-added proton and all other protons in the nucleus and (iii) a different spin-orbit coupling energy, leading to an electromagnetic shift. Since the electromagnetic forces are weak compared to the nuclear interaction, they wilf be treated by perturbation theory. It will be shown that the binding energy difference between two ~orres~ond~ng analogue states can be very well calculated with first-order perturbation theory.

2.1. SINGLE-PARTICLE

COULOMB

DISPLACEMENT

ENERGIES

The EIamihonian describing a single-particle state, taken as a core plus an extra nucleon, will be separated into a nuclear and an electromagnetic part:

12

R. J. DE MEIJER

et al.

where Hnuc, is charge independent. The wave functions for a proton, &,, and a neutron, 4,, with the same quantum numbers in the field of the same core differ slightly due to: (i) the smaller mass of the proton, and (ii) the electric repulsion between the core and the proton. Let A4 = qbp-4” describe the difference between the two wave functions, where &, and $,, satisfy the Schrodinger equations H(Z+

1, NM,)

H(Z,

N+ 1)lM

After subtraction and multiplication the binding energy difference AE = [I Since (4,lAcj)

AE = (#GG,(Z+l,

1, N)IQ

= E(Z, N+ 1)W.

by (&I

+(~,lA~>l-l;(~,lH,,(z+

is of the order of lo-‘-

= E(Z+

(2.2)

one finds, with the use of eq. (2.1), for

1, W-f&&

N+ W#Q.

10e3, eq. (2.3) is approximated N)--&(Z,

N+

l)lA>.

(2.3) by

(2.4)

The effect of the proton-neutron mass difference and that of the approximation made in eq. (2.4) is estimated ‘) to be at most 50 keV for the A = 16-40 region. This effect can be compensated for in the calculations by the choice of suitable parameters. The long-range electrical force gives rise to the Coulomb phase shift, which can be important for unbound proton states. Since the Coulomb energy varies smoothly from the bound to the unbound region ‘) it is assumed that the phase-shift effects are negligible. The electromagnetic interactions with the core differ for the proton and the neutron in the following aspects: (i) The different g-factors lead to the electromagnetic shift AE,, = E,“,- E,“,, where E,“, and E& are the electromagnetic energies for the proton and neutron state, respectively. This shift will be discussed in subsect. 2.2. (ii) Th e e 1ec t r o static interaction gives rise to the electrostatic shift AEes = ($,I V,,l@,), where V,, = H,,(Z+ 1, N)- H&Z, N+ 1) represents the electrostatic field of the core. This shift will be discussed in subsect. 2.3. Eq. (2.4) can be written as AE = AE,,+AE,,. w5) 2.2. ELECTROMAGNETIC

SHIFT

The shift due to the different spin-orbit coupling energies for proton was discussed by Inglis ‘); a brief review is given in the thesis of Lebon orbit coupling energy of a single-nucleon state can be written as “)

and neutron “). The spin-

where ze represents the nucleon charge, gs the spin gyromagnetic ratio of the nucleon, nz the nucleon mass and V,,(r) the electrostatic potential of the spherical core (Z, N) with radius R.

COULOMB

For a homogeneously

DISPLACEMENT

charged core the electromagnetic dE,,

= 0.269 $

13

ENERGIES

shift, &&‘,,, is given by

(2.7)

I +s MeV * fm3.

For this expression the numerical value (z- g,), - fz- s&, = - 8.413 has been used. 2.3. THE ELECTROSTATIC

SHIFT FOR SINGLE-PARTICLE

STATES

As pointed out in subsect. 2.1 the electrostatic shift is given by AE,, = (g$,I YJ&,). For a pure single-particle state in the nucleus (Z, N+ 1) the separation energy and the neutron radial wave function are determined by the nuclear part of the Hamiltonian IYnucltwhich is represented by a kinetic energy term and a potential well of depth Y_ From the requirement that the neutron separation energy, Es, as cdcufated from this potential weli, be equal to the experimenta separation energy, the well-depth parameter is determined and the neutron radial wave function can be calculated. Example. The method of calculation will be illustrated for the mirror nuclei 13C and 13N. The states at 0 (P = +-), 3.09 (.I” = 4’) and 3.85 MeV (J” = 3*) in 13C can be described as a doubly closed shell 12C!(O)core plus a Ip,, 2s, and Id% neutron, respectively. The binding energy difference between two mirror states is given by the electromagnetic shift LIE,,, as described in subsect. 2.2, plus the electrostatic shift AE,, = ($,I V,,&>, where a),, represents the radial wave function for the neutron in the nuclear field of the 12C(0) core, with a separation energy E, equal to Es = E;, [12C(0)]-E~f’3C(O)]-Ex[13C] = 4947 keV-E,[‘3C]. The binding energy Eb is taken to be a negative quantity. 2.4. COULOMB

DISPLACEMENT

ENERGIES

FOR MIXED

STATES

La the present model a state is described by a sum over orthonormal metrized con~gurat~ons:

with
~i~m{y~~(~)

; d >

=

a&m 61,

h.,

&3,

bit‘

The expectation value of the Coulomb energy operator Hc, defined as

in the state $r is given by

Since H, is a two-body operator, k and m may differ by at most two.

antisym-

14

R. J. DE MEIJER et al.

In order to obtain an estimate of the magnitude of non-diagonal matrix elements, the many-particle matrix element in eq. (2.8) will be reduced “) to a sum of twoparticle matrix elements. Thus one has to caIculate (pcrl&lpa) ,

for

k = p12,a f y, P # 6,



(p21iTcloz>, (02{w,lpz)

for

W-4

= 1,

for

/k-ml

= 2.

A further reduction can be made since the operator Z& works on protons only; matrix elements containing a neutron state vanish. The two-particle matrix elements can be calculated with the use of the integrals given in ref. “). In the mass regions studied and with the wave functions used, the non-diagonal terms (k # na or k = m, rx SC;y, j3 # S) are smaller than a few percent of the diagonal terms. They are omitted in the calculation and in eq. (2.8) only the diagonal terms are retained. The total Coulomb displacement energy is calculated as a weighted sum over the differences in Coulomb energy for the constituent configurations in the proton and neutron states. If one has the configurations l$(Z’, T,)) = alA)+fiIB) and Ill/@“,T,- 1)) = ylC)+4lD>, where the states IA>, IB), IC} and ID) do not possess sharply defined isospin and a, p, y and 6 are isospin coupling Clebsch-Gordan coefficients, and if these con~gurations are connected by the isospin-lowering operator t$fT,

G-1)) =

kT-tW”,

G)>,

where k = ~(T+Tz)(7’-T,+l)]-*,

then the Coulomb energy difference is given by

Here Vr denotes the expectation value of the Coulomb energy operator in the state Z, i.e. Y; = (ZlZZ,lZ>. No off-diagonal matrix elements of Hc are present. The terms like Yc-- V, can be written, after recoupling of 1C) and [A) as a sum of single-particle Coulomb displacement energies, where the nucleon is coupled to an excited core. Example. For the J” = 4- level in 13C at E, = 3.68 MeV, which cannot be described as a single-particle state, it is assumed that the configuration

is a component

of the wave function.

The configuration

can be written as:

COULOMB

-&A)+J$$3),

DISPLACEMENT

ENERGIES

where the states /A} and ]B) are given by IA) =~~~4~nJ’hl’l3M>I~;‘>~,m;t, /B> = ;lp;

-+IP:)Jw;

‘>+,m;-1_, +IP$,

I, I ’

MS;1, o 1

They can be represented by the diagrams 7L

n

v

____~.z

and

0

v

--9

xx O1

respectively. Here particles are represented by a cross, holes by a circle. For the corresponding T = -T, = 4 state in the mirror nucleus 13N, the configuration can be written as IP;~[P;]J~=o;

T,=~>+,M;+,

-+

= &C>-&D>,

where the states ]C> and ]D) are given by ]C> =~~~t~J’~‘13-~>lP;1>,,,:,,~lP~>~~.~:

I, -i 3

iD> = ~~B~J’M’IPM>~P;‘>~,,; +, -+f~$v, M’;I, 0% They can be represented by the diagrams n: xx ~0

n V

1’ and

--I

xx 0

respectively. In eq. (2.9) (C]r-]A) vanishes for these configurations. In order to calculate the term Yc-- V,, the states jB> and \C> should be recoupled such that they are described by a proton-proton hole system plus an extra neutron and proton, respectively. This yields:

16

R. J. DE MEIJER

et al.

For Hc one may write (2.11)

Hc = HR -I-Koupl 3

where HR gives the Coulomb interaction for the particles in the core and Hcoupl the Coulomb interaction of the extra nucleon with the core:

H COUP1

e’(t - Ci))(* - h(P)> I’j-Y,I ’

=&

wherep represents the extra nucleon. With eq. (2.11) all core terms cancel in eq. (2.10) since they appear both in 13N and 13C. It is to be expected that in the expansion of Hcoup,in terms of Legendre polynomials the monopole term is largest. The higher-order terms will be neglected for the evaluation of eq. (2.10). This yields: Vc-V,

= &C(25’+1) .J’

x [
o; o, o[vP&,~-; +, +IHcouprl[~~; ‘UP&, o; o, O[VP&, 3; 4, +)

l~~tl~,, o; I, o[~~tlt_, t; 3, -tlHcouprl[~~; 1~~+1~,,o; I, o[~P&,+; 3, +>

- ([~P+‘~P&,

o; I, oCVP~I,,*; t,t_lHcoup,lC~~;‘~~tl~,, o; I, o[vPI]+,+; +,+>I. (2.12)

The difference in Coulomb energy in eq. (2.12) between the square brackets can be calculated as that of an extra nucleon outside a core with spin J’ and isospin T’ = 0 or 1 as described in subsect. 2.3. The terms Vu- VAand Vu- V, can be calculated in the same way as V,- V, and thus the Coulomb displacement energy of the state is written as a sum of single-particle Coulomb displacement energies. With the assumption of an extra nucleon moving in a potential caused by an (excited) core the antisymmetrization of the extra particle with respect to the core is not strictly taken into account. The model is applied consistently, however, and the results show a remarkable agreement with the experimental data as will be seen in the next section. 3. Calculations

and results

For the explicit calculations the nuclear potential is taken to be described by a Woods-Saxon potential V(r) = -

v,

[f(r)4 ‘, %f(r)(L * S)] )

(3.1)

where V, is the well-depth parameter andf(r) = (1 + exp[(r -R)/a]) - ’ ; the reduced mass ,u of the core plus nucleon is related to C by C = (h/2pc)‘; the nuclear radius R is given by R = r,A* and CIdenotes the diffuseness parameter. The second term at the r.h.s. of eq. (3.1) represents the spin-orbit potential, where v is the coupling parameter.

COULOMB

DISPLACEMENT

ENERGIES

11

In subsect. 2.3 the problem of the separation energy of a neutron and an excited core has already been mentioned. In eq. (2.12) it has been shown how one can calculate single-particle Coulomb displacement energies for systems with a nucleon coupled to an excited core. The Coulomb displacement energy is strongly dependent on the quantum state of the extra nucleon and varies only weakly with the separation energy of the neutron. The excitation energies of higher excited core states are known in only a few cases. As the wave functions are known in even fewer cases, the determination of the proper core state generally cannot be unique. Therefore it is difficult, if not impossible, to calculate the matrix elements of HcoUplin general. Since they only weakly depend on the neutron separation energy they are taken equal to the matrix element for which the core is most strongly bound. The range of values found for I’, is V, = 45-55 MeV, except for a few cases where the neutron is bound to a very highly excited core. The values of the parameters r0 and a were determined as the best values for singleparticle states. The two sets of parameters used were r0 = 1.28 fm, a = 0.63 fm for A = 13-28 and r0 = 1.26 fm, a = 0.63 fm for A = 29-40. The Coulomb displacement energy is very sensitive to small changes in r,; for Ar, = 0.01 fm one finds AE x 30 keV. In this respect and in view of the rather large mass region covered, the r. and a-values found above are surprisingly constant. From elastic electron scattering and muonic X-ray data “) it is known that the charge distribution of the nucleus can be approximated by a Fermi distribution. These data were obtained for only a few nuclei in the mass region studied. Therefore it was preferred to describe the charge distribution by a homogeneously charged sphere rather than to introduce new Fermi distribution parameters. From the wave functions used only the configurations with an intensity larger than 5 % were taken into account. For the calculation of the total Coulomb displacement energy the wave function was renormalized. A comparison between the experimental and calculated Coulomb displacement energies and excitation energies is given in table 1; Coulomb displacement energies and excitation energies of levels that have not yet been observed experimentally, have also been calculated and are presented in table 2. In table 3 mass excesses have been presented, which are experimentally unknown but are calculated in this model. The rounding-off criteria used in the tables and figures are the same as in ref. ‘). The results of the calculations and a comparison with the experimental data for the A = 13 nuclei are presented in fig. 1. The energies given in square brackets are the binding energies relative to 13C(0). The dashed line at 13Cc,lc presents the 0 MeV line relative to 13C(0),,,. The levels presented in 13Ncalc are calculated from the wave functions and positions of the corresponding 13CexP levels. In the same way ‘%& and 130calc are calculated from the wave functions and the positions of the 13B and 13N levels, respectively. The fact that the Coulomb displacement energy for the J” = 4’ level is about 700 keV smaller than for the J” = :- level is well reproduced in the calculation.

18

R. J. DE MEIJER et al. 1

TABLE

Comparison

between experimental

and calculated Coulomb energies ex; a) PECOUl (kP’i) 2 457 7

Q-2) (Z,!lCl)

Jr,

3/2-,?/2 l/2-,lj2 1/2+,1/2 3/2--,I/2 5/2+,1/2 3/2-,?/2 3/2-,3/2 3+,

13 13c

13N - 130 14c - 14x

15 112 15s

5 5

6 09; 3 6583 9 S 728 6@4 9 7012 5 7 337 9 8313 5 2313 8061 2 8617 3 3 90-f 9 172 2 9 509 3 10 432 7 0 747 0 5 270 5 299 6 323 7 300 7 563 i) 120 295 3Yd 12 7& 12 960 13 o90 10 13 250 10 0 374 ia 18 536 19

1 i -, 1 0 +, 1 3 -, ! 0 -, 1 2 +, 1 2 -, 1 2 i, 1 o+, 1 1 -, 1 o+. I

1

-

lhry

145

1

8

15:

- 15N

l5N - 150

16~

160

lnl

-

IGO

- 16~

-

170

170 -1.v

lea -1m

16F

-

-

19N.z

200

- 19sa - 8OF

20F

-2oNe

19%

3 3 11 12 12

87: 058 846 082 471 950

[I31 [l51 [I51 [I51 [l51 1151

LIeI [181 [181

3 -; 1 2 +, 1 2 -, 1 2 c, 1 1/2+,j/2 5/2+,3/2 lj2-.I/2 5;2+,1/2 1/2+,1/2 3,(2-,1/Z j/2+,1/2 7/2+,1/2 2-, 1 o-, 1 3 -, 1 1 -, 1 0 -, 1 2 -, 'i 1 -. 1 3, l/2-,3/2 3/Z-,33/2 l/2+,3/2

201k

-

2ma

2lF

-

am

6 10 10 11

510 275 860 080 0 270

[ISI 1191 [I91

7/2-,112 J/2-,3/2 3,'2-,3/2 l/2+,3@ 0 +, 2 +, 4 c,

I 1 I

2

3 7

I281 l/2+,3/2 [281 $+J$

40 30

5,2::1/2 [291 3/2+,3/2 5/2+.3/Z [24] 0 +, 2 [24] 2 +, 2

30 5 10 *0 40

;

3 614 : 7' ';; 3&O 3 215 3 310 3 341 3 210 3 275 2 624 27% 3 542 3 5i2 3 436

z 40 13 25 13 30 30 5 13 2 13 11

15 126 -67 2 395 2 3 406 3 561 3 15 079 0 7: 2 431 2 8 0s; e 611 8 900 e 773 9 210 9 464 13 429 -46 5 213 5 753 6 250

2

2 I II 12 5 5 6 7 7 13 12 13 13

11 12 12 13

1p-* 1:r

6

1

568 737 079 C63 270 2% 220 3 le.3 2 673 814 573 577 459

2 4 j 7 75 3 :_ 13 'i 50 6 10

1j2+,1/2

4

[24] [24]

97 1 469 3 162 0 19e 7 6ko 7 470 0 672

2 471 2 936 3 003 2 312 2 283 2 725 2e2e 2 710 26% 2 $70 2 si7 3 468 3 470 3 05.1 2 939 2 592 d 615 2 660 2 654 ?a05 2 703 2 562 2 530 27% 2 e24 753

1

1eNe

190 - 19F

19F

1 1 851 2

T

displacement

[29]

0 +, 2

5/2++.3/2 l/2+,3/2

E 693 7 2 664 552 31 305 216 102 007 234 4 015 4 815 6 197 056 5 20 100 400 loo 290 lot 680 187 15 510 20 900 20 650 20 2 5:: 5 7 i3: 5 6 366

: li 12 6j4 13 003 273 3 157 4 695 4 600 4 968 115

1

: 4 041 4 030 3 805 3 921 3 44 3 557 3 358 3 261 4188 4 196 4 w2 4 781 3 h 3 515 3 9go '3;z; 4 033 4 435 4 3Sl 4 392 3 945 3 997

4 053 3 954 4064 3884 3 520 3 530 3297 3215 4021 4 061 3980 4 530 3480 3 381 4o28 3 957 4 010 3 970 4 420 4490 4 360 3 954 3 970

11 11 11 11 30 40 7 9 3 4: ti 40 15 7 11 20 40 40 70 70 7 40

6 11

t 675 11

3 3 3 7 7 e 10

4ji 317 653 525 690 863 459 167 373 7 640 -150 6 617 e 21i 10 237 10 853 11 015 16 790 5:: 780 e 847 9 170

11 11 11 Ir

energies and excitation

15 112

5

2 3 3 15

36: 509 547 c66 0 2 313 e 061 e 617 e 907 8800 9 172 9 508 10 432

3 3 50 2

4 12 25 12 30

5 17: 5 905 6 290 2 :: 7780 11 615 12 520

': 10

5 24: 5 195 6160 6 860 7 2@i 12 960 12 780 13 250 13 090 42: 252 711

13 10 4 :: 10

10

10 10

11082 12 471 12 950 13 640 0 500 3 loo 3 860 II 198 12 552 ': 3 4 4 4

zz 053 651 741 961 0 1887

15 15 15 6 'i

3 20 20 41 51 9 10 10 10 10 10

2

30 40

0

238 7 620

5 5 40 40 40 50 4:

6 8 lo 10 11 16

2: 29 -103 14 13 0 118

30

51: 050 275 860 oeo 732

30 10 5 10 20 2

65: 750 8 856 9 139

50 '2 6

7 2 4 3 7 i 70 3 I:,

COULOMB

DISPLACEMENT

ENERGIES

19

TABLE 1 (continued) c*c

a,b)

E

+I)

,N+l)-(Z+l,h’)

J7,

DE

T

CO”1

(kev) 21w

-

4 327

aNa 350

[421

1 747

7/2+,1/2 1/2+,1/2

Lb21

E 'g

9/2+,1/2 5/2+,1/2 5/2+,3/2 l/2+,3/2 5/2+,3/2 l/2+,3/2 0 +, 1 2 +, 1

[421 (421 I401 I401 I441 [441

3 735 8 856 21Na

- 21Mg

22Ne -22Na

9 217 0

22?I*-2aQ

3 Ai

23Ne - 23Ne. 23iia - 2*

4 069 5 170 0 "

1 275

1 952

1; 1 1 2

[471

50

["l"i

[42]

47;

1

24Mg - 24Al

;;g 25Na 25Mg

30 3 16

[291

I

58; 975 612

083 053 702 026 026

:g 5009 4 932 4e& 5 130 5 127

zze 3 400 7 so 7 839 - 261~

: AZ?? 5 018

I& :g 4 317

26A.I-26Si

834 a79 505 275 345 954 840

5 023 5 015

1965 2 562

26~

4 4 4 4 4 4 4

5 5 4 5 5

0 n

- 2% - 25A1

306 090 435 427

4 794 4881 4 829 4 635 5 47 4 332 4 657 4 651 4 607 4 652 4 764

2 393 2 9es 7900 7 790 0 n 563 347 5 9Bo

4 4 4 4

4 82j 4 918 4 912

43; 2 078

23Mg-23iU 24Ne - 24Na 24Na - 2%

: ;:6" 4 341

3 +; 1 4 +, 1 2 +, 1

: 2;: 228 2wo

3 159 7 8% 4 191 4 515 4 595 4 613

3 10 2 10 10

[681 [68] n) [68] n)

2% 4 4 5 5 5

927 926 5% 510 438

: ::: 5 364 5 397 5 335

27&@-27A1 : ;z 4 831 4864

9840 936 3 470 3 564 3 757

1

27A1

- 2751

_.

.d 1 2 2 3 3 4

2&--28/d 28Al-261

4 4 5 5 5 5 5 5

014 211 734 678 955 054 0

3: 1 373

a03 938 555 569 552 501 491 444

5 454 5 641

1 1

I791

4 5 5 5

975 328 317 331

4 231 4 316 4304 4 340 4296 4 137 4 445 4406 lreyl 4 tvo 4281 4 303 4 351

0 8 10 30 10 10 12 12 20 40 3 3 15

:: zz 4 780

z: 70

:: ::; 4828 4 850 4Fxl2 4806 4 750 4 720 5 230 4 290 4 774 4780

4 e” 2 4 10 9 12 40 90 30 5 16

:: a:: 4 730 5149 5 119

8 9 30 7

4 768 5042 4 907 5016 5042 4866 4 981 5 032

18 18 6 7 7 7 7 9 12

'; %

1:

5 137 5188 5 014 5cA6 5006 5 035 5063 5066 5 076 5 624

13 12 2 2 2 10 10 10 lo 11

5 5 5 5

578 477 317 418

11 11 11 15

5 279 5 344 5 377 4984 5 035 4785 ;93: 4902 5518 5 528 5 531 5546 5505 454 5 437 5680 4 935 5464

11 15 15 5 11 11 12 6 15 2 7 7 9 9 10 15 20 12 4

5500 5 260

4 60

5

% 400 1 792

2eca 2 843 3 491

8 963 9 23: 230 880 925 4 063 30 220 3 2w 4 310

2

i

8 10 10 20 20

1

512 2 034 2 442 2 975 7 710 -150

50

3 270 4 380

7

8 :

20

90 4

4 z;:

1 728

2 3 3 4 4

745 501 753 027 140

kg 7 8 lo lo lo

761 598 165 198 526

a

8 12 11 2 2 2 2

s ‘99

10 556

[411 [461

-27 A 30 -20 20

3 15 50 60 70

[501 [291

Et

0

E a

c 10

81 71 -so -150 40 -127 -129 -170 -6 40

9 12 40 90 30 5 16 9 8 30

451 2 012 2 351 2904 7 790 0 5seo 9 516 9 98L 10 42 10 737 15 436 0

30 3 16 8 7

-6.57 -56 18 I591 [611

tz:;

,790 2 502 ;ag 3 440 7 916 7 9% 228 2 070 3 159 4 191 4 595 4 613

: 1, 11 11 11

5 126

10 6

17960 2 784 : ::z 3 4 4 6

842 093 138 815

7 8 10 lo 10

850 552 240 216 4% 0 780

~~

ii: 974 2 120 2 633 3 530 3 817 4 103 6 029 9180

30 10 10 12 12 20 40 3

::z;

cm

3

15 11

10

0 10 47 -10 21 -60 10 224

tz:;

949 1 613

1 1 920

386 706 692 239 go9 925 373 2108 3 171 4 156 4 514 4 474

; 30 1

439

569 956 586

69

[431 sb 7 I3

7 820 0 451

7 15:

2 2 2 3 7 7

8 973 9 217 0 2 220

't 2 Ia

10 ; % 9 es5 9w 10 731 15 480 -66 383

:

%$656 1952 4 069 0 1 240

1

7 822 80

1 iis 2 810

953 2 165 2 647 3 540 3800 4 140 5 9 9 10

989 316 382 480

2 2 3

1

2 10

3

:

10

7 7

;

10 15 20 10

60

If:

'2

119 7 -27 -130 -116 -23

7 7 7 7 9 12

3 -201

9 10

-7 -61 145

13 12 2

38

2

-:: -31 -139 -150 -36 -68

1: 10 10 10 11 11

;iz

::

-3 185

15 11

53 -42

15 15

2 46

1: 11

-75 -18 36 37 41 21 45 -14 -10 17 40 40 -136 -183 80

12 6 15 2 7 7 9 9 10 15 20 12 4 4 60

20

R. J. DE MEIJER ei al. TABLE

1

(continued)

!keV)

(kev) 2eAl .-2851

1 620

2ffii -28P

5% 9 316 9 382

2951 -29P

[791 [771 I551

1 27:

3 623

3051 - 3OP

: 39:: 0

2 235

3OP

- 3cs

3151 -

2 i;;,” 677 2 937 4 181

31P 75;

- 315

5 390 5 380 5 771 5 798 5 731 5819 5 554 5137 4880 5 702 5 727 5707 5466 6239 6 170 6 125

5 658

1 695

31P

5 394 5 392 5 742 5 731 5 716 5 727 5 569 5094 4 760 5 811 5 741 5702 5 399 6227 6 153 6 108 5 615 579

(kev) 40 12 9 9 6 8 17 e 20 8 8 13 14 13 30 30 6

; ;g 5 694

2

3 133 0

10866 [551

40 5

15 233 -29 39 -15

1

1 264

3 373 i: z

1

1

13 16 16 5 5

Z$

-

325: 32p 32P - 32.5

325 33P

- 32C1 - 33s

33s

-33cl

5 015 0

2 z:

7: 5 070 7 002 0 1 434 1848

[lo31 [IO31 [IO31

84:

1 968

2 2 3 5 345

34C1

35s

-34X 2 3 3 4 4

[lo51

313 935 221 479 0 127 304 914 072 114

2 1560

1 763

';; i: 20 12 13 20 15 15

5 901 6 459 6 479 6381 6 293 6 290 6 266

5978 6 430 6 266 6297 6 310

15 20 5 6 50

Zk 248 213 238

6 792

2 646 3 162

6 6 6 6 6

5 660 0 77: 1949 6 612 0

-

1 411

1 612

2789 4 999

1

1

16 1 2 12

[123] [I231 I1231 11241

l/2+:1/2 7/2-,1/2 5/2+,1/2 3/2-,1/2 5/2+.1/Z j;/2+;3j2

276 634 553 631

6 991

0 2 217 2 491

638

7 o27 6 137 6 658

I

;ggg

6 3

:E 6 030 6 351 6 315 6380 6 3eS 6 254

"6 AZ: 6 %X3

0

6 5999 074 zz

2 348 1 219

15 14 15 40

6 5 055 981 6 OCO 6 429 6 440 64~8 6 313 6 178

2 6 6 6

1 587

159 249 1Si 640

E$

:z 6 773

1 992

35Cl

10 10 30

(921

3 390 6 381

'i

tEi

~~~ 6300 6 837

:: 20 14

::; 6 750

:: 30 15 9 7 5 16 20 50

6 2-n

6 220 6169 6105 6 746 6 720 6 760 2:: 6700 6 250 6682 6620 6 670 7 070 6 130 6 595

z: 30 30 4 30 30

9il 2 025 2 275 2 762 2 771 5 581 213 2 248 3 331 3 938 4 072

1 246

633 179 615 881 % 1 302

1 ea8

2 3 5 4 6 7 8

2 zz;

1

~~

z g 6 610

2 2

7 057

6 893 6 981

1 3

4; 13

2

4 12 12 12 13 12 12 13

45 286 347 437 452 968

5 7 7 1,

070 002 005 984

[941

:: ::

t;:i

9 10

[971

I;:; :: t;;i [971 40 [981 [99,551 5 [lcol 4 [loll

1

1551 5 56: 6900 7 $50 ed Boo0 2 351 2686 2 848 5 560 0 2 156

4 090 4 150 0 2 100 3 3w : E 7 194

‘i

tz

8 9 20

[971 [971

2

[IO71

20

[loal

;:

t:z;

15 3

I1131

20 [lot31 1~108,1091 20 20

6 2

2

lb 16 16 16 16 20

654 291 550 358 564 4 301 510 5 -40 1w 230 30 052 59 511 4 483 16 253 h 131

2 eo3 5 124

3 3 3 3 4 4

: ;E

;zP

10 5

zg

d

2

4 175 50 2 092

5 7 7 7

;

9 4011 0

4 467 4 964 5 il0 6 953 7 027 12 ago -58 5 520 6 881 7 315 78

[901 [901

2190

'8 AZ;

7 968 9 403 7

; I?: 6 6 6 5

1831

20

2 234

loi 264 177 678

[&I

6

786 2 951 4 176 7log -12 2 174

[all

1;

3 461 4 297 5 410

1266 2 234 3 135

6 6 6 5

(he.7 1

)

1 1 :

;

13

1

196 15

17Bo 2 630 3 210 5 610 .z z:z

10 224 4 999 " 1 1380 2 278 2 174

50 20 20 20 30

2 12

I1171 [1171 [I171 [I181

10 12

LO 12

2; -15

; 6

-92 15 -43 -720 109 14

8 17 8 20 8 8

-5 -67 -12 -20 -20

13 14 13 30 30

-29 -77 -55 -I

6 2 6

7 8 -55 -26 -102 -15

11 15 15 15 I5 15

-57 15 Ji 40

15 14 15 40

-49 22 loo -58 40 20 -30 78

3 6 40 13 20 20 20 12

125 30 -76 -76 -77 30 213 88 -50 A4 -20 30 50 -10 -50 -50 -27 -15 69

13 20 16 15 15 20 5 6 20 6 20 20 14 20 40 30 15 9 7

2 110 30 20 eu -60 30 48 -30 40 -40

1t 20 50 30 30 30 30 4 30 30

L7221 5s 143 103 -25

3bi i:::;

11251

-53 53 76

lea

30 12 4 5

5

18 7 5 13

COULOMB

DISPLACEMENT

21

ENERGIES

TABLE 1 (continued) c*c

C..b)

8)

Z,N*l)-(zhl,N)

ExW,B+l)

Jn,

%xl

T

(rev) 37&-37x YPc~',~

39cl-39Ar 39*r-3% 3% -3931

4oAr--4oK 4ca -4wa

40x-40%

6770 5046

3 3

67: 761 1311

3 3 3

*I; 127 24oo 3 %9

: IO 10

127:

5

: z!TZ 3 019 3603 0 3:

1) l/2+,3/2 3j2+.3/2 2 -, 2 :s '2 4 -I o+, 1 2 +, 1 o+, 1 2 +, 1 1) 3 -, 1 3/2+,5/2

: t:222 1 [lee]

4-i 3 -, 2-, 5-a 2 -, t1361 3 -, [I361 1 -, 2 -, 11361 o-, 4 -,

2 zl 047

i m&; 1

",z 7661

2 1

For footnotes,

1

t::;

;f$p

2 070 2 103

1

1)

;$;;:y: j/2+,1/2 7/hlP

7

1

1)

1 1 1 1 1 1 1 1 1 1

[I361

[I371 I1361

(IleVI 7034 7 454 6 504 6 473 6 461

6901 7 300 6 525 6 501 6 457

13 50 9 10 10

6442 7 065

2s

::

; 7 7 6

,"g 7 320 7 270 6 430

:: K, 40 30

;Lz 7 3&

;:z 7 251

1: 12

;;z 7 375 6 750 7 ij2 7 170 7 131 6977

7261 7309 7 360 6 670 71% 71% i 1Tw 7 119 6826

12 15 20 20 4 4 9 8 5

22; 6 957 6929 7 457

7001 69@ 6979 6-13 7 445

: 5

z.zz 361 258 473

1%

talc

q+

Ep+l,N)

talc

%

(rev)

(k.eV)

6 880 150 IO 637 11 277

5 50 8 9

6 747

10

10 65: 11905

4

;; 32 350 2427 70 2240 3 710 9132 7%

9 911929 11 351 10 10 29 30 30 2& 30 3 7x) 18 9090 170 77390

4 :

26o5 3 017 3227 3 676 4 471 7704 7 732

li

7

4

1,A)

10 10 10 12 1

3 4

7

2 2 3 3

472 793 026 660 4 380 7661 7696

[1261

1: 30 30 20

0.T

*x

b&f)

reference w*veOn function

133 150 -21

13 5;

[I141 t;gi

'4" 43 223 27 70 40 -10

:," 10 12 14 30 40 30

I:,",' 11271 I1141 11141 Cl141 [1141 [I271

30 9 10 12 14

$1 11141 (1141 [I271

4 4 g

r1341 11341 ;;;m;

-26 4

5 5 5

[I341 [I341 11341

-22 47 12

1: 7

40 68 1 [1281 144 133 7 227 7 [WI 10[129,131 m[129.131 20 [I321 43 36 t1351 1:: -7

‘1200 2 p7; go 20 [I331

1 1

0

izi

see table 2.

15,126 312-30

f --j

I 1313 exp

a)

co* (rev)

i

t:E;

lq

La.

13Cexp

J L-J

--I

-0.00

5.0663/isn

3.55 3.509

5/f

2.366

112.

wi

312

I _____ri

13Ctalc

3 3 003

112

‘“Nexp

‘3N

talc

“0 exp

'30calc

Fig. 1. Calculated and experimental excitation energies of levels in A = 13 nuclei. The energies given in square brackets are the binding energies relative to i3C(0),,,. The dashed line at 13C_lc presents the 0 MeV line relative to 13C(0),,, . The levels presented in i3N_is are calculated from the wave functions and excitation energies of the corresponding 13C,,, levels. In the same way 13CEslf and 130,,,, are calculated from 13B and 13N, respectively. Only the levels of interest have been presented.

In fig. 2 the results of the calculations and the comparison with experimental data for T = 1 states in A = 14 are given. The calculated excitation energy of the lowest J” = O+, T = 1 state in 14N is 118 keV higher than the experimental value.

-

-I&

lb

- 2ONe

- 20h

2OF

20&

22m

-e?NNa 23Nc - 23th

2oNa--2aQ

- 2OF

1pNa

203

19N.a -

1gF -1pr.c

190 - 19F

-1BF

lb

170 - 17; 17F - 17Ke

150 - 15F

14N

140 15N - 150

(Z.N+l )-(ZCl ,I?)

._

1

Jn.

10 259 10 56 11 508 7 620 85& 9 579 10 118 10 l&J 11 457 3 568 4 065 4 446 13o9 0 050 IO 046 10 470 10 e.27 11 456 16 732 18 256 20 170 20 522 20 845 6 520 7 960 9 790 10 070 IO 360 4 473 1 018 1 703 2 314 3 433 3 PM

9 7er

I311 1) [29j 1) 1) 1) 1)

2 3;

': 7 8 8 8

40

2 40

1

t::; [511 t511 1511 [511

1) 1) lj

1)

; [2:] 5 [241 '1 [241 [351 10 I341 1, 1) 1, 1) 12 1) 5 1) 2 (381 11 1) 12 1) 12 1) 1) :z 1)

10 13 4

[&I [271 1311 1) 1)

4 30 5 9 12

1 IBSI

T

1

0)

;

2+,2 0+,2 0+,2 2+,2 4+,2 2+,2 0 +, 2 2 l, 1 l/2+,3/2 3/2+,3/2 5(2+,3/2 3/2+.3/2 3/2+,3/2

2+:2 0+,2

f 2

EZ 255 260 324 150 406 449 376

z

'G 813

379 444 527 151 191

g

548 309 432

398 312 241 1) 1) 200 862 1) 1) 759 1) 655 1) 552 492 3so 175 [421 177 I421 n) 159 !451 1451 'io97 [451 3971

I341 1) 1) 1)

4 4 4 4

2 916

(kev 1

%ul

c&z

8 005

1 34: 3 060 3 240 3 460 5 170 8 791 9 475 10 c67 11128 11559

;zz 9 790 10 070 10 360

2 440 2 330 3860 10 046 10 470 10 a27 11 456 18 256 20 170 20 522 20 845 1 110

1 840

8ji

11 457 70

10 118 10 1%

I

"9;z 10 259 11 508 684 2 949 7 470 8552 9 579

12; 1657 2 486 6 201 6 240 8 120 4 964 4 750

13 777

1 1:

11 22; 12 270

5 850

kev)

Ex(Z+l,IIj

ca1c

ii

9 9

70 IO

12 13 40 40 40 40 40 40

11 1, 12 5 11

10 20 8 10 20 9 15 a

11 6

50 3 5 10

function

[2Ol

wave

Predicted Coulomb displacement

P/2+,3/2 5/2+,3/2 ,3:,' l/2-,33/2 1) 3/2+,3/2 [291 l/2+,3/2 1)

8800 50 9-n 8 313 I, 615 12 520 10 11 223 5 1) 1) 12 270 10 13 640 5 [191 4 [21,221 11 198 12 552 5 f21.221 13 059 9 [211 13 777 5 090 6 5 di 5 368 10 1241 , 120 20 6 201 6 1) 6 240 10 1) 8 120 8 12:1' 2 372 20 11 [28] 7 [281

(lie’; )

E.p,N+l)

81

TABLE

27/U - 2751

27Mg-27A1

26.&I-2651

26t.Q-26.u

25~1 - 2551

25,@-25.U

25Na - 25&

24.U-2451

24M& - 24Al

24lia- 24MB

24NC - 24ria

2ag-23Al

23Ra - 2% 670 912 791 475 067 128

(kev)

6857 7 202 1 692 3loP 3484

1 OE 2 201 8 791

j26

20 20 20

i

10 10

10

3 10 10 10 10

10

2 10

6 6 12 12 2

5

2 9

;

47:: 310 oca 19Q 010 974 740 20 20 9 9 550 20 10 380 20

:; 17 19 19 20 5 7

:z 4 762 79w 9 710 P&O 10 650 10 42

'A ::z 9 320 9 814 10 %23 11 128 19%

3 3 8 9 10 11

“1’;

1)

1)

[521 f521 II

3/2+, 3/2 l/2+,3/2 3/2+,3/2 5/2+,3/2 3/2+,3/2 3/2+, 3/2 2 +, 2 2 c, 2 4 +, 2 0 +, 2 2 c, 2 2 +, 2 4 +. 2 0 +; 2 2+, 1 1 +, 1 0 +, 2 2 +, 2 2 +, 2 4 +. 2 0 +; 2

l/2 l/2 :$::;g :~~::;$ 3/2+, 5/2+,

energies and excitation energies

2

:g 942 026 010

Et 697 670 601 099 028 555 556 z:; 077 947 764 932 332 157 344 826 860 864 784 :$ 436 420 342 359 228

: 5 4 4 4 5 5 5 4 4 4 4 : 5 5 5 5 5

031 035 620 511 404 : 4 4 4 5 5 s 5

5 5 5 5 5

:s

4 e52 4 230 4 151 4 201 4 197 4 670 4 559 4607 4 636 5 029

L 953

: 4 5 5

4 761 4 683 4 585

::: 230 639 791 855 847 841 0

I:E 0 029

',"z 2 849 4 253 4 345

& 1 819 3 868 4 515 6 857 7 202 4 606 6 162 6 694 f3394 9&o 10 179

; 4 7 8 9 8 9

0

1660

: ;

;

9

;;

3 10 10

5

10 2 2 2

10

10 10

3 10 15 11

; 9 12 12

30 30 30 30

330 80 1 80080 2 760 ea 2 960 80 'I900 20 710 20 850 20 10 650 20 17 310 20 19 ow 20 19lgil 20 20 010 20 437 10 170 10 974 9 740 20 320 20 550 20 10 380 20

s 8714

3

m) q) m) m)

m)

-2&s

-3m

- 31P

- 315

3op

31s

3lP

3m"‘3or

2%P

2&t-2@

2&a".2w

oanp,-afm

2751 - 27p

27h.t- 27s~

~zJ+l)-ml,n)

e&&N+11

8)

2 +. 2

3 -” 2 1 c, 2 o+, 1 3 +, t

2 *, 2

t

i

5

721 96s w

..I

1;

20

4 15

-

-

37Ar

3GK

- 36&

37Ar -37x

3m.

:;f

364x

:; 20 28 8 11 14

3m.

3f-s -3&l

35.a - 35K

35cf.

3ici

33cl

335

32s

32P

31s

31P

19

:: 16 16 17 17

Lz 40

15 IO

1:

13 13

Ti

h

11 1, 6 11 6 11 11 6 11 11

10 10

:

IO

517

522 411

24

2; :k

24

R. J. DE MEIJER

et al.

COULOMB

DISPLACEMENT

25

ENERGIES

TABLE 3 Calculated nucleidic mass excesses

Nucleus

Previous work (kc+)

Present work (kev) 16939+ 5 16615$- 4 17470+60 10797+ 12 3766+ 6 - 823+ 4

l5F “‘Ne z”Mg 24Si zJSi =P

18600&700 “) 3980

4333&10 - 7080120 - 9390*30 -1124o,t30 -13859-+15

2% 3%X 33Ar J5K 3%C

-9400

b,

b>

“) Refs. 9*143). b, Estimated value calculated from ref. 9).

f0.43

2:1

9.46

2-q

9.21

2:1

8.900

3-J

6.79 ,659

22'

6.77 6.61

GY:1 0:l

6.29

o-1

i:1

5.91

(0')

6.064

$170

(0

1

I 2.430 0..

'i

I

14Cev Fig. 2. Calculated

14Nexp

gs2rj

-----J 14N talc

140exp

oT

‘4OCdC

and experimental excitation energies of levels in A = 14 nuclei. The binding energies are given relative to “N(0),,,. See also fig. 1.

26

R. J. DE MEIJER et al.

This effect for the lowest J” = O+, T = 1 state appearing in all doubly odd self-conjugated nuclei will be discussed later. For the other levels, except for the E, = 5.91 and 7.78 MeV levels in 140, the deviation between the experimental and calculated value is smaller than 50 keV; in particular the change in level sequence of the J” = 3- and O-

/ [I- k!

states for the different A = 14 nuclei is explained. The E, = 5.91 MeV level in 140 has been assigned ‘) J” = (Of); however, from the calculations one would suggest J” = O-. In any case one level at about 6 MeV is missing experimentally.

1 I! 0.392

1-

0295

3.

0.120

,640

3.

042

(1-i

025

(1Yi)

0.66

3-

0.41

2‘

0.29

1-

0.02

0-

o-

16.31

2-

16Nexp

Fig. 3. Calculated

0.71

%?xp

0-

'+exp

and experimental excitation energies of levels in A = 16 nuclei. The binding energies are given relative to 16O(O),,,. See also fig. 1.

Other examples where level sequences are interchanged are the A = 16 nuclei, presented in fig. 3. The level sequence for the T = 1 states changes from 2-, O-, 3-, l- in 16N into O-, 2-, l-, 3- in 160 and probably O-, l-, 2-, 3- in 16F. It is the clearest example of change in the level sequence in In figs. 4-6 the experimental and calculated data 38, respectively. Many levels were calculated which calculated and experimental data agree except for of 38K (see sect. 4).

the are are the

mass region considered. presented for A = 27, 31 and experimentally unknown. The lowest J” = O+, T = 1 level

4. Discussion and conclusions In fig. 7 a histogram is given presenting the weighted number of levels versus the deviation between the experimental and calculated Coulomb displacement energies. For each deviation a rectangle of constant area has been drawn; the base of the rectangle equals two times the experimental error, 0. The histogram is approximated in a least-squares fit by a Gaussian curve with a mean deviation of 64 keV, which means

COULOMB

DISPLACEMENT

ENERGIES

I

1 2 ‘2 2

3

2

*‘Pew

844

!I

.

‘4 5/2‘

2kxp

2?tl

n’calc

27Si -Y?xp

Fig. 4. Calculated and experimental excitation energies of IeveIs in A = 27 nuclei. The mass of 27P is exper~ent~y unknown and is therefore represented by a dashed line, which indicates the position where z7P(0) is expected. The binding energies are given relative to z7Ai(0)~,. See also fig. 1.

that about 70 % of all levels considered show a deviation between experiment and calculation smaller than 64 keV, which is about 1 ‘A of the mean Coulomb displacement energy in the mass region studied. From this 64 keV one may subtract quadratically the internal error, due to uncertainties in the experimental input data. The remaining 56 keV arises from the assumptions made in the model itseIf and in the wave

R. J. DE MEIJER et a!.

28

k

l__---L 31

Pew

31

PcoIc

31

S ew

31

SCQIC

Fig. 5. Calculated and experimental excitation energies of levels in A = 31 nuclei. The mass of 31C1 is experimentally unknown and is therefore represented by a dashed line, which indicates the position where 31Ct(0) is expected. The binding energies are given relative to 3*P(0),,,. See also fig. 1.

COULOMB

DISPLACEMENT

ENERGIES

72

$0

ru:

13

cl*

20

7518]

3

38 Fig. 6. Calculated

K ew

and experimental excitation energies of levels in A = 38 nuclei. The binding energies are given relative to 3sK(0),,,. See also, fig. 1.

functions used. The contribution due to the incompleteness of the wave functions is obvious at the A = 19, 26 or 28 nuclei, where the wave functions ‘“) are no,t so successful in predicting the excitation energies, and where the calculations also lead to relatively large deviations in the Coulomb displacement energies. This correspondence suggests that calculations of the type presented here might be an additional test for the wave functions.

30

R. J. DE MEIJER et cd.

In the A = 39 multiplet almost all calculated Coulomb displacement energies are 140-230 keV higher than the experimental values. In the wave functions used r14*r2’), no If: or 2p2 configurations have been taken into account. The large difference in single-particle Coulomb displacement energies for Id, and If3 or 2p particles makes the Coulomb ~splacement energies for the A = 39 levels very sensitive to f2 and p2 admixtures.

0

I

/

1

--.---

Fig. 7. A histogram presenting the weighted number the calculated and experimental Coulomb displacement lowing way: for each deviation a rectangle of unit area times the uncertainty, o, in the deviation. The

AE:;:

-AE:;;g

(it&)

of levels, N/20, versus the deviation between energy. The histogram is built up in the foiis drawn: the base of the rectangle equals two best-fit Gauss curve has been drawn.

The peak at about 220 keV in the histogram originates in part from these levels. In addition it is due to J” = Of, T = 1 levels in doubly odd self-conjugated nuclei. The differences between the calculated and experimental Coulomb displacement energies for the lowest J” = Of, T = 1 levels in 14N, l*F, 22Na, 26Al, 3oP, 34C1 and 38K are 118,231,224, 145, 109,213 and 223 keV, respectively, whereas the deviations for the lowest J” = O+, T = 1 levels in the doubly even nuclei 140, ‘*Ne, “Mg, 26Si,

COULOMB

DISPLACEMENT

ENERGIES

31

3oS, 34Ar and 38Ca are -46, - 15, 30, -36, - 12, 50 and 70 keV, respectively. The other levels in doubly odd nuclei are predicted well. The exceptionally Iarge deviation for the lowest J” = 0*, T = 1 states in doubly odd self-conjugated nuclei is almost constant; this might be caused by the Coulomb pairing energy effect. Notwithstan~ng the few discrepancies discussed here, the quahty of the overall agreement justifies the c~lcuIations of the excitation energies of levels that have not yet been observed ex~rimental~y (see table 2) and of mass excesses of experimentally unknown nucIei (see table 3). The authors want to thank Professor P. M. Endt for his stimufat~~g interest. They are indebted to G. A. Timmer for valuable discussions and to Dr. P. W. M. Glaudemans and Dr. S. Maripuu for the wave functions. The help of G. de Roos with the computer work is gratefully acknowledged. References I) J. A, Nolen and J. P. Schiffer, Ann. Rev. Nucl. Sci. 19 (1969) 4’71 2) 3) 4) 5) 6) 7) 8)

9) 10) 11) 12) 13) 14) 15) 16) 17) 18) 19) 20) 21) 22) 23) 24) 25) 26) 27) 28) 29) 30) 31) 32) 33)

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32

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