Charge asymmetry of nucleon widths of the lowest T = 32 states of 13N and 13C

Nuclear Physics A161 (1971) 492-520;

@ North-HollandPublishing Co., Amsterdam

Not to be. reproduced by photoprint or microElm witbout written permission from the publisher

CHARGE ASYMMETRY OF NUCLEON WIDTHS OF THE LOWEST T = 3 STATES OF 13N AND 13C A. ARIMAt and S. YOSHIDA Department of Physics, Rutgers University, New Brunswick, New Jersey 08903 tt Received 5 October 1970 Abstract: The isospin-forbidden widths of the lowest T = .a states of 13N and 13C are calculated by including isospin mixing due to Coulomb interaction and charge dependent nuclear forces in the first-order perturbation approximation. The widths are obtained by using the Feshbach formula with the continuum-state wave function distorted by an optical potential. It is found that the widths are sensitive to the exchange character of the charge-independent nuclear potential. The absolute magnitude of the widths, and the charge asymmetry in the branching ratio to the ground and first excited states of 12C are found to be compatible with the theoretical results, and the presence of a charge-dependent nuclear force is not essential for this agreement.

1. Introduction When target nuclei with A = 22 are bombarded with protons sharp analogue resonances are observed in light nuclei. These resonances have T = 3 and are formed by an isospin-forbidden process. The decay widths to various states are measured, ‘*‘) and in some cases the widths are obtained for both members of a pair of mirror nuclei 3P4)* The branching ratios of these decays are known to be very charge asymmetric. The isospin-forbidden width is a measure of isospin impurity, which is of great interest in nuclear structure studies. However, very little theoretical analysis has been carried out thus far, and we do not know how important information might be obtained from these experimental data. In a previous paper “) we investigated the resonances in A = 13 nuclei. Reasons to choose A = 13 nuclei are that considerable experimental data, as well as good wave functions, are available. We found that the isospin mixing due to the continuum state was important and by taking this effect into account the right order of magnitude for the widths is obtained. However, we could not explain the branching ratio of mirror nuclei. In this paper we extend our previous calculation by using various types of exchange character for the central charge-independent nuclear forces. Furthermore, we vary various parameters and examine effects neglected in the previous paper. AS isospinmixing interactions we consider charge-dcpyndent nuclear forces, in addition to the Coulomb interaction. Special attention is paid to the isospin mixing within the p7 Present address: Department of Physics, University of Tokyo, Tokyo, Japan. tt Work supported in part by the National Science Foundation. 492

NUCLEON-WIDTHS

CHARGE

ASYMMETRY

493

shell by the single-particle potential. By this extended treatment we find that the observed branching ratios for the mirror nuclei 13C and 13N are compatible with theoretical predictions without a charge-dependent nuclear force. If the force is as strong as the upper limit given by other experimental data, the effect is not small. However, information about the strength is not enough to get a definite conclusion about importance of the force.

2. Feshbach’s formula for nucleon widths We derive the formula for the width according to Feshbach “). Our starting point is the following Schriidinger equation: (%-E)Y

= 0,

(2.1)

where ~8’ is the Hamiltonian for our nuclear system, E is the eigenvalue and !P is the eigenfunction of 9. The Feshbach formula includes projection operators P and Q, for which we choose the ones introduced by Shakin ‘). This choice of projection operators seems to be most convenient for the present problem. Let us first define a singleparticle (s.p.) potential U(i) which generates s.p. wave functions. This potential contains the s.p. Coulomb interaction, and is therefore charge dependent. If we denote eigenvalues and eigenfunctions for the s.p. potential U(i) by E, and Ii, a), we have the following SchrZidinger equation [T(i)+ U(i)-&Ji,

x) = 0,

(2.2)

where T(i) is the kinetic energy operator of the ith nucleon. The quantum number a stands for the spin, space and charge quantum numbers. The s.p. projection operator q(i) for the ith particle is defined as q(i)li,

a) = 0, 8, >= 0,

q(i)li,

cl> = Ii, a>, E, < 0.

(2.3)

This operator q(i) projects onto the bound state of the single particle. The operator Q is given by

Q = dM2)

- - - q(W,

(2.4)

where N is the total number of nucleons in our system. The projection operator P is &fined by P = l-Q,

(2.5)

which projects on to a contiguration which includes at least one continuum state in the s.p. wave function. This is also charge-dependent, because it is based on the potential U(i) which is charge-dependent.

494

Using these projection equations:

A. ARIMA AND S. YOSHIDA

operators

we can write eq. (2.1) as a form of coupled

(E-.@‘,,)P@J’ = =@P~QY,

(2.6a)

(E-Z,,),. = zQpw,

(2.6b)

where ZPP = PSP,

etc.

(2.7)

If we eliminate QY from (2.6a) by using a formal solution of (2.6b), we obtain

(E-H)PY = 0,

(2.8)

where

(2.9) Next we introduce a complete set in Q-space as eigenfunctions of ZPQQ, (ES-S?*&&

= 0.

(2.10)

The eigenvalues E, are all discrete, as the Hamiltonian is defined onIy in Q-space, and s is a quantum number which specifies these states. This state GS corresponds to a compound state in the Feshbach theory. By using the complete set @, in Q-space, we can write the effective Hamiltonian in the following form: (2.11) Suppose we calculate the width of a compound state characterized by quantum number S. The resonance terms which appear in eq. (2.11) can be divided into two parts, one corresponding to the resonance state s, and the other to the remaining states : (2.12) where (2.13) We introduce the eigenfunctions for H’ by (E-H’)ll/z;

= 0.

(2.14)

The quantum number a specifies the quantum state of the target nucleus, and the spin and parity of the incident nucleon, while the k is the wave vector of the incident nucleon. The (-l-) sign indicates the asymptotic boundary condition that all waves are outgoing except for the incident channel a. The partial nucleon width is given by (2.15)

~CL~ON-WIDTHS

CHARGE

495

ASYMMETRY

The integral jds2, extends over the solid angle of the direction The normalization of the continuum wave function is given by

of the vector k. (2.16)

(I,@;, I,@‘,J = 6(k-k’)6,,. 3. Calculation of isospin mixing

We now consider the isospin mixing starting from a charge-independent wave function. For the time being the nuclear forces are assumed to be charge independent, so that the isospin mixing comes from the Coulomb interaction only. The basic Hamiltonian G%?of our system is composed of three parts 2

= T-+-&l-V,,

(3.1)

where T is the kinetic energy operator, VN the charge-independent nucleon-nucleon interaction, and Vc is the Coulomb interaction which is a sum of scalar, vector and tensor parts in charge space. As usual we introduce a s.p. potential U, which is a sum of individual s.p. potentials U(i). Then the Hamiltonian (3.1) may be written as sS? = T+UfAV,

(3.2)

AV=

(3.3)

where V,+ Vc- u.

The potential d V is the residual interaction. The s.p. potential is composed of a nuclear part and a Coulomb part. However, it is more convenient to split it into isoscalar and isovector parts U = i U(i) = 5 [Uff’(i)-l- U$-j)(i)+ U:‘(i)] i=l

i=l

= U(O)+ Up’,

(3.4)

where Us) is the ch~ge-inde~ndent s.p. potential, and U&‘)(i) is the isoscalar part of the s.p. Coulomb potential given by

and UA1)(i) is its isovector part.

(3.6) In the above equations ri is the radial coordinate of the ith nucIeon, and -:-zziis the

A. ARIMA

496

AND S. YOSHIDA

third component of the isospin of the ith nucleon. The charge of our nuclear system is Z+ 1, while R, = A*r,, is the charge radius within which the charge is assumed to be uniformly distributed. Let us first calculate the compound-nucleus wave function Qi,. The Hamiltonian for this wave function is given by j’e,,

= Q(T+ U(O)+ U~‘3_dV”‘+dV”‘+dV”‘)Q,

(3.7)

where A Y(O), A V(l) and A V(‘) are the scalar, vector, and tensor parts, respectively, of the residualal interaction given by eq. (3.3): A y(O) = J$,+ vd”’ _ U’O’ Av”’ = @“_ #,

(3.8)

Ay’2’ = vd2’. The scalar, vector, and tensor parts of the two-body Coulomb interaction are given respectively by KJ”‘(l, 2) = &

(3+r,

*z2),

(3.9a)

12 vp(l,

(rlz-tz2z),

2) = - -$ 12

@‘(l, 2) = ez [rr x 22]$2’. 2,/6rr 2 In order to calculate the eigenfunction a’s we use charge-independent as a basis. These satisfy the following Schrijdinger equation (E;O’-&+))@~)

= 0,

(3.9c)

wave functions (3.10)

where Z(O) = T+ U(O)+ AAV’“‘A+BAV’o’B.

(3.11)

In the last equation we have introduced another projection operator which is chargeindependent. For 13C and 13N we define A as the projection operator which projects onto the configuration space where the lowest s-shell is filled, the next p-shell is partially filled and other higher shells are all empty. In contrast to the projection operator P, which is referred to the s.p. potential U(i), the projection operator A is referred to the charge independent potential U(‘)(i). Therefore, the wave function A@:‘) which is an eigenfunction of eq. (3.10), is obtained by diagonalizing the Hamiltonian X(‘) within p-shell configuration space which is based on the chargeindependent potential U(“)(i). As before we define also B = 1-A.

(3.12)

The best available eigenfunctions for eq. (3.10) in p-shell coniigurations are given by Cohen and Kurath *), and we shall use them. We now calculate the approximate

NUCLEON-WIDTHS

497

CHARGE ASYMMETRY

eigenfunction satisfying (2.10). Although the relation between the two Hamiltonians .X?oo and &Y(O)is not simple because of the difference of definitions of the projection operators Q and A, the following wave function is a very good approximation to the eigenfunction satisfying eq. (2.10): A’ E(O)_ &W s

U(l) +

x @O’_ (3.13) The projection operator A’ is the same as A except that it excludes the state with energy Es(‘) . In the above equation the potentials UC’)+ A V(l)+ A V(‘) which appear in Zne but are not present in .Z?(‘) are considered as the perturbation term and the first-order correction is taken into account. However, the mixing of continuum states due to A V(l)+ A V(‘) is neglected. Let us denote each term of the right-hand side of eq. (3.13) as CDs= @jO’-tCDp,A”+ @5,A”+ @,““.

(3.14)

The first term of the right-hand side of eq. (3.14) is the unperturbed wave function. The second term @t” corresponds to the mixing within the p-shell due to the s.p. isovector potential; the third term CD:”represents the same mixing, but is due to the isovector and isotensor residual interactions. The last term CD:”represents contributions from excitation above the p-shell due to the s.p. isovector potential. The calculation of @I” which includes excitation to the continuum state will now be explained in somewhat more detail. This term is calculated by operating wnh (E(“)_~(e))-lB~(‘) on the unperturbed wave function c?!‘). As the potential UC’) s is a sum of s.p. operators, we can approximate this by neglecting the effect of nuclear correlation in Q-space as follows B

U(1) =

E’o’-

E(o) B,,@

s -

s

T U”‘(i)

= T ;

U”‘(i)y

(3.15)

I

where Asi is the energy denominator for particle i. In the above approximation we assume that E;“)_&o) N (@y’lT+ U(“)/@~o’)- T_ U(O) (3.16) and we have neglected the residual interaction term A Y(O). Thus the excitation energy is the same as the s.p. excitation. The operator 1 +(B/AQ) U(l)(i) just transforms a wave function from an eigenfunction of T(i)+ U(O)(i) to one of T(i)+ U(O)(i)+ U(l)(i).

In calculating the wave function $n.. (+) which satisfies the Schriidinger eq. (2.14), we have to make a more drastic approximation, namely H’ = T + V+ T(k) + U,,,(k) + U,(k), where T+ V is the Hamiltonian

(3.17)

for the target nucleus, while the other three terms

498

A. ARJMA AND S. YOSHIDA

are operators for the incident particle which is denoted by the symbol k. Its kinetic energy operator is given by T(k), the optical potential by U,,,,,(k) and U,(k) is the s.p_ Coulomb potential which has the same form as given in eqs. (3.4) (3.5) and (3.6). The wave function I&,‘*IS a product of two functions, the target wave function xIM and that of the scattering nucleon &+. The quantum numbers I and A4 are the spin and its z component for the target nucleus, respectively. The scattering wave function &, mbmay be decomposed into partial waves

where I, m are the orbital angular momentum and its z-component for the particle, i is its total angular momentum, m, is the z-component of the spin and v is the z component of the isospin. The vector k is the wave number of the particle while f; represents the orientation, p is the reduced mass and Xjm+mgis the spin-angular function of the particle. If the form (3.18) is put into the expression of the width (2.15) we obtain the following results (3.19) Because of the above approximation the two wave functions QS and $ci are not always orthogonal. However, they should be orthogonal because they are projected onto the Q and P spaces respectively. If we use the same s.p. potential both for constructing the compound or bound state Qp,,and the scattering state $::d they are automatically orthogonal. Otherwise we have to orthogonalize them. The method used by Kerman ‘) is adopted here. The radial wave function for scattering u,j(r) is replaced by

where GU(r, r’) is the Green function for the scattering problem: - ___

- U,,,(k)-

Up’(k)

1

Glj(r, r’) = 6(r-r’),

(3.21)

and Wlj(r) is the radial wave function for the bound state which is contained in as. 4. Calculation of matrix elements In evaluating the width given by cq. (3.19) we have to calculate the matrix element ofXPQ, where .?‘Fhas three parts as shown in eq. (3.2). However, the s.p. operator T+ U vanishes for off-diagonal matrix elements; hence wc just calculate the residual

NUCLEON-WIDTHS

CHARGE

ASYMMETRY

499

interaction. By decomposing it into irreducible tensors in isospin space we need to calculate the following matrix element M”’ = (\YcsT’MTIMIV (‘)- U(t)l[y~4T~10+*jJTMdM),

(44

is the three-hole where ye3T~MTI~

state to the I60 core corresponding to the A = 13 nucleus where the antisymmetri~tion is properlytaken into account, while !Z’_roMTdoMO is the four-hole state corresponding to the 12C nucleus. The symbols ~~~‘~~~~ specify the quantum state of the A = 13 nucleus; T’ is the isospin, MT the third component; Z, the spin, M, its z component, and cj stands for any other quantum numbers which specify the state. The symbols c,TOM,,Z,MO are the corresponding quantities for the “C nucleus. The wave function JI+,I,,, describes the continuum state nucleon, which is characterized by the isospin quantum number 3, its third component v, the total anguIar momentum j, and its z-component Q. Finally the symbol P’r,rl yTzfz L.IM stands for a vector sum of two angular momenta and isospins: C%,II

c P-1MT1 T2&,I~&x~I MI1’2&IW f%121TMTIM =WT1MT2M*M* XYTlMT,llMl Y‘ TzMT~I&z

-

(44

The many-body matrix element (4.1) may be reduced to two-body matrix elements by using the two-body coefficients of fractional parentage (c.f.p) and the Racah coeflicients. Applying this reduction we arrive at M(‘) = 3JZC


T’Z(Jj; ji(T, Z,)jJ T’Z)

x
-

Z2)T,

Z0>J(2To

f

l&T,

+

1)

1)(2Za + l)W(Z, I2 Zj,; Ze Zs)(TMT tOjT’Mr) T; 7; t)sj,,

1 J2j3 + 1

(4.3)

ut”‘it~T212&1T3f3>’

where the reduced matrix element is given by x

&+T

= J2t+l <3v3

j3

JJ (-)T3-vs(T3~3~-~31t0) V,m3 m3

I~“‘- ~“‘ICT, 12IIj4lT3v3jpts>-

(4.4)

The calcufation of the particle-hole matrix element appearing in the last equation is carried out using the transformation of the nucleon operator into a particle and a hole operator: b: = a:, if CIis particle orbit, (4.5) = a;, if 0:is hole orbit. The operators a,* and a, are creation and annihilation

operators for the nucleon

500

A. ARIMA AND S. YOSBIDA

specified by a which is a shorthand notation for the third component of iisospin v,, orbital angular momentum &, total angular momentum j, and its z-component m,. The symbol b, stands for the particle and hole operator. The notation a”is to represent the time-reversed state of LX; 14) = (-)3-‘““j”-““[li-v,j,-m,). (4.6) The two-body interaction may be expressed as

If we write this in the normal form, we obtain V”’ =

C

@ =hole

6=partiole


V$

2> -I- @

3j

+.

. . .

(4.8)

The hrst term is a s.p. operator, so that t is restricted to 0 and 1. There is another s.p. potential UC’)to be calculated in the matrix element appearing in eq. (4.3). In the case of the isovector (t = 1) both matrix elements for Y{&, and U(l) vanish in the first-order perturbation approximation. In this approximation both the compound and scattering states must be unperturbed ones with 7’ = 2 and 3, respectively; In the compound state all nucleons are bound while in the scattering state one nucleon is in the continue state. Hence the s-p. potential must act on this nucleon, and the matrix element contains the overlap between “C (T = 0) and the corresponding part of the compound state which must have T = 1. Therefore the matrix element vanishes. In the case of the isoscalar (t = 0) neither of the two terms V$,)2, and U(O)vanish. However, they appear in eq. (4.3) in the form of a difference, and we can choose which is the interaction between the U(O) such that it cancels the effect of ?$$‘:z’2,, nucleon in the cont~uum state and the 160 core. In a practical cal~~a~on our choice of U(O) does not fuh?ll the condition of exact cancellation of I$‘,$, and U(O), but we neglect the difference of the two. Therefore, we consider the second term V&) only in the following calculation. For the practical calculation of our problem we need to calculate the following special cases:

~CLEO~-~DTHS

501

CHARGE ASYMMETRY

x ~~~tOT”MT3iTl M&32 A TzMT~IAV”(L 2)l.hj4 T”.MT&>~.~. ,

(4.10)

where Y@)(l, 2) is a two-body potential and t is the rank of the isospin tensor. In the following the two-particle matrix elements are calculated. For the nuclear potential we assume that only a cent& force is effective, and that the radial part is of the Yukawa type. The nuclear force between nncleon 1 and 2 is given by (4.11) wherefo and fi are the interaction in isospin singlet and triplet states, respectively. These are further expressed as fo = -

3+a,

[

*(F+

‘-7-62Aoo+ 4

e-arlz 01 Qo-----9 1 gr12 (4.12)

Let us first calculate the isoscalar interaction; the antisymmetrized written in terms of the ~-a~tisymmetri~d one as (j,

_i2W@"'l.hj4T,,.

=

1 __--

matrix element is

C<.L j2W&ij4O

J~jJ(1+6j3jJ -f-C-> '+jr+jz+I(

j,j,I/fTf j,j,I>].(4.13)

The un-antisymmetrized matrix element may be calculated by using a standard method ’ o)9 which is not given here. The two-body Coulomb interaction has non-vanishing matrix elements only between T = 1 states. The matrix element for isospin space is easily calculated, and the nuclear matrix element of the Coulomb interaction becomes n/r$&@’ X
z -3J5

Z:
ji(T, 11)j~ T’I>

j,j~o~0(lj;j;(~~,)j,j2(T,~2)01,)2/(2~,+1)(2j,+1)W(~,~2U4;

10j3)

(4.14)

A. ARIMA

502

AND S. YOSHIDA

Finally, the matrix element due to the Coulomb distortion of the radial shape of the nucleon wave function is discussed. The interaction for this matrix element is the isoscalar nuclear force. As discussed in sect. 3, this is the excitation of a single particle into continuum states by the single-particle Coulomb interaction. We only take into account the lowest order in et. Then the evaluation of the nuclear matrix element is carried out by just replacing the two-particle matrix element in (4.10) by <_j,_j, TZ%,&IV’“‘(l,

2)l&

where j means replacing the charge-independent

f4

T’z&-zfA.s.

(4.15)

radial wave function by a charge-

dependent one wlp’(r) --, CwiO)(r) = w~“‘(r)+fZZAw,(r).

(4.16)

If we use the notation Aj to represent the radial wave function dwj(r), we can write

(Ajl j2 TWl, V”‘(1,2)Ij3 j4 T’O + +

+

+ exchange terms].

(4.17)

The final expression of the nuclear matrix element has the following form:

To complete the calculation of widths we need to obtain isospin mixing coefficients within the p-shell configuration. This is carried out by using the same zeroth-order charge-independent wave function as above, and the method of calculation is similar to that for the above matrix elements but less complicated, so that details are not given here. 5. Results with Coulomb interaction o&y Let us first introduce s.p. potentials. For bound-state wave functions the following charge-independent potential is assumed

NUCLEON-WIDTHS

CHARGE

503

ASYMMETRY

where x = (q-z&,)/a.

(5.2)

The last term of the right-hand side of eq. (5.1) is the isoscalar part of the one-body Coulomb potential given by eq. (3.5). All other notations are standard, and no fm-ther explanation is necessary. In calculating the charge-dependsEt part of the radial wave function dk~,(r) given by eq. (4.16), the isovector part of the one-body Coulomb potential given by eq. (3.6) is added to the potential given by eq. (5.1). Pn generating the continuum wave function the following form of optical potential is assumed: I 2% ___ dr, 1 i-e””

__.!L-iiiWGe-xez+4ia,FVD

U,,,(k) = -

l+e””

+rJ’

LS

Id

1

B. ES U,(k), r, dr, 1 + exn

(5.3)

where

A%,,)/a, xo = (r,- A%“,,)/ao XR = (Q -

xD = (r,-- A+r,,)/a,

(5.4) .

For the proton potential the Coulomb potential is included in its full form, while the neutron potential contains no Coulomb potential at all. The parameters appearing in the above potential are chosen in the following way. First we choose optical-potential parameters describing the nuclear scattering by “‘C For these, two sets are used in our calculation. One is the set given by Nodvik et at. lx), w hi6:h was obtained by x2 fit with experimental data for protons at each incidelpt energy: Unfortunately, parameters for the energies corresponding to the resonance are not available, so we use averaged parameters which are given in table 1, labeled NDM. The other set of parameters is the one given by Watson et al. I’) who tried an overall fit to the experimental data. This is also listed in table 1 with the label WSS. TABLE 1

Optical potential parameters ,

r0’OR NDM WSS

(f-m)

(2,

1.25 1.15

0.35 0.57

&)

(MeV?m2)

52.6 61.05-0.3 EC.,.

11.4 11.0

b

g) 1.25

0.25

J%D

wG

(fm) (MeV) 19.0 0

(2)

&)

1.15

0.57

(MeV)

(;z)

0 1.4 0.64 E,.,. 1.4

For the bound state the same geometrical parameters as those of the optical potential are used. The depth of the Woods-Saxon well and the strength of the spinorbit interaction are adjusted to fit the s.p. energies of pa and p4 orbits. However,

504

A. ARIMA

AND

S. YOSHIDA

there is no easy way to find the s.p. energies from experimental data. Hence we first consider 13C and r3N to consist of a single nucleon outside of the “C core, and the separation energy is assumed to be the binding energy of the p+ orbit. For the pt orbit the first +- states in 13C and 13N, whose excitation energies are 3.68 and 3.51 MeV: respectively, are assumed to be states where pt nucleon jumps to the pit orbit, and its excitation energy is assumed to be the difference in binding energies between the p+ and pi- orbits. The parameter set obtained is given in table 2 with the labels NDM or WSS indicating the origin the of geometrical parameters. TABLE 2 Bound-state potential parameters Neutron

NDM NDMl wss

ro(fm)

u(fm)

G(Mev)

1.25

0.35 0.35 0.57

42.43 53.96 50.12

1.25 1.15

U’L,(MeV- fm’) r&m) 16.41 18.33 16.58

1.40 1.40 1.40

P+ 4.95 12.62 4.94

B.E.

Proton

B.E.

p+

p+

pa

1.85 9.25 1.94

8.09 16.24 8.06

11.25 19.58 11.25

The second parameter sets are obtained by adjusting the single-particle energies for ’ 5N and ’ 5O nuclei, considering both the ground state and the first +$- state to be single hole states. The depth parameters are fixed for A = 14, which corresponds to the mass number of the remaining part of I60 by removing one nucleon and the same parameters are used for the A = 13 system by putting A = 12 for the nuclear radius parameter. The parameters and binding energies for the mass number 13 system are given in table 2 with the label NDMl. We fix the charge radius rot as 1.4 fm to give the Coulomb energy difference of the 13C and 13N ground states. The same value is also used in the optical model potential. For the nuclear wave functions of the A = 12 and 13 systems, those given by Cohen and Kurath s) are used. They made a x2 tit to the energy spectra of p-shell nuclei by considering the two-body matrix elements as unknown parameters, without explicitly introducing a two-body nuclear force. All possible configurations within the p-shell are utilized. They obtained several sets of two-body matrix elements; among them the one denoted as (8-16)2BME is used in this calculation. Wave functions which constitute the main part of the compound and target nucleus states in our calculation, are listed in table 3. They arc the lowest T = $, I = 3 state for the A = 13 system, and the lowest T = 0 and I = 0,2 states for “C. The Coulomb interaction mixes states with different isospin, whose wave functions are also obtained by using Cohen and Kurath’s two-body matrix elements. The isospin mixing is calculated using first-order perturbation theory as explained in section 4. The energy of mixed states and amplitudes are given in table 4.

NUCLEON-WIDTHS

CHARGE TABLE

505

ASYMMETRY

3

Main wave functions for residual and compound

states

12C ground state (T = 0, Z = 0) IGO>= O.63l4/(jf)4>t-O.612~~~~)~~(~)~o>+O.235S~(~)~~(~)~1> - 0.2688~(~)~>+0.3136[(~)c> izC 1st 2+ state (T = 0, Z = 2) 102> = O.6735~~(~)3>~O.2417~(~)~~(~~~~>-O.O794/(~)~3(~)~1> - 0~3340~(~)~&,>-0.5368~(~)~~>-0.1034~(~)~~~> -t 0.24521(@4>-0.0933(#)\4’> A = 13,&

= 14.67 MeV (T = g, Z = 8)

IS@ = 0.95721$(~);o>+o.1726~(~)~2~>+o.2324~($)3> Configuration

(ISt)rZIzm> w h ere H and m are the numbers of holes.

is denoted by [(p&lfln

TABLE 4

Isospin-mixing Nucleus ToTo 00

i &(MeV) G(l0)

lZC

02

12C

&tMW

%?

1

2

19.46 0.00042 16.61

&(MeV)

3.71

G @%I

0.0

0.00024 10.47

-0.01516 3.71

4

23.16

25.84

5

6

7

-0.00186 20.04

0.00162

Ei(MeV)

3

35.99

G(l2)

Gw =N

coefficients due to two-body CouIomb interaction

0.01242

-0.00007 14.36 0.04549

0.00209 17.69 -0.03672

28.61 -0.00005

33.64

35.96

0.00080

0.00054

19.83 0.01715

10.47

14.36

17.69

19.83

0.0

0.0

0.0

0.0

Mixing amplitudes Ci(TZ) for state with (7’, I) and excitation energy Et are given for the 12C ground state, the first excited state and the first To = $ In = # state in 13C and 13N, The potential parameters NDM are used.

In table 4 only the mixing amplitude via the two-body Coulomb interaction within the p-shell is listed. The mixing by s.p. Coulomb interac~o~ within the p-shell co~fi~ation is negligibly small. This will be discussed in sect. 6. However, the mixing by the one-body Coulomb interaction for continuum states is very important, and a percentage for this type of mixing is calculated. From eq. (3.13) the mixed part of the wave function is written as A@?;=Q

jp) s

B

_ &m’

(5.5)

506

A. ARTMA AND S. YOSHIDA

By taking the norm of the wave function, we obtain the mixing probability

These percentages together with those of the p-shell configuration by the two-body Coulomb interaction are given in table 5. From the table we notice that the mixing of the continuum configuration is large and does not depend much on the particular state. When nucleons are tightly bound, iike in the cast of NDMI, the continuum contribution becomes less important, while the p-shell contribution increases. TABLE 5

Isospin-mixing Potential paramctcrs -r-__-..

percentages duo to Coulomb interactions

13C?.,.

=N$.$

.~

._~

NDM

.-

..._..~

p-shell continuum total

0 0.40 0.40

0.41 0.40 0.81

0.00 0.40 0.40

0.00 0.53 0.53

p-shell continuum total

0 0.0s 0.08

0.65 0.0s 0.73

0.00 0.07 0.07

0.00 0.07 0.07

0.29 0.55 0.84

0.00 0.43 0.43

0.00 0.52 0.52

-

NDMl _wss

‘*cz + -.-

--.-__-

--..-

1°C*a.

.-

____ p-shell continuum total

.

_-_.. 0 0.55 0.55

_...-_

The charge-independent nuclear forces which enter in the yamiltionian Z?,, should be consistent with those which appear in see. However, we used CohenKurath matrix elements in calculating the bound-state wave function and we do not know which nucleon-nucleon potentials correspond to the two-body matrix eIements. AIso, for simplicity, we restricted ourselves to a central potential only having the Yukawa shape. We adopt five types of exchange character which are commonly used. For the range parameter two values, a-l = 1.414 fm and a-’ = 1 fmare chosen. The former is most commonly used in shell-model calculations and the latter was used by Satchler 13) in microscopic calculations of direct reactions. The strength of the Yukawa interaction is taken from the work by Ball and Cerny 14). In converting the strength for different range parameters, we use the relation given by Satchler that the elTect of an interaction is the same if the value of the following

NUCLEON-WIDTHS

CHARGE

507

ASYMMETRY

TABLE 6 Charge-ind~endent force parameters Exch. charact.

00 (MeV) 100

CL-~(fm)

A 00 1.0

Wigner

Rosenfeld

35.36

1.414

135 47.74

1.0 1.414

135

1.0

Serber 47.74 135

135

All

1.0

1.0

1.0

-2.78

1.0

0.6

-0.34

0.0

1.0

1.0

0.0

0.0

1.0

0.634

-0.366

0.0

1.0

0.625

0.0

1.0 1.414 1.0

True 47.74

Al0

1.434

Ferrell-Vischer 47.74

AOI

1.0

1.414 TABLE 7 Nucleon widths

Nuclear force Projection

s.p.pot.

n0

NDM

area 1.414

1.000

yes

NDMl

1.414

wss

1.414

NDM

1.414

X&

.r P?.

2’,0

W R S FV T W R S FV T W R S FV T W R S FV T W R S FV T

0.04 0.25 0.19 0.13 0.10 0.09 0.38 0.38 0.22 0.17 0.07 0.18 0.17 0.12 0.11 0.04 0.23 0.19 0.13 0.10 0.06 0.31 0.25 0.17 0.13

0.22 0.14 0.27 0.14 0.15 0.46 0.30 0.61 0.30 0.34 0.16 0.11 0.19 0.13 0.14 0.25 0.18 0.33 0.16 0.18 0.40 0.25 0.49 0.25 0.29

0.0008 0.0008 0.03 0.003 0.0001 0.006 0.002 0.07 0.0008 0.002 0.~0 0.16 0.0000 0.03 0.05 0.0001 0.004 0.03 0.002 0.0005 0.0001 0.004 0.03 0.0001 0.004

0.27 0.37 0.44 0.20 0.25 0.63 0.83 1.07 0.5 0.58 0.43 1.09 0.93 0.57 0.61 0.43 0.45 0.64 0.30 0.37 0.54 0.76 0.89 0.41 0.49

exp.

0.24

0.14

0.40

1.5:’

Exch. Char

r n2

Proton width decaying to the ground state of %, rpo to the first excited state of “C, r,, and corresponding neutrons widths I’,,, and X”2 are given in units of keV. The last row shows the observed width.

508

A. ARIMA

AND

S. YOSHIDA

integral remains the same:

s

vo

C&vo!T. ar

a3

The parameters which characterize the charge-independent nuclear forces are listed in table 6. Computing codes are constructed to calculate the nucleon widths according to the method outlined in sect. 4. Using values of parameters given above, we obtained the results listed in table 7 in which experimental widths and parameter sets used in the calculation are also given. The proton width to the ground state of “C is denoted by rPo and rP2 = rpZ,+ -tTPz,+ represents the decay width to the first excited state of “C. The neutron widths are denoted by subscripts n instead of p. The label “yes” or “no” in the first column indicates whether the projection procedure given at the end of sect. 3 is carried out or not. Looking at the table, we first notice that the branching ratio is very sensitive to the exchange character, while a change of s.p. potential parameters changes the magnitude of the widths, but affects branching ratio only slightly. All exchange characters give I’,, < r,, in agreement with experimental data, but only the Rosenfeld mixture gives rPo > rP2 in accordance with experimental data. The effect of decrease of Yukawa range by a factor J2 is seen to increase all widths by a factor of about 2. When the depth of the bound state Woods-Saxon well TABLE 8 Nucleon-width

amplitudes with Wigner and Rosenfeld forces

Exch. Comp. Res. char- nucl. nucl. j act.

o+

B

0

2’ 2”

;t $

0 0

w

o+ 2+ 2”

2 8 %

-0.0071 +O.O042i 0.0081-0.0047i 0.0259-0.01041’

o+

0

13N

2+,

2*

B

0 0

0.0035-0.00201’ -0.0058+0.00361’ -0.0006+0.0002j

0.0057-0.00311 -0.0034+0.0022i -0.0012+0.0005i

0” 2+ 2”

* 4 Q

-0.0303$-0.01791’ -0.0003+0.0001i 0.0433-0.0173i

0.0032-0.00181 -0.0049+0.0029i -0.0005+0.0002i

0 0 0

=N

0.0034-0.00201' 0.0057-0.00311 -0.0078+0.004% -0.0034+0.0022i 0.0001 -O.OOOOi -0.0012+0.0005i

0.0064-0.00393 -0.0220+0.0130i 0.0060-0.00252’

W

R 13C

0.0031-0.0018i -0.0067+0.00393 0.0001-0.000 i

0 0 0

0.0061-0.0038i -0.0192+0.0107i 0.0052-0.00213 0.0301-0.0185i -0.0168+0.0097i -0.0067+0.0027i 0.0285-0.01822’ -0.0148+0.00823 -0.0062 +O.O024i

Wigner and Rosenfeld force with Cc-l = 1.414 fm is used as charge-independent NDM, and the projection is not taken into account.

parameters are

force, potential

NUCLEON-WIDTHS

CHARGE

ASYMMETRY

509

is increased, the proton widths decrease in most cases while the neutron widths increase. The effect of the projection seems to increase all widths by a factor of about 2. In order to understand how the charge-asymmetric branching ratio arises, and how the branching ratios change with exchange character of nuclear forces, various contributions to the width amplitudes are listed in table 8. They are the matrix elements in units of MeV - fm3 calculated with <"vljXjmjXIM~PQ@s) appearing in eq. (3.19) single-particle parameters NDM, using Wigner and Rosenfeld mixtures. In the first column the compound nucleus, residual state, and angular momentum of outgoing nucleons (j) are indicated. In the second column (M,,,,) the amplitudes due to mixing in the A = 13 nucleus within the p-shell configurations are given, while amplitudes due to mixing in “C within the p-shell configurations are placed in the third column (1M,,,, ). The next column (M,) shows amplitudes coming from the two-body Coulomb interaction in .SYpQ. The last column (MB”) corresponds to amplitudes due to isospin mixing in the continuum state. As explained in sect. 3, there is no Coulomb mixing in the case of 13N. Therefore amplitude MAV13in 13N vanishes. The Coulomb matrix element MC in the case of 13C vanishes because here the neutron must change the state from a continuum to a bound one. The Coulomb matrix element Mc and MVA12are small compared to other matrix elements. Therefore for the proton only one amplitude MB, is important, while two kinds of amplitudes, M,,,, and Meu, are important for the neutron case. These two amplitudes interfere with each other: that is why the charge-asymmetric branching ratios are found. The amplitudes MB, are almost charge-symmetric; a small difference for protons and neutrons comes from the difference in the scattering wave functions. However, these amplitudes are sensitive to the exchange character of the nuclear force. The same is true for MAvl 3. For the proton MB, amplitude for “C g.s. is large compared to the 12C, 2+ state in the case of the Rosenfeld mixture, in accordance with experimental data, while the tendency is opposite for the Wigner force. However, for neutrons with the Rosenfeld mixture, the interference with MAv13 reverses the branching ratio. Parameters are varied within the limits consistent with other experimental data, and fairly good agreement with experimental data is obtained in the case of the Rosenfeld force as seen in table 8. There is, however, another possibility remaining to be explored. This is the uncertainty of energy of the T = 3, I = 3 states closest to the compound state with T = 8, I = -3. Experimentally three T = 3, I = 3 states in 13N are known. The lowest one is at 3.51 MeV excitation, corresponding to the calculated value 3.71 MeV. The 9.48 MeV state is very weakly excited by pick-up reaction and there is no corresponding state among the calculated states. The observed 11.80 MeV state corresponds to the theoretical 10.47 MeV state. The 14.26 MeV state, which gives the most important contribution to the isospin mixing, has not yet been found experimentally. Therefore we have no information about the exact excitation energy. We tried to calculate the width by changing the energy difference As between the T = 2, I = -2 compund nucleus, and the closest T = 8, I = 3 state. The results are

510

A. ARIMA AND S. YOSHIDA

shown in table 9, where As is changed from the previous calculation (de, = 0.311 MeV) to $A&,, -*As,,, -de,. WC see that AE = %A&,,with the Rose&Id mixture gives much better agreement with experimental data. A change of de only changes the amplitude Mav13 (in the case of 13C); therefore, no change is produced in the proton width. TABLE9 Change of branching ratio with change of the energy of T = 4, I = 3 state Nuclear force CL-’ (fm)

exch. chararct.

1.414

W

R

S

Aside0 .-___.

r no

Ibo

1 4 -B -- 1 1 Q - b. -1 ;

0.04 0.04 0.04 0.04 0.25 0.25 0.25 0.25 0.19 0.19 0.19 0.19

-t -1

0.22 0.22 0.22 0.22 0.14 0.14 0.14 0.14 0.27 0.27 0.27 0.27

0.0008 0.02 0.23 0.11 0.0008 0.19 1.98 0.90 0.03 0.007 0.98 0.52

0.27 0.84 0.72 0.21 0.37 1.65 2.32 0.70 0.44 1.64 1.95 0.59

Nucleon widths calculated with varied energy separation between (3.3) state and its nearest (14) state. Potential parameters NDM are used and projection is not taken into account.

6. Effects of charge-dependent

nuclear forces

So far, only the Coulomb interaction is taken into account in our calculation as the charge-dependent interaction. However, there is some evidence that the nuclear forces are slightly charge-dependent. It is also known that there exist spin-orbit interactions derived from the Coulomb interaction as relativistic corrections. This charge dependence is supposed to be very small compared to the charge independent nuclear forces, but still the effect on the isospin-forbidden nucleon width is not small, because of the nature of forbidden transitions. The charge-dependent force is assumed to have the following form:

+

(

l--is,

[Zl x 2,]~2’ c20 --

4

* rJ2 +c 21

3+5!2? 4 >I

e-arl2

Vo -.

(6.1)

art2

For simplicity, the radial dependence of the interaction is assumed to be the same as the one of the charge-independent nuclear force. The first term of the left-hand side

N~CL~~N-WI~T~S

CHARGE

ASYMMETRY

511

of (6.1) is an isovcctor and the second term is an isotensor. The Iattcr interaction is the same between neutron-neutron and proton-proton, so that it is charge symmetric. For uOthe same value as the case of charge independent force is taken. Thcrcforc the coefficients C, e, C, 1, CzO, C,, , represent the strength of these interactions in units of the charge-independent force. These coefficients are exprcsscd in terms of p, q, r, s used by Biin-Stoylc 15) as Cl0 = p-3r, C,, = p+r, GO = Jf(q - 3s), C 21 = &I+s),

(6.2)

where

The most direct evidence for the charge dependence of nuclear force comes from ‘S nucleon-nucleon scattering cxpcrirnents at low energy lG). According to these data

where VnP, VPP, and V,, are the strengths for the np, pp and nn force, respectively, and V,-, is that of the charge-independent part. The above relation gives the following restriction on C1 e and C,, : -0.004

> cze & -0.012,

(6.5)

0.003 > ci() > -0.001. There are many other experimental data indicating the c~rge-de~ndence of nuclear force, but in most cases some nuclear model is necessary to obtain q~nti~tive information on the strength parameter. Hence it may be better to use the isospinforbidden width as one of the sources to obtain information on Cij. The treatment of the charge-dependent nuclear force is exactly the same as the case of the Coulomb interaction, except for the radial matrix element and possible spin dependence. The charge-dependent force makes contributions to the isospin mixing both in the A = 13 nuclei and in 12C within the p-shell configuration. However, there is no exact cancellation for 13N matrix elements, as in the case of Coulomb forces, because the ratio of vector and tensor strength is different from the Coulomb case in general. In the interaction XPa the charge-dependent force has contributions, and this does not vanish for 13C. Finally, the isovector part of the charge-dependent force gives rise to a Hartree-Fock type contribution and produces a small charge in the original Woods-Saxon s.p. potential. This change distorts the radial shape of the nucleon wave function. As in the case of the Coulomb interaction, this change of radial

512

A. ARIMA

AND S. YOSHIDA

shape corresponds to admixing of the continuum state if the charge-independent basis is used. All these effects, except for the last one, are calculated in a similar way to the case of Coulomb interaction. However, for the last contribution, the following approximation is used. First, the single-particle energy shift due to the charge-dependent forces is calculated by the first-order perlurbation approximation. Then, the change in the Woods-Saxon well depth, which gives the same energy shift in the s.p. energy, is calculated. Next, the radial wave function is obtained using the modified WoodsSaxon well. In other words, we assume that the additional Hartree-Fock potential, coming from the isovector charge-dependent force, has exactly the same geometrical shape as the original Woods-Saxon potential. The results of numerical calculations show that the isovector charge-dependent force produces the following changes in the binding energy of single particles (p+ p+) A3 = + (0.35 Cr i, f0.30 C, , )qj within 10 “/o accuracy. In the above equation to a neutron and proton respectively. To the between the change of Woods-Saxon depth particle B is given by AB = 0.66 Therefore

the effect of charge-dependent

(6.6)

the minus and plus signs corresponds same accuracy we find that the relation U, and binding energy of the single duo.

force on the Woods-Saxon

AU, = T (0.53 C10+0.45 C,,)V,

(6.7) well depth is (6.8)

All the above results are based on the parameter set NDM. It is useful to compare the effect of a charge-dependent force with that of the known Coulomb interaction. First isospin mixing within the p-shell is examined. Numerical results of calculations show that the following strengths cr* = c, 1 = 0.007s, Czo = C,, = -0.0062 of charge-dependent forces give the same isospin mixing as the usual Coulomb interaction, within 10 ‘A accuracy. For the case of isospin mixing into the continuum state, the approximate procedure given earlier in this section shows that c,() = clr = 0.021

(6.10)

is equivalent to the s.p. Coulomb potential, Finally the values of the matrix element
(6.11)

are found to give approximately the same matrix elements as those of the Coulomb

NUCLEON-WIDTHS

CHARGE

513

ASYMMETRY

interaction. All these results are based on NDM parameters with v. = 47.74 MeV and c1- ’ = 1.414 fm, but do not depend on the exchange character of charge-dependent force, in the cases of eqs. (6.9) and (6.1 l), while their dependence is very weak in the case of eq. (6.10). From the above results it is seen that charge-dependent nuclear forces have stronger effects on the isospin mixing within the p-shell than above the p-shell. Next, the width amplitudes are written in terms of the strength of charge-dependent nuclear forces as

c

M = MO+

t=1,2

c

s=o,

(6.12)

M,,Ct,. 1

The expansion coefficients for the case of the Rosenfeld mixture with c1-1 = 1.414 fm and NDM parameters are given in table 10. From this table we can calculate the widths for any choice of charge-dependent forces. First the effect of the isovector interaction is examined within the limits given by eq. (6.5). However, the effect is small within those limits, and the change in widths by the isovector interaction is at most 30 %. TABLE

Nucleon-width Comp. nucl.

Res. nucl. o+

r3N

W

2+

0+ 2+

10

amplitudes with charge-dependent

force

j 0.0391-0.0242i & -0.0264+0.0150i 8

-0.0084+0.00361’

-3.13.I-1.8% -1.93+l.lli +7.11-3.12i

8

O.OOll-0.0019i -0.0204+0.0108i 0.0365-0.015li

-2.82+1.691’ - 1.66+0.89i 6.10-2.521

1.

3.26 - 1.97i -0.05~~0.04i -4.04+1.77i

2.96 - 1.85i -0.07-i

0.03i

-3.43.!-1.44i

Rosenfeld force with GC-* = 1.414 fm is used as charge-independent NDM, and the projection is not taken into account.

-7.76+4.63i

-5.77+2.53i

7.02-4.291 0.49 -0.26i -8.79b3.631’

-3.76+2.3li -0.84+0.46i 4.96-2.051’

force, potential parameters are

TABLE11 Nucleon widths with isotensor nuclear force

0

0

-0.00425 -0.00850 -0.01275 -0.01700 Other parameters

0 0 0 0

0.25 0.45 0.72 1.05 1.44

arc the same as used in table 10.

0.14 0.22 0.35 0.52 0.74

4.16-2.491 0.98 -0.65i

-0.56+0.31i 10.23-4.491

0.000 0.27 1.13 2.57 4.60

0.37 1.96 4.91 9.21 14.87

514

A. ARIMA

AND

S. YOSHIDA

The cffcct of the isotcnsor interaction is large, as the limits given by eq. (6.5) are not as stringent compared with the isovector case. If we assume IC,,l =< 0.002 in addition to eq. (6.9, we obtain a width as large as 20 keV in some cases. In table 11 WC present results in favorable cases where r,,e > rP2 and f,, < m2. As seen from the table it will be possible to improve agreement with experiment by choosing an appropriate strength, but this can also be achieved by other means mentioned at the end of sect. 5. We also tried to fit the experimental data by choosing CZO > 0, which is outside the limit given by eq. (6.5). However, in no case did we obtain values consistent with the relation I-,,, > TPZ. This shows that low-energy nucleon-nucleon scattering data and the branching ratio data are consistent with respect to charge-dependent forces. However, it is difhcult to draw any quantitative conclusion about charge-dependent forces from the branching ratio data in view of the accuracy and uncertainty of parameters in our calculation. So far the isospin mixing within ths p-shell configuration due to the single-particle potential U (‘I has been neglected. This potential U(I) 1s . a Hartree-Fock potential for the 160 core, and is a scalar in space and a vector in isospin space. We see that matrix elements of UC’) which appear in the second term of eq. (3.13) are diagonal, because there is JIG bound p-orbit with one node in our case. Therefore we can write, in the second quantization method (6.13)

u(l) --f ~(ut+cIlt,--a+pa_ta)E~‘~, OL where E(l) a = (v = 3, alU”‘lV is the isovector

part of the s.p. energy.

= 3, LX>

If the spin-orbit

splitting

(6.14) of the isospin

dif-

ference and the average (6.15)

are introduced quantities

and the right-hand

side of eq. (6.13)

is expressed

in term of these

then

U(l) + &[ 2

(ut+auta-uffau_*,)-

E’Pf

,=;,

(u~~a~-u~~aa-~l)]~&(l)

(6.16)

r

because terms with E(l) are proportional to T, and has no off-diagonal matrix element between different values of T. data. In the previous calculations the energy As(l) was taken from experimental We assumed that the ground states of ’ 5N and 1 ‘0 are single p+ hole states in ’ 60. Then from the difference of the two excitation energies we obtain A.$:; = 6.15-6.323

MeV = -0.17

MeV.

(6.17)

NUCLEON-WIDTHS

CHARGE

515

ASYMMETRY

With this experimental value of d&(l), for U(l) and including all the Coulomb interaction in the other perturbation terms and no other charge-dependent nuclear forces, we calculate the widths again, changing the exchange character of the central chargeindependent force. The results are shown in table 12; we cannot obtain agreement with experiment,

even with the Rosenfeld

mixture.

TABLE12 Nucleon widths with experimental n&(l) Exch. charact.

160

r,z

I’.0

I’.=

W R

0.10

s

0.44

0.50 I .08 0.96

0.02 0.31 0.16

0.11 0.15 0.18

Other parameters

1.08

are the same as used in table 10.

7. Further consideration

on the isotopic difference of the spin-orbit splitting

It is very interesting to investigate possible sources of this isotopic spin-orbit splitting. The Woods-Saxon potential with parameters r. = 1.17 fm,

a = 0.63 fm,

V, = 60.12 MeV,

difference

of the

V,” = 18.512 MeV * fm’

(7.1)

is used to generate the s.p. wave function for 160 (the value A = 15 is used in the nuclear radius parameter). This gives neutron s-p. energies EP+ = -15.67

Ed,,,= -21.83 1

MeV,

MeV.

(7.2)

As perturbations to this charge-independent system, we first consider the Coulomb interaction. The simplest way of obtaining de (I) is to evaluate the matrix element of a s.p. Coulomb interaction (rOE = 1.4 fm), which gives the following contribution A&k.

= -3.534+3.552

MeV = 0.018 MeV.

(7.3)

This resulting energy difference has the opposite sign, and its magnitude is one order smaller than the experimental one. This plus sign may be understood as a consequence of the Coulomb energy being larger if the particlc is more tightly bound. body Coulomb interaction gives almost the same results AE~&

= -3.629+

3.648 MeV = 0.019 MeV,

Use of two-

(7.4)

in which the exchange contribution is also included. Next, the s-p. Coulomb spin-orbit coupling (7.5)

516

A. ARIMA

AND S. YOSHIDA

is considered, where p, and pLpare the magnetic momenta of neutrons and protons in units of nuclear magneton, respectively. This is a relativistic correction to the Coulomb interaction, which consists of two parts, Larmor and Thomas terms. The contribution from this interaction is A&.,.

= - 0.062 MeV,

(7.6)

which has the right sign but is a factor 3 smaller than the experimental one. To first order any charge-dependent central force gives no contribution, because we cannot obtain any spin-orbit splitting from central forces for closed shell nuclei. A possible source of the first order contribution is the charge-dependent spin-orbit force, for example (Tzi+z*j)U(rij)(6i+5j)'

&j9

(7.7)

where L, is the relative angular momentum of the particle i and j. We have no information about this type of force. For the second-order perturbation contribution to the Coulomb interaction, the charge-dependent central nuclear force and the kinetic energy of nucleons yield the following energy ~&t,~.r(~) = -0.13+0.38

= 0.25 MeV.

(7.8)

The first term is a contribution from the potential energy while the second term is from the kinetic energy. The central nuclear force is taken from the work of Banerjee et al * I7 )- Although the potential contribution has the right sign, the net contribution has the wrong sign. However, this result has little meaning because the central potential cannot give spin-orbit splitting. The spin-orbit splitting comes from the two-body spin-orbit interaction and the second-order contribution of the tensor force. Therefore we may expect that the main contributions to the isotopic difference of spin-orbit splitting comes from a charge-dependent spin-orbit type interaction. This interaction must have its origin in a two-body charge-dependent spin-orbit interaction, and we have to include this effect in the case of A = 13 nuclei, which are away from the doubly closed shell. The effect of this interaction may partly cancel the effect given by (6.13). This is a possible explanation of why we cannot get good results with the experimental value of At?“.

8. Summary and discussion We have calculated the isospin-forbidden widths for the lowest T = -2,I = 9 state in 13C and 13N by using the Feshbach formula. The wave functions given by Cohen and Kurath “) are used for the compound state and the ground and first 2+ states, while the distorted waves of an optical potential are used for the scattering wave functions. The nucleon widths for decay to the ground and first 2’ states in “C are calculated and compared with the experimental data.

NUCLEON-WIDTHS

CHARGE ASYMMETRY

517

As our wave functions are charge independent, we have to take into account the isospin mixing due to the Coulomb forces and charge-dependent nuclear forces. We calculate this effect by first-order perturbation theory, and initially consider the Coulomb interaction only. The effect of isospin mixing is divided into four parts. The first is the mixing of the states within the p-shell configurations due to the single-body Coulomb interaction. This is found to be negligibly small. Next the same mixing due to the residual Coulomb interaction is considered. This is found to be one of the most important mechanisms for mixing in 13C. However, this mixing is not as important in izC, and vanishes in 13N. The third kind is the mixing of the continuum states due to the s.p. Coulomb interaction. This is another important contribution and approximately charge symmetric. The last contribution is the mixing of the continuum state by means of the residual Coulomb interaction which is believed to be small and is therefore neglected. In calculating the width, the matrix elements of the charge-independent nuclear forces and of the Coulomb interaction must be evaluated. The latter interaction, which has matrix elements between unperturbed states, has much smaller matrix elements than those of the former interaction. For the charge-independent interaction we use central nuclear forces with a Yukawa-type radial dependence. It is found that the commonly used nuclear forces yield the right order of magnitude for the isospinforbidden width. However, the branching ratios depend on the exchange character of the nuclear force, and the Rosenfeld force gives results consistent with experimental data. The charge asymmetry of the branching ratio is found to be a consequence of interference of the two important contributions in the case of 13C, and the dominance of one of the contributions in the case of i3N. The charge-dependent nuclear force is assumed to be central, and to have the same radial shape as the charge-independent force. This interaction has two types of isospin dependence, the isovector and the isotensor. The isotensor force in the spin singlet state is relatively well known from the nucleon-nucleon scattering data, and the inclusion of this effect seems to improve agreement of the branching ratio with experimental data. Our knowledge of the isotensor interaction in the spin triplet state is much less complete, but if this interaction has a strength comparable to the one in the spin singlet state, the width becomes unreasonably large. We have not examined the effect of the isovector force in great detail, because it is believed to be very weak if it exists. As stated before, the mixing of the p-shell configuration by the one-body Coulomb interaction is negligibly small. However, if the same charge asymmetry in the spinorbit splitting between the p+ and pa hole states in I60 exists also in the s-p. energy in A = 13 nuclei, this charge asymmetric term gives rise to appreciable isospin mixing in both 13C and 13N. With this mixing WCare unable to obtain a fit of the branching ratio with experimental data within our present calculation without invoking other charge-dependent nuclear forces. In solving this problem we need more information about a possible charge-dependent spin-orbit force.

A. ARIMA

518

AND S. YOSHIDA

We have succeeded in explaining the absolute magnitude and branching ratio of the widths in A = 13 nuclei, ahhough only qualitatively. In doing so, we made some approximations in the calculations, and also used parameters which have some uncertainty. In the following we should like to discuss these points. Let us start from the isospin mixing of the continuum state due to the s.p. Coulomb potential. In this calculation the particle-hole correlation effect is neglected in evaluating the energy denominators of the perturbation expansion as shown in sect. 3. However, according to Bohr, Damgaard and Mottelson “1 this effect isve~impo~ant and reduces the isospin-mixing amplitude by as much as a factor of two. No rehable estimate of this effect is available, especially for light nuclei, but it will reduce some what the isospin mixing of the continuum state. The mixing of the p-shell configuration by the residual Coulomb interaction, which is another important contribution to the widths, is greatly affected by the exact location of the third T = 3, I = -2 level, which is not yet experimentally known. If this level is removed, we obtain r,, > r,,, and the branching ratio becomes almost charge symmetric. A small change of the energy from that predicted by Cohen and Kurath improves the agreement between theory and experiment. Therefore it is very important to know the exact excitation energy of this level. Without this information it is difficult to draw definite conclusions concerning the necessity of a charge-dependent nuclear force. Next, the charge-independent forces appearing in the formula of the width are discussed. As already stated in sect. 5, this interaction should be the same as that TABLE 13 Two-body Quantum number --.. TI jij2 i;j; -___l8%

$3.

%& a& i2b

%B

84

t!% b&

r.m.s. deviation

10 12 01 03 12 01 10 01 11 12 01 02 01 10 01

matrix elements for bound state
-3.19 -0.17 -3.58 -7.23 -1.92 +3.55 -4.86 11.56 4-0.92 -0.96 -6.22 -4.00 -f 1.69 -0.26 -4.15

in units of MeV

Central forces used in our calculation W -5.30 -2.94 -4.13 -4.13 -0.81 $1.81 -1.97 $0.88 -2.34 -3.45 -5.13 -4.01 0 -3.64 -3.64 2.14

FV

T

R

S

-3.29 -0.39 -0.63 -5.57 -2.06 -+ 5.54 -2.99 i4.26 -!-1.07 -1.81 -4.98 -5.40 11.93 -1.00 -3.00

-6.09 -1.86 -3.79 -5.57 -2.59 -!-3.54 -4.15 +2.30 0.00 -3.61 -6.22 - 5.41 -to.69 -2.85 -4.22

-3.47 -0.40 -3.79 -5.57 -2.19 +3.54 -3.17 +2.30 +1.15 - 1.90 -6.22 -5.41 +0.69 -1.05 -4.22

-3.80 -1.16 -3.79 -5.57 -1.62 $3.54 -2.59 12.30 0.00 -2.26 -6.22 -5.41 f-O.69 -1.84 -4.22

1.48

1.49

0.86

1.10

-.-

NUCLEON-WIDTHS

CHARGE

ASYMMETRY

519

used in the calculation of the wave functions. However, the potential form for the matrix elements given by Cohen and Kurath “) is not available; moreover, for the sake! of simplicity we assumed a Yukaws potential with various types of exchange character. To see to what extent the interaction used in our calculation is similar to the one corresponding to the matrix element given by Cohen and Kurath, we calculated the two-body matrix element bztwcen p-shell bound states by using various types of exchange forces, and the resillts are shown in table 13. In the table the matrix efements given by Cohen and Kurath and the r.m.s. deviation of the matrix elements calculated from them are listed. As seen from the table, the central interactions used in our calculation have matrix elements which arc very similar to those given by Cohen and Kurath. Except for the Wigner force the sign of all matrix elements for the central interactions agree with those of Cohen and Kurath, and their r.m.s. deviations are about ! MeV. This good agreement gives some justification for our use of the central i~lteractions in our calculations, although we do not know how good the agreement is for the matrix elements containing a continuum state. Paramctcrs appearing in the optical model potential and Woods-Saxon well for the bound states are relatively well known, but there remains some uncertainty. WC already compared the results obtained assuming different parameter sets, and found some change in the value of the widths. The effects of the orthogonalization of the bound and continuum wave function, which should have been taken into account throughout our caluclation, change the absolute magnitude of the widths by about a factor of 2, but the effect on the branching ratio is not large. Therefore, this will not present a serious problem because of this uncertainty until we try to make a more quantitative comparison by employing more refined calculations. In conclusion, our calculation shows that the absolute magnitude of the widths is compatible with the experimental data; the branching ratios are explained by our theory, at least qua~itativc~y, despite approximations and un~rtainties provided that the third T = 3, I = _3state in 13C is located approximately at the energy predicted by Cohen and Kurath. Precise knowledge of this energy is necessary to make more quantitative arguments. The authors would like to express their sincere thanks to G. M. Temmer, who promoted their interest in this problem and gave continuous encouragement and helpful discussions during this work. They are indebted to H. Yoshida for providing a computing code of the bound state wave function, and to R. Van Bree for his optical wave function code. Conversations during this work with L. Zamick, P. J. Ellis, A. Mekjian, R. Mohan, Y. Tikochinsky and R. Van Bree were very useful. Some parts of this work were carried out at Argonne National Laboratory where one of the authors (S.Y.) was staying during the summer of 1968, and he is grateful to D. Kurath for his hospitality_

520

A. ARIMA AND S. YOSHIDA

References 1) G. M. Temmer, in Isospin in nuclear physics, ed. D. H. Wilkinson (North-Holland, 2)

3) 4) 5) 6) 7) 8) 9) 10) 11) 12) 13) 14) 15) 16) 17) 18)

Amsterdam, 1969) p. 694. D. J. Bredin, 0. Hansen, G. M. Temmer and R. Van Bree, in Isobaric Spin in Nuclear Physics, eds. J. D. Fox and D. Robson (Academic Press, New York, 1966) p. 472; G. M. Temmer, Int. Symposium on Nuclear Structure, Dubna, USSR (IAEA, Vienna, 1968) p. 249; R. Van Bree and G. M. Temmer, to be published; M. J. LeVinc and P. D. Parker, Phys, Rev, 186 (1969) 1021 A. B. McDonald, E. G. Adelberger, H. B. Mak, D. Ashery, A. P. Shukla, C. L. Cocke and C. N. Davids, Phys. Rev. Lett. 3lB (1970) 119 E. G. Adelberger, C. L. Cocke, C. N. Davids and A. B. McDonald, Phys. Rev. Lett. 22 (1969) 352 A. Arima and S. Yoshida, in Nuclear isospin, eds. Anderson, Bloom, Corny and True (Academic Press, New York, 1969) p. 73 H. Feshbach, Ann. of Phys. 19 (1962) 287 C. M. Shakin, Ann. of Phys. 22 (1963) 54 S. Cohen and D. Kurath, Nuci. Phys. 73 (1965) 17 A. K. Kerman, in Nuclear isospin, eds. Anderson, Bloom, Cerny and True, (Academic Press, New York, 1969) p. 135 A. deshaiit and I. Talmi, Nuclear shell theory (Academic Press, New York, 1963) J. S. Nodvik, C. B. Duke and M. A. Melkanoff, Phys. Rev. 125 (1962) 975 B. A. Watson, P. P. Singh and R. E. Se8e1, Phys. Rev. 182 (1969) 977 M. B. Johnson, L. W. Owen and G. R. Satchler, Phys. Rev. 142 (1966) 748 G. C. Ball and J. Cerny, Phys. Rev. 177 (1969) 1466 R. J. Blin-Stoyle, Isospin in nuclear physics, cd. D. H. Wilkinson (North-Holland, Amsterdam, 1969) p. 115 E. M. Henley, Isospin in nuclear physics, ed. D. H. Wilkinson (North-Holland, Amsterdam, 1969) p. 15 M. K. Bancrjee, D. d’ffliveira and G. L. Stephenson, Jr. Phys. Rev. 181(1969) 1404 A. Bohr, 3. Damgaard and B. R. Mottelson, in Nuclear structure, cds. A. Hossain, Harun-arRashid and M. Islam (North-Holland, Amsterdam, 1967) p. 1