Energy-weighted longitudinal sum rule for electron-nucleus scattering and nuclear forces

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Energy-weighted longitudinal sum rule for electron-nucleus scattering and nuclear forces

V.D. Efros, I.E. Zakharchenko

To cite this version:

V.D. Efros, I.E. Zakharchenko. Energy-weighted longitudinal sum rule for electron- nucleus scattering and nuclear forces. Journal de Physique, 1978, 39 (6), pp.575-583.

�10.1051/jphys:01978003906057500�. �jpa-00208789�

LE JOURNAL DE PHYSIQUE

ENERGY-WEIGHTED LONGITUDINAL SUM RULE

FOR ELECTRON-NUCLEUS SCATTERING AND NUCLEAR FORCES

V. D. EFROS and I. E. ZAKHARCHENKO I.V. Kurtchatov Atomic

Energy

Institute, Moscow, USSR

(Reçu

^le

4 juillet

^{1977, révisé le 16}

février

^1978,

accepté

^{le 22}

février 1978)

Résumé. ²⁰¹⁴ Afin d’obtenir des informations nouvelles sur l’interaction nucléon-nucléon (NN)

on étudie la règle ^{de somme} longitudinale pondérée par l’énergie de la diffusion électron-noyau.

Des formules simples permettant le calcul de la règle ^{de somme} pour ^une forme générale de l’inter- action NN sont établies. Le calcul se ramène à celui d’une valeur de type de

V>

^avec les compo- santes de l’interaction

V,

combinaisons

algébriques

^des composantes de l’interaction NN vraie.

Des résultats d’un calcul _pour les _noyaux A = 3 utilisant des formes réalistes et modèles de l’inter- action NN sont présentés ^et discutés. On observe _que des résultats expérimentaux ^{pour les noyaux} 3He et 3H pourraient permettre ^{des tests nouveaux} de l’interaction NN. On note la possibilité ^de simplifier le calcul d’opérateurs VCoul intervenant dans le calcul de l’énergie Coulombienne dans certains calculs.

Abstract. ²⁰¹⁴ In order to obtain new information on the NN-interaction, ^the energy-weighted longitudinal ^sum rule for electron-nucleus scattering is studied. Simple formulas for calculating ^the

sum rule for a NN-force of general ^{structure are} derived. The sum-rule calculation is reduced, ^{in the} general case, to calculating ^a quantity ^of the V > type ^{with the V} NN -interaction components being algebraic combinations of the components ^{of the true} NN-interaction. Results of the sum rule calculation in the A = 3 case

using

^some realistic and model NN-forces are presented and discussed.

It is noted that the determination of appropriate

experimental

^{sums for}

3He,

^3H may allow ^{a new} test of the NN-interaction to be made. The

possibility

^of

simplifying

VCoul operators ^{in some calcula-} tions is also noted.

Classification Physics ^Abstracts 21.40 ^- 25.10 ^- 25.30

1. Introduction. ^- The

possibility

^of

deriving few-nucleon-system properties starting

^{from the}

realistic

NN-interaction,

^{that is the} interaction accu-

rately describing

^{the whole set of}

^NN-data,

^{is an old}

problem.

^{There is no}

complete

^{solution at}

present

to the extent that the set of

quantities

available in

practice

^for

testing

realistic NN-interaction is rather, limited due to the

difficulty

of the nuclear

many-body problem.

^{For this} purpose in the present paper, ^we

use an additional

quantity

^{- a sum rule}

(SR)

^for

electron

scattering by

^{nuclei. The use of a SR has}

fobvious

advantages

^over the direct use of electroexci-, tation

amplitudes

^{to a}

given-state.

^{As well as the}

amplitudes,

^{the SR}

gives

information on excited nuclear states but in contrast to the

amplitudes only

an initial

(ground)-state

^wave function fs

required

for

calculating

the SR and it is much

simpler

^{to obtain}

a

ground-state

^wave function than a

highly

^excited

one. Besides the NN-force

dependence

^{of the results}

is more transparent for the SR.

The idea of

using energy-weighted

^{SR for}

studying

the NN-force was discussed

long

ago for _y-quan-

tum

absorption.

^{It was discussed}

by

^{Drell and}

Schwartz

[1] for

electron-nucleus

scattering.

^However

they

^{did not}

give

^an

adequate

^and

complete

^{SR formu-}

lation that could allow such a

study.

^{On the other}

hand,

^the accuracy in the

existing high

energy elec-

tron-experiment probably

^{allows the}

required experi-

mental sums to be obtained. There are some corres-

ponding experimental investigations

in the literature

(see

e.g. refs.

[2, 3]) although

^the

summing

^versions

used differ from those

required

^{for our} purpose.

In section 2 and in

appendix

^{1 our SR version is}

discussed. In section 3

general

formulas for calculat-

ing

^the SR for various classes of NN-forces are

derived. In section 4 numerical results for some realis- tic NN-interactions in three-nucleon case are

given.

In

appendix

^{2 is}

presented

^{a note on} calculations with a

VCoul

^force.

^(In

^the

following h

^{= c = 1, and}

NI is a nucleon

mass.).

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:01978003906057500

576

2.

Longitudinal

^{sum rules. - We use the SR-version}

described in ref.

[4].

^The

experimental

sums corres-

ponding

^{to these SR are}

where q

^{and OJ are} the three-dimensional momentum and the full _energy transferred from an electron to

a

nucleus,

^{m = e -}

e’, GE

^{is the} proton form factor

(FF), Cl

^{is the}

longitudinal

^{FF in the} differential electron-nucleus

scattering

^{cross section}

( 1)

The

CI

FF may be found

experimentally by

separat-

ing A(q, co)

^and

B(q, w)

^{FF in the linear relation}

The SR

corresponding

^{to the sums of} eqs.

(1)

^{are as}

follows

[4]

Here t/li

^{is the wave} function of the nucleus initial state,

Q

is the nuclear

charge

operator,

(tz

^{is the nuclear}

current

longitudinal component being

^determined

by

^the

continuity equation

where je is the nuclear Hamiltonian. The usual

single-particle approximation

^{is used for the}

Q-

(’) ^In equations (1), (2) ^{it is} implied ^{that the} nucleus final state

belongs ^{to a} continuous spectrum. (In calculating ^the experimental

sum rule the contribution of the excitations with natural width less than _energy resolution in m should be taken into account in the form corresponding ^to discrete levels (see also reference [5]).)

operator, the small

spin-orbital

correction

being neglected.

^Thus

(GE

instead of

GÉ

may be written in

equation (1)

^as

well.)

^The

averaging (4 11:)-1 f ^dq...

ⁱⁿ

equations (4), (6)

^is

equivalent

^{to the}

averaging

in nuclear orienta- tion

[4].

^As follows from

equations (4)

^and

(8),

^the

quantity (4)

^{reduces to the}

expectation

^{value of the}

jo(qrij)-type ^operators

^{and it}

^gives

^{the Fourier-}

component

^{of the}

charge-distribution pair-correlation-

function scalar

component.

The described SR-version thus allows this

quantity

^{to be}

directly

^extracted

from the

experiment

unlike other versions discussed in the literature.

The

properties

^{of the SR}

of eq. (3)

have been studied

previously [4].

^A

model-independent

^relation express-

ing

the total Coulomb _energy of a nucleus in terms of the sum of eq.

(3)

^{was obtained}

(’), (33).

^For

light

nuclei the sums

of eq. (3)

^were

expressed

^{in terms of the}

purely

^elastic

scattering

^{FF. Therefore}

only

^{the SR}

of eq. (5)

will be studied below.

In eqs.

(5), (6)

^we may put

Ml

⁼

Mi + Mi,

0’11 ⁼

ah

⁺

ul,

in accordance with the kinetic and

potential

energy contributions in the commutator of _eq.

(7).

The kinetic _energy contribution is uni-

versal,

where Z and N are proton ^{and neutron numbers.}

The relation

(11)

^{and some features} of the SR

(5)

are discussed in

appendix

^1. Our main task now is to

study

^the

potential

contribution

afi.

^{We note}

in addition

that,

^{as is}

pointed

^{out in ref.}

[9]

^{in the case}

of NN-force

depending

^on

velocity

^{in a} power ^not

exceeding

one, i.e. central

plus

^tensor

plus

^{lS static}

force,

^the

uv,

contributions for mirror nuclei in a

pair

are

equal.

^{For such} nuclei the

following

^model-

independent

relation is therefore valid for the sums

of eq. (3) [9] ]

(2) Relativistic (nonstatic) corrections to this relation are studied in reference [7]. Experimental results for ’Li are obtained in refe-

rence [3].

(3) The same relation as in reference [4] ^{with a less accurate SR} formulation was discussed later in O’Connell and Lightbody’s

paper [8]. ^It is clear that the meaning ^{of the} pair correlation function in the relation is well known (e.g. ^ref. [5]) ^and the consideration in ref. [8] ^{does not} contain any ^new essential points.

(Here

^{T is the}

isospin

quantum

number.)

^{Also in the}

case of

small q (the

^terms

with q4

^and

higher

powers

of q

^are

neglected)

^the relation holds ^true for _any force. In

particular

^we will obtain the

degree

^{of accu-}

racy with which _eq.

(12)

^{holds for some classes of}

realistic NN-force in the ^{case of a}

3He-3H pair.

Unlike

^the

longitudinal

^{SR of} eqs.

(5), (6)

^the

Drell-Schwartz

energy-weighted

^SR

[1]

^{includes a}

transverse

current 6t

contribution.

Non-single-particle

contributions

to 61

^are

given by

eq.

(7)

^with

Q

^{and V}

being given

but the contribution

to 6t

^{are not known}

completely,

^{which causes a} theoretical

uncertainty.

Besides, the 6t

contributions to the SR are too

compli-

cated for an exact calculation. At the same time the

experimental

summation method that

corresponds

to the SR

[1]

allows the

appropriate

theoretical summa-

tion

only

^to be carried out to a poorer

approximation

than for the SR _{of eq.}

(5),

^{which causes an additional}

uncertainty (cf.

^ref.

[4]). ^Besides,

^{in ref.}

[1]

^{the SR}

is

presented only

^for

oversimplified

^{model wave func-}

tions and

only

^static

exchange

central force contri- butions to the SR are considered

(cf. below,

^section

3)

(4),

^.

3. NN-force contribution to the

energy-weighted longitudinal

^{SR. - Let} the NN-force in _eq.

(7)

^{be a}

(4) An energy weighted ^{SR was also} considered briefly ^{in ref.} [10]

independent ^{of ref.} [4]. ^We

only’mention

^{here that} the SR formula- tion in ref. [10] ^{involves some} inaccuracies. One of them (as ^is clearly ^{seen from the} corresponding experimental ^results [11])

is that the nucleus excitation _{energy is} used as a weight factor instead of the total transferred _energy used _{in eq.} (16). ^The inaccuracy ^is due to the translational invariance property ^not being ^{taken into}

account. In ref. [10] ^{as well as in ref.} [1] averaging ⁱⁿ angular quan- tum numbers was not considered and only ^{the static} exchange force contribution to the SR was considered.

two-body force, = E ^V(kn).

^{Then as is shown in}

appendix

^{1 the}

Mi

operator mentioned above _may be

represented

^{in the}

following simple

^form

Our task now is to calculate the interaction

of eq. (14).

Let

V(kn)

^{be a} NN-interaction of the most

general

type. ^{Such an} interaction _may be

represented by

^its

radial matrix elements

(ME)

where j, l,

s, I are the total and orbital momenta, ^the

spin

^and

isospin

^{of a nucléon}

pair.

Then it is natural to represent ^the

required V(kn)

operators

by

^{a similar} type ^ME

Unlike _eq.

(16)

^{the ME}

of eq. (17)

^also

depend

^{on the}

isospin projection.

^The calculation leads to the follow-

ing expression

^{for the ME}

(17)

^{in terms of the ME}

(16),

where j,

^are

spherical

^Bessel

functions,

for even and odd Â

respectively,

In eq.

below

»

The main contribution to the

0’11 (q2)

^{sum under}

consideration comes from the interactions

of eq. (18)

with

small l,

l’. This is similar to the situation

taking

place

for the initial interaction

of eq. (16)

^{in calculat-}

ing,

^for

instance,

^the

binding energies (cf.

^ref.

[12]). As q

578

increases, higher

^values

^and,

in accordance with eq.

(21),

NN-interactions in

higher

^{orbital momentum}

states become

important

in eq.

(18).

For

light

^{nuclei it is often more} convenient to

represent ^{the most}

general

NN-interaction in the

ls-coupling form,

rather than in the form

of eq. (16).

In eq.

(22) V,(Pi)

^and

Si(Pi)

^are

space-isospin

^and

spin

^tensor operators ^of

pi rank

(pi

⁼ P2 ⁼

0,

P3 ⁼

1,

p, ⁼

2) respectively.

The operators ^{with i =}

1,

²

correspond

^to

singlet

and

triplet

central forces. The operators

V,.i(pi)

^will

be

represented by

their reduced ME

We write the operators

V(kn)

^{in a} form similar to eq.

(22)

^with

spatial

^{ME of the same} type ^as in eq.

(23), namely

The calculation leads to the

expression

where, m,)

^and

fi (qr, mI)

^{are defined}

above,

1

In the terms of _eqs.

(18)

^and

(25) containing £5;’0’

nonsymmetrized expressions

may be also used.

For the NN-interactions which for

given

^two-

nucleon states are local interactions of the

Vé5(r)

type

(as,

^for

example,

^{in the case of the} known Reid.

potential)

the interactions of _eq.

(14)

^{are local inter-}

actions

F/J"(r)

^{and eq.}

(18)

^with

holds for these

interactions,

^{the same}

taking place

in the case

of eq. (25).

The _way to obtain the above formulas is as follows.

The contributions of the operator

from _eq.

(14) to

eqs.

(17), (24)

^are calculated

directly

and

they produce

^{the terms with}

b ÀO

in eqs.

(18),

(25). (Note

^{that .}

The contribution of the

operator

is calculated

by expanding Q(kn)

^and

Q + (kn) inl Yim(q)

^and

YÂm(q).

This makes it

possible

^to

carry

out the

required integration

ⁱⁿ

dq,

the contribution

being

^{of the}

form £ ^{Q £ VQÂm}

^with

Â,m

Then

expressing ^{the V}

interaction in terms of its ME

of eq. (16)

^or eq.

(23), carrying

^{out the}

angular

^inte-

grations

^of

Y;.m(r)

^{that occur in eq.}

⁽¹⁷⁾

^{and eq.}

(24)

and

evaluating

^{the sums in}

angular

^momenta

projec-

tions we obtain formulas similar to eqs.

(18)

^and

(25)

but with the

following isospin ME,

^{in the case of}

eq.

(18)

^for

instance,

Accounting

for the conditions for which the

quantity (21)

^{is nonzero ME}

(27)

^occur

only

^with

physical

^l’

that is : 1" + s + l’ is even. Furthermore since the ME of the

operator âk

⁺

(- 1)’ an

are nonzero for the

only l’,

ml,

values,

eq.

(27)

may be

replaced by

which leads to eq.

(19).

Now let us calculate the

k’(kn)

operators ^{in the case} where the

V(kn)

interaction is

given

^{in the} operator form. We confine ourself ^to the most

popular

^interac-

tions with the

velocity

power ^not

exceeding

^two.

Performing

^the

corresponding expansion

^{of the most}

general operator

^form interaction from ref.

[13],

one obtains that the most

general

form of the NN- interaction mentioned above is

where

P,", PÎ

^are

projection operators

ⁱⁿ

given spin

^and

isospin

^{states, , 1}

p and 1 are linear and

angular

^momentum

operators

^{of a nucleon}

pair.

^{We obtain} the resultant

V(kn)

^{in the}

form similar to that of _eq.

(28),

For the

isospin singlet

interactions the results obtained ^{for the} radial

components

^of

V(kn)

interaction ^are

as

follows,

In the above formulas

The

triplet

ⁱⁿ

isospin

interactions

depend

^{on the}

isospin projection ^operator

^{mi as well as on r, qr. The}

results obtained ^{for them are as}

follows,

580

The ai

^{values are defined}

by

eq.

(33)

^and

We note some features of the formulas in the above.

The

dependence

on qr ^at

small qr

^{values reduces to}

(gr)2

^{and the static} core V in the NN-interaction cancels

partly for q

^{values which are not}

high (the

same holds for _eq.

(18)).

^{Thus the}

V(kn)

interaction is more smooth ^at small r than the

initial (kn)

interaction. The relative contribution to

V(kn)

^from

nonstatic forces of

l’, p 2

etc. type ^{increases as}

the q

values increases. This is clear since the relative

impor-

tance of such forces _grows as the _{energy of} a state increases and the most

probable

energy transfer to a nucleus increases with

increasing

q. Besides the odd NN-forces and the forces of

Vl2 and Vil-types although they

^{do not}

give significant

contributions to the

lightest

^{nucleus wave functions}

(cf.

^ref.

[12])

^contri-

bute however in an essential _way to 0’11 since

they

enter the V-force even static components ^and

they

do not vanish at 1 ⁼ 0 as well. Note that this contribu- tion is not

negligible

^for

arbitrary

^small q. The reason

is that the nucleus final states with _energy sufficient for these forces to be

appreciable

^are

effectively

attained at any q

(5).

To obtain the above formulas we write

where

(kn),

^{is the}

isotopic exchange operator, V o,1

^l

are

space-spin

operators ^{and we} present

V(kn),

eq.

(14)

in the form

The first two terms in _eq.

(38)

^are linear combinations of contributions of the commutators with the

1, ^Iv,

pu pv,

lu

operators, but the contribution of the

Il

operator appears ^to be zero. To

bring

the contribu- tion to the last term in _eq.

(38)

^from

the Ùi

^force

(5) ^We recall here that while obtaining ^{SR one} ignores ^{the kine-}

matic COnC1JlIon cu q ^{and hence the SR are not valid at the small-} eSt q "-’ PNucJ’

components ^with

p2

^{and SPP to the}

required

^form

of _eq.

(29)

^{we use the}

identity

In the case of nuclei with A ⁼ 3,4

(3.4He,

^@

3H)

^the

formulas obtained for

V(kn)

may be further

simplified by excluding

^the

mj-dependence.

^{In fact}

^(if

^one

considers

isospin

^{to be a}

good

quantum

number)

the wave function of each nucleus with A ⁼ 3,4 involves

only

^two

isospin

^functions

(I", T’)

^and

they

may be chosen so that

only

^{one of them}

belongs

^to

the I ⁼ 1 value of a

given particle pair,

Hence the

mi-averaged

interaction of the torm

T" 1 (12) ] 1" )

^would

^give

^{the same results as}

^gives

the

P(12)

interaction of the form of _eqs.

(34)-(36).

(The

^same is valid for the interactions in _eqs.

(19), (20).)

^The

corresponding averaging

ⁱⁿ eqs.

(34)-(36)

means the

replacement of (1

^-

mi) by 1

_{~ 3 _}^{and the}

repla-

cement of the factor

depending

^on

GP, GÉ in

^eq.

⁽³⁷⁾

by

in the 4He case and

by

in the

’He

case.

(For

^the

^3H

^{case the}

replacement

p ± ⁿ is

required.)

^{As a} result the

h(kn)

interactions

acquire exactly

^{the same structure as the initial}

V(kn) (a

similar method _may be

applied

^to

simplify

the nucleon Coulomb interaction operator, ^see appen- dix

2).

The relations obtained are

quite

convenient. To obtain the

required

al1 ^sum it is sufficient

only

^to substitute

17(kn)

^for

v(kn)

^{in the usual}

potential energy V >

calculation codes.

4. Numerical results for A ⁼ 3. ^- We have cal- culated the _ul, sums for

3He, ^3H using

^{several rea-}

listic NN-interactions. We used A and C versions of

the interaction of ref.

[15] (SSCA, SSCc),

^{the inter-}

action

[16] (SSCOBEP),

several versions of the interaction of ref.

[17],

^the interaction

[18] (EH).

All these interactions have the structure

given by

eq.

(28).

^The SSC and SSCOBEP. interactions involve all the components ^from eq.

(28)

except ^those pro-

portional

^to

p2

^and

_S)k.

The interactions

[17]

^involve

central,

tensor, ls- and

p2-components

^{and the EH} interaction involves

central,

^tensor and

Is-components.

^The interactions

[ 17]

^were constructed for even states

only

^{and we used}

them with the

SSCC

interaction in odd states. With the

exception

^{of EH} interaction all the interactions

mentioned, especially

the interaction

[17], give

^an

accurâte _(with

small

x2), description

of the whole set of NN-data. EH interaction is also in agreement with the NN-data in the case of small 1 values. Besides all the interactions used

yield qualitatively

^correct

(cf.

^ref.

[12])

values of the

3H, ^’He binding energies (The

deviation from the

experimental

^{values is not}

more than 1.5

MeV.)

^The

three-body problem

^with

the interactions mentioned was solved

by

^{means of a}

hyperspheriçal

^harmonic

(HH) expansion.

^Accord-

ing

^{to the}

recipe given

^{in ref.}

[19]

^we retained the

required

^{essential HH i.e. the HH obtained}

by

^means

of

symmetrization

^{from the HH} with small Jacobi orbital momenta

/Çl’ lç2

^{values. The} calculation of the

particle permutation

operator ^{ME in the HH basis} is needed for the HH

symmetrization

^{and the ME}

were calculated

by

^{means of the recurrent formula}

in the

generalized

^momentum

K (K- K+ 2)

^{of ref.}

[20].

The convenient basis of

hyperradial

^{functions and}

three-nucleon codes

developed by

^Demin

[12]

^were

used.

Complete

convergence of the results was

achieved in all cases.

The results for

3He,3Ha’l

obtained with the

help

of the formulas

(30)-(32)

and of the formulas obtained

from _eqs.

(34)-(36)

^are

given

^{in table}

1(6 ).

^{The table}

also contains the parameter K value defined

by

^the

relation

The K value shows the _accuracy

of eq. (12).

^{In the case}

of the EH interaction the non-zero K value is caused

only by

the difference between

’He

and

3H

wave

functions due to a Coulomb

potential

^{of the} type ^as in _eq.

(A. 9).

^{In all the cases} considered the ^K: value appears ^to be small _so, force components

quadratic

in

velocity violating

^{in our case} the relation

(12)

are small. Besides the realistic interactions are found to

produce

rather similar results. However this does not mean that

corresponding

^{results are} insensitive

to NN-force. To illustrate this we have carried out

calculation for the

SSCC

interaction with the odd forces put

equal

^{to zero. As is seen from} the table the results

change considerably

^{in this case. We recall}

that,

^on

the other

hand,

for the A ⁼

3,4

^cases, bound-state and

low-energy-reaction problems are’quite

^insensi-

tive to odd force contributions

(see

^ref.

[12])

^{and thus}

they

^{do not}

provide

^{a check} of the NN-interaction odd components. Furthermore the results for realistic interactions

being similar,

^{it is} of interest to examine

simple

interactions

describing

^low energy data

only.

Thus we carried out a calculation with the interaction combined of such

simple potential

^{in even states from}

ref.

[21]

^{and of the SSC}

potential

^{in odd states}

(SIMPLE

⁺

SSCc).

^The

potential [21]

used contains

(6) ^The following well-known parametrization ^of nucléon FF

wasusedGK(q2) =

(1 ⁺ q2/18.24)-2 ;

TABLE 1

The sum-rule calculation results

for

^{the A = 3 case}

Notes : 1) ^The potentials ^{5 +} 13,..., 8 + 19 ^{are even} interactions from ref. [17] (figures mark their numbers in table II of ref. [17]).

2) SSCE ^means the interaction obtained from the SSCc by retaining only ^{even forces.}

3) ^{SIMPLE means} simple ^even potential [21] plus ^{the odd} SSCc potential (see ^the texte

4) ^For the (Tn quantity ^{the rounded values are} given ^{but the h} quantity ^was calculated from accurate values.

582

even central

plus

^{tensor forces as} exponents

adjusted

to

scattering lengths,

^effective ranges and deutron

quadrupole

^{momentum. It}

gives

reasonable

3H

bind-

ing

energy

(9.5 MeV).

The results

presented

^{in the}

table

again

^differ

considerably

from those in the case

of realistic interactions even for low

energies.

The results

presented

^{show that} the determination of the

experimental energy-weighted

^sums

3He,3HOl1

allows a new

simple

but nontrivial check to be made of realistic two-nucleon interactions in the three- nucleon

problem.

^{Since the} determination of accurate solutions of more-than-three-nucleon

problems

^is

not so

simple

this fact deserves attention. In

particular

it becomes

possible

^to check the interaction odd

components

^{that have not been}

really

^{tested in the}

few-body problem

^{until now. It is} also of interest to find out

experimentally

^{if the} x value

(roughly speaking the x

value shows to what extent the interaction is

nonlocal)

^is

really

^{so small as it is in our calcula-}

tions

(’).

On the other hand

probably

^{rather accurate}

model-independent

relations available for _appro-

priate non-weighted

^sums

3He,3Hu, [4]

may be useful for

checking experimental summing

accuracy and

probably

^for

checking

^the accuracy of the

general

common

electrodesintegration problem

formulation

(without exchange

^currents,

etc.)

^{that is used.}

By using

the

experimental

^sum

3HeUI(q2)

it is also

possible

^to

obtain the

3He

Coulomb _energy

(see [3, 4, 7]).

^The

latter is of interest as far as the familiar

3He-3H

_energy

problem

is concerned and it can

probably

be obtained with an accuracy suffficient for direct detection of the

probable

deviation of NN-forces from the

charge

symmetry

(cf. [3, 22]).

Appendix

^{I. - We}

explain

the derivation of the SR of _eq.

(5).

^The transition

amplitude corresponding

^to

FF

C,

in eq.

(2)

^is

Qfi

^where

g

^{= e-iq’}

Q, R

^{is the}

centre-of-mass coordinate. Here the functions

t/I¡, t/I

^{¡ as well as}

t/I

ⁱⁿ eqs.

(3), (5)

^are internai nuclear functions i.e. functions in Jacobi coordinate

subspace.

To obtain the SR we write

(’) Recently ^we have calculated the K value for a large variety

of nonlocal rank-one and -two NN-interactions. It turned out that the _x value reaches a maximum at q N ^2.5 f -1 slowly varying ⁱⁿ the q ci ^{1.5-30 f -1 interval. lCmax N} 5-9 %.

Note also that for higher accuracy of expérimental ^summation

the extrapolation ^of spectrum ^to high ^m values is of great ^use.

(Cf. ^{G. R.} Bishop et ^{al., ref.} [2].)

Here

Jeint

^is the internal Hamiltonian. This

gives

^the

SR in the form

Let us reduce _eq.

(A. 2)

^to the form of _eqs.

(5), (11).

We may

replace 1 t/I >

in eq.

(A. 2) by |1 t/I > SR were 1 SR >

^{is an}

arbitrary

normalized function in the centre-of-mass

subspace.

^{Hence we} may ^treat the _ope- rator in _eq.

(A. 2)

^{as an} operator in the whole _space.

So the operator may be rewritten as follows

But as is _easy to

see Q + ^{Qq dq -}

^{0 and we obtain}

the SR similar to eqs.

(5)-(7)

^{but with}

nonsymmetriz-

ed operator

Q ’ [JC, Q ].

^The

symmetrization yields

eq.

(6). Performing

space inversion one may in eq.

(6) replace

^the

àt Q

^operator contribution

by

^the

(- 61 Q +)

contribution. This reduces the ME in eqs.

(5), (6)

^to the usual double-commutator form

which

directly gives

eq.

(11). Thé 1 SR )

^{factors in the}

expression

^{for the}

ul’i

contribution _may

obviously

be omitted since

Q’[PQJ ]

^{is an internal} operator.

(These

^{factors are} also omitted _{in eq.}

(5)

for the total contribution to the SR. The form of _eq.

(5)

^is

possible

if one in addition assumes that

S(R)

^{tends to the}

limit

S(R)

^{= const. and} the action of the momentum

operators

on 1 t/I¡ >

^{is the same as}

on 1 t/I¡ SR >

^with

8(R) > const.)

Eqs. (13)-(15)

^{are obtained}

by extracting

commuting

^with

V(kn)

from double commutators of the form

1(Qà

⁺

Qn+), [ P(kn), (Qk

⁺

Qn)ll

^which

correspond

^to eq.

(A. 4).

Appendix

^{II. - The Coulomb force} operator

may be

simplified

^{in the A =}

4,

^{T = 0 case and in} the

A = 3, T = 1 /2

^case.

Let A = 4,

^{T = o. On}

applying

^the

averaging

mentioned in section 3 ^to eq.

(A. 7)

^we obtain the operator

The operator ^of eq.

(A. 7)

^is

equivalent

^{to that of}

eq.

(A. 8) provided

^{that the wave function} compo- nent admixtures with T # 0

(due

^{to Coulomb}

forces)

which are

quite

^{small are}

neglected.

^But the latter operator ^{does not}

require

any

special

calculations since to add this

operator

^{means a} redefinition of

V2s+ 1,2T+

^l

components

^{of the} eq.

(28)

interaction.

In the A ⁼

3,

^{T =}

1/2,

mi ⁼

1/2

^{case the} operator of _eq.

(A. 7)

^is

equivalent

^{in a similar sense to the}

operator

and in the

A = 3,

^{T =}

1/2,

ml ^{= -}

1/2

^{case it is}

equivalent

^{to the}

operator

^of eq.

(A. 7)

^with p and n

interchanged.

^The formula of the _eq.

(A. 7)

type ^was used in ref.

[14].

The authors are

grateful

^to the Referee who _gene-

rously

contributed to

improving

^the

manuscript.

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