The effect of the continuum states on the dynamic E2 mixing in antiprotonic atoms

Nuclear Physics A495 (1989) 622-632 North-Holland. Amsterdam

THE

THE EFFECT OF THE CONTINUUM STATES ON DYNAMIC E2 MIXING IN ANTIPROTONIC ATOMS G.Q.

LIU’

and

A.M.

GREEN*

Research Institute forTheoretical Physics, University of Helsinki, Siltavuorenpenger 20 C, SF-001 70 Helsinki, Finland S. WYCECH Institute for Nuclear Studies, Hoia 69, Warsaw, Poland Received 3 October 1988 (Revised 23 November 1988) The effect of the continuum atomic states on antiprotonic atoms is studied using a Green function method. A 4% effect is found in p- ‘74Yb for the case of the last observable transitions EL= E(n,,=9, &=8, j,,=y)+ E(8,7,?) and I?,= E(9,8,7)+ E(8,7,y). This study shows that, for the E2 dynamic coupling in antiprotonic atoms, perturbation calculations with the lowest atomic level can produce >90% of the energy correction to the basic states.

Abstract:

1. Introduction The strong-interaction experiment

spin-orbit

effects in p- ‘74Yb atom were reported in a recent by the PS176 collaboration at the Low Energy Antiproton Ring (LEAR)

at CERN. In ref. I), after extracting the electromagnetic effects including finite size and vacuum polarization corrections, resultant attractive shifts of AEL = 341* 43 eV and AEU = 283 * 36 eV were obtained for the last observable transitions EL=E(n,=9,1,=8,jo=~)~E(8,7,~), EU = E(9, S,y)-

E(8,7,9),

with the widths for the Sj levels being r,= 1261*41 respectively. Theoretical calculations Z-5) with realistic

eV and TU = 1021+41 eV p-nucleus potentials pro-

duced good agreement for the widths, but all four groups 2-5) predicted the opposite sign for the shifts with respect to the experimental values. In a recent article “) by the present authors, it was shown that the sign conflict between experiment and theory can be resolved if the deformation of the ‘74Yb nucleus is taken into account. The electromagnetic quadrupole interaction between the ‘74Yb nucleus and the p was calculated to give extra attractive shifts of about 600 eV. When these were removed from the measured shifts of 341 eV and 283 eV the resulting strong interaction shifts became AEL = -255 eV and AEU = -315 eV, respectively. ’ BITNET ’ BITNET

address: address:

“GULIU@FINUHCB”. “GREEN@FINUHCB”.

0375-9474/89/$03.50 0 Elsevier Science Publishers (North-Holland Physics Publishing Division)

B.V

G.Q. Liu et al. ,J Cantjnaanzstates

The calculation discrete

p atomic

experimental (about magnetic

of ref. 6, employs

second-order

states

However,

considered.

measurements

a few hundred

from LEAR and treated

perturbation because

properly

to ensure

theory

of the great

and the smallness

eV), care must be taken

effects are included

623

with only

precision

of

of the shifts involved that all possible

so that the p-atom

electro-

data can serve

as a senstivive test for p-nucleus potentials. It is also well known in such problems as the Lamb shift ‘), in the case of nuclear polarization in muonic atoms ‘), and the quadrupole moment of muonium ‘) that confinuum atomic states may contribute significantly and in some cases even be dominant. For example, in the quadrupole moment of the muonium ground state only 12% of the theoretical result comes from the sum over discrete levels “f. in the present paper, a Green function method is used to study the effects of the continuum atomic $3states for the case of the E2 dynamic mixing considered in ref. “). Comparing the perturbation calculation of ref. “) with the full calculation in this paper, it is found that in the case of the 90% of the corrections to the shift can be p-‘74Yb last observable transitions, obtained from the perturbation calculation when including only a few of the lowest-lying p orbitals (n 5 n,+2). Although the convergence of the perturbation series is rather slow, as can be seen from the full calculation, the contribution from higher orbitals (n > no+ 2) are in comparison 2 to 4 orders of magnitude smaller, and the contribution from p continuum states is found not to be dramatic. It should be pointed out that there are other methods for solving the second-order perturbation equation (2.1) exactly. In some particular cases, the Dalgarno method lo) offers a simple and elegant approach which makes it possible to express the final result in an algebraic form. However, in the present problem there are two complications which appear to exclude this simplicity. First, evaluation of the differential equation, that needs to be solved, can no longer be done analytically because of the presence of the nuclear excitation AE. Only in the limit of AE =0 does this simplicity emerge. Secondly, the angular momentum of the system needs to be treated more carefully since three towers of atomic levels enter. At the present time it is not clear with the Dalgarno method whether or not these towers can be included separately or only in an approach involving three coupled differential equations. This method better suits the I=0 states, and strongly depends on the form of the perturbation

H,. Another

alternative

is the numerical

method

developed

by Sternheimer ‘I). This is especially useful if in H, the nuclear potential is present, or the g-‘74Yb interaction is not a point Coulomb type. In the present problem with a point Coulomb potential, which is appropriate for high values of 1, the Green function method seems to be the best choice because of its simplicity and generality. The main reason for using this method in the present calculation is as follows. It replaces the need to solve an inhomogeneous differential equation by a double integral over confluent hypergeometric functions and thus is much simpler numerically. However, the applicability is limited to point-like nucleus situations i.e. large angular momentum {or small mass) of the atomic particle. In such situations the

624

G.Q. Liu et al. / Continuum

states

appropriate Green function can be expressed in a closed form. Technical reasons limit it further to cases where the Whittaker functions, implicit in Coulomb waves, have a simple

integral

shifts and circular

representation.

is higher when the nucleus these conditions

The latter condition

(or close to circular is excited

favours

orbit) states. Finally, and the full Green

are well met in the ~-“~yb

attractive

the numerical

function

energy

precision

is to be used. All

case.

2. The Green function formulation The aim now is to solve the second-order

perturbation

equation

eq. (2.1) exactly,

( EP - HO) w, = H, w, where ZZ,,= HP+ HN. Here Hii includes hamiltonian of the nucleus approximated H, is given by

a point Coulomb potential and HN is the as a rigid rotor. The quadrupole interaction

H, = eVDef= -where

QO is the quadrupole

Wave p-hydrogen

(2.1)

e2Qo -1 2 r3 P,(cos f&v)

(2.2)

moment.

functions PO and ?P, contain both p and nuclear like orbitals by (nlj), they can be written as

parts.

Denoting

the

IP”=~nZj,z=O;FM),

(2.3)

9, = (n’l’j’, I’ = 2; FM) )

(2.4)

where (Z, Z’) are the spins of the nucleus, and (F, M) are the total angular momentum and its projection. Note that only I’ = 2 states are considered here and higher excited nuclear states with I’> 2 are ignored. The radial

equation

from eq. (2.1) is given by d2 I 2 d

-1 2/1

dr2

where

Z_Lis the reduced

energy

consisting

nucleus,

r dr

L(L+1)+2~Ze2 -++/_LE

mass of the p-nucleus

of the p atomic

C, is an angular

energy

momentum

c, = (-1)F-“2

R, =cJ R,,

r

r2

r3

system,

E = Ep- AE

Ep and the excitation

energy

(2.5) is the total AE of the

factor given by (see ref. “))

~J(2jf1)(2~~+1)(21+1)(21’+1)

(2.6) When E is not in the eigenvalue spectrum of the Ho, the full solution can be obtained from the well-known Coulomb Green’s functions’2-‘4), GL(

r, r’,

y)

=

-2p.K

T(L+ 1+ iY) fL(P<) UtL(P>) -___ T(L-tl-iY)

p<

P>

’

of eq. (2.5)

(2.7)

625

G.Q. Liu et al. / Continuum states

where

K =

p = Kr, p’ =

m,

and y = ~yZp/

Kr’,

fL and U: are related to the confluent function

K.

The regular

hypergeometric

and irregular

function

solution

and the Whittaker

by fL(P)

-=

exy’2f (Lt

P

ut(p)

_

P

1 - jy)(2p) Le’PF(L+1-jy,2L+2,-2jp),

e”“’

T(L+l+jy)

1 w

1.L

f(L+l-jy)

p

(2.8)

(2.9)

,y,L+1&2ip).

Here F is the confluent hypergeometric function and W the Whittaker function as IS “,lh). The GL of eq. (2.7) . a solution of eq. (2.5) with 6(r - r’)/r2 defined in refs on the right-hand side. A neater form of GL is obtained if ti is defined by the following

equation pZ'2

pZ2CX2

E=--=___

2n,?,

2Fi’

(2.10)

dE,

where

the relativistic correction to the p energy is ignored. Since AE is positive (76.48 keV for ‘74Yb) it follows that ji < n, and ji = jy. Therefore the Green function GL can be expressed as 2CLK r(L+ 1 + ‘) (_2jp%:)L+l e-ip GL( r, r’, fi) = ~ 2jp,p, T(L+l-n) x F(L+ The second-order

correction

l-

jy, 2L+2,

to the energy

-2ipc)

Wjy,L+IIz(-2jp,).

(2.11)

is then given by (2.12)

where Z denotes

the multidimensional

integral

n)In&. (%l”l “‘$,:I’ N.B. Since excited

in this particular

problem

all intermediate

state, there is no need to subtract

subtractions

can lead to numerical

from

states

involve

the nuclear

GL the effect of ~r&)(n,l,~.

Such

difficulties.

For the cases considered in this paper, where only circular the integral Z can be reduced to the following form

orbits

are involved,

dxx Pi+/,,-2e --1/2(r)(1+“/“,,)~L(~, x)

z=2z,

J x

X

0

dyy

LCl,,-I

e~‘/2(v)(‘+n/n,,‘F(L+

1 -

fi,

2L+2,

y)

.

(2.13)

626

where

G.Q. Liu et al. / Continuum

OL(fi, x) is the integral

representation

states

of the Whittaker

function, (2.14)

and 2

z”= with B being

4

1

( >

-P

Bn,

(2L+l)!n()(2no+l)!

(2.15)

( : -j2”-’

the Bohr radius.

The angular-momentum factor C, vanishes except for the three L-values IO+ 2, IO, and lo - 2. But for these three L-values the integral Z has very different properties due to the singularities of the Whittaker function QL as t + 0, for no - fi < 1 in the cases concerned with here. Each L is now analysed in turn. (i) L = Zo+2 (the outer tower). In this case the integrand of QL(r?, x) is regular as t + 0, and @,( ti, x) can be evaluated straightforwardly. (ii) L = 1, (the central tower). The integrand of QL(fi, x) goes like t-(‘-‘0), where (ti - lo) < 1. The integrand of QL( ti, x) is singular but integrable. This singularity can be treated by removing a gamma function from QL( fi, x), i.e. rewriting QL( 3, x) in the following

form -d,e-~,L~n[(,+~)i+n-I]

@,(fi,x)=T(L-n+l)+

(2.16)

I0 so that the integrand in eq. (2.16) is regular as t + 0. (iii) L = Lo-2 (the inner tower). In this case, the integrand of QL(fi, x) goes like t -(n-‘&2 as t + 0, hence it is not integrable although it is known physically that the integral Z should be finite. This divergence of @J ti, x) is not intrinsic to the physical problem or the approach. The integral representation of the Whittaker function is not valid in this case 15*16)due to the existence of bound states when 5 < n. A way to cure this problem a complex

number

is to use an analytic and analytically

QL(fi, x) is now written

continuation

continue

in the following

procedure,

i.e. treating

ii as

to the real axis of fi. With this in mind

form (2.17)

.,,,,x,=.,x,+Jo~dte-~tL-~[(l+~)L+n-Ll(f)], where u

0 4 X

=l+(L+n)t+(L+~)(L+A-l) X

J

2

cc

A(x) =

0

dt

e-rtL-A

u

0 f

x

.

t2

23

(2.18)

(2.19)

G.Q. Liu et al. / Continuum

Now the divergence this case the following

is in A(x). relation

627

States

Let fi be in the domain

where

Re (L-

ti) > -1.

In

can be used

r(L-fi-tk)=

--I L-n+!f ’

l L-ii+k

(2.20)

to reduce the singularity of the integrand for A(x) and then analytically continue A(x) to the real fi in the present problem. This procedure gives A(x) in the following analytic form A(x)=I’(L-n+3)

(L+n)(L+n-1)

(Lfii)

2x2

+(L-ti+2)

1

1

x+ (L-ii+2)(L-fi+l)

1

. (2.21)

3. Results and discussion 3.1. RESULTS

OF THE

The full calculation

FULL

CALCULATION

by the Green

discrete-level perturbation ‘74Yb E2 dynamic mixing,

function

calculation the integral

method

is now compared

with the

as was used in ref. “). For the case of the 2 is evaluated for the basic states (n,, /,) =

(8,7) and (9,8). Table 1 gives the Z values for these two basic levels. Note that the integral Z is to be compared with the quantity (3.1) in the discrete level perturbation calculation. The calculation of the multidimensional integral Z was done by both Simpson’s method and the Monte Carlo routine VEGAS. Energy corrections

to the basic static states (n,, I,,,jO) = (8,7, y), (8,7, y), (9,8, q),

and (9,8, ‘2) are calculated from these Z-values. To each basic state (n,,, &,j,), there are five possible angular momentum configurations entering. For example, with the basic state (8,7, y), the contributing angular momentum configurations are (L, j) = (5,q),

(5,+),

(7,?),

(7, y),

and

(9,y).

The energy

TABLE

Values of the integral

no

I0

L

8 8 8 9 9 9

7 7 7 8 8 8

5 7 9 6 8 10

Z(n,,,

correction

1

&,, L) in eq. (2.13)

Z ((MeV/fm”) 2.521 -23.691 - 1.549 0.824 -5.585 -0.502

x 10-4)

is the sum of the

628

G.Q. Liu et at. / Con~~~uu~ states

contributions

from all five (L, j) configurations.

full calculation calculation

for each basic state as compared

obtained

by summing

shows all contributing

angular

momentum

con~gurations,

SET” = -215-(-913)=698eV, discrete

of the

level perturbation

and it can be seen that

states yields

the following

shifts

for

SE:“=-211-(-913)=702eV.

level perturbation

SE Frt = 670 eV , Comparison

2 shows the results

the series to n = 100. In table 2, the second column

a full calculation including the p continuum the 174Yh last observable transition.

The corresponding

Table

with the discrete

calculation

gives

SEP,“’ = 669 eV _

of the full and the perturbative

calculations

6EF”-

SEP,“” indicates

that the continuum states contribute about 4% in this case. The splitting of the two transitions remains unaffected (-3 eV). In the appendix, a table containing the contributions from each angular momentum configuration is given. It is found that the biggest contribution from p continuum states is in the outer tower (L = 1,,+2). For the inner and central towers, continuum contributions are negligible. It should be mentioned that the numbers for SE pert correspond to the numbers given in the third column of table 1 in ref. 6), which were 596 and 598 eV respectively. The differences arise from two reasons. (i) Relativistic effects - including the spin orbit correction to the D atomic energy and the effect discussed in section 2.2 of ref. “) - are ignored here for simplicity of the calculation. And (ii), in ref. “f the lowest levels were treated more completely by diagonalizing the energy matrix, whereas here only the perturbation series is summed over. These differences are not large enough as to change the conclusion of this paper in any significant way. The final calculation of the corrections to the measured shifts is based on the numbers

of ref. “) so that the relativistic

effects are ultimately

taken

properly. This is done by adding the percentage of the continuum the corresponding perturbative results of table 1 in ref. “), i.e.

into account

contribution

to

6EL = 596 x 104% = 620 eV

(3.2)

6E,, = 598 x 104% = 622 eV .

(3.3)

and

When these corrections

are removed

from the measured TABLE

values of AEL = 341* 43 eV

2

Energy corrections to basic states (~“~,~j~,)using

eq.

(2.12)

(-LA

El:i, (ev)

EFF,(eV)

(5, ;,, (5, VI, (7, v,, (7, Y,, (9, Y, (5 r2r “) (7 17, 5) (7 97. y (9 I?,‘5) (9 ,z17) (6, ‘$1, (6, G), (8, ‘$1, (s,y~,(lo, 4,

-913

-870

(6, ‘$,, i&y),

(S,$,

(IO, yt, (10, ?$I

-913

-870

-215

-200

-211

-201

629

G.Q. Liu er al, / ~~~~~j~~~~ states

and AEL,= 283* 36eV, the final shifts from the strong

AE,=-339+36

AEL=-279+43 eV, compared

with theoretical

predictions

interaction

7-5) of ranges

become

eV.

-f80+130)eV

(3.4) and

-(60+

120)eV, respectiveiy. 3.2. CONVERGENCE

The convergence

OF THE

DISCRETE

prablem

LEVEL

PERTURBATJON

of the E2 dynamic

mixing

SERJES

can be divided

into two

parts: (i) the convergence of the discrete level perturbation series and (ii) the contribution from the jj contjnuum states. It is seen from the table in the appendix _ on rows marked “) - that the outer tower gets the biggest contribution from the p continuum. But this does not mean that the central and inner towers converge more rapidly. This is illustrated by the following calculations. In order to be able to make comparisons between different L towers, the perturbation series is summed from the sarrzirte n = nhtari for afi t lowers, i.e. nstart = n, + 2 which is where the outer tower begins. This, therefore, excludes the lowest 1eveIs in the inner and central towers. The discrete level convergence factor R( no) is now defined as the ratio of the sum to the first term (nrtnrr), i.e.

where

(3.6) It is found that for the outer and inner towers R,(n,f + 4.4 as tl’+ ax), and for the central tower R,(nJ+2,3. A fuiI convergence factor I??” can also be defined in a similar way to R,(no), namely, R1’Ull L

=

, C

AL(nslart)

A,(n’)Scontinuum

[n =%.%I,

I/

(3.7)

For the cases considered in the $“Yb atom, the outer tower Rt!z+6.9 for (no, i. = 8,7) and 5.8 for (n,,, i,, = 9,8). It is, therefore, expected from the convergence behaviour of the discrete level calculations that R’;““(inner)

* R1;u”(outer) a 7

and ~~“(centra~~ However, magnitude This makes and central

=

R,(central) R,(outer)

x RF,U”fouter) = 3.6.

the terms A(n,,,,,) are 2 (central tower) to 4 (inner tower) orders of smaller than the contributions from the lowest levels in these towers. the contribution from higher n levels and continuum states of the inner towers less si~ni~~nt than might be thought from the value of RF” of

630

the outer

G.Q. Liu et al. / Continuum

tower.

calculations

It can, therefore,

can be reliably

be concluded

used for the inner

hand the outer tower needs to be treated It is worth

noting

that >90%

only from one angular

that

discrete

and the central correction

configuration

level perturbation towers.

with care if accuracy

of the energy

momentum

states

On the other

is desired.

to the basic states comes

i.e. the (L, j) = (IO, j,) configuration

- as can be seen from rows marked “) in the table shown in the appendix. Considering that experimental measurements involve errors of - 10%) a discrete level calculation with only the (1,j) = (lo, j,) state can give quite good results. The discrete level convergence factor RL( n,) is also calculated for the outer towers with lower n, basic states (n,,, lo) = (1, 0), (2, l), . . . , (12, 11) - see fig. 1. It is found that the convergence behaviour of the discrete level perturbation series is quite stable. For practical calculations the continuum can be included by doubling the R,(n,) to get the full convergence factor R?” and then the energy correction.

5*

x

x

x

x x

x

x

x x

x 3-

X

l-

1 Fig. 1. Discrete

2

state convergence

3

I

I

I,,

L

5

6

factor

7

8

I

I

I

9

10

11

I_ 12 n,

as defined by eq. (3.5) as a function number n,.

of the principal

quantum

4. Conclusion In this paper the calculation of ref. “) is extended to include the continuum ij atomic states, using a Green function method. It is found that in the case of the p-174Yb last observable transitions the contributions from continuum p states are at the 4% level. When this 4% correction is incorporated into the calculated shifts of ref. 6), the experimental shifts become AEL= -279 and AEU = -339 eV instead of AEL = -255 and AEU = -315 eV as in ref. “). Taking into account the fact that experimental measurements usually involve errors of about 10% [ref. ‘)I, it can be

G.Q. Liu et al. / Continuum

concluded

that a simple

can give quite contrary From calculation

perturbation

satisfactory

results

calculation

631

states

with a few of the lowest p orbitals

in the case of the E2 dynamic

to the cases cited in refs. 7-9) where continuum a study

of the convergence

properties

mixing.

effects play a dominant

of the discrete

This is role.

level perturbation

it is found that the inner and outer towers have very similar

convergence

behaviours, whereas the central tower converges more rapidly. Higher levels (n 2 n,+ 1) in the inner and central towers contribute to the energy correction negligibly. It is the higher levels in the outer tower which contribute mainly to the 4% correction in the p-‘74Yb case. The above experimental values (AEL= -279 and AEU = -339 eV) tend to be ranges of -(SO+ 130) eV and somewhat larger than the published 2m5)theoretical -(60+ 120) eV. However, it should be added that the optical potential of ref. “) based on a separable NR interaction leads naturally to larger strong shifts in the range -(200+ 250) eV. This work also illustrates that the nuclear densities at large values of r are needed and these, because of the nuclear deformation, are uncertain. One of the authors (SW) wishes to thank the Research Institute for Theoretical Physics, Helsinki for their kind hospitality. Furthermore, the authors thank the referee of ref. “) for wishing the effect of continuum states to be clarified. Appendix ENERGY

CORRECTIONS

FROM

DIFFERENT

ANGULAR

MOMENTUM

STATES

Here the quantity Z is defined in eq. (2.13) and C, is given by eq. (2.6). The full energy correction E ii/, is given by the product of Z and C: as in eq. (2.12). 1. Basic state (n,,, I,,, j,,) = (8, 7, y) L, j

Z ((MeV/fm4)

x lo-“)

Cs(fm4x

10-j)

%l,

(eV)

EL:‘,, (eV)

2.52

49 1.63

124

124

,‘A 5” ,

-23.69 2.52

-911 3

-907 3

7’ &

-23.69

384.56 13.11 11.31

5,;

>z 9”( 2

-27

;‘) -27

-1.55

661.67

-103

-64 h,

2. Basic state (no, I,,, j,,) = (8,7,7) L, j 5”> 2

Z ((MeV/fm”)

Eiti, (eV) 127

E”’per, (eV)

7”

-23.69

504.74 9.90

-23

-23

9’7’&?

-23.69 -1.55

385.97 8.71

-914 -1

-910

I2 9”) >2

2.52

x 10m4) Cg (fm4 x 10-j)

127 “)

-1

-1.55

652.96

-101

-63

h,

632

G.Q. Liu et al. / Continuum states 3. Basic state (n,,, I(,,j,,) = (9, 8,y) L, j 6”> 1

6” 7 g’32 * >!.Z 2 10 >111 7

Z ((MeV/fm4)

x 10d4)

CS(fm’x

0.824 0.824 -5.586 -5.586 -5.025

lo-‘)

504.74 9.90 385.97 8.71 652.96

Flti, (eV) 38 1 -216 -5 -33

EF,,

(eV)

42 1 -215 “) -5 -24 “)

4. Basic state (n,, , I,,, j,,) = (9,8,7) L, j 6”3

2

8” 8’& 32 10 I” 7 1o’>iI2

Z ((MeV/fm4) 0.824 -5.586 -5.586 -5.025 -5.025

x

10e4)

Cs (fm4x IO-‘)

Ei:;, (eV)

Egi., (eV)

514.63 7.14 386.94 6.91 646.05

42 -4 -216 -0 ~32

43 -4 -215 .‘) -0 -23 h,

“) This contributes >90% of the total correction. ? h, The difference in El;/, - EL:‘,, is only significant

for these outer towers.

References 1) 2) 3) 4) 5) 6) 7) 8) 9) 10)

11) 12) 13) 14) 15) 16)

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