We calculate the effect of the Coulomb interaction of the ejected β ray with the bound atomic electron in the β decay of a tritium atom. The excited state probabilities of the residual helium ion are changed by at most 0.17% from the usual sudden approximation

Koonin, S. E.

Williams, R. D.

English

Caltech Authors - Main

PHYSICAL REVIEW C VOLUME 27, NUMBER 4 APRIL 1983 
Atomic final-state interactions in tritium decay 
R. D. Williams and S. E. Koonin 
W. K. Kellogg Radiation Laboratory, California Institute of Technology, Pasadena, California 91125 
(Received 8 November 1982) 
We calculate the effect of the Coulomb interaction of the ejected 13 ray with the bound atomic 
electron in the 13 decay of a tritium atom. The· excited state probabilities of the residual helium 
ion are changed by at most 0.17% from the usual sudden approximation. 
[ RADIOACTIVITY 3H; atomic fmal states, neutrino mass. I 
In a recent experiment, Lubimov eta/. 1 attemped 
to determine the mass of the electron antineutrino 
from the p-decay spectrum of a tritiated valine 
source, and obtained a value between 14 and 46 eV. 
Since this mass is comparable with atomic binding 
energies, one expects atomic effects on the spectrum 
to be important. Bergkvist2 calculated several such 
atomic and molecular effects in the usual sudden ap-
proximation, and obtained an effective decrease in 
his detector resolution. Law3 calculated these effects 
more exactly, including continuum states (total ioni-
zation of the He atom), while Fukugita and Ku-
bodera4 treated the continuum states with Coulomb 
effects. In this paper, we further refine the calcula-
tion of atomic state probabilities by including the 
Coulomb interaction of the pray with the bound 
electron. This interaction has been studied for other 
nuclei,5•6 where effects are expected to be more pro-
nounced, but not, to our knowledge, for tritium. 
The interaction of the atomic electron with the ~ 
ray is small because the~ ray wavelength 27T/k is 
small compared to the Bohr radius a= (me 2)-1, 
where m is the electron mass and e the charge (we 
use units where fi = c = 1, so that e2 == 1!7 ) • At the 
tritium end-point energy, 18.57 keV, the usual Som-
merfeld parameter..,= -me2/k = -1/ka = -0.0271 is 
small, so that our results need only be calculated to 
leading order in 'rl· 
We will consider a process in which a tritium atom 
with its atomic electron in the ground (Is) orbital de-
cays by emitting a~ ray (momentum k), leaving the 
atomic electron in an excited He atomic orbital 'rl!· 
To first order in the weak interaction Vw, the ampli-
tude for this process is 
T/1-= ('l'_i->1 Vwlcf!,) , 
where cfl1 is the initial tritium Is wave function, and 
'I' j-> is the full solution to the interacting two-
electron probhim, specified by k and n1 and by 
incoming-wave boundary conditions. This latter 
scattering wave function is a solution to the Hamil-
tonian describing two electrons in the presence of a 
27 
He nucleus, 
(E-H)'f!'j->=0 , 
E = e,+k2/2m , 
L L 2e2 2e2 e2 
H- 2m+ 2m--,----;:;--+ lr-r'l ' 
where ( r,p ) describe the atomic electron with final 
energy e1 and (r',p') describe the~ ray, whose final 
energy is tl/2m using nonrelativistic kinematics. 
A perturbative approximation to 'I' j-> accurate to 
the order required can be obtained by iterating the 
Lippmann-Schwinger equation. We partition Has 
H=H0 +V, 
n2 n'2 2e2 e2 Ho=-L--+.s;.._ ____ , 
2m 2m r r' 
e2 e2 
V=--+ I 'I . r' r -r 
where we will treat the interelectron repulsion as a 
perturbation, and we have split the p-nucleus attrac-
tion ( -2e2/r') between Ho and V such that V::::: r'-2 
as r' ..... co. The Lippmann-Schwinger equation is 
('I' _i->1- ( cflj-> I+ ('I' J->1 VGJ+> 
== (cflj->1 + (cflj->1 VGJ+> , 
where the Green's function is 
GJ+> =lim (E-H0 +i')-1 
,_o+ 
and (cflj->1 is the solution to Ho with energy E and 
the required boundary conditions. Specifically, 
(rr'lcflj->) = x,.,(r )uk(-)(r') • 
where x.,.1 is the final Z = 2 atomic orbital, bound or 
continuum, and u~-> is the Coulomb-distorted plane 
wave with incoming spherical waves, 
H( ') 4 ~·I _,.,.,F,(kr') Y*(k)Y (r') 
uk r = '7T ~~ e k , lm 1m 
1m r 
1815 @)1983 The American Physical Society 
1816 BRIEF REPORTS 27 
Here, F1 are the usual Coulomb wave functions7 for 
charge Z =I, and cr1 are the related Coulomb phase 
shifts. We thus approximate the decay amplitude as 
T T (O) + T(l) fi= fi /1 ' 
T)0> = (cllj->1 Vwl<1>1) 
T },0 = (<I> _j-> I VGJ+> V wl<l>1) 
For the allowed decay considered here, we may safely 
take V w to create the f3 ray at the nucleus with zero 
angular momentum. Thus, apart from an overall 
constant, Vw=B(r'), and so, in an obvious notation, 
T),0>=ui->*(0) Jd3rx:,<r)X;(r), 
T) 0 = J d 3r d 3r' ukH*(r')x;1(r) 
x V(r,r 1 )(rr 1 1GJ+> lx1;0) 
Upon identifying ui-)*(0) =/a°C0(71), with Co the 
usual Gamow factor, 5 we see that 
T),0> = e1a°Co( 71) J d3r x:,x, 
is the usual sudden approximation amplitude, correct-
ed for Z = 1 Coulomb distortion. To treat T1~0 , we 
use a spectral representationof GJ+> in terms of the 
Z = 2 atomic states Xn to obtain 
(rr 1 1GJ+> lx;;O) = Ixn(r )gt~>•n (r', 0) (XniX;) , 
n 
where gt+> is the Z =I single-particle Coulomb 
Green's function. Since the atomic energies En are 
small compared to the total energy, E =:::: I8 keY, we 
may set E -En = E, independent of n, and employ 
closure to obtain 
(rr1 1GJ+> lxt;O) =gt+>(r', O)Xt(r) , 
which gives for the first order amplitude, 
T)0 = J d 3rd3r1UkH*(r 1 )x:,<r) 
[ e 2 e 2 ) (+) 1 x lr-r~l-7gE (r,O)X1(r) 
Since the f3 ray has zero angular momentum, 
(-)'*( 1 ) _ ta0 Fo(kr') Uk r -e 1 , kr 
<+>(,~ 0)=2l!£_ (Go+iFoHkr 1 ) 
gE • 2 2 k I ' 1re r 
where Go is the irregular Coulomb function. We can 
safely discard the I ¢ 0 atomic final states, which 
have only order 71 2 changes in their probabilities, 
since T ),0> is zero. 8 By using the asymptotic expres-
sions for the Coulomb functions, 5 to order Tl• we can 
write 
.( ) foX I I .1[' ') Q x = 271 dx sinx e" - --
o x X 1 
=-71[--1- +~ + i(In2x +y -1)1 2x 2 
where y- 0.5772 is Euler's constant. Note that 
-717r/2- i'TI'Y is x independent. It therefore gives a 
contribution proportional to T),0> and, indeed, reflects 
the change in /"oc 0 ( 71) going from Z = I to Z = 2. 
The required matrix elements of Q can be found 
analytically for the bound states; those of 1/r can also 
be found analytically for the continuum final states. 
However, for the lnr continuum matrix element we 
TABLE I. Matrix elements and decay probabilities to various He+ states. 
State 
is 
2s 
3s 
4s 
5s 
Continuum 
Total 
(!l~li) (1j1n~ji) 
1.257 08 -0.147 35 
0. 0.61482 
-0.03135 0.09666 
-0.023 28 0.052 81 
-0.01742 0.035 29 
IT),0>1 2 IT(O)+T 0 )1 2 /1 /1 
70.23% 70.06% 
25.00% 25.17% 
1.27% 1.28% 
0.38% 0.39% 
0.17% 0.17% 
2.63% 2.62% 
99.69% 99.69% 
27 BRIEF REPORTS 1817 
used the approximation 
(Fo(kr )lkr llnr lls) = (Fo(kr )lkr Its) 
(jo(kr )I kr llnr lis) 
X-"-..,--.,---:-.,--.--.,:_ 
Uo(kr)lkrlls) 
We have also estimated the exchange contributions 
to. the decay amplitudes ({3 ray emitted into a Z = 2 
atomic orbital and original atomic electron ejected) to 
be lower by a factor 11 2 than the direct contributions 
calculated above, and thus negligible. 
Table I shows matrix elements of aIr and lnr I a 
between the Z = 1 ls state and the Z = 2 ns states, 
tv. A. Lubimov, E. G. Novikov, V. Z. Nozik, E. F. Tre-
tyakov, and V. S. Kosik, Phys. Lett. 94B, 266 (1980). 
2K. -E. Bergkvist, Phys. Scr. 1, 23 (197Q 
3J. Law, Phys. Lett. 102B, 371 (1981). 
4M. Fukugita and K.Kubodera, Z. Phys. C 9, 365 (1981). 
sE. L. Feinberg, J. Phys. (Moscow) 1, 423 (l941); R. L. 
Intemann, in Inner Shell and X-Ray Physics of Atoms and 
Solids, edited by D. J. Fabian (Plenum, New York, 1980), 
p. 295 ; I. S. Batkin, Yad. Fiz. 33, 48 (1981) [Sov. J. 
for n = 1 to 5, and the probabilities of the ns states 
and the probability for ionization, normalized so that 
the sum of the probabilities in each column is the 
same. This sum is not exactly unity since bound s 
states with n ~ 6 have been neglected. 
The corrections are of order 0.2% for the ls and 2s 
states and much smaller thereafter. In the analysis of 
the Lubimov experiment, 1 a change in the He+ 2s 
probability from zero to 0.3 changes the inferred an-
tineutrino mass by 15 eV, so that the effects of the 
electron-electron interaction are negligible until the 
antineutrino mass is known to within 0.1 eV, which 
seems far beyond present experimental capabilities. 
Nucl. Phys. 33, 25 (1981)1. 
6R. L. Intemann, Phys. Rev. A 26, 3012 (1982). 
7M. Abramowitz and I. A. Stegun, Handbook of Mathematical 
Functions (Dover, New York, 1964), Chap. 14. 
81ntemann, in Ref. 6, states that higher partial waves are im-
portant below I max= 1/ ka = -'1) = 0.0271, which is further 
support for this approximation. We find the probability of 
exciting the 2p state to be 0.01 %. 


Atomic final-state interactions in tritium decay

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Atomic final-state interactions in tritium decay

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