Atomic Structure Theory

250 8 MBPT for Matrix Elements
Table 8.6. Matrix elements of two-particle operators Breit operator for 4s and
4p states in copper (Z=29). Numbers in brackets represent powers of 10; a[−b] ≡
a × 10 −b .
Isotope Shift in Na
Term 4s 4p1/2 4p3/2
B (1) 1.880[-04] 7.015[-05] 5.140[-05]
B (2)
s 4.288[-06] 1.623[-06] 1.431[-06]
B (2)
d
B
-1.250[-05] -3.675[-06] -3.518[-06]
(2)
e -4.290[-04] -1.222[-04] -1.167[-04]
BRPA -6.993[-04] -2.020[-04] -1.310[-04]
B (3) 6.400[-05] 3.212[-05] 2.319[-05]
Btot -4.555[-04] -1.018[-04] -5.853[-05]
As a further example, let us consider correlation corrections to the isotope
shift in sodium. As discussed in Chap. 5, the isotope shift consists of three
parts: the normal mass shift NMS, the specific mass shift SMS, and the field
shift F. We separate the SMS matrix element for states of Na, designated by
P , into two-parts, P = S + T ; S being the contribution from the effective
single-particle operator and T being the normally-ordered two-particle contribution.
Second- and third-order correlation corrections S (2) and S (3) are
calculated following the procedure discussed earlier for the hyperfine operator.
The second-order two-particle contribution T (2) is obtained from the
first line of (8.32), while T (3) is obtained by linearizing the expression for the
third-order energy. A complete discussion of the evaluation of T (3) ,animposing
task, is found in Ref. [43]. The relative importance of the third-order
contributions to SMS constants is illustrated in Table 8.7. It follows from the
table that the correlation correction is largest for the 3s state; the lowestorder
value P (1) for the 3s state has the same order of magnitude as S (2) , but
has an opposite sign. The contribution S (3) for the 3s state is 17% of S (2) .
Two-particle contributions T (2) + T (3) are two to three times smaller than
one-particle contributions S (2) + S (3) for the three states listed.
Values of the MBPT contributions to the field shift operator F for 3s and
3p states in Na are given in Table 8.8. Since the field-shift operator is a singleparticle
operator, we follow the procedure discussed previously for hyperfine
constants to evaluate the first-, second-, and third-order contributions.
Finally, in Table 8.9, we compare values for the isotope shifts δν 22,23 of
n=3 states in Na with experimental data [22]. The sum of the third-order
MBPT values for the SMS and the NMS are listed in the second column of
the table. In converting the calculated field shift constants to MHz units, we
use the value δ〈r 2 〉 22,23 =-0.205(3) fm 2 obtained from (3.157). Our data for the
isotope shift for 3p − 3s transitions agrees with experiment at the 5% level.