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Fine structure inversion of the 3d excited state of sodium
Volume 55A, number 5
PHYSICS LETTERS
29 December 1975
FINE STRUCTURE INVERSION OF THE 3d EXCITED STATE OF SODIUM* H.M. FOLEY Department of Physics, Columbia University, New York, New York 10027, USA
and
R.M. STERNFIEIMER Brookhaven National Laboratory, Upton, New York 11973, USA Received 21 October 1975 The observed inversion of the fine structure of the excited 3d state of the sodium atom explained as due to a second-order effect between the electrostatic dipole exchange interaction and the magnetic spin-orbit interaction acting on the core 2p electrons.
It has been known for many years that the finestructure interval z~svof alkali-metal atoms in excited states with 1> 1 (except for Li) is inverted or is very much smaller than that predicted by any reasonable single-electron model Ll]. The first attempt to explain this property was made by Phillips [2] in 1933 for the case of the alkali-like spectrum of Mg II. However, the plausible estimate of ii’, which was obtained for Mg II 3d, is a thirdorder effect, which involves the wave function squared of a 2p hole in the closed 2p shell. This is presumably smaller than the second-order effect to be discussed here. Recent hyperfine structure (hfs) measurements of the Rb excited 4d state by Happer and coworkers [3] show similar inversions in sign for both the fine structure and the spin-dipolar magnetic hfs. proposedue that effects, leastorbital ford polaristates, areWe primarily to these exchange spinatand zation of inner p shells by the external d electron, and are properly calculated as a second-order cross term between the electrostatic dipole exchange interaction and the magnetic spin-orbit or hfs interaction. (A clue here is the absence of the effect in Li, which has no internal p electrons). We present results for the fine structure of Na 3d as a first example of this effect. The effect considered here is the second-order “cross interaction” between the spin-orbit coupling perturbation Hf = 2(,.i~/e2)(1 /r) (d V/dr) / s and ~ This work was performed under the auspices of the Energy Research and Development Administration. 276
the exchange part of the Coulomb interaction HQ = 2/r12(Rydberg units). Thus for Na 3d, the change of energy is:
(3d,2plHQ~np,3d)(np,3dIHf~2p,3d) ~-~2
=
P
np
The sum over n extends over all excited and continuum states. The calculation is much 2D simplified by considering the interaction in the state 3 512, M~= 5/2. The dipole part of HQ which is given by: 8 r< (2/r12)d~o1e= r> (2) X [Y?(l)Y?(2) 1- Y~(l)Yj’(2) + Yi~’(l)YI(2)]. In the matrix element of eq. (2), since 3d, m = 5/2 has 2P3/2’ m1 with mj == +2, 3/2,m5 via= Yj’. this Thusonly the connects matrix element
+4,
(2p
1(2)Inp 312312IY~(l)I3d512512X3d512512IYj 312312) and its inverse are the only non-vanishing ones, when one treats Hf as a single-electron operator. In eq. (1), the function: (/1p,3dIHfl2p,3d) W i~i(np,3d) (3) ~ is, apart from an angular factor, l/r times w1(2p-+p), which2 is the 2 solution of the equation: / d w 1(2p-+p) (4) =
[(J(r))2~—f(r)] u0(2p),
Volume 55A, number 5
PHYSICS LETTERS
where u0(2p) is the radial 2p function (times r) [4] for Na,f(r) (l/r) (dV/dr), and w1(2p -+p) includes all of the excitations of 2p produced by the interaction H~,i.e., both the discrete and continuum p states. We note that eq. (4) is analogous to the equation which determines the shielding and antishielding effects for the quadrupole hyperfine structure [5]. The valence wave function v0 (3d) for Na was obtained previously. 2r’ p), u(see fig. I of ref. [61).The wave functions w1 ( 0(2p), and v0(3d) for Na are shown in fig. 1. Using the wave function w1(2p —~p),eq. (1) becomes 2d~I2 ~K[2~X 1] (5) 3 L e2 = —21 l~1 l}~
f~1
f ~
L
0(2p)u0(3d)
r fr’vo(3d)wi(2p —~p)dr’ r~~
(6)
We find K = 002282. The angular factor which precedes K in eq. (5) is —8/5. We thus obtain: 2u2 =~ X 0.01826 Ryd. (7) e2a~ This must be compared with the first-order effect of Hf on the 3d 512,512 state, which is given by: +(2~/e~) 1<1 s> 3d5~25~2, I is the integral: -
2dr = 0.007 16 Ryd/a~. I
=
I
I
I
I
I
I
I
0.24 I
Na
0.20 0.16
0•6
~
0.4 0.04
~ o.~ -a ~ o.c
D.0 0.04
-
2w 10
-0.08 1(2p—p)
-0.5 0.0 0.4 0.8
1.2
1.6
I
I
I
I
2.0 2.4 2.8
I
I
I
I
0.12
I
3.2 3.6 4.0 4.4 4.8
RADIUS r(oH)
Fig. 1. The perturbed wave function
w
5 (2p p)(times 1O_2, and the unperturbed 2p and 3d functions tion of the distance r from the nucleus. for Na, as a func—
/~.E2= (2i~/e2a~) 0.00716(1—2.550) Ryd. (10) According to a theorem due to Judd [7], the same
f r’2v~(3d)X w1(2p~p)dr’]
=
I
+
+ r
where
-a
I
.0 0.8
dru
(1
x
.2
—~
4
where the and last Kfactor is the value of l~s~ in the state 2P3/23/2 is the following integral: K=
29 December 1975
0J’f(r)
(8) (9)
[v0(3d)]
The expectation value (1 s> in eq. (8) is:(1~s~) = 2 X 4 = +1. By comparing eq. (7) with eqs. (8) and (9), we thus obtain a correction factor (overshielding) of: —0.01826/0.00716 = —2.550, i.e., the total spin-orbit energy for the 3d 5~2,5~2 state is: -
3D to the entire 2D spincorrection factor can be applied orbit splitting between the 3 512 and 3 312 states. The first-order splitting is given by: 2RydX 0.00716 = 0.051 cm~. (11) ~ =~a Hence we obtain for the second-order fine structure splitting: ~2 = —2.550 X 0.052 = —0.133 cm~,and for the total splitting: (12) ~‘ = ~‘l + = 0.052 0.133 = —0.081 cm* which has the same sign but somewhat larger magnitude than the experimental value [1], namely —
—0.049 cm~. The calculation of both the first-order and the second-order effects is delicate, especially the latter, which depends on the overlap of the perturbed p function with the 3d wave function (see fig. 1). Nevertheless the proposed 2p polarization is clearly adequate to explain the observed inversion. for the sign of this second-order effect are The clear.reasons The -*
spin-orbit perturbation acts on 2P3/2 as a repulsive potential, pushing out the 2p wave function. This increases the (intrinsically negative) exchange interaction with the external 3d electron. This effect should be general in all alkali-metal atom excited d states, except for Li which contains no p electrons in the core. In fact for Li 3d, a calculation gave 277
Volume 55A, number
S
PHYSICS LETTERS
1, which is in good agreement with = +0.037 cm the observed splitting [1], namely +0.04 cm~. Similar calculations for the Rb 4d state [3] have also been carried out [8].
One of us (R.M.S.) wishes to thank Dr. M. Blume for helpful discussions. We are also indebted to Dr. R.F. Peierls for the computer programs used in ref. 6 and the present work.
References [1] C.E. Moore, Atomic Energy levels, Vols I—Ill, National Bureau of Standards Circular No. 467 (1949).
278
29 December 1975
[2] M. Phillips, Phys. Rev. 44 (1933), 644; 45 (1934) 428. 13] K.H. Liao, L.K. Lam, R. Gupa and W. Happer, Phys. Rev. Lett. 32 (1974) 1340. [41 Clementi, IBM Journal of Research 9 (1965) [51 E. R.M. Sternheimer, Phys. Rev. 84 (1951) 244; 862.(1953) 316. [6] R.M. Sternheimer, Phys. Rev. A9 (1974) 1783. [71 B.R. Judd, P2OC. Phys. Soc. (London) 82 (1963) 874. [81 T. Lee, J.E. Rodgers, T.P. Das and R.M. Sternheimer, to be published.
PHYSICS LETTERS
29 December 1975
FINE STRUCTURE INVERSION OF THE 3d EXCITED STATE OF SODIUM* H.M. FOLEY Department of Physics, Columbia University, New York, New York 10027, USA
and
R.M. STERNFIEIMER Brookhaven National Laboratory, Upton, New York 11973, USA Received 21 October 1975 The observed inversion of the fine structure of the excited 3d state of the sodium atom explained as due to a second-order effect between the electrostatic dipole exchange interaction and the magnetic spin-orbit interaction acting on the core 2p electrons.
It has been known for many years that the finestructure interval z~svof alkali-metal atoms in excited states with 1> 1 (except for Li) is inverted or is very much smaller than that predicted by any reasonable single-electron model Ll]. The first attempt to explain this property was made by Phillips [2] in 1933 for the case of the alkali-like spectrum of Mg II. However, the plausible estimate of ii’, which was obtained for Mg II 3d, is a thirdorder effect, which involves the wave function squared of a 2p hole in the closed 2p shell. This is presumably smaller than the second-order effect to be discussed here. Recent hyperfine structure (hfs) measurements of the Rb excited 4d state by Happer and coworkers [3] show similar inversions in sign for both the fine structure and the spin-dipolar magnetic hfs. proposedue that effects, leastorbital ford polaristates, areWe primarily to these exchange spinatand zation of inner p shells by the external d electron, and are properly calculated as a second-order cross term between the electrostatic dipole exchange interaction and the magnetic spin-orbit or hfs interaction. (A clue here is the absence of the effect in Li, which has no internal p electrons). We present results for the fine structure of Na 3d as a first example of this effect. The effect considered here is the second-order “cross interaction” between the spin-orbit coupling perturbation Hf = 2(,.i~/e2)(1 /r) (d V/dr) / s and ~ This work was performed under the auspices of the Energy Research and Development Administration. 276
the exchange part of the Coulomb interaction HQ = 2/r12(Rydberg units). Thus for Na 3d, the change of energy is:
(3d,2plHQ~np,3d)(np,3dIHf~2p,3d) ~-~2
=
P
np
The sum over n extends over all excited and continuum states. The calculation is much 2D simplified by considering the interaction in the state 3 512, M~= 5/2. The dipole part of HQ which is given by: 8 r< (2/r12)d~o1e= r> (2) X [Y?(l)Y?(2) 1- Y~(l)Yj’(2) + Yi~’(l)YI(2)]. In the matrix element of eq. (2), since 3d, m = 5/2 has 2P3/2’ m1 with mj == +2, 3/2,m5 via= Yj’. this Thusonly the connects matrix element
+4,
(2p
1(2)Inp 312312IY~(l)I3d512512X3d512512IYj 312312) and its inverse are the only non-vanishing ones, when one treats Hf as a single-electron operator. In eq. (1), the function: (/1p,3dIHfl2p,3d) W i~i(np,3d) (3) ~ is, apart from an angular factor, l/r times w1(2p-+p), which2 is the 2 solution of the equation: / d w 1(2p-+p) (4) =
[(J(r))2~—f(r)] u0(2p),
Volume 55A, number 5
PHYSICS LETTERS
where u0(2p) is the radial 2p function (times r) [4] for Na,f(r) (l/r) (dV/dr), and w1(2p -+p) includes all of the excitations of 2p produced by the interaction H~,i.e., both the discrete and continuum p states. We note that eq. (4) is analogous to the equation which determines the shielding and antishielding effects for the quadrupole hyperfine structure [5]. The valence wave function v0 (3d) for Na was obtained previously. 2r’ p), u(see fig. I of ref. [61).The wave functions w1 ( 0(2p), and v0(3d) for Na are shown in fig. 1. Using the wave function w1(2p —~p),eq. (1) becomes 2d~I2 ~K[2~X 1] (5) 3 L e2 = —21 l~1 l}~
f~1
f ~
L
0(2p)u0(3d)
r fr’vo(3d)wi(2p —~p)dr’ r~~
(6)
We find K = 002282. The angular factor which precedes K in eq. (5) is —8/5. We thus obtain: 2u2 =~ X 0.01826 Ryd. (7) e2a~ This must be compared with the first-order effect of Hf on the 3d 512,512 state, which is given by: +(2~/e~) 1<1 s> 3d5~25~2, I is the integral: -
2dr = 0.007 16 Ryd/a~. I
=
I
I
I
I
I
I
I
0.24 I
Na
0.20 0.16
0•6
~
0.4 0.04
~ o.~ -a ~ o.c
D.0 0.04
-
2w 10
-0.08 1(2p—p)
-0.5 0.0 0.4 0.8
1.2
1.6
I
I
I
I
2.0 2.4 2.8
I
I
I
I
0.12
I
3.2 3.6 4.0 4.4 4.8
RADIUS r(oH)
Fig. 1. The perturbed wave function
w
5 (2p p)(times 1O_2, and the unperturbed 2p and 3d functions tion of the distance r from the nucleus. for Na, as a func—
/~.E2= (2i~/e2a~) 0.00716(1—2.550) Ryd. (10) According to a theorem due to Judd [7], the same
f r’2v~(3d)X w1(2p~p)dr’]
=
I
+
+ r
where
-a
I
.0 0.8
dru
(1
x
.2
—~
4
where the and last Kfactor is the value of l~s~ in the state 2P3/23/2 is the following integral: K=
29 December 1975
0J’f(r)
(8) (9)
[v0(3d)]
The expectation value (1 s> in eq. (8) is:(1~s~) = 2 X 4 = +1. By comparing eq. (7) with eqs. (8) and (9), we thus obtain a correction factor (overshielding) of: —0.01826/0.00716 = —2.550, i.e., the total spin-orbit energy for the 3d 5~2,5~2 state is: -
3D to the entire 2D spincorrection factor can be applied orbit splitting between the 3 512 and 3 312 states. The first-order splitting is given by: 2RydX 0.00716 = 0.051 cm~. (11) ~ =~a Hence we obtain for the second-order fine structure splitting: ~2 = —2.550 X 0.052 = —0.133 cm~,and for the total splitting: (12) ~‘ = ~‘l + = 0.052 0.133 = —0.081 cm* which has the same sign but somewhat larger magnitude than the experimental value [1], namely —
—0.049 cm~. The calculation of both the first-order and the second-order effects is delicate, especially the latter, which depends on the overlap of the perturbed p function with the 3d wave function (see fig. 1). Nevertheless the proposed 2p polarization is clearly adequate to explain the observed inversion. for the sign of this second-order effect are The clear.reasons The -*
spin-orbit perturbation acts on 2P3/2 as a repulsive potential, pushing out the 2p wave function. This increases the (intrinsically negative) exchange interaction with the external 3d electron. This effect should be general in all alkali-metal atom excited d states, except for Li which contains no p electrons in the core. In fact for Li 3d, a calculation gave 277
Volume 55A, number
S
PHYSICS LETTERS
1, which is in good agreement with = +0.037 cm the observed splitting [1], namely +0.04 cm~. Similar calculations for the Rb 4d state [3] have also been carried out [8].
One of us (R.M.S.) wishes to thank Dr. M. Blume for helpful discussions. We are also indebted to Dr. R.F. Peierls for the computer programs used in ref. 6 and the present work.
References [1] C.E. Moore, Atomic Energy levels, Vols I—Ill, National Bureau of Standards Circular No. 467 (1949).
278
29 December 1975
[2] M. Phillips, Phys. Rev. 44 (1933), 644; 45 (1934) 428. 13] K.H. Liao, L.K. Lam, R. Gupa and W. Happer, Phys. Rev. Lett. 32 (1974) 1340. [41 Clementi, IBM Journal of Research 9 (1965) [51 E. R.M. Sternheimer, Phys. Rev. 84 (1951) 244; 862.(1953) 316. [6] R.M. Sternheimer, Phys. Rev. A9 (1974) 1783. [71 B.R. Judd, P2OC. Phys. Soc. (London) 82 (1963) 874. [81 T. Lee, J.E. Rodgers, T.P. Das and R.M. Sternheimer, to be published.