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Analysis of rotational and hyperfine structure in holmium monoxide
JOURNAL
OF MOLECULAR
SPECTROSCOPY 131,367-38 I (1988)
Analysis of Rotational and Hyperfine Structure in Holmium Monoxide C. LINTON AND Y.
C. LIU
Physics Department, University of New Brunswick, Fredericton, New Brunswick, Canada E3B 5A3
Doppler-limited laser excitation spectra have been recorded for six bands of HoO. For the O0, O-l, and l-l bands ofthe [17.617.5-X, 8.5 transition and the O-O band of [18.4]6.5-IV, 7.5 (R branch only), the hyperline structure is completely resolved. For the O-Obands of the [ 17.6]7.5X, 7.5 and [ 18.4]6.5-IV2 6.5 transitions, hyperfme structure is partially resolved and only at low J. Rotational and hypertine analyses have been performed for all the transitions. The hyperline structure is examined in detail and has provided information about the nature of the coupling mechanisms in the various electronic states. The hyperfme constants are used in conjunction with results of ligand field theoretical calculations to determine the individual hyperhne parameters for the 4fand 6s electrons. These parameters are compared with those of the free holmium ion in order to provide insight into the nature of the electron orbitals in the molecule. o 1988Academic Press. Inc.
INTRODUCTION
As part of our ongoing investigation into the spectroscopy and electronic structure of the rare earth monoxides, we recently presented our preliminary results on holmium monoxide, Ho0 ( I ) . Using resolved fluorescence and wavelength-selected excitation spectroscopy, we were able to link and assign four low-lying electronic states. Rotational constants were obtained using combination differences and hyperfine constants were computed using combination differences when the structure was resolved and hypermultiplet widths when the structure was not resolved. The results, even though lacking in precision, clearly showed that the low-lying states originate from the ‘Is supermultiplet which is the lowest state of the 4f*‘6s superconfiguration of the Ho’+ atomic ion. The pattern of the observed low-lying states was shown to be consistent with the predictions of ligand field theory (LFT), and the hyperhne structure was shown to be an important diagnostic tool in assigning the free ion quantum numbers (J,, J,J to the molecular electronic states. Since this work was published, LFI calculations have become more refined and detailed predictions on all the low-lying states of all the rare-earth oxides are now available (2-4). In order to examine the holmium oxide data more closely in relation to the LFT calculations, we have examined the high-resolution data in more detail placing particular emphasis on the analysis of the hyperhne structure (hfs). For the [17.6 ]7.5-Xi 8.5 transition in which the hfs of the O-O and O-1 bands wascompletely resolved (Ref. (I ), Fig. 4A), a least-squares fit to all the lines has been performed including contributions from off-diagonal terms in the hyperlme Hamiltonian. For the [17.617.5-X2 7.5 and [18.4]6.5-IV2 6.5 transitions, the hfs was not completely 367
0022-2852188 $3.00 Copyright 0 1988 by Academic Press, Inc. Au right.3of reproduction in any form rserved.
LINTON
368
AND
LIU
resolved. However, using partially resolved low-J lines and weak 0 # AJ “satellite” lines, it was possible to separate the upper and lower state hfs and determine more reliable hyperfme constants. The rotational constants were obtained from a leastsquares fit to the line centers. Only the R branch was observed for the [ 18.416.SIV, 7.5 transition. The hfs was completely resolved and was used together with the upper state information obtained from [ 18.416.SIV2 6.5 in order to obtain the lower state hyperfine constants. Rotational constants were obtained using known upper state data and hypermultiplet centers, All the hyperflne constants are examined in terms of LFT and are used to determine the contributions of the 4fand 6s electrons to the hfs. No new experiments have been performed and the experimental details are as described in a previous work ( 1) . RESULTS
AND ANALYSIS
I. The [I 7.61 7.5-X1 8.5 Transition The O-O and O-l bands of this transition were obtained at high resolution to J = 24.5. The l-l band was observed only up to R(10.5) and P(12.5). The eight M = AJ hyperfine components were clearly resolved for each rotational transition and, at the lowest values of J, several AF = 0 “satellite” lines were observed. The preliminary analysis ( 1) used only the first and last lines of each AF = AZ hypermultiplet to generate approximate constants for each band. In the present analysis, we have simultaneously fitted all the lines of the main and satellite hypermultiplets of the O-O and O-l bands including in the Hamiltonian both centrifugal distortion and offdiagonal terms. In a pure case cg basis, the diagonal matrix elements of the hypertine Hamiltonian are given by Ediag(F,
J) = (1. J)dWtJ(J
+ 111,
(1)
where (I. J) = [F(F + 1) - J(J + 1) - Z(Z + 1)]/2
(2)
and d is the magnetic hyperfine structure constant. In order to better fit the data at higher J, a phenomenological centrifugal distortion term, dJ, is often included such that the diagonal matrix element becomes E(F
9
J)=
[d-dJJ(J+
l)]Q[F(F+ l)-J(J+ 2J(J+ 1)
l)-Z(Z+
l)]
(3)
In different coupling cases, the same expression can be used for the diagonal terms. However, d is no longer a constant, but an expression which is often a complicated function of J and which is determined by the nature of the coupling and the details of the hyperfine Hamiltonian. In this transition, the hfs follows the expected case cg pattern and can be adequately described by Eq. ( 3 ) . However, the presence of satellite lines indicates that the coupling is not pure case cs and that the following off-diagonal AJ = + 1, AQ = 0 term should be added to the energy,
HYPERFINE
EodF, J) = C
369
ANALYSIS OF Ho0
1(F, Jl%fsl F, J’) I2
J'=JkI
E’(J)
- E’(J’)
(4)
’
where E’(J) and E”( J’) are the rotational energies of the successive levels J and J’. All the matrix elements necessary to calculate E,r( F, J) have been derived by Atkins ( 5) and the detailed expression has been listed by Liu (6). The final constants were calculated using an iterative least-squares procedure and it was found that inclusion of the off-diagonal terms considerably improved the quality of the fit. These terms were found to contribute up to 0.0 12 cm-’ to the line frequencies. As only three rotational hypermultiplets were observed in the R branch and five in the P branch of the l-l band, a detailed fit was not possible. The hyperfme constants and lower state rotational constants were all fixed to the values obtained in the O-O and O-l band fit, leaving only the B value and term energy of the upper, 2) = 1 state to be determined. All the constants are listed in Table I. TABLE I Constants (cm-‘) Derived from High-Resolution Excitation Spectra of Ho0
state
T0
B
10%
d
L18.416.5
18353(4)
0.3485(l)
5.3(19)
0.164(2)
[17.617.5(~=1) [17.617.5(v=O)
18334.72(2) 17546.874(l)
0.3534(l) 0.355851(l)
0.93(l)
O.l8792(8)b 0.18792(8)
1853(4)
0.3503(l)
8.8(21)
0.129(3)
w, 7.5
1130(3P
0.3357(a)
-166(22)
X, 7.5
602.935(6)
0.35601(2)
4.7(S)
O.l95(l)d
X, 8.5(v=l) X, 8.5(v=O)
841.252(l) 0
0.356186(l) 0.357713(l)
2.75(2) 2.51(2)
0.35822(a) 0.35822(8)'
W, 6.5
Band origins 117.617.5 - X,8.5 (0,O): 17546.874(l) (0,l): 16705.622(l) (1,l): 17493.47(2) L17.617.5 - X,7.5 (0,O): 16943.939(2) [18.4]6.5 - W,6.5 (0.0): 16500.439(4) [18.416.5 - W,7.5 (0,O): 17223.27(06)
aThis was determined from low resolution resolved fluorescence measurements and is accurate only to k 3 cm-'. This determines the absolute uncertainty in T, for the W,6.5 and [18.416.5 states even though their positions relative to W,7.5 are much more precisely determined. bdJ = (6.9 + 0.6) x 10-6 cm-'. =dj = - (3.7 + 0.1) x 1O-5 cm-? d
e¶Q
= 0.092 t 0.016 cm-'.
370
LINTON AND LIU
ZZ. The [I 7.61 7.5-X2 7.5 Transition
In this transition the hfs was generally not resolved (Ref. ( I ), Fig. 4B). In the earlier work (I), the approximate lower state rotational constant, B”, was obtained from combination differences using the line centers to represent the rotational frequencies. In the present analysis, the upper state constants were fixed to those obtained from the [17.617.5-X, 8.5 analysis and the lower state constants (B, D, and TO) were computed from a least-squares fit of the line centers to the rotational Hamiltonian. The results are in Table I. Because the hfs was not resolved, it was not possible to do a detailed hyperfme analysis. However, in each ofthe lowest Jlines, R(7.5), R(8.5), P( 8.5), P(9.5), there was partially resolved hfs and several weak, but resolved, satellite lines. The P-branch lines were clearly degraded toward higher frequencies (i.e., the high-F components were at higher frequency). Only the three highest F main lines were clearly resolved. There were four or five clear satellite lines on the high-frequency side of the main hypermultiplet (the other satellite lines overlapped the main hypermultiplet ). The Rbranch structure was more difficult to assign as there were six or seven resolved peaks in the main hypermultiplets but, as they were almost equally spaced, it was difficult to determine the direction of the degradation. There were five satellite lines on the low-frequency side of the main hypermultiplet. The analysis showed that a self-consistent set of assignments could be made only if the R lines were degraded toward lower frequencies. The intensities which should increase with increasing F confirm the assignments. The first R-branch hypermultiplet, R ( 7.5 ), shown in Fig. 1, clearly illustrates the intensity distribution and satellite lines.
rF
Cl763 75-X,7.5 AF=+l
RU5)
‘-5 l--l+dd
I I\
i
V
0.2 0.4
0.3 (cm-l)
AF=O k-G-l4
’
FIG. 1. First line, R(7.5), in the R branch of the [17.617.5-X2 7.5 transition of HoO. Seven of the eight hyperfme components are observed in the main ( AF = AJ) hypermultiplet together with five weaker satellite ( AF = 0) lines.
HYPERFINE
ANALYSIS
371
OF Ho0
The opposite degradation in the P and R branches shows that the hyperfme splitting of level J in the lower state is greater than that of J + 1 in the upper state, but less than that of J - 1. Thus, it would appear that, at the same J, the splittings in both states are almost equal. Observation of resolved main and satellite hyperfine lines at low J enable us to separate the hyperfme structure of the two states and determine some hyperfme splittings in each state from the relations Upper state: (R(J,
F) - RQ(J, F) = PQ(J, F + 1) - P(J, F + 1) = E’(J, F + 1) - E’(J, F))
(54
Lower state: (R(J,
F) - RQ(~, F + 1) = P~(J, F) - P(J, F + 1) = E”(J, F + 1) - E”(J, F)),
(5b)
where R( J, F), P( J, F), etc., represent the main lines ( AF = AJ) and RQ( J, F), PQ( J, F), etc., represent the satellites ( AJ = f 1, AF = 0). The resolved lines and calculated hyperfine splittings are all listed in Table II. The correctness of the assignments was confirmed as there was found to be excellent agreement between (i) the splittings obtained from the P and R lines and (ii) the splittings in the upper state and those in the O-O and O-l bands of the [ 17.617.5-X, 8.5 transition. The lower state splittings were used to calculate the hype&e constant, d, for the lower state. The upper state constants have aheady been well determined and reproduce all the measured upper state separations to within 0.003 cm -’ . The separation between successive levels in a hyperline multiplet is given, to first order, by E( J, F + 1) - E( J, F) =
dQ(F+ J(J+
1) 1) ’
(6)
As we only have data for the two lowest rotational hypermultiplets, it is not possible to determine any J-dependent corrections to d. The values of d calculated for each J, F combination in the lower state are listed in Table II and the average value is d = 0.19 1 + 0.008 cm-’ .
Close examination of the data in Table II casts some doubt on the validity of averaging the values of d. It can be seen that the values of d in each hypermultiplet appear to increase systematically with increasing F and the range of values (0.02 cm-‘) is much larger than can be accounted for on the basis of experimental uncertainty. The offdiagonal AJ = +-1 contributions were computed but did not change the calculated separations by more than 0.00 1 cm -’ and are thus insignificant. Dulick and Field ( 7) have shown that the expression for d derived from the complete Hamiltonian representing intermediate coupling and including o&liagonal AQ = _+l terms contains several J-dependent terms, but there should still be no dependence on F. Thus it
372
LINTON AND LIU TABLE II Measurements (cm-‘) of Main and Satellite Lines in Resolved Hypermultiplets in the [17.617.5-X2 7.5 Transition J
F
R(J,F)
7.5
11 10 9 8 7 6 5
16949.924 49.944 49.970 49.997 50.023 50.045 50.071
16949.709 49.749 49.790 49.835 49.879
0.215 0.195 0.180 0.162 0.144
12 11 10 9 8
50.453 50.487 50.522 50.556 50.589
50.644 50.656 50.680 50.699 50.721
0.191 0.169 0.158 0.143 0.132
8.5
J
F
8.5
12 11 10 9 8
37.955 37.903 37.869
13 12 11 10 9 8
37.216 37.185 37.161
9.5
P(J,F)
R$J,F)
PQ(J,F)
R(J,FbRQ(J,P)
P,$J,F)-P(J.F)
R(J,F)-RP(J,F+l)
d”
0.235 0.221 0.207 0.188 0.166
0.182 0.188 0.198 0.200 0.202
0.203 0.193 0.177 0.165
0.182 0.189 0.191 0.197
PP(J,F)-P(J,F+l)
d”
38.157 38.097 38.050 38.008
0.254 0.228
0.202 0.194
0.181 0.189
37.401 37.359 37.322 37.291 37.262
0.216 0.198
0.185 0.174
0.189 0.193
seems as if the hyperhne structure in the Xz 7.5 state can not be completely described in terms of a magnetic hyperfme interaction. In order to try and account for this apparent variation in d, we have examined the possibility that the electric quadrupole interaction has a significant effect on the hfs in the X2 7.5 state. The quadrupole energy is given by (8)
where f(l
J F) = [0.75c(c+ 1) - z(z+ l)J(J+ 2Z(Z - 1)(2J - 1)(2J + 3)
9 9
111
and c = F(F + 1) - J(J + 1) - Z(Z + 1). The quadrupole contribution to the energy separation in Eq. (5) is thus given by
-eqQLf(Z, J, F+ 1) -_f(Z, J, J’)l
HYPERRNE
373
ANALYSIS OF Ho0 TABLE III
Observed and Calculated’ Hypefine Separations (cm ‘) in the X2 7.5 Stateb J
F
7.5
10
E(J,Ftl)-E(J,F) Calculated Observed
Difference'
9 8 7 6
0.235 0.221 0.207 0.188 0.166
0.240 0.223 0.205 0.186 0.166
-0.005 -0.002 0.002 0.002 0.000
8.5
11 10 9 8
0.203 0.193 0.177 0.165
0.205 0.193 0.180 0.165
-0.002 0.000 -0.003 0.000
9.5
12 11
0.185 0.174
0.178 0.170
0.007 0.004
%cludes
the quadrupole term, -eqQ[f(I,J,F+l)-f(I,J,F)l.
bFrom least squares fit, d = 0.195 f 0.001 cK1, eqQ = 0.092 + 0.016 cm-: =Standard deviation, o = 0.003 cm~'.
and this must be added to the magnetic term in Eq. (6). A least-squares fit to the separations gives the following values: d = 0.195 + 0.001 cm-‘;
eqQ = 0.092 f 0.016 cm-‘.
The observed and calculated separations are compared in Table III and the standard deviation in the fit is 0.003 cm-i. Thus the quadrupole term appears to be significant and fits the data within the experimental uncertainty. III. The [l&4 ]6.5- W, 6.5 Transition As mentioned in a previous paper ( 1), this transition is weak and the hyperhne structure is not resolved in the R branch (Ref. (I ), Fig. 4D). However, the P-branch lines are much broader and the hyperhne structure is completely or partially resolved up to J - 13.5. In addition, two satellite lines on the high-frequency side of the hypermultiplet are clearly distinguishable up to J = 9.5. The R-branch satellites, although not resolved, are also observed, but on the low-frequency side of the hypermultiplet. The first lines in each branch are shown in Fig. 2. The positions of the satellites and the degradation of the hypermultiplets (to lower frequencies) clearly show that the hyperfine splitting is larger in the upper than in the lower state. The two resolved satellite lines represent AR = 0 transitions between the high-Fcomponents. The remaining AF = 0 transitions are not observed because they overlap the main hypermultiplet. By using the separation between main and satellite lines, values of d were calculated for each state. The frequencies and separations of the main and satellite lines in the resolved P-branch hypermultiplets are listed in Table IV along with the d values which were calculated using Eqs. ( 5 ) and ( 6). There was no obvious systematic variation of d with F or J and the average values are
d’ = 0.164 +- 0.002 cm-‘,
d” = 0.129 & 0.003 cm-‘.
374
LINTON AND LIU Cl 8.416.5
1t6.5)
-
W-6.5
Af=-+I L
, 3ir AF=O
‘(15)
I 0.5
AF=-1
1
I
I
I
0.4
0.3 (cm-0
0.2
0.1
FIG. 2. First lines, R(6.5) andP(7.5), in the [18.416.5-W, 6.5 transition ofHo0. The hyperline structure, including two weak satellite ( A F = 0)lines, is clearly resolved in P( 7.5) but not in R( 6.5 ) .
These hyperfme constants were then tested by comparing the measured hyperline separations, up to J = 13.5, with those calculated using the expression P(J, F) - P(J, F-
1) =
dV(F
J( J_
- 1)
1)
-
d”W’F . 1)
J(J+
The results, shown in Table V, show that the observations are all reproduced to within experimental accuracy. Thus, the first-order model adequately describes the hyperfme structure for all the hypermultiplets in which the structure is resolved. At higher J, the structure in the P lines is no longer resolved but the width of the lines continues to decrease. However, the width of the unresolved R-branch transitions appears to increase slightly with J at higher Ss. This could be an indication that the simple case cg coupling model is beginning to break down in either one or both states and that the spin, S, of the s electron is starting to uncouple from the internuclear axis as described by Dulick and Field ( 7) for PrO. In order to be more specific about the
HYPERFINE
375
ANALYSIS OF Ho0 TABLE IV
Measurements (cm -’ ) of Resolved Hypermultiplets in the [ l&416.5-W2 6.5 Transition J
F
P(J,F)
7.5
11 10 9 8 7
16495.315 95.236 95.173 95.118 95.075
12 11 10 9 8
94.571 94.513 94.461 94.416 94.378
13 12 11 10 9
93.829 93.781 93.738 93.700 93.666
10.5
14 13 12 11 10
93.046 93.010 92.976 92.945 92.917
11.5
15 14 13 12 11
92.343 92.310 92.279 92.250 92.222
12.5
16 15 14 13 12
91.597 91.568 91.540 91.513 91.487
13.5
17 16 15 14 13
90.856 90.830 90.805 90.781 90.757
8.5
9.5
%tellite
lines
PQ(J,F)a
were
d’
P@,F)-P(J,F)
P,+J,F)-P(J,Ftl)
d”
95.454 95.369
0.218 0.196
0.164 0.163
0.139 0.133
0.124 0.130
94.697 94.628
0.184 0.167
0.164 0.164
0.126 0.115
0.130 0.130
93.941 93.883
0.160 0.145
0.166 0.164
0.112 0.102
0.132 0.130
only
measured
up
to
J = 9.5.
nature of the coupling, it would be necessary to resolve all the hfs using sub-Doppler techniques and to extend observations to much higher J. To determine which state is responsible for the anomalous hfs, we can examine the widths of the unresolved hypermultiplets. Widths of individual hypermultiplets are difficult to determine without knowing the individual linewidths and lineshapes. However, differences between widths may be determined more reliably and can be used to separate the widths in the upper and lower states. If the widths of the R and P transitions are denoted by IV-&J) and WP(J) and the widths of the hypermultiplets in each state by W’(J) and IV”(J), then we find that W,(J)
-
r&(J)
= W’(J - 1) - W’(J + 1)
(9a)
376
LINTON AND LIU TABLE V Observed and Calculated Hyperfine Splittings (cm-‘) in the [18.4]6.5-W2 6.5 Transition P(J,F)-P(J.F-1) Obs. Calc.
J
P
7.5
11 10 9 8 7
0.079 0.063 0.055 0.043 0.038
8.5
12 11 10 9 8
9.5
10.5
P(J.F)-P(J,P-1) Obs. Calc.
J
P
0.073 0.064 0.056 0.047 0.038
11.5
15 14 13 12 11
0.033 0.031 0.029 0.028 0.028
0.035 0.032 0.030 0.027 0.024
0.058 0.052 0.045 0.038 0.032
0.058 0.052 0.046 0.040 0.033
12.5
16 15 14 13 12
0.029 0.028 0.027 0.026 0.021
0.031
13 12 11 10 9
0.048 0.043 0.038 0.034 0.028
0.048 0.044 0.039 0.034 0.029
13.5
17 16 15 14 13
0.026 0.025 0.024 0.024 0.023
0.028 0.026 0.024 0.022 0.020
14 13 12 11
0.036 0.034 0.031 0.028
0.041 0.037 0.034 0.030
0.029
0.026 0.024 0.021
and wp(J+
1) - W&J-
1) = W’(J-
1) - wn(J+
l),
(ob)
where W(J-
l)-
W(J+
l)=d
&+-&-&-L
J+2
1*
Thus we can separate the two states and use the calculated d values to compare the observed and the calculated difference in widths. It was found that the agreement with observation is good at all J for the upper state but not for the lower state. Because of the difficulty in measuring widths, this may not be a very reliable criterion, but it appears as if the cg coupling is breaking down in the lower state. Line centers were used for the rotational analysis and the rotational constants were computed using a least-squares technique. The constants are listed in Table I and they reproduce the observed lines to within a standard deviation of 0.008 cm-‘. The B value of the upper state is in excellent agreement with that obtained by Kaledin and Shenyavskaya (9) from absorption spectra. The centrifugal distortion constant, D, is not as well determined in this work as the data did not extend to high enough J. However, the value (with error) just falls within the error limits of Kaledin and Shenyavskaya’s value. IV. The [18.4]6.5-WI
7.5 Transition
This transition was more difficult to analyze than the others. It has already been mentioned (I ) that, because there seem to be several overlapping transitions giving
HYPERFINE
ANALYSIS
377
OF Ho0
TABLE VI Measurements
(cm-‘)
of Hypermultiplet ofthe
[l&416.5-W,
Center6 and Widths of the R-Branch
J=
R(J)
Widthb
a.5 9.5 10.5 11.5 12.5 13.5 14.5 15.5 16.5 17.5 18.5 19.5 20.5
17233.797 34.951 36.000 36.979 37.922 38.838 39.732 40.606 41.477 42.333 43.183 44.029 44.870
0.746 0.767 0.772 0.778 0.776 0.775 0.775 0.765 0.774 0.770 0.776 0.773 0.773
%alculated to r2.
from dispersed
Lines
7.5 Transition
fluorescence
0.256 0.275 0.292 0.308 0.323 0.338 0.354 0.371 0.384 0.397 0.415 0.429 0.443
measurements and uncertain
bwidth = R(J,P=J-I)-R(J,F=J+I). ‘These values incorrect.
are dependent on J and will
thus change if
J is
to fluorescence in the same region, it was only possible to obtain an excitation spectrum of the R branch. The hyperhne structure was well resolved (Ref ( 1 ), Fig. 4C) and the centers of the hypermultiplets are listed in Table VI. The quoted J values are probably uncertain to +2 as they were obtained from low-resolution resolved fluorescence measurements. A complete rotational analysis was not possible without the P branch but it was possible to calculate lower state constants using the R-branch measurements and the known upper state constants which had already been determined. The standard deviation of the fit to the data was -0.06 cm-’ which is an order of magnitude worse than for the other analyzed transitions. However, it is much less than the width of the hypermultiplets (-0.7 cm-‘) and could partly reflect the fact that the center is not the true position of the pure rotational transition. This would not, however, account for the high negative D value and it seems as if there may be a weak perturbation affecting this state. The constants which are listed in Table I are dependent, especially in the case of TO, on the correctness of the J assignments. The multiplet structure shows that the hfs is larger in the lower than in the upper state. The structure is unusual in that the multiplet splittings increase slightly with increasing J at the lowest J’s and then remain nearly constant. This is shown in Table VI where the hypermultiplet widths (i.e., the separation of the F = J + Z and F = J - Z components) are listed along with the line center frequencies. The constancy of the splittings with J indicates that the lower state splittings are changing with J by exactly the same amount as those of the upper state. The upper state splittings can be calculated using the value of d’ = 0.164 cm-’ obtained earlier and combined with the line separations R( J, F - 1) - R( J, F) to calculate a value of d” for each pair of adjacent lines. The calculations showed that, especially at lower J, the value d” was rise
378
LINTON AND LIU
constant within a hypermultiplet (i.e., there was no F dependence) but increased rapidly with J as shown in Table VI. This appears to be a clear example of the Suncoupling phenomenon described by Dulick and Field ( 7) in which the hyperfine splittings tend toward a constant value at high J. The uncertainty about the precise value of Jleads to an uncertainty in d (a change of 1 in Jleads to a change of -0.0 150.020 in d) but the increase with J is unaffected. DISCUSSION
The hyperhne structure has previously only been used (I) to assign the atomic ion quantum numbers, J, and J, , to the low-lying states on the basis that large hfs indicates J, = J, + fi and small hfs corresponds to J, = J, - f . The more detailed examination of the hfs discussed in previous sections has shown that the situation is more complex. The ground, Xi 8.5 state appears to be the only low-lying state which is well described in terms of the magnetic hyperfine interaction obeying a case cg coupling model. The coupling in X2 7.5 is more difficult to determine as the observed transition involves two states with similar hfs so that the structure is unresolved except for portions of a few low-J lines. However, even for these low-J lines there is evidence that the lower state hfs does not follow the expected Land6 pattern. The W, 7.5 hfs is clearly anomalous, even at low J, and the observation of only a single branch made analysis dithcult. The WZ6.5 state hfs was incompletely resolved and only in the P branch at lower J. The hfs appeared normal at the lower J’s but increasing hypermultiplet widths at higher J’s in the R branch indicate that the lower state hfs is beginning to depart from pure case cP The hfs in the molecular states may be used to estimate the atomic hyperfme parameters for the 4fand 6s electrons. The hyperfine Hamiltonian can be written as Hhfs = aI* J, + bras, where a is the hyperfine parameter for the 4fcore electrons and b is the Fermi contact term for the 6s electron. For the states in which the hyperfine constant, d, is reasonably well defined, it can be related to the atomic hyperfine parameters by d = a4fa + cthsb. Expressions for cy4f and ff6s have been derived and evaluated by Dulick (10) for Pro and by Hocquet (4) for HoO. For the X1, X2, and W, states, the hyperfme parameter, d, appears to be independent of J at low J and these states can therefore be used to estimate a and b. Hocquet (4) computed the eigenfunctions of all the low-lying states and used them to calculate values of (Y4fand (Ygthat are listed in Table VII. These were then combined with the experimentally determined values of d (also listed in Table VII) to solve for a and b. The least-squares solution yielded a = 0.041 + 0.003 cm-’ and b = 0.20 & 0.05 cm-‘. The values of d calculated from these constants are compared with the observed values in Table VII. The standard deviation of the fit to d is -0.025 cm-’ and the fitted values differ from the observations by 20-25% for the X2 and II’, states. This is not really surprising as both states showed anomalous hfs and d is not truly a constant and well-defined parameter for these states. The values of (Ydepend on the eigen-
HYPERFINE
379
ANALYSIS OF Ho0
TABLE VII Comparison of Observed and Calculated Hypetine Constants state
G
a:,
dobs(cm-')
d
CdC
(cm-')
X,8.5
6.13
0.50
0.3582
0.3541
x,7.5
6.16
-0.42
0.195
0.169
W,6.5
5.22
-0.25
0.129
0.164
Least squares fit of d = a4fa + a6sb gives a = 0.041 * 0.003 cm-l and b = 0.20 f 0.05 cm-' Standard deviation of fit, o = 0.025 cm-'
%alculated by Hocquet (4).
functions so that any inaccuracy in the eigenfunctions would affect the calculations. In addition, the experimental values of d were calculated from a very limited data set. As already mentioned, the hfs in the Wi 7.5 state is anomalous and the parameter, d, is a rapidly increasing function of J that reflects the uncoupling of the s electron angular momentum from the internuclear axis. From the relationship derived by Dulick and Field (7), it can be shown that at high J, d should increase by b/2Q between successive values of J. From Table VI, it can be seen that d increases by 0.014 cm-’ at the highest J’s and, as Q = 7.5, this leads to a value of b = 0.21 cm-’ in excellent agreement with the value obtained from the X1, X2, and WZstates. The above data can be used to determine the relative contributions of the 4fand 6s electrons to the hyperfme structure. For the X1 state, the f contribution is -2.5 times greater than the s contribution, for X2 it is -3 times greater, and for W,, particularly at low J, it is -4 times greater. Thus, in all these states, thef electrons make the major contribution to the hfs. In the Wi state, as a result of the s uncoupling, the Fermi contact term is dominant. It is informative to compare the a and b parameters with those of the atom. There does not appear to be published data for the Ho2+ f”s configuration. However, for thef”s configuration of Ho+, Sugar (II) determined that a = 0.03491 cm-’ and b = 0.4437 cm-‘. These may change slightly but are not expected to alter by a large amount in going to thef”s configuration of Ho2+. The Q values for the molecule and the free ion are very similar confirming that the atomic f electron core retains its properties on formation of the molecule. The b value, however, is only -47% of its free-ion value. The Fermi contact parameter, b, is proportional to the 6s charge density at the nucleus which has thus been reduced by -53% by molecule formation. This is a clear indication that the true configuration is 4f”a, where the Q orbital is an approximately equal mixture of holmium 6s and oxygen 2p orbit&. This is very similar to the result obtained by Dulick (10) for PrO. While the above analysis has indicated the nature of the orbital structure in the molecule, it must be emphasized that it is far from complete and that the hypefine
380
LINTON AND LIU
constants are not well determined. In order to obtain a clearer understanding of the precise nature of the coupling it is essential to follow resolved hfs to high Jin all the transitions. This will require Doppler-free techniques to resolve the X, and IV, states. In order to understand both the rotational and the hyperfme structure in the IV, state, it is necessary to observe the other branches at high resolution and to extend the observations to higher J. Except for the X2 7.5 state, a purely magnetic treatment of the hfs has proved adequate within the limits of experimental error. However, for the X, 7.5 state, it was necessary to introduce the quadrupole interaction in order to fit the hyperline data. The quadrupole term was found to be statistically significant even though there were data for only three rotational levels. Preliminary (unpublished) LFf calculations by Dulick have suggested that quadrupole effects should be small but significant for HoO. With completely resolved Doppler-free spectra providing, it is hoped, a larger and more precise data set, it should be possible to investigate the contribution of small quadrupole effects in all the states. While there is clear evidence for the Puncoupling phenomenon described by Dulick and Field ( 7), it is not as clear-cut for Ho0 as it was for PrO. The effect in the IV, state is very clear but in the I%‘,state it is less obvious and only appears to start at higher J. In the simplest ligand field model, these two states are both components of the samefelectron core state with the s electron angular momentum parallel (IV, 7.5) and antiparallel (IV, 6.5) to the core angular momentum. In this case, the s uncoupling would give the same hyperfine splittings in both states at high J with the ordering of the hyperfine levels regular for IV, 7.5 and inverted for IV, 6.5. This was observed in several cases for PrO (JO), but it is clearly not true in the present case as the IV, 6.5 splitting is much less than that of IV, 7.5 and is regular. In a I J,, J,, Q) basis set using the simple model, the two states would be IV, 18, 8.5, 7.5) and IV, 18, 7.5, 6.5). The difference in the hfs in the two states suggests that there must be significant mixing with other basis functions, especially for IV,. This is consistent with the calculated eigenfunctions of Hocquet (4) in which the IV, state is predominantly J, = 8.5 whereas IV, is an almost equal mixture of J, = 8.5 and 7.5 basis states which would cause some cancellation of the hyperfine splittings. This work has shown that the hyperhne structure leads to clearer insight into the detailed electronic structure and coupling mechanisms in Ho0 and also serves as a useful test of the calculated eigenfunctions. It is hoped that future sub-Doppler studies and a more detailed analysis that includes off-diagonal and possibly quadrupole terms for all states will lead to a more detailed understanding of the phenomena discussed above. ACKNOWLEDGMENTS Most of the experiments were performed at MIT. We thank Professor R. W. Field for providing us with the free use of his laboratory and for his continued interest in this work and Dr. H. Schall for his help with the experiments. The work has been funded by grants from the Natural Sciences and Engineering Research Council of Canada and the National Science Foundation (PHY83-20098).
RECEIVED:
May 27, 1988
HYPERFINE
ANALYSIS
OF Ho0
381
REFERENCES 1. 2. 3. 4. 5. 6. 7. 8.
Y. C. LIU, C. LINTON, H. SCZHALL, AND R. W. FIELD,J. Mol. Specfrosc. 104, 72-88 (1984). M. DULICK, unpublished. M. DULICK, E. MURAD, AND R. F. BARROW,J. Chem. Phys. 85,385-390 (1986). A. HOCQUET,Thesis (Docteur de 3tme Cycle), Universid des Sciences et Techniques de Lille ( I985 ) P. W. ATKINS,Proc. Roy. Sot. A. 300,487-495 (1967). Y. C. Llu, M.Sc. thesis, University of New Brunswick (1984). M. DULICK AND R. W. FIELD,J. Mol. Spectrosc. 113, 105-141 (1985). C. H. TOWNESAND A. L. SCHAWLOW,“Microwave Spectroscopy,” p. 15 1, McGraw-Hill, New York, 1955. 9. L. A. KALEDINAND E. A. SHENYAVSKAYA,J. Mol. Spectrosc. 90, 590-59 I (198 1). 10. M. DULICK, PH.D. thesis, Massachusetts Institute of Technology ( 1982 ) . 11. J. SUGAR, J. Opt. Sot. Amer. 58, 1519-1523 (1968).
OF MOLECULAR
SPECTROSCOPY 131,367-38 I (1988)
Analysis of Rotational and Hyperfine Structure in Holmium Monoxide C. LINTON AND Y.
C. LIU
Physics Department, University of New Brunswick, Fredericton, New Brunswick, Canada E3B 5A3
Doppler-limited laser excitation spectra have been recorded for six bands of HoO. For the O0, O-l, and l-l bands ofthe [17.617.5-X, 8.5 transition and the O-O band of [18.4]6.5-IV, 7.5 (R branch only), the hyperline structure is completely resolved. For the O-Obands of the [ 17.6]7.5X, 7.5 and [ 18.4]6.5-IV2 6.5 transitions, hyperfme structure is partially resolved and only at low J. Rotational and hypertine analyses have been performed for all the transitions. The hyperline structure is examined in detail and has provided information about the nature of the coupling mechanisms in the various electronic states. The hyperfme constants are used in conjunction with results of ligand field theoretical calculations to determine the individual hyperhne parameters for the 4fand 6s electrons. These parameters are compared with those of the free holmium ion in order to provide insight into the nature of the electron orbitals in the molecule. o 1988Academic Press. Inc.
INTRODUCTION
As part of our ongoing investigation into the spectroscopy and electronic structure of the rare earth monoxides, we recently presented our preliminary results on holmium monoxide, Ho0 ( I ) . Using resolved fluorescence and wavelength-selected excitation spectroscopy, we were able to link and assign four low-lying electronic states. Rotational constants were obtained using combination differences and hyperfine constants were computed using combination differences when the structure was resolved and hypermultiplet widths when the structure was not resolved. The results, even though lacking in precision, clearly showed that the low-lying states originate from the ‘Is supermultiplet which is the lowest state of the 4f*‘6s superconfiguration of the Ho’+ atomic ion. The pattern of the observed low-lying states was shown to be consistent with the predictions of ligand field theory (LFT), and the hyperhne structure was shown to be an important diagnostic tool in assigning the free ion quantum numbers (J,, J,J to the molecular electronic states. Since this work was published, LFI calculations have become more refined and detailed predictions on all the low-lying states of all the rare-earth oxides are now available (2-4). In order to examine the holmium oxide data more closely in relation to the LFT calculations, we have examined the high-resolution data in more detail placing particular emphasis on the analysis of the hyperhne structure (hfs). For the [17.6 ]7.5-Xi 8.5 transition in which the hfs of the O-O and O-1 bands wascompletely resolved (Ref. (I ), Fig. 4A), a least-squares fit to all the lines has been performed including contributions from off-diagonal terms in the hyperlme Hamiltonian. For the [17.617.5-X2 7.5 and [18.4]6.5-IV2 6.5 transitions, the hfs was not completely 367
0022-2852188 $3.00 Copyright 0 1988 by Academic Press, Inc. Au right.3of reproduction in any form rserved.
LINTON
368
AND
LIU
resolved. However, using partially resolved low-J lines and weak 0 # AJ “satellite” lines, it was possible to separate the upper and lower state hfs and determine more reliable hyperfme constants. The rotational constants were obtained from a leastsquares fit to the line centers. Only the R branch was observed for the [ 18.416.SIV, 7.5 transition. The hfs was completely resolved and was used together with the upper state information obtained from [ 18.416.SIV2 6.5 in order to obtain the lower state hyperfine constants. Rotational constants were obtained using known upper state data and hypermultiplet centers, All the hyperflne constants are examined in terms of LFT and are used to determine the contributions of the 4fand 6s electrons to the hfs. No new experiments have been performed and the experimental details are as described in a previous work ( 1) . RESULTS
AND ANALYSIS
I. The [I 7.61 7.5-X1 8.5 Transition The O-O and O-l bands of this transition were obtained at high resolution to J = 24.5. The l-l band was observed only up to R(10.5) and P(12.5). The eight M = AJ hyperfine components were clearly resolved for each rotational transition and, at the lowest values of J, several AF = 0 “satellite” lines were observed. The preliminary analysis ( 1) used only the first and last lines of each AF = AZ hypermultiplet to generate approximate constants for each band. In the present analysis, we have simultaneously fitted all the lines of the main and satellite hypermultiplets of the O-O and O-l bands including in the Hamiltonian both centrifugal distortion and offdiagonal terms. In a pure case cg basis, the diagonal matrix elements of the hypertine Hamiltonian are given by Ediag(F,
J) = (1. J)dWtJ(J
+ 111,
(1)
where (I. J) = [F(F + 1) - J(J + 1) - Z(Z + 1)]/2
(2)
and d is the magnetic hyperfine structure constant. In order to better fit the data at higher J, a phenomenological centrifugal distortion term, dJ, is often included such that the diagonal matrix element becomes E(F
9
J)=
[d-dJJ(J+
l)]Q[F(F+ l)-J(J+ 2J(J+ 1)
l)-Z(Z+
l)]
(3)
In different coupling cases, the same expression can be used for the diagonal terms. However, d is no longer a constant, but an expression which is often a complicated function of J and which is determined by the nature of the coupling and the details of the hyperfine Hamiltonian. In this transition, the hfs follows the expected case cg pattern and can be adequately described by Eq. ( 3 ) . However, the presence of satellite lines indicates that the coupling is not pure case cs and that the following off-diagonal AJ = + 1, AQ = 0 term should be added to the energy,
HYPERFINE
EodF, J) = C
369
ANALYSIS OF Ho0
1(F, Jl%fsl F, J’) I2
J'=JkI
E’(J)
- E’(J’)
(4)
’
where E’(J) and E”( J’) are the rotational energies of the successive levels J and J’. All the matrix elements necessary to calculate E,r( F, J) have been derived by Atkins ( 5) and the detailed expression has been listed by Liu (6). The final constants were calculated using an iterative least-squares procedure and it was found that inclusion of the off-diagonal terms considerably improved the quality of the fit. These terms were found to contribute up to 0.0 12 cm-’ to the line frequencies. As only three rotational hypermultiplets were observed in the R branch and five in the P branch of the l-l band, a detailed fit was not possible. The hyperfme constants and lower state rotational constants were all fixed to the values obtained in the O-O and O-l band fit, leaving only the B value and term energy of the upper, 2) = 1 state to be determined. All the constants are listed in Table I. TABLE I Constants (cm-‘) Derived from High-Resolution Excitation Spectra of Ho0
state
T0
B
10%
d
L18.416.5
18353(4)
0.3485(l)
5.3(19)
0.164(2)
[17.617.5(~=1) [17.617.5(v=O)
18334.72(2) 17546.874(l)
0.3534(l) 0.355851(l)
0.93(l)
O.l8792(8)b 0.18792(8)
1853(4)
0.3503(l)
8.8(21)
0.129(3)
w, 7.5
1130(3P
0.3357(a)
-166(22)
X, 7.5
602.935(6)
0.35601(2)
4.7(S)
O.l95(l)d
X, 8.5(v=l) X, 8.5(v=O)
841.252(l) 0
0.356186(l) 0.357713(l)
2.75(2) 2.51(2)
0.35822(a) 0.35822(8)'
W, 6.5
Band origins 117.617.5 - X,8.5 (0,O): 17546.874(l) (0,l): 16705.622(l) (1,l): 17493.47(2) L17.617.5 - X,7.5 (0,O): 16943.939(2) [18.4]6.5 - W,6.5 (0.0): 16500.439(4) [18.416.5 - W,7.5 (0,O): 17223.27(06)
aThis was determined from low resolution resolved fluorescence measurements and is accurate only to k 3 cm-'. This determines the absolute uncertainty in T, for the W,6.5 and [18.416.5 states even though their positions relative to W,7.5 are much more precisely determined. bdJ = (6.9 + 0.6) x 10-6 cm-'. =dj = - (3.7 + 0.1) x 1O-5 cm-? d
e¶Q
= 0.092 t 0.016 cm-'.
370
LINTON AND LIU
ZZ. The [I 7.61 7.5-X2 7.5 Transition
In this transition the hfs was generally not resolved (Ref. ( I ), Fig. 4B). In the earlier work (I), the approximate lower state rotational constant, B”, was obtained from combination differences using the line centers to represent the rotational frequencies. In the present analysis, the upper state constants were fixed to those obtained from the [17.617.5-X, 8.5 analysis and the lower state constants (B, D, and TO) were computed from a least-squares fit of the line centers to the rotational Hamiltonian. The results are in Table I. Because the hfs was not resolved, it was not possible to do a detailed hyperfme analysis. However, in each ofthe lowest Jlines, R(7.5), R(8.5), P( 8.5), P(9.5), there was partially resolved hfs and several weak, but resolved, satellite lines. The P-branch lines were clearly degraded toward higher frequencies (i.e., the high-F components were at higher frequency). Only the three highest F main lines were clearly resolved. There were four or five clear satellite lines on the high-frequency side of the main hypermultiplet (the other satellite lines overlapped the main hypermultiplet ). The Rbranch structure was more difficult to assign as there were six or seven resolved peaks in the main hypermultiplets but, as they were almost equally spaced, it was difficult to determine the direction of the degradation. There were five satellite lines on the low-frequency side of the main hypermultiplet. The analysis showed that a self-consistent set of assignments could be made only if the R lines were degraded toward lower frequencies. The intensities which should increase with increasing F confirm the assignments. The first R-branch hypermultiplet, R ( 7.5 ), shown in Fig. 1, clearly illustrates the intensity distribution and satellite lines.
rF
Cl763 75-X,7.5 AF=+l
RU5)
‘-5 l--l+dd
I I\
i
V
0.2 0.4
0.3 (cm-l)
AF=O k-G-l4
’
FIG. 1. First line, R(7.5), in the R branch of the [17.617.5-X2 7.5 transition of HoO. Seven of the eight hyperfme components are observed in the main ( AF = AJ) hypermultiplet together with five weaker satellite ( AF = 0) lines.
HYPERFINE
ANALYSIS
371
OF Ho0
The opposite degradation in the P and R branches shows that the hyperfme splitting of level J in the lower state is greater than that of J + 1 in the upper state, but less than that of J - 1. Thus, it would appear that, at the same J, the splittings in both states are almost equal. Observation of resolved main and satellite hyperfine lines at low J enable us to separate the hyperfme structure of the two states and determine some hyperfme splittings in each state from the relations Upper state: (R(J,
F) - RQ(J, F) = PQ(J, F + 1) - P(J, F + 1) = E’(J, F + 1) - E’(J, F))
(54
Lower state: (R(J,
F) - RQ(~, F + 1) = P~(J, F) - P(J, F + 1) = E”(J, F + 1) - E”(J, F)),
(5b)
where R( J, F), P( J, F), etc., represent the main lines ( AF = AJ) and RQ( J, F), PQ( J, F), etc., represent the satellites ( AJ = f 1, AF = 0). The resolved lines and calculated hyperfine splittings are all listed in Table II. The correctness of the assignments was confirmed as there was found to be excellent agreement between (i) the splittings obtained from the P and R lines and (ii) the splittings in the upper state and those in the O-O and O-l bands of the [ 17.617.5-X, 8.5 transition. The lower state splittings were used to calculate the hype&e constant, d, for the lower state. The upper state constants have aheady been well determined and reproduce all the measured upper state separations to within 0.003 cm -’ . The separation between successive levels in a hyperline multiplet is given, to first order, by E( J, F + 1) - E( J, F) =
dQ(F+ J(J+
1) 1) ’
(6)
As we only have data for the two lowest rotational hypermultiplets, it is not possible to determine any J-dependent corrections to d. The values of d calculated for each J, F combination in the lower state are listed in Table II and the average value is d = 0.19 1 + 0.008 cm-’ .
Close examination of the data in Table II casts some doubt on the validity of averaging the values of d. It can be seen that the values of d in each hypermultiplet appear to increase systematically with increasing F and the range of values (0.02 cm-‘) is much larger than can be accounted for on the basis of experimental uncertainty. The offdiagonal AJ = +-1 contributions were computed but did not change the calculated separations by more than 0.00 1 cm -’ and are thus insignificant. Dulick and Field ( 7) have shown that the expression for d derived from the complete Hamiltonian representing intermediate coupling and including o&liagonal AQ = _+l terms contains several J-dependent terms, but there should still be no dependence on F. Thus it
372
LINTON AND LIU TABLE II Measurements (cm-‘) of Main and Satellite Lines in Resolved Hypermultiplets in the [17.617.5-X2 7.5 Transition J
F
R(J,F)
7.5
11 10 9 8 7 6 5
16949.924 49.944 49.970 49.997 50.023 50.045 50.071
16949.709 49.749 49.790 49.835 49.879
0.215 0.195 0.180 0.162 0.144
12 11 10 9 8
50.453 50.487 50.522 50.556 50.589
50.644 50.656 50.680 50.699 50.721
0.191 0.169 0.158 0.143 0.132
8.5
J
F
8.5
12 11 10 9 8
37.955 37.903 37.869
13 12 11 10 9 8
37.216 37.185 37.161
9.5
P(J,F)
R$J,F)
PQ(J,F)
R(J,FbRQ(J,P)
P,$J,F)-P(J.F)
R(J,F)-RP(J,F+l)
d”
0.235 0.221 0.207 0.188 0.166
0.182 0.188 0.198 0.200 0.202
0.203 0.193 0.177 0.165
0.182 0.189 0.191 0.197
PP(J,F)-P(J,F+l)
d”
38.157 38.097 38.050 38.008
0.254 0.228
0.202 0.194
0.181 0.189
37.401 37.359 37.322 37.291 37.262
0.216 0.198
0.185 0.174
0.189 0.193
seems as if the hyperhne structure in the Xz 7.5 state can not be completely described in terms of a magnetic hyperfme interaction. In order to try and account for this apparent variation in d, we have examined the possibility that the electric quadrupole interaction has a significant effect on the hfs in the X2 7.5 state. The quadrupole energy is given by (8)
where f(l
J F) = [0.75c(c+ 1) - z(z+ l)J(J+ 2Z(Z - 1)(2J - 1)(2J + 3)
9 9
111
and c = F(F + 1) - J(J + 1) - Z(Z + 1). The quadrupole contribution to the energy separation in Eq. (5) is thus given by
-eqQLf(Z, J, F+ 1) -_f(Z, J, J’)l
HYPERRNE
373
ANALYSIS OF Ho0 TABLE III
Observed and Calculated’ Hypefine Separations (cm ‘) in the X2 7.5 Stateb J
F
7.5
10
E(J,Ftl)-E(J,F) Calculated Observed
Difference'
9 8 7 6
0.235 0.221 0.207 0.188 0.166
0.240 0.223 0.205 0.186 0.166
-0.005 -0.002 0.002 0.002 0.000
8.5
11 10 9 8
0.203 0.193 0.177 0.165
0.205 0.193 0.180 0.165
-0.002 0.000 -0.003 0.000
9.5
12 11
0.185 0.174
0.178 0.170
0.007 0.004
%cludes
the quadrupole term, -eqQ[f(I,J,F+l)-f(I,J,F)l.
bFrom least squares fit, d = 0.195 f 0.001 cK1, eqQ = 0.092 + 0.016 cm-: =Standard deviation, o = 0.003 cm~'.
and this must be added to the magnetic term in Eq. (6). A least-squares fit to the separations gives the following values: d = 0.195 + 0.001 cm-‘;
eqQ = 0.092 f 0.016 cm-‘.
The observed and calculated separations are compared in Table III and the standard deviation in the fit is 0.003 cm-i. Thus the quadrupole term appears to be significant and fits the data within the experimental uncertainty. III. The [l&4 ]6.5- W, 6.5 Transition As mentioned in a previous paper ( 1), this transition is weak and the hyperhne structure is not resolved in the R branch (Ref. (I ), Fig. 4D). However, the P-branch lines are much broader and the hyperhne structure is completely or partially resolved up to J - 13.5. In addition, two satellite lines on the high-frequency side of the hypermultiplet are clearly distinguishable up to J = 9.5. The R-branch satellites, although not resolved, are also observed, but on the low-frequency side of the hypermultiplet. The first lines in each branch are shown in Fig. 2. The positions of the satellites and the degradation of the hypermultiplets (to lower frequencies) clearly show that the hyperfine splitting is larger in the upper than in the lower state. The two resolved satellite lines represent AR = 0 transitions between the high-Fcomponents. The remaining AF = 0 transitions are not observed because they overlap the main hypermultiplet. By using the separation between main and satellite lines, values of d were calculated for each state. The frequencies and separations of the main and satellite lines in the resolved P-branch hypermultiplets are listed in Table IV along with the d values which were calculated using Eqs. ( 5 ) and ( 6). There was no obvious systematic variation of d with F or J and the average values are
d’ = 0.164 +- 0.002 cm-‘,
d” = 0.129 & 0.003 cm-‘.
374
LINTON AND LIU Cl 8.416.5
1t6.5)
-
W-6.5
Af=-+I L
, 3ir AF=O
‘(15)
I 0.5
AF=-1
1
I
I
I
0.4
0.3 (cm-0
0.2
0.1
FIG. 2. First lines, R(6.5) andP(7.5), in the [18.416.5-W, 6.5 transition ofHo0. The hyperline structure, including two weak satellite ( A F = 0)lines, is clearly resolved in P( 7.5) but not in R( 6.5 ) .
These hyperfme constants were then tested by comparing the measured hyperline separations, up to J = 13.5, with those calculated using the expression P(J, F) - P(J, F-
1) =
dV(F
J( J_
- 1)
1)
-
d”W’F . 1)
J(J+
The results, shown in Table V, show that the observations are all reproduced to within experimental accuracy. Thus, the first-order model adequately describes the hyperfme structure for all the hypermultiplets in which the structure is resolved. At higher J, the structure in the P lines is no longer resolved but the width of the lines continues to decrease. However, the width of the unresolved R-branch transitions appears to increase slightly with J at higher Ss. This could be an indication that the simple case cg coupling model is beginning to break down in either one or both states and that the spin, S, of the s electron is starting to uncouple from the internuclear axis as described by Dulick and Field ( 7) for PrO. In order to be more specific about the
HYPERFINE
375
ANALYSIS OF Ho0 TABLE IV
Measurements (cm -’ ) of Resolved Hypermultiplets in the [ l&416.5-W2 6.5 Transition J
F
P(J,F)
7.5
11 10 9 8 7
16495.315 95.236 95.173 95.118 95.075
12 11 10 9 8
94.571 94.513 94.461 94.416 94.378
13 12 11 10 9
93.829 93.781 93.738 93.700 93.666
10.5
14 13 12 11 10
93.046 93.010 92.976 92.945 92.917
11.5
15 14 13 12 11
92.343 92.310 92.279 92.250 92.222
12.5
16 15 14 13 12
91.597 91.568 91.540 91.513 91.487
13.5
17 16 15 14 13
90.856 90.830 90.805 90.781 90.757
8.5
9.5
%tellite
lines
PQ(J,F)a
were
d’
P@,F)-P(J,F)
P,+J,F)-P(J,Ftl)
d”
95.454 95.369
0.218 0.196
0.164 0.163
0.139 0.133
0.124 0.130
94.697 94.628
0.184 0.167
0.164 0.164
0.126 0.115
0.130 0.130
93.941 93.883
0.160 0.145
0.166 0.164
0.112 0.102
0.132 0.130
only
measured
up
to
J = 9.5.
nature of the coupling, it would be necessary to resolve all the hfs using sub-Doppler techniques and to extend observations to much higher J. To determine which state is responsible for the anomalous hfs, we can examine the widths of the unresolved hypermultiplets. Widths of individual hypermultiplets are difficult to determine without knowing the individual linewidths and lineshapes. However, differences between widths may be determined more reliably and can be used to separate the widths in the upper and lower states. If the widths of the R and P transitions are denoted by IV-&J) and WP(J) and the widths of the hypermultiplets in each state by W’(J) and IV”(J), then we find that W,(J)
-
r&(J)
= W’(J - 1) - W’(J + 1)
(9a)
376
LINTON AND LIU TABLE V Observed and Calculated Hyperfine Splittings (cm-‘) in the [18.4]6.5-W2 6.5 Transition P(J,F)-P(J.F-1) Obs. Calc.
J
P
7.5
11 10 9 8 7
0.079 0.063 0.055 0.043 0.038
8.5
12 11 10 9 8
9.5
10.5
P(J.F)-P(J,P-1) Obs. Calc.
J
P
0.073 0.064 0.056 0.047 0.038
11.5
15 14 13 12 11
0.033 0.031 0.029 0.028 0.028
0.035 0.032 0.030 0.027 0.024
0.058 0.052 0.045 0.038 0.032
0.058 0.052 0.046 0.040 0.033
12.5
16 15 14 13 12
0.029 0.028 0.027 0.026 0.021
0.031
13 12 11 10 9
0.048 0.043 0.038 0.034 0.028
0.048 0.044 0.039 0.034 0.029
13.5
17 16 15 14 13
0.026 0.025 0.024 0.024 0.023
0.028 0.026 0.024 0.022 0.020
14 13 12 11
0.036 0.034 0.031 0.028
0.041 0.037 0.034 0.030
0.029
0.026 0.024 0.021
and wp(J+
1) - W&J-
1) = W’(J-
1) - wn(J+
l),
(ob)
where W(J-
l)-
W(J+
l)=d
&+-&-&-L
J+2
1*
Thus we can separate the two states and use the calculated d values to compare the observed and the calculated difference in widths. It was found that the agreement with observation is good at all J for the upper state but not for the lower state. Because of the difficulty in measuring widths, this may not be a very reliable criterion, but it appears as if the cg coupling is breaking down in the lower state. Line centers were used for the rotational analysis and the rotational constants were computed using a least-squares technique. The constants are listed in Table I and they reproduce the observed lines to within a standard deviation of 0.008 cm-‘. The B value of the upper state is in excellent agreement with that obtained by Kaledin and Shenyavskaya (9) from absorption spectra. The centrifugal distortion constant, D, is not as well determined in this work as the data did not extend to high enough J. However, the value (with error) just falls within the error limits of Kaledin and Shenyavskaya’s value. IV. The [18.4]6.5-WI
7.5 Transition
This transition was more difficult to analyze than the others. It has already been mentioned (I ) that, because there seem to be several overlapping transitions giving
HYPERFINE
ANALYSIS
377
OF Ho0
TABLE VI Measurements
(cm-‘)
of Hypermultiplet ofthe
[l&416.5-W,
Center6 and Widths of the R-Branch
J=
R(J)
Widthb
a.5 9.5 10.5 11.5 12.5 13.5 14.5 15.5 16.5 17.5 18.5 19.5 20.5
17233.797 34.951 36.000 36.979 37.922 38.838 39.732 40.606 41.477 42.333 43.183 44.029 44.870
0.746 0.767 0.772 0.778 0.776 0.775 0.775 0.765 0.774 0.770 0.776 0.773 0.773
%alculated to r2.
from dispersed
Lines
7.5 Transition
fluorescence
0.256 0.275 0.292 0.308 0.323 0.338 0.354 0.371 0.384 0.397 0.415 0.429 0.443
measurements and uncertain
bwidth = R(J,P=J-I)-R(J,F=J+I). ‘These values incorrect.
are dependent on J and will
thus change if
J is
to fluorescence in the same region, it was only possible to obtain an excitation spectrum of the R branch. The hyperhne structure was well resolved (Ref ( 1 ), Fig. 4C) and the centers of the hypermultiplets are listed in Table VI. The quoted J values are probably uncertain to +2 as they were obtained from low-resolution resolved fluorescence measurements. A complete rotational analysis was not possible without the P branch but it was possible to calculate lower state constants using the R-branch measurements and the known upper state constants which had already been determined. The standard deviation of the fit to the data was -0.06 cm-’ which is an order of magnitude worse than for the other analyzed transitions. However, it is much less than the width of the hypermultiplets (-0.7 cm-‘) and could partly reflect the fact that the center is not the true position of the pure rotational transition. This would not, however, account for the high negative D value and it seems as if there may be a weak perturbation affecting this state. The constants which are listed in Table I are dependent, especially in the case of TO, on the correctness of the J assignments. The multiplet structure shows that the hfs is larger in the lower than in the upper state. The structure is unusual in that the multiplet splittings increase slightly with increasing J at the lowest J’s and then remain nearly constant. This is shown in Table VI where the hypermultiplet widths (i.e., the separation of the F = J + Z and F = J - Z components) are listed along with the line center frequencies. The constancy of the splittings with J indicates that the lower state splittings are changing with J by exactly the same amount as those of the upper state. The upper state splittings can be calculated using the value of d’ = 0.164 cm-’ obtained earlier and combined with the line separations R( J, F - 1) - R( J, F) to calculate a value of d” for each pair of adjacent lines. The calculations showed that, especially at lower J, the value d” was rise
378
LINTON AND LIU
constant within a hypermultiplet (i.e., there was no F dependence) but increased rapidly with J as shown in Table VI. This appears to be a clear example of the Suncoupling phenomenon described by Dulick and Field ( 7) in which the hyperfine splittings tend toward a constant value at high J. The uncertainty about the precise value of Jleads to an uncertainty in d (a change of 1 in Jleads to a change of -0.0 150.020 in d) but the increase with J is unaffected. DISCUSSION
The hyperhne structure has previously only been used (I) to assign the atomic ion quantum numbers, J, and J, , to the low-lying states on the basis that large hfs indicates J, = J, + fi and small hfs corresponds to J, = J, - f . The more detailed examination of the hfs discussed in previous sections has shown that the situation is more complex. The ground, Xi 8.5 state appears to be the only low-lying state which is well described in terms of the magnetic hyperfine interaction obeying a case cg coupling model. The coupling in X2 7.5 is more difficult to determine as the observed transition involves two states with similar hfs so that the structure is unresolved except for portions of a few low-J lines. However, even for these low-J lines there is evidence that the lower state hfs does not follow the expected Land6 pattern. The W, 7.5 hfs is clearly anomalous, even at low J, and the observation of only a single branch made analysis dithcult. The WZ6.5 state hfs was incompletely resolved and only in the P branch at lower J. The hfs appeared normal at the lower J’s but increasing hypermultiplet widths at higher J’s in the R branch indicate that the lower state hfs is beginning to depart from pure case cP The hfs in the molecular states may be used to estimate the atomic hyperfme parameters for the 4fand 6s electrons. The hyperfine Hamiltonian can be written as Hhfs = aI* J, + bras, where a is the hyperfine parameter for the 4fcore electrons and b is the Fermi contact term for the 6s electron. For the states in which the hyperfine constant, d, is reasonably well defined, it can be related to the atomic hyperfine parameters by d = a4fa + cthsb. Expressions for cy4f and ff6s have been derived and evaluated by Dulick (10) for Pro and by Hocquet (4) for HoO. For the X1, X2, and W, states, the hyperfme parameter, d, appears to be independent of J at low J and these states can therefore be used to estimate a and b. Hocquet (4) computed the eigenfunctions of all the low-lying states and used them to calculate values of (Y4fand (Ygthat are listed in Table VII. These were then combined with the experimentally determined values of d (also listed in Table VII) to solve for a and b. The least-squares solution yielded a = 0.041 + 0.003 cm-’ and b = 0.20 & 0.05 cm-‘. The values of d calculated from these constants are compared with the observed values in Table VII. The standard deviation of the fit to d is -0.025 cm-’ and the fitted values differ from the observations by 20-25% for the X2 and II’, states. This is not really surprising as both states showed anomalous hfs and d is not truly a constant and well-defined parameter for these states. The values of (Ydepend on the eigen-
HYPERFINE
379
ANALYSIS OF Ho0
TABLE VII Comparison of Observed and Calculated Hypetine Constants state
G
a:,
dobs(cm-')
d
CdC
(cm-')
X,8.5
6.13
0.50
0.3582
0.3541
x,7.5
6.16
-0.42
0.195
0.169
W,6.5
5.22
-0.25
0.129
0.164
Least squares fit of d = a4fa + a6sb gives a = 0.041 * 0.003 cm-l and b = 0.20 f 0.05 cm-' Standard deviation of fit, o = 0.025 cm-'
%alculated by Hocquet (4).
functions so that any inaccuracy in the eigenfunctions would affect the calculations. In addition, the experimental values of d were calculated from a very limited data set. As already mentioned, the hfs in the Wi 7.5 state is anomalous and the parameter, d, is a rapidly increasing function of J that reflects the uncoupling of the s electron angular momentum from the internuclear axis. From the relationship derived by Dulick and Field (7), it can be shown that at high J, d should increase by b/2Q between successive values of J. From Table VI, it can be seen that d increases by 0.014 cm-’ at the highest J’s and, as Q = 7.5, this leads to a value of b = 0.21 cm-’ in excellent agreement with the value obtained from the X1, X2, and WZstates. The above data can be used to determine the relative contributions of the 4fand 6s electrons to the hyperfme structure. For the X1 state, the f contribution is -2.5 times greater than the s contribution, for X2 it is -3 times greater, and for W,, particularly at low J, it is -4 times greater. Thus, in all these states, thef electrons make the major contribution to the hfs. In the Wi state, as a result of the s uncoupling, the Fermi contact term is dominant. It is informative to compare the a and b parameters with those of the atom. There does not appear to be published data for the Ho2+ f”s configuration. However, for thef”s configuration of Ho+, Sugar (II) determined that a = 0.03491 cm-’ and b = 0.4437 cm-‘. These may change slightly but are not expected to alter by a large amount in going to thef”s configuration of Ho2+. The Q values for the molecule and the free ion are very similar confirming that the atomic f electron core retains its properties on formation of the molecule. The b value, however, is only -47% of its free-ion value. The Fermi contact parameter, b, is proportional to the 6s charge density at the nucleus which has thus been reduced by -53% by molecule formation. This is a clear indication that the true configuration is 4f”a, where the Q orbital is an approximately equal mixture of holmium 6s and oxygen 2p orbit&. This is very similar to the result obtained by Dulick (10) for PrO. While the above analysis has indicated the nature of the orbital structure in the molecule, it must be emphasized that it is far from complete and that the hypefine
380
LINTON AND LIU
constants are not well determined. In order to obtain a clearer understanding of the precise nature of the coupling it is essential to follow resolved hfs to high Jin all the transitions. This will require Doppler-free techniques to resolve the X, and IV, states. In order to understand both the rotational and the hyperfme structure in the IV, state, it is necessary to observe the other branches at high resolution and to extend the observations to higher J. Except for the X2 7.5 state, a purely magnetic treatment of the hfs has proved adequate within the limits of experimental error. However, for the X, 7.5 state, it was necessary to introduce the quadrupole interaction in order to fit the hyperline data. The quadrupole term was found to be statistically significant even though there were data for only three rotational levels. Preliminary (unpublished) LFf calculations by Dulick have suggested that quadrupole effects should be small but significant for HoO. With completely resolved Doppler-free spectra providing, it is hoped, a larger and more precise data set, it should be possible to investigate the contribution of small quadrupole effects in all the states. While there is clear evidence for the Puncoupling phenomenon described by Dulick and Field ( 7), it is not as clear-cut for Ho0 as it was for PrO. The effect in the IV, state is very clear but in the I%‘,state it is less obvious and only appears to start at higher J. In the simplest ligand field model, these two states are both components of the samefelectron core state with the s electron angular momentum parallel (IV, 7.5) and antiparallel (IV, 6.5) to the core angular momentum. In this case, the s uncoupling would give the same hyperfine splittings in both states at high J with the ordering of the hyperfine levels regular for IV, 7.5 and inverted for IV, 6.5. This was observed in several cases for PrO (JO), but it is clearly not true in the present case as the IV, 6.5 splitting is much less than that of IV, 7.5 and is regular. In a I J,, J,, Q) basis set using the simple model, the two states would be IV, 18, 8.5, 7.5) and IV, 18, 7.5, 6.5). The difference in the hfs in the two states suggests that there must be significant mixing with other basis functions, especially for IV,. This is consistent with the calculated eigenfunctions of Hocquet (4) in which the IV, state is predominantly J, = 8.5 whereas IV, is an almost equal mixture of J, = 8.5 and 7.5 basis states which would cause some cancellation of the hyperfine splittings. This work has shown that the hyperhne structure leads to clearer insight into the detailed electronic structure and coupling mechanisms in Ho0 and also serves as a useful test of the calculated eigenfunctions. It is hoped that future sub-Doppler studies and a more detailed analysis that includes off-diagonal and possibly quadrupole terms for all states will lead to a more detailed understanding of the phenomena discussed above. ACKNOWLEDGMENTS Most of the experiments were performed at MIT. We thank Professor R. W. Field for providing us with the free use of his laboratory and for his continued interest in this work and Dr. H. Schall for his help with the experiments. The work has been funded by grants from the Natural Sciences and Engineering Research Council of Canada and the National Science Foundation (PHY83-20098).
RECEIVED:
May 27, 1988
HYPERFINE
ANALYSIS
OF Ho0
381
REFERENCES 1. 2. 3. 4. 5. 6. 7. 8.
Y. C. LIU, C. LINTON, H. SCZHALL, AND R. W. FIELD,J. Mol. Specfrosc. 104, 72-88 (1984). M. DULICK, unpublished. M. DULICK, E. MURAD, AND R. F. BARROW,J. Chem. Phys. 85,385-390 (1986). A. HOCQUET,Thesis (Docteur de 3tme Cycle), Universid des Sciences et Techniques de Lille ( I985 ) P. W. ATKINS,Proc. Roy. Sot. A. 300,487-495 (1967). Y. C. Llu, M.Sc. thesis, University of New Brunswick (1984). M. DULICK AND R. W. FIELD,J. Mol. Spectrosc. 113, 105-141 (1985). C. H. TOWNESAND A. L. SCHAWLOW,“Microwave Spectroscopy,” p. 15 1, McGraw-Hill, New York, 1955. 9. L. A. KALEDINAND E. A. SHENYAVSKAYA,J. Mol. Spectrosc. 90, 590-59 I (198 1). 10. M. DULICK, PH.D. thesis, Massachusetts Institute of Technology ( 1982 ) . 11. J. SUGAR, J. Opt. Sot. Amer. 58, 1519-1523 (1968).