Approximate first-order trigonal crystal-field-split energy levels are derived and fitted theoretically for the Sm3+-in-CaF2 (Type I) 6H7/2, 6H9/2, and 6H11/2 states of the ground multiplet involved in fluorescence. Crystal-field theory and the Stevens method are employed in this derivation. The resulting energy deviation between the first-order theoretical levels and the empirically derived levels is within 42 cm−1 for 14 of the 15 levels involved under the assumption of a constancy of the six trigonal crystal-field parameters. Tables are given of the resulting approximate values of the crystal-field parameters, level energies, and eigenfunctions.
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See Fig. 2(b) of Ref. 3.
A review of our original analysis and data3 indicates that the original level3 at 2744 cm−1, which terminates the three broad lines Nos. 22, 24, and 26 is a highly unlikely crystal-field-split level. A likely level to replace it is one at 2405 cm−1, terminated by the three lines at 15 858 cm−1 (No. 24), at 116 189 cm−1 (No. 30), and at 16 335 cm−1 (listed in the footnote a to Table I of Ref. 3 as the unaccounted-for line 6122 Å). Therefore, our original empirical energy-level scheme3 has been revised by the deletion of the level at 2744 cm−1 and the addition of the level at 2405 cm−1.
Table II
Operator-equivalent factors for Sm3+.
Operator-equivalent factor
6H7/2
State 6H9/2
6H11/2
α
1.6×10−2
9.8×10−3
7.5×10−3
β
−2.0×10−4
−8.3×10−5
γ
−1.6×10−4
−2.4×10−5
Table III
Approximate values of trigonal crystal-field parameters for Sm3+ in CaF2 (Type I). Estimated accuracya ±50%.
Parameter
Value (cm−1)
A20〈r2〉
84
A40〈r4〉
0
A60〈r6〉
0
A43〈r4〉
−8500
A63〈r6〉
−1000
A66〈r6〉
−550
A40 and A60, which were approximately equal to zero, were set exactly equal to zero for convenience. The accuracies of A20 〈r2〉, A40 〈r4〉, and A60 〈r6〉 are undetermined. Also, see note b to Table IV.
Table IV
Approximate theoretical first-order energies (cm−1) of the trigonal crystal-field-split energy levels of Sm3+ in CaF2 (Type I).
See Table I.
Computations of the theoretical energies to second order including fractional A variations are expected to reduce the figures of merit7 of these deviations to within 10% for most all levels here; for example, fractional A variations of a 25% increase of all the A’s of the 6H9/2 state would reduce the deviations of 3 of these 5 levels of 6H9/2 to within a 10% figure-of-merit. (Such a 25% increase of the A’s of a state of a ground multiplet above the ground state has been found7 to be not unlikely for a trivalent rare earth in CaF2.)
Table V
Approximate theoretical first-order trigonal crystal-field eigenfunctions of Sm3+ in CaF2 (Type I).
State
Approximate eigenfunction
Approximate energy (cm−1)
6H7/2
−131
−46
−12
188
6H9/2
−142
−63
−36
85
157
6H11/2
−152
−57
−2
36
54
119
Tables (5)
Table I
Empirical crystal-field-split trigonal energy levels (cm−1) of Sm3+ in CaF2 (Type I).
See Fig. 2(b) of Ref. 3.
A review of our original analysis and data3 indicates that the original level3 at 2744 cm−1, which terminates the three broad lines Nos. 22, 24, and 26 is a highly unlikely crystal-field-split level. A likely level to replace it is one at 2405 cm−1, terminated by the three lines at 15 858 cm−1 (No. 24), at 116 189 cm−1 (No. 30), and at 16 335 cm−1 (listed in the footnote a to Table I of Ref. 3 as the unaccounted-for line 6122 Å). Therefore, our original empirical energy-level scheme3 has been revised by the deletion of the level at 2744 cm−1 and the addition of the level at 2405 cm−1.
Table II
Operator-equivalent factors for Sm3+.
Operator-equivalent factor
6H7/2
State 6H9/2
6H11/2
α
1.6×10−2
9.8×10−3
7.5×10−3
β
−2.0×10−4
−8.3×10−5
γ
−1.6×10−4
−2.4×10−5
Table III
Approximate values of trigonal crystal-field parameters for Sm3+ in CaF2 (Type I). Estimated accuracya ±50%.
Parameter
Value (cm−1)
A20〈r2〉
84
A40〈r4〉
0
A60〈r6〉
0
A43〈r4〉
−8500
A63〈r6〉
−1000
A66〈r6〉
−550
A40 and A60, which were approximately equal to zero, were set exactly equal to zero for convenience. The accuracies of A20 〈r2〉, A40 〈r4〉, and A60 〈r6〉 are undetermined. Also, see note b to Table IV.
Table IV
Approximate theoretical first-order energies (cm−1) of the trigonal crystal-field-split energy levels of Sm3+ in CaF2 (Type I).
See Table I.
Computations of the theoretical energies to second order including fractional A variations are expected to reduce the figures of merit7 of these deviations to within 10% for most all levels here; for example, fractional A variations of a 25% increase of all the A’s of the 6H9/2 state would reduce the deviations of 3 of these 5 levels of 6H9/2 to within a 10% figure-of-merit. (Such a 25% increase of the A’s of a state of a ground multiplet above the ground state has been found7 to be not unlikely for a trivalent rare earth in CaF2.)
Table V
Approximate theoretical first-order trigonal crystal-field eigenfunctions of Sm3+ in CaF2 (Type I).