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5.4.1 Ground Multiples Hyperfine Interactions between
Two Close-Lying Electronic Singlets
5.4.1.1 Second-Order Perturbation Approach
Simple explanations of the general features of the observed high-resolution excitation spectra can be obtained from second-order perturbation theory.
As (i) the two lowest ground-state crystal-field levels, 5I8(Aj) and 5Ig(A2), are separated by 1.7 cm '1 (~ 50 GHz), and (ii) their nearest neighbouring crystal-field level, which is an E doublet, locates at 81.3 cm '1 above the second singlet, these two singlets can be treated, in the first order, as an isolated system. By considering the general forms of the C4v crystal-field wavefunctions for an Aj and an A2 singlet in the 5Ig multiplet (Sec.2.3, Ranon and Lee 1969);
I Aj) = a { | 8 , 8 ) + |8,-8)} + a 2{ | 8 , 4 > + |8,-4>} + a 3|8, 0), (5.1a)
and
| A2) = P j p s , 8) - I 8,-8)} + ß2{ I 8, 4) - I 8,-4)); (5.1b)
axial
it can be seen that these two singlets can interact via the axial (diagonal) component, H , of
M
the magnetic hyperfine interaction, : (Sec.2.4.2)
axial transverse i ✓ x
KM = KM
+ H M
= V * J = AjVz + 2 Aj(l+J-+ U +)’
<5-2>
where Aj is the magnetic hyperfine-structure constant for the ground 5I8 multiplet. Due to their
proximity, the hyperfine interaction between the two singlets becomes significant, leading to large pseudoquadrupole splittings in both singlets that can be observed in the optical transitions.
If there is no interaction between the two singlets, the total wavefunctions, including both electronic and nuclear parts, can be written as the direct product of the electronic and nuclear
A ,.m > = { “ j { I 8. 8>+ |8,-8>} + a , { |8 . 4 ) + I 8,-4)} + a ^ \8, 0 )} |y , m> (5.3 a) A2,m> = { ß j { I 8, 8> - |8 ,- 8 ) } + P2{ | 8 , 4 > - | 8 , - 4 » } | ^ m ) (5.3 b)
7 where m is the projection of the nuclear spin I along the z-axis (eigenvalue of Iz), and I = - for the natural abundant isotope ^ 5Ho. By second-order perturbation theory, the hyperfine levels of the Aj and A0 electronic states associated with the nuclear projection m, IA1? m> and IA0, m), repel one another by the amount
AE = m
a < j y
J \ z/]
m C
V 2(5.4)
where <Jz) and £ are defined by the relations <J z\ 2 = <A , - m l J z l A 2 ' m >
= <A2. mlJzlA!. m>
<A , l J zl A 2> <A2MzI A , ) = <Jz>2 I - and A A > 12’ (5.5) (5.6)and D is the crystal-field separation between the two singlets. This pseudoquadrupole splitting results in four doubly degenerate levels with separations in the ratio 3:2:1 as shown diagrammatically in Fig.5.5. The quadrupole splittings in the excited-state singlet are normally small. Hence, the ground-state pseudoquadrupole splittings give the dominant contribution to the structure in the 5I8(A2) —> 5F5(A2) and ^ ( A j ) —> 5F5(A2) optical excitation spectra shown in Figs.5.2(c) and 5.2(d). The small quadrupole splitting in the excited-state singlet adds to the ground-state splitting in one optical transition, 5Ig(A2) —» 5F5(A2), and subtracts in the other, 5I8(Aj) - 4 5F5(A2), Fig.5.5. It can then be concluded that
(a) the hyperfine lines L, M, N and O of the 5I8(A2) —> 5F5(A2) optical transition are
7 5 3 1
associated respectively with th em = ± - , ± - , ± - , and ± - hyperfine levels, and
(b) the hyperfine lines P, Q, and R of the 5I8(Aj) —» 5F 5(A2) optical transition are
3 5 7
The missing of the m = ± ^ hyperfine line from the 5Ig(Aj) —> 5F5(A2) optical spectrum is due to the hyperfine dependence of the mixing coefficients between the 5I8(Aj) and 5I8(A 2) hyperfine wavefunctions.
A, — *—
69.0 c m 1 13.5 cm 4.5 cm 81.3 cm .7 cm energy P4 m 1 ±1/2 “jr 2 ±3/2 > r i k 4 ±5/2 jL 6 ±7/2 m D I T ±7/2 6 >r t k ±5/2 4 ±3/2 % 2 ±1/2 T 2> k 4 J Li k6
±1/2 ±3/2 ±5/2 ±7/2Figure 5.5 Pseudoquadrupole splittings within the 5Ig(A 1) and 5I8(A 2) ground-state singlets as predicted by second-order perturbation theory and the assignment of hyperfine levels to the hyperfine lines in the 5I8(Aj, A2) —> 5F 5(A2) optical transitions.
The unequal intensities of the four hyperfine nuclear transitions observed in the 5I8(A) —> 5F5 (A2) and 5I8(Aj) —> 5F5 (A2) optical transitions, Figs.5.2(c) and 5.2(d), arise as a consequence of the mixing of the wavefunctions. By second-order perturbation theory, the wavefunctions for the hyperfine levels within the two electronic singlets are of the form
-1/2
mC
{ | A , . m> - F f f ) |A 2, m>}, (5.7a) and
A2> m)' =
As can be seen from Fig.5.2(c), the optical transition 5Ig(A2) -A 5F5(A2) exhibits a completely resolved hyperfine structure consisting of four hyperfine lines. Normally, in a crystal-field of C4v symmetry, the transition 5I8(Aj) — > 5F5(A2) is not allowed (Table 4.1), but here, due to
the mixing of the wavefunctions, there is a transfer of intensity from the A9 to Aj initial states making the transition 5I8(A }) —4 5F5(A2) observable, Fig.5.2(d). The intensities of transitions from the nominally Aj and A2 ground state levels to an A2 excited state arise solely from the lA->,m) component in the ground state wavefunctions. Due to the hyperfine dependence of the mixing coefficients, the transfer of intensity is dependent on the nuclear state. The loss of intensity for the 5I8(A2) —> 5F5(A2) transition is greater for the hyperfine levels with the larger m values. This is consistent with the observation in Figs.5.2(c) and 5.2(d) where there is greatest transfer of intensity from the A2 to Aj initial states for the outside lines, i.e. those displaced by the largest amounts and, hence, associated with the m = ± - hyperfine levels. The
5 3 1
transfer of intensity is then progressively less for the m = ± - , ± - , and ± - hyperfine components.
The above approach, however, gives the same set of hyperfine splittings for both electronic singlets and, thus, is not sufficient to account for the hyperfine ODMR experiments where two separate sets of hyperfine resonances are observed. This discrepancy arises partly from the negligence of the nuclear electric quadrupole interaction, Hq, (Sec.2.4.2)
Hq = P { l z2- j l ( l + 0 }. + \ n /
{| Af m>
+< # > ! *
2’ < > } (5.7 b) , , mV 1 ' 1 + ( ^ ) 2 (5.8)where P is the effective quadrupole interaction parameter. If this interaction is also taken into account, second-order perturbation theory then gives
r- , ,
O^) .
„ ( 2 21 \ „ „ .EAi(m) ---- 5
pAm 'T'*’
(5,8a)
and
( mC) / 2 2 h
EA2(m) = D + + P2(m 2 - (5.8 b)
where Pj and P0 are the effective quadrupole parameters for the and A2 states, respectively. As P[ and P^ are normally not the same, the hyperfine splittings for the two electronic singlets are now different. However, the predicted hyperfine splittings within each singlet are still in the ratio 3:2:1, whereas the experimental splitting ratios (from Table 5.2) are 2.67:1.91:1 and 2.68:1.91: 1 for the Aj and A2 singlet, respectively. Given the proximity of the two singlets, it can be concluded that, whereas the perturbation approach gives a general understanding o f the features of the observed high-resolution excitation spectra, the higher-order terms play a significant role and so it is necessary to make a full calculation.
5.4.1.2 Exact Approach
In the zero applied magnetic field, the Hamiltonian describing the two coupled ground-state singlets can be written as
H = « . + H + H ; (5.9)
free ion cf hyperfine
where (a) H free ion is the Hamiltonian for the free Ho3+ ion (Sec.2.1); (b) H cf is the C4v crystal field Hamiltonian (Secs.2.2 and 2.3); and (c) ^hyperfine describes the dipole and quadrupole contributions to the hyperfine interaction (Sec.2.4.2),
^hyperfine = 1 AjV z ^ W + ^ +) } + P{ if - } I(I+ D } • (5.10)
Using as the basis states the direct product of the electronic and nuclear wavefunctions, IA j, m) and IA0, m), the Hamiltonian matrix can be written as