O i l ) <i f i>
6.7 Hyperfine Interactions between Close-Lying Singlet and Doublet
The lower levels of the 5F5 multiplet are comprised of two singlet A2 states and two doublet E states all within a few wavenumbers (Fig.6.3). The above analysis of the two laser holebuming has shown that a specific type of mixing between these excited-state crystal-field levels has occurred, and the distinct character of this mixing is that only two out of the eight hyperfine levels have been significantly affected. What is the interaction that can cause such a mixing between these levels? The answer to this question can be obtained by considering the form of
the wavefunctions of these crystal-field levels.
According to the group theory, the crystal field of C3v symmetry will split the excited 5F5
multiplet into 7 levels:
5F5 -> A + 2 A2 + 4 E . (6.5)
The general form of the wavefunctions associated with these crystal-field levels can be obtained
(1) For a singlet of A} symmetry
5f y I Aj) = a { | 5 , 3) + I 5,-3)}. (6.6)
(2) For a singlet of A^ symmetry
5Fr |A2) = P1{ | 5 , 3 ) - | 5 , - 3 ) } + ß2|5,0>. (6.7)
(3) For a doublet of E symmetry
5F5 : |E ±>
=Y1|5,+5>±
y2|5,+2)
+ y3|5,±1)±Y4|5,±4>.
(6.8)
From the form of the above wavefunctions, it is clear that one of the possible interactions between the two singlet and two doublet states in the excited 5F5 multiplet is the magnetic hyperfine interaction
i , x axial transverse
*
m= AM+WV- + ‘- 0 = * M
+ « M
(69)
By considering the effect of the hyperfine interaction, it is found that the admixture of nearby doublet states affects all sixteen hyperfine wavefunctions of an E state by a similar amount. This even mixing mechanism can, therefore, be dismissed as the origin of the two extra hyperfine transitions. The coupling between an E and an A state, however, does gives asymmetric mixing of the wavefunctions and will be discussed below.
The effect of the dipole hyperfine interaction on an E doublet and an A singlet (A} or A2) state
separated by an energy D is considered. For simplicity, it is assumed that there is no interaction with any other state and that there is no quadrupole interaction. The total Hamiltonian, neglecting free-ion Hamiltonian (H free ion), is
axial transverse
H
= H
+ H
= H
+ K +H
TOT crystal field M crystal field M M (6.10)
The axial component of the dipole hyperfine interaction, AjIzJz, gives rise to the dominant hyperfine splitting of the E state. The transverse dipole hyperfine interaction, ^ Aj(I+J_ + I_J+),
is the term that causes the admixture of the hyperfine components of the E and A electronic states.
By assuming that the unperturbed wavefunctions are the direct products of the electronic (crystal-field) and nuclear wavefunctions, it can be seen that, for the case of trivalent holmium ions which have the nuclear spin I = — , each hyperfine state IA, m+1) ( except I A, ± - ) )
associated with the electronic singlet interacts with a pair of the doublet hyperfine states IE , m) and IE , m+2). Here m, m+1, and m+2 are the z-projection of the nuclear spin. The resultant energies are obtained by diagonalizing the Hamiltonian matrix,
IE+, m) IA, m+1) IE., m+2)
D + m A. 2 A1J(Tm )(I+m+l)
yA^ I-m X l+m +l) 2" A±V (I-m-1 )(I+m+2)
j A/i 7(I-m-l)(I+m+2) D - (m+2) A|,
(6.11)
where the matrix elements multiplied by the dipole hyperfine interaction parameters have been rewritten as effective dipole hyperfine interaction strengths Ah, A ±, and A^ defined by
A „ = A j<E A IE+ > = - V E - IJzIE-)'
Ax = A (E +IJ+IA> = A/AIJ IE+),
and
= Aj(E IJ IA) = A (AIJ+IE.).
In the case of the crystal field of C3v symmetry, an A2 singlet has the property that
whereas for an A t singlet A± = -A^.
(6.12 a)
(6.12 b)
(6.12 c)
A - A/
In solving this problem by means of non-degenerate perturbation theory (Dalgamo 1961), it is
convenient to treat the terms describing the crystal-field interaction and the axial magnetic hyperfine interaction as the unperturbed Hamiltonian, leaving only the perpendicular component
of the magnetic hyperfine interaction to be treated as the perturbing Hamiltonian:
K = H + t t “ 181 = K + A I L (6.13 a) 0 cf M cf J z z and K ' transverse 2 A /l+ J- + U +>- (6.13 b)
For the doublet hyperfine state IE+, m ), the energy corrections calculated up to fourth order and
9
keeping the terms up to order 0 (D ) are
and (0) E + (m) = D + m A(|, (1) E+ (m) = 0, (2) 1 A1 1 A±AII -3 E+ (m) = j ( I - m ) ( I + m + l ) - j j - - A m (I-m )(I+ m + l)— — + 0 (D ), (3) E+ (m) = 0, (4) E+ (m) - 32 therefore, up to 0(D~2) A2
E+(m) = D + m A ( + | ( I - m ) ( I + m + l ) y - j m (I-m )(I+ m + l) — —
1 (I-m )(I+ m + l)(I-m -l)(I+ m + 2 ) Aj 1 (I-m )(I+ m + 1 )(I-m -1 )(I+m +2) A± -3
(m + 1) A. A, ( m + 1 ) (6.14 a) (6.14 b) (6.14 c) (6.14 d) (6.14 e) (6.14 f) A D
Note that (a) the above expressions are valid for all values of m, and (b) the energy of the doublet hyperfine state IE , m > is the same as that of IE+,-m >.
There are two types of correction terms to the energies. The first type have denominators of the form Dn where D is the separation between the doublet and singlet, and n is an integer. The leading term o f this type is the familiar pseudoquadrupole contribution which is proportional to A^/D. The second term of this type is proportional to AyA^/D2 which, for A|j = 1GHz and D = 45 GHz, is at most only a few percent of the pseudoquadrupole strength.
The second type of correction terms have denominators of the form A ^ D 20 where m and n are integers. Im portantly, these terms describe the repulsion between the two E state hyperfine levels. The leading term of this type arises in fourth order and is proportional to A ^ /AhD 2. It
will be shown that when A± is larger than AM, this term can become more significant than the pseudoquadrupole contribution and quantitatively describes the observed mixing among the E state hyperfme levels.
The wavefunction corrections for IE+, m ) calculated up to second order and keeping the terms up to order 0(D '*) are
(0)
IE+, m) = IE+, m),
^ j ^ ^
IE+, m) = { y A/(I-m )(I+ m + l) - ^ } IA, m+1), and f 1 )(I-m-1 )(I+m+2)
\ \
^ { ---8(m TT)---H f i } IE-’ m+2>; (6.15 a) (6.15 b) (6.15 c)so that the wavefunctions for the doublet hyperfme states IE+, m >' take the form
E+, m)' = I E+, m) + 5(m)| A, m+1) + £(m)| E_, m+2), where
5 (m ) = { j J ( l - m ) ( l + m + \ )
(6.15 d)
(6.16)
and
e(m) = {J (I-m )(I+m +1 )(I-m -1 )(I+m+2) Ai Aj
8 (m + 1) AyD } (6.17)
The corresponding expression for 1E_, m+2 )' is
|E_, m+2)' = I E., m+2) + K(m)| A, m+1) - e(m)| E+, m ) , (6.18)
where
As can be seen from Eqn.(6.17), the expression for e(m) contains, as the energy denominator, the separation between the coupled hyperfine levels in the E state, 2(m + 1)A(|. Three pairs of E state hyperfine levels (and their complex conjugates) exhibit the above form of coupling, and the respective mixing coefficients are :
(i) IE±, + h a n d l E T, + | ) ,
r- \ \
± ^ A D '
(ii) I E±, ± i > and|ET, ± ^ ) ,
e = ± J~5 A A
/
n
2 A.D ’
and (iii) IE+, ± | ) and IE^, ± 1- )
J~2A_ A A