nuclear moment of the Ho3+ ion («= T*) and the fluorine nuclear moment (°c if.); and (d) and
dj are respectively the electronic and nuclear superhyperfine tensors associated with the ith fluorine. The term J • a i«Fti (and hence the electronic superhyperfine tensor aj), particularly for the nearest neighbours, can contain both dipolar and non-dipolar contributions (Baker 1974). The latter contribution, which is difficult to calculate, arises from the overlapping between the electron clouds of the rare-earth ion and the neighbouring ions. For this reason, a4 is normally treated as a parameter to be determined from the experiments. On the other hand, the nuclear superhyperfine interaction T»dj»F^ can always be treated as a dipole-dipole interaction. Since
nuclear electronic
H _ is generally much smaller than H , the superhyperfine resonances are
superhyperfine superhyperfine
electronic
determined mainly by H „ ,which is also known as the transferred hyperfine
su p erh yp erfin e
interaction.
If the fluorine-fluorine interaction
HF -F =
X
“ i jF *b *F ,
i ij J
(2.91)
which can always be treated as a dipole-dipole interaction and is expected to be very smalF, is
At liquid helium temperature for which the F - F’ separation in CaF2 is 2.7178 Ä (Batchelder and Sim m ons 1964), the magnitude o f a classical dipole-dipole interaction is estimated to be 2.65 kHz, and
neglected, then there is no matrix elements connecting the different fluorine nuclear spins and, hence, each component in the sum in Eqn.(2.90) can be treated separately (Ranon and Hyde
1966, Baker et al 1968).
The interaction between Ho3+ ion and neighbouring F' ions introduces additional preferred directions in space and this complicates the determination of superhyperfine splittings.
However, for the neighbouring fluorine ion located at the site which has axial symmetry about its 'bond' axis (the line joining between Ho3+ and F nuclei), the associated superhyperfine
tensors, a and d, will be diagonal in the 'primed' coordinate system (x', y', z'), where the z' axis lies along the bond direction, and will have axial symmetry in this system (Ranon and Hyde 1966, Abragam and Bleaney 1970). The neighbouring fluorines which have this axial symmetry property are the interstitial fluorine for the case of CaF2: Ho3+ C4v centres, and the on-axis fluorines for the CaF2: Ho3+ C3v centres. As the z' axis of these 'axial' fluorines coincides with the symmetry axis of the centre, the associated superhyperfine Hamiltonian will have the form
Ho-axial F electronic 3+ nuclear 3+
H _ = H (Ho - axial F ) + H (Ho - axial F )
superhyperfine superhyperfine superhyperfine
= { V z F z+ \ ( J+F- + J-F
J
} + d { l2Fz - i ( l +F- + I F +) } . (2.91)Expression for d:
The Hamiltonian describing the dipolar interaction between Ho3+ and Fnuclear moments,
P
(Ho) = Yh X
and p (F) =y h
F , (2.92) n Ho n F is given by (Goldman 1970) dipolar ft l = - T ^ Ho-F 4 71P
(Ho)«p (F) 3 (p (Ho)*r) (p (F)*r) n n n H P Ä y y J i ^ F ( A + B + C + D + E + F) (2.93)where y is the nuclear magnetogyric ratio which is related to the nuclear g-factor (gn) by
and
gn^N
y h ’ (2.94)
= (r, 0, (j>) is the vector joining 1 and f ,
A = ( l - 3 cos20 ) IzFz, (2.95 a)
B = - | ( l - 3 c o s 20 ) ( l +F + I F +), (2.95 b) C = -
1
cos0 sine e‘“1>( [ zF+ + FzI+), (2.95 c) D = - ~ cos0 sin0 e+1<^(lzF. + F J .) , (2.95 d) I-, 3 . 2 -2i<DE = - — sin 0 e I+F+, (2.95 e)
„ 3 . 2^ +2i* _ F = - 4 - s m 0 e y I F .
4 ‘ ' (2.95 f)
For axial fluorines the terms C, D, E, and F vanish (since, for these fluorines, 0 = 0°). The expression for the parameter d in Eqn.(2.91) is therefore
d h y y 'Ho'F ( 1 - 3 — p - ( l - 3 2n r (2.96)
Substituting h = 1.054 572 66 x 10*31 yHo = 5.4873 x 104 N p , and yF = 25.1662 x 104 we obtain (a) for an interstitial F ion in CaF2: Ho3+ C4v centre which has, at liquid helium temperatures under the assumption of undistorted CaF2 lattice (Batchelder and Simmons 1964), r = (2.7178 Ä, 0, 0)
d(interstitial F : C ) = - 2.31 kHz; (2.97)
4v
and (b) for the nearest axial fluorines in CaF2: Ho3+ C3v centre which has r = (2.3536 Ä, 0, 0) d(nearest axial F : C ) = - 3.56 kHz. (2.98)
Interpretation of
anand
a^: Following Ranon and Hyde (1966), by defining two new parametersas = j {a(| + 2 a ^ } , (2.99 a)
and
ap = 3" { a„ ' ai 1 ’ (2" b)
the electronic superhyperfme interaction between a Ho3+ and an axial F' ions can be written in the form
H k ^ (Ho - axial F ) = a J . F + a { 3J F - J » F }. (2.100)
The first term, asJ • ? , is an isotropic term, whereas the second term, ap{ 3JZFZ - } , has the form of a point dipolar interaction. However, as pointed out by Ranon and Hyde (1966), it is incorrect to make the physical interpretation that the Hamiltonian in Eqn.(2.100) is the sum of an isotropic interaction and a point dipolar interaction. As has been mentioned earlier, there are contributions to the electronic superhyperfme tensor (a) from both dipolar interaction and from overlap and covalency. To calculate the overlap parameters and to estimate the covalent contribution to a, precise knowledge of the Ho3+ wavefunctions and the F" displacements are required (Baker 1974). The calculation is further complicated by the fact that not only the Ho3+ 4f-orbitals, but also the outer 5s- and 5p-orbitals may contribute substantially to the bonding. The ls-, 2s-, and 2p - orbitals of fluorine ligands may take part in the bonding (the electronic configuration of F' is Is 2s 2p ). The fluorine s- and p-wavefunctions may be admixed either
through direct overlap and covalency between them and Ho3+ 4f-wavefunctions, or through overlap and covalent interaction with the outer 5s- and 5p-shells of the Ho3+ ion. The latter are subject to core polarization, due to exchange interaction with the 4f electrons, which has a
tendency to produce a relatively large covalency contribution (Baker et al 1968, Baker 1974).
The bonding through the fluorine s-orbitals is given by as; while, in addition to the dipolar interaction, ap should also account for the bonding through the other orbitals (Ranon and Hyde
The pure dipolar contribution adip, on the other hand, can be easily calculated from the known g-value and crystal structure. The electronic magnetic moment of the Ho3+ ion can be expressed as
FT (Ho) = - g (Ho)u
J
= - - t h i , (2.101)e e B Ho
where (a) ge(Ho) is the electronic Lande factor, (b)
4 , = ^ ge(Ho> (2102)
is the electronic analogy of the nuclear magnetogyric ratio, and (c) p B is the Bohr magneton (9.2740154 x 10'24 J T"1). The expression for adip is, therefore,
y H3o F ( 3c o s2 0 - 1 ) ' = h ' 2 4 4tt h ye y Ho, F ( 3 c o s 2e - l ) >* . ( 2 . 1 0 3 ) l r J [ 2tcr J
For the ground 5I8 multiplet, ge = 1.2346 (Mujaji 1992), and hence y^o = 1.08572 x 105 The value of adip is estimated to be 4.569 MHz for the interstitial F ion in CaF2: Ho3+ C4v centre, and 7.035 MHz for the two nearest axial F ions in CaF2: Ho3+ C3v centre.
The superhyperfine interactions in several rare-earth doped fluoride crystals have been studied mostly by the electron-nuclear double resonance (ENDOR) technique (see for example the reviews by Anderson 1974, Baker 1974, and Bleaney 1988), and more recently by the optically-detected magnetic resonance (ODMR) technique (Burum et al 1982, Manson et al 1992a, Martin et al 1992, Martin et al 1993a, Mujaji et al 1993). In this thesis the results of the studies of superhyperfine interactions in the two high symmetry centres, C4v and C3v, of Ho3+ in CaF2 by the ODMR technique will be reported.