The e. p. r. and optical absorption spectra of the self-trapped hole (V
k
) centres in CaF
2
, SrF
2
and BaF
2
have been measured and it is shown that the molecular axes of these centres are alined along <100>. Preferential alinement of the V
k
centres along a particular cube edge is readily produced by bleaching into the optical absorption bands of the V
k
centre at 77 K with linearly polarized light. The anisotropy created in the crystals by preferential alinement is detected in both the e.p.r. and optical spectra. The thermally induced decay of the optically alined V
k
centres is monitored by e. p. r. methods and it is shown that, in CaF
2
, linear (0°) motion of the V
k
centres predominates over the entire decay range (85 to 138 K). In SrF
2
linear motion predominates in the early stages of decay (
ca
. 110 K) but random (90°) motion sets in at higher temperatures (
ca
. 120 K). In BaF
2
only random motion is observed. The kinetics of reorientation and decay of the V
k
centre are investigated. A band observed in the X -ray fluorescence of CaF
2
(at 279 nm), SrF
2
(at 300 nm) and BaF
2
(at 310 nm) is assigned to emission of the self-trapped exciton by comparison with a similar band in the recombination luminescence of the V
k
centre (due to e
-
+ V
k
). The temperature dependence of the X -ray fluorescence intensity is described. The polarization properties of the recombination luminescence are investigated on crystals containing optically alined V
k
centres. The e. p. r. and optical absorption spectra of trapped hole centres involving other lattice defects are also described. One such centre, with its molecular axis close to <111>, involves an interstitial fluorine ion and is named the V
H
centre. Other centres with molecular axes close to <100> appear to be stabilized by impurity ions and are labelled V
KA
and V
KA'
centres. The V
H
centre occurs in CaF
2
, SrF
2
and BaF
2
, the V
KA
centre in SrF
2
and BaF
2
, and the V
KA’
centre only in BaF
2
. The kinetics of reorientation and decay of the V
KA
centre have been investigated. We give a general discussion of the properties of hole centres, including the
H
centre, in alkaline earth fluorides.
Fast optical absorption spectra calculations for periodic solid state systems