Electronic structure and transition probabilities in pure

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Electronic structure and transition probabilities in pure and Ce BaF2, an explorative study a

a

3+

J. Andriessen , P. Dorenbos & C.W.E. Van Eijk

doped

a

a

Radiation Technology Group, Department of Applied Physics, Delft University of Technology, Delft, 2629 JB, The Netherlands Published online: 23 Aug 2006.

To cite this article: J. Andriessen , P. Dorenbos & C.W.E. Van Eijk (1991): Electronic structure 3+

and transition probabilities in pure and Ce doped BaF2, an explorative study, Molecular Physics: An International Journal at the Interface Between Chemistry and Physics, 74:3, 535-546 To link to this article: http://dx.doi.org/10.1080/00268979100102401

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MOLECULAR PHYSICS, 1991, VOL. 74, No. 3, 535-546

Electronic structure and transition probabilities in pure and Ce 3+ doped BaFz, an explorative study By J. A N D R I E S S E N , P. D O R E N B O S , and C. W. E. V A N E I J K Radiation Technology Group, Department of Applied Physics, Delft University of Technology, Delft, 2629 JB, The Netherlands

Downloaded by [Bibliotheek TU Delft] at 08:16 27 May 2013

(Received 1 February 1991; revised version accepted 13 June 1991) The energy levels responsible for the cross luminescence in BaF 2 are well reproduced with the LCAO cluster approach using Gaussian basis sets. Comparative calculations on CaF 2 and SrF2 are, even if the estimated bandwidths are considered, in good agreement with band structure calculations reported in the literature. We will explain quantitatively the appearance of two cross luminescence emission components in BaF2 with a decay time of roughly 1 ns. The work on Ce 3+ has shown a substantial reduction of the 5d-4f transition rate due to shielding by the 5p shell. This may be partly responsible for the reported small transition rates measured for this ion in various hosts.

1. Introduction Pure BaF2 crystals are widely applied as scintillation crystals because of the very fast luminescence at 220nm and 195nm. These luminescences with a decay time of about 800 ps, are caused by cross transitions of an electron from the 2p F - valence band to a hole in the Ba 2+ core band [1]. Recently, we studied the intense luminescence light resulting from x-ray irradiation of Ce 3+ doped BaFz [2]. In order to understand the processes which play a role in these interesting phenomena, we have started a quantitative study of the electronic structure and transition probabilities in pure and Ce 3§ doped BaF2. We have studied the cross transitions and the 5d-4f transition of Ce 3+ in these crystals. The positions of the bands or energy levels of the system are very important, and part of this paper is devoted to this. Because of the localized character of the transitions of interest, we have chosen for a cluster approach [3] to calculate the electronic structure of the crystal. In this approach, a cluster of ions is taken to represent the local environment, and the rest of the crystal is represented by point charges. Molecular orbitals (MO) in the form of a linear combination of atomic orbitals (LCAO) are calculated with the H a r t r e e - F o c k method (fully consistent). This provides [4] detailed and reliable information about the electronic structure of localized centres. It is usually superior to the familiar band structure approach, in which severe approximations are made concerning consistency and the handling of the multicentre interactions. The cluster approach, however, requires much computer time.

2. Theoretical framework In our approach, the MeF2 crystal (Me = Ca, Sr or Ba) consists of a relatively small atomic cluster and a set of surrounding point charges to represent the rest of the crystal. The point charges are chosen in such a way that at each of the cluster sites 0026 8976/91 $3.00 9 1991 Taylor&Francis Ltd

536

J. Andriessen et al.

A


I

J

@ Ba Figure 1. The basic ionic clusters. the correct Madelung potential is obtained if we replace cluster atoms by point charges. The point charges within a radius R0 from the central atom of the cluster, typically around three times the lattice constant, are kept to the values of the ions they replace; + 2 for Me and - 1 for F. Then two external spherical shells are defined with thickness AR0, in which the value of the charges is adjusted in order to have the right Madelung potential at the fluorine and metal sites in the cluster [3]. The Madelung potential energy (Hartree) was calculated from [5] EM(Me 2+) = EM(F-)

=

-7"566/d 4.071/d,

(1)

with d the lattice constant in atomic units (Au). In our study of pure and Ce 3+ doped fluorite crystals, we have used four different clusters for the calculation of a number of ground and excited state properties. They are shown in figure 1. Cluster types A and B were used for estimating the bandwidth of the fluorine 2p and metal np level. The two nn fluorine ions in BaF2 are not equivalent, see cluster A, which plays an important role in the origin of the 2p band width. We will return to this point later. Cluster types C1 and C2 were primarily used for studying local one-ionic properties such as excitations of Ba 2+ and Ce 3+, but also for the estimates of bandwidths. Results of C1 related to those of B provide information about the Me np bandwidth. C2 gives information about the fluorine bandwidth and was used for the study of cross luminescence in Ba 2+ and 5d-4f transitions in Ce 3+ ions. 2.1. Bandwidths Experiments performed by Poole et al. [5] show that the fluorine 2p bandwidth is mainly determined by the lattice constant d and not much by the character of the Me 2+ ion. So, a cluster with point charges to represent all Me 2+ ions would be sufficient to study the fluorine bandwidth. We have found that this is valid for CaF2 and SrF2, but not so much for BaF 2 where the interaction with the Ba 5p orbitals is

537

Cross luminescence in B a F 2 Ba

'~ Ba


Figure 2. Source of fluorine 2p bandwidth in a fluorite crystal. also of great importance. It is generally assumed [6] that the main source of the broadening of the 2p levels is the overlap of the two nearest neighbour fluorine 2p wavefunctions in the region between two Me 2+ ions. This is illustrated for cluster type A in figure 2. Because the overlap has a one-dimensional structure, we have used a simple rule for the extrapolation of bandwidth calculated for cluster type A to that of B, and further to the infinitely large cluster. The overlap gives rise to bonding and antibonding levels with energies characterized by el = ~0 + [G]

(2a)

~2 = e 0 - 1 G ] ,

(2b)

where G = J ~9p,(r)Op2(r) VM(r)d?.

(3)

e0 is the orbital energy of the fluorine 2p orbitals without overlap. VMis the coulomb interaction with the two metal ions. ~p~ and ~9p2are the overlapping 2p orbitals of the two fluorine ions. We have neglected here the small orbital mixing of ~ components. If we extend the cluster to infinite size we obtain e,~ =

(4a)

~o + 21G[

e2 = e0 - 21G[.

(4b)

For cluster type B, with just three fluorine ions on a line and crossing two Me 2+ planes, we get a factor x/2 instead of 1 in (2 a) and (2 b). This means that the cluster calculation will give levels which can be used to estimate the band structure in a tight binding approximation [7]. In this way, an estimate is found for the bandwidth at the X point in the first Brillouin zone. 2.2. Basic equations Given a certain cluster of atoms and a set of point charges to represent the surrounding lattice, our calculations are based on the following Hamiltonian. =

9

+ k

rik

Z

,

(5)

j > i

where i and j number the electrons and k the point charges which, if they belong to the cluster, are real nuclear charges and outside the cluster they are the effective point charges representing the lattice. The Hamiltonian of (5) is part of an H F type [8] calculation which was used for obtaining the ground and excited states of the

538


molecular cluster. For this calculation we have used a commercial package: the Gaussian system of programs [9], published by Carnegie-Mellon University. It is installed on the VAX 8350 and the Convex 240 computers of the Delft University of Technology. This program uses as basic form for all atomic orbitals the Gaussian type: ~b,m(f) =

~ c , r e ' xi tv, q~),

(6)


i

with Y;~ normalized spherical harmonics. For the basis sets required for the cluster atoms we have used the tables of Huzinaga [10] mostly. However, in some cases, where we have reason to believe that the wavefunctions given in the tables are not good enough or are not available, we have used our own procedure. This procedure consisted of calculating the wavefunctions with the relativistic M C D F program of J.P. Desclaux [11] and fitting the result to a sum of Gaussians, see (6), with the program ANGEL written by W.J.C. Okx [12]. This applies particularly to the basis set of Ce 3§ The (split) basis set of Ce ~ from [10] was taken. However, comparing the results with those of the numerical code [11] it was found that the 4f orbitals were seriously different due to relativistic effects. The other shell orbitals were not very different. Because of the fact that the 4f orbitals are not greatly affected by the surrounding ions we used the standard practice of correcting the 4f orbital energies in order to handle rare-earth ions in a non-relativistic framework. The 5d orbital, required for the 5d excited state of Ce 3+, was obtained by fitting the numerical M C D F wavefunction [11] to the Gaussian form. For ground state calculations we have used regular Hartree-Fock. This is not useful for excited states and, as published earlier [13], it is much better to use the general valence bond option (GVB) with zero pairs. In this way, one can perform a restricted open shell calculation, which can be forced to converge to any excited state by the orbital mix option. The calculation of transition probabilities was done using the following equation given by Green and Jennison [14] in their work on radiative and Auger transitions in NaF. r x ~w

=

4/3(E~f) 3~3(XrlPIWi)2.

(7)

The expression is in atomic units (e = h = m = 1), ~ is the fine structure constant (1/137.037). F has to be multiplied by 4.134 x 10 ~6to obtain the transition rate per second, i denotes the initial and f the final state. Eir i s the corresponding energy difference. P is the n electron dipole operator:

P = Z 4.

(8)

J

The sum j is over all electrons in the cluster. To evaluate (7), the final state X as well as the initial state W are required. All single electron orbitals of W are in principle different from the corresponding ones of X, although the difference is small. In practice it is therefore best to calculate just one state. From the work of Green and Jennison it was concluded that it is best to take as reference state the initial (excited) state; errors are around 15% in the transition probability. So, for the cross-luminescence, we take the Ba 3+ (5p) 5 state and for the Ce 3+ luminescence we take the Ce 3+ (5d) I state.

Cross luminescence in BaF2

539

The emission of light polarizes the lattice. It is therefore necessary to add a factor f i n (7), which usually has the form [15]


f

=

(n2 + 2)2n/9,

(9)

where n is the refractive index of the crystal. This expression is, however, derived on speculative arguments. Particularly the Lorentz-Lorentz correction, (n 2 --[- 2)2/9, for the local electric field is over estimated, which is obvious from many experimental observations [16]. Furthermore, the polarization of the shells of the ion itself is in general not taken into account. We will consider these effects in a following paper. For the moment, we take the factorfjust equal to n, in order to take into account at least the modification of the photon density of states by the refractive index. The energy difference Eir occurring in (7) can, according to Koopmans theorem, be estimated from the difference in energy of the corresponding orbitals of the ground state. However, the ionization energies of negative ions are generally not very well approximated by the orbital energies found from the Hartree-Fock states. This is due to correlation. One calculates, for instance, for a free F ion an orbital binding energy of 4.78eV. This has to be compared with the measured electron affinity [17] of 3.448 eV. It is, therefore, necessary to correct the calculated 2p(F) energy with 1.3 eV in order to take correlation into account. For positive ions this effect is smaller and will be neglected. 3.

Results

We have performed several calculations on the BaF2 crystal. Also, calculations were made on CaF 2 and SrF2, in order to compare our calculations with the literature [6, 18, t 9] and to prove the correctness of our approach. We can then test if the trend found experimentally for the energy levels could be reproduced. Our main interests are the excited states related to fast scintillation phenomena. However, one cannot hope to do a realistic excited state calculation without having a good ground state, with enough virtual space to generate the excited states of interest. So first we handle the ground state calculations and compare the result with the literature. 3.1. Ground state calculations for CaF2, SrF2 and BaF2 It is straightforward to perform cluster calculations as described above. There are no convergence problems, because of the high symmetry and the strongly ionic character of the bonds. It is interesting to see how the cluster size influences the energy levels. Therefore, we did a number of calculations on CaF2, which has the smallest basis set. We have used all cluster types as shown in figure 1. The results are summarized in table 1. If there are a number of levels of the same orbital type, the energy of the upper and lower ones are given to provide an estimate for the bandwidth. Table 1 shows also results reported by Albert et al. [6] and by Starostin et al. [18]. There is good overall agreement with the work of Albert et al. However, substantial lack of agreement is found with the work of Starostin et al., particularly if the bandwidth is considered. An important aspect of table 1 is that all cluster types give almost the same average energy for the levels in a band. The 2s(F) and 2p(F) bandwidths depend on the cluster size, as expected. The ratio of the 2p bandwidths for clusters A and B (in the preceding section predicted to be ~/2) is found to be 1.47; the small difference is presumably due to neglect of the ~ components. Calculated

540

J. A n d r i e s s e n et al.

Table 1.

Calculated upper and lower energy levels (eV) in several bands of CaF2. Calculations were performed for all cluster types. (1 eV ~ 1-60218 x 10-19J)


Cluster type

Reference

Level

A

B

C1

C2

[6]~

[18] ~

2p(F-)

- 14'15 - 15'18

- 13'88 - 15'40

- 14.12 --

- 14'04 - 15"92

- 14.04 - 1676

-9'77 - 1853

2s(F )

- 39'05 - 39'24

- 3894 - 3940

- 38'78 --

- 39"07 - 39"59

- 39-40 --

- 36'84 - 41 0 4

3p(Ca 2+)

-31.28 -31.37

-31.32 -31.56

-31.02 -31.21

-32-16

-32.19 --

-35.45 -41.90

3s(Ca 2+)

- 55.78

- 56.03

- 55.62

- 56.68

--

- 56.33

0.79

0.73

0"73

0-57

Lowest virtual

2-99

1.28

Albert et al. b Starostin et al.

b a n d e n e r g i e s o f the t h r e e systems C a F 2 , SrF2 a n d B a F 2 f o r cluster t y p e B are listed in t a b l e 2. T h e results also s h o w s u b s t a n t i a l l a c k o f a g r e e m e n t w i t h the results d e r i v e d for x - r a y ( X P S ) a n d u l t r a v i o l e t ( U P S ) p h o t o e l e c t r o n s p e c t r a by P o o l e et al. [5]. I f the e x p e r i m e n t a l v a l u e s a r e c o r r e c t e d for the p o l a r i z a t i o n o f the lattice [5], w h i c h has b e e n c a l c u l a t e d by S t a r o s t i n et al. [18, 19], the e n e r g y o f the 2 p ( F ) b a n d b e c o m e s m u c h c l o s e r to the c a l c u l a t e d v a l u e . H o w e v e r , the m e t a l p b a n d s d o not. O u r b e l i e f is t h a t the p o l a r i z a t i o n c o r r e c t i o n was a p p l i e d in a t o o ad hoc m a n n e r , a n d is p r o b a b l y o v e r e s t i m a t e d . It m a y be t h a t t h e p o l a r i z a t i o n is m u c h s m a l l e r b e c a u s e o f screening. I n o u r view, t h e c a l c u l a t e d e n e r g y o f the 2 p ( F ) level has to be c o r r e c t e d by 1.3 eV b e c a u s e o f c o r r e l a t i o n effects. T h e difference f r o m the m e a s u r e d e n e r g i e s is t h e n r e d u c e d by r o u g h l y a f a c t o r o f t w o . T h e r e m a i n i n g difference m a y be d u e to p o l a r i z a t i o n effects.

Table 2.

Results of cluster calculations with cluster type B of C a F 2 , S r F 2 a n d B a F 2 . The lattice parameter d is in A and the energy in eV. Crystal CaF2 5.451

SrF2 5.773

BaF~ 6.189

d

Calc.

Exp.

Calc.

Exp.

Calc.

Exp.

e2pa(F) e,p(Me) A2pb(F)

- 14"6 -31.4 2.3

- 12"3 -29.9 3.0

- 14"0 -24.2 1.5

- 11"6 -23.9 2.5

- 13-3 - 18-7 0.8

- 11'0 - 18.2 2.0

Lowest virtual

0'7

--

0.4

--

0.3

~The experimental values are found from the maxima of the spectral lines given in [5] and are not corrected for the polarization of the lattice. bThe bandwidth is not easy to find from the spectra of [5] because of limited resolution. We have estimated the full width at half maximum intensity.

Cross luminescence in BaF2 Table 3.

Properties of the radiative cross-transitions in BaF2 calculated with the cluster approach.

Orb. type

e/eV

S/(AU) 2

Eif/eV

F/10 9s- l

Bonding e(g) t2(g) ai (g)

- 19-33 - 19-13 - 19.06

0.0605 1.787 1-413

5.10 5-78 5-58

0.012 0-426 0.484

- 18-60 - 18.44

1.038 0.312

6.41 6.84

0.428 0.156 1'507

Antibonding t2(g) tl(g) Total


541

We will not apply the correlation correction in our listed band energy results, however, because there is no way then to compare with other band structure calculations and also because this effect needs more study. For the metal np levels, the case of Ca 2+ is amazing. The measured 3p 'hole' energy, 29.9eV, is about 1.5eV less then the theoretical one. What is more, if one considers the energy of the fluorine 2s level and the metal 3s level the difference is much larger. Poole et al. measured binding energies which are 7 eV for 2s and 8 eV for 3 s smaller than our values, and those given by other authors [6, 18]. So it is open to discussion if the XPS spectra are well interpreted. 3.2. Cross' luminescence in BaF2 We have studied the excited state of the (Ba 3+ Fs) 5- cluster, where an electron is missing from the Ba 2+ (5p) level, in order to learn more about the quantitative aspects of the cross luminescence [1]. The excited state of the (BaFs) 5- cluster was generated with the same basis set as employed for the ground state calculation of the (BaF8) 6cluster. In order to study in detail the transition rate, we did not average over all the initial states but calculated each contribution to the 2p(F) ~ 5p(Ba) transition separately. There are 8 allowed transitions, resulting in five lines of either al (g), e(g), tl (g), or t2(g) type fluorine MO's. The lines fall into two distinct groups, which we have given the labels bonding and antibonding, in order to show their origin. The results of the cluster calculation are listed in table 3. The first column shows the 2p(F ) orbital type which belongs to the irreducible representations of the cluster group Oh. The corresponding orbital energy is shown in the second column. The absolute value is not very useful, because of a shift downward (cf. table 2) of roughly 6 eV coming from the hole in the 5p(Ba) shell. The difference in energy between the lines depends on the cluster size and so we will use the orbital energy as an adjustable parameter in comparing with experiment. The transition element S = ( X I P ] W ) 2 of equation (7) is shown in the third column. Figure 3 shows the room temperature luminescence spectrum of pure BaF2 which was recorded during x-ray irradiation. The spectrum has been corrected for the quantum efficiency of the employed photomultiplier tube; see [2] for more experimental details. The emission peaks at 5.6 eV and 6.4 eV and the shoulder peak near 6.9 eV are caused by crosstransitions. The intense luminescence peak at 4 eV is of different origin. The solid curve shows a fit of the calculated results in table 3 to the cross luminescence spectrum. In this fit, we assumed that each of the five transitions gives a Gaussian contribution with a width of 0.43 eV (FWHM). The position of each

542



aa

A ~A&A 9A s'.8o

u'.~o

~',so

5 oo

5 ~o

5 oo

s zo

s so

7 oo

7~ lo

7;oo

Energy/eV

Figure 3. x-Ray induced luminescence in BaF2 at room temperature. Luminesence intensity (arbitrary units) is plotted versus the photon energy. The calculated transition lines (solid in the figure) were artificially broadened and shifted by a small amount. The line marked with 9 is the experimental curve. contribution is given by the fitted Ei~ values in the fourth column of table 3. The intensity of each contribution is proportional to the transition element listed in the third column of table 3 and calculated with (7) and using n = 1.47 for the index of refraction. We can identify the two main cross luminescence peaks near 5.6 eV and 6'4 eV as cross transitions from the bonding and antibonding 2p orbitals, respectively. Most remarkable about the fitted curve are the correct intensity ratios of the two main peaks and the shoulder peak at 6-9 eV. The resulting average energy (main peaks)/~f is found to be 6.2 eV, which is very near the first principal value 6.7 eV read from table 2, after a correlation correction of 1.3 eV has been taken into account. The total transition rate corresponds with a luminescence decay time of about 0-7 ns, which compares very well with the 0.8 ns found experimentally. This, unfortunately, does not imply that we have explained the cross transition quantitatively enough; several effects were not accounted for. Apart from the energy adjustment mentioned earlier, we did not consider the fact that the BaZ+-F distance in the Ba 3+ F 8 cluster will be smaller than in pure BaF2. Furthermore, polarization of the surrounding ions was neglected. The polarization effect is expected to be small for transitions which extend over the entire BaF 2 cluster [15]; the exact magnitude, however, is not known, and so one has to consider the resulting rate as a rough but apparently a good estimate. 3.3. Luminescence of Ce 3+ in BaF~ The mechanisms which play a role in the observed luminescence of Ce 3+ ions in BaF2 during x-ray irradiation are rather complex. One should consider both the energy migration to the Ce 3+ ion and the 5d-4f transition rate. In this work, we have studied the levels and the 5d-4f transition rate of an isolated Ce 3+ F8 cluster with interionic distances equal to those in pure BaF2. We first calculated the 4f j ground

Cross luminescence in BaF2 Table 4.

543

Comparison of some orbital energies (eV) and radial elements (4flrl5d) (•) for the free Ce3+ ion calculated with the Gaussian package and with MCDF. e4f

~sd

eso~"

~5o2

(4flrf5d)av

Gaussian

-42.5

- 30.4

- 51-4

- 56.3

0.337

MCDF b

- 37.9 - 37.5

- 29.9 - 29.6

- 53"9 - 50.6

- 57.4 - 54.4

0-433


aThe 4fI ground state and the 5d ~excited state have different 5p orbitals, denoted by 5p~ and 5p2, respectively. bTwo orbitals energies are given corresponding with j = l _+ 1/2. state and the 5d j excited state of C e 3+ in the cluster. Subsequently, an extra F ion was added to the cluster at the nearest interstitial position in the 111 direction from the central Ce 3+ ion in order to see the influence of charge compensation. Also the C e 3 + - F distance was decreased by 5%, in this way obtaining information about level shifts caused by lattice relaxation. The 4f wavefunctions from Huzinaga's tables were not found to be very useful for calculating the 4f level. Fully relativistic wavefunctions are needed for our purpose. We therefore performed a calculation of the free Ce 3+ ion using the Desclaux code [11]. The 5d state was also calculated in this way. Results of this preparative work are listed in table 4. From the calculation we obtained a 5d wavefunction and a correction of 4.6eV for the 4f orbital energy. Unfortunately, the 4f wavefunction obtained from M C D F cannot be used in the Gaussian package. An important point is the influence of rearrangement in the 5p shell after exciting the 4f electron to the 5d orbital. The difference in total energy of the Ce 3+ ion in the ground and excited states, which from table 4 seems to be around 8 eV, is found to be about 5.4 eV, which is in better agreement with the experimental value of 4.35 eV [20]. The transition element (41qrlSd) calculated with the Gaussian basis is useless. The M C D F value of 0-433 is in good agreement with the value 0.441 quoted by Williams et al. [16]. It has, however, to be corrected for shielding effects in the Ce 3§ core. We did a rough estimate of this effect in the 5p shell and found in lowest order of polarization a reduction of the rad;al element of 50%! This important effect seems not to have been considered by Williams et al. in their attempt to explain the small measured oscillator strength of Ce 3§ in various host crystals. More detailed work, however, has to be done to treat this effect more accurately. Important results of the cluster calculations mentioned at the beginning of this section are listed in table 5 in which the orbital energy o f 4 f is corrected with 4,6 eV. The first column of table 5 places the 4f level just below the fluorine 2p band. However, the results are calculated with a Ceqv distance equal to the B a - F distance in the pure BaF2 crystal; the actual C e - F distance in BaF2 is smaller. Assuming a 5 % smaller distance, a shift of the Ce 3+ levels by 3 eV is calculated. This shift brings the 4f level above the fluorine 2p band, which is not shifted much by the smaller C e - F distance. Columns 3 and 4 in table 5 show that charge compensation by an interstitial F - ion shifts both the Ce and F levels with some 3 eV. From this we have estimated that the 4f position, relative to the conduction band, is - 1 3 eV. For 5d we find, similarly, - 6 eV and, for the fluorine 2p band centre, - 15.5 eV. The large value of the 4f-5d energy difference, 7 eV, is comparable with the value of 8 eV found for the free ion (see table 4) with the M C D F approach. So it seems best

544 Table 5.

J. Andriessen et al. Calculated energy levels (eV) of Ce 3+ in BaF2, related to the 5d-4f luminescence. Cluster type" Level

CeF8 (4f)

4f(Ce) 5p 5d 2p(F) A2p Lowest virtual

- 20.1 - 34.8 - 19.3 1.09 - 1.44

CeF8 (5d)

CeF 9(5d)

- 37.5 - 11.7 - 19.1 1.09 - 2.77

- 34.8 - 8.98 - 15.5 3.57 0' 10


aThe state of the Ce 3§ ion in the cluster is denoted by (4f) for the ground state and by (5d) for the excited state.

to work with the Ce3+(5d) ion only and place the 4f level about 4eV below the 5d level. Furthermore we used the free ion 4f wavefunction obtained with the M C D F method and the 5d wavefunction for the free ion as well as that of the cluster calculation in order to see if the radial element (5dlrl4f) changed much in going from the free ion to the ion doped in BaF2. We found a difference of only 5%. With this result, arguments given by Williams et al. [16] and Krupke [21] for explaining the small measured oscillator strengths for absorption in various host crystals, are in serious doubt.

4.

Discussion and conclusion

The ab initio calculation of the electronic structure of the fluorite crystal using the cluster approach with Gaussian basis sets gives good estimates for the properties and energy levels of both the anions and cations. In particular, the relative position of these levels, which is the most important quantity for our work, is almost independent of the cluster size. It is important to correct the energy of the fluorine levels by roughly 1.3 eV in order to take correlation into account. It is gratifying how straightforwardly the method can be applied to excited states. We have established a quantitative understanding of the two emission components in the cross luminescnece emission spectrum of pure BaF2. The calculations on the luminescence of Ce 3+ doped BaF2 show that the method has enough flexibility to handle this luminescence satisfactorily. The calculated energy of the 4f level of Ce 3+ has to be corrected by 4.6 eV, because of relativistic effects. The calculated small bandwidths, as compared with experiment, are not a result of the cluster approach. The band structure calculation of Albert et al. [6] on CaF2 and other calculations quoted in that reference also yield small widths for the valence bands. This, however disagrees with the results of the detailed study by Starostin et al. [18] on CaF2, SrF2, and BaF2, where for CaF2 a 2p(F) bandwidth of 8 eV was found! It is not possible to find an explanation for the disagreement, because information about the (tight binding) basis set and the approximations made is not available. It is quite common to argue that the reason for a large bandwidth is the neglect of three-centre interaction integrals in the calculations. Because we, in contrast to Starostin et al., do not make such an approximation, all multicentre integrals are correctly evaluated, this may explain our agreement with Alberti et al., who also evaluated them correctly. We made a rough estimate of the influence of the basis set on the bandwidth by adding new components to the fluorine basis set.


Cross luminescence in BaF2

545

However, we found no sizeable effect except in the case where we worked with unrealistic 'blown up' ions. So, no explanation for the small bandwidths can be given. Our ultimate aim is to understand and predict optical transitions related to fast scintillation phenomena. Our calculation of the crosstransition in BaF2 is similar to the work done by Green and Jennison [13, 14] on radiative transitions and Auger effects in NaF. Actually, for some transitions we repeated their work to see if our basis for the fluorine ions yielded comparable results. It might appear that we understand the transition well enough, but this in our view is not the case. The point is that the physics of the very creation of the light at the active centre is not understood. The good agreement Green and Jennison found for their satellite intensity is a relative one; important physical effects, such as polarization, cancel. This polarization, particularly of the local environment and the ion itself, is mostly not accounted for. From the result given by Williams et al. [16] this seems to be disastrous; ratios of three or more between measured and calculated oscillator strengths were found. The detailed mechanisms which are responsible for the excitation of the Ce 3+ ion to a 5d state during x-ray irradiation are not known yet. This is one of the subjects o f our current theoretical effort. We conclude this paper by a short preview of calculations we have in mind to clarify more thoroughly the processes responsible for the scintillation in BaF2 and other crystals of interest. In order to explain the measured characteristics of the scintillation light, one of us has formulated a number of possible mechanisms for the transfer of the electron-hole pair energy created by x-ray irradiation to the Ce 3+ centre. These mechanisms need quantities which can sometimes be found experimentally but in many cases must be calculated. Further, the calculation of the transition time, so crucial for finding fast scintillators, must be refined by the inclusion of polarization at the ionic level. An important challenge for this work is provided by the confusing experimental facts tabulated by Williams et al. [16], where the absorption oscillator strength depends strongly on ionic distance. If we know these mechanisms in detail, it will serve as a guide to obtain new fast scintillation crystals.

[t] [2] [3] [4] [5] [6] [7] [8] [9]

[10] [11] [12]

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