FORMATION OF LATTICE DEFECTS BY IONIZING RADIATION IN ALKALI HALIDES

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FORMATION OF LATTICE DEFECTS BY IONIZING

RADIATION IN ALKALI HALIDES

N. Itoh

To cite this version:

N. Itoh. FORMATION OF LATTICE DEFECTS BY IONIZING RADIATION IN

ALKALI HALIDES. Journal de Physique Colloques, 1976, 37 (C7), pp.C7-27-C7-37.

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1. Introduction. — Much attention has been paid decay, which leads to Frenkel pair formation are

to the formation of lattice defects in alkali halides in complementary to each other. Recently it has been recent years and several review papers have been suggested [9-11 ] that there is a path in the non-adia-published [1-5]. The particularly interesting aspect of batic potential energy surface which leads from the defect formation in alkali halides is that energy pos- X^" molecular ion, which constitutes the self trapped sessed by the electronic system is converted to energy exciton, to a situation in which there is energetic for the mass transport of the constituent ions to form Frenkel pair separated along a < 110 > direction, a Frenkel pair. Although various suggestions have been In these models the precursor of Frenkel pairs is made concerning to the mechanism of this process, taken to be a different electronic excited state of the none of them are either satisfactory to account for all self trapped exciton from that which emits lumines-the experimental data or to describe every detailed cence.

physical process involved in the de-excitation process. Another problem underlying defect formation pro-It is established, however, that single ionization or the cesses is the identification of the defects produced, formation of an exciton or an electron-hole pair causes it is now clear that a neutral Frenkel pair composed the Frenkel pair formation [2], substantially on the of an F center and an H center is the primary pro-basis of the discussion of the production yield. The duct [12]. At liquid helium temperature ionized earlier suggestions [6] that a Frenkel pair is produced Frenkel pairs composed of an a center and an I center at a site where a halogen is doubly ionized cannot are created as well as neutral Frenkel pairs. Little account for all the Frenkel defects created, even though very direct verification of the structure of the intersti-this process may indeed be operative. The salient pro- tial center produced above liquid nitrogen tempera-blem at present on the production mechanism is to tures, where the H center is unstable, has been obtain-make clear the particular excited state which is res- ed. It is, however, generally accepted [13] that a ponsible for the Frenkel pair production. Frenkel pair formed at this temperature region is Pooley [7] and Hersh [8] have first suggested a composed of the F center and a center involving two model in which a self trapped exciton, an electron interstitial atoms. The stabilization of the interstitials trapped by an X J molecular ion (X denotes a halo- by monovalent or divalent impurities is also of impor-gen), is the precursor of a Frenkel pair. They suggested tance between liquid nitrogen and dry ice temperatures, that the radiative recombination and non-radiative At still higher temperatures, the vacancies bxome

FORMATION OF LATTICE DEFECTS

BY IONIZING RADIATION IN ALKALI HALIDES

N. 1TOH

Department of Nuclear Engineering, Nagoya University, Nagoya, Japan

Résumé. — Les travaux en cours sur la formation de défauts par des rayonnements ionisants

dans les halogénures alcalins sont passés en revue. Nous discutons tout d'abord Pexciton auto-piégé (Vk e), qui mène à la formation de défauts de Frenkel. Plusieurs observations expérimentales sont présentées, qui indiquent que les (Vk e) producteurs d'émissions a et n ne sont pas les précurseurs des paires de Frenkel. Le second problème est le mouvement dynamique de l'atome d'halogène intersti-tiel possédant une énergie cinétique conférée par la recombinaison non rayonnante des (Vk e). Les résultats expérimentaux sur la dépendance de la portée du mouvement dynamique dans différentes halogénures alcalins par rapport à la température sont passés en revue, et la corrélation de cette dépendence avec celle du taux de production des défauts par rapport à la température est discutée. Enfin, nous discutons le mécanisme de la stabilisation interstitielle et ses effets sur la cinétique de la production des défauts.

Abstract. — Current works on the defect formation by ionizing radiation in alkali halides are

reviewed. In the first place discussion is made on the self-trapped exciton (Vk e) which leads to the formation of the Frenkel defects. Several expérimental observations are presented which indicate that (Vk e) producing a and ^-émissions are not the precursors of the Frenkel pairs. The second problem is on the dynamic motion of the interstitial halogen atom possessing the kinetic energy imparted through the non-radiative recombination of (Vk e). Expérimental results of the tempéra-ture dépendence of the range of the dynamic motion in various alkali halides are surveyed and its corrélation with the température dépendence of the production yields of defects is discussed. Finally the mechanism of the interstitial stabilization and its effects on the kinetics of defect production is discussed.

JOURNAL DE PHYSIQUE Colloque C7, supplément au n° 12, Tome 37, Décembre 1976, page C7-27

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mobile and the aggregated vacancies are also produced.

In view of this relatively clear understanding of the structure of the defects in each temperature region, the kinetics of the defect formation processes have been proposed [14-201. It is of importance, for the radiation damage studies in general, to understand how the microscopic features of the radiation damage processes are correlated with their macroscopic kinetics. It is hoped that the alkali halides can act as a prototype material for this type of studies.

This review emphasizes current developments in the studies of defect formation in alkali halides since the other recent reviews have been published [l-51. The precursors of the Frenkel pairs and their de- excitation processes are discussed first. The motion of interstitials in the dynamic defect formation process is also treated and compared with the recent studies of similar phenomena in semiconductors. Finally the studies of formation kinetics of Frenkel pairs are surveyed.

2. Precursors of Frenkel pairs. - It has been well established that the Frenkel defects are created in a de-excitation process of an exciton. The exciton in alkali halides has a large lattice coupling and relaxes into a self trapped state, which may be designated as an electron trapped by a self trapped hole, namely by the V, center [21].

The electronic structure of the self trapped exciton (abbreviated as (V, e)) has been studied through experiments of luminescence [22-241, magnetic circular dichroism [25-261, EPR [27-281, and optical absorption [29-301. The energy diagram of the (V, e) may be described separately for the hole state and for the electronic state. Figure 1 shows the enegy scheme for

(V, e). The ordinate of the figure represents approximately the transition energy at the equilibrium position of (V, e) for electrons and a t that of the V, center for holes for KBr. The notation of the levels has been designated in terms of the irreducible repre- sentation for D, and D,,. According to Blair et

al. [24], for D,, the X , y, z axes stand for the halogen molecular axis of V,, a

<

110

>

array of two neighbouring metal ions, and a

<

100

>

axis, respectively. The assignment of the electronic state is according to Stoneham and co-workers [31-321. These authors

have shown that the ordering of the energy level shown in the figure is valid for many alkali halides. The assignment of the hole transition (optical absorption) is in accordance with transitions in the V,

center [33]. The splitting of the l7, level, which has been observed in the V, center transition in alkali bromides and iodides has been ascribed to the spin- orbit interaction [33]. In most alkali halides, two emission bands, one, a-polarized, and the other, n-polarized and at a lower transition energy than the former, have been observed. The work on EPR and the magnetic circular dichroism indicates that the excited state

for the n-polarized emission is a triplet state : the spin-orbit mixing between the

'X,

and 317u states makes the transition partially allowed. A detailed theoretical study of the triplet state has been made by Fowler et al. [34].

For the two electron system of (V, e), the singlet and triplet spin states should be present for each configuration. Stoneham 1311 and Marrone er al. [27] has shown that the exchange interaction of the lowest state is of an order of 0.01 eV. In view of the fact that the difference between the energies of the a- and n-emissions is larger by two orders of magnitudes than the calculated exchange energy, the a-emission has been ascribed to take place from a higher A,, state [31], as described in figure 1. Based on the temperature dependence of the lifetime of the triplet (B,, ; A,,) state ( l ) by Purdy and Murray 1351, Song

and Stoneham [36] showed the existence of the singlet (B,, ; A,,) state which is populated by phonon interaction from the low-lying triplet state. The absence of the fast component in the emission from the (B,, ; A,,) state and its x character have been ascribed to that the singlet (B,, ; A,,) state is no populated directly from the higher singlet states.

cc-

polarized

t

#-polarized

B3u du

Q~

B I Q B ~ Q

FIG. 1.

-

Single electron state description of the self-trapped exciton. Arrows indicate the optical transitions which have been

observed.

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FORMATION OF LATTICE DEFECTS BY IONIZING RADIATION IN ALKALI HALIDES C7-29 although the latter has not been resolved as comple-

tely as the V, transitions. Williams [37] has made a

polarization study and assigned the electron transitions to be from (B,, ; A,,) to (B,, ; B,,) and to (B,, ; Bz,). He also found evidence for the intersystem crossing from the excited triplet state to the singlet state.

The lowest triplet (B,,; A,,) state may be further split by the spin orbit interaction into (A,,

+

B,,

+

B,,). Fishbach et al. [38] observed the

increase of the lifetime of the n-luminescence below

20 K and ascribed it to the population of the A,, state that lies below B,, and B,, states. The transition from the A,, state to the ground state is forbidden and may take place only at the lowest temperature, where the thermal excitation to the B,, and B,, is not appreciable. Karasawa and Hirai [39] have observed a simi-

lar phenomena in KBr. They have also observed a delayed population of the triplet (B,, ; A,,) state both through luminescence and absorption measurements [40]. This result proves the existence of a non-

radiative channel from a higher triplet state.

The rise time of the F-center formation in KC1

has been recently measured by Bradford et al. [41]

by employing two-photon absorption of the uv light pulses obtained from a mode-locked YAG laser at the fourth harmonic. Their results are reproduced in figure 2. The rise time at 25 K was obtained to be 11 1 9 ps. This result indicates clearly that the F center is produced in its ground state. Furthermore the lifetime of the self-trapped excitons for emitting photons are

-

1 ns for the a-emission and

-

1 ms for the n-emission [23]. The results substantiate the

former observation by Hirai ilnd co-workers [42-441,

that the delay of the F center production is smaller than the decay of luminescence. Based on this result it has been concluded that the Frenkel-pair formation does not arise from the same excited state as those producing luminescence. We must therefore assume

that the self-trapped exciton or a relaxed exciton which is converted to the Frenkel pairs, and is referred to as (V, e),, is different from the self-trapped excitons which emit c- and n-emissions and are referred to as (V, e), and (V, e),. respectively.

The conclusion that (V, e), is different from (V, e), and (V, e), may be also deduced from the following experimental observation by Hirai [45]. They measured the temperature dependence of the a - and n-emission yields and the F-center production yield of KBr below 4.2 K and obtained the results shown in figure 3.

It was found that the F-center production yield decreases as the temperature is decreased below 4.2K,

whereas the emission yields are independent of temperature. If both processes are originated from the same relaxed exciton, the deduction of the F-center formation process should be compensated by the emission process. The absence of the temperature dependence of the latter may indicate that the F-center formation process and the a- and n-emission processes have different origins. There should be another non-radiative process or unidentified emissive process which compensates the loss of the F-center production yield. Identification of this process may provide useful information concerning to the precursor of the Fren- kel defect.

Id

efficiency and FIG. 3. - Temperature dependence of the F-center formation U- and n-emission yields induced by an electron

pulse in KBr. ODv, OD20 and ODm ind~cates the optical density measured at the F-band maxlmum immediately after the pulse ; 20 ns after the pulse and after a steady state is attained. See the

l O L L--

om ' 2 0 26 original paper for detail (after Karasawa and Hirai [45]).

..

eV

oN

-

1 _i

Tanimura and Okada [46] have arrived at the same

conclusion from another observation. They studied the effect of the addition of Na+ impurity on the pro-

I

I duction yield of the F center and on a-, n- and nA-

emissions, where the nA-emission refers to the recombination luminescence of an electron trapped by a

--

20 m 4o a m YKr center (the V, center trapped by N a f ) The result

PULSE DELAY (PICOSECONDS) they observed, as shown in figure 4, is that a decrease

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Na* CONCENTRATION ( mol% )

FIG. 4. - Na+ concentration dependence of the formation efficiency of the F-center (0) and a-center (U) and of the yield of a emission (e), n-emission ( X ) and n.4-emission (A) induced by X-ray irradiation of KBr (after Tanirnura and Okada [46]).

Na' concentration up to 10-2 in fractional concen- tration. This result indicates that only the excitation process which leads to (V, e), is sensitive to the addition of Na" at this concentration.

There are some pieces of experimental evidence which indicate that the (V, e), and (V, e), are mutually correlated. Karasawa and Hirai [45,47] have observed that upon increasing the temperature from 20 K to 80 K, the production yield of the triplet exciton increases, and the increase is compensated by the decrease of the production yield of the F-center in NaCl. Tanimura [48] has pointed out that the production yield of the X-emission for several alkali halides is anti-correlated with the production yield of the F-center : at smaller S / D , where S is the space

for the interstitial and D is the radius of the halogen

atom, the production yield of the F center is smaller and that of the n-emission is larger. The yield of the U-emission depends very little on the kind of the alkali halide whenever it is observed. Apparently the relaxation into (V, e), depends little on other processes, but (V, e), and (VK e), are correlated until they finally becomes uncorrelated.

The (V, e), has been demonstrated to be at a higher energy than (V, e), again by Tanimura and co- workers [49] and Murray [35], independently. They observed that the thermal annihilation of a Frenkel pair, an F center and an H center, produces X-emissions for both KC1 and KBr. Tanimura has observed the a-emission induced by the F-H recombination. These results lead us to conclude that a stable pair of the F- and H-centers has an energy higher than the optical transition energy from (B,, ; A,,*) to the

ground state, typically about 4 eV. Recently Wil- liams [37] observed that photoexcitation by 694 nm light from the (B,, ; A,,) state of NaCl produces the F centers. Although the spectral dependence of the F-production yield has not been obtained, the result clearly indicates that (V, e), is situated above (V, e),. As pointed out by Williams, it is of great importance to use a technique such as the cascade excitation to locate the (V, e),. Unlike the direct excitation, the cascade excitation may produce the excited state rather uniformly in the crystal and also an excited state to which the transition is not allowed from the ground state can be populated at a considerable concentration. It has been also shown that the, H center is produced by the recombination of the V, center with an electron [50-511. It appears that the (V, e), state can be produced by the excitation of the lowest self trapped exciton or by the recombination of a V, center with an electron.

Some suggestion has been made for the structure of (V, e),. Toyozawa [l l ] has assumed that the 2 pc, state is the precursor of the F center. ltoh and Sai- doh [IO] have assumed that the (V: e) state, where

*

denotes an excited state, such as (B,, ; A,,) and (B1, ; A,,) states would be responsible for the F-H The self-trapped excitons in which the V, center is excited may have a configuration where the halogen-halogen distance in the V, center is rather close to the halogen-halogen distance in the lattice. Thus the relaxed exciton that has the V, center at an excited state may have a higher mobility than the relaxed exciton with the V, center at the ground state. Lushichik [52] and later Nishimura and co-wor- ker [53-541 have shown experimental results which indicate that the unrelaxed exciton is mobile. Such mobile unrelaxed exciton has been called hot exciton. Such an indication of the exciton motion might be ascribed to a higher excited state of unrelaxed exciton. Tanimura [46] has interpretted their results of the Na-concentration dependence shown in figure 4 that there is a highly mobile intermediate state before the relaxation to (V, e),.

The precursors of the Frenkel pairs are now known to be different from the relaxed excitons which give the a- and X-emissions, although the structure is not known exactly. The process of the Frenkel pair production is of great interest, since the precursor state is not the lowest relaxed state and yet de-excitation occurs by a non-radiative process in which the energy is converted to dynamic motion.

3. Recombination processes and dynamic intersti-

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FORMATION OF LATTICE DEFECTS BY I 0 lNIZING RADIATION IN ALKALI HALIDES C7-3 1

thermally activated dynamic motion and the difference between E, and E,, may be worth further analysis.

Activation energy for the dynamic and thermal motion of the interstitial halogen and for the defect production in alkali halides

Crystal - NaCl KC1 RbCl KBr RbBr CsBr K I

Activation energy (eV)

thermal dynamic defect

motion motion (') formation (*)

-

0.07 0.075 0.07 - 0.06 0.09 0.03 0.03 0.08 0.065 0.035 0.01 5 0.075 -

(a) Saidoh M. and Itoh N. (reference [56]).

( b ) Sonder E. (reference [68]).

FIG. 5.

-

Configuration coordinate diagram for the self trapped exciton (after Itoh and Saidoh [lO]).

lattice point owing to the repulsive interaction between X; and neighbouring alkali ions. Toyozawa [l l ] has suggested that an adiabatic instability is caused by the admixture of 2s and 2pa, states of the trapped electron, giving the downhill potential for the transla- tion motion of X;. A similar potential curve has been

suggested by Kabler [9]. Although the detailed potential curve has been yet unclear, it appears generally accepted that the adiabatic downhill potential may lead a halogen atom into an interstitial position. One should also note the configuration interaction between an electron-excited (VK e) state and the hole- excited (V: e) states may make the nature of the (V, e), state more complicated.

The interstitial atom in which the de-excitation energy is imparted exhibits a so-called dynamic motion. Saidoh and Itoh have measured the temperature dependence of the interaction of the interstitials with Na' impurities and with H e n t e r s during the dynamic motion of the former and have pointed out [55] that the range I of the dynamic motion of the intersti-

tial halogen may be expressed as

where E, is an activation energy and I, and I , are constants. They measured Ed for various alkali halides [56-571 and found that the values are smaller than the activation energies E,, for the thermal migra- tions as shown in table 1. Table I includes also the activation energy for the temperature dependence of tbe defect formation, on which description is made later. This temperature-dependent term may be a

Saidoh and Itoh [57] have postulated that the difference between E,, and E, is due to the interstitial halogen being in an electronically excited state during the dynamic motion. Various types of excited state may be plausible. In an earlier paper [55] Saidoh and Itoh postulated that the H center in an excited state evolves from the (V: e) state and undergoes the dyna-

mic motion. It was also pointed out [57] that a crowdion of negatively charged halogens which share a positive hole may be responsible for the dynamic motion. In both cases it has been assumed that a positive hole is shared by the n-orbitals in a linear array of halogen ions. This configuration would have smaller overlap repulsion with neighbouring alkali ions and a longer mean free crowdion distance. Although Saidoh and ltoh ascribed their experimental results of the temperature dependence of the interaction volume to that of the range of the dynamic motion of the interstitial, as described above, Dienes and Smo- luchowski [58] suggested a model based on an analysis of the stochastic aspects of the random walk and ascribed the increase of the interaction volume to the increase of the frequency of the rotational motion of the H center. Townsend [59] ascribed the temperature dependence of the interaction volume to the motion of (V, e), before the de-excitation.

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of defect motion. The experimental situation is different in two cases : multiple recombination steps are required for the defect to migrate by an appreciable distance for semi-conductors but the single annihilation of excitons is responsible for the long range migration of interstitials in alkali halides. The mechanism proposed by Saidoh and Itoh may be interpreted as a non-radiative energy transfer into a non-bonding state which is still in an electronically excited state, or as a non-radiative energy transfer through an electronically excited state which is a crowdion state involving a positive hole.

The sputtering phenomena of alkali halides by electron bombardment have been discussed in terms of the dynamic motion [62]. The several aspects have been presented which supports the supposition that the dynamic motion of interstitials induced by electron or photon irradiation causes the sputtering. First of all the sputtering yield was pointed out to be higher in crystals with smaller S / D where S is the space

for an interstitial atom and D is the radius of the

interstitial 1631. Secondly

<

110

>

ejection pattern is clearly produced by multi-photon excitation with a ruby laser of KCI, NaCl and KBr [64]. The range of the dynamic motion was deduced [62] from the sputtering yield and was shown to decrease with increasing S / D as shown in figure 6. The sputtering yield is a

monotonic function of the range of the dynamic motion. It is interesting to note that the formation yield of the F center increases as $ / D decreases to some extent [64] but decreases for small S / D [65]. This

result is easily understood since for the production of a Frenkel defects, both the range and the space are effective but for the sputtering, only the former is effective. In addition to the

<

110

>

patterns,

<

211

>

patterns were also observed, which was explained [66] in terms of defocussing of the usual

<

110

>

momentum. It is simply added without further comment that the sputtering by electronic excitation has been ascribed to the motion of exciton [60, 671.

1.5 >

K1 K a

NaBr NaCl KBr RbBr NaF RbCl KF RPF

I l l I I I I I I

RANGE

1 2

FIG. 6. -The range of the dynamic crowdion deduced from experimental data of sputtering yield as a function of SID, where S is the space between halogen ion and D is the diameter of the halogen atom. The production yield of the F-center is also

plotted (after Itoh 1611).

The temperature dependence of the range of the dynamic motion described above has been correlated to the efficiency of the production yieId of the FrenkeI defects [56]. Here only the production of the stable, o r well separated, Frenkel defects is taken into account. The experimental production yield has been obtained directly from the growth curves or from the destruction rate of the M-center. The measurement of the destruction rate of the M-center yields a value of the production yield free from the effect of thermal annihilation since the thermal annihilation of the interstitial to the M-center, which is much less numerous than the F-center, can be disregarded [68, 691. Sonder [69] has demonstrated that the production yield obtained with the two methods is the same. The values of the activation energies for the defect production are listed in table I. KBr is the only crystal in which both activation energies for the defect production and the dynamic motion are obtained. The activation energies for the two processes in KBr agree well. Saidoh and Itoh [57] have argued that if the range of a dynamic sequence exceeds a critical value r,, a stable Frenkel

pair is formed. According to this assumption, the pro- bability that a Frenkel pair is formed is given by

P = exp r 0

[lo

+

1, exp(- E , k n

1 .

The comparison of the experimental production yield with eq. (2) for KC1 is shown in figure 7. It is seen clearly that eq. (2) explain the production yield rea- sonably well for the temperature range between 80 K

and 200 K.

-

fn TEMPERATURE ( K

FIG. 7. - A best-fit of the temperature dependence of the production yield of the F-center with eq. (2) (after Saidoh and

Itoh [57]).

Sonder [68-691 and Guillot and conlorkers [70]i

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FORMATION OF LATTICE DEFECTS BY IONIZING RADIATION IN ALKALI HALIDES C7-33 model, corresponds to the difference between the

heights of the potential barriers for escape and for recombination. This model, however, explains only a part of the temperature dependence such as shown in figure 7. Furthermore as shown in figure 7, the plot according to eq.

(2)

follows approximately the Arrehenius equation. Consequently one is tempted to ascribe the increase of the production yield with temperature to the increase in the range of the dynamic motion. As seen from table I the comparison of the activation energy has been made only for KBr, for which good agreement has been obtained. It is of interest to extend both types of measurements to other alkali halides where only one type of measurement has been made.

4. Stabilization of interstitials and kinetics of defect formation.

-

The stabilization of the interstitials at the temperature, where the H-center is mobile has been discussed by several authors. H-centers trapped by monovalent alkali impurities [71] and by divalent alkaline earth impurities [72] have been identified. The structure of the former has been deter- mined clearly, and called the HA center. It has been shown [73] that there is similarities between the Hayes-Nichols band, which is due to the paramagnetic center created by irradiation of Ca++-doped KBr, and the H, band in Na-doped KBr in their growth during irradiation at liquid helium temperature and during thermal annealing. It was also shown that the interaction of the interstitials with a pair of the divalent impurity and a vacancy during the dynamic motion is analogous to that with the Na' impurity in KBr. These results support the original model of the Hayes-Nichols center : the H center trapped by the impurity pair ; although several alternatives have been suggested. This center is referred to as the H, center.

The V, center is apparently of the most simple structure among the centers involving two H centers and is stable only below about 200 K. It has been shown that the center has a symmetry axis along a

<

100

>

direction [74]. Owing to the attractive nature [75], the two interacting H centers may col- lapse into a molecular center. Green and White [76] has proposed that the V, center is a halogen molecule, of which each atom is situated in the neighbouring cubes formed by four alkali and halogen ions. Dil- ler [75] has shown that

X;

molecular ion oriented along a

<

11 1

>

direction is more stable than the X, molecule. Hobbs et al. [77-791 have suggested that two interacting H centers create a dislocation loop and that the final form of two H centers is a halogen molecule situated in a pair of positive and negative ion vacancies. In view of the symmetry axis of the V, center and of the result that the V, center is dissociated into H centers by photoexcitation at low temperature 1801. the model of the V4 center proposed by

Green and White appears to be more appropriate.

The Hobbs model certainly corresponds to other V centers created at higher temperatures.

The mechanism of the stabilization of the interstitial appears to be due to the elastic interaction. The interaction between the H center and other defects has been calculated by Dienes et al. [81], by Bach- mann and Peisl [82], and by Diller [75]. The volume of interaction between the interstitial under dynamic motion and the other defects has been obtained by Saidoh et al. 1831. For KBr, the interaction volume of the dynamic crowdion with the Na impurity was about 1/10 of that with the H center ; this result is in accordance with the difference in the elastic interaction energy. The result indicates that the dynamic motion of the replacement sequence is inhibited by the elastic distortion around lattice defects. The H-center formed at the end of the path of the crowdion is apparently stabilized by the defects if the crowdion motion is prohibited by the defect.

The comparison of the interaction volume was also made in the process of thermal annealing. Saidoh

and coworker [84] measured the production ratio of

the HA center to the V, center in Na-doped KBr, and found that the production ratio is proportional to the concentration ratio of the Na impurity to the H center contained prior to the annealing. The ratio of the interaction volume was derived from the relation. The result is shown in table 11. It is seen that the interaction volume of the thermally migrating interstitial with the H-center is 40 times larger than that with the Na impurity. The fact that this interaction volume ratio is larger than that for the dynamic motion has been ascribed to the interaction volumes for the thermal and dynamic motion being proportional to the cube and square of the interaction radius, respectively.

Ratio of the interaction oolzrme for the impurity-interstitial interaction

and for the di-interstitial formation

Ratio -

( a ) Saidoh M. and Itoh N . (reference [84]).

(9 Hoshi J., Saidoh M. and Itoh N. (reference [73]). The observation that the di-interstitial formation has larger cross-section than the impurity trapping has been made in Ca++-doped KBr by Hoshi and

coworkers [73]. They showed that the concentration

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at liquid helium temperatures. Therefore it was concluded that the D, center is the di-interstitial center stabilized by a complex of the divalent impurity and positive ion vacancy. The ratio between the interaction volume for an H-center to form a D, center and to form the H, center derived from the quadratic relation is shown in table 11. Again the interaction volume to form a di-interstitial is much larger than that for an H-center to be stabilized by an impurity complex.

The interstitial stabilization is of primary importance in understanding the defect formation kinetics in the temperature range where the interstitial is mobile. In this paper kinetics of defect formation below 200 K is discussed. Extensive works on the growth curve of the F-center at liquid nitrogen temperature has been made by Pinard and coworkers [16,

70, 861 and by Sonder [14]. The former authors used the irradiation with low energy electrons, the small penetration depth of which makes it possible to determine the optical density at high F center concentrations. It was found [70] that the F-center concentration is proportional to near 10t9 cm-3, where t is the irradiation time. They also found that the growth curve shows a to.' dependence in doped crystal. It has been also shown that the dose rate dependence of the defect formation at liquid nitrogen temperature is very small [14]. Figure 8 shows the result obtained by Sonder, which includes the growth curve obtained with dose rates different by five orders of magnitude.

Energy absorbed

l ~ e ~ / c m ' l -

--

FIG. 8.

-

F-center production curve : by electron irradiation. Irradiation dose rate : 1.6 X 1018 eV cm-3 S-l for sample 50,

2 X 1016 9 X 1017 eV cm3 S-l for sample 53 and

2 X 1 0 2 2 eV cm-3 S-1 for sample 71. (after Sonder [14]).

The theory of the growth curve above the temperature, where the H-center becomes mobile, has been given by Farge and coworkers [15, 871, by Sonder [14], Agullo-Lopez and Jaque [l71 and Guillot [16]. These authors have assumed simple kinetics, which involves the interstitial-vacancy annihilation and interstitial stabilization. The formation of the interstitial clusters

has been taken into account by Agullo-Lopez and by Guillot. The experimental growth curves around liquid nitrogen temperature is not in accordance with the result of theoretical analysis with the simple models. The solution of the kinetic equations with parameters derived from the experimental results on thermal motion of interstitials lead to strong dose- rate dependent growth curves [14].

According to the discussions on dynamic motion and interstitial stabilization, the kinetic equation for the defect formation at liquid nitrogen temperature should explain the following features : (i) that the F-centers and V, centers constitute Frenkel pairs and (ii) that the growth rate of the F-center does not depend on the irradiation dose rate. The fact that the H center is unstable at this temperature region indicates that the equilibrium concentration of H-centers is governed by the dose rate and the average life time of the H- center ; implying a strong dose rate dependence [14]. Furthermore the optical absorption bands due to the V, center have a well-identified character, although a slight dependence of the width on the dose, dose rate and temperature has been observed [88]. Since the transition energy of the molecular halogen center depends critically on the number of atoms involved in the center [89], the convergency to a single V-type band (namely the V, band) indicates that the interstitial center formed at this temperature range involves two H-centers and the presence of a stable interstitial

center involving three H-centers is improbable. It has been shown by Hobbs and Hughes [78] that the clusters of interstitials are formed even at liquid nitrogen temperature, where the interstitial centers show exclusively the V, absorption band.

Since the interaction volume for di-interstitial production is much larger than that for interstitial trapping, it is of interest to solve the kinetic equation by employing the interaction rate obtained experimentally. According to the studies of interstitial formation by pulsed electron-beam irradiation, the interstitial motion may be divided into dynamic and thermal types. The interaction rate with a defect during dynamic motion may be proportional to the concentration dk of the defect and that during the thermal motion may be proportional to the product of the concentration dk of the defect and that i of the interstitial. Thus the kinetic eauatjon mav be described as

where the concentrations are denoted by fractions, a is the production rate per second, and

P,

and y:

are the reaction rates during the dynamic motion and the thermal motion, respectively. It is more conve- nient to use a reduced unit for the reaction rate and time

d i

(10)

FORMATION OF LATTICE DEFECTS BY IONIZING RADIATION IN ALKALI HALIDES (3-35

where y, = Yk*/ci and D =

displacements produced written as dna.

at. Here D is the number of by halogen site and often

. . - - - -

- -

- - -r--

The kinetic equation for the reactions :

P . L . @ H

+

V H + H + H 2

H

+

I -, H.1 H.1

+

H -, H,.I

were constructed. Here P. L. denotes the perfect lattice, H the H-center and I an impurity, or a defect that traps an interstitial. In this kinetics, the annihilation of the interstitial by vacancy, di-interstitial formation and the trapping by impurity has been taken into account. It is also assumed that the di-interstitial centers associated with impurities are produced. The parameters

p,

have been deduced by Saidoh and coworkers. For the random recombination

Yz

may be given by [90]

"

e-EJkT

Yk*

= Zk v i = Zk 0 (4)

where z, is the number of sites around a defect from which a jump of interstitial towards the defect means certain annihilation and v, and E are the characteris- tic frequency and the activation energy for the interstitial migration. Saidoh and Itoh have estimated

P,

for Na' in KBr to be about 150 times the volume of the cube [83] formed by four alkali and halogen ions.

/lk for the recombination of the interstitial with the F-center was taken to be 300 in view of the fact that

the saturation value of the F.center growth curve is nearly 0.3

%

[91]. The value of

P,

for the H-center is taken to be 1500 according to Saidoh and Itoh. The

ratio of the z, for the interaction with impurities to the H-center shown in table I1 was used. The ratio of

z, for the F-center to zk for the impurity was tentatively taken to be 2.

The results of the numerical solution of the kinetic equation for an impurity concentration of 10-6 is shown in figure 9, where the dose rate 1 corresponds to

FIG. 9. - Calculated growth curve. Number in the figure shows the relative value of y (see text).

an interaction rate y = 3 X 10t4 in the reduced unit

for the interaction between the H-centers. The other coefficients were changed by the same ratio. In order to evaluate the absolute value of y*, the result of annealing experiment by Saidoh [84] may be used. Accord- ing to their result, the value of y* for the interaction between the H-centers at the annealing temperature 43.5 K is 8 X 10' S-'. Taking the value of the acti-

vation energy of 0.09 eV for the H-center migration, the value of y* at liquid nitrogen temperature was evaluated to be 5 X 107 S-'. It follows that curve 1

represents the growth curve for a = 1.6 X 10-'

S-' at liquid nitrogen temperature, which is nearly equal t o the dose rate of normal X-ray irradiation. It is seen from figure 8, that the production of the F- center saturates at nearly 10-5, which disagrees with the experimental results. Furthermore, the formation curve depends strongly on the dose rate.

In order to solve the difficulty, the following model is invoked. We suppose that an H-center is temporarily trapped by the V, center already produced. This trapping may be caused by the strain field around the V, center. It was tentatively assumed that the lifetime of the trapped H-center is 1 second. The introduction of such a trap may explain the experimental results that the nucleus of the interstitial cluster is formed by low temperature irradiation [77]. Kinetic equations were constructed to include clusters of di- interstitial centers of considerable size. It was assumed that the values of p and y of a di-interstitial are the same as those of the Na' impurity. It is also assumed that the values are not changed, even though other di-interstitials are formed in its proximity. This is certainly an underestimate of the cluster growth. The results for the dose rate of 105 and 10 in the prescribed units are shown in figure 9. It is clear that the dose rate dependence is reduced considerably. It appears that the saturation value of the F-center growth curve reaches as high as 10-3. The maximum size of the di- interstitial was about 50 and the average size was 22 for lower dose rate, whereas the average size for the

FIG. 10. -Calculated growth curve. Number in the figure

(11)

higher dose rate was only 8. The dependency of the F-center concentration on the dose at high doses is smaller than the power of 0.75. This result may be substantially improved if the dependence of the interaction volume on the cluster size is taken into account, as has been made by Guilliot et al. [16].

There are other significant qualitative features concerning the result shown in figure 10. An increase in the temperature results in the increase in a and y*. As already described, the temperature dependence of y* is stronger than that of a. Thus the increase in the temperature results in an increase of y. Therefore the result indicates that the number of nucleation sites become smaller as the temperature increases.

Finally a few comments are made on the kinetics at liquid helium temperature. In the model described

above, the kinetic equation at liquid helium temperature does not include the term for y's. It turns out that the solution of such equations leads to a proportional relation between the F-center concentration and the dose closely to the saturation. Experimental results show a considerable deviation from the proportio- nality. Some of the secondary effects, such as the conversion to the a-I Frenket pairs 1191 and tunneling recombination [20] and athermal annealing [l81

may be the cause of this discrepancy.

Acknowledgments.

-

The author would like to express his gratitude to A. M. Stoneham and A. E. Hug- hes for valuable discussions and for reading the manus- cript. Thanks are also due to L. W. Hobbs, M. Saidoh, P. W. Tasker and K. Tanimura for valuable discussions.

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FORMATION OF LATTICE DEFECTS BY IONIZING RADIATION IN ALKALI HALIDES C7-37 I511 HALL, T. P. P., HUGHES, A. E. and POOLEY, D., J. Phys. C 9

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DISCUSSION

G. GUILLOT. - How the decrease of production on both temperature and defect concentration : could efficiency of Frenkel pairs below 4 K can be explained ? the (a

+

I) system not interact with the (F

+

H)

N. ITOH.

-

I could speculate that the (V,e)F state has a small potential minimum against the motion of the X,- t o the

<

110

>

direction, such as shown in figure 5 of the paper. It is conceivable that such a minimum exists since the motion is certainly inhibited by the overlap interaction with neigh- boring alkali ions.

F. LUTY.

-

I n your discussion you mentioned only the neutral Frenkel defects (H- and F-centers) without any consideration of the charged Frenkel pairs (a- and I-centers), which a t low temperature are much more dominant in number. Even if the cc- a n d I-centers may not be the primary radiation defects, can one really neglect them s o completely ? As the a- and I-production and annealing behavior depends strongly

. -

systems in the processes you described ?

N. ITOH. - I don't think much progress has been

made o n the mechanism of the production of the charged Frenkel pairs since you have suggested that they are produced by the tunneling of electrons from F