A mechanism of radiation-induced emission of Schottky defects from extended defects proposed originally for metals has recently been applied to ionic crystals, where it is based on interactions of excitons with extended defects such as dislocations and colloids. Exciton trapping and decay at colloids may result in the emission of F centers and consequent shrinkage of the colloid. In the present paper, the radiation-induced emission of F centers is taken into account in the modeling of nucleation and growth of sodium colloids and chlorine bubbles in NaCl exposed to electron or gamma irradiation. The evolution of colloid and bubble number densities and volume fractions with increasing irradiation dose is modeled in the framework of a modified rate theory and compared with experimental data. Experimental values of the colloid volume fractions and number densities have been estimated on the basis of latent heat of melting of metallic Na obtained with combined differential scanning calorimetry experiments and atomic force microscopy investigations of metallic clusters.

Abyzov, A.S.,

Dubinko, V.I.,

Hartog, H.W. den,

Sugonyako, A.V.,

Turkin, A.A.,

Vainshtein, D.I.,

English

University of Groningen Digital Archive

phys. stat. sol. (c) 2, No. 1, 438–443 (2005) / DOI 10.1002/pssc.200460202 
© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim 
Nucleation and growth of sodium colloids in NaCl under  
irradiation: theory and experiment 
V. I. Dubinko
1, A. A. Turkin
1, A. S. Abyzov
1, A. V. Sugonyako
2, D. I. Vainshtein
2,  
and H. W. den Hartog
*2  
1  National Science Center, Kharkov Institute of Physics and Technology, 310108 Kharkov, Ukraine 
2  Solid State Physics Laboratory, University of Groningen, Nijenborgh 4, NL-9747 AG Groningen, 
The Netherlands 
Received 11 July 2004, revised 24 September 2004, accepted 6 October 2004 
Published online 20 January 2005 
PACS 61.46.+w, 61.80.Ed, 61.80.Fe, 61.82.Ms, 82.50.Kx 
A mechanism of radiation-induced emission of Schottky defects from extended defects proposed origi-
nally for metals has recently been applied to ionic crystals, where it is based on interactions of excitons 
with extended defects such as dislocations and colloids. Exciton trapping and decay at colloids may result 
in the emission of F centers and consequent shrinkage of the colloid. In the present paper, the radiation-
induced emission of F centers is taken into account in the modeling of nucleation and growth of sodium 
colloids and chlorine bubbles in NaCl exposed to electron or gamma irradiation. The evolution of colloid 
and bubble number densities and volume fractions with increasing irradiation dose is modeled in the 
framework of a modified rate theory and compared with experimental data. Experimental values of the 
colloid volume fractions and number densities have been estimated on the basis of latent heat of melting 
of metallic Na obtained with combined differential scanning calorimetry experiments and atomic force 
microscopy investigations of metallic clusters. 
© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim 
1  Introduction 
Exposure of alkali halides to ionizing radiation results in the formation of nano-sized halogen “bubbles” 
by agglomeration of H centers and of complementary inclusions of metallic “colloids” formed by ag-
glomeration of F centers. H and F centers are the primary radiation-induced Frenkel pairs in the halide 
sub-lattice created in the crystal bulk due to decay of self-trapped excitons [1]. The H center is a hole 
trapped at an interstitial halide ion, and an F center is a vacancy in the halide sub-lattice with a trapped 
electron. The cation sub-lattice is not damaged during primary displacement processes in the crystal bulk 
[1]. However, early experimental results on electron irradiated alkali halides such as KCl and NaCl [2] 
provided evidence for the formation of perfect dislocation loops, which require agglomeration of intersti-
tial ions in both anion and cation sub-lattices. The first explanation for this phenomenon was given by 
Hobbs et al. [2], and it was based on the production of a molecular center (a halogen molecule occupying 
a stoichiometric divacancy) due to interaction between dislocations and two H centers. In 1977, Jain and 
Lidiard [3] have formulated a model, according to which, the dislocation bias for H centers was the driv-
ing force for the colloid growth in alkali halides in the same way as it is for void growth in metals under 
irradiation. The mechanism of dislocation climb [2] used in the Jain and Lidiard model, requires two H 
centers meeting each other at the dislocation core, and subsequently, the model was modified by 
Dubinko et al. [4], who suggested that a dislocation can climb due to the production of a VF center (self-
 
 
* Corresponding author: e-mail: h.w.den.hartog@phys.rug.nl, Phone: +3150-363-4789, Fax: +3150-363-4879 phys. stat. sol. (c) 2, No. 1 (2005) / www.pss-c.com  439 
© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim 
trapped hole neighboring a cation vacancy) induced by interactions between a dislocation and only one H 
center. Subsequent diffusion of VF centers away from dislocations and recombination with F centers 
results in the formation of stoichiometric divacancies thus providing a driving force for the formation 
and growth of vacancy voids observed experimentally. 
In this model, the void growth rate is determined by the flux of VF centers and by the net flux of F cen-
ters (i.e. the difference between F and H center fluxes), which depends on the difference between the 
void bias and the mean bias of the system for the absorption of F centers. This difference is positive for 
void radii lager than a critical radius, 
crit
V R , which is determined by the ratio of material constants related 
to the void and dislocation biases [4]: 
  () b
b
R
d
im
crit
V 2 1÷ ≈ =
δ
α   (1) 
where b  is the atomic spacing and 
im α  is the constant associated with image interaction: 
  () ,
im im im
HF aa a =-       ()
1/3 2 2
,
, 3
1
36 (1 )
FH im
FH
B kT b
mn W
a
n
È˘ +
= Í˙ ◊- Î˚
,
  (2) 
d δ is the dislocation bias for the absorption of H centers, which is determined by the ratio of the relaxa-
tion volumes associated with H and F centers,  F H Ω Ω / : 
 
⎟ ⎟
⎠
⎞
⎜ ⎜
⎝
⎛
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
Ω
Ω
=
H H F
H
d k L
2
ln ln δ ,            ( ) 1
3( 1 )
HH
b
L
kT
mn
W
pn
+
=
-
  (3) 
where ν is the Poisson ratio, µ is the shear modulus of the host matrix, kH is the square root of the total 
sink strength for H centers, kT has its usual meaning. 
The critical void radius RV
crit is given by the same equation (1) as the critical colloid radius, and it is 
smaller than the radius of void nuclei produced by bubble-colloid collisions [4, 5]. Accordingly, one 
would expect all void nuclei to grow to visible sizes. The number density of the void nuclei, NV
0, can be 
estimated as the product of the bubble number density, NB ≈ 10
18 ÷ 10
19 cm
–3 and the colloid volume 
fraction, VC ≈ 10
–2 ÷ 10
–1 : NV
0 ≈ NB × VC ≈ 10
16 ÷ 10
17 cm
–3. However, the observed void number densities 
are about 10
13 ÷ 10
14 cm
–3 , which means that only a very small part of void nuclei survive and grow to 
visible sizes. The physical reason for this low “survival” rate remained unaccounted in the previous 
model. 
In a recent paper [6], an alternative mechanism of void and dislocation loop growth in alkali halides was 
considered, which was based on production of divacancies (Schottky defects) by interactions between 
dislocations and excitons, as was first suggested by Seitz in 1954 [7]. Voids can also emit divacancies 
due to the interaction with excitons, and so the balance between absorption and emission of divacancies 
determines the void evolution in this model, which has been shown to be in good agreement with ex-
perimental observations. 
In the present paper, the exciton-induced emission of F-centers from colloids is taken into account in the 
calculation of colloid nucleation and growth rates in the framework of the modified rate theory [8]. The 
driving force of colloid growth is the large bias of halogen bubbles for absorption of H centers rather 
than the dislocation bias employed in the previous models [3, 4]. Thus, radiolysis (separation of chlorine 
and sodium into bubbles and colloids) and the development of stoichiometric defects (voids and disloca-
tions) occur simultaneously but relatively independently. The only important “bridge” between these 
subsystems is bubble nucleation, which starts from recombination of H centers and divacancies since the 
homogeneous nucleation rate due to recombination of H centers is negligible. 440  V. I. Dubinko et al.: Nucleation and growth of sodium colloids in NaCl 
© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim 
2  Excitonic production of Frenkel and Schottky defects 
According to Seitz [7], an exciton may cause movement of nearby dislocation jogs, resulting in the crea-
tion of equal numbers of anion and cation vacancies - Schottky defects (SD). Owing to the heat gener-
ated locally during exciton annihilation, the jog can be displaced while a divacancy arises in the lat-
tice. It a previous paper [8] , we have incorporated the excitonic production mechanism of both Frenkel and 
Schottky defects in rate theory and evaluated so called local equilibrium concentrations of SD, 
eq
sS c , (almost) 
spherical (colloids and voids), cylindrical (dislocations) and planar (grain boundaries) sinks, where subscript 
“s” designates the type of SD (s= v for divacancies, s=F for F centers) and “S” is the type of the sink (S= V 
for voids, S= F for colloids, S= B for bubbles and S= D for dislocations). 
In absence of the radiation field, one has 
th
sS
eq
sS c c = , which can be obtained from thermodynamics. Un-
der irradiation, 
eq
sS c  is generally given by the sum of 
th
sS c  and 
irr
sS c  , which is due to the radiation-induced 
emission of SD’s. Here, 
irr
sS c  has a purely kinetic origin and can be expressed via the radiation-induced 
local emission rate of SD, 
irr
sS K  [8]: 
02
2
D
irr FP ex ex
vD
FP D
KZ l
K
Pr b p
= ,   
() D D
D
ex r k
Z
π
π
ln
2
= , 
v
irr
vD
irr
vD D
b
K c
2
× =
  (4) 
V
ex
FP
FP irr
vV bR
l
P
K
K
2 0
= ,   
v
irr
vV
irr
vV D
b
K c
2
× =
  (5) 
where  ex l  is the mean free diffusion path of the exciton i.e. the average displacement of the exciton 
before it decays,  FP P  is the probability of the formation of a Frenkel pair (FP) in the anion sub-lattice 
due to exciton decay in the bulk, 
0
FP K is the FP production rate in the bulk crystal,  
D
ex Z  and  D r  are the 
dislocation capture efficiency and capture radius for excitons and  v D is the divacancy diffusion coeffi-
cient. 
The difference between the mean divacancy concentration,  v C , and the local equilibrium concentrations at 
voids is the driving force for the void growth (or shrinkage), and its rate is given by [6] 
  ( )
eq Vv
vv V
V
dR D
CC
dt R
=-
  (6) 
The colloid growth rate is determined by the difference between the bias-induced absorption [4] and the 
exciton-induced emission [8] of F centers: 
  ( )
1
() () ()
CC C e q C
FCF F HCH H FCF F C
C
dR
ZRD C ZRD C ZRD C
dt R
=- -
  (7) 
where  , FH CC are the mean concentrations of F and H centers, respectively, which are determined by the 
usual rate equations, and  , ()
C
FH C Z R  are the colloid capture efficiencies for F and H centers [4]: 
 
,
, () 1
im
FH C
FH C
C
b
ZR
R
a ◊
=+
  (8) 
Bubble growth is due to the difference between the bias-induced absorption of H centers, since the  
emission of H centers requires higher energies than for F centers and therefore it has not been taken  phys. stat. sol. (c) 2, No. 1 (2005) / www.pss-c.com  441 
© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim 
into account in the present model. Accordingly, we find the following expression for the bubble growth 
rate  
  ( )
1
() ()
BB B
HBH H FBF F
B
dR
Z RD C ZRD C
dt R
=-
  (9) 
  () ( )
2
,, ,
, 2 1
im d
FH FH FH B
FH B
B B
b b
ZR
R R
m aa a -
=+ +
  (10) 
where  ,
d
FH a  and  , FH
m a  are the constants associated with the stress-induced diffusion anisotropy and the 
modulus  interaction, respectively [4]. 
The bubble growth rate increases with decreasing bubble radius, and so they can grow from molecular 
centers formed by recombination of H centers and divacancies.  On the other hand, the growth rates of 
colloids and voids decrease with decreasing size and they are negative below some critical radius, which 
determines the rate of their nucleation. The critical colloid radius is small (see eq. (1)), and so they can 
nucleate homogeneously at a rather high rate [9]. At the same time, the critical void radius is compara-
tively large [6] 
  () b
Z
r
R D
ex
D crit
V 4 3
2
÷ ≈ =
π   (11) 
and so they can be formed only from sub-critical void nuclei produced as a result explosive recombina-
tion of colloids and bubbles [6]. The present model provides a possibility to evaluate nucleation and the 
growth rate of colloids, bubbles and voids in the framework of rate theory [8]. The results of the model 
calculations are shown in Fig. 1 together with experimental data available up to date. 
 
 
Fig. 1  Temperature dependence of the volume fraction and the number density of extended defects in  NaCl elec-
tron irradiated up to different dose levels at a dose rate of 200 Mrad/hour. 442  V. I. Dubinko et al.: Nucleation and growth of sodium colloids in NaCl 
© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim 
3 Conclusion 
According to the present model, chlorine bubbles are the most finely dispersed ED in the system (the 
inter-bubble spacing is typically a few nm) implying that they start to collide with growing voids first, 
filling them with chlorine gas (fig. 2) The chlorine molecules in the “bubbles” are in a solid or liquid 
state due to a super high pressure (in the GPa range), but after collision with a void it is in the vapor 
state. Chlorine accumulation in voids provides a very important possibility for explosive back reactions 
with metallic sodium when growing voids start hitting colloids, which results in void-crack production 
and ultimately in explosive fracture of the material [5]. 
 
 
 
 
Fig. 2  Illustration of the distribution of chlorine bubbles, sodium colloids and voids in irradiated NaCl crystals. 
The bubble size is ~1-2 nm; the colloid size ~ inter-bubble spacing ~5-10 nm; the void size is about inter-colloid 
spacing ~20-30 nm, when collisions with colloids start. 
 
 
From a more general perspective, the mechanisms of void and dislocation loop growth and colloid 
shrinkage in alkali halides considered above are characteristic manifestations of radiation-induced emis-
sion of Schottky defects, which may be responsible for a number of various radiation effects such as the 
void dissolution and swelling saturation in irradiated metals [10], irradiation creep [11] and possibly the 
void ordering [12]. 
 
Acknowledgements   This study has been supported by the STCU grant # 1761 and by a NWO-NATO grant # NB 
67-319. 
References 
 [1] Ch.B. Lushchik, Creation of Frenkel Defect Pairs by Excitons in Alkali Halides, in: Physics of Radiation 
Effects in Crystals, R.A. Johnson and A.N. Orlov, Eds. (Elsevier Science Publishers B.V., 1986). pp. 474-
525. 
  [2]  L.W. Hobbs and A. E. Hughes, Radiation Damage in Diatomic Materials at High Doses, Harwell Report AERE 
– R 8092 (1975); L. W. Hobbs A. E. Huges, and Pooley, Proc Roy. Soc. (Lond.) A 332, 167 (1973). 
  [3]  Uma Jain and A. B. Lidiard, Philos. Mag. 35, 245 (1977). 
  [4]  V. I. Dubinko, A. A. Turkin, D. I. Vainshtein, and H. W. den Hartog, J. Nucl. Mater. 227, 184 (2000); 289, 86 
(2001). 
 [5] V. I. Dubinko, A. A. Turkin, D. I. Vainshtein, and H. W. den Hartog, Radiat. Eff. Defects Solids 156, 27-31 
(2001). 
 [6] D V. I. Dubinko, A. A. Turkin, D. I. Vainshtein, and H. W. den Hartog, Mechanism of void growth in irradi-
ated NaCl based on exciton-induced formation of divacancies at dislocations, COSIRES 2004, Helsinki, June 
28 – July 2 (2004). 
 
Cl2 
Void 
Chlorine bubble 
Sodium colloid phys. stat. sol. (c) 2, No. 1 (2005) / www.pss-c.com  443 
© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim 
  [7]  F. Seitz, Rev. Mod. Phys. 26 7 (1954) 
  [8]  V. I. Dubinko, D. I. Vainshtein, and H. W. den Hartog, Radiat. Eff. Defects Solids 158, 705 (2003). 
  [9]  V. I. Dubinko, A. A. Turkin, D. I. Vainshtein, and H. W. den Hartog, J. Nucl. Mater. 304, 117 (2002). 
[10] V. I. Dubinko and N. P. Lazarev, Effect of the radiation-induced vacancy emission from voids on the void 
evolution, COSIRES 2004, Helsinki, June 28 – July 2 (2004). 
[11] V. I. Dubinko, New mechanism of irradiation creep based on the radiation-induced vacancy emission from 
dislocations, submitted to Radiation Effects and Defects in Solids 
[12] V. I. Dubinko, New principles of radiation damage and recovery based on the radiation induced emission of 
Schottky defects, arXiv/cond-mat/0212154 

Creation of Frenkel Defect Pairs by Excitons in Alkali Halides, in:

Defects Solids 158,

Effect of the radiation-induced vacancy emission from voids on the void evolution, COSIRES

Mechanism of void growth in irradiated NaCl based on exciton-induced formation of divacancies at dislocations, COSIRES

New mechanism of irradiation creep based on the radiation-induced vacancy emission from dislocations, submitted to Radiation Effects and Defects

New principles of radiation damage and recovery based on the radiation induced emission of Schottky defects,

Radiation Damage in Diatomic Materials at High Doses,

Nucleation and growth of sodium colloids in NaCl under irradiation: theory and experiment

Crossref

A mechanism of radiation-induced emission of Schottky defects from extended defects proposed originally for metals has recently been applied to ionic crystals, where it is based on interactions of excitons with extended defects such as dislocations and colloids. Exciton trapping and decay at colloids may result in the emission of F centers and consequent shrinkage of the colloid. In the present paper, the radiation-induced emission of F centers is taken into account in the modeling of nucleation and growth of sodium colloids and chlorine bubbles in NaCl exposed to electron or gamma irradiation. The evolution of colloid and bubble number densities and volume fractions with increasing irradiation dose is modeled in the framework of a modified rate theory and compared with experimental data. Experimental values of the colloid volume fractions and number densities have been estimated on the basis of latent heat of melting of metallic Na obtained with combined differential scanning calorimetry experiments and atomic force microscopy investigations of metallic clusters

Dubinko, V.I.

Turkin, A.A.

Abyzov, A.S.

Sugonyako, A.V.

Vainshtein, D.I.

Hartog, H.W. den

University of Groningen

   University of GroningenNucleation and growth of sodium colloids in NaCl under irradiationDubinko, V.I.; Turkin, A.A.; Abyzov, A.S.; Sugonyako, A.V.; Vainshtein, D.I.; Hartog, H.W. denPublished in:Physica Status Solidi (C)DOI:10.1002/pssc.200460202IMPORTANT NOTE: You are advised to consult the publisher's version (publisher's PDF) if you wish to cite fromit. Please check the document version below.Document VersionPublisher's PDF, also known as Version of recordPublication date:2005Link to publication in University of Groningen/UMCG research databaseCitation for published version (APA):Dubinko, V. I., Turkin, A. A., Abyzov, A. S., Sugonyako, A. V., Vainshtein, D. I., & Hartog, H. W. D. (2005).Nucleation and growth of sodium colloids in NaCl under irradiation: theory and experiment. Physica StatusSolidi (C), 2(1), 438-443. https://doi.org/10.1002/pssc.200460202CopyrightOther than for strictly personal use, it is not permitted to download or to forward/distribute the text or part of it without the consent of theauthor(s) and/or copyright holder(s), unless the work is under an open content license (like Creative Commons).Take-down policyIf you believe that this document breaches copyright please contact us providing details, and we will remove access to the work immediatelyand investigate your claim.Downloaded from the University of Groningen/UMCG research database (Pure): http://www.rug.nl/research/portal. For technical reasons thenumber of authors shown on this cover page is limited to 10 maximum.Download date: 12-11-2019phys. stat. sol. (c) 2, No. 1, 438–443 (2005) / DOI 10.1002/pssc.200460202 © 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim Nucleation and growth of sodium colloids in NaCl under  irradiation: theory and experiment V. I. Dubinko1, A. A. Turkin1, A. S. Abyzov1, A. V. Sugonyako2, D. I. Vainshtein2,  and H. W. den Hartog*2  1  National Science Center, Kharkov Institute of Physics and Technology, 310108 Kharkov, Ukraine 2 Solid State Physics Laboratory, University of Groningen, Nijenborgh 4, NL-9747 AG Groningen, The Netherlands Received 11 July 2004, revised 24 September 2004, accepted 6 October 2004 Published online 20 January 2005 PACS 61.46.+w, 61.80.Ed, 61.80.Fe, 61.82.Ms, 82.50.Kx A mechanism of radiation-induced emission of Schottky defects from extended defects proposed origi-nally for metals has recently been applied to ionic crystals, where it is based on interactions of excitons with extended defects such as dislocations and colloids. Exciton trapping and decay at colloids may result in the emission of F centers and consequent shrinkage of the colloid. In the present paper, the radiation-induced emission of F centers is taken into account in the modeling of nucleation and growth of sodium colloids and chlorine bubbles in NaCl exposed to electron or gamma irradiation. The evolution of colloid and bubble number densities and volume fractions with increasing irradiation dose is modeled in the framework of a modified rate theory and compared with experimental data. Experimental values of the colloid volume fractions and number densities have been estimated on the basis of latent heat of melting of metallic Na obtained with combined differential scanning calorimetry experiments and atomic force microscopy investigations of metallic clusters. © 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim 1 Introduction Exposure of alkali halides to ionizing radiation results in the formation of nano-sized halogen “bubbles” by agglomeration of H centers and of complementary inclusions of metallic “colloids” formed by ag-glomeration of F centers. H and F centers are the primary radiation-induced Frenkel pairs in the halide sub-lattice created in the crystal bulk due to decay of self-trapped excitons [1]. The H center is a hole trapped at an interstitial halide ion, and an F center is a vacancy in the halide sub-lattice with a trapped electron. The cation sub-lattice is not damaged during primary displacement processes in the crystal bulk [1]. However, early experimental results on electron irradiated alkali halides such as KCl and NaCl [2] provided evidence for the formation of perfect dislocation loops, which require agglomeration of intersti-tial ions in both anion and cation sub-lattices. The first explanation for this phenomenon was given by Hobbs et al. [2], and it was based on the production of a molecular center (a halogen molecule occupying a stoichiometric divacancy) due to interaction between dislocations and two H centers. In 1977, Jain and Lidiard [3] have formulated a model, according to which, the dislocation bias for H centers was the driv-ing force for the colloid growth in alkali halides in the same way as it is for void growth in metals under irradiation. The mechanism of dislocation climb [2] used in the Jain and Lidiard model, requires two H centers meeting each other at the dislocation core, and subsequently, the model was modified by Dubinko et al. [4], who suggested that a dislocation can climb due to the production of a VF center (self-  * Corresponding author: e-mail: h.w.den.hartog@phys.rug.nl, Phone: +3150-363-4789, Fax: +3150-363-4879 phys. stat. sol. (c) 2, No. 1 (2005) / www.pss-c.com 439 © 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim trapped hole neighboring a cation vacancy) induced by interactions between a dislocation and only one H center. Subsequent diffusion of VF centers away from dislocations and recombination with F centers results in the formation of stoichiometric divacancies thus providing a driving force for the formation and growth of vacancy voids observed experimentally. In this model, the void growth rate is determined by the flux of VF centers and by the net flux of F cen-ters (i.e. the difference between F and H center fluxes), which depends on the difference between the void bias and the mean bias of the system for the absorption of F centers. This difference is positive for void radii lager than a critical radius, critVR , which is determined by the ratio of material constants related to the void and dislocation biases [4]:  ( )bbRdimcritV21÷≈=δα  (1) where b  is the atomic spacing and imα  is the constant associated with image interaction:  ( ),im im imH Fa a a= -      ( )1/32 2,, 3136 (1 )F HimF HBk T bm n WanÈ ˘+= Í ˙◊ -Î ˚, (2) dδ is the dislocation bias for the absorption of H centers, which is determined by the ratio of the relaxa-tion volumes associated with H and F centers, FHΩΩ / :  ⎟⎟⎠⎞⎜⎜⎝⎛⎟⎟⎠⎞⎜⎜⎝⎛ΩΩ=HHFHdkL2lnlnδ,           ( )13 (1 )H HbLkTm nWp n+=- (3) where ν is the Poisson ratio, µ is the shear modulus of the host matrix, kH is the square root of the total sink strength for H centers, kT has its usual meaning. The critical void radius RVcrit is given by the same equation (1) as the critical colloid radius, and it is smaller than the radius of void nuclei produced by bubble-colloid collisions [4, 5]. Accordingly, one would expect all void nuclei to grow to visible sizes. The number density of the void nuclei, NV0, can be estimated as the product of the bubble number density, NB ≈ 1018 ÷ 1019 cm–3 and the colloid volume fraction, VC ≈ 10–2 ÷ 10–1 : NV0 ≈ NB × VC ≈ 1016 ÷ 1017 cm–3. However, the observed void number densities are about 1013 ÷ 1014 cm–3 , which means that only a very small part of void nuclei survive and grow to visible sizes. The physical reason for this low “survival” rate remained unaccounted in the previous model. In a recent paper [6], an alternative mechanism of void and dislocation loop growth in alkali halides was considered, which was based on production of divacancies (Schottky defects) by interactions between dislocations and excitons, as was first suggested by Seitz in 1954 [7]. Voids can also emit divacancies due to the interaction with excitons, and so the balance between absorption and emission of divacancies determines the void evolution in this model, which has been shown to be in good agreement with ex-perimental observations. In the present paper, the exciton-induced emission of F-centers from colloids is taken into account in the calculation of colloid nucleation and growth rates in the framework of the modified rate theory [8]. The driving force of colloid growth is the large bias of halogen bubbles for absorption of H centers rather than the dislocation bias employed in the previous models [3, 4]. Thus, radiolysis (separation of chlorine and sodium into bubbles and colloids) and the development of stoichiometric defects (voids and disloca-tions) occur simultaneously but relatively independently. The only important “bridge” between these subsystems is bubble nucleation, which starts from recombination of H centers and divacancies since the homogeneous nucleation rate due to recombination of H centers is negligible. 440 V. I. Dubinko et al.: Nucleation and growth of sodium colloids in NaCl © 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim 2 Excitonic production of Frenkel and Schottky defects According to Seitz [7], an exciton may cause movement of nearby dislocation jogs, resulting in the crea-tion of equal numbers of anion and cation vacancies - Schottky defects (SD). Owing to the heat gener-ated locally during exciton annihilation, the jog can be displaced while a divacancy arises in the lat-tice. It a previous paper [8] , we have incorporated the excitonic production mechanism of both Frenkel and Schottky defects in rate theory and evaluated so called local equilibrium concentrations of SD, eqsSc , (almost) spherical (colloids and voids), cylindrical (dislocations) and planar (grain boundaries) sinks, where subscript “s” designates the type of SD (s= v for divacancies, s=F for F centers) and “S” is the type of the sink (S= V for voids, S= F for colloids, S= B for bubbles and S= D for dislocations). In absence of the radiation field, one has thsSeqsS cc = , which can be obtained from thermodynamics. Un-der irradiation, eqsSc  is generally given by the sum of thsSc  and irrsSc  , which is due to the radiation-induced emission of SD’s. Here, irrsSc  has a purely kinetic origin and can be expressed via the radiation-induced local emission rate of SD, irrsSK  [8]: 0 22Dirr FP ex exvDFP DK Z lKP r bp= ,   ( )DDDexrkZππln2=, virrvDirrvDDbKc2×= (4) VexFPFPirrvVbRlPKK20=,   virrvVirrvVDbKc2×= (5) where exl  is the mean free diffusion path of the exciton i.e. the average displacement of the exciton before it decays, FPP  is the probability of the formation of a Frenkel pair (FP) in the anion sub-lattice due to exciton decay in the bulk, 0FPK is the FP production rate in the bulk crystal,  DexZ  and Dr  are the dislocation capture efficiency and capture radius for excitons and vD is the divacancy diffusion coeffi-cient. The difference between the mean divacancy concentration, vC , and the local equilibrium concentrations at voids is the driving force for the void growth (or shrinkage), and its rate is given by [6]  ( )eqV v v vVVdR D C Cdt R= - (6) The colloid growth rate is determined by the difference between the bias-induced absorption [4] and the exciton-induced emission [8] of F centers:  ( )1 ( ) ( ) ( )C C C eqC F C F F H C H H F C F FCCdR Z R D C Z R D C Z R D Cdt R= - - (7) where ,F HC C  are the mean concentrations of F and H centers, respectively, which are determined by the usual rate equations, and ,( )CF H CZ R  are the colloid capture efficiencies for F and H centers [4]:  ,,( ) 1imF HCF H CCbZ RRa ◊= + (8) Bubble growth is due to the difference between the bias-induced absorption of H centers, since the  emission of H centers requires higher energies than for F centers and therefore it has not been taken  phys. stat. sol. (c) 2, No. 1 (2005) / www.pss-c.com 441 © 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim into account in the present model. Accordingly, we find the following expression for the bubble growth rate   ( )1 ( ) ( )B BB H B H H F B F FBdR Z R D C Z R D Cdt R= - (9)  ( )( ) 2, , ,, 21im dF H F H F HBF H BB Bb bZ RR Rma a a-= + + (10) where ,dF Ha  and ,F Hma  are the constants associated with the stress-induced diffusion anisotropy and the modulus  interaction, respectively [4]. The bubble growth rate increases with decreasing bubble radius, and so they can grow from molecular centers formed by recombination of H centers and divacancies.  On the other hand, the growth rates of colloids and voids decrease with decreasing size and they are negative below some critical radius, which determines the rate of their nucleation. The critical colloid radius is small (see eq. (1)), and so they can nucleate homogeneously at a rather high rate [9]. At the same time, the critical void radius is compara-tively large [6]  ( )bZrRDexDcritV432÷≈=π  (11) and so they can be formed only from sub-critical void nuclei produced as a result explosive recombina-tion of colloids and bubbles [6]. The present model provides a possibility to evaluate nucleation and the growth rate of colloids, bubbles and voids in the framework of rate theory [8]. The results of the model calculations are shown in Fig. 1 together with experimental data available up to date.   Fig. 1 Temperature dependence of the volume fraction and the number density of extended defects in  NaCl elec-tron irradiated up to different dose levels at a dose rate of 200 Mrad/hour. 442 V. I. Dubinko et al.: Nucleation and growth of sodium colloids in NaCl © 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim 3 Conclusion According to the present model, chlorine bubbles are the most finely dispersed ED in the system (the inter-bubble spacing is typically a few nm) implying that they start to collide with growing voids first, filling them with chlorine gas (fig. 2) The chlorine molecules in the “bubbles” are in a solid or liquid state due to a super high pressure (in the GPa range), but after collision with a void it is in the vapor state. Chlorine accumulation in voids provides a very important possibility for explosive back reactions with metallic sodium when growing voids start hitting colloids, which results in void-crack production and ultimately in explosive fracture of the material [5].     Fig. 2 Illustration of the distribution of chlorine bubbles, sodium colloids and voids in irradiated NaCl crystals. The bubble size is ~1-2 nm; the colloid size ~ inter-bubble spacing ~5-10 nm; the void size is about inter-colloid spacing ~20-30 nm, when collisions with colloids start.   From a more general perspective, the mechanisms of void and dislocation loop growth and colloid shrinkage in alkali halides considered above are characteristic manifestations of radiation-induced emis-sion of Schottky defects, which may be responsible for a number of various radiation effects such as the void dissolution and swelling saturation in irradiated metals [10], irradiation creep [11] and possibly the void ordering [12].  Acknowledgements  This study has been supported by the STCU grant # 1761 and by a NWO-NATO grant # NB 67-319. References  [1] Ch.B. Lushchik, Creation of Frenkel Defect Pairs by Excitons in Alkali Halides, in: Physics of Radiation Effects in Crystals, R.A. Johnson and A.N. Orlov, Eds. (Elsevier Science Publishers B.V., 1986). pp. 474-525.  [2] L.W. Hobbs and A. E. Hughes, Radiation Damage in Diatomic Materials at High Doses, Harwell Report AERE – R 8092 (1975); L. W. Hobbs A. E. Huges, and Pooley, Proc Roy. Soc. (Lond.) A 332, 167 (1973).  [3] Uma Jain and A. B. Lidiard, Philos. Mag. 35, 245 (1977).  [4] V. I. Dubinko, A. A. Turkin, D. I. Vainshtein, and H. W. den Hartog, J. Nucl. Mater. 227, 184 (2000); 289, 86 (2001).  [5] V. I. Dubinko, A. A. Turkin, D. I. Vainshtein, and H. W. den Hartog, Radiat. Eff. Defects Solids 156, 27-31 (2001).  [6] D V. I. Dubinko, A. A. Turkin, D. I. Vainshtein, and H. W. den Hartog, Mechanism of void growth in irradi-ated NaCl based on exciton-induced formation of divacancies at dislocations, COSIRES 2004, Helsinki, June 28 – July 2 (2004).  Cl2 Void Chlorine bubble Sodium colloid phys. stat. sol. (c) 2, No. 1 (2005) / www.pss-c.com 443 © 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim  [7] F. Seitz, Rev. Mod. Phys. 26 7 (1954)  [8] V. I. Dubinko, D. I. Vainshtein, and H. W. den Hartog, Radiat. Eff. Defects Solids 158, 705 (2003).  [9] V. I. Dubinko, A. A. Turkin, D. I. Vainshtein, and H. W. den Hartog, J. Nucl. Mater. 304, 117 (2002). [10] V. I. Dubinko and N. P. Lazarev, Effect of the radiation-induced vacancy emission from voids on the void evolution, COSIRES 2004, Helsinki, June 28 – July 2 (2004). [11] V. I. Dubinko, New mechanism of irradiation creep based on the radiation-induced vacancy emission from dislocations, submitted to Radiation Effects and Defects in Solids [12] V. I. Dubinko, New principles of radiation damage and recovery based on the radiation induced emission of Schottky defects, arXiv/cond-mat/0212154 

Nucleation and growth of sodium colloids in NaCl under irradiation:theory and experiment

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 University of Groningen
Nucleation and growth of sodium colloids in NaCl under irradiation
Dubinko, V.I.; Turkin, A.A.; Abyzov, A.S.; Sugonyako, A.V.; Vainshtein, D.I.; Hartog, Hermien
Published in:
Physica Status Solidi (C)
DOI:
10.1002/pssc.200460202
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Dubinko, V. I., Turkin, A. A., Abyzov, A. S., Sugonyako, A. V., Vainshtein, D. I., & Hartog, H. W. D. (2005).
Nucleation and growth of sodium colloids in NaCl under irradiation: theory and experiment. Physica Status
Solidi (C), 2(1), 438-443. DOI: 10.1002/pssc.200460202
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phys. stat. sol. (c) 2, No. 1, 438–443 (2005) / DOI 10.1002/pssc.200460202 
© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim 
Nucleation and growth of sodium colloids in NaCl under  
irradiation: theory and experiment 
V. I. Dubinko1, A. A. Turkin1, A. S. Abyzov1, A. V. Sugonyako2, D. I. Vainshtein2,  
and H. W. den Hartog*2  
1
  National Science Center, Kharkov Institute of Physics and Technology, 310108 Kharkov, Ukraine 
2
 Solid State Physics Laboratory, University of Groningen, Nijenborgh 4, NL-9747 AG Groningen, 
The Netherlands 
Received 11 July 2004, revised 24 September 2004, accepted 6 October 2004 
Published online 20 January 2005 
PACS 61.46.+w, 61.80.Ed, 61.80.Fe, 61.82.Ms, 82.50.Kx 
A mechanism of radiation-induced emission of Schottky defects from extended defects proposed origi-
nally for metals has recently been applied to ionic crystals, where it is based on interactions of excitons 
with extended defects such as dislocations and colloids. Exciton trapping and decay at colloids may result 
in the emission of F centers and consequent shrinkage of the colloid. In the present paper, the radiation-
induced emission of F centers is taken into account in the modeling of nucleation and growth of sodium 
colloids and chlorine bubbles in NaCl exposed to electron or gamma irradiation. The evolution of colloid 
and bubble number densities and volume fractions with increasing irradiation dose is modeled in the 
framework of a modified rate theory and compared with experimental data. Experimental values of the 
colloid volume fractions and number densities have been estimated on the basis of latent heat of melting 
of metallic Na obtained with combined differential scanning calorimetry experiments and atomic force 
microscopy investigations of metallic clusters. 
© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim 
1 Introduction 
Exposure of alkali halides to ionizing radiation results in the formation of nano-sized halogen “bubbles” 
by agglomeration of H centers and of complementary inclusions of metallic “colloids” formed by ag-
glomeration of F centers. H and F centers are the primary radiation-induced Frenkel pairs in the halide 
sub-lattice created in the crystal bulk due to decay of self-trapped excitons [1]. The H center is a hole 
trapped at an interstitial halide ion, and an F center is a vacancy in the halide sub-lattice with a trapped 
electron. The cation sub-lattice is not damaged during primary displacement processes in the crystal bulk 
[1]. However, early experimental results on electron irradiated alkali halides such as KCl and NaCl [2] 
provided evidence for the formation of perfect dislocation loops, which require agglomeration of intersti-
tial ions in both anion and cation sub-lattices. The first explanation for this phenomenon was given by 
Hobbs et al. [2], and it was based on the production of a molecular center (a halogen molecule occupying 
a stoichiometric divacancy) due to interaction between dislocations and two H centers. In 1977, Jain and 
Lidiard [3] have formulated a model, according to which, the dislocation bias for H centers was the driv-
ing force for the colloid growth in alkali halides in the same way as it is for void growth in metals under 
irradiation. The mechanism of dislocation climb [2] used in the Jain and Lidiard model, requires two H 
centers meeting each other at the dislocation core, and subsequently, the model was modified by 
Dubinko et al. [4], who suggested that a dislocation can climb due to the production of a VF center (self-
 
 
*
 Corresponding author: e-mail: h.w.den.hartog@phys.rug.nl, Phone: +3150-363-4789, Fax: +3150-363-4879 
phys. stat. sol. (c) 2, No. 1 (2005) / www.pss-c.com 439 
© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim 
trapped hole neighboring a cation vacancy) induced by interactions between a dislocation and only one H 
center. Subsequent diffusion of VF centers away from dislocations and recombination with F centers 
results in the formation of stoichiometric divacancies thus providing a driving force for the formation 
and growth of vacancy voids observed experimentally. 
In this model, the void growth rate is determined by the flux of VF centers and by the net flux of F cen-
ters (i.e. the difference between F and H center fluxes), which depends on the difference between the 
void bias and the mean bias of the system for the absorption of F centers. This difference is positive for 
void radii lager than a critical radius, crit
V
R , which is determined by the ratio of material constants related 
to the void and dislocation biases [4]: 
 ( )b
b
R
d
im
crit
V
21÷≈=
δ
α  (1) 
where b  is the atomic spacing and imα  is the constant associated with image interaction: 
 ( ),im im imH Fa a a= -      
( )
1/32 2
,
, 3
1
36 (1 )
F Him
F H
Bk T b
m n W
a
n
È ˘+
= Í ˙
◊ -Î ˚
,
 (2) 
d
δ is the dislocation bias for the absorption of H centers, which is determined by the ratio of the relaxa-
tion volumes associated with H and F centers, 
FH
ΩΩ / : 
 
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
Ω
Ω
=
HHF
H
d
kL
2
lnlnδ
,           
( )1
3 (1 )H H
bL
kT
m n
W
p n
+
=
-
 (3) 
where ν is the Poisson ratio, µ is the shear modulus of the host matrix, kH is the square root of the total 
sink strength for H centers, kT has its usual meaning. 
The critical void radius RVcrit is given by the same equation (1) as the critical colloid radius, and it is 
smaller than the radius of void nuclei produced by bubble-colloid collisions [4, 5]. Accordingly, one 
would expect all void nuclei to grow to visible sizes. The number density of the void nuclei, NV0, can be 
estimated as the product of the bubble number density, NB ≈ 1018 ÷ 1019 cm–3 and the colloid volume 
fraction, VC ≈ 10–2 ÷ 10–1 : NV0 ≈ NB × VC ≈ 1016 ÷ 1017 cm–3. However, the observed void number densities 
are about 1013 ÷ 1014 cm–3 , which means that only a very small part of void nuclei survive and grow to 
visible sizes. The physical reason for this low “survival” rate remained unaccounted in the previous 
model. 
In a recent paper [6], an alternative mechanism of void and dislocation loop growth in alkali halides was 
considered, which was based on production of divacancies (Schottky defects) by interactions between 
dislocations and excitons, as was first suggested by Seitz in 1954 [7]. Voids can also emit divacancies 
due to the interaction with excitons, and so the balance between absorption and emission of divacancies 
determines the void evolution in this model, which has been shown to be in good agreement with ex-
perimental observations. 
In the present paper, the exciton-induced emission of F-centers from colloids is taken into account in the 
calculation of colloid nucleation and growth rates in the framework of the modified rate theory [8]. The 
driving force of colloid growth is the large bias of halogen bubbles for absorption of H centers rather 
than the dislocation bias employed in the previous models [3, 4]. Thus, radiolysis (separation of chlorine 
and sodium into bubbles and colloids) and the development of stoichiometric defects (voids and disloca-
tions) occur simultaneously but relatively independently. The only important “bridge” between these 
subsystems is bubble nucleation, which starts from recombination of H centers and divacancies since the 
homogeneous nucleation rate due to recombination of H centers is negligible. 
440 V. I. Dubinko et al.: Nucleation and growth of sodium colloids in NaCl 
© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim 
2 Excitonic production of Frenkel and Schottky defects 
According to Seitz [7], an exciton may cause movement of nearby dislocation jogs, resulting in the crea-
tion of equal numbers of anion and cation vacancies - Schottky defects (SD). Owing to the heat gener-
ated locally during exciton annihilation, the jog can be displaced while a divacancy arises in the lat-
tice. It a previous paper [8] , we have incorporated the excitonic production mechanism of both Frenkel and 
Schottky defects in rate theory and evaluated so called local equilibrium concentrations of SD, eqsSc , (almost) 
spherical (colloids and voids), cylindrical (dislocations) and planar (grain boundaries) sinks, where subscript 
“s” designates the type of SD (s= v for divacancies, s=F for F centers) and “S” is the type of the sink (S= V 
for voids, S= F for colloids, S= B for bubbles and S= D for dislocations). 
In absence of the radiation field, one has thsS
eq
sS cc = , which can be obtained from thermodynamics. Un-
der irradiation, eqsSc  is generally given by the sum of 
th
sS
c  and irr
sS
c  , which is due to the radiation-induced 
emission of SD’s. Here, irr
sS
c  has a purely kinetic origin and can be expressed via the radiation-induced 
local emission rate of SD, irrsSK  [8]: 
0 2
2
D
irr FP ex ex
vD
FP D
K Z lK
P r bp
= ,   
( )
DD
D
ex
rk
Z
π
π
ln
2
=
, 
v
irr
vD
irr
vD
D
b
Kc
2
×=
 (4) 
V
ex
FP
FPirr
vV
bR
l
P
K
K
20
=
,   
v
irr
vV
irr
vV
D
b
Kc
2
×=
 (5) 
where 
ex
l  is the mean free diffusion path of the exciton i.e. the average displacement of the exciton 
before it decays, 
FP
P  is the probability of the formation of a Frenkel pair (FP) in the anion sub-lattice 
due to exciton decay in the bulk, 0
FP
K is the FP production rate in the bulk crystal,  D
ex
Z  and 
D
r  are the 
dislocation capture efficiency and capture radius for excitons and 
v
D is the divacancy diffusion coeffi-
cient. 
The difference between the mean divacancy concentration, vC , and the local equilibrium concentrations at 
voids is the driving force for the void growth (or shrinkage), and its rate is given by [6] 
 ( )eqV v v vV
V
dR D C C
dt R
= -
 (6) 
The colloid growth rate is determined by the difference between the bias-induced absorption [4] and the 
exciton-induced emission [8] of F centers: 
 ( )
1 ( ) ( ) ( )C C C eqC F C F F H C H H F C F FC
C
dR Z R D C Z R D C Z R D C
dt R
= - -
 (7) 
where ,F HC C  are the mean concentrations of F and H centers, respectively, which are determined by the 
usual rate equations, and 
,
( )CF H CZ R  are the colloid capture efficiencies for F and H centers [4]: 
 
,
,
( ) 1
im
F HC
F H C
C
b
Z R
R
a ◊
= +
 (8) 
Bubble growth is due to the difference between the bias-induced absorption of H centers, since the  
emission of H centers requires higher energies than for F centers and therefore it has not been taken  
phys. stat. sol. (c) 2, No. 1 (2005) / www.pss-c.com 441 
© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim 
into account in the present model. Accordingly, we find the following expression for the bubble growth 
rate  
 ( )
1 ( ) ( )B BB H B H H F B F F
B
dR Z R D C Z R D C
dt R
= -
 (9) 
 ( )
( ) 2, , ,
, 21
im d
F H F H F HB
F H B
B B
b b
Z R
R R
m
a a a-
= + +
 (10) 
where 
,
d
F Ha  and ,F Hma  are the constants associated with the stress-induced diffusion anisotropy and the 
modulus  interaction, respectively [4]. 
The bubble growth rate increases with decreasing bubble radius, and so they can grow from molecular 
centers formed by recombination of H centers and divacancies.  On the other hand, the growth rates of 
colloids and voids decrease with decreasing size and they are negative below some critical radius, which 
determines the rate of their nucleation. The critical colloid radius is small (see eq. (1)), and so they can 
nucleate homogeneously at a rather high rate [9]. At the same time, the critical void radius is compara-
tively large [6] 
 ( )b
Z
r
R
D
ex
Dcrit
V
43
2
÷≈=
π  (11) 
and so they can be formed only from sub-critical void nuclei produced as a result explosive recombina-
tion of colloids and bubbles [6]. The present model provides a possibility to evaluate nucleation and the 
growth rate of colloids, bubbles and voids in the framework of rate theory [8]. The results of the model 
calculations are shown in Fig. 1 together with experimental data available up to date. 
 
 
Fig. 1 Temperature dependence of the volume fraction and the number density of extended defects in  NaCl elec-
tron irradiated up to different dose levels at a dose rate of 200 Mrad/hour. 
442 V. I. Dubinko et al.: Nucleation and growth of sodium colloids in NaCl 
© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim 
3 Conclusion 
According to the present model, chlorine bubbles are the most finely dispersed ED in the system (the 
inter-bubble spacing is typically a few nm) implying that they start to collide with growing voids first, 
filling them with chlorine gas (fig. 2) The chlorine molecules in the “bubbles” are in a solid or liquid 
state due to a super high pressure (in the GPa range), but after collision with a void it is in the vapor 
state. Chlorine accumulation in voids provides a very important possibility for explosive back reactions 
with metallic sodium when growing voids start hitting colloids, which results in void-crack production 
and ultimately in explosive fracture of the material [5]. 
 
 
 
 
Fig. 2 Illustration of the distribution of chlorine bubbles, sodium colloids and voids in irradiated NaCl crystals. 
The bubble size is ~1-2 nm; the colloid size ~ inter-bubble spacing ~5-10 nm; the void size is about inter-colloid 
spacing ~20-30 nm, when collisions with colloids start. 
 
 
From a more general perspective, the mechanisms of void and dislocation loop growth and colloid 
shrinkage in alkali halides considered above are characteristic manifestations of radiation-induced emis-
sion of Schottky defects, which may be responsible for a number of various radiation effects such as the 
void dissolution and swelling saturation in irradiated metals [10], irradiation creep [11] and possibly the 
void ordering [12]. 
 
Acknowledgements  This study has been supported by the STCU grant # 1761 and by a NWO-NATO grant # NB 
67-319. 
References 
 [1] Ch.B. Lushchik, Creation of Frenkel Defect Pairs by Excitons in Alkali Halides, in: Physics of Radiation 
Effects in Crystals, R.A. Johnson and A.N. Orlov, Eds. (Elsevier Science Publishers B.V., 1986). pp. 474-
525. 
 [2] L.W. Hobbs and A. E. Hughes, Radiation Damage in Diatomic Materials at High Doses, Harwell Report AERE 
– R 8092 (1975); L. W. Hobbs A. E. Huges, and Pooley, Proc Roy. Soc. (Lond.) A 332, 167 (1973). 
 [3] Uma Jain and A. B. Lidiard, Philos. Mag. 35, 245 (1977). 
 [4] V. I. Dubinko, A. A. Turkin, D. I. Vainshtein, and H. W. den Hartog, J. Nucl. Mater. 227, 184 (2000); 289, 86 
(2001). 
 [5] V. I. Dubinko, A. A. Turkin, D. I. Vainshtein, and H. W. den Hartog, Radiat. Eff. Defects Solids 156, 27-31 
(2001). 
 [6] D V. I. Dubinko, A. A. Turkin, D. I. Vainshtein, and H. W. den Hartog, Mechanism of void growth in irradi-
ated NaCl based on exciton-induced formation of divacancies at dislocations, COSIRES 2004, Helsinki, June 
28 – July 2 (2004). 
 
Cl2 
Void 
Chlorine bubble 
Sodium colloid 
phys. stat. sol. (c) 2, No. 1 (2005) / www.pss-c.com 443 
© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim 
 [7] F. Seitz, Rev. Mod. Phys. 26 7 (1954) 
 [8] V. I. Dubinko, D. I. Vainshtein, and H. W. den Hartog, Radiat. Eff. Defects Solids 158, 705 (2003). 
 [9] V. I. Dubinko, A. A. Turkin, D. I. Vainshtein, and H. W. den Hartog, J. Nucl. Mater. 304, 117 (2002). 
[10] V. I. Dubinko and N. P. Lazarev, Effect of the radiation-induced vacancy emission from voids on the void 
evolution, COSIRES 2004, Helsinki, June 28 – July 2 (2004). 
[11] V. I. Dubinko, New mechanism of irradiation creep based on the radiation-induced vacancy emission from 
dislocations, submitted to Radiation Effects and Defects in Solids 
[12] V. I. Dubinko, New principles of radiation damage and recovery based on the radiation induced emission of 
Schottky defects, arXiv/cond-mat/0212154 


NARCIS 

   University of GroningenNucleation and growth of sodium colloids in NaCl under irradiationDubinko, V.I.; Turkin, A.A.; Abyzov, A.S.; Sugonyako, A.V.; Vainshtein, D.I.; Hartog, H.W. denPublished in:Physica Status Solidi (C)DOI:10.1002/pssc.200460202IMPORTANT NOTE: You are advised to consult the publisher's version (publisher's PDF) if you wish to cite fromit. Please check the document version below.Document VersionPublisher's PDF, also known as Version of recordPublication date:2005Link to publication in University of Groningen/UMCG research databaseCitation for published version (APA):Dubinko, V. I., Turkin, A. A., Abyzov, A. S., Sugonyako, A. V., Vainshtein, D. I., & Hartog, H. W. D. (2005).Nucleation and growth of sodium colloids in NaCl under irradiation: theory and experiment. Physica StatusSolidi (C), 2(1), 438-443. https://doi.org/10.1002/pssc.200460202CopyrightOther than for strictly personal use, it is not permitted to download or to forward/distribute the text or part of it without the consent of theauthor(s) and/or copyright holder(s), unless the work is under an open content license (like Creative Commons).The publication may also be distributed here under the terms of Article 25fa of the Dutch Copyright Act, indicated by the “Taverne” license.More information can be found on the University of Groningen website: https://www.rug.nl/library/open-access/self-archiving-pure/taverne-amendment.Take-down policyIf you believe that this document breaches copyright please contact us providing details, and we will remove access to the work immediatelyand investigate your claim.Downloaded from the University of Groningen/UMCG research database (Pure): http://www.rug.nl/research/portal. For technical reasons thenumber of authors shown on this cover page is limited to 10 maximum.Download date: 27-04-2023phys. stat. sol. (c) 2, No. 1, 438–443 (2005) / DOI 10.1002/pssc.200460202 © 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim Nucleation and growth of sodium colloids in NaCl under  irradiation: theory and experiment V. I. Dubinko1, A. A. Turkin1, A. S. Abyzov1, A. V. Sugonyako2, D. I. Vainshtein2,  and H. W. den Hartog*2  1  National Science Center, Kharkov Institute of Physics and Technology, 310108 Kharkov, Ukraine 2 Solid State Physics Laboratory, University of Groningen, Nijenborgh 4, NL-9747 AG Groningen, The Netherlands Received 11 July 2004, revised 24 September 2004, accepted 6 October 2004 Published online 20 January 2005 PACS 61.46.+w, 61.80.Ed, 61.80.Fe, 61.82.Ms, 82.50.Kx A mechanism of radiation-induced emission of Schottky defects from extended defects proposed origi-nally for metals has recently been applied to ionic crystals, where it is based on interactions of excitons with extended defects such as dislocations and colloids. Exciton trapping and decay at colloids may result in the emission of F centers and consequent shrinkage of the colloid. In the present paper, the radiation-induced emission of F centers is taken into account in the modeling of nucleation and growth of sodium colloids and chlorine bubbles in NaCl exposed to electron or gamma irradiation. The evolution of colloid and bubble number densities and volume fractions with increasing irradiation dose is modeled in the framework of a modified rate theory and compared with experimental data. Experimental values of the colloid volume fractions and number densities have been estimated on the basis of latent heat of melting of metallic Na obtained with combined differential scanning calorimetry experiments and atomic force microscopy investigations of metallic clusters. © 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim 1 Introduction Exposure of alkali halides to ionizing radiation results in the formation of nano-sized halogen “bubbles” by agglomeration of H centers and of complementary inclusions of metallic “colloids” formed by ag-glomeration of F centers. H and F centers are the primary radiation-induced Frenkel pairs in the halide sub-lattice created in the crystal bulk due to decay of self-trapped excitons [1]. The H center is a hole trapped at an interstitial halide ion, and an F center is a vacancy in the halide sub-lattice with a trapped electron. The cation sub-lattice is not damaged during primary displacement processes in the crystal bulk [1]. However, early experimental results on electron irradiated alkali halides such as KCl and NaCl [2] provided evidence for the formation of perfect dislocation loops, which require agglomeration of intersti-tial ions in both anion and cation sub-lattices. The first explanation for this phenomenon was given by Hobbs et al. [2], and it was based on the production of a molecular center (a halogen molecule occupying a stoichiometric divacancy) due to interaction between dislocations and two H centers. In 1977, Jain and Lidiard [3] have formulated a model, according to which, the dislocation bias for H centers was the driv-ing force for the colloid growth in alkali halides in the same way as it is for void growth in metals under irradiation. The mechanism of dislocation climb [2] used in the Jain and Lidiard model, requires two H centers meeting each other at the dislocation core, and subsequently, the model was modified by Dubinko et al. [4], who suggested that a dislocation can climb due to the production of a VF center (self-  * Corresponding author: e-mail: h.w.den.hartog@phys.rug.nl, Phone: +3150-363-4789, Fax: +3150-363-4879 phys. stat. sol. (c) 2, No. 1 (2005) / www.pss-c.com 439 © 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim trapped hole neighboring a cation vacancy) induced by interactions between a dislocation and only one H center. Subsequent diffusion of VF centers away from dislocations and recombination with F centers results in the formation of stoichiometric divacancies thus providing a driving force for the formation and growth of vacancy voids observed experimentally. In this model, the void growth rate is determined by the flux of VF centers and by the net flux of F cen-ters (i.e. the difference between F and H center fluxes), which depends on the difference between the void bias and the mean bias of the system for the absorption of F centers. This difference is positive for void radii lager than a critical radius, critVR , which is determined by the ratio of material constants related to the void and dislocation biases [4]:  ( )bbRdimcritV21÷≈=δα  (1) where b  is the atomic spacing and imα  is the constant associated with image interaction:  ( ),im im imH Fa a a= -      ( )1/32 2,, 3136 (1 )F HimF HBk T bm n WanÈ ˘+= Í ˙◊ -Î ˚, (2) dδ is the dislocation bias for the absorption of H centers, which is determined by the ratio of the relaxa-tion volumes associated with H and F centers, FHΩΩ / :  ⎟⎟⎠⎞⎜⎜⎝⎛⎟⎟⎠⎞⎜⎜⎝⎛ΩΩ=HHFHdkL2lnlnδ,           ( )13 (1 )H HbLkTm nWp n+=- (3) where ν is the Poisson ratio, µ is the shear modulus of the host matrix, kH is the square root of the total sink strength for H centers, kT has its usual meaning. The critical void radius RVcrit is given by the same equation (1) as the critical colloid radius, and it is smaller than the radius of void nuclei produced by bubble-colloid collisions [4, 5]. Accordingly, one would expect all void nuclei to grow to visible sizes. The number density of the void nuclei, NV0, can be estimated as the product of the bubble number density, NB ≈ 1018 ÷ 1019 cm–3 and the colloid volume fraction, VC ≈ 10–2 ÷ 10–1 : NV0 ≈ NB × VC ≈ 1016 ÷ 1017 cm–3. However, the observed void number densities are about 1013 ÷ 1014 cm–3 , which means that only a very small part of void nuclei survive and grow to visible sizes. The physical reason for this low “survival” rate remained unaccounted in the previous model. In a recent paper [6], an alternative mechanism of void and dislocation loop growth in alkali halides was considered, which was based on production of divacancies (Schottky defects) by interactions between dislocations and excitons, as was first suggested by Seitz in 1954 [7]. Voids can also emit divacancies due to the interaction with excitons, and so the balance between absorption and emission of divacancies determines the void evolution in this model, which has been shown to be in good agreement with ex-perimental observations. In the present paper, the exciton-induced emission of F-centers from colloids is taken into account in the calculation of colloid nucleation and growth rates in the framework of the modified rate theory [8]. The driving force of colloid growth is the large bias of halogen bubbles for absorption of H centers rather than the dislocation bias employed in the previous models [3, 4]. Thus, radiolysis (separation of chlorine and sodium into bubbles and colloids) and the development of stoichiometric defects (voids and disloca-tions) occur simultaneously but relatively independently. The only important “bridge” between these subsystems is bubble nucleation, which starts from recombination of H centers and divacancies since the homogeneous nucleation rate due to recombination of H centers is negligible. 440 V. I. Dubinko et al.: Nucleation and growth of sodium colloids in NaCl © 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim 2 Excitonic production of Frenkel and Schottky defects According to Seitz [7], an exciton may cause movement of nearby dislocation jogs, resulting in the crea-tion of equal numbers of anion and cation vacancies - Schottky defects (SD). Owing to the heat gener-ated locally during exciton annihilation, the jog can be displaced while a divacancy arises in the lat-tice. It a previous paper [8] , we have incorporated the excitonic production mechanism of both Frenkel and Schottky defects in rate theory and evaluated so called local equilibrium concentrations of SD, eqsSc , (almost) spherical (colloids and voids), cylindrical (dislocations) and planar (grain boundaries) sinks, where subscript “s” designates the type of SD (s= v for divacancies, s=F for F centers) and “S” is the type of the sink (S= V for voids, S= F for colloids, S= B for bubbles and S= D for dislocations). In absence of the radiation field, one has thsSeqsS cc = , which can be obtained from thermodynamics. Un-der irradiation, eqsSc  is generally given by the sum of thsSc  and irrsSc  , which is due to the radiation-induced emission of SD’s. Here, irrsSc  has a purely kinetic origin and can be expressed via the radiation-induced local emission rate of SD, irrsSK  [8]: 0 22Dirr FP ex exvDFP DK Z lKP r bp= ,   ( )DDDexrkZππln2=, virrvDirrvDDbKc2×= (4) VexFPFPirrvVbRlPKK20=,   virrvVirrvVDbKc2×= (5) where exl  is the mean free diffusion path of the exciton i.e. the average displacement of the exciton before it decays, FPP  is the probability of the formation of a Frenkel pair (FP) in the anion sub-lattice due to exciton decay in the bulk, 0FPK is the FP production rate in the bulk crystal,  DexZ  and Dr  are the dislocation capture efficiency and capture radius for excitons and vD is the divacancy diffusion coeffi-cient. The difference between the mean divacancy concentration, vC , and the local equilibrium concentrations at voids is the driving force for the void growth (or shrinkage), and its rate is given by [6]  ( )eqV v v vVVdR DC Cdt R= - (6) The colloid growth rate is determined by the difference between the bias-induced absorption [4] and the exciton-induced emission [8] of F centers:  ( )1( ) ( ) ( )C C C eqC F C F F H C H H F C F FCCdRZ R D C Z R D C Z R D Cdt R= - - (7) where ,F HC C  are the mean concentrations of F and H centers, respectively, which are determined by the usual rate equations, and , ( )CF H CZ R  are the colloid capture efficiencies for F and H centers [4]:  ,, ( ) 1imF HCF H CCbZ RRa ◊= + (8) Bubble growth is due to the difference between the bias-induced absorption of H centers, since the  emission of H centers requires higher energies than for F centers and therefore it has not been taken  phys. stat. sol. (c) 2, No. 1 (2005) / www.pss-c.com 441 © 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim into account in the present model. Accordingly, we find the following expression for the bubble growth rate   ( )1( ) ( )B BB H B H H F B F FBdRZ R D C Z R D Cdt R= - (9)  ( )( ) 2, , ,, 21im dF H F H F HBF H BB Bb bZ RR Rma a a-= + + (10) where ,dF Ha  and ,F Hma  are the constants associated with the stress-induced diffusion anisotropy and the modulus  interaction, respectively [4]. The bubble growth rate increases with decreasing bubble radius, and so they can grow from molecular centers formed by recombination of H centers and divacancies.  On the other hand, the growth rates of colloids and voids decrease with decreasing size and they are negative below some critical radius, which determines the rate of their nucleation. The critical colloid radius is small (see eq. (1)), and so they can nucleate homogeneously at a rather high rate [9]. At the same time, the critical void radius is compara-tively large [6]  ( )bZrRDexDcritV432÷≈=π  (11) and so they can be formed only from sub-critical void nuclei produced as a result explosive recombina-tion of colloids and bubbles [6]. The present model provides a possibility to evaluate nucleation and the growth rate of colloids, bubbles and voids in the framework of rate theory [8]. The results of the model calculations are shown in Fig. 1 together with experimental data available up to date.   Fig. 1 Temperature dependence of the volume fraction and the number density of extended defects in  NaCl elec-tron irradiated up to different dose levels at a dose rate of 200 Mrad/hour. 442 V. I. Dubinko et al.: Nucleation and growth of sodium colloids in NaCl © 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim 3 Conclusion According to the present model, chlorine bubbles are the most finely dispersed ED in the system (the inter-bubble spacing is typically a few nm) implying that they start to collide with growing voids first, filling them with chlorine gas (fig. 2) The chlorine molecules in the “bubbles” are in a solid or liquid state due to a super high pressure (in the GPa range), but after collision with a void it is in the vapor state. Chlorine accumulation in voids provides a very important possibility for explosive back reactions with metallic sodium when growing voids start hitting colloids, which results in void-crack production and ultimately in explosive fracture of the material [5].     Fig. 2 Illustration of the distribution of chlorine bubbles, sodium colloids and voids in irradiated NaCl crystals. The bubble size is ~1-2 nm; the colloid size ~ inter-bubble spacing ~5-10 nm; the void size is about inter-colloid spacing ~20-30 nm, when collisions with colloids start.   From a more general perspective, the mechanisms of void and dislocation loop growth and colloid shrinkage in alkali halides considered above are characteristic manifestations of radiation-induced emis-sion of Schottky defects, which may be responsible for a number of various radiation effects such as the void dissolution and swelling saturation in irradiated metals [10], irradiation creep [11] and possibly the void ordering [12].  Acknowledgements  This study has been supported by the STCU grant # 1761 and by a NWO-NATO grant # NB 67-319. References  [1] Ch.B. Lushchik, Creation of Frenkel Defect Pairs by Excitons in Alkali Halides, in: Physics of Radiation Effects in Crystals, R.A. Johnson and A.N. Orlov, Eds. (Elsevier Science Publishers B.V., 1986). pp. 474-525.  [2] L.W. Hobbs and A. E. Hughes, Radiation Damage in Diatomic Materials at High Doses, Harwell Report AERE – R 8092 (1975); L. W. Hobbs A. E. Huges, and Pooley, Proc Roy. Soc. (Lond.) A 332, 167 (1973).  [3] Uma Jain and A. B. Lidiard, Philos. Mag. 35, 245 (1977).  [4] V. I. Dubinko, A. A. Turkin, D. I. Vainshtein, and H. W. den Hartog, J. Nucl. Mater. 227, 184 (2000); 289, 86 (2001).  [5] V. I. Dubinko, A. A. Turkin, D. I. Vainshtein, and H. W. den Hartog, Radiat. Eff. Defects Solids 156, 27-31 (2001).  [6] D V. I. Dubinko, A. A. Turkin, D. I. Vainshtein, and H. W. den Hartog, Mechanism of void growth in irradi-ated NaCl based on exciton-induced formation of divacancies at dislocations, COSIRES 2004, Helsinki, June 28 – July 2 (2004).  Cl2 Void Chlorine bubble Sodium colloid phys. stat. sol. (c) 2, No. 1 (2005) / www.pss-c.com 443 © 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim  [7] F. Seitz, Rev. Mod. Phys. 26 7 (1954)  [8] V. I. Dubinko, D. I. Vainshtein, and H. W. den Hartog, Radiat. Eff. Defects Solids 158, 705 (2003).  [9] V. I. Dubinko, A. A. Turkin, D. I. Vainshtein, and H. W. den Hartog, J. Nucl. Mater. 304, 117 (2002). [10] V. I. Dubinko and N. P. Lazarev, Effect of the radiation-induced vacancy emission from voids on the void evolution, COSIRES 2004, Helsinki, June 28 – July 2 (2004). [11] V. I. Dubinko, New mechanism of irradiation creep based on the radiation-induced vacancy emission from dislocations, submitted to Radiation Effects and Defects in Solids [12] V. I. Dubinko, New principles of radiation damage and recovery based on the radiation induced emission of Schottky defects, arXiv/cond-mat/0212154 

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University of GroningenNucleation and growth of sodium colloids in NaCl under irradiationDubinko, V.I.; Turkin, A.A.; Abyzov, A.S.; Sugonyako, A.V.; Vainshtein, D.I.; Hartog, H.W. denPublished in:Physica Status Solidi (C)DOI:10.1002/pssc.200460202IMPORTANT NOTE: You are advised to consult the publisher's version (publisher's PDF) if you wish to cite fromit. Please check the document version below.Document VersionPublisher's PDF, also known as Version of recordPublication date:2005Link to publication in University of Groningen/UMCG research databaseCitation for published version (APA):Dubinko, V. I., Turkin, A. A., Abyzov, A. S., Sugonyako, A. V., Vainshtein, D. I., & Hartog, H. W. D. (2005).Nucleation and growth of sodium colloids in NaCl under irradiation: theory and experiment. Physica StatusSolidi (C), 2(1), 438-443. https://doi.org/10.1002/pssc.200460202CopyrightOther than for strictly personal use, it is not permitted to download or to forward/distribute the text or part of it without the consent of theauthor(s) and/or copyright holder(s), unless the work is under an open content license (like Creative Commons).The publication may also be distributed here under the terms of Article 25fa of the Dutch Copyright Act, indicated by the “Taverne” license.More information can be found on the University of Groningen website: https://www.rug.nl/library/open-access/self-archiving-pure/taverne-amendment.Take-down policyIf you believe that this document breaches copyright please contact us providing details, and we will remove access to the work immediatelyand investigate your claim.Downloaded from the University of Groningen/UMCG research database (Pure): http://www.rug.nl/research/portal. For technical reasons thenumber of authors shown on this cover page is limited to 10 maximum.Download date: 24-12-2025phys. stat. sol. (c) 2, No. 1, 438–443 (2005) / DOI 10.1002/pssc.200460202 © 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim Nucleation and growth of sodium colloids in NaCl under  irradiation: theory and experiment V. I. Dubinko1, A. A. Turkin1, A. S. Abyzov1, A. V. Sugonyako2, D. I. Vainshtein2,  and H. W. den Hartog*2  1  National Science Center, Kharkov Institute of Physics and Technology, 310108 Kharkov, Ukraine 2 Solid State Physics Laboratory, University of Groningen, Nijenborgh 4, NL-9747 AG Groningen, The Netherlands Received 11 July 2004, revised 24 September 2004, accepted 6 October 2004 Published online 20 January 2005 PACS 61.46.+w, 61.80.Ed, 61.80.Fe, 61.82.Ms, 82.50.Kx A mechanism of radiation-induced emission of Schottky defects from extended defects proposed origi-nally for metals has recently been applied to ionic crystals, where it is based on interactions of excitons with extended defects such as dislocations and colloids. Exciton trapping and decay at colloids may result in the emission of F centers and consequent shrinkage of the colloid. In the present paper, the radiation-induced emission of F centers is taken into account in the modeling of nucleation and growth of sodium colloids and chlorine bubbles in NaCl exposed to electron or gamma irradiation. The evolution of colloid and bubble number densities and volume fractions with increasing irradiation dose is modeled in the framework of a modified rate theory and compared with experimental data. Experimental values of the colloid volume fractions and number densities have been estimated on the basis of latent heat of melting of metallic Na obtained with combined differential scanning calorimetry experiments and atomic force microscopy investigations of metallic clusters. © 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim 1 Introduction Exposure of alkali halides to ionizing radiation results in the formation of nano-sized halogen “bubbles” by agglomeration of H centers and of complementary inclusions of metallic “colloids” formed by ag-glomeration of F centers. H and F centers are the primary radiation-induced Frenkel pairs in the halide sub-lattice created in the crystal bulk due to decay of self-trapped excitons [1]. The H center is a hole trapped at an interstitial halide ion, and an F center is a vacancy in the halide sub-lattice with a trapped electron. The cation sub-lattice is not damaged during primary displacement processes in the crystal bulk [1]. However, early experimental results on electron irradiated alkali halides such as KCl and NaCl [2] provided evidence for the formation of perfect dislocation loops, which require agglomeration of intersti-tial ions in both anion and cation sub-lattices. The first explanation for this phenomenon was given by Hobbs et al. [2], and it was based on the production of a molecular center (a halogen molecule occupying a stoichiometric divacancy) due to interaction between dislocations and two H centers. In 1977, Jain and Lidiard [3] have formulated a model, according to which, the dislocation bias for H centers was the driv-ing force for the colloid growth in alkali halides in the same way as it is for void growth in metals under irradiation. The mechanism of dislocation climb [2] used in the Jain and Lidiard model, requires two H centers meeting each other at the dislocation core, and subsequently, the model was modified by Dubinko et al. [4], who suggested that a dislocation can climb due to the production of a VF center (self-  * Corresponding author: e-mail: h.w.den.hartog@phys.rug.nl, Phone: +3150-363-4789, Fax: +3150-363-4879 phys. stat. sol. (c) 2, No. 1 (2005) / www.pss-c.com 439 © 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim trapped hole neighboring a cation vacancy) induced by interactions between a dislocation and only one H center. Subsequent diffusion of VF centers away from dislocations and recombination with F centers results in the formation of stoichiometric divacancies thus providing a driving force for the formation and growth of vacancy voids observed experimentally. In this model, the void growth rate is determined by the flux of VF centers and by the net flux of F cen-ters (i.e. the difference between F and H center fluxes), which depends on the difference between the void bias and the mean bias of the system for the absorption of F centers. This difference is positive for void radii lager than a critical radius, critVR , which is determined by the ratio of material constants related to the void and dislocation biases [4]:  ( )bbRdimcritV21÷≈=δα  (1) where b  is the atomic spacing and imα  is the constant associated with image interaction:  ( ),im im imH Fa a a= -      ( )1/32 2,, 3136 (1 )F HimF HBk T bm n WanÈ ˘+= Í ˙◊ -Î ˚, (2) dδ is the dislocation bias for the absorption of H centers, which is determined by the ratio of the relaxa-tion volumes associated with H and F centers, FHΩΩ / :  ⎟⎟⎠⎞⎜⎜⎝⎛⎟⎟⎠⎞⎜⎜⎝⎛ΩΩ=HHFHdkL2lnlnδ,           ( )13 (1 )H HbLkTm nWp n+=- (3) where ν is the Poisson ratio, µ is the shear modulus of the host matrix, kH is the square root of the total sink strength for H centers, kT has its usual meaning. The critical void radius RVcrit is given by the same equation (1) as the critical colloid radius, and it is smaller than the radius of void nuclei produced by bubble-colloid collisions [4, 5]. Accordingly, one would expect all void nuclei to grow to visible sizes. The number density of the void nuclei, NV0, can be estimated as the product of the bubble number density, NB ≈ 1018 ÷ 1019 cm–3 and the colloid volume fraction, VC ≈ 10–2 ÷ 10–1 : NV0 ≈ NB × VC ≈ 1016 ÷ 1017 cm–3. However, the observed void number densities are about 1013 ÷ 1014 cm–3 , which means that only a very small part of void nuclei survive and grow to visible sizes. The physical reason for this low “survival” rate remained unaccounted in the previous model. In a recent paper [6], an alternative mechanism of void and dislocation loop growth in alkali halides was considered, which was based on production of divacancies (Schottky defects) by interactions between dislocations and excitons, as was first suggested by Seitz in 1954 [7]. Voids can also emit divacancies due to the interaction with excitons, and so the balance between absorption and emission of divacancies determines the void evolution in this model, which has been shown to be in good agreement with ex-perimental observations. In the present paper, the exciton-induced emission of F-centers from colloids is taken into account in the calculation of colloid nucleation and growth rates in the framework of the modified rate theory [8]. The driving force of colloid growth is the large bias of halogen bubbles for absorption of H centers rather than the dislocation bias employed in the previous models [3, 4]. Thus, radiolysis (separation of chlorine and sodium into bubbles and colloids) and the development of stoichiometric defects (voids and disloca-tions) occur simultaneously but relatively independently. The only important “bridge” between these subsystems is bubble nucleation, which starts from recombination of H centers and divacancies since the homogeneous nucleation rate due to recombination of H centers is negligible. 440 V. I. Dubinko et al.: Nucleation and growth of sodium colloids in NaCl © 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim 2 Excitonic production of Frenkel and Schottky defects According to Seitz [7], an exciton may cause movement of nearby dislocation jogs, resulting in the crea-tion of equal numbers of anion and cation vacancies - Schottky defects (SD). Owing to the heat gener-ated locally during exciton annihilation, the jog can be displaced while a divacancy arises in the lat-tice. It a previous paper [8] , we have incorporated the excitonic production mechanism of both Frenkel and Schottky defects in rate theory and evaluated so called local equilibrium concentrations of SD, eqsSc , (almost) spherical (colloids and voids), cylindrical (dislocations) and planar (grain boundaries) sinks, where subscript “s” designates the type of SD (s= v for divacancies, s=F for F centers) and “S” is the type of the sink (S= V for voids, S= F for colloids, S= B for bubbles and S= D for dislocations). In absence of the radiation field, one has thsSeqsS cc = , which can be obtained from thermodynamics. Un-der irradiation, eqsSc  is generally given by the sum of thsSc  and irrsSc  , which is due to the radiation-induced emission of SD’s. Here, irrsSc  has a purely kinetic origin and can be expressed via the radiation-induced local emission rate of SD, irrsSK  [8]: 0 22Dirr FP ex exvDFP DK Z lKP r bp= ,   ( )DDDexrkZππln2=, virrvDirrvDDbKc2×= (4) VexFPFPirrvVbRlPKK20=,   virrvVirrvVDbKc2×= (5) where exl  is the mean free diffusion path of the exciton i.e. the average displacement of the exciton before it decays, FPP  is the probability of the formation of a Frenkel pair (FP) in the anion sub-lattice due to exciton decay in the bulk, 0FPK is the FP production rate in the bulk crystal,  DexZ  and Dr  are the dislocation capture efficiency and capture radius for excitons and vD is the divacancy diffusion coeffi-cient. The difference between the mean divacancy concentration, vC , and the local equilibrium concentrations at voids is the driving force for the void growth (or shrinkage), and its rate is given by [6]  ( )eqV v v vVVdR DC Cdt R= - (6) The colloid growth rate is determined by the difference between the bias-induced absorption [4] and the exciton-induced emission [8] of F centers:  ( )1( ) ( ) ( )C C C eqC F C F F H C H H F C F FCCdRZ R D C Z R D C Z R D Cdt R= - - (7) where ,F HC C  are the mean concentrations of F and H centers, respectively, which are determined by the usual rate equations, and , ( )CF H CZ R  are the colloid capture efficiencies for F and H centers [4]:  ,, ( ) 1imF HCF H CCbZ RRa ◊= + (8) Bubble growth is due to the difference between the bias-induced absorption of H centers, since the  emission of H centers requires higher energies than for F centers and therefore it has not been taken  phys. stat. sol. (c) 2, No. 1 (2005) / www.pss-c.com 441 © 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim into account in the present model. Accordingly, we find the following expression for the bubble growth rate   ( )1( ) ( )B BB H B H H F B F FBdRZ R D C Z R D Cdt R= - (9)  ( )( ) 2, , ,, 21im dF H F H F HBF H BB Bb bZ RR Rma a a-= + + (10) where ,dF Ha  and ,F Hma  are the constants associated with the stress-induced diffusion anisotropy and the modulus  interaction, respectively [4]. The bubble growth rate increases with decreasing bubble radius, and so they can grow from molecular centers formed by recombination of H centers and divacancies.  On the other hand, the growth rates of colloids and voids decrease with decreasing size and they are negative below some critical radius, which determines the rate of their nucleation. The critical colloid radius is small (see eq. (1)), and so they can nucleate homogeneously at a rather high rate [9]. At the same time, the critical void radius is compara-tively large [6]  ( )bZrRDexDcritV432÷≈=π  (11) and so they can be formed only from sub-critical void nuclei produced as a result explosive recombina-tion of colloids and bubbles [6]. The present model provides a possibility to evaluate nucleation and the growth rate of colloids, bubbles and voids in the framework of rate theory [8]. The results of the model calculations are shown in Fig. 1 together with experimental data available up to date.   Fig. 1 Temperature dependence of the volume fraction and the number density of extended defects in  NaCl elec-tron irradiated up to different dose levels at a dose rate of 200 Mrad/hour. 442 V. I. Dubinko et al.: Nucleation and growth of sodium colloids in NaCl © 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim 3 Conclusion According to the present model, chlorine bubbles are the most finely dispersed ED in the system (the inter-bubble spacing is typically a few nm) implying that they start to collide with growing voids first, filling them with chlorine gas (fig. 2) The chlorine molecules in the “bubbles” are in a solid or liquid state due to a super high pressure (in the GPa range), but after collision with a void it is in the vapor state. Chlorine accumulation in voids provides a very important possibility for explosive back reactions with metallic sodium when growing voids start hitting colloids, which results in void-crack production and ultimately in explosive fracture of the material [5].     Fig. 2 Illustration of the distribution of chlorine bubbles, sodium colloids and voids in irradiated NaCl crystals. The bubble size is ~1-2 nm; the colloid size ~ inter-bubble spacing ~5-10 nm; the void size is about inter-colloid spacing ~20-30 nm, when collisions with colloids start.   From a more general perspective, the mechanisms of void and dislocation loop growth and colloid shrinkage in alkali halides considered above are characteristic manifestations of radiation-induced emis-sion of Schottky defects, which may be responsible for a number of various radiation effects such as the void dissolution and swelling saturation in irradiated metals [10], irradiation creep [11] and possibly the void ordering [12].  Acknowledgements  This study has been supported by the STCU grant # 1761 and by a NWO-NATO grant # NB 67-319. References  [1] Ch.B. Lushchik, Creation of Frenkel Defect Pairs by Excitons in Alkali Halides, in: Physics of Radiation Effects in Crystals, R.A. Johnson and A.N. Orlov, Eds. (Elsevier Science Publishers B.V., 1986). pp. 474-525.  [2] L.W. Hobbs and A. E. Hughes, Radiation Damage in Diatomic Materials at High Doses, Harwell Report AERE – R 8092 (1975); L. W. Hobbs A. E. Huges, and Pooley, Proc Roy. Soc. (Lond.) A 332, 167 (1973).  [3] Uma Jain and A. B. Lidiard, Philos. Mag. 35, 245 (1977).  [4] V. I. Dubinko, A. A. Turkin, D. I. Vainshtein, and H. W. den Hartog, J. Nucl. Mater. 227, 184 (2000); 289, 86 (2001).  [5] V. I. Dubinko, A. A. Turkin, D. I. Vainshtein, and H. W. den Hartog, Radiat. Eff. Defects Solids 156, 27-31 (2001).  [6] D V. I. Dubinko, A. A. Turkin, D. I. Vainshtein, and H. W. den Hartog, Mechanism of void growth in irradi-ated NaCl based on exciton-induced formation of divacancies at dislocations, COSIRES 2004, Helsinki, June 28 – July 2 (2004).  Cl2 Void Chlorine bubble Sodium colloid phys. stat. sol. (c) 2, No. 1 (2005) / www.pss-c.com 443 © 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim  [7] F. Seitz, Rev. Mod. Phys. 26 7 (1954)  [8] V. I. Dubinko, D. I. Vainshtein, and H. W. den Hartog, Radiat. Eff. Defects Solids 158, 705 (2003).  [9] V. I. Dubinko, A. A. Turkin, D. I. Vainshtein, and H. W. den Hartog, J. Nucl. Mater. 304, 117 (2002). [10] V. I. Dubinko and N. P. Lazarev, Effect of the radiation-induced vacancy emission from voids on the void evolution, COSIRES 2004, Helsinki, June 28 – July 2 (2004). [11] V. I. Dubinko, New mechanism of irradiation creep based on the radiation-induced vacancy emission from dislocations, submitted to Radiation Effects and Defects in Solids [12] V. I. Dubinko, New principles of radiation damage and recovery based on the radiation induced emission of Schottky defects, arXiv/cond-mat/0212154 

   University of GroningenNucleation and growth of sodium colloids in NaCl under irradiationDubinko, V.I.; Turkin, A.A.; Abyzov, A.S.; Sugonyako, A.V.; Vainshtein, D.I.; Hartog, H.W. denPublished in:Physica Status Solidi (C)DOI:10.1002/pssc.200460202IMPORTANT NOTE: You are advised to consult the publisher's version (publisher's PDF) if you wish to cite fromit. Please check the document version below.Document VersionPublisher's PDF, also known as Version of recordPublication date:2005Link to publication in University of Groningen/UMCG research databaseCitation for published version (APA):Dubinko, V. I., Turkin, A. A., Abyzov, A. S., Sugonyako, A. V., Vainshtein, D. I., & Hartog, H. W. D. (2005).Nucleation and growth of sodium colloids in NaCl under irradiation: theory and experiment. Physica StatusSolidi (C), 2(1), 438-443. https://doi.org/10.1002/pssc.200460202CopyrightOther than for strictly personal use, it is not permitted to download or to forward/distribute the text or part of it without the consent of theauthor(s) and/or copyright holder(s), unless the work is under an open content license (like Creative Commons).The publication may also be distributed here under the terms of Article 25fa of the Dutch Copyright Act, indicated by the “Taverne” license.More information can be found on the University of Groningen website: https://www.rug.nl/library/open-access/self-archiving-pure/taverne-amendment.Take-down policyIf you believe that this document breaches copyright please contact us providing details, and we will remove access to the work immediatelyand investigate your claim.Downloaded from the University of Groningen/UMCG research database (Pure): http://www.rug.nl/research/portal. For technical reasons thenumber of authors shown on this cover page is limited to 10 maximum.Download date: 01-02-2024phys. stat. sol. (c) 2, No. 1, 438–443 (2005) / DOI 10.1002/pssc.200460202 © 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim Nucleation and growth of sodium colloids in NaCl under  irradiation: theory and experiment V. I. Dubinko1, A. A. Turkin1, A. S. Abyzov1, A. V. Sugonyako2, D. I. Vainshtein2,  and H. W. den Hartog*2  1  National Science Center, Kharkov Institute of Physics and Technology, 310108 Kharkov, Ukraine 2 Solid State Physics Laboratory, University of Groningen, Nijenborgh 4, NL-9747 AG Groningen, The Netherlands Received 11 July 2004, revised 24 September 2004, accepted 6 October 2004 Published online 20 January 2005 PACS 61.46.+w, 61.80.Ed, 61.80.Fe, 61.82.Ms, 82.50.Kx A mechanism of radiation-induced emission of Schottky defects from extended defects proposed origi-nally for metals has recently been applied to ionic crystals, where it is based on interactions of excitons with extended defects such as dislocations and colloids. Exciton trapping and decay at colloids may result in the emission of F centers and consequent shrinkage of the colloid. In the present paper, the radiation-induced emission of F centers is taken into account in the modeling of nucleation and growth of sodium colloids and chlorine bubbles in NaCl exposed to electron or gamma irradiation. The evolution of colloid and bubble number densities and volume fractions with increasing irradiation dose is modeled in the framework of a modified rate theory and compared with experimental data. Experimental values of the colloid volume fractions and number densities have been estimated on the basis of latent heat of melting of metallic Na obtained with combined differential scanning calorimetry experiments and atomic force microscopy investigations of metallic clusters. © 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim 1 Introduction Exposure of alkali halides to ionizing radiation results in the formation of nano-sized halogen “bubbles” by agglomeration of H centers and of complementary inclusions of metallic “colloids” formed by ag-glomeration of F centers. H and F centers are the primary radiation-induced Frenkel pairs in the halide sub-lattice created in the crystal bulk due to decay of self-trapped excitons [1]. The H center is a hole trapped at an interstitial halide ion, and an F center is a vacancy in the halide sub-lattice with a trapped electron. The cation sub-lattice is not damaged during primary displacement processes in the crystal bulk [1]. However, early experimental results on electron irradiated alkali halides such as KCl and NaCl [2] provided evidence for the formation of perfect dislocation loops, which require agglomeration of intersti-tial ions in both anion and cation sub-lattices. The first explanation for this phenomenon was given by Hobbs et al. [2], and it was based on the production of a molecular center (a halogen molecule occupying a stoichiometric divacancy) due to interaction between dislocations and two H centers. In 1977, Jain and Lidiard [3] have formulated a model, according to which, the dislocation bias for H centers was the driv-ing force for the colloid growth in alkali halides in the same way as it is for void growth in metals under irradiation. The mechanism of dislocation climb [2] used in the Jain and Lidiard model, requires two H centers meeting each other at the dislocation core, and subsequently, the model was modified by Dubinko et al. [4], who suggested that a dislocation can climb due to the production of a VF center (self-  * Corresponding author: e-mail: h.w.den.hartog@phys.rug.nl, Phone: +3150-363-4789, Fax: +3150-363-4879 phys. stat. sol. (c) 2, No. 1 (2005) / www.pss-c.com 439 © 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim trapped hole neighboring a cation vacancy) induced by interactions between a dislocation and only one H center. Subsequent diffusion of VF centers away from dislocations and recombination with F centers results in the formation of stoichiometric divacancies thus providing a driving force for the formation and growth of vacancy voids observed experimentally. In this model, the void growth rate is determined by the flux of VF centers and by the net flux of F cen-ters (i.e. the difference between F and H center fluxes), which depends on the difference between the void bias and the mean bias of the system for the absorption of F centers. This difference is positive for void radii lager than a critical radius, critVR , which is determined by the ratio of material constants related to the void and dislocation biases [4]:  ( )bbRdimcritV21÷≈=δα  (1) where b  is the atomic spacing and imα  is the constant associated with image interaction:  ( ),im im imH Fa a a= -      ( )1/32 2,, 3136 (1 )F HimF HBk T bm n WanÈ ˘+= Í ˙◊ -Î ˚, (2) dδ is the dislocation bias for the absorption of H centers, which is determined by the ratio of the relaxa-tion volumes associated with H and F centers, FHΩΩ / :  ⎟⎟⎠⎞⎜⎜⎝⎛⎟⎟⎠⎞⎜⎜⎝⎛ΩΩ=HHFHdkL2lnlnδ,           ( )13 (1 )H HbLkTm nWp n+=- (3) where ν is the Poisson ratio, µ is the shear modulus of the host matrix, kH is the square root of the total sink strength for H centers, kT has its usual meaning. The critical void radius RVcrit is given by the same equation (1) as the critical colloid radius, and it is smaller than the radius of void nuclei produced by bubble-colloid collisions [4, 5]. Accordingly, one would expect all void nuclei to grow to visible sizes. The number density of the void nuclei, NV0, can be estimated as the product of the bubble number density, NB ≈ 1018 ÷ 1019 cm–3 and the colloid volume fraction, VC ≈ 10–2 ÷ 10–1 : NV0 ≈ NB × VC ≈ 1016 ÷ 1017 cm–3. However, the observed void number densities are about 1013 ÷ 1014 cm–3 , which means that only a very small part of void nuclei survive and grow to visible sizes. The physical reason for this low “survival” rate remained unaccounted in the previous model. In a recent paper [6], an alternative mechanism of void and dislocation loop growth in alkali halides was considered, which was based on production of divacancies (Schottky defects) by interactions between dislocations and excitons, as was first suggested by Seitz in 1954 [7]. Voids can also emit divacancies due to the interaction with excitons, and so the balance between absorption and emission of divacancies determines the void evolution in this model, which has been shown to be in good agreement with ex-perimental observations. In the present paper, the exciton-induced emission of F-centers from colloids is taken into account in the calculation of colloid nucleation and growth rates in the framework of the modified rate theory [8]. The driving force of colloid growth is the large bias of halogen bubbles for absorption of H centers rather than the dislocation bias employed in the previous models [3, 4]. Thus, radiolysis (separation of chlorine and sodium into bubbles and colloids) and the development of stoichiometric defects (voids and disloca-tions) occur simultaneously but relatively independently. The only important “bridge” between these subsystems is bubble nucleation, which starts from recombination of H centers and divacancies since the homogeneous nucleation rate due to recombination of H centers is negligible. 440 V. I. Dubinko et al.: Nucleation and growth of sodium colloids in NaCl © 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim 2 Excitonic production of Frenkel and Schottky defects According to Seitz [7], an exciton may cause movement of nearby dislocation jogs, resulting in the crea-tion of equal numbers of anion and cation vacancies - Schottky defects (SD). Owing to the heat gener-ated locally during exciton annihilation, the jog can be displaced while a divacancy arises in the lat-tice. It a previous paper [8] , we have incorporated the excitonic production mechanism of both Frenkel and Schottky defects in rate theory and evaluated so called local equilibrium concentrations of SD, eqsSc , (almost) spherical (colloids and voids), cylindrical (dislocations) and planar (grain boundaries) sinks, where subscript “s” designates the type of SD (s= v for divacancies, s=F for F centers) and “S” is the type of the sink (S= V for voids, S= F for colloids, S= B for bubbles and S= D for dislocations). In absence of the radiation field, one has thsSeqsS cc = , which can be obtained from thermodynamics. Un-der irradiation, eqsSc  is generally given by the sum of thsSc  and irrsSc  , which is due to the radiation-induced emission of SD’s. Here, irrsSc  has a purely kinetic origin and can be expressed via the radiation-induced local emission rate of SD, irrsSK  [8]: 0 22Dirr FP ex exvDFP DK Z lKP r bp= ,   ( )DDDexrkZππln2=, virrvDirrvDDbKc2×= (4) VexFPFPirrvVbRlPKK20=,   virrvVirrvVDbKc2×= (5) where exl  is the mean free diffusion path of the exciton i.e. the average displacement of the exciton before it decays, FPP  is the probability of the formation of a Frenkel pair (FP) in the anion sub-lattice due to exciton decay in the bulk, 0FPK is the FP production rate in the bulk crystal,  DexZ  and Dr  are the dislocation capture efficiency and capture radius for excitons and vD is the divacancy diffusion coeffi-cient. The difference between the mean divacancy concentration, vC , and the local equilibrium concentrations at voids is the driving force for the void growth (or shrinkage), and its rate is given by [6]  ( )eqV v v vVVdR DC Cdt R= - (6) The colloid growth rate is determined by the difference between the bias-induced absorption [4] and the exciton-induced emission [8] of F centers:  ( )1( ) ( ) ( )C C C eqC F C F F H C H H F C F FCCdRZ R D C Z R D C Z R D Cdt R= - - (7) where ,F HC C  are the mean concentrations of F and H centers, respectively, which are determined by the usual rate equations, and , ( )CF H CZ R  are the colloid capture efficiencies for F and H centers [4]:  ,, ( ) 1imF HCF H CCbZ RRa ◊= + (8) Bubble growth is due to the difference between the bias-induced absorption of H centers, since the  emission of H centers requires higher energies than for F centers and therefore it has not been taken  phys. stat. sol. (c) 2, No. 1 (2005) / www.pss-c.com 441 © 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim into account in the present model. Accordingly, we find the following expression for the bubble growth rate   ( )1( ) ( )B BB H B H H F B F FBdRZ R D C Z R D Cdt R= - (9)  ( )( ) 2, , ,, 21im dF H F H F HBF H BB Bb bZ RR Rma a a-= + + (10) where ,dF Ha  and ,F Hma  are the constants associated with the stress-induced diffusion anisotropy and the modulus  interaction, respectively [4]. The bubble growth rate increases with decreasing bubble radius, and so they can grow from molecular centers formed by recombination of H centers and divacancies.  On the other hand, the growth rates of colloids and voids decrease with decreasing size and they are negative below some critical radius, which determines the rate of their nucleation. The critical colloid radius is small (see eq. (1)), and so they can nucleate homogeneously at a rather high rate [9]. At the same time, the critical void radius is compara-tively large [6]  ( )bZrRDexDcritV432÷≈=π  (11) and so they can be formed only from sub-critical void nuclei produced as a result explosive recombina-tion of colloids and bubbles [6]. The present model provides a possibility to evaluate nucleation and the growth rate of colloids, bubbles and voids in the framework of rate theory [8]. The results of the model calculations are shown in Fig. 1 together with experimental data available up to date.   Fig. 1 Temperature dependence of the volume fraction and the number density of extended defects in  NaCl elec-tron irradiated up to different dose levels at a dose rate of 200 Mrad/hour. 442 V. I. Dubinko et al.: Nucleation and growth of sodium colloids in NaCl © 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim 3 Conclusion According to the present model, chlorine bubbles are the most finely dispersed ED in the system (the inter-bubble spacing is typically a few nm) implying that they start to collide with growing voids first, filling them with chlorine gas (fig. 2) The chlorine molecules in the “bubbles” are in a solid or liquid state due to a super high pressure (in the GPa range), but after collision with a void it is in the vapor state. Chlorine accumulation in voids provides a very important possibility for explosive back reactions with metallic sodium when growing voids start hitting colloids, which results in void-crack production and ultimately in explosive fracture of the material [5].     Fig. 2 Illustration of the distribution of chlorine bubbles, sodium colloids and voids in irradiated NaCl crystals. The bubble size is ~1-2 nm; the colloid size ~ inter-bubble spacing ~5-10 nm; the void size is about inter-colloid spacing ~20-30 nm, when collisions with colloids start.   From a more general perspective, the mechanisms of void and dislocation loop growth and colloid shrinkage in alkali halides considered above are characteristic manifestations of radiation-induced emis-sion of Schottky defects, which may be responsible for a number of various radiation effects such as the void dissolution and swelling saturation in irradiated metals [10], irradiation creep [11] and possibly the void ordering [12].  Acknowledgements  This study has been supported by the STCU grant # 1761 and by a NWO-NATO grant # NB 67-319. References  [1] Ch.B. Lushchik, Creation of Frenkel Defect Pairs by Excitons in Alkali Halides, in: Physics of Radiation Effects in Crystals, R.A. Johnson and A.N. Orlov, Eds. (Elsevier Science Publishers B.V., 1986). pp. 474-525.  [2] L.W. Hobbs and A. E. Hughes, Radiation Damage in Diatomic Materials at High Doses, Harwell Report AERE – R 8092 (1975); L. W. Hobbs A. E. Huges, and Pooley, Proc Roy. Soc. (Lond.) A 332, 167 (1973).  [3] Uma Jain and A. B. Lidiard, Philos. Mag. 35, 245 (1977).  [4] V. I. Dubinko, A. A. Turkin, D. I. Vainshtein, and H. W. den Hartog, J. Nucl. Mater. 227, 184 (2000); 289, 86 (2001).  [5] V. I. Dubinko, A. A. Turkin, D. I. Vainshtein, and H. W. den Hartog, Radiat. Eff. Defects Solids 156, 27-31 (2001).  [6] D V. I. Dubinko, A. A. Turkin, D. I. Vainshtein, and H. W. den Hartog, Mechanism of void growth in irradi-ated NaCl based on exciton-induced formation of divacancies at dislocations, COSIRES 2004, Helsinki, June 28 – July 2 (2004).  Cl2 Void Chlorine bubble Sodium colloid phys. stat. sol. (c) 2, No. 1 (2005) / www.pss-c.com 443 © 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim  [7] F. Seitz, Rev. Mod. Phys. 26 7 (1954)  [8] V. I. Dubinko, D. I. Vainshtein, and H. W. den Hartog, Radiat. Eff. Defects Solids 158, 705 (2003).  [9] V. I. Dubinko, A. A. Turkin, D. I. Vainshtein, and H. W. den Hartog, J. Nucl. Mater. 304, 117 (2002). [10] V. I. Dubinko and N. P. Lazarev, Effect of the radiation-induced vacancy emission from voids on the void evolution, COSIRES 2004, Helsinki, June 28 – July 2 (2004). [11] V. I. Dubinko, New mechanism of irradiation creep based on the radiation-induced vacancy emission from dislocations, submitted to Radiation Effects and Defects in Solids [12] V. I. Dubinko, New principles of radiation damage and recovery based on the radiation induced emission of Schottky defects, arXiv/cond-mat/0212154 

   University of GroningenNucleation and growth of sodium colloids in NaCl under irradiationDubinko, V.I.; Turkin, A.A.; Abyzov, A.S.; Sugonyako, A.V.; Vainshtein, D.I.; Hartog, H.W. denPublished in:Physica Status Solidi (C)DOI:10.1002/pssc.200460202IMPORTANT NOTE: You are advised to consult the publisher's version (publisher's PDF) if you wish to cite fromit. Please check the document version below.Document VersionPublisher's PDF, also known as Version of recordPublication date:2005Link to publication in University of Groningen/UMCG research databaseCitation for published version (APA):Dubinko, V. I., Turkin, A. A., Abyzov, A. S., Sugonyako, A. V., Vainshtein, D. I., & Hartog, H. W. D. (2005).Nucleation and growth of sodium colloids in NaCl under irradiation: theory and experiment. Physica StatusSolidi (C), 2(1), 438-443. https://doi.org/10.1002/pssc.200460202CopyrightOther than for strictly personal use, it is not permitted to download or to forward/distribute the text or part of it without the consent of theauthor(s) and/or copyright holder(s), unless the work is under an open content license (like Creative Commons).The publication may also be distributed here under the terms of Article 25fa of the Dutch Copyright Act, indicated by the “Taverne” license.More information can be found on the University of Groningen website: https://www.rug.nl/library/open-access/self-archiving-pure/taverne-amendment.Take-down policyIf you believe that this document breaches copyright please contact us providing details, and we will remove access to the work immediatelyand investigate your claim.Downloaded from the University of Groningen/UMCG research database (Pure): http://www.rug.nl/research/portal. For technical reasons thenumber of authors shown on this cover page is limited to 10 maximum.Download date: 21-06-2022phys. stat. sol. (c) 2, No. 1, 438–443 (2005) / DOI 10.1002/pssc.200460202 © 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim Nucleation and growth of sodium colloids in NaCl under  irradiation: theory and experiment V. I. Dubinko1, A. A. Turkin1, A. S. Abyzov1, A. V. Sugonyako2, D. I. Vainshtein2,  and H. W. den Hartog*2  1  National Science Center, Kharkov Institute of Physics and Technology, 310108 Kharkov, Ukraine 2 Solid State Physics Laboratory, University of Groningen, Nijenborgh 4, NL-9747 AG Groningen, The Netherlands Received 11 July 2004, revised 24 September 2004, accepted 6 October 2004 Published online 20 January 2005 PACS 61.46.+w, 61.80.Ed, 61.80.Fe, 61.82.Ms, 82.50.Kx A mechanism of radiation-induced emission of Schottky defects from extended defects proposed origi-nally for metals has recently been applied to ionic crystals, where it is based on interactions of excitons with extended defects such as dislocations and colloids. Exciton trapping and decay at colloids may result in the emission of F centers and consequent shrinkage of the colloid. In the present paper, the radiation-induced emission of F centers is taken into account in the modeling of nucleation and growth of sodium colloids and chlorine bubbles in NaCl exposed to electron or gamma irradiation. The evolution of colloid and bubble number densities and volume fractions with increasing irradiation dose is modeled in the framework of a modified rate theory and compared with experimental data. Experimental values of the colloid volume fractions and number densities have been estimated on the basis of latent heat of melting of metallic Na obtained with combined differential scanning calorimetry experiments and atomic force microscopy investigations of metallic clusters. © 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim 1 Introduction Exposure of alkali halides to ionizing radiation results in the formation of nano-sized halogen “bubbles” by agglomeration of H centers and of complementary inclusions of metallic “colloids” formed by ag-glomeration of F centers. H and F centers are the primary radiation-induced Frenkel pairs in the halide sub-lattice created in the crystal bulk due to decay of self-trapped excitons [1]. The H center is a hole trapped at an interstitial halide ion, and an F center is a vacancy in the halide sub-lattice with a trapped electron. The cation sub-lattice is not damaged during primary displacement processes in the crystal bulk [1]. However, early experimental results on electron irradiated alkali halides such as KCl and NaCl [2] provided evidence for the formation of perfect dislocation loops, which require agglomeration of intersti-tial ions in both anion and cation sub-lattices. The first explanation for this phenomenon was given by Hobbs et al. [2], and it was based on the production of a molecular center (a halogen molecule occupying a stoichiometric divacancy) due to interaction between dislocations and two H centers. In 1977, Jain and Lidiard [3] have formulated a model, according to which, the dislocation bias for H centers was the driv-ing force for the colloid growth in alkali halides in the same way as it is for void growth in metals under irradiation. The mechanism of dislocation climb [2] used in the Jain and Lidiard model, requires two H centers meeting each other at the dislocation core, and subsequently, the model was modified by Dubinko et al. [4], who suggested that a dislocation can climb due to the production of a VF center (self-  * Corresponding author: e-mail: h.w.den.hartog@phys.rug.nl, Phone: +3150-363-4789, Fax: +3150-363-4879 phys. stat. sol. (c) 2, No. 1 (2005) / www.pss-c.com 439 © 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim trapped hole neighboring a cation vacancy) induced by interactions between a dislocation and only one H center. Subsequent diffusion of VF centers away from dislocations and recombination with F centers results in the formation of stoichiometric divacancies thus providing a driving force for the formation and growth of vacancy voids observed experimentally. In this model, the void growth rate is determined by the flux of VF centers and by the net flux of F cen-ters (i.e. the difference between F and H center fluxes), which depends on the difference between the void bias and the mean bias of the system for the absorption of F centers. This difference is positive for void radii lager than a critical radius, critVR , which is determined by the ratio of material constants related to the void and dislocation biases [4]:  ( )bbRdimcritV21÷≈=δα  (1) where b  is the atomic spacing and imα  is the constant associated with image interaction:  ( ),im im imH Fa a a= -      ( )1/32 2,, 3136 (1 )F HimF HBk T bm n WanÈ ˘+= Í ˙◊ -Î ˚, (2) dδ is the dislocation bias for the absorption of H centers, which is determined by the ratio of the relaxa-tion volumes associated with H and F centers, FHΩΩ / :  ⎟⎟⎠⎞⎜⎜⎝⎛⎟⎟⎠⎞⎜⎜⎝⎛ΩΩ=HHFHdkL2lnlnδ,           ( )13 (1 )H HbLkTm nWp n+=- (3) where ν is the Poisson ratio, µ is the shear modulus of the host matrix, kH is the square root of the total sink strength for H centers, kT has its usual meaning. The critical void radius RVcrit is given by the same equation (1) as the critical colloid radius, and it is smaller than the radius of void nuclei produced by bubble-colloid collisions [4, 5]. Accordingly, one would expect all void nuclei to grow to visible sizes. The number density of the void nuclei, NV0, can be estimated as the product of the bubble number density, NB ≈ 1018 ÷ 1019 cm–3 and the colloid volume fraction, VC ≈ 10–2 ÷ 10–1 : NV0 ≈ NB × VC ≈ 1016 ÷ 1017 cm–3. However, the observed void number densities are about 1013 ÷ 1014 cm–3 , which means that only a very small part of void nuclei survive and grow to visible sizes. The physical reason for this low “survival” rate remained unaccounted in the previous model. In a recent paper [6], an alternative mechanism of void and dislocation loop growth in alkali halides was considered, which was based on production of divacancies (Schottky defects) by interactions between dislocations and excitons, as was first suggested by Seitz in 1954 [7]. Voids can also emit divacancies due to the interaction with excitons, and so the balance between absorption and emission of divacancies determines the void evolution in this model, which has been shown to be in good agreement with ex-perimental observations. In the present paper, the exciton-induced emission of F-centers from colloids is taken into account in the calculation of colloid nucleation and growth rates in the framework of the modified rate theory [8]. The driving force of colloid growth is the large bias of halogen bubbles for absorption of H centers rather than the dislocation bias employed in the previous models [3, 4]. Thus, radiolysis (separation of chlorine and sodium into bubbles and colloids) and the development of stoichiometric defects (voids and disloca-tions) occur simultaneously but relatively independently. The only important “bridge” between these subsystems is bubble nucleation, which starts from recombination of H centers and divacancies since the homogeneous nucleation rate due to recombination of H centers is negligible. 440 V. I. Dubinko et al.: Nucleation and growth of sodium colloids in NaCl © 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim 2 Excitonic production of Frenkel and Schottky defects According to Seitz [7], an exciton may cause movement of nearby dislocation jogs, resulting in the crea-tion of equal numbers of anion and cation vacancies - Schottky defects (SD). Owing to the heat gener-ated locally during exciton annihilation, the jog can be displaced while a divacancy arises in the lat-tice. It a previous paper [8] , we have incorporated the excitonic production mechanism of both Frenkel and Schottky defects in rate theory and evaluated so called local equilibrium concentrations of SD, eqsSc , (almost) spherical (colloids and voids), cylindrical (dislocations) and planar (grain boundaries) sinks, where subscript “s” designates the type of SD (s= v for divacancies, s=F for F centers) and “S” is the type of the sink (S= V for voids, S= F for colloids, S= B for bubbles and S= D for dislocations). In absence of the radiation field, one has thsSeqsS cc = , which can be obtained from thermodynamics. Un-der irradiation, eqsSc  is generally given by the sum of thsSc  and irrsSc  , which is due to the radiation-induced emission of SD’s. Here, irrsSc  has a purely kinetic origin and can be expressed via the radiation-induced local emission rate of SD, irrsSK  [8]: 0 22Dirr FP ex exvDFP DK Z lKP r bp= ,   ( )DDDexrkZππln2=, virrvDirrvDDbKc2×= (4) VexFPFPirrvVbRlPKK20=,   virrvVirrvVDbKc2×= (5) where exl  is the mean free diffusion path of the exciton i.e. the average displacement of the exciton before it decays, FPP  is the probability of the formation of a Frenkel pair (FP) in the anion sub-lattice due to exciton decay in the bulk, 0FPK is the FP production rate in the bulk crystal,  DexZ  and Dr  are the dislocation capture efficiency and capture radius for excitons and vD is the divacancy diffusion coeffi-cient. The difference between the mean divacancy concentration, vC , and the local equilibrium concentrations at voids is the driving force for the void growth (or shrinkage), and its rate is given by [6]  ( )eqV v v vVVdR DC Cdt R= - (6) The colloid growth rate is determined by the difference between the bias-induced absorption [4] and the exciton-induced emission [8] of F centers:  ( )1( ) ( ) ( )C C C eqC F C F F H C H H F C F FCCdRZ R D C Z R D C Z R D Cdt R= - - (7) where ,F HC C  are the mean concentrations of F and H centers, respectively, which are determined by the usual rate equations, and , ( )CF H CZ R  are the colloid capture efficiencies for F and H centers [4]:  ,, ( ) 1imF HCF H CCbZ RRa ◊= + (8) Bubble growth is due to the difference between the bias-induced absorption of H centers, since the  emission of H centers requires higher energies than for F centers and therefore it has not been taken  phys. stat. sol. (c) 2, No. 1 (2005) / www.pss-c.com 441 © 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim into account in the present model. Accordingly, we find the following expression for the bubble growth rate   ( )1( ) ( )B BB H B H H F B F FBdRZ R D C Z R D Cdt R= - (9)  ( )( ) 2, , ,, 21im dF H F H F HBF H BB Bb bZ RR Rma a a-= + + (10) where ,dF Ha  and ,F Hma  are the constants associated with the stress-induced diffusion anisotropy and the modulus  interaction, respectively [4]. The bubble growth rate increases with decreasing bubble radius, and so they can grow from molecular centers formed by recombination of H centers and divacancies.  On the other hand, the growth rates of colloids and voids decrease with decreasing size and they are negative below some critical radius, which determines the rate of their nucleation. The critical colloid radius is small (see eq. (1)), and so they can nucleate homogeneously at a rather high rate [9]. At the same time, the critical void radius is compara-tively large [6]  ( )bZrRDexDcritV432÷≈=π  (11) and so they can be formed only from sub-critical void nuclei produced as a result explosive recombina-tion of colloids and bubbles [6]. The present model provides a possibility to evaluate nucleation and the growth rate of colloids, bubbles and voids in the framework of rate theory [8]. The results of the model calculations are shown in Fig. 1 together with experimental data available up to date.   Fig. 1 Temperature dependence of the volume fraction and the number density of extended defects in  NaCl elec-tron irradiated up to different dose levels at a dose rate of 200 Mrad/hour. 442 V. I. Dubinko et al.: Nucleation and growth of sodium colloids in NaCl © 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim 3 Conclusion According to the present model, chlorine bubbles are the most finely dispersed ED in the system (the inter-bubble spacing is typically a few nm) implying that they start to collide with growing voids first, filling them with chlorine gas (fig. 2) The chlorine molecules in the “bubbles” are in a solid or liquid state due to a super high pressure (in the GPa range), but after collision with a void it is in the vapor state. Chlorine accumulation in voids provides a very important possibility for explosive back reactions with metallic sodium when growing voids start hitting colloids, which results in void-crack production and ultimately in explosive fracture of the material [5].     Fig. 2 Illustration of the distribution of chlorine bubbles, sodium colloids and voids in irradiated NaCl crystals. The bubble size is ~1-2 nm; the colloid size ~ inter-bubble spacing ~5-10 nm; the void size is about inter-colloid spacing ~20-30 nm, when collisions with colloids start.   From a more general perspective, the mechanisms of void and dislocation loop growth and colloid shrinkage in alkali halides considered above are characteristic manifestations of radiation-induced emis-sion of Schottky defects, which may be responsible for a number of various radiation effects such as the void dissolution and swelling saturation in irradiated metals [10], irradiation creep [11] and possibly the void ordering [12].  Acknowledgements  This study has been supported by the STCU grant # 1761 and by a NWO-NATO grant # NB 67-319. References  [1] Ch.B. Lushchik, Creation of Frenkel Defect Pairs by Excitons in Alkali Halides, in: Physics of Radiation Effects in Crystals, R.A. Johnson and A.N. Orlov, Eds. (Elsevier Science Publishers B.V., 1986). pp. 474-525.  [2] L.W. Hobbs and A. E. Hughes, Radiation Damage in Diatomic Materials at High Doses, Harwell Report AERE – R 8092 (1975); L. W. Hobbs A. E. Huges, and Pooley, Proc Roy. Soc. (Lond.) A 332, 167 (1973).  [3] Uma Jain and A. B. Lidiard, Philos. Mag. 35, 245 (1977).  [4] V. I. Dubinko, A. A. Turkin, D. I. Vainshtein, and H. W. den Hartog, J. Nucl. Mater. 227, 184 (2000); 289, 86 (2001).  [5] V. I. Dubinko, A. A. Turkin, D. I. Vainshtein, and H. W. den Hartog, Radiat. Eff. Defects Solids 156, 27-31 (2001).  [6] D V. I. Dubinko, A. A. Turkin, D. I. Vainshtein, and H. W. den Hartog, Mechanism of void growth in irradi-ated NaCl based on exciton-induced formation of divacancies at dislocations, COSIRES 2004, Helsinki, June 28 – July 2 (2004).  Cl2 Void Chlorine bubble Sodium colloid phys. stat. sol. (c) 2, No. 1 (2005) / www.pss-c.com 443 © 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim  [7] F. Seitz, Rev. Mod. Phys. 26 7 (1954)  [8] V. I. Dubinko, D. I. Vainshtein, and H. W. den Hartog, Radiat. Eff. Defects Solids 158, 705 (2003).  [9] V. I. Dubinko, A. A. Turkin, D. I. Vainshtein, and H. W. den Hartog, J. Nucl. Mater. 304, 117 (2002). [10] V. I. Dubinko and N. P. Lazarev, Effect of the radiation-induced vacancy emission from voids on the void evolution, COSIRES 2004, Helsinki, June 28 – July 2 (2004). [11] V. I. Dubinko, New mechanism of irradiation creep based on the radiation-induced vacancy emission from dislocations, submitted to Radiation Effects and Defects in Solids [12] V. I. Dubinko, New principles of radiation damage and recovery based on the radiation induced emission of Schottky defects, arXiv/cond-mat/0212154 

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Nucleation and growth of sodium colloids in NaCl under irradiation: theory and experiment

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