The document summarizes research on modeling the annealing kinetics of latent tracks in solid state nuclear track detectors (SSNTDs). It describes the historical development and limitations of existing models. A single activation energy (SAE) model is proposed to overcome limitations. The SAE model relates the annealing rate to time and temperature using an empirical equation involving a single activation energy, Ea, which is a material-dependent property. Experimental validation of the SAE model involves isothermal and isochronal annealing experiments to determine Ea and other parameters. The SAE model provides a better description of annealing in SSNTDs compared to previous models.

1. Heavy Ion Radiation Damage
Annealing in SSNTDs and Single
Activation Energy Model
H.S. Virk
Visiting Professor, SGGS World
University, Fatehgarh Sahib (Punjab)

2. Historical Review of SSNTD
 Nuclear Track Society of India is
organizing SSNTD-15 at HNB Garhwal
University at Tehri with RK Ramola as its
Convener; the first meeting was held in
BARC, Trombay in 1979 by RH Iyer as
Convener.
 INTS will organize 24th
Int. Conference at
Bologna in Italy in 2008; starting with
first meeting in Strasbourg, France in
1957.
 RL Fleischer, one of the founding fathers,
predicted a Plateau for SSNTD research in
one of his Reports; but he has been
proved wrong!

3. SSNTD Plateau

4. SSNTD Trends in Report
 Average global rate of production of
research papers in SSNTD during
1970-90 = 280+- 60
 SSNTD applications in Nuclear,
Space and FT Dating research are
showing a downward trend.
 Applications in Radon Monitoring
and Heavy Ion Materials research
are showing an upward trend.

5. History of SSNTD Research in Guru
Nanak Dev University, Amritsar
 Starting in 1979, almost all areas of SSNTD
applications have been covered.
 FT Dating, Inclusion Dating, Annealing.
 Uranium estimation, and its exploration.
 Radon monitoring in soil, air and homes.
 Earthquake monitoring using radon/helium
in soil and groundwater.
 Ion Track Applications in diverse fields.
 Heavy Ion effects in a variety of Polymers.

6. Radon Survey in Groundwater at
Palampur (Virk, Randhawa & Ramola)

7. INTRODUCTION
 Passage of a heavy ion in an SSNTD
creates intense radiation damage
which results in a series of point
defects and extended defects along
the latent track. Various Models
have been proposed to explain track
formation in SSNTDs. An equally
cumbersome explanation has been
given for track removal in SSNTDs,
better known as radiation damage
annealing of tracks.

8. ANNEALING of LATENT TRACKS
 Track annealing is dominantly a
diffusive process in which
interstitially displaced atoms
thermally penetrate an activation
barrier to recover their initial lattice
positions. Thus one is led to an
Arrhenius type relation based on
Boltzmann equation.

10. NATURE of DEFECTS
 Tombrello et al. suggest that
extended defects are generated by
atomic K-shell excitations in the
heavier elements of the SSNTD.
HREM reveals that latent tracks are
constituted of extended defects,
separated by gap zones loaded with
point defects.

11. Annealing of Defects in a Solid
 Annealing rate or mobility of point
defects increases rapidly with rise of
temperature. It may occur by three
different mechanisms:
 Random diffusion to sinks.
 Recombination of vacancies and
interstitials.
 Annealing of defects by interaction
with impurities.

12. Single Activation Energy Concept
 If the annealing of a defect occurs by a single
activated process with a constant activation
energy Ea, then rate of change of concentration
of the defect is describable by the equation:
dn/dt = -F(n)K = -F(n)Ko exp(-Ea /kT ), (1)
 Where n is the fractional concentration of the
defect, F(n) is any continuous function of n,
and K is the rate constant involving a
Boltzmann factor, exp(-Ea/kT), for its
dependence on annealing temperature T.
 It is implicitly implied by equation (1) that
activation energy Ea is independent of n.

13. Determination of Ea
 There are several methods for
determination of activation energy
from annealing-data curves: (i)
method of cross-cut, (ii) ratio of
slopes, (iii) constant rate of heating,
and (iv) combination of isochronal
and isothermal anneal. We discuss
here only the method of cross-cut
because of its simplicity and ease of
performance compared with the
other methods.

15. TRACK ANNEALING MODELS
 Track annealing models are classified into three
categories according to their mathematical
formulation, viz., logarithmic model, linear model
and exponential model.
 Most of the earlier authors used logarithmic model
for annealing of fission tracks in minerals and
glasses using the Arrhenius equation:
t exp(- Ea /kT ) = constant,
Where is Ea effective activation energy, k is
Boltzmann constant, t and T represent annealing
time and temperature, respectively. This model
gives a spectrum of activation energies at different
annealing temperatures.

16. Limitations of Arrhenius Equation
 (i) The Arrhenius equation is applicable
under constant temperature conditions. It
necessitates approximations as soon as the
fading temperature varies with time.
 (ii) Most models are based on an 'a priori'
assumption that the latent track anneals as
a whole. . Hence it is not justifiable to
correlate the residual lengths or diameters
of the partially annealed tracks with
annealing temperature and time.
 (iii) The activation energy is a function of
the degree of track loss in a given
temperature-time plane which results in
fanning of the Arrhenius plots.

17. Single Activation Energy (SAE) Model:
Conceptual Formulation
 Ionisation rate or energy loss dE/dx
in a material varies continuously
along the track profile.
 As a consequence, etching rate also
varies along the track profile.
 Annealing rate must also vary along
the track profile.
 Chemical etching destroys much of
useful information (physics).

18. Bimolecular Reaction Model
 To resolve the contradictions of Arrhenius
approach, Modgil and Virk proposed the
Single Activation Energy Model on the
assumption that the activation energy is a
material dependent property. The
empirical formulation of this model relates
the instantaneous annealing velocity
Va = dl/dt or dD/dt,
explicitly with time and temperature, a
crude justification for which has been
provided by the assumption of a
bimolecular reaction model.

19.  Author’s group studied radiation damage
annealing kinetics in SSNTDs, viz., minerals,
polymers and glasses, in great details and
proposed an empirical formulation:
Va = At-n
exp (-Ea / kT),
Where Va is annealing rate, Ea is activation
energy, k is Boltzmznn constant, t and T are
annealing time and temperature, A is
proportionality constant and n is exponent of
annealing time, t.
 The advantage of this new approach is that it
yields single activation energy of annealing
which is an intrinsic property of a given
SSNTD, independent of the ion beam used
and annealing time and temperature.

20. Experimental Approach to SAE Model
Two sets of experiments are performed to
test this model:
 Isothermal Experiments: Annealing rate,
Va, is studied by varying t and keeping T
constant. Plot of log Va versus log t will
yield the value of n.
 Isochronal Experiments: Keeping t
constant and varying T, Va is determined.
Plot of log Va versus 103
/T yield the value
of Ea, the activation energy of SSNTD.

21. Determination of annealing rate,Va

23. Determination of Ea

25. Modification of SAE by Price Group
 Salamon et al. replaced the annealing
velocity by the etch rate reduction of
annealed latent tracks and found that the
activation energy, Ea and other
parameters, i.e. n and A, are also
constants. Price et al. have found an
application of our model in their annealing
experiments using phosphate glass
detectors for recording of relativistic
cosmic ray nuclei tracks in Space Shuttle
‘IONS Experiment’.

26. Modification : A Final Version of SAE
 To overcome the shortcomings of our
earlier formulation and that
proposed by Salamon et al., Bhatia &
Virk proposed the new formulation
replacing the instantaneous annealing
velocity, Va, by the instantaneous
track etch velocity, Vt: d/dta(Vt) = At-n
exp (-Ea / kT),
which gives a better fit to annealing
data in SSNTDs

32. Special features of SAE model
 (i) It predicts a single activation energy of
annealing for all heavy ions as required
by the Arrhenius equation.
 (ii) It may be used for revealing the
thermal history of track recording SSNTDs
(minerals, meteorites and lunar rocks).
 (iii) It explains the partial fading of tracks
due to environmental annealing.
 (iv) It has a universal application for all
SSNTDs (both crystalline and amorphous)
using a variety of heavy ion beams and
fission fragments.

33. Missed Opportunities & Lessons
 Experimental work need to be
supplemented by Theoretical
analysis of data.
 Publish the data in top rank journals
otherwise your work may be
ignored by peers.
 Our SAE model lost its IMPACT
because of the above reasons and
we felt almost cheated!