fractal kinetics bruyères-le-châtel
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David L. Griscom
impactGlass research international Mexico City Paris Tokyo Washington
Commissariat à l’Energie Atomique, Bruyères-le-Châtel, France
1 December 2005
An
Ideal Wedding of a Mathematical Formalism
often written off as “just another method of curve fitting”
with a Remarkable Body of Data
which has defied simple mathematical description,
thus severely limiting its utility for its intended purposes
Acknowledgements
The Experimental data for the Ge-doped-silica fibers were
recorded at the Naval Research Laboratory by E.J. Friebele
with important assistance from M.E. Gingerich, M. Putnum,
G.M. Williams, and W.D. Mack.
A full account of this work has been published:
D.L. Griscom, Phys. Rev. B64, 174201 (2001)
Classical Kinetic Solutions for Radiolytic Defect Creation with
Thermal Decay: Dependencies on Dose Rate
103
104
105
106
107
108
0.01
0.1
1
10
100
1000
Classical Kinetic Solutions:
Red Curves: 2nd
-Order; Small Circles: 1st-Order
340 rad/s
17 rad/s
0.45 rad/s
Experimental Data:
17 rad/s
0.45 rad/s
340 rad/s
Ind
uced
Lo
ss (
dB
/km
)
Dose (rad)
Slope = 1.0
Radiation-Induced Absorption (1.3 m) in Ge-Doped-Silica-Core Optical Fibers:
Failure of Classical Kinetics to Fit Data as Functions of Dose Rate
103
104
105
106
107
108
0.01
0.1
1
10
100
1000
Classical Kinetic Solutions:
Red Curves: 2nd
-Order; Small Circles: 1st-Order
340 rad/s
17 rad/s
0.45 rad/s
Experimental Data:
17 rad/s
0.45 rad/s
340 rad/s
Ind
uced
Lo
ss (
dB
/km
)
Dose (rad)
Slope = 1.0
It is Impossible to Fit
These Data with These Solutions!
•Gottfried von Leibnitz (1695): “Thus it follows that d½x will be equal to xdx:x,
… from which one day useful consequences will be drawn.”
What is Fractal (Fractional) Kinetics?
•I.M. Sokovov, J. Klafter, A. Blumen, Physics Today, November, 2002, p. 48:
“Equations built on fractional derivatives describe the anomalously slow diffusion
observed in systems with a broad distribution of relaxation times.”
•R. Kopelman, Science 241, 1620 (1988).
•Science 297, 1268 (2002): News article on “Tsallis entropy”.
(q 1)
Fractal Kinetics in Brief
Fractal spaces differ from Euclidian spaces by having fractal dimensions df
such that
df < d,
where d is the dimension of the Euclidian space in which the fractal is embedded.
Each fractal also possesses a spectral dimension ds (< df < d), defined by the
probability P of a random walker returning to its point of origin after a time t:
P(t) t-ds/2.
The present work introduces a parameter, ds/2. Thus, for many amorphous
materials, values of 2/3 might be expected ...
– which
serves as a prototype for many amorphous materials.
It is known that ds 4/3 for the entire class of random fractals embedded in
Euclidian spaces of dimensions d 2 , including the percolation cluster
Supercomputer Simulations of Fractal Kinetics Raoul Kopelman, Science 241, 1620 (1988)
A + B AB
Sierpinski “gasket”: df =1.585, ds = 1.365 Percolation Cluster: df =1.896, ds = 1.333
First-order growth kinetics with thermally activated decay.
The classical rate equation for this situation can be written
dN(t)/dt = KDN* - RN,
and its solution is given by
N(t) = Nsat{1 - exp[-Rt]},
where K and R are constants, D is the dose rate, N* is a
number of unit value and dimensions of number density
(e.g., cm-3), and
Nsat = (KD/R)N*.
Rate Equations for Defect Creation under Irradiation
•
•
•
Result of Change in Dimensionless Variable kt (kt)
First-order growth kinetics with thermally activated decay.
The fractal rate equation for this situation can be written
dN((kt))/d(kt) = (KD/R) N* - N
0 < <1 k = R
with solution
N((kt)) = Nsat{1 - exp[-(kt)]},
where Nsat = (KD/R) N*. •
•
Second-order growth kinetics with thermally activated decay.
The classical rate equation for this situation can be written
dN(t)/dt = KDN* - RN2/N*,
and its solution is given by
N(t) = Nsattanh(kt),
where Nsat = (KD/R)1/2N* and k = (KDR)1/2.
Rate Equations for Defect Creation under Irradiation
•
• •
Result of Change in Dimensionless Variable kt (kt)
Second-order growth kinetics with thermally activated decay.
The fractal rate equation for this situation can be written
dN((kt))/d(kt) = (KD/R) /2N* - (R/KD) /2N2/N*
0 < <1 k = (KDR)1/2
with solution
N((kt)) = Nsattanh[(kt)],
where Nsat = (KD/R) /2N*. •
• •
•
Three Fitting
Parameters
Experimental Curves Fitted by Fractal First-Order Solutions Multi-Mode Ge-Doped-Silica-Core Fibers, =1300 nm
1000 10000 100000 10000000.1
1
10
100
0.0009 rad/s
=0.82
17 rad/s
=0.79
0.45 rad/s
=0.94
340 rad/s
=0.71
In
du
ce
d L
oss (
dB
/km
)
Dose (rad)
(Reactor Irradiation)
γ Irradiation
1E-3 0.01 0.1 1 10 100
10
100
(c)
Slope =
Slope = /2
Satu
ration L
oss (
dB
/km
)
Dose Rate (rad/s)
1E-3 0.01 0.1 1 10 10010
-8
10-7
10-6
10-5
10-4
10-3
Linear!!!
(b)Classical 1st-Order Kinetics
Classical 2nd-Order
Kinetics
Rate
Coeff
icie
nt (s
-1)
1E-3 0.01 0.1 1 10 100
0.7
0.8
0.9(a)
Stretched 2nd Order Kinetics
Stretched 1st-Order Kinetics
Pow
er-
Law
Exponent
Fractal-Kinetic Fitting Parameters: Both Kinetic Orders Multi-Mode Ge-Doped-Silica-Core Fibers, =1300 nm
Slope = 1
Classical 1st Order
Classical 2nd
Order Fractal 1st & 2nd Order (Slope = 1) Empirical!!!
Fractal 1st & 2nd Order
(Slope Variable)
Annoying Cusp
Annoying
k
Nsat
Dose Rate (rad/s)
Classical 1st Order
Slope = ½
Classical 2nd Order
1000 10000 100000 10000000.1
1
10
100
0.0009 rad/s
=1.0
17 rad/s
=0.70
0.45 rad/s
=1.00
340 rad/s
=0.62
Ind
uce
d L
oss (
dB
/km
)
Dose (rad)
Experimental Curves Fitted by Fractal Solutions (Second Order) Multi-Mode Ge-Doped-Silica-Core Fibers, =1300 nm
“Population B” Included in Fits
Population B (included in all four curve fits)
Dose-Rate-Independent
N.B. The dominant “Population A” comprises
all defects with thermally activated decays
Fractal-Kinetic Fitting Parameters (Second Order) Multi-Mode Ge-Doped-Silica-Core Fibers, =1300 nm
Parameters for
dose-rate-
independent
“Population B”
included in fits
for all four dose
rates
1E-3 0.01 0.1 1 10 1001
10
100
(c)
Slope =
Slope = /2
Satu
ration L
oss (
dB
/km
)
Dose Rate (rad/s)
1E-3 0.01 0.1 1 10 10010
-9
10-8
10-7
10-6
10-5
10-4
10-3
Linear
(b)
Rate
Coeff
icie
nt (s
-1)
1E-3 0.01 0.1 1 10 1000.6
0.7
0.8
0.9
1.0
(a)
Pow
er-
Law
Exponent
No More
Annoying Cusp
Here!
Approximately
Straight Line Here
Intended Result of
Introducing
“Population B”:
Rate Coefficient
Is More Perfectly
Linear than Before !!!
k
Nsat
Dose Rate (rad/s)
Slope=/2
Empirical!!!
Happy Colateral
Consequences:
Fractal-Kinetic Fitting Parameters (Second Order) Multi-Mode Ge-Doped-Silica-Core Fibers, =1300 nm
Proposed
“Canonical Form”
for
Classical Fractal
Phase Transition
1E-3 0.01 0.1 1 10 100
1
10
100
Slope = 1/2
(c)
Slope = /2
Sa
tura
tio
n L
oss (
dB
/km
)
Dose Rate (rad/s)
1E-3 0.01 0.1 1 10 100
10-7
10-6
10-5
10-4
10-3
Slope = 1/2
(b)
Slope = 1
Ra
te C
oeff
icie
nt
(s-1)
1E-3 0.01 0.1 1 10 1000.6
0.7
0.8
0.9
1.0
(a)
Classical Fractal
Po
we
r-La
w E
xp
on
en
t
Here “Population B”
was contrived to give
classical behavior
below the point where
= 1
k
Nsat
Slope=/2
Slope=1
Slope=1/2
Slope=1/2
Classical Fractal
Dose Rate (rad/s)
1000 10000 100000 10000000.1
1
10
100
C ( = 0.66)B
0.0009 rad/s
=1.0
17 rad/s
=0.61
0.45 rad/s
=0.94
340 rad/s
=0.46
In
du
ce
d L
oss (
dB
/km
)
Dose (rad)
Experimental Curves Fitted by Fractal Solutions (Second Order) Single-Mode Ge-Doped-Silica-Core Fibers, =1300 nm
Dose-Rate-Independent “Populations B and C” Included in Fits
(Reactor Irradiation)
Populations B and C
1E-3 0.01 0.1 1 10 1000.1
1
10
Slope = 1/2
Slope = /2(c)
Satu
ration L
oss (
dB
/km
)
Dose Rate (rad/s)
1E-3 0.01 0.1 1 10 1001E-8
1E-7
1E-6
1E-5
1E-4
1E-3
Slope = 1/2
Slope = 1(b)
Rate
Coeff
icie
nt (1
/s)
1E-3 0.01 0.1 1 10 100
0.5
0.6
0.7
0.8
0.9
1.0
Single-Population Fits
Fits Including Effects
of Populations B & C(a)
Exponent
Fractal-Kinetic Fitting Parameters (Second Order) Single-Mode Ge-Doped-Silica-Core Fibers, =1300 nm
However… All fits are constrained
by the (questionable)
assumption that the
dosimetry for reactor-
irradiation is equivalent
to that for γ irradiation
vis-à-vis the induced
optical absorption. γ-Rays
Reactor
Irradiation Dose Rate (rad/s)
k
Nsat
Slope=1/2
Slope=1/2
Inclusion of Populations
B & C does not alter the
fundamental result:
There still seems to be a
classical fractal
transition.
Experimental Curves Fitted by Fractal Solutions (Second Order) Single-Mode Ge-Doped-Silica-Core Fibers, =1300 nm
Dose-Rate-Independent “Populations B and C” Included in Fits
1000 10000 100000 10000000.1
1
10
100
C
B
0.011 rad/s
=1.017 rad/s
=0.66
0.45 rad/s
=0.85340 rad/s
=0.52
In
duced
Lo
ss (
dB
/km
)
Dose (rad)
( Irradiation)
Fractal-Kinetic Fitting Parameters (Second Order) Single-Mode Ge-Doped-Silica-Core Fibers, =1300 nm
Now All Fits
Pertain to
γ-Irradiated
Fibers Only.
Classical Fractal Transition
0.01 0.1 1 10 100
1
10
Slope = /2
(c)
Satu
ration L
oss (
dB
/km
)
Dose Rate (rad/s)
0.01 0.1 1 10 10010
-8
10-7
10-6
10-5
10-4
10-3
Slope = 1
(b)
Rate
Coeffic
ient (1
/s)
0.01 0.1 1 10 100
0.5
0.6
0.7
0.8
0.9
1.0
Fits Including Influences
of Populations B and C
Single-Population Fits(a)
Exponent
No Data for Reactor-
Irradiated Fibers Are
Included. Empirical Result
of Fractal Kinetics!
Slope=1
Slope=/2
Dose Rate (rad/s)
k
Nsat
But Caution: May Now Be
Asymptotic to 1.0
as the Dose Rate
Approaches Zero.
? ?
Fractal Kinetics of Defect Creation in Ge-Doped-Silica Glasses:
What Have We Learned by Simulation of the Growth Curves?
==========================================================
Parameters
__________________________________________________
First-Order Solution Second-Order Solution ==========================================================
Specified by k = R k = (KDR)½ New Formalisms
Nsat = (KD/R) Nsat = (KD/R) /2 ______________________________________________________________
Empirically R D R D1/2
Inferred in This Work
K D½ K D1/2
==========================================================
Note:
In classical cases
(=1), both K and
R are constants.
In fractal cases
(0<<1), both K
and R are dose-
rate dependent.
•
• •
• •
• •
Empirically
Post-Irradiation Thermal Decay Curves and Fractal-Kinetic Fits
for γ-Irradiated Ge-Doped-Silica Core Fibers
1 10 100 1000 10000 100000 10000000
10
20
30
40
50
60
70
80
90
SM Fiber Data
MM Fiber Data
Naive Fractal Second-Order
Prediction from Growth-Curve Fit
(=0.62)
Fractal Second-Order
Best Fit (=0.51)
Fractal 1st-Order
Best Fit
(=0.44)
Fractal Secnd-Order
Best Fit (=0.54)
Naive Fractal
First-Order
Prediction from
Growth-Curve Fit
(=0.71)
Ind
uce
d L
oss (
dB
/km
)
Time (s)
Non-Decaying Component
Fractal Second-Order
Best Fit (=0.54)
Fractal Second-Order
Best Fit (=0.51)
(Equal to
Cumulative
Populations
B and C
Used in
Fitting the
Growth
Curves!)
101
102
103
104
105
106
Time (s)
= 0.66
Fractal 2nd
-Order
Solution
Fractal
2nd
-Order
Solution
(Kohlrausch
Function)
102
103
104
105
106
107
1
10
100
= 0.66
Fractal
1st-Order
Solution
Fractal
2nd
-Order
Solution
Ind
uce
d L
oss (
dB
/km
)
Dose (rad)
Idealized Fractal Kinetics of Radiation-Induced Defect
Formation and Decay in Amorphous Insulators
During Irradiation After Cessation of Radiation
1st
101
102
103
104
105
106
Time (s)
= 0.66
Fractal 2nd
-Order
Solution
Fractal
2nd
-Order
Solution
(Kohlrausch
Function)
102
103
104
105
106
107
1
10
100
= 0.66
Fractal
1st-Order
Solution
Fractal
2nd
-Order
Solution
Ind
uce
d L
oss (
dB
/km
)
Dose (rad)
During Irradiation After Cessation of Radiation
1st
Slope
Slope -
Idealized Fractal Kinetics of Radiation-Induced Defect
Formation and Decay in Amorphous Insulators
100
101
102
103
104
105
106
Factor of 4
No-Adjustable-Parameters
Prediction Based on
Fitted Growth Curve
Time (s)
101
102
103
104
105
106
107
1
10
Data
Fitted Decaying Part
Fitted Non-Decaying
Parts
In
du
ce
d L
oss (
dB
/km
)
Dose (rad)
Fractal Kinetics of Radiation-Induced Defect
Formation and Decay in Amorphous Insulators: The Reality
C
B
B+C
100
101
102
103
104
105
106
Factor of 4
No-Adjustable-Parameters
Prediction Based on
Fitted Growth Curve
Time (s)
101
102
103
104
105
106
107
1
10
Data
Fitted Decaying Part
Fitted Non-Decaying
Parts
In
du
ce
d L
oss (
dB
/km
)
Dose (rad)
C
B
B+C
N.B. These data
prove the existence
of (non-decaying)
dose-rate independent
components.
Fractal Kinetics of Radiation-Induced Defect
Formation and Decay in Amorphous Insulators: The Reality
Fractal kinetics
of optical bands
in pure silica
glass…
0
5000
10000
15000
20000
250006,470 s
102 rad/s 15.3 rad/s
33
120
240
480
960 s 0
Ind
uce
d A
bsorp
tio
n (
dB
/km
)
400 500 600 700 800 900 1000 1100 1200 1300 1400 15000
500
1000
1500
2000
2500
Wavelength (nm)
N.B. These
bands appear
to arise from
self- trapped
holes.
Note absorption in all
three communications
windows.
Growth and Disappearance of “660- and 760-nm” Bands:
Optical Spectroscopy D.L. Griscom, Appl. Phys. Lett. 71 (1997) 175
High-purity, low-OH,
low-Cl, pure-silica-core
fiber (KS-4V) under γ
irradiation for 240 s at
1.0 Gy/s
NBOHCs
760 nm
N.B. These results are
remarkably similar to
those for a low-OH, low-Cl
F-doped silica-core fiber
measured simultaneously.
660 nm
(Bands near 660, 760, and 900 nm are due to self-trapped holes.)
t-1
It appears that
the material is
“reconfigured”
by long-term,
low-dose-rate
irradiation in
such a way that
color centers
(STHs) are no
longer formed,
even when the
irradiation
continues
Loss at 760
nm during
γ irradiation
in the dark
at 1 Gy/s,
T=27 oC
Experimenter-Introduced
“Mid-Course” Transients
Growth and Disappearance of “660- and 760-nm” Bands:
Overview of Kinetics D.L. Griscom, Appl. Phys. Lett. 71 (1997) 175
KS-4V, 760 nm
t-1
It appears that
the material is
“reconfigured”
by long-term,
low-dose-rate
irradiation in
such a way that
color centers
(STHs) are no
longer formed,
even when the
irradiation
continues
– or
is repeated at a
later time.
Growth and Disappearance of “660- and 760-nm” Bands:
Overview of Kinetics D.L. Griscom, Appl. Phys. Lett. 71 (1997) 175
Same Fiber Re-Irradiated
KS-4V, 760 nm
Growth and Disappearance of “660- and 760-nm” Bands:
Dose-Rate Dependence D.L. Griscom, Phys. Rev. B64 (2001) 174201
100
1000
Stretched 2nd Order, =0.96
Stretched 2nd Order: =0.90
Kohlrausch: =0.95
Stretched 2nd Order: =0.53
Kohlrausch: =0.60102 rad/s
15.3 rad/s
102 10
3 10
4 10
5 10
6 10
7
In
duced L
oss (
dB
/km
)
Time (s)
…Dependent
Only on Time
(Not Dose Rate)
at Long Times!
Radiation
Bleaching
Large Initial
Dose-Rate
Dependence
Two Lengths of
Virgin Fiber,
Irradiated
Separately
KS-4V, 900 nm
Growth and Disappearance of “660- and 760-nm” Bands:
Optical Bleaching During Irradiation D.L. Griscom, Phys. Rev. B64 (2001) 174201
100 10001000
10000
(a) Light On
F-Doped
KS-4V
Ind
uce
d L
oss (
dB
/km
)
Time (s)
100 1000
= 670 nm
(b) Light Off
535 rad/s
25 rad/s
Time (s)
is independent of dose rate
in case of KS-4V core fiber.
depends strongly on dose
rate but is independent of
the type of silica in the core. F-doped is slightly different.
Growth and Disappearance of “660- and 760-nm” Bands:
Isothermal Fading (Radiation Interupted), Regrowth D.L. Griscom, Phys. Rev. B64 (2001) 174201
100 1000 10000
1000
10000
(a)
Stretched 2nd Order, =0.71
Kohlrausch, =0.52
In
du
ce
d L
oss (
dB
/km
)
Time after Irradiation (s)
100 1000
(b)
F-Doped-Silica-Core Fiber,
Dose Rate = 102 rad/s
760 nm
Best Fits:
Stretched 2nd Order: =0.45
Kohlrausch: =0.53
660 nm
Best Fits:
Stretched 2nd Order: =0.60
Kohlrausch: =0.69
Irradiation Time (s)
Data Points for
t=0 were Used
in Fitting These
Data.
Fitted Values of
Are Independent
of Wavelength.
Fitted Values of
Are Strongly
Dependent on
Wavelength.
Fading Regrowth
Fractal-Kinetic Fitting Parameters (Both Orders) Multi-Mode Low-OH, Low-Cl Pure-Silica-Core Fibers During Irradiation
Data due to
Nagasawa et al.
(1984) pertain to
a silicone-clad
pure-silica core
fiber.
Gaussian
resolutions were
performed to
extract intensities
of the 660- and
760-nm bands
separately.
My data for F-
doped-silica-clad
pure-silica-core
fiber with an Al
jacket.
Measurements
were made at
fixed wavelengths
of 670 and 900
nm (no Gaussian
resolutions)
The same fiber
was subjected to
the 3 different
dose rates in
progression
beginning with
the lowest. 10 100 1000
1000
10000
Slope=1/2
Polymer-Clad
Silica-Core
Fiber KS-4V Silica-Core
Fiber, Aluminum
Jacketed
Slope=/2
Slope=
(c)
Satu
ration L
oss (
dB
/km
)
Dose Rate (rad/s)
10 100 1000
10-4
10-3
10-2
11/9/00 16:18:58
Weighted Contributions
of Overlapping Bands
660-nm Band
Only
760-nm Band
Only
Slope=0.78(b)
Rate
Coeff
icie
nt (1
/s)
10 100 1000
0.5
0.6
0.7
0.8
0.9
1.0
=670 nm
Initial
Response
Recovery from
Optical Bleaching
Initial Response, =900 nm(a)
Exponent
•
•
•
Slope=0.78
k
Nsat
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