The Migration of Oxygen during the
Anodic Oxidation of Tantalum
J. P. S. Pringle*
Chalk R i v e r Nuclear Laboratories, Atomic Energy o~ Canada Limited, Chalk River, Ontario, Canada
ABSTRACT
Oxygen-18 concentration profiles in thin oxide films h a v e been m e a s u r e d
b y combining a sectioning t e c h n i q u e for locating t h e isotope w i t h a n u c l e a r
m e t h o d for detecting it. The system studied was the anodic oxidation of t a n t a lum, and the oxide was sectioned b y slow dissolution in c o n c e n t r a t e d H F
almost s a t u r a t e d w i t h NH4F; the 1sO p r e s e n t was detected b y b o m b a r d i n g
t h e oxide w i t h 3.042 MeV protons and counting neutrons e m i t t e d in 1sO (p,n) 18F
reaction. On anodizing first in 180 e l e c t r o l y t e and t h e n in ~80 electrolyte, t h e
180 i n c o r p o r a t e d last was found outside the leo l a y e r i n c o r p o r a t e d first. T h e r e
was, however, a small m i x i n g of the populations at t h e b o u n d a r y b e t w e e n
them, and the d e g r e e of m i x i n g was found to be p r o p o r t i o n a l to the square
root of the thickness a d d e d in t h e 180 electrolyte. T h e results could be a n a lyzed v e r y well in t e r m s of forced diffusion ( u n d e r the influence of the electric
field d u r i n g anodization) from a constant source (the electrolyte) into a
semi-infinite m e d i u m (the oxide), thus confirming that o x y g e n i n d e e d m i g r a t e s d u r i n g the anodic oxidation of t a n t a l u m .
The sectioning technique described in p r e v i o u s
papers (1, 2) has b e e n combined w i t h a least squares
fit analysis (3) a n d used to s t u d y t r a n s p o r t processes
in the anodic oxidation of t a n t a l u m (4). Radioisotopes of t h e noble gases w e r e found to a p p r o x i m a t e
quite closely to t h e inert i m m o b i l e m a r k e r s r e q u i r e d
for t r a n s p o r t n u m b e r experiments, a n d it was then
possible to show t h a t the oxide thickens l a r g e l y b y
o x y g e n migration. The n e x t step, therefore, was to
find out h o w the i n d i v i d u a l o x y g e n atoms m o v e d
t h r o u g h the oxide film, and this r e q u i r e d a t r a c e r for
the o x y g e n itself. Results o b t a i n e d in such e x p e r i m e n t s are described below.
The configuration of a t r a c e r e x p e r i m e n t depends
on t h e m e t h o d s used to incorporate, detect, and locate
the t r a c e r species. Thus the noble gas atoms used in
the previous s t u d y (4) w e r e i n c o r p o r a t e d b y ion i m plantation, detected b y m e a n s of t h e i r radioactivity,
and located using t h e sectioning technique. No l o n g lived radioisotopes of o x y g e n exist, and so one of
the r a r e stable isotopes, 1~O or 1so, m u s t be used
instead. These can b e d e t e c t e d either b y m e a n s of
n u c l e a r reactions (5, 6) or b y mass s p e c t r o m e t r y (7),
b u t t h e detection efficiency of these methods is m u c h
less t h a n t h a t for radioactivity. Accordingly, m u c h
m o r e t r a c e r is required, and since i m p l a n t a t i o n s at
high fiuence a l t e r t h e p r o p e r t i e s of t h e anodic oxide
(2), this was an u n d e s i r a b l e m e t h o d for t h e incorpor a t i o n of the o x y g e n isotopes. The only a l t e r n a t i v e
was to anodize in w a t e r enriched w i t h the a p p r o p r i a t e
isotope.
In t h e i r p i o n e e r i n g s t u d y on t h e m i g r a t i o n of o x y g e n
d u r i n g anodic oxidation, A m s e l and S a m u e l (6) i n c o r p o r a t e d 180 t r a c e r b y anodizing in H2180, d e t e c t e d
it b y counting a p a r t i c l e s e m i t t e d from t h e n u c l e a r
reaction 180 (p, a)15N, and located it b y t a k i n g a d v a n t a g e of the e x t r e m e l y n a r r o w 1165 keV p r o t o n
r e s o n a n c e in this reaction. Such n u c l e a r m e t h o d s of
location denend on m e a s u r i n g t h e e n e r g y lost b y
c h a r g e d particles in t r a v e r s i n g the solid, and the resolution is l i m i t e d by the e n e r g y straggling t h a t occurs.
This s t r a g g l i n g is quite sufficient to m a s k the effect of
small v a r i a t i o n s in t h e concentration profile, so
t h a t gross changes alone can be observed (8). Using
specimens p r e p a r e d in these laboratories, Evans and
P e m s l e r (7) h a v e detected and located 1so t r a c e r s b y
" Electrochemical Society Active M e m b e r .
K e y w o r d s : t a n t a l u m , anodic oxidation, oxygen tracer, n u c l e a r
microanalysis, oxygen migration.
m e a n s of ion m i c r o p r o b e mass spectrometry, in w h i c h
the oxide is g r a d u a l l y s p u t t e r e d a w a y w i t h 2 k e V
argon ions. By feeding the s p u t t e r e d a t o m s to a mass
spectrometer, the 180 isotope could be detected and
located as a function of the s p u t t e r i n g time. The
resolution is, of course, l i m i t e d b y the u n i f o r m i t y of
the s p u t t e r i n g and a p p e a r s to be l i t t l e b e t t e r t h a n
t h a t for the n u c l e a r method.
In the present experiments, t s o was i n c o r p o r a t e d b y
anodizing in H21sO electrolyte, detected b y counting
neutrons from the 180(p, n ) l S F reaction, and located
b y m e a n s of the sectioning technique described in p r e vious p a p e r s (1, 2). The use of s e p a r a t e techniques
for detection and location i m p r o v e d the resolution b y
about an order of magnitude.
The Principle of the Tracer Experiments
As in the previous p a p e r (4), an ideal oxidation
system will b e assumed; one in w h i c h t h e oxide is
homogeneous, of u n i f o r m thickness, and g r o w n on a
p l a n e m e t a l surface. The e x p e r i m e n t s w e r e p e r f o r m e d b y anodizing to an initial thickness, hi, in
o r d i n a r y H2160 electrolyte, and t h e n to a final t h i c k ness, hf, in identical e l e c t r o l y t e m a d e u p w i t h H21sO.
U n d e r these circumstances, t h r e e basic possibilities
for the 180 profile exist, as i l l u s t r a t e d in Fig. 1.
I. If the oxide thickens e n t i r e l y b y m e t a l migration,
the l s o atoms will be i n c o r p o r a t e d on top of the 160
atoms, and populations wi]l not mix. Since the 160
l a y e r is of e x t r e m e l y u n i f o r m thickness (1), the 180
concentration will fall a b r u p t l y to the n a t u r a l a b u n dance at the b o u n d a r y b e t w e e n the layers.
If the oxide thickens e n t i r e l y b y o x y g e n migration,
t h e r e are two e x t r e m e possibilities.
II. The o x y g e n atoms could move in succession as
space becomes available, thus p r e s e r v i n g t h e i r o r d e r
in queues. Due to the statistical n a t u r e of the motion,
some queues wi]l m o v e faster t h a n others, a n d so some
of the l e a d i n g 180 atoms will o v e r t a k e some of t h e
t r a i l i n g 160 atoms. The p o p u l a t i o n m i x i n g t h a t results
serves to distinguish this possibility from the previous
one.
III. The other extreme is to suppose that once the
oxygen atoms are incorporated in the oxide, they
m o v e right scross until they are stoDped at the metal/
oxide interface. Under these conditions, the oxygen
order will be reversed, since the IsO atoms incorporated ]ast will be found underneath the 160 layer
formed first.
1391
Downloaded on 2016-09-18 to IP 130.203.136.75 address. Redistribution subject to ECS terms of use (see ecsdl.org/site/terms_use) unless CC License in place (see abstract).
1392
October 1973
J. Electrochem. Soc.: S O L I D - S T A T E S C I E N C E A N D T E C H N O L O G Y
I
II
III
METAL MIGRATION
OXYGEN IMMOBILE
SUCCESSIVE
SINGLE
ONE JUMP
ACROSS OXIDE
0
Fig. 1. The three basic zsO
concentration profiles in anodic
tantalum oxide when the oxide is
formed initially to a thickness,
hi, in H2180, and then to bf in
H21sO. The lower curves show the
corresponding neutron count profiles when the films are analyzed
by the 180(p, n)lSF method described in the text.
,
0 'e
JUMPS
0 0 0
/
erfc
/
/
......
" ....
i!
016
016
::~:
8 : .
I
Z
:
0
e-
hf
hi
~%fc
hf
ELECTROLYTE ~These three profiles are the simplest and most easily
interpretable, b u t m a n y others are possible. Thus,
profiles i n t e r m e d i a t e to II and III will occur if the
atoms move several interatomic distances at a time,
instead of one as assumed in II, or all as in III. W h e n
both metal and oxygen move, with a n e w oxide being
formed at the interfaces, the lso distribution will be
a combination of I with II or III, or some i n t e r m e d i ate thereof. If a new oxide is formed w i t h i n the existing oxide, the ~sO profile will depend on the location
of this new oxide as well. Certain types of profile
could, therefore, be r a t h e r difficult to interpret.
Fortunately, most of these possibilities can be eliminated by the evidence a l r e a d y available. T r a n s p o r t
n u m b e r m e a s u r e m e n t s (4) show that both t a n t a l u m
and oxygen migrate, and that the n e w oxide is formed
at the m e t a l / o x i d e a n d oxide/electrolyte interfaces
only. The possible profiles, therefore, are those in
which I is combined with II or III, or with some i n t e r mediate b e t w e e n II and III.
Measurement of the ~sO Concentration Profiles
Incorporation of the zsO.--Tantalum foils, 35 •
10
)<0.37 m m ~, were cut from 0.015 in. sheet supplied by
the Fansteel Metallurgical Corporation. The middle
28 m m on each side was divided into seven areas, each
4 • I0 ram, b y m e a n s of scratches r u n n i n g across the
foil. A 0.5 m m (0.020 in.) diameter t a n t a l u m wire
was spot welded on one corner, and the foils were
then chemically polished as described previously (1).
After a 2 rain dip in the NH4F-HF stripping solution
to remove a n y film left b y the polishing (9), the foils
were anodized to the desired initial thickness using
0.01M K I in o r d i n a r y water as the electrolyte. This
particular electrolyte was chosen to avoid dilution of
the enriched oxygen electrolytes; the anion contains
no oxygen, and the salt does not crystallize with w a t e r
of hydration.
Two H21sO samp]es were obtained from YEDA,
Israe], and KI, dried at 150~ for 2 hr, was dissolved
in them to m a k e up approximately 0.01M solutions;
26.10 mg were dissolved in 15 m l of 11.43% HelsO
(0.0105M), a n d 2.60 mg in 2 m l of 97.56% H ~ t s O
hi
E
hf
hi
METAL
(0.0087M). The t a n t a l u m foils were then reanodized
to the final thickness in one or other of these electrolytes, using specially designed Lucite cells. A platin u m wire served as cathode, and because the anodizing circuit (10) r e q u i r e d a reference electrode, a silver
wire was added for the purpose. The i n t e n d e d A g / A g I
electrode, however, did not have the necessary impedance, and so the other end of the silver wire was
electrically connected to the usual glass electrode by
dipping both in a b e a k e r of o r d i n a r y 0.01M KI.
Anodization behavior with this a r r a n g e m e n t was
normal, as d e t e r m i n e d from a n g s t r o m s / v o l t and seco n d s / v o l t measurements. The use of 0.01M K I in place
of 0.1M H2SO4 did not noticeably affect the thickness
calibration (1) for the t a n t a l u m oxide. No t e m p e r a ture control was provided for the anodizing cell, with
the result that the heat liberated d u r i n g anodization
raised the t e m p e r a t u r e of the small q u a n t i t y of electrolyte present; indeed, the t e m p e r a t u r e rose sufficiently to produce considerable condensation on the
walls of the cell above the electrolyte. At 10 m A / c m 2,
the heating was so pronounced that the final thickness
was very much greater than that intended, and so
m u c h h y d r o g e n was discharged at the cathode that
the specimen was anodized in a froth.
Detection of the zsO.--Several charged particle
nuclear reactions can be used for the detection of 1sO.
If the reaction product is itself a charged particle, as
in the case of the 1sO (p, a)I~N reaction used b y Amsel
and Samuel (6), the detector must be placed inside
the v a c u u m system of the accelerator. Such apparatus
was not available at the time these e x p e r i m e n t s were
performed, and so reaction products that would pass
through the acce]erator wall were needed. These
could only be n e u t r o n s or ~ rays, and n e u t r o n s from
the 1sO(p, n)lSF reaction were in fact used. The other
product of this reaction, radioactive ~SF, has been
employed b y Thompson (11) to measure the 1sO content of n a t u r a l anodic oxide on t a n t a l u m .
The ~sO(p, n)lSF reaction has a proton threshold
energy of 2.574 MeV (12), above which there are a
series of resonances (13). To optimize the conditions
Downloaded on 2016-09-18 to IP 130.203.136.75 address. Redistribution subject to ECS terms of use (see ecsdl.org/site/terms_use) unless CC License in place (see abstract).
VoL 120, No. 10
20
I
[
MIGRATION OF OXYGEN
r
I
I
I
I
I
I
I
I
3.030
3.040
0
~16
'-
f'-~O,,
/
(') I 2
Z
r~
t--D
ILl
I
ffMAX
~
8
FWHM
-
51 keY
f
AT 3 . 0 3 7 5 MeV
0
I
3.000
r
3.010
I
I
3.020
I
3.050
PROTON ENERGY IN MeV
Fig. 2. The 3.037 MeY resonance in the ]SO(p, n)18F reaction
measured with an anodic Ta21SO5 sample 490.~, thick in proton
bombardment ii; 3 MeV protons lose approximately 2 keV in traversing this thickness of oxide. The scale along the abscissa is a working scale, which has been calibrated only approximately; it appears
to be about 5 keV low. The random standard error in each energy
measurement is known to be about 1 keV, and the standard error in
each count is 2.1 times ~ / (observed count). Error ellipses have
been drawn at two standard errors from the points actually measured
to show the 95% confidence limits. The maximum energy lost in an
oxide film during the course of these experiments is indicated by
(~D18.X.
for n e u t r o n counting, it was necessary to consider
t hr ee factors.
First, neutrons can be produced by p, n reactions
on nuclei other than 1so. In this respect, it was f o r t u nate that the lO, n t h r e s h o l d for leO is 17.4 MeV, and
t h a t lSlTa, w i t h its high Z, has a la r g e coulomb b a r rier. Tests w i t h c h e m i c a l l y polished t a n t a l u m showed
that the b a c k g r o u n d count decreased b y an o r d er of
m a g n i t u d e for e v e r y MeV reduction in the proton
e n e r g y b el o w 5.5 MeV; this observation is in a g r e e m e n t w i t h published data for th e 181Ta(p,n)lSlW
COLLI MATOR
cross section (14). To keep the b a c k g r o u n d down,
therefore, the proton energy had to be as small as
possible.
The second r e q u i r e m e n t was to m a x i m i z e the n e u tron yield f r o m the lsO, and this m e a n t using a resonance in the l s O ( p , n ) l S F reaction. A cco r d i n g to the
l i t e r a t u r e (13), the lowest resonance o c c u r s . a t 2.649
___ 0.005 MeV; w h e n tested here, this gave acceptable
n e u t r o n counting rates. It was not used for the actual
experiments, h o w e v e r , because it could not m e e t the
third r e q u i r e m e n t ; that the p, n cross section be constant at all points in the oxide film.
The m a x i m u m thickness of the 1sO profiles studied
here was about 2700A. F r o m e n e r g y loss data (15), it
can be calculated that 3 MeV protons lose a p p r o x i m a t e l y 13 keV in t r a v e r s i n g this thickness of t a n t a l u m
pentoxide, so that a proton incident on one side w i t h
an e n e r g y 6.5 keV above resonance w o u l d e m e r g e
from the other w i t h an e n e r g y 6.5 keV below resonance. The third r e q u i r e m e n t meant, therefore, that
the p, n cross section had to be a p p r o x i m a t e l y constant for 6.5 keV on either side of the resonance peak.
Since the full w i d t h at half m a x i m u m for the 2.649
MeV resonance was only 8 keV, this resonance was
obviously much too narrow.
The n e x t resonance in the l S O ( p , n ) l S F reaction
occurs at 3.037 MeV, w i t h an F W H M r e p o r t e d as 33
• 3 keV (13); the l a t t e r was confirmed here, as
d e m o n s t r a t e d in Fig. 2. This resonance is the broadest
available b el o w 3.5 MeV (13), and t h er ef o r e had to
be used in the present experiments. Th e incident
proton e n e r g y was fixed at 3.042 MeV, 5 keV above
resonance, to keep the a v e r a g e cross section for a ny
oxide film up to 2700A in thickness as constant as
possible; the residual v ar i at i o n was estimated to lie
within a range of 4%.
Location of the l s O . - - T h e anodized foils w e r e
t h i n n ed by means of the sectioning t e c h n i q u e (1) in
such a w ay as to l eav e a series of steps in the oxide
surface, as indicated schematically in Fig. 3. The area
of each step, 4 • 10 mm, was defined by the scratch
m ar k s on the t a n t a l u m surface, and the oxide thickness
at each step adjusted to a preselected v al u e by controlling the t i m e spent in the N H 4 F - H F stripping
solution. Once the desired thickness had been reached,
HOLES 3ram DIA.
[]
hf
[]
p+ BEAM 2mm DIA.
- - ;.~.::4.:C!;,I:;(,.:{ f}.7:.:,;.;...~-.
/
STEPS 4 mm WIDE
I
hi
:-i."/ :: ":
1393
'7: 7.:.":.:.-: :-.::"..: -:.
9 " -. . :1
--
-
-
Fig. 3. Schematic diagram illustrating the measurement of an
180 concentration profile by
means of the 1SO(p, n)leF reaction. The film was anodized initially to h~ in H2100, and then to
hf in H2180; one step was left at
hi', and the others stripped back
to give a series of thicknesses
about hi. The scratch marks on
the tantalum surface defined the
area of each step.
Downloaded on 2016-09-18 to IP 130.203.136.75 address. Redistribution subject to ECS terms of use (see ecsdl.org/site/terms_use) unless CC License in place (see abstract).
1394
J. Electrochem. Soc.: S O L I D - S T A T E S C I E N C E A N D T E C H N O L O G Y
the step was covered with Apiezon N grease to prevent f u r t h e r attack as t h i n n i n g continued elsewhere;
the grease was s u b s e q u e n t l y dissolved in trichlorethylene. Comparison of the i n t e n d e d thicknesses with
those actually measured by the spectrophotometric
technique showed that the difference was rarely more
t h a n 10A and f r e q u e n t l y less than 5A. The u n i f o r m i t y
of the t h i n n i n g and the errors in the thickness m e a surements have been discussed elsewhere (1, 3, 4).
Two types of oxide samples were prepared for the
present study. Those i n t e n d e d to investigate the gross
features of the 180 concentration profile were anodized
first to 880A in 160 electrolyte, a n d t h e n to more t h a n
3000A in 180 electrolyte; steps of 150A or more were
then placed in the oxide surface to cover the whole
range of thicknesses from 3000 to about 700A.
These samples showed that the 180 profile was of the
form II in Fig. 1, a n d so the second type of sample was
designed to measure the m i x i n g at the 160/lso b o u n d ary. They therefore required m u c h smaller steps of
10-20A, centered about the initial thickness, n o r m a l l y
880A, of 160 oxide. This thickness was chosen as
standard because it is close to the bottom of the range
that can be continuously measured with spectrophotometer (1).
The composite 160/lso oxides were thus t h i n n e d
down, as illustrated in Fig. 3, to leave seven different
thicknesses on each side of the t a n t a l u m foil, for a
total of fourteen different thicknesses in all. By b o m b a r d i n g each thickness with the same dose of 3.042
MeV protons, the relative a m o u n t of 180 b e n e a t h a
given depth in the composite oxide could be determined. The results, in the form of n e u t r o n counts vs.
oxide thickness remaining, described the integrated
concentration profile, from which the 180 profile itself
could be obtained b y differentiation.
Proton bombardments.--Four t a n t a l u m foils prepared as above were m o u n t e d l o n g i t u d i n a l l y in an
a l u m i n u m target holder. The latter was electrically
insulated, so that the proton charge deposited in the
t a n t a l u m foils could be passed to a c u r r e n t integrator
as a m e a s u r e of the proton dose. Collimating plates
were placed about a centimeter away on each side of
the foils, and t w e n t y - e i g h t 3 m m diameter holes
bored in each plate at positions opposite the i n d i v i d u a l
steps on each foil (Fig. 3). The whole assembly was
then inserted into the target c h a m b e r of a High Voltage E n g i n e e r i n g EM Van der Graaff accelerator.
To make a measurement, the proton beam was first
focused to a spot 2 m m in diameter on a quartz viewing screen placed in front of the target assembly. By
m e a n s of a rack a n d pinion, the target assembly could
be moved so that a n y one of the t w e n t y - e i g h t positions could be selected for b o m b a r d m e n t ; the quartz
was then removed, and the protons passed through
the collimating holes to strike the oxide surface. N e u trons emitted in the 1 8 0 ( p , n ) lSF reaction peak
strongly in the forward direction, and were detected
b y means of a conventional long counter placed as
close as possible behind the target chamber. The
a m o u n t of m a t e r i a l b e t w e e n the oxide film and the
long counter was minimized to reduce n e u t r o n scattering; n e u t r o n s knocked out of the 1sO nuclei passed
first t h r o u g h the u n d e r l y i n g t a n t a l u m foil 0.015 in.
thick, and then through the 1/16 in. a l u m i n u m end
plate of the Van de Graaff v a c u u m system. The scatt e r i n g was in a n y case small; on one occasion a 1/2 in.
steel plate was i n a d v e r t e n t l y left u n d e r the end plate,
a n d reduced the n e u t r o n counting rate b y about 10%.
Measurements on a series of foils were made as
follows. A standard sample, containing 490A of 97.56%
1sO oxide, was b o m b a r d e d first to establish the position of the resonance with respect to the energy scale
of the accelerator; this sample was about 2 keV
"thick" to 3.042 MeV protons. The energy region
October I973
about 3.037 MeV was scanned in 10 keV steps, and,
once the peak had been located, the proton energy
was fixed 5 keV above, as illustrated in Fig. 2. The
oxide steps on the prepared targets were tl~en b o m barded successively, going down one side of the target holder, rotating it t h r o u g h 180 ~, and back up the
other side. The target assembly was then removed
and replaced by its twin, carrying a n e w load of t a r gets which were measured in the same way. Finally,
w h e n all positions on all prepared targets had been
bombarded, the standard sample was again inserted
in the proton beam to make sure that conditions had
not changed in the course of the e x u e r i m e n t s ; no
significant discrepancies were in fact observed.
The proton dose was 150 ~coulombs at a beam curr e n t of about 1 ~A; this corresponds to an i m p l a n t a tion fluence of about 3 • 1016 1H+/cm~ over the area
implanted. A single close of this m a g n i t u d e had no
discernible effect on the appearance of the foils, b u t
the repeated doses applied to the s t a n d a r d changed
the interference colors in such a way as to indicate a
thickening. This thickening has been a t t r i b u t e d to
the deposition of carbon from traces of p u m p oil in
the v a c u u m system (2), a n d had no effect on the n e u tron counting rate.
The total n e u t r o n count varied from a b a c k g r o u n d
of about 100 to a m a x i m u m of 90,000, and was subject to the following errors: (i) the u s u a l statistical
error, equal to the square root of the n u m b e r of
counts; (ii) inaccuracies in the m e a s u r e m e n t of the
proton dose, due partly to c o n t a m i n a t i o n of the proton beam by other ions or b y n e u t r a l h y d r o g e n atoms,
and partly to fluctuations in secondary electron emission b y the target oxides; (iii) fluctuations in the efficiency of the n e u t r o n counter; (iv) fluctuations in
the average proton energy a n d / o r the energy spread,
leading to fluctuations in n e u t r o n yield; and (v) v a r iation of the beam position w i t h i n the collimator
holes, leading to variations in the n u m b e r of n e u t r o n s
generated in the foils, and also in the collimator itself. The collimator m a t e r i a l was therefore of some
importance, and originally gold was chosen on the
ground that it is the only metal of high atomic n u m ber which does not support a surface oxide at room
temperature. Experience showed, however, that the
gold collimator gave a r a t h e r high background, a n d
so it was renlaced with one made of tantalum. The
latter was chemically polished, and t h e n treated with
the HF-NH~F r e a c e n t to remove as much surface oxide as possible (9). This t a n t a l u m collimator gave a
m u c h lower n e u t r o n background, and proved satisfactory.
Duplicate m e a s u r e m e n t s on several different samples showed that the average error was 2.1 times the
statistical error [see (i) above] and so the former
was t a k e n as the standard error for the purposes of
least squares fit analysis. There was, however, considerable variation b e t w e e n i n d i v i d u a l samples, due
p r e s u m a b l y to the effects of ( i i ) - ( v ) .
Originally, it had been i n t e n d e d to b o m b a r d one
side of the foils only. When the holder was modified
to allow b o m b a r d m e n t of both sides, t h e two o r i e n t a tions presented different geometries to the n e u t r o n
counter. By b o m b a r d i n g the seven steps of a particular oxide film in both orientations, a conversion factor of 0.825 _ 0.006 (standard error) could be calculated. Since the oxide film was t h i n n e d a l t e r n a t e l y on
each side of the foil, two s e v e n - p o i n t samples could
in principle be obtained; by applying the conversion
factor, these could be combined in a single fourteenpoint sample, which proved more suitable for a n a l y sis. On reversing the foil in the target holder, a second f o u r t e e n - p o i n t sample was obtained for comparison with the first; w i t h i n e x p e r i m e n t a l error, two
such samples agreed.
Downloaded on 2016-09-18 to IP 130.203.136.75 address. Redistribution subject to ECS terms of use (see ecsdl.org/site/terms_use) unless CC License in place (see abstract).
VoL 120, No. 10
MIGRATION OF OXYGEN
Results
iON
The e x p e r i m e n t a l results were obtained as a series
of n e u t r o n counts vs. oxide thickness remaining, their
general form being illustrated in Fig. 4 and 5. These
plots correspond to the integrated 180 distribution in
the oxide, and reference to Fig. 1 will show that they
are most closely related to condition II or a c o m b i n a tion of I and II. I n order to e l i m i n a t e the possibility
of condition I alone, it is necessary to show that the
m i x i n g of the 160 and 180 populations in Fig. 5 is
real, and not just an artifact due to u n e v e n stripping
of the oxide layers. The u n i f o r m i t y of the oxide stripping process has b e e n studied in detail (1), and the
m a g n i t u d e of the irregularities is definitely not great
enough (4) to account for the effects observed. Migration of the oxygen is therefore confirmed, and it
most p r o b a b l y occurs via mechanism II; that is, the
oxygen atoms all migrate in succession as space becomes available, so that their order is largely conserved. The system can then be regarded as a form
of b u l k diffusion, and has been analyzed m a t h e m a t i cally in the Appendix.
The analysis predicts that the 1sO concentration
profile should be an error function complement (erfc),
and hence that the n e u t r o n count profile actually observed should be an i n t e g r a t e d error function comp l e m e n t (ierfc). As indicated in Fig. 6, these f u n c tions involve five parameters, whose i n d i v i d u a l significance is as follows:
C(1), expressed in n e u t r o n c o u n t s / a n g s t r o m of oxide/150 #coulombs of proton charge, is a proportionality
constant relating n e u t r o n counts to 1sO enrichment.
C(2), m e a s u r e d i n angstroms from the m e t a l / o x i d e
interface, defines the midpoint, or mode of the error
function complement w i t h i n the oxide film.
[
(/3
t - 8o
z
=)
0
,j
z 60
0(.r
I.iE)
L.al
z 40
I
\
~
L
L
t
I
l
[
t
J
I
I
I
I
hi :870-I" 2
I160
1080
I000
OXIDE THICKNESS
920
REMAINING
840
760
(ANGSTROMS)
Fig. 5. Ten times magnification of the 160/180 boundary region in
Fig. 4, illustrating the fit of an integrated error function complement (ierfc) to the data points in the vicinity of hi.
INTEGRATED
ERROR
FUNCTION
(5)
Y(I)
C,)
I,,2
0
0
Z
0
r,,1-bU
Z
%
C(I)+C~4] , ,
,
" ~
"C{5)1 ERROR
1
r~
FUNCTION
C(O
~o
c,!
I
Ct2'J
Fig. 6. The relationship between the error function complement
y -- ( C ( 1 ) / ~ / 2 ) . e r f c [ - - ( X - - C ( 2 ) ) / ~ / 2 " C ( 3 ) ]
+
C(4)
and its integral
Y =
(C(1) "C(3)/~/2) .ierfc[--(X--C(2))/~/2"C(3)
] +
C(4)'X + C(5)
%
20
o\
I
l
I
I
i
I
0
5200
2400
1600
4000
OXIDE THICKNESS REMAINING
I
~Z 4
C(4), in the same u n i t s as C(1), relates n e u t r o n
counts to the n a t u r a l 180 background.
I00
j
ierfc
C(3), also in angstroms, is the standard deviation
associated with the error function complement, and
measures its dispersion about the mode.
C(5), in n e u t r o n counts/150 ~coulombs of proton
charge, are those due to impurities in the oxide and
the u n d e r l y i n g t a n t a l u m .
i
1395
LC(21:873-+5
[
II ~"4.LI
800
I
0
(ANGSTROMS)
Fig. 4. Neutron counting profile obtained with the trial specimen
in the first set of proton bombardments. Initial anodization to 870.&
in H~).BO, final anodization to 3622.~ in H2tsO, both at 1 mA/cm 2
in 0.01M KI. The error ellipses define the 95% confidence limits
around the measured points, and the straight line has been fitted
through the five points from 1254 and 2893.~. The small square in
the lower right hand corner defines the area covered by Fig. 5.
The relationships b e t w e e n these p a r a m e t e r s are
defined by a series of equations ( [ A - 1 ] - [ A - 1 9 ] ) in
the Appendix, and these were tested e x p e r i m e n t a l l y
in three sets of proton b o m b a r d m e n t s . The first set,
i, was used to establish the best conditions for the
experiments, which were t h e n reproduced in sets ii
and iii. The reproduction was not, however, completely accurate; the b o m b a r d m e n t energy in set ii
appeared to be slightly different, while the n e u t r o n
counts in set iii were all 10% low due to the presence
of the 89 in. steel plate.
The relationships tested were as follows:
(i) The n e u t r o n counts obtained from a n oxide film
formed entirely in o r d i n a r y water should be proportional to the oxide thickness r e m a i n i n g (Eq. [A-16]).
A sample of this kind was b o m b a r d e d twice d u r ing set i, and once d u r i n g set iii when, to save time,
Downloaded on 2016-09-18 to IP 130.203.136.75 address. Redistribution subject to ECS terms of use (see ecsdl.org/site/terms_use) unless CC License in place (see abstract).
1396
J. Electrochem. Soc.: S O L I D - S T A T E S C I E N C E A N D T E C H N O L O G Y
Table I. Least squares fit of the straight line Y b ( I ) =
C(4) " X ( I ) + C(5) to the neutron counts vs. oxide
thickness data for anodic tantalum oxide
formed entirely in ordinary aqueous
electrolyte (natural 1sO abundance
0.204%). Same sample used in
both runs.
Run
Parameter
C (4)
(counts/A)
Parameter
C(5)
(counts)
Number
of d a t a
points fitted
N
Fitting
probability
p(X 2) .v-~
i
0.061 "4- 0.006
64 ----- 10
28
0.064
iii
0.047 • 0.013
66 -- 24
9
0,008
only nine of the fourteen positions were bombarded;
the results are given in Table I. I n view of the u n certainty in the calculation of the fitting probabilities
(see A p p e n d i x ) , no good evidence against a linear
relationship has been obtained, and so this has been
assumed. I n subsequent tests, these values of C(4)
and C(5) have been used to correct the observed data
for background, those for set ii being assumed identical to those for set i.
(ii) For an oxide formed first in o r d i n a r y water,
and then in H2180, the n e u t r o n counts due to the enr i c h m e n t should, for oxide thicknesses m u c h greater
than the initial 160 thickness, be linearly dependent
on the oxide thickness r e m a i n i n g (Eq. [A-18]).
(iii) F u r t h e r m o r e , this linear relationship should
extrapolate to zero counts w h e n the r e m a i n i n g oxide
thickness equals the initial thickness of 160 oxide (Eq.
[A-14] and [A-18]).
A trial specimen was prepared for the first set of
proton b o m b a r d m e n t s by anodizing a t a n t a l u m foil to
870 ___ 2A in o r d i n a r y aqueous electrolyte, and then
to 3622 ___ 6A in 97.56% H2180. One side was stripped
in approximately 400A steps to leave seven thicknesses between 3622 and 1254A, while the other was
stripped to leave a series of thicknesses about 870A.
The observed n e u t r o n counting profiles for the two
sides are illustrated in Fig. 4 and 5.
After correction for background, a straight line
was fitted to the seven points on the first side, and
the fitting probability found to be well below 0.001.
The fit was therefore decidedly not acceptable, and
reference to Fig. 4 shows why; the two points at
the largest oxide thickness fall below an extension of
the straight line d r a w n through the other five. The
same sample was r e - a n a l y z e d twice more in set if,
w h e n this decrease in the n e u t r o n counts at the
largest oxide thicknesses was not observed; all seven
points could now be fitted to a straight line (fitting
probabilities 0.041 and 0.064). The p h e n o m e n o n was
therefore peculiar to set i, and was almost certainly
due to an erroneous setting for the incident proton
energy. If this is too low, protons n e a r i n g the end
of their track in a thick 180 oxide will have a n energy
substantially below resonance, and so the n e u t r o n
yield from these protons will be reduced.
W i t h i n e x p e r i m e n t a l error, therefore, the results
were consistent with a linear relationship b e t w e e n
n e u t r o n counts and oxide thickness, and it but rem a i n e d to obtain estimates for C(1) and C(2). Due
to the variation in the average (p, n) cross section
with oxide thickness, it was decided to fit a straight
line reueatedly to the e x p e r i m e n t a l data, dropving
the point with largest oxide thickness b e t w e e n each
fit, and to take C(1) and C(2) from that fit which
gave the greatest fitting probability. "Best" estimates
for C(1) and C(2) thus obtained are listed in Table
II, from which it can be seen that C(2) is, w i t h i n
twice the combined i n d i v i d u a l standard errors, identical to t h e initial 180 oxide thickness, hi. Similar b u t
October 1973
Table II. Least squares fit of the straight line Y e ( I ) =
C(I)'(X(I)
- - C(2)) to the neutron counts vs. oxide
thickness data for an oxide film formed first in
ordinary water (lsO abundance 0.204%) to hi
and then in water enriched with lsO to an
abundance of 97.56%. Neutron counts
due to the enrichment (97.36%) were
fitted only at oxide thicknesses greater
than hi -b 3.C(3); see Appendix for
details.
Proton
bombardmerit
Parameter
C(1)
(counts/A)
Parameter
C(2)
(A)
Number
of d a t a
points fitted
N
Fitting
probability
p ( X~) ~-2
F i r s t s p e c i m e n w i t h hi = 870 • 2A
i
33.53 "+- 0.16
873 "4- 5
ii
33,03 -+- 0.34
865 ~ 11
5
5
33.11 ~ 0.23
5
0.288
0.622
10
0.078
ii a
Average
668 ~ 8
0.835
33,2 ___0.2
S e c o n d s p e c i m e n w i t h hi = 904 • 2A
iiJ.
30.16 "4- 0.34
923 ~ 10
a Configuration in specimen holder reversed.
less accurate results were obtained using the 180 enr i c h m e n t of 11.23%.
Predictions (ii) and (iii) are thus verified.
(iv) For oxide thicknesses approximately equal to
the original 160 thickness, the n e u t r o n counting profile should have the form of an i n t e g r a t e d error function complement (Eq. [A-17]).
(v) This ierfc function should have its mode at hi
(Eq. [A-14] and [A-17]).
For the purpose of fitting Eq. [A-17], C(1) was
taken from Table II as 33.2 -~ 0.2 (standard error)
for sets i and if, and 30.2 ___ 0.3 for set iii; C(2) was
t a k e n as hi with standard error ~/ [(e(hi)) ~ -t(0.00144
h i ) 2 ] , where e(hi) is the r a n d o m m e a s u r e m e n t error in hi (1), while (0.00144 hi) is the standard
error due to the u n c e r t a i n t y in the refractive index
(1,4). Reference to Table III will show that all b u t
two of the t w e n t y separate data samples had fitting
probabilities greater than 0.025, and that the two exceptions were duplicated b y acceptable fits. W i t h i n
e x p e r i m e n t a l error, therefore, the ierfc function, with
C(2) = hi, does adequately fit the e x p e r i m e n t a l data,
and so predictions (iv) and (v) have been confirmed
experimentally.
(vi) The standard deviation of the ierfc function
should be proportional to the square root of the thickness added in the H2180 electrolyte, provided that
the latter is not too small (Eq. [A-19] and discussion
in the A p p e n d i x ) .
I n testing for the r e l a t i o n b e t w e e n C(3) and (hf -hi), three of the eighteen acceptable fits in Table III
were eliminated from consideration. The sample anodized by 26A in the lso electrolyte was eliminated because (hf -- hi) ~ 5.C(3); and both samples from the
specimen anodized b e t w e e n 256 and 1041A were eliminated because the thickness m e a s u r e m e n t s were atypical, for reasons described in the next section. A least
squares fit of Eq. [A-19] to the r e m a i n i n g fifteen samp!es gave the constant of proportionality, 3, as 1.725
+__ 0.026A'/2 (standard error), w i t h p(x2)14 as 0.845.
The latter is more t h a n adequate, and so prediction
(vi) is established.
(vii) The standard deviation of the ierfc function
should be i n d e p e n d e n t of hi provided hi is not too
small (Eq. [A-19] a n d discussion in A p p e n d i x ) .
This follows from the fit of Eq. [A-19] to the data
in T a b l e III, section B.
The oxide thicknesses for the first sample in section
B had to be measured in first order. As noted elsewhere (1), first order m i n i m a are rather broad and
Downloaded on 2016-09-18 to IP 130.203.136.75 address. Redistribution subject to ECS terms of use (see ecsdl.org/site/terms_use) unless CC License in place (see abstract).
Vot. 120, No. 10
MIGRATION
Table III. Leest squares (it of the integrated error function
complement Ye(I) = (C(1)'C(3)/~/2)'ierfc
[--(X(I)--C(2})/
N / 2 ' C ( 3 ) ] to the neutron counting profile with C(2) -I- 2-C(31
X(I) ~ C(2) - - 2'C(3), The variation of C(1) and C(2) was
restricted as described in the Appendix. When two results are
quoted for the same specimen, the second was obtained by
reversing the orientation in the target holder. Unless
otherwise stated, the anodizing conditions were 1 mA/cm '~
in 0.01N KI at about room temperature.
Number
Final
Initial
1sO + laO
~BO
thickness thickness
hf (A)
hi (A)
A.
907
949
989
1093
1345
1795
2227
2732
3506
3753
3757
5734
Thickness
a d d e d i n ~O
electrolyte
(ht -- h i ) a (A)
Dependence
881
881
881
881
882
879
880
877
908
892
883
904
Standard
deviation
C(3) (A)
of d a t a
points
fitted
N
o n t h i c k n e s s a d d e d i n tsO e l e c t r o l y t e
26
144-6
10
664-4
144-6
9
1084-4
234-9
8
2124-4
314-6
9
4634-4
404-3
11
9164-4
534-3
II
13474-5
654-4
11
18554-5
764-6
11
2597__.64
854-8
11
2863___64
854-6
12
28634-64
884-4
12
874-9
12
48304-9
1214-6
12
1224-12
12
Fitting
probability
p(X2)~'-a
0.089
0.197
0.126
0.114
0.849
0.866
0.793
0.242
0.083
0.337
0.858
0,003
0.250
O.0OO
B. D e p e n d e n c e o n i n i t i a l t h i c k n e s s f o r m e d i n 1nO e l e c t r o l y t e
10~1
256
765
89___5 c
11
0.197
454-6
1O
0.531
1795
879
9164-4
534-3
tl
0.866
2706
1762
9444-5
504-8
11
0.097
3680
2645
1034-----74
554-7
11
0.408
C. D e p e n d e n c e o n c u r r e n t d e n s i t y
10 m A / c m ~ ( w i t h heating)
4161
881
3281 4- 6b
109 4- 6
1 mA/cm~
3506
908
2597 __ 64
85 4- 8
0.1 m A / c m 2
3633
872
2761 4- 7
88 4- 6
10
0.232
11
0.083
11
0.558
a Errors quoted only for those samples u s e d t o c a l c u l a t e t h e
b r o a d e n i n g p a r a m e t e r , lL
b Figures not additive because individual values have been
r o u n d e d to t h e n e a r e s t a n g s t r o m .
r F i r s t f i g u r e o b t a i n e d f r o m t h i c k n e s s m e a s u r e m e n t s i n f i r s t order, the second using t h i c k n e s s m e a s u r e m e n t s c o n v e r t e d f r o m seco n d order; see t e x t for details.
difficult to measure accurately; furthermore, the range
of thicknesses that can be measured in the satisfactory
300-400 m~ region is limited to 197-330N. As a n added
check, therefore, the oxide steps were incorporated
from the starting thickness of 1041A, so that their
m a g n i t u d e could be measured in second order. The
stepped oxide was t h e n t h i n n e d down in such a w a y
as to place the middle step as close as possible to 256A.
If the oxide thins uniformly, the step heights m e a sured in second order can be combined with the m e a sured first order thickness for the middle step to calculate the final thickness for all the other steps. The
estimates for C(3) obtained from the two sets of
m e a s u r e m e n t s do not differ significantly from each
other, or from tho'se i n the rest of section B, Table IIL
All six relationships involving the 1sO tracer have
thus been confirmed experimentally. Three additional
factors were considered.
(viii) C u r r e n t density.
F r o m Eq. [A-4] and [A-5], the standard deviation
of the ierfc function would appear to v a r y as the reciprocal of the c u r r e n t density, i. However, it also
varies directly as the diffusion constant, D, and if D
is l i n e a r l y proportional to i, the ratio D/i will be constant. I n that case, the standard deviation, C(3),
would be i n d e p e n d e n t of the c u r r e n t density d u r i n g
the anodization process.
The data in Table III, section C, were used for the
fit of Eq. [A-19], and the adequacy of the fit shows
that C(3) does not v a r y significantly with c u r r e n t
density.
(ix) Isotopic ratios i n oxide a n d electrolyte.
1397
O F OXYGEN
If there is no isotope effect during the incorporation
of oxygen, the isotopic ratio in the oxide should be
the same as that in the electrolyte.
From Eq. [A-14] and [A-15], C(1) and C(4) are
proportional to the 1sO e n r i c h m e n t and the n a t u r a l
1sO abundance, respectively. A comparison of n e u t r o n
counting rates vs. 1sO e n r i c h m e n t of the electrolyte
is given in Table IV. Least squares fits of straight
lines passing through the origin gave a slope of 0.34
for sets i and ii, and 0.32 for set iii, but p(x2)2 was
far less t h a n 0.025 in both cases. The best that can be
said, therefore, is that the isotopic ratios are approximately proportional.
(x) K i r k e n d a l l Effect.
The differing masses of 160 and lsO might lead to
a difference in migration rate, and hence to a type
of K i r k e n d a l l effect (16). This would be manifest as
a n e u t r o n count profile significantly different from an
ierfc function, for which there is no evidence.
Discussion
Nature o~ the oxygen migration.--The e x p e r i m e n t a l
results show quite clearly that the oxygen atoms move,
and that they all move. Since the analysis is based
on a statistical system in which a large n u m b e r of
atoms all make a series of small jumps, it follows
that the m e a n distance traveled by each oxygen atom
during each charge transfer event must be small compared to the total distance migrated. Just how small
can be estimated, very crudely, from the following
analysis.
Suppose that the m e a n distance b e t w e e n adjacent
oxygen sites is a, and that the "high field a p p r o x i m a tion" (17) applies. It follows that the oxygen j u m p s
will cluster about the axis of the field and in the forward direction; that is, towards the m e t a l / o x i d e interface. The average distance j u m p e d in the direction
of the field will then be close to a; for simplicity, it
will be assumed constant and equal to a. If the oxide
thickens by an a m o u n t (hf -- hi), the leading 1sO
atoms travel a distance to.(hf -- hi), in an average
of n jumps so that
n . a = to" (/If -- hi)
A second equation including these quantities can
be obtained by considering the standard deviations.
That of the left hand side is a.x/n, as discussed for
the derivation of Eq. [19] in Ref. (4); that of the
right hand side is in fact C(3), because it can be
shown (18) that Eq. [A-11] describes the broadening of an infinitesimally thin layer of 1sO. S u b s t i t u t ing for C(3) b y m e a n s of Eq. [A-19]
a . x / n = /~'x/(hf -- hi)
Solving for a
a = ~2/to = 3.9A
Nearest neighbor oxygen distances in the crystalline forms of t a n t a l u m oxide r a n g e from 2.02 (19) to
3.19A (20) with an average that appears to be about
2.7A; the observed figure is therefore somewhat
greater. Considering the crudity of the analysis, however, the agreement is quite reasonable. On this
basis, then, it would appear that oxygen migrates b y
j u m p i n g from one site to the next.
Table IV. Comparison of neutron counting rates per angstrom of
anodic oxide with the 180 enrichment of the electrolyte, as quoted
by the supplier (YEDA)
1sO
enrichment
e(%)
R u n s i a n d ii
C(1) o r C(4)
Run iii
C(1) or C(4)
0.204
11.23
97.36
0.061 "~ 0.006
4,15 ~-- 0.08
33.2
+---0,2
0.047 • 0.013
3.69 -4- 0.04
30.2
"4- 0.3
Downloaded on 2016-09-18 to IP 130.203.136.75 address. Redistribution subject to ECS terms of use (see ecsdl.org/site/terms_use) unless CC License in place (see abstract).
1398
The r e a l situation is, however, m u c h m o r e complex.
Results r e p o r t e d e l s e w h e r e (4) show t h a t i m p l a n t e d
noble gases b e h a v e in a v e r y similar fashion to o x y gen; t h a t is, the noble gas a t o m s move, a n d it m a y
r e a s o n a b l y be i n f e r r e d that t h e y all move. W h e n the
preceding analysis was a p p l i e d to them, however, the
results were absurd, a n d it was then concluded t h a t
t h e i r m o v e m e n t was a form of B r o w n i a n motion (4).
If so, the m i g r a t i o n of t h e oxygen m a y well proceed
b y two processes: the forced d i r e c t i o n a l m i g r a t i o n
discussed so far, and a m o r e or less r a n d o m diffusion
due to a B r o w n i a n - t y p e motion.
This second process w i l l h a v e little or no effect on
the 1sO profile mode, b u t it will c o n t r i b u t e to the
s t a n d a r d deviation. A n e s t i m a t e for this contribution
m a y be o b t a i n e d b y assuming t h a t t h e b r o a d e n i n g
p a r a m e t e r , p, associated w i t h the m o t i o n of the noble
gases varies w i t h mass. E x t r a p o l a t i o n of the d a t a in
Table I I I of Ref. (4) to oxygen of a v e r a g e mass 17
gives /~ as 1.37A~'% w h i c h m a y be c o m p a r e d w i t h the
observed value of 1.725 __+ 0.026AV~. If the forced m i g r a t i o n and B r o w n i a n m o t i o n a r e c o m p l e t e l y i n d e p e n dent processes, the s q u a r e of the o b s e r v e d ~ will be
equal to t h e sum of the squares of its components;/~ for
t h e forced m i g r a t i o n alone will t h e n be 1.OSAV~, so t h a t
a w o u l d be 1.46A. This calculation is, however, p r o b a b l y not justified, since t h e B r o w n i a n process is almost
c e r t a i n l y a consequence of the forced migration, an(]
t h e r e f o r e not i n d e p e n d e n t of it.
The n a t u r e of the o x y g e n m i g r a t i o n is thus r a t h e r
obscure, b u t t h e p r e s e n t results are not inconsistent
w i t h the hypothesis p u t f o r w a r d in the previous p a p e r
(4), that c h a r g e is t r a n s f e r r e d d u r i n g t h e anodic o x idation of t a n t a l u m via the simultaneous m o v e m e n t
of a group of atoms, both t a n t a l u m and oxygen. Such
a process would r e q u i r e some m i x i n g of t h e 260 and ~so
populations, b u t w o u l d g e n e r a l l y conserve their order,
as is observed.
Conclusions
1. Concentration profiles for o x y g e n isotopes in
anodic t a n t a l u m oxide can be m e a s u r e d w i t h e x c e l lent precision b y combining the sectioning technique
for locating the isotope w i t h a n u c l e a r m e t h o d for
detecting it.
2. This scheme r e q u i r e s that the detection efficiency
shall be i n d e p e n d e n t of the oxide thickness. O x y gen-18 was d e t e c t e d b y m e a n s of n e u t r o n s p r o d u c e d
in the 1sO(p, n ) l S F reaction; b o m b a r d m e n t w i t h 3.042
MeV protons ensured t h a t the detection efficiency
was sensibly constant.
3. On anodizing first in zsO e l e c t r o l y t e and then
in ~so, it was found that the o x y g e n o r d e r was largely,
b u t not completely, conserved.
4. The m i x i n g of the leO and ~so populations across
the b o u n d a r y b e t w e e n t h e m could be analyzed v e r y
w e l l in t e r m s of forced diffusion f r o m a constant
source into a semi-infinite medium, thus confirming
that o x y g e n m i g r a t e s u n d e r the influence of the electric field.
5. T h e j u m p distance associated w i t h the o x y g e n
m i g r a t i o n was found to be of the o r d e r of 4A, w h i c h
is little m o r e t h a n t h e e x p e c t e d n e a r e s t n e i g h b o r o x y g e n distance.
Acknowledgments
The a u t h o r is grateful to J. B u t l e r for advice on
neutron counting, to J. D e n h a r t o g and G. H e a r n for
assistance w i t h the proton b o m b a r d m e n t s , a n d to
T. A. E a s t w o o d and W. D. M a c k i n t o s h for r e v i e w i n g
the manuscript.
M a n u s c r i p t s u b m i t t e d Dec. 20, 1972; r e v i s e d m a n u script received M a y 28, 1973.
A n y discussion of this p a p e r will a p p e a r in a Discussion Section to be p u b l i s h e d in the J u n e 1974
JOURNAL.
October 1973
d. Electrochem. Soe.: S O L I D - S T A T E S C I E N C E A N D T E C H N O L O G Y
APPENDIX
Theory o~ the migration.raThe oxide film m a y be
r e g a r d e d as a semi-infinite medium, b o u n d e d b y a
p l a n e corresponding to t h e o x i d e / e l e c t r o l y t e interface
b u t otherwise e x t e n d i n g to infinity. This concept is
valid in discussing the zso m i g r a t i o n p r o v i d e d t h a t
t h e m e t a l / o x i d e interface is sufficiently far f r o m the
z80/*so b o u n d a r y that it does not influence the m i x ing of the populations; this w o u l d a p p e a r true for the
e x p e r i m e n t s r e p o r t e d here. A r g u m e n t s p r e s e n t e d elsew h e r e (4) suggest t h a t t h e r m a l diffusion is negligible
at r o o m t e m p e r a t u r e , and hence t h a t the oxygen
moves solely u n d e r the influence of the electric field
p r e s e n t d u r i n g anodization; that is, it is a forced m i gration, w i t h the e l e c t r o l y t e acting as a constant
source of ~so. The diffusion system is one, therefore,
of forced diffusion f r o m a constant source into a s e m i infinite medium.
The kinetics of forced diffusion have been discussed
in d e t a i l b y Boltaks (18), w h o t r e a t s the p r o b l e m as
free diffusion coupled w i t h a s u p e r i m p o s e d drift v e locity due to the a p p l i e d field. F o r forced diffusion
from a constant source into a semi-infinite m e d i u m ,
he cites the following equation
Co
-+exP(DZ).
V2~/(2Dt)
(l_erf
(
x+vt
)
))]
w h e r e C is the concentration of the diffusing species
at time t and distance x into t h e medium, Co is the
s t e a d y - s t a t e concentration of the diffusing species
at the b o u n d a r y p l a n e x -- o, v is the a v e r a g e v e locity of the diffusing p a r t i c l e s in t h e direction of
the a p p l i e d field, and D is the diffusion constant of
the free component that arises in consequence of the
concentration g r a d i e n t set up b y t h e forced migration.
The second t e r m of Eq. [A-1] is effectively a p e r t u r b a t i o n due to the presence of the b o u n d a r y plane;
as t increases, it tends to zero. C o m p u t a t i o n s w e r e
m a d e to find out how r a p i d l y it a p p r o a c h e d zero,
using a p p r o p r i a t e values of v and D c a l c u l a t e d f r o m
the experiments. It t u r n e d out t h a t the second t e r m
was c o n s i d e r a b l y less t h a n 1% of the first for all
values of x once vt was g r e a t e r than 5-X/(2Dt); that
is, once the m e a n distance t r a v e l e d b y t h e leading
particles, vt, was m o r e t h a n five times t h e i r s t a n d a r d
deviation, V ( 2 D t ) , about t h a t mean. This condition
was m e t in t h e p r e s e n t experiments, and so Boltaks
equation reduces to
Co
C(x,t)
y[1--erf
(
x--vt
[A-2]
---~/~2~(2-Dt) / A p p l i c a t i o n of Eq. [A-2] to the m i g r a t i o n of o x y gen in anodic oxides r e q u i r e s some changes in format,
and these a r e i l l u s t r a t e d s c h e m a t i c a l l y in Fig. 7. S u p pose t h e r e a n o d i z a t i o n in H2~sO be p e r f o r m e d for a
time, t, at constant c u r r e n t density, i, and t h a t t h e
oxide thickens from an initial thickness, hi, to a final
thickness, hr. If the c u r r e n t efficiency is 1(}0% (17),
the charge passed will be r e l a t e d to t h e oxide
f o r m e d b y F a r a d a y ' s law; t h a t is
it : ( p F / Q ) - ( h f - - hi)
[A-3]
w h e r e p is the oxide density, F the F a r a d a y constant,
and Q the e q u i v a l e n t w e i g h t of Ta20~. The factor
f ( 2 D t ) in t h e e r r o r function of Eq. [A-2] can then
be r e p l a c e d b y flx/(hf -- hi) w h e r e fl is given b y
p -- ~/(2DpF/iQ)
[A-4]
E x a m i n a t i o n of Fig. 7 will show that b y c h a n g i n g
the r e f e r e n c e p l a n e from the o x i d e / e l e c t r o l y t e i n t e r face to the m e t a l / o x i d e interface, and r e v e r s i n g t h e
sense of the axes, the q u a n t i t y ( x -- vt) in Eq. [A-2]
can be r e p l a c e d b y - - ( h -- hi), w h e r e h is a v a r i a b l e
m e a s u r e d r e l a t i v e to h -- o at the m e t a l / o x i d e i n t e r face.
E q u a t i o n [A-2] then becomes
Co
C(h, h f - - h l ) : - ~ - - [ 1 - - e r f (
--(h--hl)
[A-S]
Downloaded on 2016-09-18 to IP 130.203.136.75 address. Redistribution subject to ECS terms of use (see ecsdl.org/site/terms_use) unless CC License in place (see abstract).
Vol. I20, No. I0
1399
MIGRATION OF OXYGEN
C(x,t)
FORCED
DIFFUSION
Least squares fitting procedure.--These erfc and
ierfc functions are the first and second integrals of
a n o r m a l distribution. Using the n o m e n c l a t u r e of
previous papers (3, 4) the p a r e n t n o r m a l distribution
m a y be w r i t t e n
Y (I) -- C (1) / (~/ (2n) .C (3) ) . exp [ - - ( - ( X ( I )
--C(2))/~/2.C(3))
0
vt x
--
"=X
ANODIC
OXIDATION
=f-h i )
/
h.,
h' h i h
hf
0
Fig. 7. Diagram illustrating the relation between the quantities
used in the theory of forced diffusion and those used in the anodic
oxidation.
Substituting
: f l ~ / ( h f - - hi)
[A-6]
Eq. [A-5] becomes
C(h, hf -- hi) ---- (Co/2) . e r f c ( - - (h -- hi)/~/2.~)
[A-7]
Equation [A-7] describes the 180 profile to be
expected if the oxygen migrates by forced diffusion.
The n e u t r o n counts actually observed, however, will
be proportional to the integral of this equation; thus
if the oxide is t h i n n e d from he to a certain thickness
h', the n e u t r o n counts, K, will be given b y
~
oh 9
~O
P" (Co/2) . e r f c ( - - (h -- h i ) / ~ / 2 . a ) .dh
[A-8]
where P is a constant of proportionality, d e t e r m i n e d
b y the cross section for the 1sO(p, n)~SF reaction and
the efficiency of the n e u t r o n counter. The integration
is taken from the m e t a l / o x i d e interface at h ---- o to
the existing oxide surface at h' (Fig. 7). If the 1so
concentration in the metal is assumed zero, the lower
limit m a y be changed to - - ~ without affecting the
value o2 K ( h ' ) . Replacing - - ( h -- hi)/~/2.= b y y, and
m a k i n g the appropriate substitutions, Eq. [A-8] becomes
K(h') :
[A-11]
where the p a r a m e t e r C(1) is a normalizing constant,
C(2) the modal value, and C(3) the s t a n d a r d deviation. I n t e g r a t i o n of this equation gives
Y(I) = (C(1)/2).erfc[--(X(I)
C(h,
K(h') :
2]
f:
(h'-- hl)/X/2.~
P" (Co/2)
9e r f c ( y ) . ( - - ~ / 2 . ~ ) . d y
[A-9]
-.(h'-h~)/x/2.e e r f e ( y ) . d y
= (eP Co/X/2).ierfc(-- (h' -- hi)/X/2.e)
-
C(2))/~/2.C(3)] q- C(4)
[A-10]
Equation [A-10] describes the expected variations
in n e u t r o n counts as a function of distance, h', from
the m e t a l / o x i d e interface. For reasons given in the
discussion of Eq. [ A - l ] , neither Eq. [A-7] nor [A-10]
is strictly correct when (hf -- hi) is small, for then
the p e r t u r b i n g effect of the oxide/electrolyte b o u n d ary cannot be ignored. There must also be a p e r t u r b ing effect due to the m e t a l / o x i d e b o u n d a r y when hi
is very small, and although no calculations have been
made, it seems reasonable to suppose that the two
p e r t u r b a t i o n s would be similar. Equations [A-7] and
[A-10] should therefore be valid provided both (hr -hi) and hi are greater t h a n about 5~--,/hf -- hi).
[A-12]
where C(4) is a constant of integration. On integrating again
Y(1) -- (C(1).C(3)/~/2).ierfc[--(X(1)
-- C(2))/~/2.C(3)] q- C(4).X(1) -~ C(5)
[A-13]
where C(5) is another constant of integration. The
relationship between Eq. [ A - I l l and [A-12] has been
illustrated in Fig. 5 of Ref. (3), while that between
Eq. [A-12] and [A-13] is illustrated in Fig. 6 of the
present paper.
Equation [A-13] is the function that must be fitted
through the experimental points I _ 1, 2. . . . . . .
N.
The first term on the right hand side corresponds to
Eq. [A-10], so that
C(1) - P'Co
C(2) -= hi
C(3) -- ~
[A-14]
X ( I ) =- h'
C(4) accounts for the 1sO present in n a t u r a l a b u n dance, Cn, in the 180 layer, so that
C(4) -- P ' C n
[A-15]
while C(5) accounts for n e u t r o n s produced from the
u n d e r l y i n g t a n t a l u m a n d a n y impurities that may be
present.
Equation [A-13] contains five adjustable p a r a m eters, which is too m a n y to be characterized successfully with a m a x i m u m of only fourteen points per
sample. Fortunately, two of them, C(4) and C(5),
can readily be eliminated. If a t a n t a l u m specimen is
anodized solely in o r d i n a r y water, the n e u t r o n counting profile due to the n a t u r a l 1sO will be
Yb(I) : C ( 4 ) . X ( I ) + C(5)
[A-16]
The least squares fit of a straight line t h r o u g h the
e x p e r i m e n t a l points from such a sample will then
provide estimates for C(4) and C(5), so that by subtracting Eq. [A-16] from Eq. [A-13]
Y e ( I ) : Y ( I ) -- Yb(I)
-- (C(1) . C ( 3 ) / ~ / 2 ) . i e r f c ( - - ( X ( I )
--
Reversing the limits of the integration
K ( h ' ) = (~P Co/~/2)"
-
C(2) ) / ~ / 2 . C ( 3 ) )
[A-17]
where the Ye(I) are the n e u t r o n counts due to the
180 enrichment.
When X(1) < < C(2), ierfc [ - - ( X ( I ) -- C ( 2 ) )
~ / 2 . C ( 3 ) ] tends to zero, and so Y e ( I ) also tends to
zero. When X ( I ) > > C(2), however, ierfc [ - - ( X ( I )
-C ( 2 ) ) / ~ / 2 . C ( 3 ) ] tends to 2 . ( X ( I ) -- C ( 2 ) ) /
~ / 2 . C ( 3 ) , so that Eq. [A-17] becomes
Y e ( I ) = C(1)" ( X ( I ) -- C ( 2 ) )
[A-18]
which is a straight line. For least squares fitting p u r poses, Eq. [A-18] was indistinguishable from [A-17]
once X ( I ) > C(2) + 3 . C ( 3 ) . If Eq. [A-18] is fitted
to the appropriate data, estimates for C(1) and C(2)
can be obtained; the latter is expected, from Eq.
[A-14], to equal the initial 180 oxide thickness, hi.
In m a k i n g the least squares fit of a function with
several parameters, it f r e q u e n t l y happens that i n formation on some of the parameters is already available, as in the case of C(2) above. Such information
is wasted if the least squares fit is used to obtain
Downloaded on 2016-09-18 to IP 130.203.136.75 address. Redistribution subject to ECS terms of use (see ecsdl.org/site/terms_use) unless CC License in place (see abstract).
1400
J. Electrochem. Soc.: S O L I D - S T A T E S C I E N C E A N D T E C H N O L O G Y
additional estimates for these p a r a m e t e r s ; i.e., if the
p a r a m e t e r s are a l l o w e d to v a r y freely. By fixing the
p a r a m e t e r s at k n o w n values, or b y allowing t h e m to
v a r y w i t h i n limits dictated b y k n o w n s t a n d a r d errors,
such information can be i n c o r p o r a t e d in the least
squares fitting program. The m o r e the v a r i a t i o n of
the p a r a m e t e r s is restricted, the less accommodating
does the p r o c e d u r e become, and the g r e a t e r the difficulty in obtaining a satisfactory fit. The most s t r i n gent goodness-of-fit test, therefore, is obtained b y
restricting the v a r i a t i o n as m u c h as possible.
This principle has been a p p l i e d in the least squares
fit of Eq. [A-17]. It was assumed t h a t C(2) should be
equal to hi, and this assumption was then incorpor a t e d in the fitting p r o c e d u r e b y r e s t r i c t i n g the v a r i a tion of C(2) about hi to the s t a n d a r d error in hi.
Values of C(1) together w i t h t h e i r errors w e r e o b tained from least squares fits of Eq. [A-18], and these
w e r e i n c o r p o r a t e d in similar fashion. Equation [A-17]
was fitted to d a t a w i t h C(2) + 2-C(3) > X ( I ) >
C(2) -- 2 - C ( 3 ) , while Eq. [A-18] was fitted only to
d a t a w i t h X ( I ) > C(2) + 3 . C ( 3 ) ; the i n f o r m a t i o n
used to obtain C(1) was not t h e r e f o r e used again
in fitting Eq. [A-17].
Operationally, the least squares fitting was p e r f o r m e d as described in Ref. (4), w i t h the error,
E X ( I ) , in X ( I ) t a k e n p r o p o r t i o n a l to 0.067 times the
difference b e t w e e n the "kick" w a v e l e n g t h s (1), a n d
the error, E Y ( I ) in Y ( I ) as 2.1 times the statistical
counting error. As noted earlier, the multiplication
factor in E Y ( I ) tends to v a r y from s a m p l e to sample,
and since t h e fitting probabilities are e x t r e m e l y
sensitive to the m a g n i t u d e s of the e x p e r i m e n t a l error,
their correct values a r e in some doubt. F u r t h e r m o r e ,
the use of restricted p a r a m e t e r s raises a p r o b l e m in
connection with t h e n u m b e r of degrees of freedom,
which is of i m p o r t a n c e for ca]culating the ~2 p r o b a b i l i t y (3). Since t h e r e is no obvious a n s w e r to this,
the most u n f a v o r a b l e case was assumed; that is, the
n u m b e r of degrees of freedom in t h e fit of Eq. [A-17]
was r e d u c e d to a m i n i m u m b y t r e a t i n g C(1) and
C ( 2 ) , for this purpose only, as v a r y i n g freely. The
actual m a g n i t u d e of the i n d i v i d u a l ~2 probabilities is
f o r t u n a t e l y not of g r e a t significance; w h a t does m a t ter is w h e t h e r t h e y are, in general, g r e a t e r or less
than t h e chosen acceptance level of 0.025 (3). A n d
for that purpose, the p r e s e n t computations were good
enough.
October 1973
By combining Eq. [A-6] and [A-14]
C(3) : / ~ / ( h f -- hi)
[A-19]
so t h a t C(3) is expected to be p r o p o r t i o n a l to the
square root of the oxide thickness f o r m e d in the 1so
electrolyte.
REFERENCES
1. J. P. S. Pringle, This Journal, 119, 482 (1972).
2. J. P. S. Pringle, S u b m i t t e d to This Journal.
3. J. P. S. Pringle, Accepted for publication in This
JournaZ.
4. J. P. S. Pringle, ibid., 120, 398 (1973).
5. R. W. Ollerhead, E. Almqvist, and J. A. Kuehner,
J. Appl. Phys., 37, 2440 (1966).
6. G. A m s e l a n d D. Samuel, J. Phys. Chem. Solids, 23,
1707 (1962).
7. C. A. Evans and J. P. Pemsler, Anal. Chem., 42,
1060 (1970).
8. C. Ortega and J. Siejka, A b s t r a c t No. 86, page 221,
Electrochem. Soc. E x t e n d e d Abstracts, F a l l M e e t ing, Cleveland, Ohio, Oct. 3-7, 1971.
9. B. Maurel, D. Dieumegard, and G. Amsel, This
Journal, 119, 1715 (1972).
10. D. Walker, A t o m i c E n e r g y of C a n a d a Ltd. R e p o r t
AECL-2502, C h a l k River, Canada, 1965.
11. B. A. Thompson, Anal. Chem., 33, 583 (1961).
12. R. O. Bondelid and J. W. Butler, Nucl. Phys., 53,
618 (1964).
13. H. A. Hill and J. M. Blair, Phys. Rev., 104, 198
(1956).
14. L. F. Hansen, R. C. JoDson, H. Mark, and C. D.
Swift, ibid., 30, 389 (1962).
15. W. Whaling, " H a n d b u c h der Physik," S. Flugge,
Editor, Vol. 34, pp. 193-217, S p r i n g e r - V e r l a g ,
Berlin (1958).
16. E. O. K i r k e n d a l l , Trans. Am. Inst. Mining Met.
Engrs., 147, 104 (1942).
17. L. Young, "Anodic Oxide Films," A c a d e m i c Press,
London and New Y o r k (1961).
18. B. I. Boltaks, "Diffusion in Semiconductors," A c a demic Press, New Y o r k (1963).
19. N. C. Stevhen~con and R. S. Roth, J. Solid State
Chem., 3, 145 (1971).
20. N. C. Stephenson a n d R. S. Roth, Acta Crg~., B27,
1037 (1971).
The Choice and Evaluation of Phosphorsfor
Application to Lamps with Improved Color Rendition
J. J. Opstelten, D. Radielovi~, and W. L. Wanmaker*
N. V. Philips" Gloeilampenfabrieken, Eindhoven, The Netherlands
ABSTRACT
The o p t i m u m color r e n d e r i n g i n d e x was calculated both for low- and h i g h pre.ssure m e r c u r y v a p o r l a m p s b y adding spectral lines or emission bands of
various w a v e l e n g t h s to the r e l e v a n t m e r c u r y spectrum. The o p t i m u m w a v e lengths w e r e calculated. In fluorescent l a m p s t h e y d e p e n d both on t h e t y p e of
l a m p a n d on t h e luminous efficacy. In h i g h - p r e s s u r e m e r c u r y - v a p o r l a m p s it
is desirable to add both a blue and a red emission to t h e m e r c u r y spectrum.
F r o m these specifications a certain choice of phosphors to be used can be made.
The merits of these phosphors are e v a l u a t e d and some suggestions are given
concerning t h e design of l a m p s w i t h a n i m p r o v e d color rendition.
The choice of the phosphors to b e used in fluorescent
l a m p s w i t h i m p r o v e d color r e n d i t i o n can be m a d e
b y constructing l a m p s w i t h various phosphor blends
and e v a l u a t i n g the color r e n d e r i n g indices. This is a
r a t h e r complicated and t i m e - c o n s u m i n g method, h o w ever. Accordingly, w h e n fluorescent l a m p s w i t h i m * E l e c t r o c h e m i c a l Society A c t i v e Me~nber.
K e y words: color rendition, fluorescent lamps, phosphors.
p r o v e d color r e n d i t i o n w e r e introduced in an e a r l y
stage, calculations w e r e c a r r i e d out with "theoretical"
l a m p s (1, 2). F o r instance K r u i t h o f and O u w e l t j e s (1)
studied the color r e n d i t i o n (using both t h e spectral
band method and the b e h a v i o r of test samples as a
criterion) of m i x t u r e s of calcium halophosphate, m a g nesium arsenate, and w i l l e m i t e phosphors. W h e n using
these phosphor blends t h e y found from t h e i r calcula-
Downloaded on 2016-09-18 to IP 130.203.136.75 address. Redistribution subject to ECS terms of use (see ecsdl.org/site/terms_use) unless CC License in place (see abstract).