Scripta M E T A L L U R G I C A
Vol. 9, pp. 1211-1218, 1975 Printed in the United States
Pergamon Press,
Inc.
AN 5 7 F e AND 5 7 C o M O S S B A U E R STUDY OF V A C A N C Y - I M P U R I T Y B I N D I N G IN A L U M I N I U M
J. B. W a r d Physics Department, U n i v e r s i t y of C a n t e r b u r y C h r i s t c h u r c h 4, New Zealand. (Received August
i.
21, 1975)
Introduction
V a c a n c y - i m p u r i t y b i n d i n g in A£ and its alloys has been e x t e n s i v e l y studied, because of its i m p o r t a n c e in the a g e - h a r d e n i n g process. The methods of c a l c u l a t i n g and m e a s u r i n g b i n d i n g e n e r g i e s in these alloys have been r e v i e w e d by M a r c h and R o u s s e a u (I). T r a d i t i o n a l methods, such as m e a s u r e m e n t of the e l e c t r i c a l r e s i s t i v i t y of the q u e n c h e d alloys, include c o n t r i b u t i o n s from all types of defects p r e s e n t in the sample, both a s s o c i a t e d and single, and can t h e r e f o r e be d i f f i c u l t to interprete. A m e t h o d of studying a single type of defect w o u l d be very valuable. In c e r t a i n c i r c u m s t a n c e s v a c a n c y - i m p u r i t y b i n d i n g can be studied by M o s s b a u e r s p e c t r o s c o p y (MS). Such a case could occur if the impurity w e r e a M 6 s s b a u e r atom, since a v a c a n c y n e a r e s t n e i g h b o u r (nn) to such an a t o m w o u l d change that atom's MS(2). W i t h v a c a n c y c o n c e n t r a t i o n s C v of 10 -3 to 10 -6 , typical of m a n y m e t a l s q u e n c h e d from high temperatures, the fraction F of Mossbauer atoms with one v a c a n c y nn w o u l d be the small q u a n t i t y F =
(Civ/C i) = zC v ,
if we assume the v a c a n c i e s are r a n d o m l y distributed. Here Civ is the fractional c o n c e n t r a t i o n of M ~ s s b a u e r a t o m impurities w h i c h have one v a c a n c y nn, C i the total fractional c o n c e n t r a t i o n of M 6 s s b a u e r atom~impurity, and z the alloys c o o r d i n a t i o n number. Because of the small value of Cv, F w i l l be e x t r e m e l y small unless a p o s i t i v e b i n d i n g energy E~v exists b e t w e e n the vacancy and the M ~ s s b a u e r atom. If this second c o n d i t i o n is fulfilled F can be increased to sizeable values by annealing, as will be shown in Section 4. 2. The effects of nn v a c a n c i e s on the MS of 57Fe atoms in a metal. We will assume that the 57Fe impurity atoms are d i s s o l v e d s u b s t i t u t i o n a l l y at r a n d o m lattice sites in a n o n - m a g n e t i c cubic metal host and that their c o n c e n t r a t i o n C i is small enough so that all 57Fe atoms have host nn's. In this case we will get a single line MS of the natural line w i d t h (for a thin source and absorber). V a c a n c i e s i n t r o d u c e d into this lattice will a f f e c t the MS of those 57Fe atoms they are nn to. So far, there has been no q u a n t i t a t i v e p u b l i s h e d treatment of the effect on the MS of a v a c a n c y or an interstitial a t o m being nn to the M ~ s s b a u e r atom. H o w e v e r one w o u l d expect the f o l l o w i n g changes in the case of the 57Fe a b s o r b t i o n spectrum, a s s u m i n g a s i n g l e - l i n e 57Co source is used.
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M O S S B A U E R STUDY OF A L U M I N I U M
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(i) R e c o i l e s s f r a c t i o n f: this should d e c r e a s e since the 57Fe atom will have its v i b r a t i o n a m p l i t u d e increased. (ii) Isomer shift 6: there will be two effects here. R e m o v i n g a n n will ~ake the 57Fe atom's S w a v e f u n c t i o n s expand, r e d u c i n g [~o 12. On t h e j o t h e r ana, there may be some d÷s e l e c t r o n t r a n s f e r w h i c h will increase J~o 2 . The first effect is p r o b a b l y g r e a t e r than the second, as is the case w h e n Fe metal is c o m p r e s s e d (3). Thus J~oJ 2 should d e c r e a s e and ~ increase. (iii) Q u a d r u p o l e S p l i t t i n g AEQ: the l o w e r i n g of s y m m e t r y by the p r e s e n c e of the nn v a c a n c y should cause a q u a d r u p o l e s p l i t t i n g to appear. In the case of an i n t e r s t i t i a l a t o m nn to the 57Fe atom, effects (i) and (ii) above should by a n a l a g o u s r e a s o n i n g be reversed, i.e., f should i n c r e a s e and 6 decrease. E f f e c t (iii) should also occur. L a c k i n g m e a s u r e d or c a l c u l a t e d values for effects (i) - (iii) in the case of v a c a n c y nn's, it can h o w e v e r be a s s u m e d that they will be of about the same m a g n i t u d e as the changes p r o d u c e d by i n t e r s t i t i a l nn's. Such effects were d i s c o v e r e d by M a n s e l et al (4), who i r r a d i a t e d an A~ source foil, c o n t a i n i n g 57Co as a dilute impurity, w i t h n e u t r o n s at 4K, and found a new c o m p o n e n t in the MS w i t h a change of isomer shift A6 = + 0.40±0.01 m m s -I w i t h respect to the o r i g i n a l single line, and a q u a d r u p o l e s p l i t t i n g ~EQ = 0.10±0.02 m m s -I (note that A6>0 for a source ~ A6<0 for an absorber). This c o m p o n e n t a p p e a r e d d u r i n g the i r r a d i a t i o n and i n c r e a s e d during that stage of the a n n e a l i n g in w h i c h A£ i n t e r s t i t i a l atoms are known to be mobile; it was t h e r e f o r e a t t r i b u t e d to those 57Co atoms that had c a p t u r e d i n t e r s t i t i a l A£ atoms. For a thin a b s o r b e r the % dip in a MS line is a p p r o x i m a t e l y p r o p o r t i o n a l to the c o n c e n t r a t i o n of 57Fe atoms c o n t r i b u t i n g to the line. We w o u l d therefore expect that, if a f r a c t i o n F of the M ~ s s b a u e r atoms capture vacancies, the o r i g i n a l single line MS % dip should be r e d u c e d by a f r a c t i o n of about F, and this change may be m e a s u r a b l e in cases w h e r e the new lines are too w e a k to observe directly. 3. T h e o r e t i c a l c a l c u l a t i o n s on v a c a n c y - i m p u r i t y b i n d i n g in A£: s t a r t i n g values of the parameters. Since a p r e l i m i n a r y e x p e r i m e n t a l study (5) i n d i c a t e d the effects of q u e n c h i n g on the MS of 57Fe in AZ w e r e v e r y small, a t h e o r e t i c a l study was u n d e r t a k e n to d e t e r m i n e the e x p e r i m e n t a l c o n d i t i o n s w h i c h m a x i m i s e d F. The f o l l o w i n g symbols, and p a r a m e t e r values w h e r e quoted, w e r e used Fraction defect concentrations: Cv
: single v a c a n c i e s
C. : i m p u r i t i e s (57Fe atoms) 1 A c t i v a t i o n energies:
C2v
: divacancies
C. : i m p u r i t y - v a c a n c y nn pairs iv
E m : single v a c a n c y m o t i o n a l energy = 0.62eV v E~v: d i v a c a n c y m o t i o n a l energy = 0.50eV
(6)
E~v: d i v a c a n c y b i n d i n g energy
(6)
= 0.17eV
(6)
ET v a c a n c y - i m p u r i t y b i n d i n g energy iv Other p a r a m e t e r s v = atomic v i b r a t i o n f r e q u e n c y = 3.66 x 1012s -I R = sink d e n s i t y factor.
(7)
Vol.
9, No.
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M~SSBAUER
STUDY OF ALUMINIUM
1213
It is necessary to consider the effects of divacancy formation, breakup and annealing since these defects form in considerable numbers, especially after quenches from high temperatures. The following reactions were considered: vacancy + vacancy ~ divacancy vacancy + impurity ~ vacancy-impurity pair vacancy + sink divacancy ÷ sink The defects were assumed to be randomly distributed and to anneal by diffusion of vacancies and divacancies, these defects finally disappearing at randomly distributed sinks. In these circumstances, the differential equations for Cv, C2v and Civ are (8): (dCiv/dt)
= 849C v(Ci-Civ)exp(-E~/kT) -7VCivexp (- (E~+E~v)/kT)
(dCv/dt)
i.
= -84~C v (Ci-Civ) exp (-E$/kT) +7~CiveXp(-(E~+E~v)/kT) -84 ~CSexp (-E~/kT)+I 4~C2veXp (-(E~+E~v)/kT) -RgCvexp(-E~/kT)
(dC2v/dt)
2.
= 42~C2exp(-Em/kT) v v b -7~C2veXp(-(E~+E2v)/kT)-R~C2veXp(-E~v/kT)
3.
These equations, solved simultaneously, will give the defect concentrations at any time for a sample at temperature T, provided that the right starting values for C.v, Cv and C2v are inserted. These starting values were obtained in the following way. O0
O0
O0
The equilibrium concentrations Civ , C v and C2v at the temperature T O (the temperature from which the sample is to be quenched) are given by the foIlowing equations (9): cOO=v A exp(-E~/kTQ) C°°/(C iv-" i- C °O` iv J = 12C °° v exp(E~ v/kTQ ) COO2v= 6(C$ O)2exp(E~v/kT Q) f where A is 12.6 and Ev, the vacancy formation energy, values of E b. and E~. used in this paper, C~O and C ~ V ~V were accordingly se~ V to zero.
is 0.76eV (i0). For the are both <0.01 C °O and v
A quench from T O down to some lower temperature T O with infinite speed would hold C °0 constant, however this is not attainable and quench speeds of around 10 3 tov i0 4 K s- 1 are the fastest one can achieve in practice. During the quench, vacancies pair up into divacancies and vacancy-impurity pairs. The amount of this pairing that takes place was calculated following Doyama (section 2 of ref.10), thus arriving at the correct starting valuesl cO'v C~v and C~v'e-Th following a quench from TQ. Quench speeds of 103 K s- were assumed. quench bath temperature T O was assumed to be -40C. The parameter R, the sink density factor, has to be assigned a value also; R controls the speed with which vacancies and divacancies disappear from the sample on annealing. R is known to depend strongly on TO, vacancy supersaturations annealing faster (at a fixed annealing temperature TA) , the higher the value of TQ. Data from Federighi (ii), given in Table i, was used to calculate R for TQ = 600C. Federighi (ii) has reported the resistivity values of pure aluminium quenched from 600C and annealed at various fixed temperatures TA; the T½(exp)
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M ~ S S B A U E R STUDY OF A L U M I N I U M
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row of Table i, d e r i v e d f r o m his data, is the time taken for the r e s i s t i v i t y a n n e a l i n g to be 50% completed. The T½(calc) row was o b t a i n e d by i n t e g r a t i n g e q u a t i o n s 2 and 3 (this paper) w i t h s t a r t i n g values of C@ and C~v a p p r o p r i a t e to a q u e n c h from T O = 600C. C~v and C i w e r e taken as zero (pure sample) and R was given an a r b i t r a r y value. T A was i n i t i a l l y taken as -71C, and the time T ½ ( c a l c ) f o r the total v a c a n c y c o n c e n t r a t i o n (Cv + 2C2v ) to halve was noted. TABLE 1 M e a s u r e d and c a l c u l a t e d r e c o v e r y times T% for v a c a n c i e s in aluminium. TQ = 600C. Sink density f~ctor r = 1.03x10 -4. Annealing temperature T A, °C
-71
-60
-40
-18
T½(exp),
300
60
3.5
0.65
300
62.0
5.0
0.51
T~(calc), ~minutes
reference
(ii),
this p a p e r
The value of R was then a d j u s t e d and the i n t e g r a t i o n s r e p e a t e d until the c a l c u l a t e d and e x p e r i m e n t a l T% v a l u e s w e r e the same. Y~ was then c a l c u l a t e d u s i n g the same s t a r t i n g parameters, and the d e r i v e d value of R, for T A = -60C, -40C and -18C. As table 1 shows, one value of R ( R = I . 0 3 x I 0 -4) gave good a g r e e m e n t at all four a n n e a l i n g t e m p e r a t u r e s . The values of R for T Q = 4 9 0 C and 550C w e r e c a l c u l a t e d in the same way. E q u a t i o n s 1 - 3 w e r e next s i m u l t a n e o u s l y i n t e g r a t e d w i t h a w i d e v a r i e t y of values of Eib , C i and TA, and w i t h s t a r t i n g d e f e c t c o n c e n t r a t i o n s and R values a p p r o p r i a t e ~o TQ = 490C and 550C. The i n t e g r a t i o n s w e r e done on a PDP-II c o m p u t e r using a R u n g e - K u t t a method. The b e h a v i o u r of F = (Civ/Ci) is summarised in the next section. 4.
Theoretical Calculations
: results.
In all the data, F was found to rise d u r i n g the anneal to some m a x i m u m value Fma x in some c h a r a c t e r i s t i c time T, and then to decay. The d e p e n d e n c e of Fma x and T on the q u e n c h p a r a m e t e r s is d e s c r i b e d next. (i) Effect of a n n e a l i n ~ t e m p e r a t u r e T AF i g u r e 1 shows that Fma x and T both i n c r e a s e as T A decreases. A n n e a l s at low t e m p e r a t u r e are thus favoured, the limit b e i n g set by the time a v a i l a b l e for the experiment. (ii) E f f e c t of the b i n d i n g ener@y E b iv" b Figure 2 shows the d e p e n d e n c e of Fma x and T on the value of Eiv for a low t e m p e r a t u r e anneal. B e c a u s e of l i m i t a t i o n s on the m e a s u r e m e n t a c c u r a c y of M ~ s s b a u e r parameters, there will be a value of E~v b e l o w w h i c h b i n d i n g c a n n o t be d e t e c t e d by this method. Both the v a l u e of Fma x, and the time T, can be used to d e d u c e E~v (iii) Effect of impurity c o n c e n t r a t i o n C iF i g u r e 3 shows a low t e m p e r a t u r e anneal w i t h two d i f f e r e n t C i values. There is only a small e f f e c t on Fma x and T w h e n C i is c h a n g e d by a factor of 102 . (iv) E f f e c t of quench t e m p e r a t u r e TQ. This is shown in figure 4. I n c r e a s i n g TQ p r o d u c e s an increase in Fma x w i t h o u t m u c h change in T, however the decay rate of F also increases.
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MOSSBAUER
STUDY
OF A L U M I N I U M
5. E x p e r i m e n t a l Part A : E f f e c t of Q u e n c h i n g a b s o r b t i o n MS of 57Fe in A l u m i n i u m
1215
and the
The absorbers, c o n t a i n i n g a n o m i n a l atomic c o n c e n t r a t i o n of ixl0 -4 Fe, were made by h e a t i n g t o g e t h e r 99.999% p u r i t y zone r e f i n e d A£ t o g e t h e r w i t h Fe e n r i c h e d to 89.7 atomic % 57Fe in p r e c l e a n e d 5 g r a p h i t e crucible. The initial m e l t i n g took p l a c e under a v a c u u m of i-3x10torr, and the samples were held at 8 0 0 - 9 0 0 C for 12 hours to ensure the Fe d i s s o l v e d in the m o l t e n A£. The ingots w e r e then c o o l e d rapidly, c l e a n e d and rolled down to a thickness of 0.71±0.02mm. The foils were a n n e a l e d in a g r a p h i t e c r u c i b l e for 10-14 days at 625C, after w h i c h time the MS c o n s i s t e d of a single line of w i d t h F = 0.25mm s -I. The long a n n e a l i n g time is n e c e s s a r y b e c a u s e of the rather low d i f f u s i o n c o e f f i c i e n t of Fe in A£ (12). An a n a l y s i s (13) of the p r e p a r e d a b s o r b e r gave the total Fe atomic c o n c e n t r a t i o n as (0.87±0.10)xi0 -4. The samples were held inside a v e r t i c a l tubular q u e n c h i n g funace by light spring p r e s s u r e on the end of a steel rod, and were q u e n c h e d by b e i n g thrust into a tube of aqueous CaCI 2 s o l u t i o n at -30C or -40C. Q u e n c h speeds were m e a s u r e d with a 0.002 inch d i a m e t e r c h r o m e l - a l u m e l t h e r m o c o u p l e j u n c t i o n sandw i c h e d inside a d u m m y sample; they v a r i e d b e t w e e n 2 x 10 -3 and 6 x 103 OK s -I. All MS m e a s u r e m e n t s were taken w i t h the samples at dry-ice t e m p e r a t u r e (-78.5C) in a cryostat. The s p e c t r o m e t e r was of the c o n s t a n t a c c e l e r a t i o n type, and the source was 1.25 mc i 57Co in Pd. The MS c o l l e c t i o n time was 1-3 days and the c o u n t s per channel were (1.3-4.0) x 105 .
After
The
following
Run
i:
Quenched Annealed Annealed Annealed
from for for for
Run
2:
Quenched Annealed Annealed
from T O = 490C to T O = -40C for 3 days at - 7 8 . 5 C for i0 min at 0C, 14C, 28C,
each anneal
quenching
and a n n e a l i n g
TQ = 3 days 25 min i0 min
the MS line
programmes
were
carried
out:-
625C to T o = -30C. at -78.5C at +IC at +I00C
% dip,
width
42C and 56C.
and p o s i t i o n
were measured.
Results of Runs 1 and 2. No new lines a p p e a r e d in the MS, and no s y s t e m a t i c changes were o b s e r v e d in line width, p o s i t i o n or % dip. The last p a r a m e t e r had a m e a s u r e d value of (7.75 ± 0.20)% for all spectra taken d u r i n g the a n n e a l i n g process. Using the results of section 4, an upper limit on the b i n d i n g energy E ~ v ~ 0.06eV, for v a c a n c y b i n d i n g to an Fe atom in A£, is implied. 6.
E x p e r i m e n t a l , Part B : E f f e c t of Q u e n c h i n g the source s p e c t r u m of L57Co in A£
on
This part of the work was c a r r i e d out at P o r t l a n d State U n i v e r s i t y Physics Department. The 0.5 mC i source of 57Co in A£ was p r e p a r e d by d i f f u s i o n in an a t m o s p h e r e of H2; it was q u e n c h e d from TQ = 490C to T O =-40C in the same m a n n e r as the a b s o r b e r foils. The source was then p l a c e d in liquid N 2 and put into a v a c u u m c r y o s t a t w i t h o u t letting it w a r m up; it was then a n n e a l e d for a total time of 12 hours at -ii0 ± 2C and for 5 hours at -75 ± 0.8C. During the anneals, the r e c o i l l e s s f r a c t i o n f of the source was m o n i t o r e d ' c o n t i n u o u s l y u s i n g the "black absorber" technique. A p a r t from changes due to small t e m p e r a t u r e variations, f r e m a i n e d c o n s t a n t to w i t h i n 0.5%; thus no v a c a n c y - 57Co a t o m b i n d i n g was d e t e c t e d in the experiment. L a c k i n g k n o w l e d g e of the e f f e c t of nn v a c a n c y on f, a lower limit to E~v c a n n o t be set; h o w e v e r it can be r e a s o n a b l y i n f e r r e d that E~. is again small for this system, since a s i z e a b l e c h a n g e in f w oul d be expected.
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MOSSBAUER STUDY OF ALUMINIUM
7.
Vol.
9, No.
ii
Conclusions.
The optimum conditions for observing vacancy binding to impurity atoms in A£ using the MSssbauer effect have been analysed; for a given quench temperature To, low anneal temperatures are favoured, but the impurity atom concentration is not critical. A study of the MS quenched A£ samples containing 57Fe and 57Co showed no evidence of vacancy binding; the upper limit of E~v for Fe in A£ was estimated to be 0.06eV. h E[v for Fe in Ai has been reported as 0.18eV (14). No value for this quantity is known for Co in A£. 8.
Acknowledgements.
The author would like to express appreciation of the friendship and hospitality extended by members of the Portland State University Physics Department during his study leave in 1971; and to especially acknowledge helpful discussions and advice from Drs. D. G. Howard and R. H. Nussbaum. References 1.
N.H. March and J.S. Rousseau,
Crystal Lattice Defects 2:2-46
2.
C.K. Wertheim, North-Holland,
3.
W.H. Southwell, D.L. Decker and H.B. Vanfleet, Phys.Rev 171: 354-360 (1968); J.A. Moyzis, Jr.,and H.G. Drickamer, ibid 1 7 1 : 3 8 9 - 3 9 2 (1968); J.A. Moyzis, Jr., G. DePasquali and H.G. Drickamer, ibid 1 7 2 : 6 6 5 - 6 7 0 (1968)
4.
W. Mansel,
5.
D.S. Butcher, Christchurch,
6.
J.S. Koehler, in: Vacancies and Interstitials in Metals, A. Seeger, D. Schumacher, W. Schilling and J. Diehl, eds, North-Holland, Amsterdam (1970), page 175.
7.
A. Seeger and H. Mehrer,
8.
A.C. Damask and G.J. Dienes, Point Defects in Metals, New York 1963, Chapter 2 sections 4,6.
9.
M. Doyama, Phys.Rev.148:681-694
A. Hausmann and W. Sander, Defects in Crystalline Solids, Amsterdam (1971)
G. Vogl and W. Koch, Phys. Rev.Letters M.Sc.thesis, Physics Department, New Zealand (1970)
31:359-361
(1973)
University of Canterbury,
ibid, page 8.
& R.W. Balluffi,
R.O. Simmons
ii.
T. Federighi, in : Lattice Defects in Quenched Metals, et al, eds., Academic Press (1965).
Phys.Rev.
12.
G.M. Hood, Phil.Mag.
21:305-328
Gordon & Breach,
(1966)
10.
13. Tektronix,
(1971)
117:52-61(1960) R.M.J. Cotterill
(1970)
Inc, October 1971
14. A.J. Perry and K.M. Entwistle, J.Inst.Metals 96:344-349(1968); E.S.D. Das and K.I.Vasu, Scripta Metallurgica ~ 291-293(1970), Dwarakandasa, ibid 6:187-190(1972)
K.S. Raman, E.S.
Vol.
9, No.
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M O S S B A U E R STUDY OF A L U M I N I U M
I
i
1
I
I
i
I
I
!
1217
I
i
0.0
Ci=10s
F
E'r::-:: :i eV
0-4
~~
-,0° /
0"2
1 2
/
I
10
I
10
I_~ 10
I
I
l
1
_I
I
103
I
I
10
I 10 4
I
IO s
l 2
10
time, minutes FIG.
1
F r a c t i o n a l capture F of v a c a n c i e s by impurities in q u e n c h e d and a n n e a l e d AZ: d e p e n d e n c e of F on anneal t e m p e r a t u r e T A. 0"00
I
r
r
I
I
I
I
J
i 10 ~
i
i 104
i
Ci;l{) 4
- ,o:.c 004
F 005 eV
0.02
_ ~ ~
0.04 eV
003 0"00
10
i 50
l 10 2
eV lC
time, minutes FIG.
2
D e p e n d e n c e of F on v a c a n c y - i m p u r i t y
binding
energy
E~ . iv
1218
MOSSBAUER
I
0.2
T o-
STUDY
OF A L U M I N I U M
1
Vol.
1
490¢
I
~
CcI~6
F 0.1
1 5 x lO:
I 1os
I 5 xlO s
i 1o4
S x 10 4
time, minutes FIG. Dependence
04 .......
|
3
of F on i m p u r i t y
|
f
I
concentration
I
C i.
I
~ T a =
I
1
Ci-10 -4 T,=-75c • :st0 eV
SSOc 0-4
F 0'3
0"0
I
I t0z
i
I 10:
I
I 104
I
time, minutes FIG. Dependence
4
of F on q u e n c h
temperature
TQ.
I 10'
9, No.
Ii