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Vacancy formation in iron investigated by positron annihilation in thermal equilibrium
Scripta
METALLURGICA
VACANCY
Vol. ii, pp. 8 0 3 - 8 0 9 , 1977 P r i n t e d in the U n i t e d S t a t e s
FORMATION
IN
ANNIHILATION
H.-E.Schaefer
(a),
IRON IN
K.Maier
A.Seeger
INVESTIGATED
THERMAL
(a),
(a,b),
BY
Press,
Inc.
POSITRON
EQUILIBRIUM
M.Weller
and
Pergamon
J.Diehl
(b),
D.Herlach
(a)
(b)
I n s t i t u t fur T h e o r e t i s c h e und Angewandte P h y s i k der U n i v e r s i t ~ t Stuttgart (a) and M a x - P l a n c k - l n s t i t u t ffir M e t a l l f o r s c h u n g (b), Stuttgart, Germany (Received
I.
June
24,
1977)
Introduction
In the s t u d y of d e f e c t s in bee m e t a l s iron o c c u p i e s a c e n t r a l p o s i t i o n . On the one h a n d ~ - F e is t h e b a s e m e t a l of a h u g e n u m b e r of t e c h n o l o g i c a l l y important a l l o y s . For m a n y a p p l i c a t i o n s a d e t a i l e d k n o w l e d g e of the p r o p e r t i e s of d e f e c t s in ~ - F e is h i g h l y d e s i r a b l e . On the o t h e r h a n d f e r r o m a g n e t i c techniques offer experimental possibilities for s t u d y i n g d e f e c t s in ~ - F e that are not a v a i l a b l e for any o t h e r b c c m e t a l . In s p i t e of this, v e r y l i t t l e has d e f i n i t e l y b e e n est a b l i s h e d on a t o m i c d e f e c t s in bec iron. In p a r t i c u l a r , the v i e w s on the m i g r a t i o n e n t h a l p y of m o n o v a c a n c i e s ' H MI V ' and on the i n t e r p r e t a t i o n of some of the r e c o v e r y s t a g e s o b s e r v e d in a - F e h a v e b e e n q u i t e c o n t r o v e r s i a l (I-4). T h i s is due p a r t l y to the e x t r e m e s e n s i t i v i t y of a - i r o n to the p r e s e n c e of i m p u r i t i e s , in p a r t i c u l a r FIAs ( f o r e i g n i n t e r s t i t i a l a t o m s ) , p a r t l y to the fact t h a t b e t w e e n T a y = I]84K and Ty6 = 1665K the fee y - p h a s e s e p a r a t e s the bcc ~- and ~ - p h a s e s of p u r e iron f r o m e a c h o t h e r , p r o v i d i n g additional difficulties for b o t h q u e n c h i n g and h i g h - t e m p e r a t u r e equilibrium experiments. A further complication, not as w i d e l y r e c o g n i z e d , is that r e c o v e r y e x p e r i m e n t s are u s u a l l y c a r r i e d out in the c o m p l e t e l y or a l m o s t c o m p l e t e l y o r d e r e d f e r r o m a g n e t i c state, whereas hightemperature experiments r e f e r to the p a r a m a g n e t i c or a l m o s t c o m p l e t e l y d i s o r d e r e d ferromagnetic states. The p r e s e n t p a p e r is p a r t of a c o n c e r t e d e f f o r t to o v e r c o m e the a b o v e mentioned difficulties and to o b t a i n r e l i a b l e d a t a on t h e a t o m i c d e f e c t s in airon. T h e p r e c e d i n g p a p e r (5) r e p o r t s on an a t t e m p t to o b t a i n r e l i a b l e s e l f d i f f u s i o n d a t a on ferromagnetic i r o n and in p a r t i c u l a r a c c u r a t e v a l u e s for the s u m H ~ v + H { v of the m o n o v a c a n c y migration and f o r m a t i o n e n t h a l p i e s . The present p a p e r e m p l o y s p o s i t r o n t r a p p i n g at v a c a n c i e s ( s t u d i e d by D o p p l e r b r o a d e n i n g of t~e 2y p o s i t r o n - a n n i h i l a t i o n line) to o b t a i n the m o n o v a c a n c y formation enthalpy HIV. This a l l o w s us to d e r i v e a r e l i a b l e v a l u e for the m i g r a t i o n e n t h a l p y H~V. as w e l l as an e s t i m a t e for the m o n o v a c a n c y diffusion coefficient, and to d i s c r i m i n a t e b e t w e e n the t w o m a j o r v i e w p o i n t s on the r e c o v e r y s t a g e a s s o c i a t e d w i t h monovacancy migration. It w i l l b e c o m e e v i d e n t that the r e c o v e r y s t a g e at a b o u t 2 2 0 K , constituting the a n a l o g u e of S t a g e III in fcc m e t a l s , c a n n o t be due to t h e long-range m i g r a t i o n of m o n o v a c a n c i e s . The s u b s e q u e n t p a p e r (6) a n a l y z e s the r e c o v e r y e x p e r i m e n t s pertaining to the m i g r a t i o n of v a c a n c i e s in a - i r o n in some d e t a i l . It c o m e s to the c o n c l u s i o n that t h e y are i n d e e d in g o o d a g r e e m e n t w i t h the r e s u l t s d e d u c e d f r o m the h i g h temperature experiments, p r o v i d e d the r o l e of FIAs is p r o p e r l y t a k e n into a c c o u n t . This c o n c l u s i o n is f u r t h e r s t r e n g t h e n e d in a f o u r t h p a p e r (7), in w h i c h it is s h o w n t h a t the r e s u l t s on the i n t e r a c t i o n b e t w e e n m o n o v a c a n c i e s and c a r b o n or n i t r o g e n in ~ - F e o b t a i n e d in the third p a p e r (6) l e a d s to a d e t a i l e d understanding of the Snoek-- K S s t e r ( c o l d - w o r k ) relaxation and of c a r b i d e and nitride formation following electron irradiation.
803
804
VACANCY FORMATION IN IRON
2. E x p e r i m e n t a l
Vol, ii, No, 9
Procedures
2.1 S p e c i m e n : A c y l i n d r i c a l s e a l e d - s o u r c e s p e c i m e n (8,9) was p r e p a r e d f r o m J o h n s o n - M a t t h e y h i g h - p u r i t y i r o n w i t h a t o t a l m e t a l l i c i m p u r i t y c o n t e n t of ~10 at ppm. B e c a u s e of the h i g h e v a p o r a t i o n rate of iron a b o v e 1550K [0.1 ~ m / s at 1730K ( 1 0 , 1 1 ) ] the w a l l of the s p e c i m e n tube was c h o s e n I m m t h i c k . A f t e r m a c h i n ing and e t c h i n g , the s p e c i m e n w a s a n n e a l e d in dry h y d r o g e n for 63h at I020K. This r e s u l t e d in a r e s i d u a l r e s i s t i v i t y r a t i o in a m a g n e t i c f i e l d of H = 500 Oe F ( 3 O O K / h K ) = 1 8 0 0 , a n d C and N c o n t e n t s of less t h a n I at p p m as d e t e r m i n e d by internal friction measurements. A f t e r i n s e r t i n g the 6 ~ C i 2 2 N a C I p o s i t r o n s o u r c e ( e v a p o r a t e d on a h i g h p u r i t y iron foil) the s p e c i m e n was e l e c t r o n - b e a m w e l d e d and f i n a l l y d e g a s s e d (2h, IIOOK, IO-~Pa) in o r d e r to r e m o v e r e s i d u a l h y d r o g e n . The r e s i s t i v i t y r a t i o of a d u m m y h e a t e d t o g e t h e r w i t h the s p e c i m e n d u r i n g the e n t i r e t e m p e r a t u r e prog r a m d e c r e a s e d f r o m F = 1270 b e f o r e to F = 640 a f t e r the h i g h - t e m p e r a t u r e treatment. This may i n d i c a t e that d u r i n g the h i g h - t e m p e r a t u r e m e a s u r e m e n t s the imp u r i t y c o n t e n t of the p o s i t r o n - a n n i h i l a t i o n s p e c i m e n i n c r e a s e d s l i g h t l y due to i m p u r i t i e s p i c k e d up f r o m the l o w - p r e s s u r e a t m o s p h e r e . 2.2 E x p e r i m e n t a l s e t - u p : B e t w e e n 400K and 1 7 5 0 K the m e a s u r e m e n t s w e r e p e r f o r m e d in a r e s i s t a n c e - h e a t e d h i g h - v a c u u m ( 1 0 - ~ P a ) f u r n a c e w i t h a Pt-- Pt 10%Rh t h e r m o c o u p l e for t e m p e r a t u r e m e a s u r e m e n t and c o n t r o l . The t e m p e r a t u r e g r a d i e n t a l o n g the s p e c i m e n was e s t i m a t e d to be less t h a n IOK n e a r the m e l t i n g point. The lowt e m p e r a t u r e c r y o s t a t , the y - s p e c t r o m e t e r for m e a s u r i n g the D o p p l e r b r o a d e n i n g of the p o s i t r o n - a n n i h i l a t i o n l i n e - s h a p e , and the d e t e r m i n a t i o n of the W p a r a m e t e r ( w h i c h is p a r t i c u l a r l y s e n s i t i v e to the p r o b a b i l i t y that t h e r m a l i z e d p o s i t r o n s a n n i h i l a t e w i t h e l e c t r o n s in the ion cores and w h i c h h e n c e is a g o o d i n d i c a t o r of p o s i t r o n t r a p p i n g at v a c a n c i e s ) w e r e d e s c r i b e d e a r l i e r (12). 3._ E x p e r i m e n t a l
Results
T h e t e m p e r a t u r e d e p e n d e n c e of the l i n e - s h a p e p a r a m e t e r W ( F i g . l ) , r e v e r s i b l e t h r o u g h o u t , shows the f o l l o w i n g c h a r a c t e r i s t i c f e a t u r e s :
W
#,
bcc// 0.24
/// ,,// /
/ f fcc ,/ /
(]26
.......'~
.
13
..." .... & A...A.~,.,b.~t,~Z" A~J~rA
.......
I
I
:*~-TI
Tc''
i
Fig.l:
was
Fe
0.23
o27,
which
i
'
s'oo
i
Tpa Tm
~g ,
5oo
I
i
'
rrKl
T e m p e r a t u r e d e p e n d e n c e of the p o s i t r o n - a n n i h i l a t i o n line-shape p a r a m e t e r W in Fe. The d o t t e d line b e t w e e n 4 . 2 K and IOOOK i n d i c a t e s the l i n e a r t h e r m a l e x p a n s i o n (15) n o r m a l i z e d to the e x p e r i m e n t a l data at 4.2K and IO00K. The full line g i v e s the fit of E q . ( 5 ) to the m e a s u r e d d a t a w i t h the v a l u e s of case (6) in Sect. h.2a.
Vol.
11,
i
No.
9
Between
ii
VACANCY FORMATION IN IRON
5K and 3 5 0 K
the
W parameter
A b o v e 350K the W p a r a m e t e r d e c r e a s e s an i n f l e c t i o n p o i n t at a b o u t 950K.
iii
with
independent.
increasing
temperature,
showing
If c h a r a c t e r i z e d by s t r a i g h t - l i n e a p p r o x i m a t i o n s (13,14) as i n d i c a t e d the thin l i n e s in F i g . l , the h i g h - t e m p e r a t u r e d e c r e a s e sets in at T 3 = I070K, s l i g h t l y a b o v e the C u r i e t e m p e r a t u r e T c = 10h3K.
iv
by
At the s t r u c t u r a l p h a s e t r a n s i t i o n s ~-y and y-6 the W p a r a m e t e r is d i s c o n t i n u o u s . It is h i g h e r in the fcc p h a s e than in the bcc p h a s e , i n d i c a t i n g that v a c a n c y t r a p p i n g is m o r e p r o n o u n c e d in the bcc phase.
v) The
W parameter
tends
to
saturate
4. A n a l y s i s 4.1
is t e m p e r a t u r e
805
Qualitative
near
the
of the
melting
point
T m = 1811K.
Data
analysis
The " M a c K e n z i e t e m p e r a t u r e " T~ = 1070K fits w e l l into the e m p i r i c a l T 3 - T m r e l a t i o n s h i p v a l i d for fcc and h c p S m e t a l s and for Ta (14). It s u g g e s t s that the m o n o v a c a n c y f o r m a t i o n e n t h a l p y H~V in ~ - F e lies b e t w e e n 1.3 eV and 1.6 eV. The t e m p e r a t u r e v a r i a t i o n b e l o w T 3 c a n n o t be e x p l a i n e d by t h e r m a l e x p a n s i o n . This may be seen from Fig.1 , w h e r e the d o t t e d line shows the t h e r m a l e x p a n s i o n (15) n o r m a l i z e d at 4K and at 1000K. The m e a s u r e d l i n e - s h a p e p a r a m e t e r e x h i b i t s the t e n d e n c y to f o r m the p l a t e a u b e l o w T 3 c h a r a c t e r i s t i c of m e t a s t a b l e selft r a p p i n g of p o s i t r o n s (16,17). A s i m i l a r t e m p e r a t u r e v a r i a t i o n was f o u n d in In (18) and Ta (14) and s u c c e s s f u l l y a n a l y z e d in terms of the s e l f - t r a p p i n g m o d e l . In the case of s - i r o n the n u m e r i c a l v a l u e s d e r i v e d from this m o d e l ( S e c t . 4 . 2 ) are s u b j e c t to u n c e r t a i n t i e s r e s u l t i n g f r o m the u n k n o w n e f f e c t s of m a g n e t i c o r d e r on p o s i t r o n a n n i h i l a t i o n . H o w e v e r , t h e s e e f f e c t s s h o u l d be v i r t u a l l y t e m p e r a t u r e i n d e p e n d e n t up to a few lOOK b e l o w T c and w i l l l e a v e u n a f f e c t e d the q u a l i t a t i v e f e a t u r e s w h i c h f a v o u r the s e l f - t r a p p i n g m o d e l over the t h e r m a l - e x p a n s i o n model. 4.2
Quantitative
analysis
~ ] _ ~ i ~ ~ In the s i m p l e s t m o d e l of m e t a s t a b l e t e m p e r a t u r e d e p e n d e n c e of the W p a r a m e t e r is g i v e n by Wf-fst[W f - (Tf/Tst) w(T)
Wst]
=
~ W](T)
I - fst
where above
= [1 + A -I
(]6,17)
(1)
,
T 3/2
e x p ( e o / k B T ) ] -I
lifetimes fractions
,
for a n n i h i l a t i o n of p o s i t r o n s in
(2>
k means Boltzmann's constant, 6 the e n e r g y of the s e l f - t r a p p e d o t~e " c o n d u c t i o n - b a n d m i n i m u m " of the p o s i t r o n s and m+k B 3/2 ~t m + k B 3/2 A~.
A -1 ~ VA(2~2)
the
[I - ( ~ f / T s t ) ]
w h e r e W~, W t' Tf, T t d e n o t e the W p a r a m e t e r s or the in the ~ r e e S o r s e l f - ~ r a p p e d s t a t e s _ r e s p e c t i v e l y . The the s e l f - t r a p p e d s t a t e m a y be e x p r e s s e d as fst
self-trapping
H ~. = VA ( ~ ) J
~
(] + ~. W~). a
state
(S)
a
w i t h the a t o m i c v o l u m e V A and the e f f e c t i v e mass of the p o s i t r o n s m+. In E q . ( 3 ) ~v~ ~ ~ - v. d e n o t e s thN d i f f e r e n c e in the f r e q u e n c i e s of the jth v i b r a t i o n a l m o ~ e s i~ the 8 p r e s e n c e of a s e l f - t r a p p e d (v ) a n d ~ a free (~) p o s i t r o n . The n u m e r i cal f a c t o r s a p p e a r i n g in E q . ( 3 ) are ( m + k ~ / 2 w ~ 2 ) 3/2 = 2 . 4 1 4 . 1 0 2 1 m ~ 3 K -3/2 (with m + = m o = f r e e e l e c t r o n m a s s ) , V A = 1 2 . 1 . 1 0 - ~ 0 m 3 (for ~-Fe). B e c a u s e of the c h a n g e in f e r r o m a g n e t i c o r d e r w i t h t e m p e r a t u r e it is d i f f i cult to o b t a i n all the p a r a m e t e r s a p p e a r i n g iD Egs (I-3) f r o m a fit to W ( T ) . We c h o o s e ~ h e r ~ f o r e (e) H ( v J / V j ) = I O ( A = 3 . 5 - I 0 O K 3 / 2 ) , or (8) ~ ( v ~ / v ~ ) = 3 0 ( A = 1 . 1 7 " I 0 6 K 3 / 2 ) , and o b t a i n w i t h T f / T s t = 0.9 the v a l u e s Wf = ~ . 2 ~ 6 and (~) eo=O.32eV, Wst = 0 . 2 5 1 6 , or (~) 6o=0.29eV, W s t = 0 . 2 4 5 3 .
806
VACANCY
~_~K~_~_~£~B£~± terms of the p o s i t r o n
eq clv
=
FORMATION
IN IRON
Vol,
The q u a n t i t a t i v e a n a l y s i s of the data t r a p p i n g by the e q u i l i b r i u m c o n c e n t r a t i o n
exp(S~Vlk~ exp(-H~v/k B T)
(S~v = e n t r o p y of f o r m a t i o n of m o n o v a c a n c i e s ) the bcc p h a s e the m i d d l e and in the fcc phase S" are not a c c e s s i b l e e x p e r i m e n t a l l y . The fit eq WI(T) + WIV GTf CIV W(T) = eq I + GTf CIV
ii,
No,
9
above T c in of m o n o v a c a n c i e s ,
(~)
,
is h a n d i c a p p e d by the fact that in the ends of the " h i g h - t e m p e r a t u r e of the h i g h - t e m p e r a t u r e data to (12) (5)
with t e m p e r a t u r e - i n d e p e n d e n t v a l u e s of S F , H F , Wst ( p o s i t r o n l i f e t i m e in the . . 1 Vv a c a n c y c o n c e n t r a t i o n ) , s e l f - t r a p p e d state), if(trapping rate per 1V unl~ and WI(T) from Eq.(1) gives for the vacancy formation enthalpy in the paramagnetic bcc phase HIVF = (I .53 +- 0 . 1 5 ) e V
(6)
and a s i m i l a r value for the fcc phase (eomp. T a b l e I). In Eq.(6) the error limits r e s u l t from the ( s t a t i s t i c a l etc.) s c a t t e r of the data. The p o s i t r o n a n n i h i l a t i o n p a r a m e t e r s f o l l o w i n g from the fit are g i v e n in T a b l e
I.
If t e m p e r a t u r e v a r i a t i o n s o~T ±I/2 are a l l o w e d for, the v a c a n c y f o r m a t i o n e n t h a l p i e s are c h a n g e d b Y ~ v = ± 0.08 eV w i t h o u t c h a n g i n g the q u a l i t y of the fit. A t e m p e r a t u r e d e p e n d e n c e of the s a t u r a t i o n p a r a m e t e r WIV of the form ~ W I v / d T ) W ~ v = 4 . 1 0 - 5 K -I as o b t a i n e d in (12) may be c o m p e n s a t e d by a slight v a r i a t i o n of the a b s o l u t e v a l u e s of Wst and WIV w i t h o u t c h a n g i n g H~V. We see that the above a n a l y s i s leads to m o n o v a c a n c y f o r m a t i o n e n t h a l p i e s H Fiv that are in r e a s o n a b l e a g r e e m e n t w i t h the e x p e c t a t i o n s from the e m p i r i c a l "'H FIV -T 3 and. H FIV -T m r e l a t i o n s h i p s (Fig '2 of ref • 14) • We take this as e v i d e n c e that it was i n d e e d a good a p p r o x i m a t i o n to n e g l e c t p o s i t r o n d e t r a p p i n g from v a c a n c i e s (19-21) and d i v a c a n c y c o n t r i b u t i o n s (21) in the above a n a l y s i s . 5. D i s c u s s i o n 5.1
Positron
mobility
The e f f e c t s of v a c a n c i e s in t h e r m a l e q u i l i b r i u m on p o s i t r o n t r a p p i n g are s t r o n g e r in bcc iron than in fcc iron (Fig.l). The p o s i t r o n l i f e t i m e s are exp e c t e d to vary only little w i t h c r y s t a l s t r u c t u r e (see b e l o w ) , h e n c e the d i f f e r e n c e b e t w e e n the two c r y s t a l s t r u c t u r e s in T a b l e I must be m a i n l y due to a l a r g e r value of G e x p ( S ~~v)/_k in the bcc s t r u c t u r e . The e n t r o p y of m o n o v a c a n e y f o r m a t i o n may w e l l be l a r g e r in the bcc s t r u c t u r e than in the fcc s t r u c t u r e , but it is u n l i k e l y that S ~ v ( b c c ) - S ~ v ( f C c ) e x c e e d s k B. It f o l l o w s that ~ ( b c c ) / ~ ( f c c ) ~3, i n d i c a t i n g that e i t h e r the p o s i t r o n m o b i l i t y or the rate of c a p t u r e of p o s i t r o n s by v a c a n c i e s is h i g h e r in bcc iron than in fcc iron. (For the r e l a t i o n ship of m o b i l i t y and c a p t u r e to p o s i t r o n t r a p p i n g by v a c a n c i e s see ref. 17.) The h i g h e r v a l u e s Wst and WIV (which are e q u i v a l e n t to s m a l l e r v a l u e s of the corres p o n d i n g p o s i t r o n l i f e t i m e s ) in the bcc than in the fcc phase may be r e l a t e d to the i n c r e a s e of the i n t e r a t o m i c d i s t a n c e s going from the bcc to the fcc phase (22),
5.2 E q u i l i b r i u m
vacancy
concentrations
If we a s s u m e S~V = (1.O ± O . 5 ) k = we o b t a i n from E q s . ( 4 , 6 ) for the e q u ~ i b r i u m v a c a n c y c o n c e n t r a t i o n at T a. = 1184~: C ~ ( 1 1 8 h K ) = 7.7.10-7±°.9. V a c a n c y c o n c e n v t r a t i o n s of the o r d e r of m a g¥ n l•t u d ~~ 10- 6 cannot be u n a m b l g o u s l y d e t e c t e d in quenching ~xperiments (3) u n l e s s the c o n c e n t r a t i o n of FIAs is lower than 0.2 at ppm (6). HiV = 1.53 eV, S[V = k B plus an e s t i m a t e d 20% c o n t r i b u t i o n of d i v a c a n cies g~Kes us for the t o t a l v a c a n c y c o n c e n t r a t i o n at the m e l t i n g point c~q(T m) = 1.8"10 >.3 M i g r a t i o n
enthalpy
and
diffusion
coefficient
The d i f f u s i o n c o e f f i c i e n t of m o n o v a c a n c i e s , self-diffusion c o e f f i c i e n t , D T, a c c o r d i n g to
of m o n o v a c a n c i e s DIV,
is r e l a t e d
in ~ - i r o n to the
tracer
VOI.
II,
No.
9
VACANCY
D T = fIV In (7) it has b e e n fIV = 0 . 7 2 d e n o t e s d i f f u s i o n d a t a for DIV
DIV
FORMATION
IN I R O N
exp( S iF v / k B) exp( _ H [ v / k B T
807
(7)
)
a s s u m e d that o n l y m o n o v a c a n c i e s c o n t r i b u t e to s e l f - d i f f u s i o n ; the m o n o v a c a n c y c o r r e l a t i o n f a c t o r . I n s e r t i o n of the selfp a r a m a g n e t i c a - i r o n (5) gives us for T>T c
= 0.5
exp(-O.92e~kBT)Cm
2 -I s
(8)
The c o m p a r i s o n w i t h r e c o v e r y m e a s u r e m e n t s r e q u i r e s an e s t i m a t e of the e f f e c t of f e r r o m a g n e t i c o r d e r on H~V, s i n c e the p o s i t r o n a n n i h i l a t i o n t e c h n i q u e is not c a p a b l e of g i v i n g i n f o r m a t i o n on v a c a n c y f o r m a t i o n b e l o w T c. A p p l i c a t i o n of the t h e o r y of Ruch et al. (24) [ c o m p . ( 5 ) ] g i v e s us as the m o n o v a c a n c y f o r m a t i o n e n t h a l p y at a b o u t 570K F = (I • 60 ± 0 . 15) eV H1V
(9)
If the e x t r a p o l a t i o n of the s e l f - d i f f u s i o n c o e f f i c i e n t to low t e m p e r a t u r e s is p e r f o r m e d by means of Zene~'s t h e o r y (5,25) one finds~ with the help of Fig, 3 of ref. (5) 2 -I DIV(573K) = 0.5 exp ( - 1 . 2 8 e V / k B T ) C m s (10) This
estimate
makes
use
of the
assumption
TABLE
S[V
=
kB
1
E ~ t h a l p i e s of M o n o v a c a n c y F o r m a t i o n H~V, M i g r a t i o n as w e l l as P o s i t r o n A n n i h i l a t i o n P a r a m e t e r s ( b a s e d M H]v(eV)
H[v(eV) bcc
paramag.
fcc
QSD(
H I, VM and S e l f - D i f f u s i o n Q S D on Case ~ in Sect. 4.2a).
F eV) ~ T f e x p ( g i V / k B)
Wf
WIV
Wst
1.53±0.15
0.92
2.45
(5)
106
-
0.2453
0.226
1.54±0.15
1.26
2.80(23)
105
-
0.2441
0.220
1.60±0.15
1.28
2.88
-
0.266
bcc
ferromag.
5.L
Consequences
for
the
(5)
interpretation
of a n n e a l ± n 5
sta~es
B e c a u s e of the u n c e r t a i n t i e s in the e x t r a p o l a t i o n of the h i g h - t e m p e r a t u r e d a t a to l o w e r t e m p e r a t u r e s it is d i f f i c u l t to give d e f i n i t i v e and m e a n i n g f u l e r r o r l i m i t s to E q . ( 1 0 ) . N e v e r t h e l e s s it is p o s s i b l e to d e c i d e u n a m b i g o u s l y the m a i n c o n t r o v e r s y on m o n o v a c a n c i e s in a - i r o n ( c o m p . S e c t . 1 ) . E v e n if all u n c e r t a i n ties are a s s u m e d to w o r k in the d i r e c t i o n of a low e n t h a l p y of m o n o v a c a n c y m i g r a t i o n , H~V c a n n o t be l o w e r than 1.0 eV. An e s t i m a t e b a s e d on the m o s t l i k e l y c o r r e c t i o n s is the v a l u e of E q . ( 1 0 ) M HIV An
upper
limit
: ( 1 •28±0 .2 5 ) e V of ~1.4
eV is o b t a i n e d
( II ) in the
subsequent
paper
(6).
This
means:
A) The r e c o v e r y s t a g e o b s e r v e d at a b o u t 220K a f t e r l o w - t e m p e r a t u r e cold-work (26) and i r r a d i a t i o n (27), for w h i c h the enthalpy of m i g r a t i o n has b e e n f o u n d to be (26) H Ill = ( 0 . 5 3 ± 0 . 0 3 ) e V (12) c a n n o t be due to m o n o v a c a n c y m i g r a t i o n . m i g r a t i o n of an i n t r i n s i c a t o m i c d e f e c t a n a l o g u e to S t a g e III in fcc m e t a l s ; we
(This s t a g e is c l e a r l y due to l o n g - r a n g e ( 2 6 , 2 7 ) and forms in m a n y r e s p e c t s the s h a l l d e n o t e it h e n c e f o r t h as S t a g e III.)
B) S i n c e it is u n l i k e l y that in bcc m e t a l s m u l t i p l e v a c a n c i e s are s u b s t a n t i a l l y more m o b i l e t h a n monovacancies, S t a g e Ill in a - i r o n must be due to the l o n g r a n g e m i g r a t i o n of an i n t e r s t i t i a l - t y p e d e f e c t . This is f u r t h e r s u p p o r t e d by the fact that the d e f e c t s a n n e a l i n g out in S t a g e III h a v e b e e n s h o w n to give rise to
808
V A C A N C Y FORMATION
IN IRON
Vol,
II, No,
9
relaxation effects (28-30)', but that in spite of the s e n s i t i v i t y of the magnetic measurements the c o r r e s p o n d i n g after-effects have never been found after quenching from high temperatures (28), C) The increase of positron trapping observed during Stage-IIl annealing (31,32) cannot be due to the formation of vacancy clusters. It shows u n a m b i g u o u s l y that clustering of s e l f - i n t e r s t i t i a l s may result in the formation of efficient positron traps in metals. This invalidates the conclusions on vacancy m i g r a t i o n drawn by Mantl and T r i f t s h g u s e r (33), Eldrup, Mogensen, and Evans (34), and by Hinode, T a n i g a w a and Doyama (35). The present experimental data do not permit to decide with certainty whether the i n t e r s t i t i a l - t y p e defects m i g r a t i n g in Stage III are m o n o i n t e r s t i t i a l s or small interstitial elustersas suggested by M i n i e r - C a s s a y r e (36) and Moser (28) (e.g. di-interstitials). The close analogy to the fcc metals," however, is in favour of the first possibility, in agreement with the view of Nihoul (37). Since there is strong evidence for long-range s e l f - i n t e r s t i t i a l migration in e-iron at about 130K, this would mean that in e-iron a t w o - i n t e r s t i t i a l model is applicable, too. Conclusions I ) Iron shows positron trapping at vacancies in thermal e q u i l i b r i u m both in the bcc and the fcc phase with d i s c o n t i n u i t i e s at the phase transitions, 2) The vacancy formatlon enthalples are HIV F • . F = (1.53+0.15)eV (bcc) and sH~Ta=e(.1"54-+0"15)eV p~ (fcc). The t r a p p i n g rate is higher in the bcc than in the fcc 3) Combination with s e l f - d i f f u s i o n data, taking into c o n s i d e r a t i o n the influence of the ferromagnetic o r d e r - - d i s o r d e r transition, gives a vacancy m i g r a t i o n enthalpy HMv = (1.28-+0.25)eV in e-iron. It follows that m o n o v a c a n c y migration occurs above room temperature (450K-600K). 4) The defect U n d e r g o i n g long-range migration at about 220K (which appears to give rise to the analogue of Stage III in fcc metals) is of i n t e r s t i t i a l type. 5) The temperature dependence of the line-shape parameter at low and intermediate temperatures may be ascribed to the m e t a s t a b l e s e l f - t r a p p i n g of positrons. Acknowledgements The authors appreciate helpful discussions with Dr.H.Mehrer. Financial support of the B u n d e s m i n i s t e r i u m fGr Forschung und Technologic is g r a t e f u l l y acknowledged. References I. 2. 3. h. 5. 6. 78. 9. 10. 11. ]2. 13. 14. 15.
W . G l a e s e r and H.Wever, phys.stat.sol. 35, 367 (1969). F. Ichikawa, K.yamakawa, and F.E.Fujita, Scripta Met. 6, 929 (1972). H.Wever and W.Seith, p h y s . s t a t . s o l . ( a ) 28, 187 (1975). Y.Ikeda, T.Gotoh, K.Abiko, and H.Kimura, Crystal Lattice Defects 5, 163 (1974). G.Hettich, H.Mehrer, and K.Maier, Scripta Met.) 795~ this issue. J.Diehl, U . M e r b o l d , and M.Weller, Seripta Met ~ 8 1 ~ this i$$~e, M.Mondino and A.Seeger, Scripta Met., 8]7, this ]$$~e~ D.Herlaeh and K.Maier, Appl.Phys. 11, 199 (1976). K.Maier, H.Metz, D.Herlach, and H.-E.Schaefer, to be p u b l i s h e d in the Proc. Int. Conf. on "Atomic Defects in Metals", Argonne, Iii., USA, 1976. H.Jehn in "Gase und K o h l e n s t o f f in Metallen", E.Fromm and E . G e b h a r d t , E d s . , Springer, Berlin-- Heidelberg-- NewYork, 1976, p.306. E.Hultgren, P.D.Desai, D.T.Hawkins, M.Gleiser, K.K.KelIy, D.D.Wagnlan, Selected Values of the T h e r m o d y n a m i c Properties of the Elements, ASM 1973. D.Herlachl H.Stoll, W.Trost, H.Metz, T.E.Jackman, K.Maier, H . - E . S c h a e f e r , and A.Seeger, Appl.Phys. 12, 59 (1977). l . K . M a e K e n z i e and P . C . L i e h t e n b e r g e r , Appl.Phys. 9, 331 (1976). K.Maier, H.Metz, D.Herlach, H.-E.Sehaefer, and A.Seeger, Phys.Rev.Letters,in pri~ T h e r m o p h y s i c a l Properties of Matter, Vol. 12, Y.S.Touloukian, R.B.Kirby, R.E.Taylor, and P.D.Desai, Eds., Plenum Press, New York,. 1975.
Vol,
]6. ]7. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29.
30. 31. 32.
33. 34. 35. 36. 37.
ii~ No,
9
VACANCY FORMATION
IN IRON
809
A.Seeger, Appl.Phys. 7, 85 (1975). A.Seeger, Appl.Phys. 7, 257 (1975). D.Segers, L . D o r i k e n s - V a n p r a e t , and M.Dorikens, Appl.Phys. 13, 5 I (]977), A . N . G o l a n d and T.M.HalI, Phys.Letters 45A, 397 (1973). W.Frank and A.Seeger, Appl.Phys. 3, 61 (1974). A.Seeger, Appl.Phys. 4, ]83 (1974). Z.S.Basinski, W . H u m e - R o t h e r y , and A.L.Sutton, Proc.Roy.Soc. London A 229, 459 (1955). D . M . G r a h a m and D.H.Tomlin, Phil.Mag. 8, ]581 (1963). L.Ruch, D.R.Sain, H.L.Yeh, and L.A.Girifalco, J.Phys.Chem.Sol. 37, 649 (1976). C.Zener, Imperfections in Nearly Perfect Crystals, W.Shockley, J.H.Hollomon, R.Maurer, and F°Seitz, Eds. , Wiley, New York, 1952, p.289. L.J.Cuddy, Acta Met. 16, 23 (~968). J.C.Leveque, T . A n a g n o s t o p o u l o s , H.Bilger, and P.Moser, phys.stat.sol. 31, K 47 (1969). P.Moser, Mem.Sci.Rev.Met. 63, I (1966). H.-E.Schaefer, D.Butteweg, and W°Dander, P r o c . l n t . C o n f . o n "Fundamental Aspects of Radiation Damage in Metals", Gatlinburg, Tenn., USA, M . T . R o b i n s o n and F.W.Young, Eds., 1975, p.463. V.Hivert, R.Pichon, H.Bilger, P.Bichon, J.Verdone, D.Dautreppe, and P.Moser, J.~hys.Chem.Sol. 31, 1843 (1970) M.Weller, J.Diehl, and W.Triftsh~user, Solid State Comm. ]7, 1223 (1975). H.-E.Schaefer, D.Herlach, H.Metz, and K.Maier, A r b e i t s t a g u n g "Nukleare F e s t k 6 r p e r f o r s c h u n g - Verbund 4002: K e r n p h y s i k a l i s c h e Methoden angewandt auf F e s t k 6 r p e r p r o b l e m e " , Ebermannstadt, 1976. S.Mantl and W . T r i f t s h & u s e r , P h y s . R e v . L e t t e r s 34, 1554 (1975). M.Elcrup, 0.E.Mogensen, and J.H.Evans, J.Phys. F 6, 499 (1976). K.Hinode, S.Tanigawa, and M.Doyama, rad.effects 32, 73 (1977). C.Minier-Cassayre, Th$se Universit$ de Grenoble, 1965. J.~ihoul in "Vacancies and Interstitials in Metals", A.Seeger, D.Schumacher, W.Schilling, and J.Diehl, Eds. , North Holland Publ. Comp., Amsterdam, 1970, p.839.
METALLURGICA
VACANCY
Vol. ii, pp. 8 0 3 - 8 0 9 , 1977 P r i n t e d in the U n i t e d S t a t e s
FORMATION
IN
ANNIHILATION
H.-E.Schaefer
(a),
IRON IN
K.Maier
A.Seeger
INVESTIGATED
THERMAL
(a),
(a,b),
BY
Press,
Inc.
POSITRON
EQUILIBRIUM
M.Weller
and
Pergamon
J.Diehl
(b),
D.Herlach
(a)
(b)
I n s t i t u t fur T h e o r e t i s c h e und Angewandte P h y s i k der U n i v e r s i t ~ t Stuttgart (a) and M a x - P l a n c k - l n s t i t u t ffir M e t a l l f o r s c h u n g (b), Stuttgart, Germany (Received
I.
June
24,
1977)
Introduction
In the s t u d y of d e f e c t s in bee m e t a l s iron o c c u p i e s a c e n t r a l p o s i t i o n . On the one h a n d ~ - F e is t h e b a s e m e t a l of a h u g e n u m b e r of t e c h n o l o g i c a l l y important a l l o y s . For m a n y a p p l i c a t i o n s a d e t a i l e d k n o w l e d g e of the p r o p e r t i e s of d e f e c t s in ~ - F e is h i g h l y d e s i r a b l e . On the o t h e r h a n d f e r r o m a g n e t i c techniques offer experimental possibilities for s t u d y i n g d e f e c t s in ~ - F e that are not a v a i l a b l e for any o t h e r b c c m e t a l . In s p i t e of this, v e r y l i t t l e has d e f i n i t e l y b e e n est a b l i s h e d on a t o m i c d e f e c t s in bec iron. In p a r t i c u l a r , the v i e w s on the m i g r a t i o n e n t h a l p y of m o n o v a c a n c i e s ' H MI V ' and on the i n t e r p r e t a t i o n of some of the r e c o v e r y s t a g e s o b s e r v e d in a - F e h a v e b e e n q u i t e c o n t r o v e r s i a l (I-4). T h i s is due p a r t l y to the e x t r e m e s e n s i t i v i t y of a - i r o n to the p r e s e n c e of i m p u r i t i e s , in p a r t i c u l a r FIAs ( f o r e i g n i n t e r s t i t i a l a t o m s ) , p a r t l y to the fact t h a t b e t w e e n T a y = I]84K and Ty6 = 1665K the fee y - p h a s e s e p a r a t e s the bcc ~- and ~ - p h a s e s of p u r e iron f r o m e a c h o t h e r , p r o v i d i n g additional difficulties for b o t h q u e n c h i n g and h i g h - t e m p e r a t u r e equilibrium experiments. A further complication, not as w i d e l y r e c o g n i z e d , is that r e c o v e r y e x p e r i m e n t s are u s u a l l y c a r r i e d out in the c o m p l e t e l y or a l m o s t c o m p l e t e l y o r d e r e d f e r r o m a g n e t i c state, whereas hightemperature experiments r e f e r to the p a r a m a g n e t i c or a l m o s t c o m p l e t e l y d i s o r d e r e d ferromagnetic states. The p r e s e n t p a p e r is p a r t of a c o n c e r t e d e f f o r t to o v e r c o m e the a b o v e mentioned difficulties and to o b t a i n r e l i a b l e d a t a on t h e a t o m i c d e f e c t s in airon. T h e p r e c e d i n g p a p e r (5) r e p o r t s on an a t t e m p t to o b t a i n r e l i a b l e s e l f d i f f u s i o n d a t a on ferromagnetic i r o n and in p a r t i c u l a r a c c u r a t e v a l u e s for the s u m H ~ v + H { v of the m o n o v a c a n c y migration and f o r m a t i o n e n t h a l p i e s . The present p a p e r e m p l o y s p o s i t r o n t r a p p i n g at v a c a n c i e s ( s t u d i e d by D o p p l e r b r o a d e n i n g of t~e 2y p o s i t r o n - a n n i h i l a t i o n line) to o b t a i n the m o n o v a c a n c y formation enthalpy HIV. This a l l o w s us to d e r i v e a r e l i a b l e v a l u e for the m i g r a t i o n e n t h a l p y H~V. as w e l l as an e s t i m a t e for the m o n o v a c a n c y diffusion coefficient, and to d i s c r i m i n a t e b e t w e e n the t w o m a j o r v i e w p o i n t s on the r e c o v e r y s t a g e a s s o c i a t e d w i t h monovacancy migration. It w i l l b e c o m e e v i d e n t that the r e c o v e r y s t a g e at a b o u t 2 2 0 K , constituting the a n a l o g u e of S t a g e III in fcc m e t a l s , c a n n o t be due to t h e long-range m i g r a t i o n of m o n o v a c a n c i e s . The s u b s e q u e n t p a p e r (6) a n a l y z e s the r e c o v e r y e x p e r i m e n t s pertaining to the m i g r a t i o n of v a c a n c i e s in a - i r o n in some d e t a i l . It c o m e s to the c o n c l u s i o n that t h e y are i n d e e d in g o o d a g r e e m e n t w i t h the r e s u l t s d e d u c e d f r o m the h i g h temperature experiments, p r o v i d e d the r o l e of FIAs is p r o p e r l y t a k e n into a c c o u n t . This c o n c l u s i o n is f u r t h e r s t r e n g t h e n e d in a f o u r t h p a p e r (7), in w h i c h it is s h o w n t h a t the r e s u l t s on the i n t e r a c t i o n b e t w e e n m o n o v a c a n c i e s and c a r b o n or n i t r o g e n in ~ - F e o b t a i n e d in the third p a p e r (6) l e a d s to a d e t a i l e d understanding of the Snoek-- K S s t e r ( c o l d - w o r k ) relaxation and of c a r b i d e and nitride formation following electron irradiation.
803
804
VACANCY FORMATION IN IRON
2. E x p e r i m e n t a l
Vol, ii, No, 9
Procedures
2.1 S p e c i m e n : A c y l i n d r i c a l s e a l e d - s o u r c e s p e c i m e n (8,9) was p r e p a r e d f r o m J o h n s o n - M a t t h e y h i g h - p u r i t y i r o n w i t h a t o t a l m e t a l l i c i m p u r i t y c o n t e n t of ~10 at ppm. B e c a u s e of the h i g h e v a p o r a t i o n rate of iron a b o v e 1550K [0.1 ~ m / s at 1730K ( 1 0 , 1 1 ) ] the w a l l of the s p e c i m e n tube was c h o s e n I m m t h i c k . A f t e r m a c h i n ing and e t c h i n g , the s p e c i m e n w a s a n n e a l e d in dry h y d r o g e n for 63h at I020K. This r e s u l t e d in a r e s i d u a l r e s i s t i v i t y r a t i o in a m a g n e t i c f i e l d of H = 500 Oe F ( 3 O O K / h K ) = 1 8 0 0 , a n d C and N c o n t e n t s of less t h a n I at p p m as d e t e r m i n e d by internal friction measurements. A f t e r i n s e r t i n g the 6 ~ C i 2 2 N a C I p o s i t r o n s o u r c e ( e v a p o r a t e d on a h i g h p u r i t y iron foil) the s p e c i m e n was e l e c t r o n - b e a m w e l d e d and f i n a l l y d e g a s s e d (2h, IIOOK, IO-~Pa) in o r d e r to r e m o v e r e s i d u a l h y d r o g e n . The r e s i s t i v i t y r a t i o of a d u m m y h e a t e d t o g e t h e r w i t h the s p e c i m e n d u r i n g the e n t i r e t e m p e r a t u r e prog r a m d e c r e a s e d f r o m F = 1270 b e f o r e to F = 640 a f t e r the h i g h - t e m p e r a t u r e treatment. This may i n d i c a t e that d u r i n g the h i g h - t e m p e r a t u r e m e a s u r e m e n t s the imp u r i t y c o n t e n t of the p o s i t r o n - a n n i h i l a t i o n s p e c i m e n i n c r e a s e d s l i g h t l y due to i m p u r i t i e s p i c k e d up f r o m the l o w - p r e s s u r e a t m o s p h e r e . 2.2 E x p e r i m e n t a l s e t - u p : B e t w e e n 400K and 1 7 5 0 K the m e a s u r e m e n t s w e r e p e r f o r m e d in a r e s i s t a n c e - h e a t e d h i g h - v a c u u m ( 1 0 - ~ P a ) f u r n a c e w i t h a Pt-- Pt 10%Rh t h e r m o c o u p l e for t e m p e r a t u r e m e a s u r e m e n t and c o n t r o l . The t e m p e r a t u r e g r a d i e n t a l o n g the s p e c i m e n was e s t i m a t e d to be less t h a n IOK n e a r the m e l t i n g point. The lowt e m p e r a t u r e c r y o s t a t , the y - s p e c t r o m e t e r for m e a s u r i n g the D o p p l e r b r o a d e n i n g of the p o s i t r o n - a n n i h i l a t i o n l i n e - s h a p e , and the d e t e r m i n a t i o n of the W p a r a m e t e r ( w h i c h is p a r t i c u l a r l y s e n s i t i v e to the p r o b a b i l i t y that t h e r m a l i z e d p o s i t r o n s a n n i h i l a t e w i t h e l e c t r o n s in the ion cores and w h i c h h e n c e is a g o o d i n d i c a t o r of p o s i t r o n t r a p p i n g at v a c a n c i e s ) w e r e d e s c r i b e d e a r l i e r (12). 3._ E x p e r i m e n t a l
Results
T h e t e m p e r a t u r e d e p e n d e n c e of the l i n e - s h a p e p a r a m e t e r W ( F i g . l ) , r e v e r s i b l e t h r o u g h o u t , shows the f o l l o w i n g c h a r a c t e r i s t i c f e a t u r e s :
W
#,
bcc// 0.24
/// ,,// /
/ f fcc ,/ /
(]26
.......'~
.
13
..." .... & A...A.~,.,b.~t,~Z" A~J~rA
.......
I
I
:*~-TI
Tc''
i
Fig.l:
was
Fe
0.23
o27,
which
i
'
s'oo
i
Tpa Tm
~g ,
5oo
I
i
'
rrKl
T e m p e r a t u r e d e p e n d e n c e of the p o s i t r o n - a n n i h i l a t i o n line-shape p a r a m e t e r W in Fe. The d o t t e d line b e t w e e n 4 . 2 K and IOOOK i n d i c a t e s the l i n e a r t h e r m a l e x p a n s i o n (15) n o r m a l i z e d to the e x p e r i m e n t a l data at 4.2K and IO00K. The full line g i v e s the fit of E q . ( 5 ) to the m e a s u r e d d a t a w i t h the v a l u e s of case (6) in Sect. h.2a.
Vol.
11,
i
No.
9
Between
ii
VACANCY FORMATION IN IRON
5K and 3 5 0 K
the
W parameter
A b o v e 350K the W p a r a m e t e r d e c r e a s e s an i n f l e c t i o n p o i n t at a b o u t 950K.
iii
with
independent.
increasing
temperature,
showing
If c h a r a c t e r i z e d by s t r a i g h t - l i n e a p p r o x i m a t i o n s (13,14) as i n d i c a t e d the thin l i n e s in F i g . l , the h i g h - t e m p e r a t u r e d e c r e a s e sets in at T 3 = I070K, s l i g h t l y a b o v e the C u r i e t e m p e r a t u r e T c = 10h3K.
iv
by
At the s t r u c t u r a l p h a s e t r a n s i t i o n s ~-y and y-6 the W p a r a m e t e r is d i s c o n t i n u o u s . It is h i g h e r in the fcc p h a s e than in the bcc p h a s e , i n d i c a t i n g that v a c a n c y t r a p p i n g is m o r e p r o n o u n c e d in the bcc phase.
v) The
W parameter
tends
to
saturate
4. A n a l y s i s 4.1
is t e m p e r a t u r e
805
Qualitative
near
the
of the
melting
point
T m = 1811K.
Data
analysis
The " M a c K e n z i e t e m p e r a t u r e " T~ = 1070K fits w e l l into the e m p i r i c a l T 3 - T m r e l a t i o n s h i p v a l i d for fcc and h c p S m e t a l s and for Ta (14). It s u g g e s t s that the m o n o v a c a n c y f o r m a t i o n e n t h a l p y H~V in ~ - F e lies b e t w e e n 1.3 eV and 1.6 eV. The t e m p e r a t u r e v a r i a t i o n b e l o w T 3 c a n n o t be e x p l a i n e d by t h e r m a l e x p a n s i o n . This may be seen from Fig.1 , w h e r e the d o t t e d line shows the t h e r m a l e x p a n s i o n (15) n o r m a l i z e d at 4K and at 1000K. The m e a s u r e d l i n e - s h a p e p a r a m e t e r e x h i b i t s the t e n d e n c y to f o r m the p l a t e a u b e l o w T 3 c h a r a c t e r i s t i c of m e t a s t a b l e selft r a p p i n g of p o s i t r o n s (16,17). A s i m i l a r t e m p e r a t u r e v a r i a t i o n was f o u n d in In (18) and Ta (14) and s u c c e s s f u l l y a n a l y z e d in terms of the s e l f - t r a p p i n g m o d e l . In the case of s - i r o n the n u m e r i c a l v a l u e s d e r i v e d from this m o d e l ( S e c t . 4 . 2 ) are s u b j e c t to u n c e r t a i n t i e s r e s u l t i n g f r o m the u n k n o w n e f f e c t s of m a g n e t i c o r d e r on p o s i t r o n a n n i h i l a t i o n . H o w e v e r , t h e s e e f f e c t s s h o u l d be v i r t u a l l y t e m p e r a t u r e i n d e p e n d e n t up to a few lOOK b e l o w T c and w i l l l e a v e u n a f f e c t e d the q u a l i t a t i v e f e a t u r e s w h i c h f a v o u r the s e l f - t r a p p i n g m o d e l over the t h e r m a l - e x p a n s i o n model. 4.2
Quantitative
analysis
~ ] _ ~ i ~ ~ In the s i m p l e s t m o d e l of m e t a s t a b l e t e m p e r a t u r e d e p e n d e n c e of the W p a r a m e t e r is g i v e n by Wf-fst[W f - (Tf/Tst) w(T)
Wst]
=
~ W](T)
I - fst
where above
= [1 + A -I
(]6,17)
(1)
,
T 3/2
e x p ( e o / k B T ) ] -I
lifetimes fractions
,
for a n n i h i l a t i o n of p o s i t r o n s in
(2>
k means Boltzmann's constant, 6 the e n e r g y of the s e l f - t r a p p e d o t~e " c o n d u c t i o n - b a n d m i n i m u m " of the p o s i t r o n s and m+k B 3/2 ~t m + k B 3/2 A~.
A -1 ~ VA(2~2)
the
[I - ( ~ f / T s t ) ]
w h e r e W~, W t' Tf, T t d e n o t e the W p a r a m e t e r s or the in the ~ r e e S o r s e l f - ~ r a p p e d s t a t e s _ r e s p e c t i v e l y . The the s e l f - t r a p p e d s t a t e m a y be e x p r e s s e d as fst
self-trapping
H ~. = VA ( ~ ) J
~
(] + ~. W~). a
state
(S)
a
w i t h the a t o m i c v o l u m e V A and the e f f e c t i v e mass of the p o s i t r o n s m+. In E q . ( 3 ) ~v~ ~ ~ - v. d e n o t e s thN d i f f e r e n c e in the f r e q u e n c i e s of the jth v i b r a t i o n a l m o ~ e s i~ the 8 p r e s e n c e of a s e l f - t r a p p e d (v ) a n d ~ a free (~) p o s i t r o n . The n u m e r i cal f a c t o r s a p p e a r i n g in E q . ( 3 ) are ( m + k ~ / 2 w ~ 2 ) 3/2 = 2 . 4 1 4 . 1 0 2 1 m ~ 3 K -3/2 (with m + = m o = f r e e e l e c t r o n m a s s ) , V A = 1 2 . 1 . 1 0 - ~ 0 m 3 (for ~-Fe). B e c a u s e of the c h a n g e in f e r r o m a g n e t i c o r d e r w i t h t e m p e r a t u r e it is d i f f i cult to o b t a i n all the p a r a m e t e r s a p p e a r i n g iD Egs (I-3) f r o m a fit to W ( T ) . We c h o o s e ~ h e r ~ f o r e (e) H ( v J / V j ) = I O ( A = 3 . 5 - I 0 O K 3 / 2 ) , or (8) ~ ( v ~ / v ~ ) = 3 0 ( A = 1 . 1 7 " I 0 6 K 3 / 2 ) , and o b t a i n w i t h T f / T s t = 0.9 the v a l u e s Wf = ~ . 2 ~ 6 and (~) eo=O.32eV, Wst = 0 . 2 5 1 6 , or (~) 6o=0.29eV, W s t = 0 . 2 4 5 3 .
806
VACANCY
~_~K~_~_~£~B£~± terms of the p o s i t r o n
eq clv
=
FORMATION
IN IRON
Vol,
The q u a n t i t a t i v e a n a l y s i s of the data t r a p p i n g by the e q u i l i b r i u m c o n c e n t r a t i o n
exp(S~Vlk~ exp(-H~v/k B T)
(S~v = e n t r o p y of f o r m a t i o n of m o n o v a c a n c i e s ) the bcc p h a s e the m i d d l e and in the fcc phase S" are not a c c e s s i b l e e x p e r i m e n t a l l y . The fit eq WI(T) + WIV GTf CIV W(T) = eq I + GTf CIV
ii,
No,
9
above T c in of m o n o v a c a n c i e s ,
(~)
,
is h a n d i c a p p e d by the fact that in the ends of the " h i g h - t e m p e r a t u r e of the h i g h - t e m p e r a t u r e data to (12) (5)
with t e m p e r a t u r e - i n d e p e n d e n t v a l u e s of S F , H F , Wst ( p o s i t r o n l i f e t i m e in the . . 1 Vv a c a n c y c o n c e n t r a t i o n ) , s e l f - t r a p p e d state), if(trapping rate per 1V unl~ and WI(T) from Eq.(1) gives for the vacancy formation enthalpy in the paramagnetic bcc phase HIVF = (I .53 +- 0 . 1 5 ) e V
(6)
and a s i m i l a r value for the fcc phase (eomp. T a b l e I). In Eq.(6) the error limits r e s u l t from the ( s t a t i s t i c a l etc.) s c a t t e r of the data. The p o s i t r o n a n n i h i l a t i o n p a r a m e t e r s f o l l o w i n g from the fit are g i v e n in T a b l e
I.
If t e m p e r a t u r e v a r i a t i o n s o~T ±I/2 are a l l o w e d for, the v a c a n c y f o r m a t i o n e n t h a l p i e s are c h a n g e d b Y ~ v = ± 0.08 eV w i t h o u t c h a n g i n g the q u a l i t y of the fit. A t e m p e r a t u r e d e p e n d e n c e of the s a t u r a t i o n p a r a m e t e r WIV of the form ~ W I v / d T ) W ~ v = 4 . 1 0 - 5 K -I as o b t a i n e d in (12) may be c o m p e n s a t e d by a slight v a r i a t i o n of the a b s o l u t e v a l u e s of Wst and WIV w i t h o u t c h a n g i n g H~V. We see that the above a n a l y s i s leads to m o n o v a c a n c y f o r m a t i o n e n t h a l p i e s H Fiv that are in r e a s o n a b l e a g r e e m e n t w i t h the e x p e c t a t i o n s from the e m p i r i c a l "'H FIV -T 3 and. H FIV -T m r e l a t i o n s h i p s (Fig '2 of ref • 14) • We take this as e v i d e n c e that it was i n d e e d a good a p p r o x i m a t i o n to n e g l e c t p o s i t r o n d e t r a p p i n g from v a c a n c i e s (19-21) and d i v a c a n c y c o n t r i b u t i o n s (21) in the above a n a l y s i s . 5. D i s c u s s i o n 5.1
Positron
mobility
The e f f e c t s of v a c a n c i e s in t h e r m a l e q u i l i b r i u m on p o s i t r o n t r a p p i n g are s t r o n g e r in bcc iron than in fcc iron (Fig.l). The p o s i t r o n l i f e t i m e s are exp e c t e d to vary only little w i t h c r y s t a l s t r u c t u r e (see b e l o w ) , h e n c e the d i f f e r e n c e b e t w e e n the two c r y s t a l s t r u c t u r e s in T a b l e I must be m a i n l y due to a l a r g e r value of G e x p ( S ~~v)/_k in the bcc s t r u c t u r e . The e n t r o p y of m o n o v a c a n e y f o r m a t i o n may w e l l be l a r g e r in the bcc s t r u c t u r e than in the fcc s t r u c t u r e , but it is u n l i k e l y that S ~ v ( b c c ) - S ~ v ( f C c ) e x c e e d s k B. It f o l l o w s that ~ ( b c c ) / ~ ( f c c ) ~3, i n d i c a t i n g that e i t h e r the p o s i t r o n m o b i l i t y or the rate of c a p t u r e of p o s i t r o n s by v a c a n c i e s is h i g h e r in bcc iron than in fcc iron. (For the r e l a t i o n ship of m o b i l i t y and c a p t u r e to p o s i t r o n t r a p p i n g by v a c a n c i e s see ref. 17.) The h i g h e r v a l u e s Wst and WIV (which are e q u i v a l e n t to s m a l l e r v a l u e s of the corres p o n d i n g p o s i t r o n l i f e t i m e s ) in the bcc than in the fcc phase may be r e l a t e d to the i n c r e a s e of the i n t e r a t o m i c d i s t a n c e s going from the bcc to the fcc phase (22),
5.2 E q u i l i b r i u m
vacancy
concentrations
If we a s s u m e S~V = (1.O ± O . 5 ) k = we o b t a i n from E q s . ( 4 , 6 ) for the e q u ~ i b r i u m v a c a n c y c o n c e n t r a t i o n at T a. = 1184~: C ~ ( 1 1 8 h K ) = 7.7.10-7±°.9. V a c a n c y c o n c e n v t r a t i o n s of the o r d e r of m a g¥ n l•t u d ~~ 10- 6 cannot be u n a m b l g o u s l y d e t e c t e d in quenching ~xperiments (3) u n l e s s the c o n c e n t r a t i o n of FIAs is lower than 0.2 at ppm (6). HiV = 1.53 eV, S[V = k B plus an e s t i m a t e d 20% c o n t r i b u t i o n of d i v a c a n cies g~Kes us for the t o t a l v a c a n c y c o n c e n t r a t i o n at the m e l t i n g point c~q(T m) = 1.8"10 >.3 M i g r a t i o n
enthalpy
and
diffusion
coefficient
The d i f f u s i o n c o e f f i c i e n t of m o n o v a c a n c i e s , self-diffusion c o e f f i c i e n t , D T, a c c o r d i n g to
of m o n o v a c a n c i e s DIV,
is r e l a t e d
in ~ - i r o n to the
tracer
VOI.
II,
No.
9
VACANCY
D T = fIV In (7) it has b e e n fIV = 0 . 7 2 d e n o t e s d i f f u s i o n d a t a for DIV
DIV
FORMATION
IN I R O N
exp( S iF v / k B) exp( _ H [ v / k B T
807
(7)
)
a s s u m e d that o n l y m o n o v a c a n c i e s c o n t r i b u t e to s e l f - d i f f u s i o n ; the m o n o v a c a n c y c o r r e l a t i o n f a c t o r . I n s e r t i o n of the selfp a r a m a g n e t i c a - i r o n (5) gives us for T>T c
= 0.5
exp(-O.92e~kBT)Cm
2 -I s
(8)
The c o m p a r i s o n w i t h r e c o v e r y m e a s u r e m e n t s r e q u i r e s an e s t i m a t e of the e f f e c t of f e r r o m a g n e t i c o r d e r on H~V, s i n c e the p o s i t r o n a n n i h i l a t i o n t e c h n i q u e is not c a p a b l e of g i v i n g i n f o r m a t i o n on v a c a n c y f o r m a t i o n b e l o w T c. A p p l i c a t i o n of the t h e o r y of Ruch et al. (24) [ c o m p . ( 5 ) ] g i v e s us as the m o n o v a c a n c y f o r m a t i o n e n t h a l p y at a b o u t 570K F = (I • 60 ± 0 . 15) eV H1V
(9)
If the e x t r a p o l a t i o n of the s e l f - d i f f u s i o n c o e f f i c i e n t to low t e m p e r a t u r e s is p e r f o r m e d by means of Zene~'s t h e o r y (5,25) one finds~ with the help of Fig, 3 of ref. (5) 2 -I DIV(573K) = 0.5 exp ( - 1 . 2 8 e V / k B T ) C m s (10) This
estimate
makes
use
of the
assumption
TABLE
S[V
=
kB
1
E ~ t h a l p i e s of M o n o v a c a n c y F o r m a t i o n H~V, M i g r a t i o n as w e l l as P o s i t r o n A n n i h i l a t i o n P a r a m e t e r s ( b a s e d M H]v(eV)
H[v(eV) bcc
paramag.
fcc
QSD(
H I, VM and S e l f - D i f f u s i o n Q S D on Case ~ in Sect. 4.2a).
F eV) ~ T f e x p ( g i V / k B)
Wf
WIV
Wst
1.53±0.15
0.92
2.45
(5)
106
-
0.2453
0.226
1.54±0.15
1.26
2.80(23)
105
-
0.2441
0.220
1.60±0.15
1.28
2.88
-
0.266
bcc
ferromag.
5.L
Consequences
for
the
(5)
interpretation
of a n n e a l ± n 5
sta~es
B e c a u s e of the u n c e r t a i n t i e s in the e x t r a p o l a t i o n of the h i g h - t e m p e r a t u r e d a t a to l o w e r t e m p e r a t u r e s it is d i f f i c u l t to give d e f i n i t i v e and m e a n i n g f u l e r r o r l i m i t s to E q . ( 1 0 ) . N e v e r t h e l e s s it is p o s s i b l e to d e c i d e u n a m b i g o u s l y the m a i n c o n t r o v e r s y on m o n o v a c a n c i e s in a - i r o n ( c o m p . S e c t . 1 ) . E v e n if all u n c e r t a i n ties are a s s u m e d to w o r k in the d i r e c t i o n of a low e n t h a l p y of m o n o v a c a n c y m i g r a t i o n , H~V c a n n o t be l o w e r than 1.0 eV. An e s t i m a t e b a s e d on the m o s t l i k e l y c o r r e c t i o n s is the v a l u e of E q . ( 1 0 ) M HIV An
upper
limit
: ( 1 •28±0 .2 5 ) e V of ~1.4
eV is o b t a i n e d
( II ) in the
subsequent
paper
(6).
This
means:
A) The r e c o v e r y s t a g e o b s e r v e d at a b o u t 220K a f t e r l o w - t e m p e r a t u r e cold-work (26) and i r r a d i a t i o n (27), for w h i c h the enthalpy of m i g r a t i o n has b e e n f o u n d to be (26) H Ill = ( 0 . 5 3 ± 0 . 0 3 ) e V (12) c a n n o t be due to m o n o v a c a n c y m i g r a t i o n . m i g r a t i o n of an i n t r i n s i c a t o m i c d e f e c t a n a l o g u e to S t a g e III in fcc m e t a l s ; we
(This s t a g e is c l e a r l y due to l o n g - r a n g e ( 2 6 , 2 7 ) and forms in m a n y r e s p e c t s the s h a l l d e n o t e it h e n c e f o r t h as S t a g e III.)
B) S i n c e it is u n l i k e l y that in bcc m e t a l s m u l t i p l e v a c a n c i e s are s u b s t a n t i a l l y more m o b i l e t h a n monovacancies, S t a g e Ill in a - i r o n must be due to the l o n g r a n g e m i g r a t i o n of an i n t e r s t i t i a l - t y p e d e f e c t . This is f u r t h e r s u p p o r t e d by the fact that the d e f e c t s a n n e a l i n g out in S t a g e III h a v e b e e n s h o w n to give rise to
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relaxation effects (28-30)', but that in spite of the s e n s i t i v i t y of the magnetic measurements the c o r r e s p o n d i n g after-effects have never been found after quenching from high temperatures (28), C) The increase of positron trapping observed during Stage-IIl annealing (31,32) cannot be due to the formation of vacancy clusters. It shows u n a m b i g u o u s l y that clustering of s e l f - i n t e r s t i t i a l s may result in the formation of efficient positron traps in metals. This invalidates the conclusions on vacancy m i g r a t i o n drawn by Mantl and T r i f t s h g u s e r (33), Eldrup, Mogensen, and Evans (34), and by Hinode, T a n i g a w a and Doyama (35). The present experimental data do not permit to decide with certainty whether the i n t e r s t i t i a l - t y p e defects m i g r a t i n g in Stage III are m o n o i n t e r s t i t i a l s or small interstitial elustersas suggested by M i n i e r - C a s s a y r e (36) and Moser (28) (e.g. di-interstitials). The close analogy to the fcc metals," however, is in favour of the first possibility, in agreement with the view of Nihoul (37). Since there is strong evidence for long-range s e l f - i n t e r s t i t i a l migration in e-iron at about 130K, this would mean that in e-iron a t w o - i n t e r s t i t i a l model is applicable, too. Conclusions I ) Iron shows positron trapping at vacancies in thermal e q u i l i b r i u m both in the bcc and the fcc phase with d i s c o n t i n u i t i e s at the phase transitions, 2) The vacancy formatlon enthalples are HIV F • . F = (1.53+0.15)eV (bcc) and sH~Ta=e(.1"54-+0"15)eV p~ (fcc). The t r a p p i n g rate is higher in the bcc than in the fcc 3) Combination with s e l f - d i f f u s i o n data, taking into c o n s i d e r a t i o n the influence of the ferromagnetic o r d e r - - d i s o r d e r transition, gives a vacancy m i g r a t i o n enthalpy HMv = (1.28-+0.25)eV in e-iron. It follows that m o n o v a c a n c y migration occurs above room temperature (450K-600K). 4) The defect U n d e r g o i n g long-range migration at about 220K (which appears to give rise to the analogue of Stage III in fcc metals) is of i n t e r s t i t i a l type. 5) The temperature dependence of the line-shape parameter at low and intermediate temperatures may be ascribed to the m e t a s t a b l e s e l f - t r a p p i n g of positrons. Acknowledgements The authors appreciate helpful discussions with Dr.H.Mehrer. Financial support of the B u n d e s m i n i s t e r i u m fGr Forschung und Technologic is g r a t e f u l l y acknowledged. References I. 2. 3. h. 5. 6. 78. 9. 10. 11. ]2. 13. 14. 15.
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