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Heat of mixing in liquid alloys
Volume 23, number 9
PHYSICS
in the magnesium spectrum, the lines at masses 25 and 26 contain a 10°/o contribution of the isotopes 25Mg and 26Mg respectively. The MgHmolecular ion should be produced by a method slmilar to Bell- and the same propertles should hold for this ion. Besides the molecular ion contribution in both spectra, peaks appear at the mass of the element, so that these may be interpreted as the negative ions of the element itself. One may assume that the Be- ion exists m the predicted 2S state [3]. Since, in the case of magnesium, the calculation, even for the 2S state, gave negative electron affinity, the Mg- ion may be produced in a metastable state. In this case the lifetime must be much longer than that of He-, which exists m a metastable ls2s2p 4p state wlth a lifetlme of about 10-5 sec [10]. The He- ion could not be extracted from the ion source under investigation. The maximum currents obtained over more
Table 2
Current (nAp
Be-
Bell-
24Mg-
24MgH- + 25Mg-
30
200
300
700
LETTERS
28 November 1966
t h a n one h o u r f o r t h e a b o v e m e n t i o n e d i o n s a r e g i v e n in t a b l e 2. T h e s e c u r r e n t s a r e l a r g e enough for the ions to be a c c e l e r a t e d in a t a n d e m V a n de G r a a f f a c c e l e r a t o r .
We w o u l d like t o t h a n k P r o f . D r . O. H a x e l f o r h i s c o n t i n u o u s i n t e r e s t in o u r w o r k .
References 1. N. S Bu~elnlkowa, F o r t s c h r Physik 8 (1960) 626. 2 L.M. Brandscomb, Atomic and molecular p r o c e s ses, ed D R Bates (Academm P r e s s , New York, 1962) p 136. 3. B L. Molseiwltsch, Advances m atomic and molecular physics, eds D R Bates and I E s t e r m a n (Academm P r e s s , New York, 1965) p 63. 4. G. Glockler, Phys. Rev. 46 (1934) 111 5 D R. Bates, P r o c Roy. Irish Acad. Scl 51 (1947) 151. 6. H O Pritchard, Chem Rev 52 (1953) 529 7. B. Edl~n, J Chem. Phys 33 (1960) 98 8. H Baumann, K. Bethge and E. Heinicke, Nucl. Instr. Methods (1966), in p r e s s 9 E.Hemieke, K Bethge and H. Baumann, Nucl Instr. Methods, m preparation. 10 D.R Sweetman, P r o c Phys Soc (London) 76 (1960) 998.
HEAT OF MIXING IN LIQUID ALLOYS Shigeru TAMAKIand Ichiro SHIOTA Research Institute for Iron, Steel and Other Metals, Tohoku University, Sendai, Japan Received 7 November 1966
A calculation of the heat of mixing in liquid alkali alloys is p r e s e n t e d The two-body correlation function ls used for the calculation The theoretical value ls satisfactorily comparable to the experimental one in the Na-K s y s t e m
In t h e s t u d y of l i q u i d m e t a l s , t h e t w o - b o d y c o r r e l a t i o n f u n c t i o n a(q) i s a n i m p o r t a n t f a c t o r f o r t h e i n t e r p r e t a t i o n of p h y s i c a l p r o p e r t i e s [ 1 - 3 ] . J o h n s o n e t ~1, h a v e f o u n d t h e e f f e c t i v e i o n - i o n i n t e r a c t i o n p o t e n t i a l of l i q u i d m e t a l s f r o m the B o r n - G r e e n and P e r c u s - Y e v i c k equation in which the m e a s u r e d r a d i a l p a i r d i s t r i b u t i o n f u n c t i o n g(r) w a s i n s e r t e d [4]. In t h i s p a p e r , a c a l c u l a t i o n of t h e h e a t of
mixing in liquid Na-K alloys is given using a (q). Christman and Huntington have calculated the heat of mixing in liquid Na-K alloys [5]. The advantage of their treatment is the absence of the structure factor in the diagonal matrix element of the APW Harniltonian. On the contrary, our method for the calculation is rather phenomenological, but is most practical. We proceed to evaluate the total ener543
Volume 23. n u m b e r 9
PHYSICS
LETTERS
gy of t h e s y s t e m of i o n s a n d e l e c t r o n s of a l i q u i d a l l o y . A f t e r H a r r i s o n , t h e t o t a l e n e r g y of a s y s t e m d i v i d e d b y t h e n u m b e r of i o n s i s w r i t t e n a s
[6]:
28 November 1966
16~r~ ×
Eto t =E d+Eel+Ees,
Eel =
[2eo/8=3]
f
%3
E(k) dR,
+ ~ . 2.21
"s(rNa-K)2"
"s(rNa}2" -
-2
k
w h e r e VA- B m e a n s t h e e f f e c t i v e i o n - i o n i n t e r a c tion between A and B ion in A-B alloys.
-- ~-
h2k 2
E(k) = - ~ - - + +~
a(q)
qi0
- t
~
%
~
~0
(~2/2~)g(~ [k+q]=)
(l)
'
O f%
where W is the pseudo-potential of an ion. The first term on the right-hand side of eq. (1) is altered by the consideration of electron-electron interactions. By the Hartree-Fock approximation and the treatment of Nozi~res-Pines, the first term is attributed to [7]:
/
"b %
w
E r e = 2.21/r~s - 0 . 9 1 6 / r s + E c , w h e r e E c ~ - 0 . 1 1 5 + 0.031 l n r s. A c c o r d i n g t o Harrison, the third term on the right-hand side of eq. (1) ( r e f e r r e d t o a s E b s ) i s e q u a l t o :
.z Ebs =-
~ q/0
a(q)"
~°q2 -87re -~
](k+qlW°lk)t 2
6(q)- 1
c~-(~-- '
.4 .~ A t ~ c {r~et,~
.g
N•
Fig 1. Heat of mlxmg.
(2) w h e r e W ° i s e q u a l t o E(q). W. T h e s u m of E d a n d Ebs is equal to the effective ion-ion interaction which is obtained from the self-consistent field m e t h o d [8]. In l i q u i d N a - K a l l o y s , t h e h e a t of m i x i n g AENa-K is expressed by:
w h e r e E b i n d m e a n s t h e b i n d i n g e n e r g y of m e t a l or alloy, which is determined by Eto t and the i o n i z a t i o n e n e r g y . If t h e c o n t r i b u t i o n s of A E f r o m t h e p a r t s of 1/rs, lnrs, E e s a r e s e c o n d o r d e r , then the main contributions are from E d + Ebs ,
i/~s 2 and
a(q) in
544
For the pseudo-potential in the effective ioni o n i n t e r a c t i o n p o t e n t i a l , we c a n u s e t h e m o d e l p o t e n t i a l g i v e n b y H a r r i s o n [9]. T h e c a l c u l a t e d c u r v e f o r N a - K i s s h o w n i n fig. 1. T h e t h e o r e t i cal curve is satisfactorily comparable to the exp e r i m e n t a l one w h i c h w a s o b s e r v e d b y Y o k o k a w a a n d K l e p p a [10]. 1 J . M Zlman, Phil. Mag. 6 (1961) 1013. 2 C. Bradley, T. Faber, E. G, Wilson and J M Ziman, Phzl. Mag 7 (1962) 865. 3. M. Watabe, M. Tanaka, H. Endo and B. K. Jones, Phil. Mag. 12 (1965) 347. 4. M.D. Johnson, P. Hutchinson and N H. March, Proc Roy Soc 282A (1964) 283. 5. J R C h r i s t m a n and H B. Huntington, Phys. Rev. 139A (1965) 83. 6 W.A. Harrison, Pseudo-potential m the theory of m e t a l s (W A. Benlamm, Inc , New York, 1966). 7. P. NozlSres and D. Pines, Phys. Rev 111 (1958)442 8 S. Tamaki and I Shiota, to be published 9. W A Harrison, Phys Rev 131 (1963) 2433. 10. T Yokokawa and O J Kleppa, private c o m m u n i c a tion to Prof. H. Endo
PHYSICS
in the magnesium spectrum, the lines at masses 25 and 26 contain a 10°/o contribution of the isotopes 25Mg and 26Mg respectively. The MgHmolecular ion should be produced by a method slmilar to Bell- and the same propertles should hold for this ion. Besides the molecular ion contribution in both spectra, peaks appear at the mass of the element, so that these may be interpreted as the negative ions of the element itself. One may assume that the Be- ion exists m the predicted 2S state [3]. Since, in the case of magnesium, the calculation, even for the 2S state, gave negative electron affinity, the Mg- ion may be produced in a metastable state. In this case the lifetime must be much longer than that of He-, which exists m a metastable ls2s2p 4p state wlth a lifetlme of about 10-5 sec [10]. The He- ion could not be extracted from the ion source under investigation. The maximum currents obtained over more
Table 2
Current (nAp
Be-
Bell-
24Mg-
24MgH- + 25Mg-
30
200
300
700
LETTERS
28 November 1966
t h a n one h o u r f o r t h e a b o v e m e n t i o n e d i o n s a r e g i v e n in t a b l e 2. T h e s e c u r r e n t s a r e l a r g e enough for the ions to be a c c e l e r a t e d in a t a n d e m V a n de G r a a f f a c c e l e r a t o r .
We w o u l d like t o t h a n k P r o f . D r . O. H a x e l f o r h i s c o n t i n u o u s i n t e r e s t in o u r w o r k .
References 1. N. S Bu~elnlkowa, F o r t s c h r Physik 8 (1960) 626. 2 L.M. Brandscomb, Atomic and molecular p r o c e s ses, ed D R Bates (Academm P r e s s , New York, 1962) p 136. 3. B L. Molseiwltsch, Advances m atomic and molecular physics, eds D R Bates and I E s t e r m a n (Academm P r e s s , New York, 1965) p 63. 4. G. Glockler, Phys. Rev. 46 (1934) 111 5 D R. Bates, P r o c Roy. Irish Acad. Scl 51 (1947) 151. 6. H O Pritchard, Chem Rev 52 (1953) 529 7. B. Edl~n, J Chem. Phys 33 (1960) 98 8. H Baumann, K. Bethge and E. Heinicke, Nucl. Instr. Methods (1966), in p r e s s 9 E.Hemieke, K Bethge and H. Baumann, Nucl Instr. Methods, m preparation. 10 D.R Sweetman, P r o c Phys Soc (London) 76 (1960) 998.
HEAT OF MIXING IN LIQUID ALLOYS Shigeru TAMAKIand Ichiro SHIOTA Research Institute for Iron, Steel and Other Metals, Tohoku University, Sendai, Japan Received 7 November 1966
A calculation of the heat of mixing in liquid alkali alloys is p r e s e n t e d The two-body correlation function ls used for the calculation The theoretical value ls satisfactorily comparable to the experimental one in the Na-K s y s t e m
In t h e s t u d y of l i q u i d m e t a l s , t h e t w o - b o d y c o r r e l a t i o n f u n c t i o n a(q) i s a n i m p o r t a n t f a c t o r f o r t h e i n t e r p r e t a t i o n of p h y s i c a l p r o p e r t i e s [ 1 - 3 ] . J o h n s o n e t ~1, h a v e f o u n d t h e e f f e c t i v e i o n - i o n i n t e r a c t i o n p o t e n t i a l of l i q u i d m e t a l s f r o m the B o r n - G r e e n and P e r c u s - Y e v i c k equation in which the m e a s u r e d r a d i a l p a i r d i s t r i b u t i o n f u n c t i o n g(r) w a s i n s e r t e d [4]. In t h i s p a p e r , a c a l c u l a t i o n of t h e h e a t of
mixing in liquid Na-K alloys is given using a (q). Christman and Huntington have calculated the heat of mixing in liquid Na-K alloys [5]. The advantage of their treatment is the absence of the structure factor in the diagonal matrix element of the APW Harniltonian. On the contrary, our method for the calculation is rather phenomenological, but is most practical. We proceed to evaluate the total ener543
Volume 23. n u m b e r 9
PHYSICS
LETTERS
gy of t h e s y s t e m of i o n s a n d e l e c t r o n s of a l i q u i d a l l o y . A f t e r H a r r i s o n , t h e t o t a l e n e r g y of a s y s t e m d i v i d e d b y t h e n u m b e r of i o n s i s w r i t t e n a s
[6]:
28 November 1966
16~r~ ×
Eto t =E d+Eel+Ees,
Eel =
[2eo/8=3]
f
%3
E(k) dR,
+ ~ . 2.21
"s(rNa-K)2"
"s(rNa}2" -
-2
k
w h e r e VA- B m e a n s t h e e f f e c t i v e i o n - i o n i n t e r a c tion between A and B ion in A-B alloys.
-- ~-
h2k 2
E(k) = - ~ - - +
a(q)
qi0
- t
~
%
~
~0
(~2/2~)g(~ [k+q]=)
(l)
'
O f%
where W is the pseudo-potential of an ion. The first term on the right-hand side of eq. (1) is altered by the consideration of electron-electron interactions. By the Hartree-Fock approximation and the treatment of Nozi~res-Pines, the first term is attributed to [7]:
/
"b %
w
E r e = 2.21/r~s - 0 . 9 1 6 / r s + E c , w h e r e E c ~ - 0 . 1 1 5 + 0.031 l n r s. A c c o r d i n g t o Harrison, the third term on the right-hand side of eq. (1) ( r e f e r r e d t o a s E b s ) i s e q u a l t o :
.z Ebs =-
~ q/0
a(q)"
~°q2 -87re -~
](k+qlW°lk)t 2
6(q)- 1
c~-(~-- '
.4 .~ A t ~ c {r~et,~
.g
N•
Fig 1. Heat of mlxmg.
(2) w h e r e W ° i s e q u a l t o E(q). W. T h e s u m of E d a n d Ebs is equal to the effective ion-ion interaction which is obtained from the self-consistent field m e t h o d [8]. In l i q u i d N a - K a l l o y s , t h e h e a t of m i x i n g AENa-K is expressed by:
w h e r e E b i n d m e a n s t h e b i n d i n g e n e r g y of m e t a l or alloy, which is determined by Eto t and the i o n i z a t i o n e n e r g y . If t h e c o n t r i b u t i o n s of A E f r o m t h e p a r t s of 1/rs, lnrs, E e s a r e s e c o n d o r d e r , then the main contributions are from E d + Ebs ,
i/~s 2 and
a(q) in
544
For the pseudo-potential in the effective ioni o n i n t e r a c t i o n p o t e n t i a l , we c a n u s e t h e m o d e l p o t e n t i a l g i v e n b y H a r r i s o n [9]. T h e c a l c u l a t e d c u r v e f o r N a - K i s s h o w n i n fig. 1. T h e t h e o r e t i cal curve is satisfactorily comparable to the exp e r i m e n t a l one w h i c h w a s o b s e r v e d b y Y o k o k a w a a n d K l e p p a [10]. 1 J . M Zlman, Phil. Mag. 6 (1961) 1013. 2 C. Bradley, T. Faber, E. G, Wilson and J M Ziman, Phzl. Mag 7 (1962) 865. 3. M. Watabe, M. Tanaka, H. Endo and B. K. Jones, Phil. Mag. 12 (1965) 347. 4. M.D. Johnson, P. Hutchinson and N H. March, Proc Roy Soc 282A (1964) 283. 5. J R C h r i s t m a n and H B. Huntington, Phys. Rev. 139A (1965) 83. 6 W.A. Harrison, Pseudo-potential m the theory of m e t a l s (W A. Benlamm, Inc , New York, 1966). 7. P. NozlSres and D. Pines, Phys. Rev 111 (1958)442 8 S. Tamaki and I Shiota, to be published 9. W A Harrison, Phys Rev 131 (1963) 2433. 10. T Yokokawa and O J Kleppa, private c o m m u n i c a tion to Prof. H. Endo