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r spin correlation
Volume25A, number 8
PHYSICS L E T T E R S
23 October 1967
3. A. E. Hughes and W.A. Runciman, Proc. Phys. Soc. 86 (1965) 615. 4. P.Duval, J.Gareyte, Y. Merle d'Aubigne, Physics Letters 22 (1966) 67. 5. A.E. Hughes, Oxford Thesis 1966. 6. F. Lanzl, Phys. Letters 23A (1967) $10.
References 1. R.H.Silsbee, Phys. Rev. 138 (1965) 180. 2. F. Lanzl and W. vonder Osten, Phys. Letters 15 (1965) 208.
CRITICAL MAGNETIC SCATTERING OF NEUTRONS A N D T H E i s i n K 2 r t //r S P I N C O R R E L A T I O N J. KOCI~SKI Warsaw Technical University, Institute of Physics, Warsaw, Poland and B. MRYGOI~ Institute of Physics of the Polish Academy of Sciences, Warsaw, Poland Received 21 September 1967
Experimental data on critical magnetic scattering of neutrons are explained in terms of the ] sinK2r Vr spin correlation, and the possibility of detection of the predicted diffraction minima is pointed out.
The e x p e r i m e n t a l l y found dependence of the i n v e r s e i n t e n s i t y of s c a t t e r e d n e u t r o n s on the s q u a r e of the s c a t t e r i n g angle in i r o n and nickel will be explained in t e r m s of the s c a t t e r i n g c r o s s - s e c t i o n , calculated with spin c o r r e l a t i o n : [1-3] (SoZ(0) Sz(0))To = ] s i n K 2 r I//492r
(1)
which has been a l r e a d y applied by Stump and M a i e r in the i n t e r p r e t a t i o n of the detected angul a r dependence of the m a x i m u m of c r i t i c a l s c a t t e r i n g in nickel [4]. The c r o s s - s e c t i o n d e r i v e d with the c o r r e l a tion (1) [c.f. 2 eq. (26)], has been n u m e r i c a l l y calculated for i r o n and nickel, on the a s s u m p t i o n of a m o n o c h r o m a t i c i n c o m i n g wave ko equal to 4.75 ~ or 4.28 ~,, to allow c o m p a r i s o n with exp e r i m e n t a l data [5-7]. The v a l u e s of the other r e l e v a n t p a r a m e t e r s a r e the l a t t i c e constant a = 2 . 8 6 A , T o = I043°K, K2 = 5.01/a, A = = 1.99 × 10 -3 c m 2 / s e c for i r o n , and a = 3 . 5 2 .~, T c = 631°K, K2 = 6.37/a, A = 0.91 × 10 -3 c m 2 / s e c for nickel, as d e t e r m i n e d p r e v i o u s l y [1-3]. Typical r e s u l t s for i r o n a r e r e p r e s e n t e d in fig. 1. In drawing the t h e o r e t i c a l c u r v e s we have a s s u m e d that the c a l c u l a t e d v a l u e s for the i n v e r s e i n t e n s i t y which c o r r e s p o n d to the s q u a r e d angles 600
02 = 9 × 10 -4 and 02 = 16 × 10 -4 coincide with those on the r e c t i l i n e a r p a r t of the line drawn through the e x p e r i m e n t a l points. F o r a s a m p l e t e m p e r a t u r e of about 10 ° above T c or higher, a change AT = ± 2 ° in the fluctuation t e m p e r a t u r e i n f l u e n c e s but little the shape of an i n t e n s i t y curve. It follows that in i r o n the fluctuation t e m p e r a t u r e is equal to about T c - 6°, for the s a m p l e t e m p e r a t u r e in the vicinity of the c r i t i c a l point, and that it may slightly i n c r e a s e with growing s a m p l e t e m p e r a t u r e . Better a g r e e m e n t with the data of Passel1 then those of J a c r o t is connected with a b r o a d e r i n c o m i n g s p e c t r u m in the e x p e r i m e n t of the l a t t e r , unaccounted for in the calculation. In iron, the range of c o r r e l a t i o n R, d e t e r m i n e d f r o m eq. (17) of ref. 2, with T = = T c - 6 ° is equal to 30.4 A at the c r i t i c a l point, and to 7.1 A at T c + 50 °. In nickel, the single r e p o r t e d i n t e n s i t y c u r v e at the Curie point [6], is r e p r o d u c e d with T = T c - 4.5 ° approximately. The range of correlation at T c = 631°K is equal to 25.8 A. The c r o s s - s e c t i o n g e n e r a l l y d e c r e a s e s with i n c r e a s i n g s c a t t e r i n g angle at constant s a m p l e t e m p e r a t u r e , however, c o n t r a r y to the Van Hove theory [8], the d e c r e a s e is not monotonic and diffraction m i n i m a appear at the angles
Volume 25A, n u m b e r 8
PHYSICS
30
T
23 October 1967
W i l k i n s o n a n d S h u l l [9] h a v e s t o p p e d t h e i r measurements for a sample temperature T o = = T c + 20 ° n e a r t h e f i r s t m i n i m u m . J a c r o t e t al. [5] a n d P a s s e l l e t al. [7] h a v e n o t m e a s u r e d a t large enough angles for the long incoming waves, w h i c h t h e y h a v e u s e d . S p o o n e r a n d A v e r b a c h [10] h a v e b e e n e x p l o r i n g t h e r e g i o n of t h e f i r s t s i d e maximum, but the applied collimation has been insufficient for detecting more than a slowly changing small intensity. The first side maximum has been already detected by Mi/nster and Sagel for the X-ray critical scattering in the AI-Zn binary alloy [11, 12] [c.f. 12 fig. 2 p. 233] b u t it h a s n o t b e e n explained.
,~Ox~" 4 0
/
I....
0
LETTERS
fO 20 JO~10"~ [5CATTERING ANOLEJ~ in RAOIAN.T~
Fig. 1. The i n v e r s e s c a t t e r e d intensity data for iron (points), a c c o r d i n g to J a c r o t et al. [5] (sample t e m p e r a t u r e T O = T c + 10°) and P a s s e l l et al. [7] (sample t e m p e r a t u r e T o = T c + 2 ° and T O = T e + 200), with the t h e o r e t i c a l c u r v e s in the elastic approximation (. . . . ), and with the inelasticity accounted for ( ). The fluctuation t e m p e r a t u r e T is indicated at the right ends of the curves.
References
1. 2. 3. 4. 5.
On = n X o / R
n integer
(2) 6.
f o r t h e v a l u e s of R g i v e n b y R = mnK~. 1
7. m integer
(3) 8. 9.
[c.f. 2 eq. (30)]. T h e t e m p e r a t u r e of t h e s a m p l e f o r w h i c h eq. (3) h o l d s m a y b e c a l c u l a t e d f r o m eq. (17) of r e f . 2, s i n c e t h e f l u c t u a t i o n t e m p e r a ture has been determined. For iron, inspection of t h e c o r r e s p o n d i n g s c a t t e r e d i n t e n s i t y c u r v e s leads to the conclusion, that the first side maximum should be detectable.
10. 11. 12.
J. Kocifiski, J. Phys. Chem. Solids 26 (1965) 895. J. Kocifiski, Aeta Phys. Polon. 30 (1966) 591. J. Kocifiski, Aeta Phys. Polon. 24 (1963) 273. N. Stump and G. Maier, Phys. L e t t e r s 24A (1967) 625. B. J a c r o t , J. Konstantinovie, G. P a r e t t e and D. C r i b i e r , Syrup. on Inelastic s c a t t e r i n g of neutrons, Chalk R i v e r (1962). B. J a c r o t , in: The interaction of radiation with solids (North-Holland, Am s t e r d a m , 1964). L. P a s s e l l , K. Blinowski, T. Brun and P. Nielsen, Phys. Rev. 139 (1965) A1866. L. Van Hove, Phys. Rev. 95 (1954) 1374. M. K. Wilkinson and C. G. Shull, Phys. Rev. 103 (1956) 516. S, Spooner and B. L. Averbach, Phys. Rev. 142 (1966) 291. A. MUnster and K. Sagel, Mol. Phys. 1 (1958) 23. A. MUnster in Fluctuation phenomena in solids (Academic P r e s s , New York and London 1965).
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601
PHYSICS L E T T E R S
23 October 1967
3. A. E. Hughes and W.A. Runciman, Proc. Phys. Soc. 86 (1965) 615. 4. P.Duval, J.Gareyte, Y. Merle d'Aubigne, Physics Letters 22 (1966) 67. 5. A.E. Hughes, Oxford Thesis 1966. 6. F. Lanzl, Phys. Letters 23A (1967) $10.
References 1. R.H.Silsbee, Phys. Rev. 138 (1965) 180. 2. F. Lanzl and W. vonder Osten, Phys. Letters 15 (1965) 208.
CRITICAL MAGNETIC SCATTERING OF NEUTRONS A N D T H E i s i n K 2 r t //r S P I N C O R R E L A T I O N J. KOCI~SKI Warsaw Technical University, Institute of Physics, Warsaw, Poland and B. MRYGOI~ Institute of Physics of the Polish Academy of Sciences, Warsaw, Poland Received 21 September 1967
Experimental data on critical magnetic scattering of neutrons are explained in terms of the ] sinK2r Vr spin correlation, and the possibility of detection of the predicted diffraction minima is pointed out.
The e x p e r i m e n t a l l y found dependence of the i n v e r s e i n t e n s i t y of s c a t t e r e d n e u t r o n s on the s q u a r e of the s c a t t e r i n g angle in i r o n and nickel will be explained in t e r m s of the s c a t t e r i n g c r o s s - s e c t i o n , calculated with spin c o r r e l a t i o n : [1-3] (SoZ(0) Sz(0))To = ] s i n K 2 r I//492r
(1)
which has been a l r e a d y applied by Stump and M a i e r in the i n t e r p r e t a t i o n of the detected angul a r dependence of the m a x i m u m of c r i t i c a l s c a t t e r i n g in nickel [4]. The c r o s s - s e c t i o n d e r i v e d with the c o r r e l a tion (1) [c.f. 2 eq. (26)], has been n u m e r i c a l l y calculated for i r o n and nickel, on the a s s u m p t i o n of a m o n o c h r o m a t i c i n c o m i n g wave ko equal to 4.75 ~ or 4.28 ~,, to allow c o m p a r i s o n with exp e r i m e n t a l data [5-7]. The v a l u e s of the other r e l e v a n t p a r a m e t e r s a r e the l a t t i c e constant a = 2 . 8 6 A , T o = I043°K, K2 = 5.01/a, A = = 1.99 × 10 -3 c m 2 / s e c for i r o n , and a = 3 . 5 2 .~, T c = 631°K, K2 = 6.37/a, A = 0.91 × 10 -3 c m 2 / s e c for nickel, as d e t e r m i n e d p r e v i o u s l y [1-3]. Typical r e s u l t s for i r o n a r e r e p r e s e n t e d in fig. 1. In drawing the t h e o r e t i c a l c u r v e s we have a s s u m e d that the c a l c u l a t e d v a l u e s for the i n v e r s e i n t e n s i t y which c o r r e s p o n d to the s q u a r e d angles 600
02 = 9 × 10 -4 and 02 = 16 × 10 -4 coincide with those on the r e c t i l i n e a r p a r t of the line drawn through the e x p e r i m e n t a l points. F o r a s a m p l e t e m p e r a t u r e of about 10 ° above T c or higher, a change AT = ± 2 ° in the fluctuation t e m p e r a t u r e i n f l u e n c e s but little the shape of an i n t e n s i t y curve. It follows that in i r o n the fluctuation t e m p e r a t u r e is equal to about T c - 6°, for the s a m p l e t e m p e r a t u r e in the vicinity of the c r i t i c a l point, and that it may slightly i n c r e a s e with growing s a m p l e t e m p e r a t u r e . Better a g r e e m e n t with the data of Passel1 then those of J a c r o t is connected with a b r o a d e r i n c o m i n g s p e c t r u m in the e x p e r i m e n t of the l a t t e r , unaccounted for in the calculation. In iron, the range of c o r r e l a t i o n R, d e t e r m i n e d f r o m eq. (17) of ref. 2, with T = = T c - 6 ° is equal to 30.4 A at the c r i t i c a l point, and to 7.1 A at T c + 50 °. In nickel, the single r e p o r t e d i n t e n s i t y c u r v e at the Curie point [6], is r e p r o d u c e d with T = T c - 4.5 ° approximately. The range of correlation at T c = 631°K is equal to 25.8 A. The c r o s s - s e c t i o n g e n e r a l l y d e c r e a s e s with i n c r e a s i n g s c a t t e r i n g angle at constant s a m p l e t e m p e r a t u r e , however, c o n t r a r y to the Van Hove theory [8], the d e c r e a s e is not monotonic and diffraction m i n i m a appear at the angles
Volume 25A, n u m b e r 8
PHYSICS
30
T
23 October 1967
W i l k i n s o n a n d S h u l l [9] h a v e s t o p p e d t h e i r measurements for a sample temperature T o = = T c + 20 ° n e a r t h e f i r s t m i n i m u m . J a c r o t e t al. [5] a n d P a s s e l l e t al. [7] h a v e n o t m e a s u r e d a t large enough angles for the long incoming waves, w h i c h t h e y h a v e u s e d . S p o o n e r a n d A v e r b a c h [10] h a v e b e e n e x p l o r i n g t h e r e g i o n of t h e f i r s t s i d e maximum, but the applied collimation has been insufficient for detecting more than a slowly changing small intensity. The first side maximum has been already detected by Mi/nster and Sagel for the X-ray critical scattering in the AI-Zn binary alloy [11, 12] [c.f. 12 fig. 2 p. 233] b u t it h a s n o t b e e n explained.
,~Ox~" 4 0
/
I....
0
LETTERS
fO 20 JO~10"~ [5CATTERING ANOLEJ~ in RAOIAN.T~
Fig. 1. The i n v e r s e s c a t t e r e d intensity data for iron (points), a c c o r d i n g to J a c r o t et al. [5] (sample t e m p e r a t u r e T O = T c + 10°) and P a s s e l l et al. [7] (sample t e m p e r a t u r e T o = T c + 2 ° and T O = T e + 200), with the t h e o r e t i c a l c u r v e s in the elastic approximation (. . . . ), and with the inelasticity accounted for ( ). The fluctuation t e m p e r a t u r e T is indicated at the right ends of the curves.
References
1. 2. 3. 4. 5.
On = n X o / R
n integer
(2) 6.
f o r t h e v a l u e s of R g i v e n b y R = mnK~. 1
7. m integer
(3) 8. 9.
[c.f. 2 eq. (30)]. T h e t e m p e r a t u r e of t h e s a m p l e f o r w h i c h eq. (3) h o l d s m a y b e c a l c u l a t e d f r o m eq. (17) of r e f . 2, s i n c e t h e f l u c t u a t i o n t e m p e r a ture has been determined. For iron, inspection of t h e c o r r e s p o n d i n g s c a t t e r e d i n t e n s i t y c u r v e s leads to the conclusion, that the first side maximum should be detectable.
10. 11. 12.
J. Kocifiski, J. Phys. Chem. Solids 26 (1965) 895. J. Kocifiski, Aeta Phys. Polon. 30 (1966) 591. J. Kocifiski, Aeta Phys. Polon. 24 (1963) 273. N. Stump and G. Maier, Phys. L e t t e r s 24A (1967) 625. B. J a c r o t , J. Konstantinovie, G. P a r e t t e and D. C r i b i e r , Syrup. on Inelastic s c a t t e r i n g of neutrons, Chalk R i v e r (1962). B. J a c r o t , in: The interaction of radiation with solids (North-Holland, Am s t e r d a m , 1964). L. P a s s e l l , K. Blinowski, T. Brun and P. Nielsen, Phys. Rev. 139 (1965) A1866. L. Van Hove, Phys. Rev. 95 (1954) 1374. M. K. Wilkinson and C. G. Shull, Phys. Rev. 103 (1956) 516. S, Spooner and B. L. Averbach, Phys. Rev. 142 (1966) 291. A. MUnster and K. Sagel, Mol. Phys. 1 (1958) 23. A. MUnster in Fluctuation phenomena in solids (Academic P r e s s , New York and London 1965).
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601