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Note on dynamical scaling
Volume 32A. number 5
PHYSICS LETTERS
NOTE
ON DYNAMICAL
10 August 1970
SCALING
H. SCHMIDT P h y s ~ k - D e p a r t r n e n t d e r Techn~schen Hochschule Munchen, Munchen , G e r m a n y
P. SZ]~PFALUSY * lns~tut Max yon L a u e - Paul Langez,~n, Garch~ng b. M~inchen, G e r m a n y
Received 13 June 1970
A simple relation between the coefficients of the characteristic equation for the critical mode is proposed based upon measurements on RbMnF 3. It yields for HeII a correlation length considerably larger than prevmus estimates but agreeing with the observed heahng length.
The concept of d y n a m i c a l s c a h n g [1,2] has p r o v e n to be useful in u n d e r s t a n d i n g and r e l a t i n g v a r i o u s c r i t i c a l p h e n o m e n a n e a r phase t r a n s i t i ons of second o r d e r . Many of the p r e d i c t i o n s c o n c e r n i n g the c r i t i c a l exponents b a s e d on the d y n a m i c a l s c a l i n g t h e o r i e s have been t e s t e d and g e n e r a l l y v e r i f i e d by e x p e r i m e n t s [3-8]. The o r i g i n a l f o r m u l a t i o n [1] of d y n a m i c a l s c a l i n g by i m p o s i n g m a t c h i n g c o n d i t i o n s b e t w e e n the c r l h c a l r e g i o n and the h y d r o d y n a m l c a l r e g i o n s not only r e l a t e d the c r i t i c a l e x p o n e n ts but a l s o made p r e d i c t i o n s for the absolute v a l u e s of c r i t i c a l q u a n t i t i e s by a s s u m i n g that the c r i t i c a l mode stops p r o p a g a t i n g at the c r i t i c a l t e m p e r a t u r e . A n a l y s i n g new e x p e r i m e n t a l r e s u l t s on the a n t l f e r r o m a g n e h c phase t r a n s i t i o n [8] one finds that the c r i t i c a l mode does not exhibit th~s b e h a v i o u r but i n s t e a d r e m a i n s p r o p a g a t i n g e v e n in the c r i t i c a l r e g i o n r i g h t at T c. In this note we point out that a s i m p l e new a s s u m p t i o n can account f o r this f e a t u r e quantitatively. The s a m e a s s u m p t m n when applied to the k - t r a n s i t i o n in liquid h e l i u m t o g e t h e r with the m e a s u r e d c r l h c a l damping of second sound in HeII g i v e s a c o h e r ence length, the magnitude of which is in a c c o r d with m e a s u r e m e n t s by Henkel, Smith and Reppy [10] c o n c e r n i n g the healing length m s u p e r f l u i d hehum. As in ref. [1] our s t a r t i n g point is the c h a r a c t e r i s t i c equation for the e i g e n f r e q u e n c t e s ¢o of the c r i t i c a l mode in the h y d r o d y n a m i c a l r e g i o n of the o r d e r e d stateco 2 + iDk 2 - Wk:2 = O, w h e r e k i s the w a v e n u m b e r under c o n s i d e r a t i o n . D k 2 a n d ¢ok then d e t e r m i n e the p o l e s of the r e s p o n s e functrans. In ref. [1] the m e r g i n g f r o m the hydro326
d y n a m i c s below T c (propagahon) and above T c (diffusion) was a s s u m e d to o c c u r m such a way that the wavelike p r o p a g a h o n has c o m p l e t e l y stopped just at T c for all v a l u e s of k, which m e a n s that the c h a r a c t e r i s t i c f r e q u e n c i e s o)(k) = +~/'w-~- (½Dk2) 2 - ½1Dk 2 = ~: ft k - i F k a r e p u r e l y i m a g i n a r y , yielding the condition 2o9 k = D k 2 m the c r i t i c a l region. R e c e n t e x p e r i m e n t s by Lau et al.[8] do not b e a r out this c o n l e c t u r e . Instead they show c l e a r e v i d e n c e for p r o p ag at i o n of the c r i t i c a l /node of an a n t l f e r r o m a g n e t at T c. T h i s b e h a v i o u r can be accounted for by postulating the condition for the c r i t i c a l mode at T c to be of the simple form ¢ok = O k 2
(1)
T h i s condition e n s u r e s a n a t u r a l p a s s i n g into the h y d r o d y n a m l c a l r e g i o n below and above the c r i t i cal t e m p e r a t u r e , w h e r e the leading t e r m s in the c r i t i c a l mode for s m a l l k a r e w k ~ c k and D k 2, respectively. This yields n
= ¢3r~
(2)
r e p r o d u c i n g the o b s e r v e d r a t i o n [8] for RbMnF 3 ~ k / r k ~ 1.66 within 5%. The new condition (1) does obviously not change the value of the c r i t i c a l exponent of any quantity but a f f e c t s only the c r i t i c a l c o e f f i c i e n t s in the h y d r o d y n a m i c a l region. The s u c c e s s of the application of scal i n g laws f o r v a r i o u s s y s t e m s s u p p o r t s the g e n e r a l a s s u m p tion that the r e l a t i o n s among d i f f e r e n t c r i t i c a l q u a n t i t i e s do not depend on the d e t a i l e d n a t u r e of
* On leave from Roland Et~tv6s University, Budapest.
Volume 32A, n u m b e r 5
PHYSICS
LETTERS
s h t , if o n e t a k e s i n t o a c c o u n t t h a t the h e a h n g l e n g t h i s l a r g e r t h a n t h e c o r r e l a t m n l e n g t h by a g e o m e t r i c a l f a c t o r of a b o u t 4 2 [10].
t h e s y s t e m . In p a r t i c u l a r t h e y s h o u l d b e t h e s a m e f o r a l l s y s t e m s w h i c h h a v e t h e s a m e s t r u c t u r e of t h e h y d r o d y n a m i c s , a s IS t h e c a s e w i t h a n h f e r r o m a g n e t s [9] a n d l i q m d h e h u m . T h i s m e a n s t h a t eq. (1) a l s o h a s to h o l d f o r t h e c r i t i c a l m o d e m t h e c r i t i c a l r e g i o n of l i q u i d He. This condition has ~mmedmte consequences f o r t h e a b s o l u t e v a l u e of t h e c o r r e l a t m n l e n g t h ~. If we m a k e u s e of t h e m a t c h i n g c o n d i t i o n b e t w e e n t h e c h a r a c t e r i s t i c f r e q u e n c i e s of t h e c r i t i c a l m o d e in t h e h y d r o d y n a m i c r e g i o n of HeII a n d in t h e c r i t i c a l r e g i o n f o r t h e w a v e n u m b e r k ~ ~-1 we f i n d f r o m eq. (2): c~ -1 ~ ½ 4 3 D ~ - 2 . H e r e we h a v e i n s e r t e d ~I~ = ck w h e r e c a n d D a r e t h e measured velocity and damping coefficient, resp e c t i v e l y of s e c o n d s o u n d . Us.rig t h e o b s e r v e d v a l u e s of c [11] a n d D [3] o n e f i n d s = ~o(1 - T / T x ) -2/3, w h e r e ~o i s a b o u t 3A. T h i s m a g n i t u d e of ~o i s l a r g e r b y a n o r d e r of m a g m tude than the coherence length deduced from the p h a s e f l u c t u a t i o n s [1] of t h e o r d e r p a r a m e t e r , a n d l a r g e r by a f a c t o r of 2 t h a n t h e c o h e r e n c e l e n g t h e s t i m a t e d f r o m t h e f l u c t u a t i o n s of t h e m a g n i t u d e of t h e o r d e r p a r a m e t e r [12]. H o w e v e r . it i s m a c c o r d w i t h t h e m e a s u r e d v a l u e of t h e h e a h n g l e n g t h ~s = ~o (1 - T / T x ) - 2/3, a s o b t a i n e d f r o m t h e a n a l y s i s of s u p e r f l m d flow t h r o u g h a
OPTICAL
RADIATION
M. HOLLSTEIN
References [1] R. A. F e r r e l l , N. Menyhgrd, H. Schmidt, F. Schwabl and P. Szgpfalusy, Phys. Rev. L e t t e r s 18 (1967)891: Ann. Phys. (N.Y.) 47 (1968) 565. [2] B. I. Halperin and P. C. Hohenberg. Phys. Rev. L e t t e r s 19 (1967) 700; Phys. Rev. 177 (1969) 952. [3] J. A. Tyson, Phys. Rev. L e t t e r s 21 (1968) 1235. [4] M. Archibald. J . M . Mochel and L. Weaver, Phys. Rev. L e t t e r s 21 (1968) 1156. [5] G. A h l e r s . Phys. Rev. L e t t e r s 21 (1968) 1159. [6] M. F. Collins. V . J . Mmkiewmz. R. Nathans. L. P a s s e l l and G. Shlrane, Phys. Rev. 179 (1969) 417. [7] V. J. Mmkmwicz. M. F. C o l h n s , R. Nathans and G. Shirane, Phys. Rev. 182 (1968) 629. [8] H. Y. Lau. L.M. C o r l i s s , A. Delapalme, J . M . Hastings. R. Nathans and A. T u c c l a r o n e , Phys. Rev. L e t t e r s 23 (1969) 1125. [9] B. I. Halperm and P. C. Hohenberg, Phys. Rev. 188 (1969) 898. [10] R. P. Henkel, E. N. Smith and J. D. Reppy, Phys. Rev. L e t t e r s 23 (1969) 1216. [11] Jo A. Tyson and D. H. Douglass J r . . Phys. Rev. L e t t e r s 21 (1968) 1308. Zh. Eksp. I. Teor. F~z. 52 (1967) [12] Yu. G. Mamaladze, 729; Soy. Phys. JETP 25 (1967) 479.
PRODUCED I N VA, R I O U S $, A. SALOP,
10 August 1970
B Y 1100 e V H e GASES **
J.R. PETERSON
+
,
He
*
,
AND
He °
and D. C. LORENTS
Stanford Research Institute, Menlo Park. Cal~forma, USA Received 24 April 1970
Optmal s p e c t r a a r e o b s e r v e d a s s o c i a t e d with c o l h m o n a l excitation by He +, He*, He °, Penning lomzatlon by He*, and charge t r a n s f e r by He +. Metastable He* is colhmonally excited to higher electron states, producing HeI e m i s s i o n s , much m o r e readily than is ground state He °.
In c o n j u n c t i o n w i t h m e a s u r e m e n t s on e n e r g y t r a n s f e r [1] a n d P e n n i n g i o n i z a t i o n r e a c t i o n c r o s s s e c t i o n s [2], we h a v e u s e d a n i m a g e - t u b e s p e c t r o g r a p h to o b t a i n r e f o r m a t i o n on t h e e x c i tation resulting from collisions between metast a b l e h e l i u m a t o m (He*) b e a m s a n d v a r i o u s g a s e s . A s h o r t s u r v e y w a s m a d e of t h e o p t i c a l s p e c t r a p r o d u c e d b y 1100 e V He* b e a m s in H e , A r , Ne, H 2 a n d N 2. S i m i l a r s t u d i e s w e r e m a d e u s i n g h e l i u m i o n (He + ) a n d g r o u n d s t a t e h e l i u m a t o m (He °) b e a m s .
The basic apparatus has been described elsew h e r e [3]. A n e u t r a l b e a m i s f o r m e d f r o m a m a s s a n a l y z e d He + b e a m by c h a r g e t r a n s f e r in e i t h e r a l k a l i v a p o r , l e a d i n g to He* a s t h e p r i n c i p a l c o m p o n e n t (> 50%) o r m h e l i u m g a s l e a d i n g to He °. ** Supported m p a r t by the Advanced R e s e a r c h P r o j e c t s Agency, through the U.S. Army R e s e a r c h Office Durham. J P r e s e n t A d d r e s s : D o r m e r GmbH, F r i e d r m h s h a f e n (Bodensee), West Germany.
327
PHYSICS LETTERS
NOTE
ON DYNAMICAL
10 August 1970
SCALING
H. SCHMIDT P h y s ~ k - D e p a r t r n e n t d e r Techn~schen Hochschule Munchen, Munchen , G e r m a n y
P. SZ]~PFALUSY * lns~tut Max yon L a u e - Paul Langez,~n, Garch~ng b. M~inchen, G e r m a n y
Received 13 June 1970
A simple relation between the coefficients of the characteristic equation for the critical mode is proposed based upon measurements on RbMnF 3. It yields for HeII a correlation length considerably larger than prevmus estimates but agreeing with the observed heahng length.
The concept of d y n a m i c a l s c a h n g [1,2] has p r o v e n to be useful in u n d e r s t a n d i n g and r e l a t i n g v a r i o u s c r i t i c a l p h e n o m e n a n e a r phase t r a n s i t i ons of second o r d e r . Many of the p r e d i c t i o n s c o n c e r n i n g the c r i t i c a l exponents b a s e d on the d y n a m i c a l s c a l i n g t h e o r i e s have been t e s t e d and g e n e r a l l y v e r i f i e d by e x p e r i m e n t s [3-8]. The o r i g i n a l f o r m u l a t i o n [1] of d y n a m i c a l s c a l i n g by i m p o s i n g m a t c h i n g c o n d i t i o n s b e t w e e n the c r l h c a l r e g i o n and the h y d r o d y n a m l c a l r e g i o n s not only r e l a t e d the c r i t i c a l e x p o n e n ts but a l s o made p r e d i c t i o n s for the absolute v a l u e s of c r i t i c a l q u a n t i t i e s by a s s u m i n g that the c r i t i c a l mode stops p r o p a g a t i n g at the c r i t i c a l t e m p e r a t u r e . A n a l y s i n g new e x p e r i m e n t a l r e s u l t s on the a n t l f e r r o m a g n e h c phase t r a n s i t i o n [8] one finds that the c r i t i c a l mode does not exhibit th~s b e h a v i o u r but i n s t e a d r e m a i n s p r o p a g a t i n g e v e n in the c r i t i c a l r e g i o n r i g h t at T c. In this note we point out that a s i m p l e new a s s u m p t i o n can account f o r this f e a t u r e quantitatively. The s a m e a s s u m p t m n when applied to the k - t r a n s i t i o n in liquid h e l i u m t o g e t h e r with the m e a s u r e d c r l h c a l damping of second sound in HeII g i v e s a c o h e r ence length, the magnitude of which is in a c c o r d with m e a s u r e m e n t s by Henkel, Smith and Reppy [10] c o n c e r n i n g the healing length m s u p e r f l u i d hehum. As in ref. [1] our s t a r t i n g point is the c h a r a c t e r i s t i c equation for the e i g e n f r e q u e n c t e s ¢o of the c r i t i c a l mode in the h y d r o d y n a m i c a l r e g i o n of the o r d e r e d stateco 2 + iDk 2 - Wk:2 = O, w h e r e k i s the w a v e n u m b e r under c o n s i d e r a t i o n . D k 2 a n d ¢ok then d e t e r m i n e the p o l e s of the r e s p o n s e functrans. In ref. [1] the m e r g i n g f r o m the hydro326
d y n a m i c s below T c (propagahon) and above T c (diffusion) was a s s u m e d to o c c u r m such a way that the wavelike p r o p a g a h o n has c o m p l e t e l y stopped just at T c for all v a l u e s of k, which m e a n s that the c h a r a c t e r i s t i c f r e q u e n c i e s o)(k) = +~/'w-~- (½Dk2) 2 - ½1Dk 2 = ~: ft k - i F k a r e p u r e l y i m a g i n a r y , yielding the condition 2o9 k = D k 2 m the c r i t i c a l region. R e c e n t e x p e r i m e n t s by Lau et al.[8] do not b e a r out this c o n l e c t u r e . Instead they show c l e a r e v i d e n c e for p r o p ag at i o n of the c r i t i c a l /node of an a n t l f e r r o m a g n e t at T c. T h i s b e h a v i o u r can be accounted for by postulating the condition for the c r i t i c a l mode at T c to be of the simple form ¢ok = O k 2
(1)
T h i s condition e n s u r e s a n a t u r a l p a s s i n g into the h y d r o d y n a m l c a l r e g i o n below and above the c r i t i cal t e m p e r a t u r e , w h e r e the leading t e r m s in the c r i t i c a l mode for s m a l l k a r e w k ~ c k and D k 2, respectively. This yields n
= ¢3r~
(2)
r e p r o d u c i n g the o b s e r v e d r a t i o n [8] for RbMnF 3 ~ k / r k ~ 1.66 within 5%. The new condition (1) does obviously not change the value of the c r i t i c a l exponent of any quantity but a f f e c t s only the c r i t i c a l c o e f f i c i e n t s in the h y d r o d y n a m i c a l region. The s u c c e s s of the application of scal i n g laws f o r v a r i o u s s y s t e m s s u p p o r t s the g e n e r a l a s s u m p tion that the r e l a t i o n s among d i f f e r e n t c r i t i c a l q u a n t i t i e s do not depend on the d e t a i l e d n a t u r e of
* On leave from Roland Et~tv6s University, Budapest.
Volume 32A, n u m b e r 5
PHYSICS
LETTERS
s h t , if o n e t a k e s i n t o a c c o u n t t h a t the h e a h n g l e n g t h i s l a r g e r t h a n t h e c o r r e l a t m n l e n g t h by a g e o m e t r i c a l f a c t o r of a b o u t 4 2 [10].
t h e s y s t e m . In p a r t i c u l a r t h e y s h o u l d b e t h e s a m e f o r a l l s y s t e m s w h i c h h a v e t h e s a m e s t r u c t u r e of t h e h y d r o d y n a m i c s , a s IS t h e c a s e w i t h a n h f e r r o m a g n e t s [9] a n d l i q m d h e h u m . T h i s m e a n s t h a t eq. (1) a l s o h a s to h o l d f o r t h e c r i t i c a l m o d e m t h e c r i t i c a l r e g i o n of l i q u i d He. This condition has ~mmedmte consequences f o r t h e a b s o l u t e v a l u e of t h e c o r r e l a t m n l e n g t h ~. If we m a k e u s e of t h e m a t c h i n g c o n d i t i o n b e t w e e n t h e c h a r a c t e r i s t i c f r e q u e n c i e s of t h e c r i t i c a l m o d e in t h e h y d r o d y n a m i c r e g i o n of HeII a n d in t h e c r i t i c a l r e g i o n f o r t h e w a v e n u m b e r k ~ ~-1 we f i n d f r o m eq. (2): c~ -1 ~ ½ 4 3 D ~ - 2 . H e r e we h a v e i n s e r t e d ~I~ = ck w h e r e c a n d D a r e t h e measured velocity and damping coefficient, resp e c t i v e l y of s e c o n d s o u n d . Us.rig t h e o b s e r v e d v a l u e s of c [11] a n d D [3] o n e f i n d s = ~o(1 - T / T x ) -2/3, w h e r e ~o i s a b o u t 3A. T h i s m a g n i t u d e of ~o i s l a r g e r b y a n o r d e r of m a g m tude than the coherence length deduced from the p h a s e f l u c t u a t i o n s [1] of t h e o r d e r p a r a m e t e r , a n d l a r g e r by a f a c t o r of 2 t h a n t h e c o h e r e n c e l e n g t h e s t i m a t e d f r o m t h e f l u c t u a t i o n s of t h e m a g n i t u d e of t h e o r d e r p a r a m e t e r [12]. H o w e v e r . it i s m a c c o r d w i t h t h e m e a s u r e d v a l u e of t h e h e a h n g l e n g t h ~s = ~o (1 - T / T x ) - 2/3, a s o b t a i n e d f r o m t h e a n a l y s i s of s u p e r f l m d flow t h r o u g h a
OPTICAL
RADIATION
M. HOLLSTEIN
References [1] R. A. F e r r e l l , N. Menyhgrd, H. Schmidt, F. Schwabl and P. Szgpfalusy, Phys. Rev. L e t t e r s 18 (1967)891: Ann. Phys. (N.Y.) 47 (1968) 565. [2] B. I. Halperin and P. C. Hohenberg. Phys. Rev. L e t t e r s 19 (1967) 700; Phys. Rev. 177 (1969) 952. [3] J. A. Tyson, Phys. Rev. L e t t e r s 21 (1968) 1235. [4] M. Archibald. J . M . Mochel and L. Weaver, Phys. Rev. L e t t e r s 21 (1968) 1156. [5] G. A h l e r s . Phys. Rev. L e t t e r s 21 (1968) 1159. [6] M. F. Collins. V . J . Mmkiewmz. R. Nathans. L. P a s s e l l and G. Shlrane, Phys. Rev. 179 (1969) 417. [7] V. J. Mmkmwicz. M. F. C o l h n s , R. Nathans and G. Shirane, Phys. Rev. 182 (1968) 629. [8] H. Y. Lau. L.M. C o r l i s s , A. Delapalme, J . M . Hastings. R. Nathans and A. T u c c l a r o n e , Phys. Rev. L e t t e r s 23 (1969) 1125. [9] B. I. Halperm and P. C. Hohenberg, Phys. Rev. 188 (1969) 898. [10] R. P. Henkel, E. N. Smith and J. D. Reppy, Phys. Rev. L e t t e r s 23 (1969) 1216. [11] Jo A. Tyson and D. H. Douglass J r . . Phys. Rev. L e t t e r s 21 (1968) 1308. Zh. Eksp. I. Teor. F~z. 52 (1967) [12] Yu. G. Mamaladze, 729; Soy. Phys. JETP 25 (1967) 479.
PRODUCED I N VA, R I O U S $, A. SALOP,
10 August 1970
B Y 1100 e V H e GASES **
J.R. PETERSON
+
,
He
*
,
AND
He °
and D. C. LORENTS
Stanford Research Institute, Menlo Park. Cal~forma, USA Received 24 April 1970
Optmal s p e c t r a a r e o b s e r v e d a s s o c i a t e d with c o l h m o n a l excitation by He +, He*, He °, Penning lomzatlon by He*, and charge t r a n s f e r by He +. Metastable He* is colhmonally excited to higher electron states, producing HeI e m i s s i o n s , much m o r e readily than is ground state He °.
In c o n j u n c t i o n w i t h m e a s u r e m e n t s on e n e r g y t r a n s f e r [1] a n d P e n n i n g i o n i z a t i o n r e a c t i o n c r o s s s e c t i o n s [2], we h a v e u s e d a n i m a g e - t u b e s p e c t r o g r a p h to o b t a i n r e f o r m a t i o n on t h e e x c i tation resulting from collisions between metast a b l e h e l i u m a t o m (He*) b e a m s a n d v a r i o u s g a s e s . A s h o r t s u r v e y w a s m a d e of t h e o p t i c a l s p e c t r a p r o d u c e d b y 1100 e V He* b e a m s in H e , A r , Ne, H 2 a n d N 2. S i m i l a r s t u d i e s w e r e m a d e u s i n g h e l i u m i o n (He + ) a n d g r o u n d s t a t e h e l i u m a t o m (He °) b e a m s .
The basic apparatus has been described elsew h e r e [3]. A n e u t r a l b e a m i s f o r m e d f r o m a m a s s a n a l y z e d He + b e a m by c h a r g e t r a n s f e r in e i t h e r a l k a l i v a p o r , l e a d i n g to He* a s t h e p r i n c i p a l c o m p o n e n t (> 50%) o r m h e l i u m g a s l e a d i n g to He °. ** Supported m p a r t by the Advanced R e s e a r c h P r o j e c t s Agency, through the U.S. Army R e s e a r c h Office Durham. J P r e s e n t A d d r e s s : D o r m e r GmbH, F r i e d r m h s h a f e n (Bodensee), West Germany.
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