Laser-phase transition analogy: II

Volume 35A, number5

PHYSICS LETTERS

LASER-PHASE

TRANSITION

ANALOGY:

28 June 1971

II

J. C. GOLDSTEIN ~ , M. O. SCULLY Department of Physics and Optical Sciences Center, University of Artzona, Tucson, Arizona, USA and Department of Physics and Materials Science Center Massachusetts Institute of Technology, Cambridge, Massachusetts, USA

and P . A . LEE Department of Physics, Yale University, New Haven, Connecticut, USA Received 10May1971

A dynamical theory of an Ising ferromagnet coupled to a thermal reservoir is presented. Beginning with microscopic, reversible Hamiltonian dynamics, we proceed to an irreversible equation of motion for the magnetization probability density P(m, t), in analogy with the laser analysis.

In a r e c e n t p u b l i c a t i o n [1], it was d e m o n s t r a t e d that the b e h a v i o r of a l a s e r n e a r t h r e s h old i s v e r y analogous to a second o r d e r p h a s e t r a n s i t i o n **. It was shown that the e l e c t r i c field E, population i n v e r s i o n a (threshold i n v e r s i o n ~t), and a s y m m e t r y - b r e a k i n g e x t e r n a l signal S satisfy the s a m e equation of state a s the m a g n e t i z a t i o n m, t e m p e r a t u r e T ( c r i t i c a l TC) and a s y m m e t r y - b r e a k i n g e x t e r n a l field Ho of a f e r r o m a g n e t i c s y s t e m t r e a t e d in a m e a n field a p p r o x i mation. The l a s e r e l e c t r i c field p r o b a b i l i t y d e n sity d i s t r i b u t i o n P(E ) was noted to be the s t e a d y state solution of a F o k k e r - P l a n c k equation which in t u r n was obtained f r o m a m i c r o s c o p i c theory. It was suggested in ref. [1] that a s i m i l a r n o n e q u i l i b r i u m a n a l y s i s y i e l d i n g the m a g n e t i z a t i o n p r o b a b i l i t y density d i s t r i b u t i o n P ( m ) a s the s t e a d y - s t a t e r e s u l t of a d y n a m i c a l m i c r o s c o p i c theory would be of i n t e r e s t . F u r t h e r m o r e , r e c e n t a d v a n c e s in e x p e r i m e n t a l t e c h n i q u e s of light s c a t t e r i n g s p e c t r o s copy [3] (use of l a s e r s and photon counting s t a t i s t i c s ) make d i r e c t o b s e r v a t i o n of the d i s t r i b u tion function p o s s i b l e . In view of its c o n n e c t i o n

with the photon counting e x p e r i m e n t s , we a r e n a t u r a l l y led to study the full d i s t r i b u t i o n function of the s y s t e m of i n t e r e s t r a t h e r than its lowest o r d e r m o m e n t s or c o r r e l a t i o n functions, as was done in other r e c e n t s t u d i e s [4] *. Motivated by the c o n s i d e r a t i o n s of the p r e ceeding two p a r a g r a p h s , we h e r e outline a d y n a m i c a l theory of a n Ising f e r r o m a g n e t [5] coupled to a t h e r m a l r e s e r v o i r u s i n g t e c h n i q u e s s i m i l a r to those u s e d in the l a s e r a n a l y s i s . Beginning with m i c r o s c o p i c , r e v e r s i b l e H a m i l t o n i a n dyn a m i c s , we p r o c e e d to an i r r e v e r s i b l e equation of motion for the m a g n e t i z a t i o n p r o b a b i l i t y d e n sity P ( m , t). T h i s equation of motion d e s c r i b e s the r e l a x a t i o n of the s y s t e m to e q u i l i b r i u m when it is i n i t i a l l y in a n o n e q u i l i b r i u m state **. Thus, let u s c o n s i d e r the s y s t e m of i n t e r e s t S (i.e., magnet) to be in contact with a t h e r m a l r e s e r v o i r R (e.g., phonons, spin i m p u r i t i e s , etc.). The i n t e r a c t i o n with the t h e r m a l r e s e r v o i r will c a u s e t r a n s i t i o n s which tend to b r i n g the s y s t e m of i n t e r e s t into e q u i l i b r i u m . We c o n s i d e r S to be d e s c r i b e d by the u n i f o r m coupling Ising H a m i l tonian

Work supported in part by the U.S. Air Force Office of Scientific Research and in part by the Advanced Research Projects Agency. ~ Based on a Ph.D. thesis submitted to the Massachusetts Institute of Technology. * John Simon Guggenheim Fellow. *~. Similar conclusions have been reached by Graham and Haken [2].

* Note that knowledge of all of the moments of the distribution function we discuss is necessary to determine the distribution itself, and that correlation functions may be computed once the equation of motion P(m,t) is known. ** Although we will here restrict our attention to ferromagnetism, extensions of the present work to ferroelectricity and superconductivity are being investigated and will be published elsewhere. 317

V o l u m e 35A, n u m b e r 5

N HS = _½j ~

z z -.Ho sis

i j=l

PHYSICS

N

z

j=l

which r e p r e s e n t s N spin one half a n g u l a r m o menta, each having magnetic m o m e n t p, i n t e r a c ting u n i f o r m l y with each other by m e a n s of a f e r romagnetic ( J >0) i n t e r a c t i o n and with a u n i f o r m e x t e r n a l field Ho in the z direction. The s y s t e m i s then coupled to a r e s e r v o i r having H a m i l t o n i a n H R by the i n t e r a c t i o n V given by

j=l ot where S] (S~) is

with which the s y s t e m is in t h e r m a l contact. To i l l u s t r a t e this point we introduce two types of r e s e r v o i r s : (a) a boson (phonon) r e s e r v o i r in which the o p e r a t o r s O~ and O~ obey boson c o m mutation r e l a t i o n s ; (b) a . s p i n r e s e r v o i r in which the o p e r a t o r s O a and O~T obey spin one half c o m m u t a t i o n r e l a t i o n s . This equations of motion for the m e a n m a g n e t i z a t i o n when the s y s t e m is coupled to r e s e r v o i r s of type (a) or (b), a s i m plied by eq. (1) ' a r e then r e s p e c t i v e l y [7-9]

(I/7(b)

the spin lowering (raising) ope r a t o r for the j t h spin and Oot is a r e s e r v o i r o p e r a t o r for the j th phonon mode or spin i m p u r i t y which is coupled to the j t h spin with a s t r e g t h

g~ (U)"

Using a d e n s i t y m a t r i x technique s i m i l a r to that d i s c u s s e d in ref. [1] and noting each spin i n t e r a c t s equally with all other spins, we p r o c e e d by a s s u m i n g each spin weakly i n t e r a c t s with the t h e r m a l r e s e r v o i r independently of the o t h e r s and that the r e s e r v o i r h a s a sufficient n u m b e r of d e g r e e s of f r e e d o m to imply M a r k offian b e h a v i o r of the s y s t e m . We obtain an equation of motion for P(m, t ) which i s defined [6] a s P(m,t) = WrS, R [ P ( t ) 5(m - ~ - - 1 S ~ ) ] , where p i t ) is the total density o p e r a t o r of the spin p l u s r e s e r v o i r s y s t e m and m is the ( d i m e n s i o n l e s s ) c - n u m b e r magnetization. T h i s e q u a tion of motion is found to be / ~ ( m , t ) = r 2 ( m - 1 ) [ ½ N - ( m - 1 ) ] P ( m - 1, t ) 1

[r2(m) (½N - m) + r l ( m ) (yN + m)] P ( m , t ) +

rl(m +l)(½N+m+ l)P(m+ l,t)

(1)

where the a s s o c i a t e d damping c o n s t a n t F 1 is given by (2)

292 Ref t'< ~ O?a(t) ~ O~(t') >R exp{ iw21(t - t')} -oo

and F2(m) is given by a s i m i l a r e x p r e s s i o n found by r e p l a c i n g all O ' s (O ~ 's) by O t ' s (O 's) and w21 by -~21. We have e m p h a s i z e d that r l and r 2 depend on m. T h i s is a c o n s e q u e n c e of the fact that the energy difference ~ 2 1 between the two p o s s i b l e o r i e n t a t i o n s of a single spin depends on ~.

The specific f o r m s of the damping c o n s t a n t s F 1 and F 2 depend upon the type of r e s e r v o i r 318

28 J u n e 1971

(1/~(a)d

N

rl(m) = t

LETTERS

= 1 -(re(t))

coth[r(h+(m(t)))]

d ( m(t )> = t a n h [ r (h + ( re(t)))]

- ( m(t ))

where 7(a) [T(b)] is a decay rate c h a r a c t e r i s t i c of the phonon [spin] r e s e r v o i r and h = Ho/(kN~t/f~); ~"= Tc/T; kBTC =kNf~2/~, with the W e i s s i n t e r n a l field c o n s t a n t and ~ is the v o l u m e of the magnet. We have n e g l e c t e d f l u e tuations in eqs. (3) and (4) by w r i t i n g the a v e r a g e of a product as the product of a v e r a g e s . The s t e a d y - s t a t e solutions of both eqs. (3) and (4) a r e identical (independent of the d e t a i l s of the r e s e r v o i r system) and a r e the m o l e c u l a r field equation of state. The s t e a d y - s t a t e solution ~ of eq. (1) is S i m i l a r l y independent of the d e t a i l s of the r e s e r v o i r and is given by

P(m)

=

~[(½N +m)l(½N -re)l] -1 exp[~m

(5)

(½Jm + ~ H o ) ]

where ~7 is a n o r m a l i z a t i o n c o n s t a n t (depending u p o n N , ~, He, J and ( k B T ) - I =fl). We see that F (m) = l n [ P (m)/~ ] i s p r e c i s e l y the f r e e energy of the magnetic s y s t e m computed f r o m the B r a g g W i U i a m s a p p r o x i m a t i o n [10]. We a l s o note that for t e m p e r a t u r e s T < TC, P(m) s a t i s f i e s a Compare Suzuki and Kubo [7]. A more complete discussion of the physical significance of these results, including the possible dependence of y(a) and T {b) on m, wilibe g i v e n i n ref.[9]; e.g. y(a) cc m i f m i f m to be odd in m. Langer [8] makes the interesting observation that '... it is probably necessary to consider specific models of the disipative mechanism... [but] it is possible that many of the most interesting physically observable irreversible effects may be nearly insensitive to the precise model of irreversibility which is used.' The two mean equations of motion here are very different for the two reservoirs considered and it is clear the former possibility applies. ~:~ Defining an entropy in terms of P(m, t) an 'H' theorem which states that it is a monotonically increasing function of time which achieves its maximum value when P (m ,t ) reaches steady state (t-~+ oo) can be proven.

Volume 35A, number 5

PHYSICS

F o k k e r - P l a n c k type equation (obtained f r o m eq. (1)) whose s t e a d y - s t a t e solution g i v e s F ( m ) a s the Landau f r e e e n e r g y (which may also be obtained f r o m eq. (5)), thus f u r t h e r e m p h a s i z i n g the analogy of this s y s t e m to the l a s e r n e a r t h r e s h o l d as t r e a t e d in ref. [1]. In c o n c l u s i o n , we e m p h a s i z e that the d y n a m ical b e h a v i o r of the magnetic s y s t e m - a s e x p r e s s e d in the equation of motion of the m a g n e t i zation p r o b a b i l i t y d i s t r i b u t i o n - d e p e n d s upon the d e t a i l s of the r e s e r v o i r s y s t e m to which it is coupled. That is, the damping c o n s t a n t s r I and r 2 a r e different for different r e s e r v o i r s , and a s one c o n s e q u e n c e the m e a n eqs. of motion (3) and (4) a r e quite different for the two types of r e s e r v o i r s c o n s i d e r e d h e r e . Some e x p e r i m e n t a l i m p l i c a t i o n s c o n c e r n i n g photon counting s t a t i s t i c a l e x p e r i m e n t s on n o n e q u l i b r i u m magnetic s y s t e m s will be c o n s i d e r e d in a future p u b l i c a tion [9]. The a u t h o r s wish to acknowledge s t i m u l a t i n g and helpful d i s c u s s i o n s with P. C. M a r t i n and H. E. Stanley.

_l~efe'F81q,ces [1] V. DeGiorgio and M. O. Seully, Phys. Rev. A2 (1970) 1170; M.I.T. Solid state and molecular

LETTERS

28 June 1971

Theory Group Semi-Annual Progress Report Number 71 {1969) p. 80. [2] R. Graham and H. Haken, Z. Physik 237 (1970) 31; See also H. Haken, .Proceedings of the Third International Conference On Quantum Electronics, Kyoto, Japan, to be published. [3] H. Z. Cummins, Laser light scatteringSpectroscopy, in; Quantum optics, Proc. Intern. School of Physics 'Enrico FermV, Course XLH, ed. R. J. Glauber {Academic Press, New York 1969); E. R. Pike, Photon Statistics, Inaugural Conf. of the European Physical Society, Florence, Italy, April 1969. [4] K.Kawasaki, Ann. Phys. (N.Y.) 61 (1970) 1; B. J. Halperin and P. C. Hohenburg, Phys. Rev. 177 {1969) 952; Phys.Rev. 188 (1969) 898. [5] R.J. Glauber, J. Math. Phys. 4 (1963) 294. [6] M. Lax and H. Yuen, Phys. Rev. 172 (1968) 362; W. H. Louisell, in: Quantum optics, Proc. Intern. School of Physics 'Enrico Fermi', Course XLII, ed. R. J. Glauber {Academic Press, New York,

(1969) ;

W.H. Louisell, in: The physics of quantum electronics, eds. S. F. Jacobs and J. B. Mandelbaum (Optical Sciences Center Technical Report 31, University of Arizona, Tucson, Arizona, 1968), Vo[. H,p. 311; J . P . Gordon, Phys.Rev. 161 (1967) 367. [7] M. Suzuki and R.Kubo, J. Phys. Sec.Japan 24 (1968) 51. [8] J. S. Langer, Phys. Rev. 184 (1969) 219. [9] J. C. Goldstein and M. O. Seutly, to be published. [10] R. Kubo, Statistical mechanics (North-Holland, Amsterdam, 1965) p. 305.

AUGER TRANSITION AND SUPPLEMENTARY ELECTRON IN THE IONIZATION OF ALUMINUM BY E L E C T R O N

EJECTION IMPACT

R. A B O U A F Laboratoire de Collisions Electroniques (associ~ au CNRS), Facult~ des Sciences, 91 -Orsay, France Received 6 May 1971

It is shown that the Auger transitions L2 ~-MM(AI2+), L2,3MMM and L2,3M-MMM(A[3+) are in the AI2 + and Al3 + production., -

p r e d o m i n a n• t

Ionization efficiency curves (ionic current plotted against the electron energy) of multiply charged atomic ions frequently show irregularities such as breaks or displacement of the maxim u m . These structures are generally due to inner shell ionization followed by Auger reorganization [1-5]. K a n e k o ' s r e s u l t s [5] on the Mg2+ i o n i z a t i o n efficiency c u r v e have shown an i m p o r tant b r e a k n e a r 55 eV due to the p r o d u c t i o n of

Mg2+ by the 2 p - 3 s 3 s Auger t r a n s i t i o n [2]. We have investigated the A1z+ c u r v e in o r d e r to show the o c c u r r e n c e of the s a m e t r a n s i t i o n . The 2p shell of A1 l i e s at 7 7 + 2 e V [ 6 ] and the a p p e a r ance p o t e n t i a l s of AI2+ and A1if+ a r e r e s p e c t i v e l y 24.81 and 53.25 e V [ 7 ] , so it i s e n e r g e t i c a l l y p o s sible to produce AI~+ and A13+ i o n s by an Auger p r o c e s s following the c r e a t i o n of a 2p vacancy. Atomic a l i m i n u m vapour i s produced in a 319