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Ferromagnetic transitions in a short range vector field model
Volume 34A, number 2
PHYSICS LETTERS
8 February 1971
FERROMAGNETIC TRANSITIONS IN A S H O R T R A N G E V E C T O R FIELD MODEL
*t
G. K A L M A N ~ and S. T. LAI $~ Ast~.ophysics I , stitute , Department of Physics, Brandeis Unive~'sity , Waltham, Massachusetts 02154, USA Received 12 January 1971
Hartree-Fock calculations of a short range vector field system show that ferromagnetism is always favored at intermediate densities. For relativistic Fermi energy and strong coupling, it is also favored at high densities.
The motivation of this problem stems from interests not only in statistical mechanics but also in high density astrophysics [1-4]#. W e consider a cold F e r m i gas interacting through a massive vector field [6]; such a system will reflect certain features of crushed nuclear matter. Electrons are assumed captured and are not considered. For a particle of m o m e n t u m k and m a s s m in the system, the Hartree-Fock (HF) energy E(k) is given by E(k)
~2k2
2~g2
=---~-- +
~2
f
dk'
Ik,i
4~g 2
Ik-k'f2+~2 (I)
high so that the location of rain [E(~) 1 m o v e s away f r o m ~ = 0. Thus the condition f o r v a r t i a l m a g n e t i z a t i o n i s aE/a~ I~_-o = a2E/a~2 [~=0 = 0. Total m a g n e t i z a t i o n i s attained when min[E(~)] is l o c a t e d at ~ = I , i.e. aE/a~[~= I = o and
a2s/a~2 [~=1 >
O.
W e do not reproduce here the resulting complicated explicit algebraic expressions; instead w e refer to fig. 1 where the boundaries of partial and total magnetizations are plotted together with the curveE(~ = 1) = 0. Fig. 2 depicts the energy difference between the ferromagnetic and the normal states for the various regions defined in fig. 1. Ferromagnetism is favored at intermediate densities because in the low density regime where ~t"1 is small compared to the inter-
w h e r e ~t-1 is the r a n g e of the field, and g is the coupling constant. The H a r t r e e t e r m is spin i n dependent and h en ce n e e d not be c o n s i d e r e d . The exchange t e r m is always n e g a t i v e and is r e s p o n s i b l e f o r spin a l i g n m e n t and hence f o r f e r r o m a g n e t i s m [8]. If the spin populations N± a r e unequal, the s y s t e m is f e r r o m a g n e t i c with m a g n e t i z a t i o n ~ = (AT+ - N_)/(N++N). B y comparing
the average energy E(~) per particle for ~ = 0 and ~ ~ 0, the lowest energy state at a given density can be determined. Starting with low densities the lowest energy state is at ~ = 0 until the density is sufficiently * Supported by Contract AFCRL-19628-C-0074. Based on part of a Ph.D. thesis (Brandeis University, (1970) submitted by S. T. LaL Now at Physics Department, Boston College, Chestnut Hill, Massachusetts 02167. $$ Now at Physics Department, University of Windsor, WIndsor U , Ontario, Canada. # See for news of discovery of the first intense magnetic white dwarf, ref. [5].
1,0
I.~
2.0
2.5
X
Flg._l. Magnetization boundaries plotted against a --=2mgz/~ ~rhz and x ---2kF/~t. In Va and VHa the energy curves E(~) have two extrema so that our boundary c r l terla [eqs. (2) and (3).] do not apply. These regions are, however, very small (the maximum width Ax < 0.01) and the dotted boundaries are exaggerated and not to scale.
75
Volume 34A, number 2
I
PHYSICS L E T T E R S
°
Fig. 2. Energy difference E (~) -E (0) plotted against magnetization ~ for various regions defined in fig. 1. p a r t i c l e distance d, the s y s t e m b e h a v e s like one with 5 - f u n c t i o n i n t e r a c t i o n while at high d e n s i t i e s where p-1 >> d the s y s t e m r e s e m b l e s a Coulomb s y s t e m . Continuous phase t r a n s i t i o n of the second kind o c c u r s along the b o u n d a r i e s II-IV, while the (demagnetization) t r a n s i t i o n at high d e n s i t y is abrupt and is of the f i r s t kind. At e x t r e m e l y high d e n s i t i e s the F e r m i energy b e c o m e s c o m p a r a b l e to the r e s t energy and a l though the potential and plane wave functions r e m a i n unchanged, the kinetic energy b e c o m e s [(~kc) 2 + m2c4] 1/2. With this modification, the k i n e t i c e n e r g y has the s a m e density dependence as the exchange e n e r g y and it can no longer c o m p ~ e with the l a t t e r , provided the p a r a m e t e r V (gZ//~crr) > 1 (in p a r t i c u l a r , if it c o r r e s p o n d s t o that of s t r o n g i n t e r a c t i o n : V ~ 5). Hence, f e r r o m a g n e t i s m is favored throughout the e n t i r e high d e n s i t y r e g i m e . F o r V < 1, however, t h e r e is no s i g n i f i c a n t change in the s t r u c t u r e of fig. 1. Actual f e r m i o n s a r e endowed with an e l e c t r o -
76
8 February 1971
magnetic dipole m o m e n t which g e n e r a t e s a m a g netic field: the r e s u l t i n g magnetic dipole i n t e r action [8] r e p r e s e n t s (in the H a r t r e e a p p r o x i m a tion) an additional energy t e r m - 4 ~ n e 2 ~ 2 / 3 m2~. This t e r m can compete with the exchanj~e t e r m only at e x t r e m e l y high d e n s i t i e s (~ 1 0 2 1 g / c m 3 ) . Although d y n a m i c a l c o r r e l a t i o n s and c o r r e l a tion induced phase t r a n s i t i o n s a r e obviously absent in the HF model, a HF type g a s - l i q u i d phase t r a n s i t i o n , exists [9] in s c r e e n e d or u n s c r e e n e d Coulomb s y s t e m s , in the p r e s e n t case, the p r e s ence of a H a r t r e e t e r m m a i n t a i n s dp/dnpositive, both in the n o r m a l and f e r r o m a g n e t i c state, signifying that no g a s - l i q u i d phase t r a n s i t i o n s u p e r s e d e s the f e r r o m a g n e t i c t r a n s i t i o n s . S. T. Lai is grateful to P r o f e s s o r G. Szamosi for postdoctoral r e s e a r c h support at the U n i v e r sity of Windsor.
References [1] V. Canuto and H.-Y. Chiu, Phys. Rev. 173 (1968} 1210, 1220, 1229. [2] D. H. Bronwell and J. Callaway, Nuovo Cimento 60B {1969) 169. [3] S. D. Silverstein, Phys. Rev. Letters 23 (1969) 139. [4] J. M. Pearson and G. Saunier, Phys. Hev. Letters 24 (1969) 325. [5] J. C. Kemp et al., Ap. J. Letters 161 (1970) L77 and Time Magazine, Sept. 14 (1970). [6] Ya. B. Zeldovich, Soviet Physics JETP 14 (1962} 1143; G. Kalman, Phys. Rev. 158 (1967) 144; S. A. Bludman and M. A. Ruderman, Phys. Rev. 170 (1968) 1176; M. A. Ruderman, Phys. Rev. 172 (1968) 1286. [7] C. Herring, Magnetism, Vol.. 4, eds. G. T. Rado and H. Suhl (Academic Press, New York, 1966). [8] H. Bethe and E. E. Salpeter, Quantum mechanics of one and two electron atoms (Springer Verlag, Berlin, 1957). [9] S. Gartenhaus and G. Stranahan, Phys. Rev. Letters 14 (1965} 341; Errata 14 (1965) 621.
PHYSICS LETTERS
8 February 1971
FERROMAGNETIC TRANSITIONS IN A S H O R T R A N G E V E C T O R FIELD MODEL
*t
G. K A L M A N ~ and S. T. LAI $~ Ast~.ophysics I , stitute , Department of Physics, Brandeis Unive~'sity , Waltham, Massachusetts 02154, USA Received 12 January 1971
Hartree-Fock calculations of a short range vector field system show that ferromagnetism is always favored at intermediate densities. For relativistic Fermi energy and strong coupling, it is also favored at high densities.
The motivation of this problem stems from interests not only in statistical mechanics but also in high density astrophysics [1-4]#. W e consider a cold F e r m i gas interacting through a massive vector field [6]; such a system will reflect certain features of crushed nuclear matter. Electrons are assumed captured and are not considered. For a particle of m o m e n t u m k and m a s s m in the system, the Hartree-Fock (HF) energy E(k) is given by E(k)
~2k2
2~g2
=---~-- +
~2
f
dk'
Ik,i
4~g 2
Ik-k'f2+~2 (I)
high so that the location of rain [E(~) 1 m o v e s away f r o m ~ = 0. Thus the condition f o r v a r t i a l m a g n e t i z a t i o n i s aE/a~ I~_-o = a2E/a~2 [~=0 = 0. Total m a g n e t i z a t i o n i s attained when min[E(~)] is l o c a t e d at ~ = I , i.e. aE/a~[~= I = o and
a2s/a~2 [~=1 >
O.
W e do not reproduce here the resulting complicated explicit algebraic expressions; instead w e refer to fig. 1 where the boundaries of partial and total magnetizations are plotted together with the curveE(~ = 1) = 0. Fig. 2 depicts the energy difference between the ferromagnetic and the normal states for the various regions defined in fig. 1. Ferromagnetism is favored at intermediate densities because in the low density regime where ~t"1 is small compared to the inter-
w h e r e ~t-1 is the r a n g e of the field, and g is the coupling constant. The H a r t r e e t e r m is spin i n dependent and h en ce n e e d not be c o n s i d e r e d . The exchange t e r m is always n e g a t i v e and is r e s p o n s i b l e f o r spin a l i g n m e n t and hence f o r f e r r o m a g n e t i s m [8]. If the spin populations N± a r e unequal, the s y s t e m is f e r r o m a g n e t i c with m a g n e t i z a t i o n ~ = (AT+ - N_)/(N++N). B y comparing
the average energy E(~) per particle for ~ = 0 and ~ ~ 0, the lowest energy state at a given density can be determined. Starting with low densities the lowest energy state is at ~ = 0 until the density is sufficiently * Supported by Contract AFCRL-19628-C-0074. Based on part of a Ph.D. thesis (Brandeis University, (1970) submitted by S. T. LaL Now at Physics Department, Boston College, Chestnut Hill, Massachusetts 02167. $$ Now at Physics Department, University of Windsor, WIndsor U , Ontario, Canada. # See for news of discovery of the first intense magnetic white dwarf, ref. [5].
1,0
I.~
2.0
2.5
X
Flg._l. Magnetization boundaries plotted against a --=2mgz/~ ~rhz and x ---2kF/~t. In Va and VHa the energy curves E(~) have two extrema so that our boundary c r l terla [eqs. (2) and (3).] do not apply. These regions are, however, very small (the maximum width Ax < 0.01) and the dotted boundaries are exaggerated and not to scale.
75
Volume 34A, number 2
I
PHYSICS L E T T E R S
°
Fig. 2. Energy difference E (~) -E (0) plotted against magnetization ~ for various regions defined in fig. 1. p a r t i c l e distance d, the s y s t e m b e h a v e s like one with 5 - f u n c t i o n i n t e r a c t i o n while at high d e n s i t i e s where p-1 >> d the s y s t e m r e s e m b l e s a Coulomb s y s t e m . Continuous phase t r a n s i t i o n of the second kind o c c u r s along the b o u n d a r i e s II-IV, while the (demagnetization) t r a n s i t i o n at high d e n s i t y is abrupt and is of the f i r s t kind. At e x t r e m e l y high d e n s i t i e s the F e r m i energy b e c o m e s c o m p a r a b l e to the r e s t energy and a l though the potential and plane wave functions r e m a i n unchanged, the kinetic energy b e c o m e s [(~kc) 2 + m2c4] 1/2. With this modification, the k i n e t i c e n e r g y has the s a m e density dependence as the exchange e n e r g y and it can no longer c o m p ~ e with the l a t t e r , provided the p a r a m e t e r V (gZ//~crr) > 1 (in p a r t i c u l a r , if it c o r r e s p o n d s t o that of s t r o n g i n t e r a c t i o n : V ~ 5). Hence, f e r r o m a g n e t i s m is favored throughout the e n t i r e high d e n s i t y r e g i m e . F o r V < 1, however, t h e r e is no s i g n i f i c a n t change in the s t r u c t u r e of fig. 1. Actual f e r m i o n s a r e endowed with an e l e c t r o -
76
8 February 1971
magnetic dipole m o m e n t which g e n e r a t e s a m a g netic field: the r e s u l t i n g magnetic dipole i n t e r action [8] r e p r e s e n t s (in the H a r t r e e a p p r o x i m a tion) an additional energy t e r m - 4 ~ n e 2 ~ 2 / 3 m2~. This t e r m can compete with the exchanj~e t e r m only at e x t r e m e l y high d e n s i t i e s (~ 1 0 2 1 g / c m 3 ) . Although d y n a m i c a l c o r r e l a t i o n s and c o r r e l a tion induced phase t r a n s i t i o n s a r e obviously absent in the HF model, a HF type g a s - l i q u i d phase t r a n s i t i o n , exists [9] in s c r e e n e d or u n s c r e e n e d Coulomb s y s t e m s , in the p r e s e n t case, the p r e s ence of a H a r t r e e t e r m m a i n t a i n s dp/dnpositive, both in the n o r m a l and f e r r o m a g n e t i c state, signifying that no g a s - l i q u i d phase t r a n s i t i o n s u p e r s e d e s the f e r r o m a g n e t i c t r a n s i t i o n s . S. T. Lai is grateful to P r o f e s s o r G. Szamosi for postdoctoral r e s e a r c h support at the U n i v e r sity of Windsor.
References [1] V. Canuto and H.-Y. Chiu, Phys. Rev. 173 (1968} 1210, 1220, 1229. [2] D. H. Bronwell and J. Callaway, Nuovo Cimento 60B {1969) 169. [3] S. D. Silverstein, Phys. Rev. Letters 23 (1969) 139. [4] J. M. Pearson and G. Saunier, Phys. Hev. Letters 24 (1969) 325. [5] J. C. Kemp et al., Ap. J. Letters 161 (1970) L77 and Time Magazine, Sept. 14 (1970). [6] Ya. B. Zeldovich, Soviet Physics JETP 14 (1962} 1143; G. Kalman, Phys. Rev. 158 (1967) 144; S. A. Bludman and M. A. Ruderman, Phys. Rev. 170 (1968) 1176; M. A. Ruderman, Phys. Rev. 172 (1968) 1286. [7] C. Herring, Magnetism, Vol.. 4, eds. G. T. Rado and H. Suhl (Academic Press, New York, 1966). [8] H. Bethe and E. E. Salpeter, Quantum mechanics of one and two electron atoms (Springer Verlag, Berlin, 1957). [9] S. Gartenhaus and G. Stranahan, Phys. Rev. Letters 14 (1965} 341; Errata 14 (1965) 621.