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A remark on localised states in metals
PHYSICS
Volume 6, number 3
A REMARK
ON LOCALISED
15 September 1963
LETTERS
STATES
IN
METALS
*
Takeo IZUYAMA ** Materials
Theory Group, Department of Electrical Engineering, Massachusetts Institute of Technology, Cambridge 39, Massachusetts Received 26 August 1963
The origin of the localised moments I) observed in metals has been extensive1 discussed by Anderson 2), Wolf 2) and Clogston %). The latter two authors treated the problem by introducing an impurity potential at the position of Fe into the Hartree-Fock potential of conduction electrons, while Anderson assumed the presence of an additional wave function at the impurity atom. One of the main results of Anderson’s theory is that the Pauli susceptibility of such a system in its non-magnetic state shows an anomalous behaviour when the non-magnetic system is near its transition to the magnetic state. The purpose of this note is to show that this result can be confirmed under quite general circumstances ; it does not depend on the speciai model nor the Hartree-Fock approximation. The ground state energy E(n+,n_) of the system is a function of n+ and n_, respectively, the average occupation numbers of up spin localised electrons and of down spin electrons. E(n+,n_) p @(n+,n_), x\t(n+,n_)) should be understood as the minimum energy of the system containing an impurity under the following subsidiary conditions : Wn+?L),
n+ Q(~+P_))
= n+ ,
(an+,=_),
n_ Mn+,n_))
=n_ ,
where ~0 is the number operator of the localised state with u spin, and 3~is the total Hamiltonian of the system. Instead of using the variables n, and n_, we use 6 = n++n_ and r] = n+- n_. The energy depends parametrically on the Coulomb energy U between localised electrons in the same shell, the s-d admixture integral V, and other fundamental quantities which characterise the system. As a function of n, the energy in the non-magnetic case will have a form such as indicated in fig. 1, where 5 is fixed at the value 5, determined by @E/2 ,$) rl=o = 0. In the magnetic case the energy is something like that indicated in fig. 2, where .$ and ?I are determined by
$
(5, 7) = $f (E, 7) = 0,
Fig. 2
Fig. 1
Neglecting the interaction between impurities, total energy may be expressed as
the
E=Eo+Nif(S;U,V,...)112+Nig(5;U,V;...)~4,
where Ni is the number of impurities and g is (1) always positive. We are considering here the neighbourhood of the critical point where ri appears. Therefore, higher order terms with respect to 7) than n4 are unnecessary in the expression (1). When the parameters U, V, . . . are in the non-magnetic region, f is positive at 5 = E_o. On the other hand, when the parameters are in the magnetic region, f should be negative at least at 5 = 5. Thus, the critical point at the transition from non-magnetic to magnetic situation is determined by fko;
Uc,Vc,
.-*)=O
’
(2)
Suppose that an external magnetic field His applied to the system with the parameters in their nonmagnetic region. The energy in this case is E@)
= E - -$i g/+ilq
- M,H
,
where MC is the magnetisation of the conduction electrons. MC and ?Jmay be regarded as independent variables. The induced moment of a localised state is denoted by 7~. Then the total susceptibility at T = 0 is given by X = XP + NiVH/H 9 where Xp is the Pauli susceptibility tion electrons. In the non-magnetic
of the conduccase, the in-
etc.,
i.e., they give the minimum energy at fixed values of the parameters U, V, . . .
* This work was supported by the O.N.R. and A.R.P.A. ** On leave from the Department of Physics, Nagoya University, Nagoya, Japan. 245
Volume 6, number 3 duced moment, small quantity. to H. Then we qHdetermined
PHYSICS
LETTERS
VH, is proportiona. to H and is a Similarly, 5 - to is also proportional may safely replace 5 inf by 5,. Thus by (&!?(H)/~Q) = 0 is __ ‘H-4f(eo;
15 September 1963
Finally, I would like to express my sincere thanks to Professor G. W. Pratt, Jr. for reading the manuscript.
Therefore, as the parameters U, V, . . . approach to the critical values, ~H/H becomes extremely large on account of the condition (2). Consequently X becomes anomalously large in this case. (Unfortunately, the above result can not in any way be related to the observed giant magnetic moment at Fe in the alloys in the neighbourhood of Pd. If the giant magnetic moment itself is proportional
ENERGY
__~-
to H, the macroscopic magnetisation induced by H in the parama netic region of temperature is proportional to H 5 and, accordingly, is very small. )
gClBH u, v,. . .)’
*‘*
____
References 1) A.M. Clogston et al., Phys. Rev. 125 (1962) 541. Other references to experimental work can be found in this paper. 2) P. W. Anderson, Phys. Rev. 124 (1961) 41.1. 3) P.A.Wolf, Phys.Rev. 124 (1961) 1030. 4) A. M.Clogston, Phys.Rev. 125 (1962) 439. *
*
*
DENSITY IN A DISPERSIVE MEDIUM FORMULATED IN CONSTITUTIVE AND “ELECTROMAGNETIC” PARAMETERS
TERMS
OF
J. NEUFELD Cak Ridge National Laboratory,
Oak Ridge,
Tennessee
*
Received 23 August 1963 Extensive literature published within recent years deals with various distinctions between freouencv dispersive and space dispersive media 1). It is shown in this note that by using a modified phenomenological formulition one can eliminate these distinctions and provide a unified treatment which is applicable to both types of dispersive media. This unified treatment leads to a new formulation for the energy density in a space dispersive medium. Consider an electromagnetic field interacting with a non-magnetic dispersive medium. The phenomenological behaviour of such a field can be described by Maxwell’s equations :
[k x Q, J =&, relating the field
w; [k x ‘&I = - ;+,
w,
(1)
quantities Ek, w (electric field intensity), Bk w (magnetic induction), nk, w (magnetic intensity), &, w (electric displacement) and by two constitutive equations : Dk,w
=aE
k,o’
* Hk w =Bk
w,
(2)
which are determined by thi structire of the dispersive medium. The parameter P represents the dielectric constant and subscripts k and o in the expressions such as Dk, W, Ek, +,, etc. indicate that each of these field quantities 1s a function of the wave vector k and frequency w. The relationships 246
(2) are based on the expressions for the Lorentz force and Newton’s second law of motion. In deriving these relationships, Maxwell’s equations are now used 2). A medium described by (1) and (2) may be frequency dispersive or space dispersive. The distinction between these two media is based on the formulation of the “constitutive dielectric parameter” ( defined by (2). Closer discussion of equations (2) shows that in a frequency dispersive medium the parameter ( is a function of w and independent of k, i.e., f s Ed. On the other hand, in a space dispersive medium the parameter F is a function of both k and w, i.e., ( q ‘k w. Combining the constitutive equations (2) with I&xwell’s equations (l), pne obtains the dispersion relationship F(w, k) = 0 which yields expressions of the type k = k,ts) in which k appears as an explicit function of W. These expressions describe various oscillatory modes in the dispersive medium. (The symbol “s” enumerates these modes. ) In this investigation new macroscopic parameters are introduced which are designated as “elecAn electromagnetic parameter for tromagnetic”. any selected oscillatory mode can be obtained from the corresponding constitutive parameter by sub* Operated by Union Carbide Corporation for the U.S Atomic Energy Commission.
Volume 6, number 3
A REMARK
ON LOCALISED
15 September 1963
LETTERS
STATES
IN
METALS
*
Takeo IZUYAMA ** Materials
Theory Group, Department of Electrical Engineering, Massachusetts Institute of Technology, Cambridge 39, Massachusetts Received 26 August 1963
The origin of the localised moments I) observed in metals has been extensive1 discussed by Anderson 2), Wolf 2) and Clogston %). The latter two authors treated the problem by introducing an impurity potential at the position of Fe into the Hartree-Fock potential of conduction electrons, while Anderson assumed the presence of an additional wave function at the impurity atom. One of the main results of Anderson’s theory is that the Pauli susceptibility of such a system in its non-magnetic state shows an anomalous behaviour when the non-magnetic system is near its transition to the magnetic state. The purpose of this note is to show that this result can be confirmed under quite general circumstances ; it does not depend on the speciai model nor the Hartree-Fock approximation. The ground state energy E(n+,n_) of the system is a function of n+ and n_, respectively, the average occupation numbers of up spin localised electrons and of down spin electrons. E(n+,n_) p @(n+,n_), x\t(n+,n_)) should be understood as the minimum energy of the system containing an impurity under the following subsidiary conditions : Wn+?L),
n+ Q(~+P_))
= n+ ,
(an+,=_),
n_ Mn+,n_))
=n_ ,
where ~0 is the number operator of the localised state with u spin, and 3~is the total Hamiltonian of the system. Instead of using the variables n, and n_, we use 6 = n++n_ and r] = n+- n_. The energy depends parametrically on the Coulomb energy U between localised electrons in the same shell, the s-d admixture integral V, and other fundamental quantities which characterise the system. As a function of n, the energy in the non-magnetic case will have a form such as indicated in fig. 1, where 5 is fixed at the value 5, determined by @E/2 ,$) rl=o = 0. In the magnetic case the energy is something like that indicated in fig. 2, where .$ and ?I are determined by
$
(5, 7) = $f (E, 7) = 0,
Fig. 2
Fig. 1
Neglecting the interaction between impurities, total energy may be expressed as
the
E=Eo+Nif(S;U,V,...)112+Nig(5;U,V;...)~4,
where Ni is the number of impurities and g is (1) always positive. We are considering here the neighbourhood of the critical point where ri appears. Therefore, higher order terms with respect to 7) than n4 are unnecessary in the expression (1). When the parameters U, V, . . . are in the non-magnetic region, f is positive at 5 = E_o. On the other hand, when the parameters are in the magnetic region, f should be negative at least at 5 = 5. Thus, the critical point at the transition from non-magnetic to magnetic situation is determined by fko;
Uc,Vc,
.-*)=O
’
(2)
Suppose that an external magnetic field His applied to the system with the parameters in their nonmagnetic region. The energy in this case is E@)
= E - -$i g/+ilq
- M,H
,
where MC is the magnetisation of the conduction electrons. MC and ?Jmay be regarded as independent variables. The induced moment of a localised state is denoted by 7~. Then the total susceptibility at T = 0 is given by X = XP + NiVH/H 9 where Xp is the Pauli susceptibility tion electrons. In the non-magnetic
of the conduccase, the in-
etc.,
i.e., they give the minimum energy at fixed values of the parameters U, V, . . .
* This work was supported by the O.N.R. and A.R.P.A. ** On leave from the Department of Physics, Nagoya University, Nagoya, Japan. 245
Volume 6, number 3 duced moment, small quantity. to H. Then we qHdetermined
PHYSICS
LETTERS
VH, is proportiona. to H and is a Similarly, 5 - to is also proportional may safely replace 5 inf by 5,. Thus by (&!?(H)/~Q) = 0 is __ ‘H-4f(eo;
15 September 1963
Finally, I would like to express my sincere thanks to Professor G. W. Pratt, Jr. for reading the manuscript.
Therefore, as the parameters U, V, . . . approach to the critical values, ~H/H becomes extremely large on account of the condition (2). Consequently X becomes anomalously large in this case. (Unfortunately, the above result can not in any way be related to the observed giant magnetic moment at Fe in the alloys in the neighbourhood of Pd. If the giant magnetic moment itself is proportional
ENERGY
__~-
to H, the macroscopic magnetisation induced by H in the parama netic region of temperature is proportional to H 5 and, accordingly, is very small. )
gClBH u, v,. . .)’
*‘*
____
References 1) A.M. Clogston et al., Phys. Rev. 125 (1962) 541. Other references to experimental work can be found in this paper. 2) P. W. Anderson, Phys. Rev. 124 (1961) 41.1. 3) P.A.Wolf, Phys.Rev. 124 (1961) 1030. 4) A. M.Clogston, Phys.Rev. 125 (1962) 439. *
*
*
DENSITY IN A DISPERSIVE MEDIUM FORMULATED IN CONSTITUTIVE AND “ELECTROMAGNETIC” PARAMETERS
TERMS
OF
J. NEUFELD Cak Ridge National Laboratory,
Oak Ridge,
Tennessee
*
Received 23 August 1963 Extensive literature published within recent years deals with various distinctions between freouencv dispersive and space dispersive media 1). It is shown in this note that by using a modified phenomenological formulition one can eliminate these distinctions and provide a unified treatment which is applicable to both types of dispersive media. This unified treatment leads to a new formulation for the energy density in a space dispersive medium. Consider an electromagnetic field interacting with a non-magnetic dispersive medium. The phenomenological behaviour of such a field can be described by Maxwell’s equations :
[k x Q, J =&, relating the field
w; [k x ‘&I = - ;+,
w,
(1)
quantities Ek, w (electric field intensity), Bk w (magnetic induction), nk, w (magnetic intensity), &, w (electric displacement) and by two constitutive equations : Dk,w
=aE
k,o’
* Hk w =Bk
w,
(2)
which are determined by thi structire of the dispersive medium. The parameter P represents the dielectric constant and subscripts k and o in the expressions such as Dk, W, Ek, +,, etc. indicate that each of these field quantities 1s a function of the wave vector k and frequency w. The relationships 246
(2) are based on the expressions for the Lorentz force and Newton’s second law of motion. In deriving these relationships, Maxwell’s equations are now used 2). A medium described by (1) and (2) may be frequency dispersive or space dispersive. The distinction between these two media is based on the formulation of the “constitutive dielectric parameter” ( defined by (2). Closer discussion of equations (2) shows that in a frequency dispersive medium the parameter ( is a function of w and independent of k, i.e., f s Ed. On the other hand, in a space dispersive medium the parameter F is a function of both k and w, i.e., ( q ‘k w. Combining the constitutive equations (2) with I&xwell’s equations (l), pne obtains the dispersion relationship F(w, k) = 0 which yields expressions of the type k = k,ts) in which k appears as an explicit function of W. These expressions describe various oscillatory modes in the dispersive medium. (The symbol “s” enumerates these modes. ) In this investigation new macroscopic parameters are introduced which are designated as “elecAn electromagnetic parameter for tromagnetic”. any selected oscillatory mode can be obtained from the corresponding constitutive parameter by sub* Operated by Union Carbide Corporation for the U.S Atomic Energy Commission.