Competition of Two Terms in Exchange Hamiltonian for magnetic substances

The unjustifiable or wrong in the previous magnetism theories has been indicated in this paper. For a N electrons system with Heisenberg exchange integral, the correct exchange Hamilton should be of two terms, not only one as in the previous magnetism theories. The role of the minor term in the exchange Hamilton was considered. Based on the principle of superposition of state, the eigenstate of the system with Heisenberg exchange integral, the sum of a positive term and a negative term, and the energy (relative to exchange interaction) eigenvalue were attained. When the positive term is equal to the absolute value of the negative term, the system is in the spin glass state, the probabilities of parallel and antiparallel arrange for every pair of spins of electron of nearest neighbours in the system are equal. When the positive term is not equal to the absolute value of the negative term,the probabilities are not equal, and there coexist the ferromagnetic states and spin glass or antiferromagnetic state and spin glass,when ferromagnetic states and spin glass or antiferromagnetic state and spin glass coexist, the energy of the system is lower than that when only ferromagnetic states or antiferromagnetic state exists as in previous theory. Weiss ferromagnetic state or Neel anti ferromagnetic state is just a special state as the positive term is equal to zero or negative term is equal to zero.

Most of ferromagnetic substances have two magnetic transition temperatures, the paramagnetic Curie temperature and the ferromagnetic Curie temperature. The paramagnetic Curie temperature can be found experimentally according to the Curie-Weiss law and the ferromagnetic Curie temperature can be measured experimentally. The paramagnetic Curie temperature of a substance is higher than the ferromagnetic Curie temperature. All the theories could give the ferromagnetic Curie temperature T c determined by the overall exchange energy A = A 1 + A 2 (A 1 >0，A 2 <0), which unfortunately corresponds to the paramagnetic Curie temperature θ p experimentally measured. It means that theory can not give correct ferromagnetic Curie temperatures of the magnetic substances. This is a big contradiction within the frame of these theories. 2) It was found that the temperature dependence of the reciprocal magnetic susceptibility (i.e., 1/χ ~ T curve) of ferromagnetic materials shows an upward winding near Curie temperature 5,6 . It has been ascribed to the existence of the short -range order. But in practice, the short -range order leads to that the matter is magnetized more easily and thus must result in a downward winding of the 1/χ ~ T curve near the Curie temperature. There must be another reason for this upward winding. The disorder freezing of the spins in the systems of spin glasses has been challenging to all theories in modern physics. It is also hard to find a good theoretical model to explain the re-entrant phenomenon in the spin glasses systems, such , Heisenberg -Dirac exchange Hamiltonian [8] for N localizes spins can denoted by . The first terms of the two formulas which are positive definite favours ferromagnetic coupling, the second terms which are always negative definite favours the antiferromagnetic state. For convenience, in this paper we denote the first term and the second term as A 1 (>0) and A 2 (<0), respectively.
The spontaneous ordering of the spins or the atomic moments originated from the dominant term and minor term in the exchange integrals. But the role of the minor term in the exchange integrals for magnetic matters was not considered in the previous theories of magnetism. The role of the minor term in the exchange integrals was studied in ref. 10. The independent physical role of the two exchange energy A 1 and A 2 was considered, a unique phenomenological theory for ferromagnetism, antiferromagnetism and spin glass was developed. In this phenomenological theory the magnetic behaviors of matters was determined by the competition among the thermal motion, A 1 and A 2 , such as two Curie temperatures in ferromagnetic materials, the freezing of the spin glasses and the re-entrant phenomenon in the spin glasses systems observed in experiments were given. When A 1 = 2 A , A = 0, the matters is in the spin glass state; when A 1 > 2 A ，there is the coexistence of FM and SG; When 2 A >A 1 , there is the coexistence of AFM and SG. Therefore only considering the overall interaction of first term and second term in Heisenberg exchange Hamiltonian is unjustifiable, it is just the reason for the inconsistent of the previous theories with experiments. This object will be studied continuously from the first principle in this paper. In order to consider simultaneously the physical function of the first term and the second term, in this paper we propose a new exchange Hamiltonian involving the first term and the second term instead of the old exchange Hamiltonian only involving the overall interaction.
The remainder of this paper is arranged as follows. The Heisenberg exchange Hamiltonian and quantum state of the electrons system with exchange integral 2 1 A A A + = ( A 1 >0, A 2 <0) will be described in Sec. II and III. The Energy of the electrons system will be calculated in Sec. IV. Section V is the summry. II Heisenberg exchange Hamiltonian of the system with exchange    Heisenberg -Dirac exchange Hamiltonian [8] for N localizes spins can denoted by which is always negative definite favours the antiferromagnetic state; the potential exchange between spins of two different atoms (first term of eqn (2)) which is positive definite favours ferromagnetic coupling, the kinetic exchange term (second term of eqn (2)) which is is always negative definite favours the antiferromagnetic state. For convenience, in this paper we denote the first term and the second term of eqn (1) or (2) as A 1 (>0) and A 2 (<0), respectively.
In the previous magnetism theories，magnetism of substances has been studyed only based on the overall exchange Hamiltonian ) in eqn (2) , but this point of view is unjustifiable! Because the first term of eqn (1) and (2) favour the ferromagnetic coupling, the second term of eqn (1) and (2) favour the antiferromagnetic coupling, the first and the second terms have independent and different physical function respectively, and there is the competition of the two terms. Only considering overall interaction is equivalent to only considering the first term or the second term of eqn (1) and (2). If the overall interaction is positive, only considering overall interaction is equivalent to only considering the function of the first term of eqn (1) and (2); If the overall interaction is negative, only considering overall interaction is equivalent to only considering the function of the second term of eqn (1) and (2). So only considering the overall interaction of first term and second term is not justifiable, it is just the reason for the inconsistent of the previous theories with experiments. In order to consider simultaneously the physical function of the first term and the second term, we propose a new exchange Hamiltonian involving the first term and the second term instead of the old exchange Hamiltonian of the overall interaction. The new exchange Hamiltonian can be expressed as III. Quantum state of the electrons system with exchange integral For the electrons system with Heisenberg exchange integral which has been used in previous magnetism theories and have completely different physical meaning. When A>0, Eq. (3) is just equivalent to the first term in Eq. (4), and when A<0, it is just equivalent to the second term in Eq. 4). Therefore the exchange Hamilton is only correct for this system and it will be used in this paper.
When the spins of all electrons in the system arrange parallelly，the state of the system is expressed with Dirac symbol 1 [11] ，the magnetic moment M of system is an observable, the value is M 1 ；when the spins of all electrons arrange antiparallelly，the state of the system is expressed with Dirac symbol 2 , M is also an observable, the value is M 2 . So 1 and 2 are the eigenstates belonging to the different eigenvalues M 1 and M 2 for the same dynamical variable M. According to Dirac [11] ， 1 and 2 are orthogonal each other. For a system of two electrons [12]  A . So every pair of spins is partly in the quantum state of 1 and partly in the 2 and there only exist the two situations (similar to the situation of a photon passing through a crystal of tourmaline, which polarized obliquely to the the optic axis). Therefore the eigenstate of the system is neither 1 nor 2 , but a linearity superposition of them according to the general principle of quantum mechanics. It can be represented as

IV.
Energy of the electrons system with exchange integral where E is the eigenvalue of ex H , i.e. the energy relative to the exchange energy for the system, The left terms of equal-sign in Eq. (7) give Applying 1 ex H to 2 turns the direction of 2 towards the direction of 1 for an angle β [11] , and applying 2 ex H to 1 turns the direction of 1 towards the direction of 2 for an angle α . Then the ex H 1 = Nz 2 A (sinα 2 + cosα 1 ) The ratio between the absolute values of the weights of 1 in Eq. (9) and the weights of 2 in Eq. (10) must be equal to 1 From Eq. (7) we attain And formula (11) gives It can be seen from Formula (12)