Материал: Маслов ИНТРОДУЦТИОН ТО ПХЫСИЦС ОФ СЕЦОНД-ОРДЕР МАГНЕТИЦ 2015

Внимание! Если размещение файла нарушает Ваши авторские права, то обязательно сообщите нам

This is the sum of diagonal elements of 2 ×2 matrix. In other words Q =Tr (AN ). So, knowing the matrix A, we should find the matrix AN

and add up its diagonal elements. Taking into account that the trace of square n-by-n matrix A is Tr (Ak )= ∑λik , where λi are the eigenvalues

of A, the Q can described as the eigenvalues sum, i.e., as the sum of two elements. Let us find them

−

T −λ

= 0;

det

−

−λ

λ2 −λe

−λe

−

+ e T −e−

T = 0;

−

λ2 −λeT

eT + e T

+ e T

−e

= 0;

(23.9)

−

− 2λeT cosh

+ e T

−e

= 0;

cosh2

−

= eT cosh

−1

+ e

1,2

Let us

analyze

the

latter

expression. Since

0 <1−e− T <1

( J > 0,

−e

−4J

0 <T < +∞) and

cosh

≥ 0 , therefore 1

< cosh

. So, the

term

under

the

square

root

positive.

Next,

cosh2

−1+ e−

< cosh2

, therefore the both λ

and λ

are pos-

N 1, for the Q we ob-

itive. Considering that λ1 ≡ λ+

> λ2

≡ λ−

and

tain

Q = λN

+ λN = λN

λ2

(23.10)

≈ λN .

Therefore the Helmholtz free energy is equal to

F = −T ln Q = −TN ln l1 =

−

= −TN ln eT

cosh

−1

+ e

= (23.11)

−

= −NJ −TN ln cosh

cosh

−1+ e

T .

Let us find the magnetization. Recall that the expression for the differ-

ential of free energy is

H = Hez ,

then

M = Mez

= −MdH.

well. So, the magnetization can be described as

M = −

dF = −µB dF .

(23.12)

Nµ2

V dH

Since M

x , see Eq. (12.6), the order parameter takes the form

dF .

x = −

(23.13)

N dh

Considering the Eq. (23.11), let us calculate it

cosh

sinh

TeT

sinh

−

d (ln l )

cosh

−1

+ e

x =T

−4J

cosh

−1+ e T

+ e

−

sinh

cosh

−1

+ e

−

2 h

−1+ e

−

cosh

−1

cosh

Finally, for the order parameter we obtain

sinh

x =

(23.14)

sinh

+ e

−

Let us analyze the resulting expression.

1. Without the interactions between the magnetic moments ( J = 0 )

	h
we can rewrite the formula for the order parameter as	x = tanh	.

	T

This result we already know. The Curie law follows from it.

2. If J ≠ 0 and h →0 the order parameter is equal to zero at any temperature T > 0 .

Therefore the ferromagnetic transition in the one-dimensional Ising model with interactions only between the nearest neighbors is absent! In other words the one-dimensional chain of magnetic moments with interactions only between the nearest neighbors is paramagnet at any temperature T ≠ 0 and exchange integral J ≠ 0 . So, the mean-field theory gives a qualitatively wrong result. The reason is the small number of the nearest neighbors in the chain z = 2 , so, the fluctuations are very high. Finally, note that if we apply long-range exchange potential between magnetic moments in the one-dimensional Ising model, the ferromagnetic phase transition will be occur.

24. Short-range and long-range orders. Correlations. Fluctuations

In the Ising model, the order parameter is defined as a thermodynamic average of projection of the dimensionless atomic magnetic moment

σi = µi µB on z-axis, σiz T (see § 13). In the case of ferromagnetic ordering the value of σiz T ≡ x does not depend on i, vanishing at T > θ and being nonzero at T < θ. As for the average value of the product

σiz σjz T , in the mean field approximation (that is, neglecting fluctua-

							, so that
tions), we had σiz σjz = σiz		σjz	T + σiz	T σjz − σiz T		σjz T	, so that

				0,	T > θ;	(24.1)
σiz σjz		= σiz	σjz	=	, T < θ	(24.1)
	T		T	T x2	, T < θ

for any pair (i, j ≠ i) of magnetic moments.

Let us look on this issue from somewhat different point of view. To gain insight into the behavior of σiz σjz T at T < θ, consider for clarity

sake the limiting case T = 0. In this limit, the state of the system is doubly degenerate (see § 11), all magnetic moments being directed either

“up” or		“down”, so that				for any i and hence
					σiz T = +1 or –1
		= 1 for any (i, j), in accordance with the mean field result,
σiz σjz	T
Eq. (24.1). In terms of				probabilities P↑↑ , P↑↓ ,		P↓↑ , P↓↓ of magnetic
moments		orientations		we	have either P↑↑ =1,	P↓↓ = P↑↓ =P↓↑ = 0 or
P↓↓ =1,	P↑↑ = P↑↓ =P↓↑ = 0 , and the value of
			= (+1)(+1)P↑↑		+ (−1)(−1)P↓↓ + (+1)(−1)P↑↓ + (−1)(+1)P↓↑
σiz σjz
			T

equals to 1 in both cases. The physical reason is that at T < θ the value

of		is not only nonzero but always equals to the value of
	σjz T		σiz T .

If the value of one physical quantity ( σjz T in our case) depends on the value of another physical quantity ( σiz T in our case), those quantities

are said to be correlated. So, in the ferromagnetic state there is an intimate correlation between the orientations of different magnetic mo-

ments. It is worth noting that below the Curie temperature σiz σjz T ≠ 0

even if the distance Ri − R j between i-th and j-th moments greatly exceeds the interatomic spacing. Hence, there is a long-range order in ori-

entation of magnetic moments. Since the value of		for the
	σiz σjz T

nearest moments is also nonzero, there is a short-range order as well. Here is a close analogy with arrangement of atoms in the crystalline lattice.

Let us turn to the case T > θ. Now σiz T = 0 for any i because of equal probabilities of “up” and “down” orientation of every magnetic moment, P↑ = P↓ = 12 (no ferromagnetic ordering). As for a pair of dis-

tant moments, in the absence of long-range order four types of their mutual orientation can occur with equal probabilities 1/4, so that

σiz σjz T = (+1)(+1) 14 + (−1)(−1) 14 + (+1)(−1) 14 + (−1)(+1) 14 = 0

for any (i, j ≠ i), in accordance with the mean field result, Eq.(24.1). It

may appear at first sight that at T > θ there are no correlations between the nearest moments as well. Note however that once in a ferromagnet the exchange energy Jij for the nearest neighbor atoms is positive, it is energetically favorable for two nearest magnetic moments to be oriented in one direction. Hence, we should have

P↑↑ > 14 , P↓↓ > 14 , P↑↓ < 14 , P↓↑ < 14 ,

that is, σiz σjz T ≠ 0 . In other words, one might expect that at T > θ, in

spite of the absence of long-range ferromagnetic order, there are nevertheless short-range ferromagnetic correlations which the mean field approximation fails to describe. Below we shall see that this is so indeed.

25. Correlation function of a ferromagnet in the Ising model

Now our purpose is to study ferromagnetic correlations in the Ising model beyond the mean field approximation. Let us consider two mag-

netic moments, µ1 and µ2 (not necessarily nearest neighbors) situated at atoms “1” and “2” with coordinates R1 and R2 respectively. The

Источник: https://studfile.net/preview/16708691/

Смотрите также:

_1966(~1
_РГР № 1 (2)
"Логистическая технология JIT"
"Особенности правовой системы Российской Федерации" по дисциплине "Теория государства и права"
«Анализ ликвидности и платежеспособности предприятия» (на материалах ОАО Московская кондитерская фабрика «Красный Октябрь»)
«Дизайнерские дети» как профилактика инвалидизации общества
«ЗЕМЕЛЬНЫЕ РЕСУРСЫ КРАСНОДАРСКОГО КРАЯ»
«Методы учета затрат на производство продукции, работ, услуг»Курсовая работа
[2 курс] Определения
00338