ising model in the zeroth approximation done by ghassan m. masa’deh
DESCRIPTION
Ising model in the zeroth approximation Done by Ghassan M. Masa’deh. Introduction:. In principle the Ising model is not a very good approximation for any temperature range. However it has the advantage of starting directly from the energy levels. - PowerPoint PPT PresentationTRANSCRIPT
![Page 1: Ising model in the zeroth approximation Done by Ghassan M. Masa’deh](https://reader036.vdocuments.us/reader036/viewer/2022062520/568159fb550346895dc74794/html5/thumbnails/1.jpg)
Ising model in the Ising model in the zeroth approximationzeroth approximation
Done by Done by Ghassan M. Masa’dehGhassan M. Masa’deh
![Page 2: Ising model in the zeroth approximation Done by Ghassan M. Masa’deh](https://reader036.vdocuments.us/reader036/viewer/2022062520/568159fb550346895dc74794/html5/thumbnails/2.jpg)
Introduction:In principle the Ising model is not a very good approximation for any temperature range. However it has the advantage of starting directly from the energy levels
and skipping all the steps that lead to
them from the Hamiltonian, in other methods .
![Page 3: Ising model in the zeroth approximation Done by Ghassan M. Masa’deh](https://reader036.vdocuments.us/reader036/viewer/2022062520/568159fb550346895dc74794/html5/thumbnails/3.jpg)
This convenient short cut makes it possible to concentrate on the details of the statistical
mechanics .
![Page 4: Ising model in the zeroth approximation Done by Ghassan M. Masa’deh](https://reader036.vdocuments.us/reader036/viewer/2022062520/568159fb550346895dc74794/html5/thumbnails/4.jpg)
Therefore, the ising model is very widely used in a variety of other problems, more than in ferromagnetism for which it was originally developed .
![Page 5: Ising model in the zeroth approximation Done by Ghassan M. Masa’deh](https://reader036.vdocuments.us/reader036/viewer/2022062520/568159fb550346895dc74794/html5/thumbnails/5.jpg)
The Ising model
In 1928 Gorsky attempted a statistical study of order - disorder transitions in binary alloys on the basis of the assumption that the work expended in transferring an atom from an order position to a disordered one is directly proportional to the degree of order prevailing in the system
![Page 6: Ising model in the zeroth approximation Done by Ghassan M. Masa’deh](https://reader036.vdocuments.us/reader036/viewer/2022062520/568159fb550346895dc74794/html5/thumbnails/6.jpg)
This idea was further developed by Bragg and Williams, who, for the first time introduced the concept of long range order
![Page 7: Ising model in the zeroth approximation Done by Ghassan M. Masa’deh](https://reader036.vdocuments.us/reader036/viewer/2022062520/568159fb550346895dc74794/html5/thumbnails/7.jpg)
The basic assumption in the Bragg-Williams approximation is " the energy of an individual atom in the given system is determined by the average degree of order prevailing in the entire system rather than by the fluctuating
configurations of the neighboring atoms. ".
![Page 8: Ising model in the zeroth approximation Done by Ghassan M. Masa’deh](https://reader036.vdocuments.us/reader036/viewer/2022062520/568159fb550346895dc74794/html5/thumbnails/8.jpg)
Define: along range parameter 'L' is given by:
= N+ - N-/N …….(1) -1<L<1
Where: σi = +1 for an up spin - = 1 for adown spin
N+ = total number of up spins N- = total number of down spins
L
iNL /1
![Page 9: Ising model in the zeroth approximation Done by Ghassan M. Masa’deh](https://reader036.vdocuments.us/reader036/viewer/2022062520/568159fb550346895dc74794/html5/thumbnails/9.jpg)
Where N = N+ + N-
Substitute in (1) we get
L = (2N+/N) -1
So, N+=N/2(1+L)
And
N-= N/2 (1-L)
![Page 10: Ising model in the zeroth approximation Done by Ghassan M. Masa’deh](https://reader036.vdocuments.us/reader036/viewer/2022062520/568159fb550346895dc74794/html5/thumbnails/10.jpg)
The magnetization M is given by :
M = (N+- N-)µ =µ NL ; -Nµ<M<Nµ
For Ising model we can write the Hamiltonian by:
H{σi} = -J ∑σI σ j - µB∑σi ………………(2)
Where : - µB is the potential energy - Jσi σj is the kinetic energy
![Page 11: Ising model in the zeroth approximation Done by Ghassan M. Masa’deh](https://reader036.vdocuments.us/reader036/viewer/2022062520/568159fb550346895dc74794/html5/thumbnails/11.jpg)
H{σi} = -J (1/2 q σ) ∑σi - µB∑σi ……………(3)
We can find the total configurational energy of the system is given by
E= -1/2 (qJL) NL –( µB) NL ……….(4)
And
<E< = U = -1/2 qJNL - µBNL………..(5)2
![Page 12: Ising model in the zeroth approximation Done by Ghassan M. Masa’deh](https://reader036.vdocuments.us/reader036/viewer/2022062520/568159fb550346895dc74794/html5/thumbnails/12.jpg)
Define: ∆ɛ is the difference between the over all configurational energy of an up spin and the over all configurational energy of down spin , the energy expended in changing any up spin into a down one is
given by :
∆ɛ = -J(qσ) ∆σ - µB∆σ ……………..(6)
= 2µ (qJσ/(µ+B) ) ; ∆σ= -2……….(7)
![Page 13: Ising model in the zeroth approximation Done by Ghassan M. Masa’deh](https://reader036.vdocuments.us/reader036/viewer/2022062520/568159fb550346895dc74794/html5/thumbnails/13.jpg)
The relative values of the equilibrium numbers N+ and N- then follow from the Boltzmann
principle: N-/N+ = exp (-∆ɛ/KT)
= exp (-2µ(B'+B)/KT) ………(8)
Where B' the internal molecular field and given by:
B' = qJM/Nµ2
![Page 14: Ising model in the zeroth approximation Done by Ghassan M. Masa’deh](https://reader036.vdocuments.us/reader036/viewer/2022062520/568159fb550346895dc74794/html5/thumbnails/14.jpg)
)1-L)/(1+L = (exp [-2 (qJL+B)/KT] ……..(11)
)qJL+B /(KT =1/2 ln [(1+L)/(1-L)]
= tanh L ………(12)
L= tanh [(qJL+B)/KT ] ………………(13)
Let B =0 =< L0 = tanh [(qJL0)/KT] ……..(14)
This is called the possibility of spontaneous magnetization
-1
![Page 15: Ising model in the zeroth approximation Done by Ghassan M. Masa’deh](https://reader036.vdocuments.us/reader036/viewer/2022062520/568159fb550346895dc74794/html5/thumbnails/15.jpg)
We obtain a Tc below which the system can acquire a non zero spontaneous
magnetization and above which it can not. We can identify the Tc with the Curi temperature. The temperature that marks a transition from the ferromagnetic to the paramagnetic behavior of the system or
vice verse .
![Page 16: Ising model in the zeroth approximation Done by Ghassan M. Masa’deh](https://reader036.vdocuments.us/reader036/viewer/2022062520/568159fb550346895dc74794/html5/thumbnails/16.jpg)
From equation (14) :
L0(T) {3(1-T/Tc)} ;(T T≃ ≲ c,B→0)…(15)
At T→0 =< L0 → 1
L0(T) 1-2 exp(-2T≃ c/T) ; (T/Tc 1)….(16)≪
1/2
![Page 17: Ising model in the zeroth approximation Done by Ghassan M. Masa’deh](https://reader036.vdocuments.us/reader036/viewer/2022062520/568159fb550346895dc74794/html5/thumbnails/17.jpg)
The configrational energy of the system is given by:
U0(T) = -1/2 qJNL0 ……………(17)
And the specific heat is
C0(T) = -qJNL0 dL0/dT) = NKL0)] / (T/Tc)/(1- L0) - (T/Tc)... [(18(
At T<Tc =< U0(T) = C0(T) = 0`
2
2 2 2
![Page 18: Ising model in the zeroth approximation Done by Ghassan M. Masa’deh](https://reader036.vdocuments.us/reader036/viewer/2022062520/568159fb550346895dc74794/html5/thumbnails/18.jpg)
The specific heat at the transition temperature Tc is:
C0(T) = lim {(NK* 3x)/[[(1-x)/(1-3x)] – (1-x)]} = 3/2 NK ……(19) And at T→ 0
C0(T) 4NK (T≃ c/T) exp(-2Tc/T) (20)
2
2
![Page 19: Ising model in the zeroth approximation Done by Ghassan M. Masa’deh](https://reader036.vdocuments.us/reader036/viewer/2022062520/568159fb550346895dc74794/html5/thumbnails/19.jpg)
Note that the vanishing of the configurational energy and the specific heat of the system at temperature above Tc is directly related to the configurational order prevailing in the system at lower temperatures is completely wiped out
as T→ Tc.
We note that all the microstate are equally likely to occur is related to the fact that for T ≥ Tc there is no configurational order in the system
![Page 20: Ising model in the zeroth approximation Done by Ghassan M. Masa’deh](https://reader036.vdocuments.us/reader036/viewer/2022062520/568159fb550346895dc74794/html5/thumbnails/20.jpg)
Define : X is the magnetic susceptibility of the system
and given by: X(B,T) = (dM/dT)T = Nµ(dL/dB)T
)= Nµ /K)](1-L(B,T)]/(T-Tc{1-L(B,T)} [[ For L 1 we obtain the Curi – weiss law≪ X0(T) (Nµ /K)/ (T-T≃ c) (T T≳ c ,B→ 0 )
2 2 2
2
![Page 21: Ising model in the zeroth approximation Done by Ghassan M. Masa’deh](https://reader036.vdocuments.us/reader036/viewer/2022062520/568159fb550346895dc74794/html5/thumbnails/21.jpg)
For T <Tc we get.
X0(T) (Nµ /2K)/ (T≃ c-T) (T T≲ c , B→ 0 )
Experimentally the Curi – Weiss law is satisfied with considerable accuracy except that the empirical value of Tc thus obtained is always some what larger than the true transition temperature of the material.
2
![Page 22: Ising model in the zeroth approximation Done by Ghassan M. Masa’deh](https://reader036.vdocuments.us/reader036/viewer/2022062520/568159fb550346895dc74794/html5/thumbnails/22.jpg)
As T → 0 the law field susceptibility vanishes in accordance with the formula
X0(T) (4 Nµ /KT) exp (-2T≃ c/T)
Finally , the relation ship between L and B at T=Tc and use tanh x x + x /3 we have≃:
L (3 µB/KT) (T =T≃ c ,B→ 0 )
2
-1 3
1/3
![Page 23: Ising model in the zeroth approximation Done by Ghassan M. Masa’deh](https://reader036.vdocuments.us/reader036/viewer/2022062520/568159fb550346895dc74794/html5/thumbnails/23.jpg)
Thank you