**1. Introduction**

Ernst Ising (**Figure 1**) was born on May 10, 1900, in Loe Koln. He started schooling in 1907 and obtained his diploma at the gymnasium there in the year 1918. After brief military training, he studied mathematics and physics at Gottingen University in the year 1919. After a short gap, he continued his studies and learnt astronomy apart of other subjects. He got focused to theoretical physics at the suggestion of Professor W. Lenz. He started investigating ferromagnetism under supervision of W. Lenz by the end of the year 1922. Ising published short paper in 1925 as a summary of his doctoral thesis [1, 2]. He exactly calculated partition function for one-dimensional lattice system of spins. Ising had first proven that no phase transition to a ferromagnetic ordered state occurs in one dimension at any temperature.

His argument in the favor of his mathematical note was very simple. Suppose, if one of the spins get flipped at a random position because of thermal agitation, there is no force available, which can stop the neighboring spins to flip in the same direction. And this process will go on and on, and completely ordered state will not remain stable at a finite temperature. Thus no phase transition will occur at a finite temperature. Ideally speaking, any ordered state will always remain like a metastable state at finite temperature and nothing more. Molecular motion seizes at absolute zero temperature. So, one may expect that no spin fluctuations may occur at absolute zero temperature. Henceforth, the stable ordered state is a natural outcome at absolute zero temperature. But, it cannot be said to be a critical temperature in true sense. The existence of phase transition at this temperature has no physical meaning, because there is no temperature below it. After going through some approximate calculations, Ising purportedly showed that his model could not exhibit a phase transition in two and three dimensions, either. Latter, his conclusion was proven to be erroneous [1, 2] (**Figure 2**).

are more scientists in the list, whose contribution to Lenz-Ising model or simply say Ising model must be cited here, because of their historical relevance. They have greatly enriched and contributed to this new model. This list includes scientists like Gorskly (1928), R. H. Fowler (1930), Bragg and Williams (1934), R. Peierls (1936), J. G. Krikwood (1938), Hens Bethe (1939), Kramers and Wannier (1941), and Onsager (1942). They further extended Ising model to a new class of problems.

Ising model has been extensively used for solving a variety of problems [3–18].

Ising model was first exploited for investigating spontaneous magnetization in ferromagnetic film (i.e. magnetization in the absence of external magnetic field). An example case of Ising model using metropolis algorithm is shown in **Figure 3**. Transition temperature depends on the strength of the inter-spin exchange coupling; the dominating term governs the kinetics, when long-range interactions are introduced in the calculations. Latter, it was used to study phase separation in binary alloys and liquid-gas phase transitions (i.e., condensation of molecule in one region of space of the box). Binary alloys constitute of two different atoms. At temperature T = 0, Zn-Cu alloy; known as brass, gets completely ordered. This state is said to be β-brass. In β-brass state, each Zn atom is surrounded by eight copper

*Variation in critical temperature vs. next nearest exchange coupling for a bcc lattice (reproduced with*

Some of the problems are discussed, here, with appropriate examples.

**2. Application of Ising model**

*Random spin flipping in one-dimensional system.*

*The Ising Model: Brief Introduction and Its Application DOI: http://dx.doi.org/10.5772/intechopen.90875*

**Figure 2.**

**Figure 3.**

**117**

*permission from Singh [3]).*

**2.1 Phase separation and wetting/dewetting**

Barry Simon has quoted it very well "This model was suggested to Ising by his thesis advisor, Lenz. Ising solved the one-dimensional model, and on the basis of the fact that the one-dimensional model had no phase transition; he asserted that there was no phase transition in any dimension. As we shall see, this is false. It is ironic that on the basis of an elementary calculation and erroneous conclusion, Ising's name has become among the most commonly mentioned in the theoretical physics literature. But history has had its revenge. Ising's name, which is correctly pronounced "E-zing", is almost universally mispronounced "I-zing"."

Ising's paper credited Wilhelm Lenz for his original idea, who had first proposed it in the year 1920. W. Lenz was Ising's research supervisor. It has been often rendered as Lenz-Ising model in many citations. Lenz suggested that dipolar atoms in crystals are free to rotate in quantized manner. He proposed quantum treatment of dipole orientations, though in its classical version, Ising considered only two spin states, i.e., S = ½. Ising discussed his results with Professor Lenz and Dr. Wolfgang Pauli, who was teaching at Hamburg at that time. Ising's work was first cited by famous contemporary scientist Heisenberg. Heisenberg was first one to realize the failure of Lenz-Ising model. In order to explain ferromagnetism, he developed his own theory, using complicated interactions of spins. There

**Figure 1.** *Ernst (Ernest) Ising (May 10, 1900–May 11, 1998).*

*The Ising Model: Brief Introduction and Its Application DOI: http://dx.doi.org/10.5772/intechopen.90875*

**Figure 2.** *Random spin flipping in one-dimensional system.*

is no force available, which can stop the neighboring spins to flip in the same direction. And this process will go on and on, and completely ordered state will not remain stable at a finite temperature. Thus no phase transition will occur at a finite temperature. Ideally speaking, any ordered state will always remain like a metastable state at finite temperature and nothing more. Molecular motion seizes at absolute zero temperature. So, one may expect that no spin fluctuations may occur at absolute zero temperature. Henceforth, the stable ordered state is a natural outcome at absolute zero temperature. But, it cannot be said to be a critical temperature in true sense. The existence of phase transition at this temperature has no physical meaning, because there is no temperature below it. After going through some approximate calculations, Ising purportedly showed that his model could not exhibit a phase transition in two and three dimensions, either. Latter, his conclusion

*Solid State Physics - Metastable, Spintronics Materials and Mechanics of Deformable…*

Barry Simon has quoted it very well "This model was suggested to Ising by his thesis advisor, Lenz. Ising solved the one-dimensional model, and on the basis of the fact that the one-dimensional model had no phase transition; he asserted that there was no phase transition in any dimension. As we shall see, this is false. It is ironic that on the basis of an elementary calculation and erroneous conclusion, Ising's name has become among the most commonly mentioned in the theoretical physics literature. But history has had its revenge. Ising's name, which is correctly

Ising's paper credited Wilhelm Lenz for his original idea, who had first proposed

pronounced "E-zing", is almost universally mispronounced "I-zing"."

it in the year 1920. W. Lenz was Ising's research supervisor. It has been often rendered as Lenz-Ising model in many citations. Lenz suggested that dipolar atoms

treatment of dipole orientations, though in its classical version, Ising considered only two spin states, i.e., S = ½. Ising discussed his results with Professor Lenz and Dr. Wolfgang Pauli, who was teaching at Hamburg at that time. Ising's work was

first one to realize the failure of Lenz-Ising model. In order to explain ferromagnetism, he developed his own theory, using complicated interactions of spins. There

in crystals are free to rotate in quantized manner. He proposed quantum

first cited by famous contemporary scientist Heisenberg. Heisenberg was

was proven to be erroneous [1, 2] (**Figure 2**).

**Figure 1.**

**116**

*Ernst (Ernest) Ising (May 10, 1900–May 11, 1998).*

are more scientists in the list, whose contribution to Lenz-Ising model or simply say Ising model must be cited here, because of their historical relevance. They have greatly enriched and contributed to this new model. This list includes scientists like Gorskly (1928), R. H. Fowler (1930), Bragg and Williams (1934), R. Peierls (1936), J. G. Krikwood (1938), Hens Bethe (1939), Kramers and Wannier (1941), and Onsager (1942). They further extended Ising model to a new class of problems.
