Why are some materials more ferromagnetic than others? Neodymium magnets are a lot stronger than iron, copper and nickel. Neodymium is a mixture of different chemical elements. What is the reason that this material creates a lot more magnetic field strength than others?
Is there a justification by a mathematical model or equation that describes why ferromagnetism is in some metals a whole lot stronger than other metals? Maybe is there an explanation with the Ising model or do I have to consider more advanced models for describing how ferromganetism depends on the composition of metals?
 A: 
Neodym magnets are a lot stronger than iron, copper and nickel.Neodym is a mixture of different chemical elements. What is the reason that this material creates a lot more magnetic field strength than others?

How you answer this question depends on what exactly you mean by "stronger." In the colloquial sense of the word, there are a few important material properties which determine the strength of a permanent magnet:


*

*When a ferromagnetic material is magnetized by an external field, the microscopic magnetic moments present in the material tend to align with the field. The saturation magnetization, $M_s$, is the magnetization of the material when the external field is strong enough to align all the moments. Magnetic materials with high $M_s$ can produce strong magnetic fields when all their magnetic moments are aligned.

*The remanence, $M_r$, is the magnetization after a magnet has been saturated by an external field, and the field has subsequently been reduced to zero. Materials with high $M_r$ retain their magnetization when they are not in a magnetic field. In other words, the magnetic moments will stay aligned in the same direction when the saturating field is removed.

*The coercivity, $H_c$, is the magnetic field required to switch half the magnetic moments in a magnet. Materials with high $H_c$ are resistant to unwanted demagnetization (and magnetization) by external fields.
Here's a typical hysteresis curve with the above quantities labeled:

Rare earth magnets are strong in the sense that they have a great balance of all three of these properties. However, these materials aren't the strongest in terms of $M_s$ $(\sim 1300 \; \mathrm{emu/cm^3}$ for NdFeB). Elemental Fe has a higher saturation magnetization $(\sim 1700 \; \mathrm{emu/cm^3})$. NdFeB doesn't have the highest recorded coercivity either (SmCo is higher). And lots of systems have $M_r = M_s$, which is highly desirable for permanent magnets.

Is there a justification by a mathematical model or equation that describes why ferromagnetism is in some metals a whole lot stronger than other metals? Maybe is there an explaination with the Ising model or do I have to consider more advanced models for describing how ferromganetism depends on the composition of metals?

This is a really difficult question that a lot of people are trying to answer, because the ability to predict strong magnetic materials would be really useful for device applications. In general $M_s$ is influenced by electronic structure, as others have already mentioned. $H_c$ and $M_r$ tend to depend on anisotropy, defects, pinning sites, and other properties. If you are interested in $M_s$, $M_r$, or $H_c$, maybe it would be best addressed in a separate question?
A: Ferromagnetism comes from the atoms having multiple valence electrons with parallel spins. Metals typically have valences on multiple shells, which increases the electrical interactions. 
Neodymium has an absurdly high amount, 7 to be exact, of unpaired electrons, which gives it it's magnetic properties.
