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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?

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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:

  1. 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.

  2. 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.

  3. 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: ML

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?

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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.

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  • $\begingroup$ TL;DR more unpaired electrons = more magnetic $\endgroup$ – mcchucklezz Nov 25 '17 at 16:48
  • $\begingroup$ Can dependence of magnetic field strength on amount of unpaired electrons be mathematically formulated ? $\endgroup$ – kryomaxim Nov 25 '17 at 17:06
  • $\begingroup$ @kryomaxim from a quantum mechanical perspective, the amount of electrons in the system would make the math wildly unwieldy, you'd be in a better position by just going off of the number of unpaired electrons. $\endgroup$ – mcchucklezz Nov 25 '17 at 17:07
  • $\begingroup$ Neodymium has only 4 electrons in the $4f$ shell. Neodymium metal is not ferromagnetic. $\endgroup$ – Pieter Nov 25 '17 at 20:58
  • $\begingroup$ @Pieter it also has unpaired electrons in the d orbitals, and how is it not ferromagnetic $\endgroup$ – mcchucklezz Nov 25 '17 at 21:05
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The strength of permanent magnets is mostly due to the coercive field, which is the strength of the external field that can reverse the magnetization direction of a magnetic material. It varies by many orders of magnitude for different materials. It is small for magnetically soft materials like iron in transformer cores, mu-metal, etc. (The intensity of magnetization is quite similar for all iron based magnets including Nd$_2$Fe$_{14}$B, about 2 tesla.)

The size of the coercive field depends on how easy it is to move domain walls. This depends on two factors: on the metallurgical structure and on the width of the domain walls. It is easier to pin thin domain walls.

The width of domain walls depends on the magnetocrystalline anisotropy. The anisotropy is generally larger (domain walls are thin) for crystal structures with uniaxial symmetry than for cubic crystal structures. And because of spin-orbit interaction, it is stronger for the $4f$ electrons of the rare earth metals than for the $3d$ electrons of iron, cobalt or nickel. The rôle of the rare earths in ${\rm Nd}_2{\rm Fe}_{14}$B or SmCo$_6$ is to couple the spin direction of the magnetic moment of the transition $3d$ shell more strongly to the crystal lattice.

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