The accurate expression for the dispersion relation of a polycrystalline bulk metal depends on several factors, including:

Therefore, there is no single "accurate" expression that fits all polycrystalline bulk metals. However, here are some options depending on your specific needs:

1. Simple models:

• Debye model: This is the simplest model, assuming a single, isotropic elastic medium. It can be used as a first approximation for metals with cubic symmetry and large grain sizes.
• Free electron model: This model is applicable for metals with delocalized electrons and can be used to describe the low-frequency acoustic phonons.

2. More complex models:

• Discrete lattice models: These models explicitly consider the atomic positions and interactions within the crystal lattice. They can be used to calculate the dispersion relation for any crystal structure and grain size, but they require more computational resources.
• Effective medium theories: These theories treat the polycrystal as an equivalent homogeneous medium with averaged elastic properties. They can be useful for understanding the overall behavior of the system without the complexity of a full lattice model.

3. Experimental data fitting:

• If you have experimental data for the frequency vs. resonance field, you can fit it to a theoretical model using a least-squares method. This approach allows you to account for the specific properties of your sample, such as the crystal structure, grain size, and texture.

Ultimately, the best way to determine the accurate dispersion relation for your specific polycrystalline bulk metal depends on the available information and your desired level of accuracy.