Solid State | Lattice constant | BCC -> HCP

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SUMMARY

Sodium undergoes a phase transformation from body-centered cubic (BCC) to hexagonal close-packed (HCP) structure at approximately 23K, maintaining a constant density. Given the BCC lattice constant of 4.23 Å, the ideal c/a ratio for HCP is 1.633, which allows for the calculation of the HCP lattice constant 'a'. The relationship between the unit cell shapes and the number of atoms per unit cell is crucial for deriving the necessary equations to determine the HCP lattice constant.

PREREQUISITES
  • Understanding of crystal structures, specifically BCC and HCP.
  • Knowledge of lattice constants and their significance in solid-state physics.
  • Familiarity with unit cell geometry and volume calculations.
  • Basic principles of density and phase transformations in materials science.
NEXT STEPS
  • Research the geometric relationships between BCC and HCP unit cells.
  • Learn how to derive the volume of a unit cell based on lattice constants.
  • Study the implications of density conservation during phase transformations.
  • Explore the significance of the c/a ratio in determining the stability of HCP structures.
USEFUL FOR

Students and professionals in materials science, solid-state physics, and crystallography, particularly those interested in phase transformations and lattice structure calculations.

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Homework Statement


Sodium transforms from bcc to hcp at about T=23K. Assuming that the density remains fixed and the c/a ratio is ideal (1.633), calculate the hcp lattice constant a, given that the lattice constant a'=4.23 Angstrom in the cubic phase


Homework Equations


I can't find any equations regarding this


The Attempt at a Solution


We haven't gone over this in class so I don't even know where to begin.
 
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Using your text or the internet, you should be able to find the following information for bcc and hcp:

1. Shape of a unit cell.

2. Number of atoms per unit cell.

3. Volume of a unit cell in terms of the lattice constant(s). (If you can't find formulas for this, see if you can derive the volumes using geometry and the known shape of the unit cell.)

Think about the condition required for the density not to change in going from bcc to hcp.
 

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