Cubic Lattice Atom Diameter Calculation

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SUMMARY

The discussion focuses on calculating the atomic radius of barium in a body-centered cubic (BCC) lattice, given its density of 3.50 g/cm³ and molar mass of 137.3 g/mol. The calculation involves determining the mass of the unit cell, its volume, and subsequently the cell side length, which is found to be 506.9 pm. The diagonal length of the cell is derived using the relationship 4r = √3a, leading to an atomic radius of 219.5 pm. The conversation also touches on the geometric relationships in different crystal structures.

PREREQUISITES
  • Understanding of body-centered cubic (BCC) lattice structure
  • Familiarity with density and molar mass calculations
  • Knowledge of unit cell volume and side length calculations
  • Basic geometry involving triangles and parallelograms
NEXT STEPS
  • Study the geometric relationships in face-centered cubic (FCC) and hexagonal close-packed (HCP) structures
  • Learn about the calculation of atomic radii in different crystal systems
  • Explore the implications of atomic radius on material properties
  • Investigate the relationship between density and crystal structure in metals
USEFUL FOR

Students in materials science, chemistry, and physics, particularly those studying crystallography and solid-state physics, will benefit from this discussion.

Bob Ho
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Homework Statement


Barium metal crystallizes in a body-centred cubic lattice. The density of the metal is 3.50gcm^-3. Calculate the radius(in pm) of a barium atom.

M(Ba)=137.3g/mol , NA = 6.022x10^23/mol

The Attempt at a Solution


For 1 unit cell: m=2x137.3/6.022x10^23
=4.56x10^-22g

V=m/p
=4.56x10^-22/3.5
=1.303x10^8pm^3

Cell side length a=v^(1/3)=506.9pm

From here I am confused, the answer is that the cell diagonal length =4r=√3a
Atomic radius r=√3/4a=219.5pm

Can anyone please explain to me how the cell diagonal length varies with different structures, or anything that can help me, thanks.
 
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It is a (relatively) simple geometry. General approach is to find a plane that goes through atoms centers, draw the atoms and see what is known. You will usually end with some mix of triangles and rectagles or parallelograms where you know some of the distances and you have to find others.
 

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