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What is the atomic packing factor for hcp?
HCP is one of the most common structures for metals. HCP has 6 atoms per unit cell, lattice constant a = 2r and c = (4√6r)/3 (or c/a ratio = 1.633), coordination number CN = 12, and Atomic Packing Factor APF = 74%.
How do you find the atomic packing factor?
Explanation: The atomic packing factor is defined as the ratio of the volume occupied by the average number of atoms in a unit cell to the volume of the unit cell. for FCC a = 2√2 r where a is side of the cube and r is atomic radius.
What is the packing fraction in hcp structure?
In hcp structure, the packing fraction is 0.74.
How do you find the atomic radius of hcp?
The APF and coordination number of the HCP structure is the same as the FCC structure, that is, 0.74 and 12 respectively. An isolated HCP unit cell has a total of 6 atoms per unit cell. = a3 (FCC and BCC) a = 2R√2 (FCC); a = 4R/√3 (BCC) where R is the atomic radius.
What is the packing efficiency of HCP?
What is the packing efficiency in HCP and CCP structure? The Packing efficiency of Hexagonal close packing (hcp) and cubic close packing (ccp) is 74%.
What is hcp unit cell?
In a unit cell, an atom’s coordination number is the number of atoms it is touching. The hexagonal closest packed (hcp) has a coordination number of 12 and contains 6 atoms per unit cell. The face-centered cubic (fcc) has a coordination number of 12 and contains 4 atoms per unit cell.
What is packing fraction formula?
The equation for packing fraction is: Packing fraction = (N atoms) x (V atom) / V unit cell. N atoms is the number of atoms in a unit cell. V atom is the volume of the atom, and V unit cell is the volume of a unit cell. Substitute the number of atoms per unit cell into the equation.
What is the coordination number for HCP?
In the hexagonal closest packed (hcp) each ion has 12 neighboring ions hence it has a coordination number of 12 and contains 6 atoms per unit cell.
What is HCP structure?
Hexagonal close packed (hcp) refers to layers of spheres packed so that spheres in alternating layers overlie one another. Hexagonal close packed is a slip system, which is close-packed structure. The hcp structure is very common for elemental metals, including: Beryllium. Cadmium.
How do you find the number of atoms in HCP?
Total number of atoms in one unit cell =(3×1)+(2×21)+(12×61)=3+1+2=6.
Which is the correct formula for radius of atom in hexagonal packing HCP unit cell?
C. 12√2r3.
How do you calculate HCP volume?
Volume =63 ×r2× 4r=242 r3. Was this answer helpful?
What is the full form of HCP in chemistry?
Hexagonal Close Packed (hcp) (ABAB…layer sequence) (e.g. Zn) Unit cell of Zn Full 3D structure of Zn. HCP can be represented as a layer structure. Other Hexagonal Close Packed structures include Mg, Ti, Co, Cd and Be. Cubic Closed Packed (ccp) Sometimes called face centred cubic (fcc) (ABCABC…layer sequence)
What is the atomic packing factor of bcc?
Body-centered cubic (bcc): 0.68.
How many atoms are there in the hexagonal close packed hcp unit cell?
The hexagonal closest packed (hcp) has a coordination number of 12 and contains 6 atoms per unit cell.
What is the volume of HCP unit cell?
Volume =63 ×r2× 4r=242 r3. Was this answer helpful?
What is hcp unit cell?
In HCP, there are 6 corner atoms in the top layer and 6 corner atoms in the bottom layer, so 12 atoms in the unit cell. The contribution of each atom in the unit cell is one-sixth. 2. 2 atoms are present at two face centres. The contribution of each atom in the unit cell is one-half.
show that the atomic packing factor for hcp is 0.74.
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- Summary of article content: Articles about show that the atomic packing factor for hcp is 0.74. Show that the atomic packing factor for HCP is 0.74. This problem calls for a demonstration that the APF for HCP is 0.74. Again, the. APF is the ratio of the … …
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Hexagonal Close-Packed (HCP) Unit Cell – Materials Science & Engineering
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- Most searched keywords: Whether you are looking for Hexagonal Close-Packed (HCP) Unit Cell – Materials Science & Engineering Updating The Hexagonal Close-Packed (HCP) unit cell can be imagined as a hexagonal prism with an atom on each vertex, and 3 atoms in the center. It can also be imagined as stacking 3 close-packed hexagonal layers such that the top layer and bottom layer line up. HCP is one of the most common structures for metals. HCP has 6 atoms per unit cell, lattice constant a = 2r and c = (4√6r)/3 (or c/a ratio = 1.633), coordination number CN = 12, and Atomic Packing Factor APF = 74%. HCP is a close-packed structure with AB-AB stacking.
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[Solved] What is the atomic packing factor for BCC and FCC, respectiv
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The atomic packing factor is defined as the ratio of the volume occupied by the average number of atoms in a unit cell to the volume of the unit c
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show that the atomic packing factor for hcp is 0.74.
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show that the atomic packing factor for hcp is 0.74.
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What is Atomic Packing Factor (and How to Calculate it for SC, BCC, FCC, and HCP)? – Materials Science & Engineering
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- Most searched keywords: Whether you are looking for What is Atomic Packing Factor (and How to Calculate it for SC, BCC, FCC, and HCP)? – Materials Science & Engineering What is Atomic Packing Factor (and How to Calculate it for SC, BCC, FCC, and HCP)?. Atomic Packing Factor (APF) tells you what percent of an object is made of atoms vs empty space. You can think of this as a volume density, or as an indication of how tightly-packed the atoms are. Calculating the atomic packing factor for a crystal is simple: for some repeating volume, calculate the volume of the atoms inside and divide by the total volume.
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What is Atomic Packing Factor (and How to Calculate it for SC, BCC, FCC, and HCP)? – Materials Science & Engineering
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- Summary of article content: Articles about What is Atomic Packing Factor (and How to Calculate it for SC, BCC, FCC, and HCP)? – Materials Science & Engineering 3.6Show that the atomic packing factor for HCP is 0.74.SolutionThe APF is just the total sphere volume-unit cell volume ratio.For HCP, there are the … …
- Most searched keywords: Whether you are looking for What is Atomic Packing Factor (and How to Calculate it for SC, BCC, FCC, and HCP)? – Materials Science & Engineering 3.6Show that the atomic packing factor for HCP is 0.74.SolutionThe APF is just the total sphere volume-unit cell volume ratio.For HCP, there are the … Atomic Packing Factor (APF) tells you what percent of an object is made of atoms vs empty space. You can think of this as a volume density, or as an indication of how tightly-packed the atoms are. Calculating the atomic packing factor for a crystal is simple: for some repeating volume, calculate the volume of the atoms inside and divide by the total volume.
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SOLVED:1. Show that the atomic packing factor for HCP is 0.74
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- Summary of article content: Articles about SOLVED:1. Show that the atomic packing factor for HCP is 0.74 So here I’m from their body center QB um crystal. We have that for edge, the eight edge of MMA cubic. Um That will be a 1/8 of the atom in age uh in each … …
- Most searched keywords: Whether you are looking for SOLVED:1. Show that the atomic packing factor for HCP is 0.74 So here I’m from their body center QB um crystal. We have that for edge, the eight edge of MMA cubic. Um That will be a 1/8 of the atom in age uh in each … VIDEO ANSWER:Okay in this question we’re asked to show that the atomic banking sector. But hcps buying 275. I started buying some and poor as we know that the atomic picking picture is the total volume sphere. Do you not send volume? But SCP there are ah 60 spare parts and we can write the Volume of this very six into fall by our cube, divide by three. Yes this is the volume of the sphere. So we get eight by our cube at the bottom of this fear. No the unit cell volume is just the protector of the base area time the satellite see. So this base area is just three times the area of the barrel. A pipe. A C. D. E. And which is given a step in the pipe is given us. This is a this is the parent arrived. Mhm. We are given the 25 days and Mhm. Sorry. This is a real pipe. A. A. C. Indeed E. And if we draw a perpendicular And 2 9 e. And this is B. And the angle farm here is 60°. And if this angle is 30 degree. And okay, this distance is which is the length of the atomic same. Which is it went to to our And from here to here Again it is equal to two. And this distance is so that this land is also equal to tour. I saw the area of a C. D. Is just a blend of city times the hide Bc. So but city is just here C. D. Is equal to uh which is equal to to us. So therefore Bc. Mhm. Is equal to who are of course 30°. So we can write B. C. Equal to To an road three Out of or two. So the base area base area is this area is equal to three into c. d. Into umbc. So we can write it as three into to uh into two and do three out of Or two. And this is it will too six are scared and root three. Yes we know that C. Is equal to 1.33 1.6338. So putting the value of in a heroic mission we get six 1.633 years into. But you are yes we know that the volume of the unit cell is equal to area into. Let’s see and therefore we see equal to six are scarce. E. Into and Route three and we get we see will do six alive scared. And route three multiplied by two multiplied by one x 633. And we get the volumes as it were true 12:03 into 1.633 A cube. And therefore the atomic packing factor. Our fraction is equal to. We is out of. We see as we have calculated bss aired by thank you, divide by 12 and three Into 1.633. Our cube and simplifying this evaluation. We get The atomic taking picture is 0574
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Hexagonal Close-Packed (HCP) Unit Cell – Materials Science & Engineering
The Hexagonal Close-Packed (HCP) crystal structure is one of the most common ways for atoms to arrange themselves in metals. The HCP crystal structure is based on the Bravais lattice of the same name, with 1 atom per lattice point at each corner of the hexagonal prism, and 3 inside the prism. HCP is one of the most stable crystal structures and has the highest packing density. The hexagonal close-packed cell belongs to space group #194 or P6 3 /mmc, Strukturbericht A3, and Pearson symbol hP2. Mg is the prototype for FCC.
The Hexagonal Close-Packed (HCP) unit cell can be imagined as a hexagonal prism with an atom on each vertex, and 3 atoms in the center. It can also be imagined as stacking 3 close-packed hexagonal layers such that the top layer and bottom layer line up. HCP is one of the most common structures for metals. HCP has 6 atoms per unit cell, lattice constant a = 2r and c = (4√6r)/3 (or c/a ratio = 1.633), coordination number CN = 12, and Atomic Packing Factor APF = 74%. HCP is a close-packed structure with AB-AB stacking.
Don’t worry, I’ll explain what those numbers mean and why they’re important later in the article. For now, let’s talk about which materials actually exist as face-centered cubic.
Common Examples of Hexagonal Close-Packed Materials
Since HCP is one of the most common crystal structures, there are many materials to choose from! Scandium, titanium, cobalt, zinc, yttrium, zirconium, technetium, ruthenium, cadmium, gadolinium, terbium, dysprosium, holmium, erbium, thulium, lutetium, hafnium, rhenium, osmium, and thallium all have an HCP structure at standard pressure and temperature. This list is not comprehensive; HCP can also be found in high temperature/pressure phases, or in alloys.
HCP is common because of its high coordination number.
Hexagonal Close-Packed Coordination Number
Coordination Number (CN) is the number of nearest neighbors that each atom has.
As a close-packed structure, the HCP crystal has the number of nearest-neighbors (NN): 12. Each of these NNs contributes a bond, giving the crystal structure very high stability.
Hexagonal Close-Packed Lattice Constants
The hexagonal close-packed lattice is a hexagonal prism with an atom on each vertex and three in center. Using the hard sphere model, which imagines each atom as a discrete sphere, the HCP crystal has each atom touch along the top and bottom of the prism.
You can also visualize the primitive HCP cell, which has an atom at each of 8 corners and another one near the center.
The HCP cell is defined by two lattice constants a and c, which correspond to two side lengths in the primitive cell (there is also a 3rd side length, b, but that’s the exact same as a).
is the distance between any two nearest atoms, which is also the length of each side of the hexagon.
Since is the distance between two touching atoms, a must be .
Calculating is a bit trickier, but it can be done with trigonometry.
Click here to see the math! If you look at the central atom in the primitive cell, you can see that it has a distance between the atoms in the plane above and in the plane below. If you projected the atom into one of those planes, it would be exactly in the middle of 3 atoms. This position is the center of the equilateral triangle. Let’s draw a line between the center of the triangle and one of its corners. We can call this . Because the angles of an equilateral triangle are all 60°, the angle between and is 30°.
So
Now we can make another triangle, between , , and .
Which means that Or
And remembering that ,
Now we can make another triangle, between , , and .
Which means that Or
And remembering that ,
If you go through the math, you’ll find that c should be about 1.6a in a perfect HCP crystal.
If you wanted to describe the hexagonal close-packed crystal with math, you would describe the cell with the vectors:
These are actually primitive vectors, which you can read about in the section below.
Hexagonal Close-Packed Atomic Packing Factor
The Atomic Packing Factor (APF) is essentially the density of the unit cell. Since we use the hard sphere model, each point inside the cell is either part of an atom, or part of the void.
APF is basically the fraction of atoms to void. For a full article explaining APF, check out this link.
APF is the
The total volume of the unit cell is the area of one hexagon, multiplied by the height of the prism.
Click here to see the math! Start by breaking this into parts. The volume of the hexagonal prism will be the area of the hexagon * the height of the prism. The area of the hexagon is just 6 equilateral triangles. Let’s start by calculating the area of a single triangle. Any triangle’s area is Each side of the triangle has a length , so let’s use that as our base. Now we need to find the height of the triangle. Once again, the pythagorean theorem saves the day! We can make right triangle between , , and the height .
Which means
So the area of the triangle is
And since there are 6 equilateral triangles per hexagon,
Multiplying this area by the height gives
and using and ,
Now we need to count how many atoms are in each unit cell. It may look like there are 17 atoms in the HCP unit cell, but that’s actually the number of different atoms that intersect the unit cell. Most of those atoms are only partially inside the cell. If you count the portion of atoms in the cell, ⅙ of each vertex atom would count. Since there are 12 vertex atoms, . There is also a half atom on the top and bottom faces, which adds to 1 more whole atom. Finally, there are 3 atoms fully inside the HCP unit cell. Thus, there are 6 atoms per unit cell. If we write everything in terms of the radius of an atom, you can see that every face-centered cubic crystal will have the same packing factor regardless of the actual element. The volume of a sphere is . We previously established that the area of the whole cell is , so the APF of HCP is
As you can see, hexagonal close-packed crystals have 74% packing. That is exactly the same value as face-centered cubic (FCC) crystals, because both HCP and FCC are close-packed structures with the maximum possible APF (although it is possible to have higher packing if you use multiple kinds of atoms with different sizes).
In addition to FCC and HCP, it is possible to have other close-packed structures such as the close-packed rhombohedral structure found in samarium. It all comes down to stacking order.
Primitive Hexagonal Close-Packed Unit Cell
Advanced topic, click to expand! I already mentioned this before, but both the conventional and primitive HCP cells are commonly used. The conventional cell has advantages because it is highly symmetric and easy for humans to understand. The primitive cell is smaller which can make mathematical manipulation easier. When dealing with mathematical descriptions of crystals, it’s often easier to describe the unit cell in the smallest form possible (that’s the definition of a primitive cell). If you are interested in primitive cells, you can read all about them in this article. You’ve already seen the HCP primitive cell, but in case you were skipping to this section: Here are the primitive vectors for the HCP unit cell
Interstitial Sites in Hexagonal Close-Packed
Interstitial sites are the spaces inside a crystal where another kind of atom could fit. HCP has two types of interstitial sites: octahedral and tetrahedral. (Technically trigonal sites are also possible, but they are not practically useful).
HCP has 6 octahedral sites, which means that a small interstitial atom could fit in 6 positions such that it is equally surrounded by 6 HCP lattice atoms.
These octahedral interstitial sites have a radius of 0.414R, where r is the radius of the lattice atoms.
HCP also has 12 tetrahedral sites, which means that a small interstitial atom could fit in 12 positions such that it is equally surrounded by 4 HCP lattice atoms.
These tetrahedral atoms can be 0.225R, where r is the radius of the lattice atoms.
Slip Systems in Hexagonal Close-Packed
Advanced topic, click to expand! Slip systems are the way that atoms slide past each other when deforming. Slip systems determine many mechanical properties of materials, and is the main reason why a material will be ductile or brittle. To understand slip system directions, you will need to be familiar with Miller Indices notation, (we’re preparing an article about it). I’ll be using the notation for a primitive cell (3 numbers) instead of the conventional HCP cell (4 numbers). The HCP close-packed planes are {001}, so those are the slip planes. Within the {001} planes, the slip direction (close-packed direction) is <100>. You can see in this (001) plane, there are 3 slip directions: [100], [110], and [010]. However, since [110] can be made by the linear combination of [100] and [010], there are actually only 2 independent slip systems. These are called the basal slip systems in HCP. Name of a
Slip System Number of
Slip Systems Slip Plane Slip Direction Basal 3 Prismatic 3 Pyramidal 12 Slip Systems in HCP There are also prismatic and pyramidal slip systems in HCP. These systems, however, are not necessarily close-packed, and may need to be thermally activated. Because not all HCP metals have all at least 5 independent slip systems active at room temperature, not all HCP metals are ductile at room temperature. In some metals, the atoms don’t have the ideal c/a ratio, which deactivates slip systems. In other metals, the slip systems can be thermally activated to provide ductility. For example, titanium is ductile at room temperature, but zinc will only become ductile after it’s heated.
Final Thoughts
The Hexagonal Close-Packed (HCP) crystal structure is one of the most common ways that atoms can be arranged in pure solids. HCPis close-packed, which means it has the maximum APF of 0.74. Because HCP structures are often imperfect (specifically, they don’t have the c/a ratio we calculated), they may not have enough slip systems active to allow ductility. That’s why HCP metals can have dramatically different properties.
Here is a summary chart of all HCP crystal properties:
Crystal Structure Hexagonal Close-Packed Unit Cell Type Hexagonal Relationship Between Cube Edge
Length a and the Atomic Radius R a = 2R Close-Packed Structure Yes Atomic Packing Factor (APF) 74% Coordination Number 12 Number of Atoms per Unit Cell 6 Number of Octahedral Interstitial Sites 6 Number of Tetrahedral Interstitial Sites 12 Size of Octahedral Voids r = 0.414R Size of Tetrahedral Voids r = 0.225R
References and Further Reading
If you want to know more about the basics of crystallography, check out this article about crystals and grains.
If you weren’t sure about the difference between crystal structure and Bravais lattice, check out this article.
I also mentioned atomic packing factor (APF) earlier in this article. This is an important concept in your introductory materials science class, so if you want a full explanation of APF, check out this page.
If you’re interested in advanced crystallography or crystallography databases, you may want to check out the AFLOW crystallographic library.
For a great reference for all crystal structures, check out “The AFLOW Library of Crystallographic Prototypes.”
Single-Element Crystal Structures and the 14 Bravais Lattices
If you want to learn about specific crystal structures, here is a list of my articles about Bravais lattices and some related crystal structures for pure elements.
1. Simple Cubic
2. Face-Centered Cubic
2a. Diamond Cubic
3. Body-Centered Cubic
4. Simple Hexagonal
4a. Hexagonal Close-Packed
4b. Double Hexagonal Close-Packed (La-type)
5. Rhombohedral
5a. Rhombohedral Close-Packed (Sm-type)
6. Simple Tetragonal
7. Body-Centered Tetragonal
7a. Diamond Tetragonal (White Tin)
8. Simple Orthorhombic
9. Base-Centered Orthorhombic
10. Face-Centered Orthorhombic
11. Body-Centered Orthorhombic
12. Simple Monoclinic
13. Base-Centered Monoclinic
14. Triclinic
Other articles in my crystallography series include:
Introduction to Bravais Lattices
What is the Difference Between “Crystal Structure” and “Bravais Lattice”
Atomic Packing Factor
How to Read Miller Indices
How to Read Hexagonal Miller-Bravais Indices
Close-Packed Crystals and Stacking Order
Interstitial Sites
Primitive Cells
How to Read Crystallography Notation
What are Point Groups
List of Point Groups
What are Space Groups
List of Space Groups
The 7 Crystal Systems
[Solved] What is the atomic packing factor for BCC and FCC, respectiv
Explanation:
The atomic packing factor is defined as the ratio of the volume occupied by the average number of atoms in a unit cell to the volume of the unit cell.
Mathematically, Atomic Packing Factor (APF):
APF \( = \frac{{{N_{atoms}} ~\times ~{V_{atoms}}}}{{{V_{unit\;cell}}}}\) …(1)
Characteristics of various types of structures are shown in the table below:
Characteristics BCC FCC HCP a to r relation \(a = \frac{{4r}}{{√ 3 }}\) \(a = 2√ 2 r\) \(a = 2r\) The average number of atoms 2 4 6 Co-ordination number 8 12 12 APF 0.68 0.74 0.74 Examples Na, K, V, Mo, Ta, W Ca, Ni, Cu, Ag, Pt, Au, Pb, Al Be, Mg, Zn, Cd, Te
For Cubic Unit Cell
Nav = \(N_c\over 8\) + \(N_f\over 2\) + \(N_i \over 1\)
Nav = Average no of atoms in unit cell, Nc = No of corner atoms, Ni = No of interior atoms, Nf = No of face centre atoms
Calculation:
No of atoms in f.c.c unit cell = 4
\(APF = \frac{{{N_{atoms}}{V_{atom}}}}{{{V_{crystal}}}} = \frac{{4\left( {\frac{4}{3}} \right)\pi {r^3}}}{{{{\left( {a } \right)}^3}}}= \frac{{4\left( {\frac{4}{3}} \right)\pi {r^3}}}{{{{\left( {{{2√2r}}{}} \right)}^3}}}\)
for FCC a = 2√2 r where a is side of the cube and r is atomic radius.
APF = 0.74
For BCC:
Nav = \(8\over 8\) + 0 + \(1\over 1\) = 2
√3a = 4r
Put all values in equation 1
(APF)BCC = 0.68
MATSE 259 Solutions to homework # For the HCP crystal structure, show
MATSE 259
Solutions to homework #
For the HCP crystal structure, show that the ideal c/a ratio is 1.
We are asked to show that the ideal c/a ratio for HCP is 1. A sketch of one
third of an HCP unit cell is shown below.
c
a
a
J
M
K
L
Consider the tetrahedron labeled as JKLM, which is reconstructed as
J K
L
M
H
The atom at point M is midway between the top and bottom faces of the unit cell
that is MH
__
= c/2. And, since atoms at points J, K, and M, all touch one another,
JM
__
= JK
__
= 2R = a
where R is the atomic radius. Furthermore, from triangle JHM,
(JM
__
)
2
= (JH
__
)
2
+ (MH
__
)
2
, or
a
2
= (JH
__
)
2
+
⎝
⎜
⎛
⎠
⎟
c⎞
2
2
Now, we can determine the JH
__
length by consideration of triangle JKL, which is
an equilateral triangle,
J
L
K
H
a/
30
cos 30° =
a/
JH
=
3
2
, and
JH
__
=
a
3
Substituting this value for JH
__
in the above expression yields
a
2
=
⎝
⎜
⎛
⎠
⎟
a⎞
3
2
+
⎝
⎜
⎛
⎠
⎟
c⎞
2
2
=
a
2
3
+
c
2
4
and, solving for c/a
c
a
=
8
3
= 1.
Thus, the area of the basal plane = 6 ×
4
3
a
2 =
2
3 3
a
2 .
Further, as can be seen from the figure of the basal plane, a = 2R. Therefore, the
base area = 6R
2 3
Now, using the result of Problem 1, given above, the relation between the unit
cell height, c, and the basal plane edge length, a, is given as:
1.
a
c
=
Thus, c = 1 = 3
The unit cell volume can now be calculated as:
2 3
C
V =c×base area=3×10 =33
Thus, 0.
33
8 R
V
V
APF
3
3
C
S =
π
= =
Rhenium has an HCP crystal structure, an atomic radius of 0 nm, and a
c/a ratio of 1. Compute the volume of the unit cell for Re.
This problem asks that we calculate the unit cell volume, VC, for Re which has an
HCP crystal structure. Now, VC = c × base area, and the base area has been
calculated in Problem 2 above as 6R
2 3. Thus,
VC = c × 6R
2
3
In this problem, c = 1, and a = 2R. Therefore
V
C
= 1 × 2 × 6 3 × R
3
= 1 × 2 × 6 3 × (1 x 10
–
cm)
3
= 8 x 10
–
cm
3
= 8 x 10
–
nm
3
So you have finished reading the show that the atomic packing factor for hcp is 0.74. topic article, if you find this article useful, please share it. Thank you very much. See more:
