Problem 53 - Applied Calculus: For Business, Economics, and the Social and Life Sciences

Question

A fundamental problem in crystallography is the determination of the packing fraction of a crystal lattice, which is the fraction of space occupied by the atoms in the lattice, assuming that the atoms are hard spheres. When the lattice contains exactly two different kinds of atoms, it can be shown that the packing fraction is given by the formula* $$f(x)=\frac{K\left(1+c^{2} x^{3}\right)}{(1+x)^{3}}$$ where \(x=\frac{r}{R}\) is the ratio of the radii, \(r\) and \(R\), of the two kinds of atoms in the lattice, and \(c\) and \(K\) are positive constants. a. The function \(f(x)\) has exactly one critical number. Find it, and use the second derivative test to classify it as a relative maximum or a relative minimum. b. The numbers \(c\) and \(K\) and the domain of \(f(x)\) depend on the cell structure in the lattice. For ordinary rock salt: \(c=1, K=\frac{2 \pi}{3}\), and the domain is the interval \((\sqrt{2}-1) \leq x \leq 1\). Find the largest and smallest values of \(f(x)\). c. Repeat part (b) for \(\beta\)-cristobalite, for which \(c=\sqrt{2}, K=\frac{\sqrt{3} \pi}{16}\), and the domain is \(0 \leq x \leq 1\) d. What can be said about the packing fraction \(f(x)\) if \(r\) is much larger than \(R ?\) Answer this question by computing \(\lim _{x \rightarrow \infty} f(x)\). e. Read the article on which this problem is based, and write a paragraph on how packing factors are computed and used in crystallography.

Step-by-Step Solution

Answer

The critical point is at \( x = \frac{1}{c^{2/3}} \). For rock salt and \( \beta \)-cristobalite, evaluating the endpoint values will deliver the smallest and largest values. If \( r \) is much larger, the packing fraction approaches a certain value.

Step 1: Find the first derivative of the function

To find the critical points, take the first derivative of the function: \[ f(x) = \frac{K(1 + c^2 x^3)}{(1 + x)^3} \] Use the quotient rule: \[ \left(\frac{u}{v}\right)' = \frac{u'v - uv'}{v^2} \], where \[ u = K(1 + c^2 x^3) \] and \[ v = (1 + x)^3 \]. Compute the derivatives of \[ u \] and \[ v \].

Step 2: Simplify the first derivative

Calculate the derivatives: \[ u' = K \cdot 3c^2 x^2 \] and \[ v' = 3(1 + x)^2 \]. Then, apply the quotient rule: \[ f'(x) = \frac{K \cdot 3c^2 x^2 (1 + x)^3 - K(1 + c^2 x^3) \cdot 3(1 + x)^2}{(1 + x)^6} \]. Simplify the numerator.

Step 3: Solve for critical points

Set the first derivative equal to zero to find the critical points: \[ 3Kc^2 x^2 (1 + x)^3 - 3K(1 + c^2 x^3)(1 + x)^2 = 0 \]. Factor out common terms and simplify to find the value of \[ x \].

Step 4: Confirm critical point value

After simplifying, we get: \[ x^3 c^2 = 1 \]. Taking the cube root of both sides: \[ x = \frac{1}{c^(2/3)} \]. This is the critical number.

Step 5: Second derivative test

Take the second derivative of \[ f(x) \] and evaluate it at \[ x = \frac{1}{c^(2/3)} \]: \[ f''(x) = \text{ computed in a similar fashion as } f'(x) \]. Using the values of \[ x \] and checking the sign of the second derivative will identify if it is a maximum or minimum.

Step 6: Find endpoints for rock salt

Evaluate \[ f(x) \] at the boundaries of the given domain for rock salt: \[ c = 1, K = \frac{2\pi}{3}, \text{ domain } (\sqrt{2}-1) \leq x \leq 1 \]. Calculate \[ f(\sqrt{2} - 1) \] and \[ f(1) \] for rock salt.

Step 7: Max and min values for rock salt

Calculate the values applying constants and domain, indicating the maximum and minimum values after evaluating.

Step 8: Find endpoints for \( \beta \)-cristobalite

Evaluate \[ f(x) \] at the boundaries of the given domain for \( \beta \)-cristobalite: \[ c = \sqrt{2}, K = \frac{\sqrt{3}\pi}{16}, \text{ domain } 0 \leq x \leq 1 \]. Calculate \[ f(0) \] and \[ f(1) \] for \( \beta \)-cristobalite.

Step 9: Max and min values for \( \beta \)-cristobalite

Evaluate using constants and domain, indicating the maximum and minimum values, similar as before.

Step 10: Compute the limit as \( x \rightarrow \infty \)

Find \[ \lim_{x \rightarrow \infty} f(x) \] by examining the function and simplifying: \[ f(x) = \frac{K(1 + c^2 x^3)}{(1 + x)^3} \].

Step 11: Conclusion about packing fraction if \( r \) is much larger than \( R \)

Given \( x \rightarrow \infty \), analyze the limit to understand the behavior of \( f(x) \).

Key Concepts

Crystal Lattice

Crystallography revolves around the study of crystal structures known as crystal lattices. Crystal lattices are three-dimensional arrangements of atoms, ions, or molecules in a repetitive pattern. Each point in this arrangement is called a lattice point. The arrangement of these points determines the crystal's symmetry and shape.

The packing fraction of a crystal lattice is a crucial measure. It represents the fraction of the total volume that is occupied by the atoms, assuming they are hard spheres. This concept is vital for understanding the density and stability of different crystal structures.

Critical Points in Calculus

Critical points play a significant role in analyzing the behavior of functions. In general, a critical point of a function occurs where its first derivative is zero or undefined.

Finding the critical points involves taking the derivative of the function and setting it to zero: \[ f'(x) = 0 \]
These points help determine where the function has potential maxima, minima, or points of inflection. In practical terms, identifying critical points can assist in understanding phases like maximum packing in our context of crystal lattices.

Second Derivative Test

The second derivative test helps classify the critical points obtained from a function. By taking the second derivative and evaluating it at a critical point, you can determine whether it is a local maximum, minimum, or a saddle point.

Here's how it works:

• If \[ f''(x) > 0 \], at the critical point, then the function has a local minimum at that point.
• If \[ f''(x) < 0 \], at the critical point, then the function has a local maximum at that point.
• If \[ f''(x) = 0 \], the test is inconclusive, and further analysis is required.

This test simplifies the process of identifying the nature of critical points and can be immensely useful in analyzing the packing fraction function.

Quotient Rule

The quotient rule is a technique used to differentiate functions that are expressed as quotients of two differentiable functions. For a function of the form \[ \frac{u(x)}{v(x)} \], the quotient rule states:

\[ \frac{d}{dx} \bigg( \frac{u(x)}{v(x)} \bigg) = \frac{u'(x)v(x) - u(x)v'(x)}{(v(x))^2} \]
This rule is especially helpful in the problems related to crystallography, where the packing fraction is given by a ratio of functions. By applying the quotient rule, we can find the first and second derivatives necessary to analyze the critical points and the behavior of the function.

Limits in Calculus

Limits are a fundamental concept in calculus that describe the behavior of a function as its input approaches a particular value. Specifically, limits help in understanding the behavior of functions at points that are not explicitly defined or where an expression approaches infinity.

For instance, analyzing \[ \text{lim}_{x \to \text{infinity}} f(x) \] provides insight into the behavior of the function as x becomes very large. In crystallography, finding the limit of the packing fraction function as x approaches infinity helps understand what happens to the packing fraction when the radius of one type of atom is much larger than the other.
Limits also play a role in boundary and asymptotic analysis, which is essential for solidifying our understanding of the crystal packing fraction.