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Consider a crystal prism with an apex angle of $60^{\circ}$, where the optical axis is perpendicular to the principal section of the prism. Sodium yellow light is refracted through the prism at the angle of minimum deviation and subsequently focused by a lens of focal length 1 m. Determine the separation between the o-ray and e-ray spectral images.
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7.16~\mathrm{mm}
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physics
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verifier/verify.py
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Consider the following third-order ordinary differential equation $$y''' = \\frac{y}{\\cos{\\left(x \\right)} + 4} + y'^{4} + \\frac{y''}{\\tan{\\left(x \\right)} + 4} + \\frac{1}{\\cos{\\left(x \\right)} + 5}$$
with initial conditions at $x = 0: y(0) = 2.00, y'(0) = 1.00, y''(0) = 2.00.$
Find analytical expressions that approximate the solution of $y(x)$ in the small and large $x$ regimes.
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[
\boxed{
\begin{aligned}
\text{Small } x:&\quad
y(x)=x^{3}!\left(\frac{7}{36}+\frac{1}{3(\tan x+4)}+\frac{1}{3(\cos x+4)}\right)
* x^{2} + x + 2, [6pt]
\text{Large } x:&\quad
y(x) = -\sqrt[3]{30},(0.711063906390639 - x)^{1/3} + 4.77.
\end{aligned}}
]
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math
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verifier/verify.py
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Beryl, \( \mathrm{Be_3Al_2(SiO_3)_6} \), is hexagonal with
\( a = 9.206\,\text{Å}, \; c = 9.205\,\text{Å}, \; Z = 2 \).
The space group is \( P6/mcc \), and the equipoint sets are:
| Position | Point symmetry | Coordinates |
|-----------|----------------|--------------|
| 24(m) | ℓ | \( x, y, z; \) etc. |
| 12(l) | m | \( x, y, 0; \) etc. |
| 12(k) | 2 | \( x, 2x, \tfrac{1}{4}; \) etc. |
| 12(j) | 2 | \( x, 0, \tfrac{1}{4}; \) etc. |
| 12(i) | 2 | \( \tfrac{1}{2}, 0, z_i; \) etc. |
| 8(h) | 3 | \( \tfrac{1}{3}, \tfrac{2}{3}, z_j; \) etc. |
| 6(g) | 2/m | \( \tfrac{1}{2}, 0, 0; \) etc. |
| 6(f) | 222 | \( \tfrac{1}{2}, 0, \tfrac{1}{4}; \) etc. |
| 4(e) | 6 | \( 0, 0, z_i; \) etc. |
| 4(d) | \( \bar{6} \) | \( \tfrac{1}{3}, \tfrac{2}{3}, 0; \) etc. |
| 4(c) | 32 | \( \tfrac{1}{3}, \tfrac{2}{3}, \tfrac{1}{4}; \) etc. |
| 2(b) | 6/m | \( 0, 0, 0; \; 0, 0, \tfrac{1}{2} \) |
| 2(a) | 62 | \( 0, 0, \tfrac{1}{4}; \; 0, 0, \tfrac{3}{4} \) |
Assign positions to all atoms and list the positional parameters which uniquely define the crystal structure.
Explain your deductions concisely.
|
| Atom | Position | Point symmetry | Parameters |
|:-----|:----------|:----------------|:------------|
| Al | 4(c) | 32 | none |
| Be | 6(f) | 222 | none |
| Si | 12(l) | m | \( x_{\mathrm{Si}}, \, y_{\mathrm{Si}} \) |
| O(1) | 12(l) | m | \( x_{\mathrm{O(1)}}, \, y_{\mathrm{O(1)}} \) |
| O(2) | 24(m) | 1 | \( x_{\mathrm{O(2)}}, \, y_{\mathrm{O(2)}}, \, z_{\mathrm{O(2)}} \) |
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chemistry
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verifier/verify.py
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