MikaStars39/nano-eval huggingface dataset:
The combination of binding energies of Ne in bcc and fcc structures can be calculated using the Lennard-Jones potential. The ratio of the binding energies of Ne in bcc to fcc structures is____. The lattice sums in bcc structure are:
$$
\sum_{j} {}^{\prime} P_{i_{j}} {}^{-12}=9.11418, \ \ \sum_{i} {}^{\prime} P_{i_{j}} {}^{-8}=12.2533
$$

Options:
A. $$
\beta\approx0. 9 9 8
$$
B. $$
\beta\approx1. 2 0 3
$$
C. $$
\beta\approx1. 0 3 2
$$
D. $$
\beta\approx1. 0 8 9
$$
E. $$
\beta\approx0. 8 5 0
$$
F. $$
\beta\approx0. 9 5 5
$$
G. $$
\beta\approx1. 0 4 0
$$
H. $$
\beta\approx0. 7 7 5
$$
I. $$
\beta\approx1. 1 5 0
$$
J. $$
\beta\approx1. 3 0 1
$$
Answer: G
The vibration of atoms in metals can be approximated using a Debye frequency model. In simple metals, the ions, with mass M and charge e, are dispersed in a uniform sea of conduction electrons. It is assumed that when an ion is at its normal lattice position, it is in stable equilibrium. If an ion is displaced a small distance $r$ from its equilibrium position, the restoring force mainly comes from the charge within the sphere centered at the equilibrium position with a radius $r$. Assuming the number density of ions (or conduction electrons) is 3/ $4 \pi R^{3}$, this defines R. An approximate estimation of sodium's frequency value is ____.

Options:
A. $$
\approx1. 1 1 \times1 0^{1 5} \mathrm{H z}
$$
B. $$
\approx9. 8 7 \times1 0^{1 2} \mathrm{H z}
$$
C. $$
\approx3. 4 5 \times1 0^{1 4} \mathrm{H z}
$$
D. $$
\approx8. 9 9 \times1 0^{1 2} \mathrm{H z}
$$
E. $$
\approx6. 5 4 \times1 0^{1 3} \mathrm{H z}
$$
F. $$
\approx1. 2 3 \times1 0^{1 2} \mathrm{H z}
$$
G. $$
\approx2. 5 3 \times1 0^{1 3} \mathrm{H z}
$$
H. $$
\approx7. 6 5 \times1 0^{1 4} \mathrm{H z}
$$
I. $$
\approx5. 6 7 \times1 0^{1 3} \mathrm{H z}
$$
J. $$
\approx4. 5 6 \times1 0^{1 1} \mathrm{H z}
$$
Answer: G
Consider a two-dimensional square lattice with a crystal potential given by
$$
U(x, y) = -4U \cos \left( \frac{2\pi x}{a} \right) \cos \left( \frac{2\pi y}{a} \right).
$$
Using the fundamental equations, the energy gap at the corner of the Brillouin zone $\left( \frac{\pi}{a}, \frac{\pi}{a} \right)$ is approximately ().

Options:
A. $$
4 U
$$
B. $$
0
$$
C. $$
\frac{3U}{2}
$$
D. $$
U
$$
E. $$
\frac{U}{2}
$$
F. $$
\frac{U}{4}
$$
G. $$
-2 U
$$
H. $$
2 U
$$
I. $$
-4 U
$$
J. $$
3 U
$$
Answer: H
The monovalent four-angle structure within the metal features open orbit connections relative to the Brillouin zone boundary. The opposing interface distance is $\mathbf{G}=2 \times 10^{8} \mathrm{cm}^{-1}$. The magnetic field $B=10^{3} \mathrm{gauss}=10^{-1} \mathrm{tesla}$, perpendicular to the open orbit plane. What is the order of magnitude of the motion period in k-space? Assume $\upsilon \approx 10^{8} \mathrm{cm/sec}$.

Options:
A. $$
8. 7 2 \times7 0^{-1 3} (sec)
$$
B. $$
4. 5 7 \times2 0^{-1 2} (sec)
$$
C. $$
3. 9 5 \times8 0^{-1 0} (sec)
$$
D. $$
7. 8 2 \times5 0^{-1 3} (sec)
$$
E. $$
6. 8 3 \times3 0^{-1 1} (sec)
$$
F. $$
1. 3 1 \times1 0^{-1 0} (sec)
$$
G. $$
0. 2 9 \times6 0^{-1 2} (sec)
$$
H. $$
5. 1 2 \times4 0^{-1 2} (sec)
$$
I. $$
9. 1 0 \times9 0^{-1 4} (sec)
$$
J. $$
2. 6 4 \times1 0^{-1 1} (sec)
$$
Answer: F
Derive the leading term of the heat capacity of a three-dimensional ferromagnet at low temperatures $k_{B}T<<J$ using the formula of dispersion relation of magnon $\omega = A k^{2}$, which is ().

Options:
A. $$
\approx0. 2 0 1 k_{B} \Big( 2k_{B} T / \hbar A \Big)^{2 / 5}
$$
B. $$
\approx0. 0 9 8 k_{B} \Big( k_{B} T / \hbar A \Big)^{5 / 2}
$$
C. $$
\approx0. 2 2 6 k_{B} \Big( 2k_{B} T / \hbar A \Big)^{1 / 2}
$$
D. $$
\approx0. 0 1 9 k_{B} \Big( k_{B} T / \hbar A \Big)^{1 / 5}
$$
E. $$
\approx0. 1 5 5 k_{B} \Big( 3k_{B} T / \hbar A \Big)^{3 / 5}
$$
F. $$
\approx0. 1 3 7 k_{B} \Big( 3k_{B} T / \hbar A \Big)^{2 / 3}
$$
G. $$
\approx0. 0 8 2 k_{B} \Big( k_{B} T / \hbar 2A \Big)^{2 / 3}
$$
H. $$
\approx0. 1 0 9 k_{B} \Big( 0.5 k_{B} T / \hbar A \Big)^{4 / 3}
$$
I. $$
\approx0. 1 1 3 k_{B} \Big( k_{B} T / \hbar A \Big)^{3 / 2}
$$
J. $$
\approx0. 1 7 3 k_{B} \Big( 1.5 k_{B} T / \hbar A \Big)^{1 / 3}
$$
Answer: I
If the interaction energy of a crystal can be expressed as
$$
u(r) = -\frac{\alpha}{r^m} + \frac{\beta}{r^n}
$$
Given $m = 2, n = 10, r_0 = 3 \, \text{Å}, \, W = 4 \, \text{eV}$, determine the values of $\alpha$ and $\beta$.

Options:
A. $$
\alpha=
4 5 \, \mathrm{e V} \cdot\AA,
$$
$$
\beta=5. 9 0 \times1 0^{-1 6} \mathrm{\, e V} \cdot\AA
$$
B. $$
\alpha=
6 4 \, \mathrm{e V} \cdot\AA,
$$
$$
\beta=1. 8 5 \times1 0^{-1 7} \mathrm{\, e V} \cdot\AA
$$
C. $$
\alpha=
5 5 \, \mathrm{e V} \cdot\AA,
$$
$$
\beta=8. 1 8 \times1 0^{-1 7} \mathrm{\, e V} \cdot\AA
$$
D. $$
\alpha=
9 0 \, \mathrm{e V} \cdot\AA,
$$
$$
\beta=3. 2 4 \times1 0^{-1 5} \mathrm{\, e V} \cdot\AA
$$
E. $$
\alpha=
3 2 \, \mathrm{e V} \cdot\AA,
$$
$$
\beta=6. 0 2 \times1 0^{-1 7} \mathrm{\, e V} \cdot\AA
$$
F. $$
\alpha=
1 0 \, \mathrm{e V} \cdot\AA,
$$
$$
\beta=9. 5 2 \times1 0^{-1 5} \mathrm{\, e V} \cdot\AA
$$
G. $$
\alpha=
2 8 \, \mathrm{e V} \cdot\AA,
$$
$$
\beta=4. 9 3 \times1 0^{-1 6} \mathrm{\, e V} \cdot\AA
$$
H. $$
\alpha=
1 3 \, \mathrm{e V} \cdot\AA,
$$
$$
\beta=7. 4 5 \times1 0^{-1 6} \mathrm{\, e V} \cdot\AA
$$
I. $$
\alpha=
7 7 \, \mathrm{e V} \cdot\AA,
$$
$$
\beta=8. 7 6 \times1 0^{-1 6} \mathrm{\, e V} \cdot\AA
$$
J. $$
\alpha=
8 9 \, \mathrm{e V} \cdot\AA,
$$
$$
\beta=2. 3 0 \times1 0^{-1 5} \mathrm{\, e V} \cdot\AA
$$
Answer: A
Given one-dimensional composite lattice $m=5 \times 1.67 \times 10^{-24} \, \mathrm{g}, \, \frac{M}{m}=4, \beta=1.5 \times 10^{1} \, \mathrm{N/m}$, $(1.5 \times 10^{4} \, \mathrm{dyn/cm})$, find the average number of phonons at 300K, which is ().

Options:
A. 0.123,0.398,0.765
B. 0.367,0.433,0.897
C. 0.234,0.309,0.765
D. 0.415,0.358,0.912
E. 0.231,0.291,0.782
F. 0.341,0.274,0.863
G. 0.452,0.219,0.813
H. 0.222,0.279,0.876
I. 0.289,0.194,0.654
J. 0.345,0.265,0.678
Answer: H
In a one-dimensional lattice with a lattice constant of 2.5 Å, the time required for an electron to move from the bottom of the energy band to the top when subjected to external electric field of $10^2 V/m$ is estimatedly (); the time is estimatedly () when the external electric field is $10^7 V/m$.

Options:
A. $$
7. 3 1 \times1 0^{-8} \, \mathrm{s}
$$
$$
7. 3 1 \times1 0^{-13} \, \mathrm{s}
$$
B. $$
5. 4 7 \times1 0^{-8} \, \mathrm{s}
$$
$$
5. 4 7 \times1 0^{-13} \, \mathrm{s}
$$
C. $$
9. 1 5 \times1 0^{-8} \, \mathrm{s}
$$
$$
9. 1 5 \times1 0^{-13} \, \mathrm{s}
$$
D. $$
8. 6 3 \times1 0^{-8} \, \mathrm{s}
$$
$$
8. 6 3 \times1 0^{-13} \, \mathrm{s}
$$
E. $$
8. 2 7 \times1 0^{-8} \, \mathrm{s}
$$
$$
8. 2 7 \times1 0^{-13} \, \mathrm{s}
$$
F. $$
6. 8 9 \times1 0^{-8} \, \mathrm{s}
$$
$$
6. 8 9 \times1 0^{-13} \, \mathrm{s}
$$
G. $$
8. 0 1 \times1 0^{-8} \, \mathrm{s}
$$
$$
8. 0 1 \times1 0^{-13} \, \mathrm{s}
$$
H. $$
7. 9 2 \times1 0^{-8} \, \mathrm{s}
$$
$$
7. 9 2 \times1 0^{-13} \, \mathrm{s}
$$
I. $$
6. 2 4 \times1 0^{-8} \, \mathrm{s}
$$
$$
6. 2 4 \times1 0^{-13} \, \mathrm{s}
$$
J. $$
9. 5 8 \times1 0^{-8} \, \mathrm{s}
$$
$$
9. 5 8 \times1 0^{-13} \, \mathrm{s}
$$
Answer: E
For a thermodynamic system undergoing a state change, the relationship between pressure and volume is given by \( p V^{1.3} = \text{constant} \). If the initial state of the thermodynamic system is \( p_{1} = 600 \ \mathrm{kPa}, \ V_{1} = 0.3 \ \mathrm{m}^{3} \), what is the amount of work done by the system when the volume expands to \( V_{2} = 0.5 \ \mathrm{m}^{3} \)?

Options:
A. $$
1 9. 8 7 k J
$$
B. $$
4 5. 4 4 k J
$$
C. $$
4 1. 9 3 k J
$$
D. $$
2 3. 7 1 k J
$$
E. $$
2 8. 1 9 k J
$$
F. $$
3 2. 8 5 k J
$$
G. $$
3 7. 6 2 k J
$$
H. $$
3 4. 3 6 k J
$$
I. $$
4 7. 0 7 k J
$$
J. $$
2 5. 5 0 k J
$$
Answer: H
A fixed quantity of gas absorbs 50 kJ of heat, and at the same time, the thermodynamic energy increases by 80 kJ. How much external work is done?

Options:
A. $$
1 0 k J
$$
B. $$
0 k J
$$
C. $$
2 0 k J
$$
D. $$
3 0 k J
$$
E. $$
5 0 k J
$$
F. $$
-4 0 k J
$$
G. $$
-2 0 k J
$$
H. $$
4 0 k J
$$
I. $$
-3 0 k J
$$
J. $$
-1 0 k J
$$
Answer: I
If the specific heat capacity is a monotonically increasing function of temperature $t$, when $t_{2} > t_{1}$, which among the average specific heat capacities $c \mid_{0}^{t_{1}}$, $c \mid_{0}^{t_{2}}$, $c \mid_{t_{1}}^{t_{2}}$ is the largest? Which one is the smallest?

Options:
A. \( $C \big|_{0}^{t_{2}}$ \) is enhanced,
\( $C \big|_{0}^{t_{1}}$ \) is reduced.
B. \( $C \big|_{t_{1}}^{t_{2}}$ \) is maximized,
\( $C \big|_{0}^{t_{1}}$ \) is minimized.
C. \( $C \big|_{t_{1}}^{t_{2}}$ \) is stabilized,
\( $C \big|_{0}^{t_{1}}$ \) is maximized.
D. \( $C \big|_{0}^{t_{1}}$ \) is maximized,
\( $C \big|_{t_{1}}^{t_{2}}$ \) is minimized.
E. \( $C \big|_{t_{1}}^{t_{2}}$ \) is minimized,
\( $C \big|_{0}^{t_{2}}$ \) is maximized.
F. \( $C \big|_{t_{1}}^{t_{2}}$ \) is optimized,
\( $C \big|_{0}^{t_{2}}$ \) is minimized.
G. \( $C \big|_{0}^{t_{2}}$ \) is altered,
\( $C \big|_{0}^{t_{1}}$ \) is stabilized.
H. \( $C \big|_{0}^{t_{1}}$ \) is stabilized,
\( $C \big|_{t_{1}}^{t_{2}}$ \) is altered.
I. \( $C \big|_{0}^{t_{2}}$ \) is maximized,
\( $C \big|_{t_{1}}^{t_{2}}$ \) is minimized.
Answer: B
The molar mass of a certain gas is 29 $\times 10^{-3}$ kg/mol, and it is heated at constant volume from $t_{1}=320 \ \ \mathbf{C}$ to $t_{2}=940 \ \ \mathrm{C}$. Given the change in thermodynamic energy during the process $\Delta u=700 \ \ \mathrm{kJ} / \mathrm{kg}$, calculate the changes in enthalpy and entropy assuming it is an ideal gas

Options:
A. $$
1 3 8 9. 0 6 J / ( K. k g )
$$
$$
1 3 8 9. 0 6 J / ( K. k g )
$$
B. $$
1 4 7 8. 8 8 J / ( K. k g )
$$
$$
1 4 7 8. 8 8 J / ( K. k g )
$$
C. $$
1 3 9 0. 6 0 J / ( K. k g )
$$
$$
1 3 9 0. 6 0 J / ( K. k g )
$$
D. $$
1 4 0 5. 4 4 J / ( K. k g )
$$
$$
1 4 0 5. 4 4 J / ( K. k g )
$$
E. $$
1 4 9 2. 9 4 J / ( K. k g )
$$
$$
1 4 9 2. 9 4 J / ( K. k g )
$$
F. $$
1 4 4 5. 1 8 J / ( K. k g )
$$
$$
1 4 4 5. 1 8 J / ( K. k g )
$$
G. $$
1 5 2 0. 3 6 J / ( K. k g )
$$
$$
1 5 2 0. 3 6 J / ( K. k g )
$$
H. $$
1 4 1 5. 7 2 J / ( K. k g )
$$$$
898.00kJ / ( K. k g )
$$
I. $$
1 5 1 5. 5 5 J / ( K. k g )
$$
$$
1 5 1 5. 5 5 J / ( K. k g )
$$
J. $$
1 4 3 2. 5 0 J / ( K. k g )
$$
$$
1 4 3 2. 5 0 J / ( K. k g )
$$
Answer: H
An ideal gas absorbed 3,349 kJ of heat in a constant pressure process, given that $c_{V}=0.741 \ \mathrm{kJ} / (\mathrm{kg} \cdot\mathrm{K})$ and $R_{g}=0.297 \ \mathrm{kJ} / (\mathrm{kg} \cdot\mathrm{K})$. Find the work done on the surroundings and the change in thermodynamic energy

Options:
A. $$
9 6 2 \operatorname{K J}
$$
B. $$
9 8 0 \operatorname{K J}
$$
C. $$
9 5 8 \operatorname{K J}
$$
D. $$
9 4 0 \operatorname{K J}
$$
E. $$
9 7 5 \operatorname{K J}
$$
F. $$
9 1 5 \operatorname{K J}
$$
G. $$
9 3 0 \operatorname{K J}
$$
H. $$
9 2 5 \operatorname{K J}
$$
I. $$
9 6 8 \operatorname{K J}
$$
J. $$
8 9 8 \operatorname{K J}
$$
Answer: C
During the compression of gas in the cylinder, work is done on the gas by the external system amounting to $200 kJ/kg$. The change in the thermodynamic energy of the gas is $60 kJ/kg$, and the change in entropy is -$0.274 kJ/(kg·K)$. The gas can exchange heat with the surrounding environment at a temperature of $20^{\circ} \ \mathrm{C}$. Determine the entropy generation per kilogram of gas compressed

Options:
A. $$
0. 2 5 2 k J / k g \cdot K
$$
B. $$
0. 1 7 4 k J / k g \cdot K
$$
C. $$
0. 2 3 0 k J / k g \cdot K
$$
D. $$
0. 1 5 0 k J / k g \cdot K
$$
E. $$
0. 2 6 0 k J / k g \cdot K
$$
F. $$
0. 2 1 0 k J / k g \cdot K
$$
G. $$
0. 3 5 8 k J / k g \cdot K
$$
H. $$
0. 2 0 4 k J / k g \cdot K
$$
I. $$
0. 1 6 8 k J / k g \cdot K
$$
J. $$
0. 1 9 8 k J / k g \cdot K
$$
Answer: H
The known temperature of the room walls is $16^{\circ} \ \mathrm{C}$. If the indoor air temperature is $21^{\circ} \ \mathrm{C}$, what is the maximum relative humidity allowed for the air to prevent condensation from occurring on the walls?

Options:
A. 80%
B. 70%
C. 77%
D. 73%
E. 65%
F. 72%
G. 75%
H. 78%
I. 60%
J. 68%
Answer: D
For a certain gas turbine device, the intake conditions are $p_{1}=0.1 \ \text{MPa}$ and $t_{1}=27^{\circ} \ \text{C}$, with a cycle pressure ratio $\pi=4$. The amount of heat added in the combustion chamber is $333 \ \text{kJ/kg}$, and it undergoes adiabatic expansion to $0.1 \ \text{MPa}$. Assuming a constant specific heat capacity, find the maximum temperature of the cycle and the thermal efficiency of the cycle

Options:
A. $$
9 0 0. 1 {{K}}
$$
$$
2 5. 6 \%
$$
B. $$
7 3 0. 4 {{K}}
$$
$$
3 0. 8 \%
$$
C. $$
7 9 5. 2 {{K}}
$$
$$
2 9. 3 \%
$$
D. $$
8 5 0. 7 {{K}}
$$
$$
3 1. 9 \%
$$
E. $$
6 8 0. 6 {{K}}
$$
$$
3 3. 2 \%
$$
F. $$
7 7 7. 3 {{K}}
$$
$$
3 2. 7 \%
$$
G. $$
6 5 0. 5 {{K}}
$$
$$
4 5. 1 \%
$$
H. $$
8 9 0. 2 {{K}}
$$
$$
2 7. 4 \%
$$
I. $$
7 1 5. 9 {{K}}
$$
$$
3 5. 2 \%
$$
J. $$
8 0 0. 8 {{K}}
$$
$$
2 8. 9 \%
$$
Answer: F
The hot junction of the thermocouple can be approximated as spherical, with a known diameter of $d = 0.5 \, \mathrm{mm}$. The material's density is $\rho = 8500 \, \mathrm{kg/m}^3$, and the specific heat capacity is $c = 400 \, \mathrm{J/(kg \cdot K)}$. The initial temperature of the thermocouple is $25^\circ \mathrm{C}$. It is suddenly placed in an air stream at $120^\circ \mathrm{C}$, where the surface heat transfer coefficient between the thermocouple surface and the air stream is $90 \, \mathrm{W/(m}^2 \cdot \mathrm{K})$. Find: 1) the time constant of the thermocouple

Options:
A. $$
2 8. 9 \, \mathrm{s e c} \, o n d s
$$
B. $$
9 5. 7 \, \mathrm{s e c} \, o n d s
$$
C. $$
4 0. 1 \, \mathrm{s e c} \, o n d s
$$
D. $$
1 0. 5 \, \mathrm{s e c} \, o n d s
$$
E. $$
1 8. 3 \, \mathrm{s e c} \, o n d s
$$
F. $$
5 8. 0 \, \mathrm{s e c} \, o n d s
$$
G. $$
4 5. 8 \, \mathrm{s e c} \, o n d s
$$
H. $$
6 5. 6 \, \mathrm{s e c} \, o n d s
$$
I. $$
7 2. 4 \, \mathrm{s e c} \, o n d s
$$
J. $$
3 5. 2 \, \mathrm{s e c} \, o n d s
$$
Answer: F
The transmission coefficient of the window glass in a certain greenhouse for radiation with wavelengths in the range of 0.4 to 2.5 micrometers is approximately 0.95, while for other wavelengths, it is approximately 0. The sun can be approximated as a black body with a temperature of 5800K, and the objects inside the greenhouse can be considered as black bodies with a temperature of 30°C. Try to calculate the portion of total radiation from both solar radiation and indoor object radiation that passes through the window glass

Options:
A. $$
90.0\%,10\%
$$
B. $$
75.0\%,1\%
$$
C. $$
85.7\%,8\%
$$
D. $$
66.5\%,5\%
$$
E. $$
60.0\%,0\%
$$
F. $$
82.3\%,3\%
$$
G. $$
50.9\%,2\%
$$
H. $$
79.9\%,0\%
$$
I. $$
95.2\%,6\%
$$
J. $$
70.1\%,4\%
$$
Answer: H
The wall of a cold storage room is constructed from three layers of materials from inside to outside: steel plate, slag wool, and asbestos board, with corresponding thicknesses of 0.8 mm, 150 mm, and 10 mm. The thermal conductivities of these materials are $45 \cdot \mathbf{W} / (\mathbf{m} \cdot \mathbf{K})$, $0.07 \cdot \mathbf{W} / (\mathbf{m} \cdot \mathbf{K})$, and $0.1 \cdot \mathbf{W} / (\mathbf{m} \cdot \mathbf{K})$, respectively. The temperatures inside and outside the cold storage room are$-2^{\circ} \ \mathrm{C}$ and $30^{\circ} \ \mathrm{C}$, respectively. The surface heat transfer coefficients on the inner and outer wall surfaces are $2 \mathrm{W} / (\mathrm{m}^2 \cdot \mathrm{K})$ and $3 \mathrm{W} / (\mathrm{m}^2 \cdot \mathrm{K})$. To maintain a constant temperature inside the cold storage, determine the amount of heat the refrigeration equipment needs to remove from the cold storage per hour

Options:
A. $$
3 7 4 4 8. 7 s J
$$
B. $$
4 8 9 4 9. 0 s J
$$
C. $$
6 1 7 2 7. 1 s J
$$
D. $$
4 1 2 9 4. 5 s J
$$
E. $$
4 5 8 3 0. 2 s J
$$
F. $$
3 2 5 8 1. 6 s J
$$
G. $$
2 7 1 7 6. 3 s J
$$
H. $$
3 8 6 2 5. 8 s J
$$
I. $$
5 5 4 1 3. 9 s J
$$
J. $$
2 8 9 9 0. 4 s J
$$
Answer: A
The substance with a tetragonal structure, has a relative atomic mass $A_{\mathrm{r}} = 114.82$, an atomic radius $r = 0.1625 \ \mathrm{nm}$, and lattice constants
$$
a = 0.3252 \, \mathrm{nm}, \, c = 0.4946 \, \mathrm{nm}
$$
with a density $\rho = 7.286 \, \mathrm{g/cm}^{3}$. What is the compactness of ln?

Options:
A. 0.6852
B. 0.6915
C. 0.6934
D. 0.6921
E. 0.6897
F. 0.6938
G. 0.6889
H. 0.6865
I. 0.6904
J. 0.6873
Answer: J
$\mathsf{M g O}$ has a NaCl-type structure. The ionic radius of $\mathbf{M} g^{2+}$ is $0.078 \ \mathrm{n m}$, and the ionic radius of $\mathrm{O}^{2-}$ is $0.132 \ \mathrm{n m}$. The density $(\rho)$ and packing density $(K)$ of MgO is ().

Options:
A. $$
3. 5 7 2 \, ( \mathbf{g} / \mathbf{c m}^{3} )
$$
$$
0. 6 3 8
$$
B. $$
3. 7 1 1 \, ( \mathbf{g} / \mathbf{c m}^{3} )
$$
$$
0. 7 6 5
$$
C. $$
3. 9 4 3 \, ( \mathbf{g} / \mathbf{c m}^{3} )
$$
$$
0. 8 0 9
$$
D. $$
3. 4 9 8 \, ( \mathbf{g} / \mathbf{c m}^{3} )
$$
$$
0. 5 5 3
$$
E. $$
3. 8 2 9 \, ( \mathbf{g} / \mathbf{c m}^{3} )
$$
$$
0. 7 2 1
$$
F. $$
3. 6 1 3 \, ( \mathbf{g} / \mathbf{c m}^{3} )
$$
$$
0. 6 2 7
$$
G. $$
3. 6 8 7 \, ( \mathbf{g} / \mathbf{c m}^{3} )
$$
$$
0. 7 0 3
$$
H. $$
3. 1 5 4 \, ( \mathbf{g} / \mathbf{c m}^{3} )
$$
$$
0. 4 9 6
$$
I. $$
4. 1 0 5 \, ( \mathbf{g} / \mathbf{c m}^{3} )
$$
$$
0. 9 4 2
$$
J. $$
2. 9 8 0 \, ( \mathbf{g} / \mathbf{c m}^{3} )
$$
$$
0. 5 2 0
$$
Answer: F
Assume the vibration frequency of the atoms surrounding a vacancy in Cu is $10^{13} \, \mathrm{s}^{-1}$, $\Delta E$ is $0.15 \times 10^{-18} \mathrm{J}, \operatorname{exp}(\Delta S_{w} / k)$ is approximately 1. What is the vacancy migration frequency at 700K and room temperature ($27^\circ C$) ?

Options:
A. $$
1. 8 9 4 \times1 0^{9} ( s^{-1} )
$$
$$
1. 9 7 5 \times1 0^{-6} ( s^{-1} )
$$
B. $$
2. 7 9 3 \times1 0^{7} ( s^{-1} )
$$
$$
2. 8 1 4 \times1 0^{-1} ( s^{-1} )
$$
C. $$
1. 3 6 5 \times1 0^{5} ( s^{-1} )
$$
$$
1. 4 5 9 \times1 0^{-10} ( s^{-1} )
$$
D. $$
3. 4 1 8 \times1 0^{6} ( s^{-1} )
$$
$$
3. 5 6 9 \times1 0^{-3} ( s^{-1} )
$$
E. $$
5. 2 6 1 \times1 0^{8} ( s^{-1} )
$$
$$
5. 3 2 5 \times1 0^{-9} ( s^{-1} )
$$
F. $$
2. 1 6 5 \times1 0^{7} ( s^{-1} )
$$
$$
2. 2 0 7 \times1 0^{-2} ( s^{-1} )
$$
G. $$
9. 4 2 8 \times1 0^{2} ( s^{-1} )
$$
$$
7. 1 3 8 \times1 0^{-8} ( s^{-1} )
$$
H. $$
8. 1 4 5 \times1 0^{3} ( s^{-1} )
$$
$$
3. 6 2 9 \times1 0^{-5} ( s^{-1} )
$$
I. $$
6. 5 1 6 \times1 0^{4} ( s^{-1} )
$$
$$
6. 2 3 8 \times1 0^{-7} ( s^{-1} )
$$
J. $$
4. 7 3 2 \times1 0^{5} ( s^{-1} )
$$
$$
4. 8 4 1 \times1 0^{-4} ( s^{-1} )
$$
Answer: F
When the temperature decreases from 600°C to 300°C, the equilibrium concentration of vacancies in the Ge crystal decreases by six orders of magnitude. What is the vacancy formation energy in the Ge crystal?

Options:
A. $$
1. 9 8 ( \mathrm{e V} )
$$
B. $$
2. 1 2 ( \mathrm{e V} )
$$
C. $$
1. 2 3 ( \mathrm{e V} )
$$
D. $$
1. 5 0 ( \mathrm{e V} )
$$
E. $$
1. 4 9 ( \mathrm{e V} )
$$
F. $$
1. 6 4 ( \mathrm{e V} )
$$
G. $$
3. 0 0 ( \mathrm{e V} )
$$
H. $$
1. 7 5 ( \mathrm{e V} )
$$
I. $$
2. 9 5 ( \mathrm{e V} )
$$
J. $$
2. 5 6 ( \mathrm{e V} )
$$
Answer: A
Carburizing steel in a carbon-rich environment can harden its surface. It is known that during this carburizing heat treatment at $1000 \mathrm{~C}$, the carbon content decreases from $x=5\%$ to $x=4\%$ at a location $1 \sim 2 \, \mathrm{mm}$ from the steel's surface. What is the influx of carbon atoms $J \left( \mathrm{atoms} / \mathrm{m}^{2} \, \mathrm{s} \right)$ in the near-surface region? (The density of $\gamma$-Fe at 1000°C is $7.63 \, \mathrm{g/cm^{3}}$, the diffusion coefficient of carbon in $\gamma-\mathrm{Fe}$ is $D_{0}=2.0 \times 10^{-5} \, \mathrm{m}^{2}/\mathrm{s}$, and the activation energy $Q = 142 \, \mathrm{kJ/mol}$.)

Options:
A. $$
2. 3 5 \times1 0^{1 9} \mathrm{a t} / ( \mathrm{m}^{2} \mathrm{s} )
$$
B. $$
2. 0 5 \times1 0^{1 9} \mathrm{a t} / ( \mathrm{m}^{2} \mathrm{s} )
$$
C. $$
2. 2 5 \times1 0^{1 9} \mathrm{a t} / ( \mathrm{m}^{2} \mathrm{s} )
$$
D. $$
2. 4 0 \times1 0^{1 9} \mathrm{a t} / ( \mathrm{m}^{2} \mathrm{s} )
$$
E. $$
1. 8 0 \times1 0^{1 9} \mathrm{a t} / ( \mathrm{m}^{2} \mathrm{s} )
$$
F. $$
3. 0 0 \times1 0^{1 9} \mathrm{a t} / ( \mathrm{m}^{2} \mathrm{s} )
$$
G. $$
2. 6 5 \times1 0^{1 9} \mathrm{a t} / ( \mathrm{m}^{2} \mathrm{s} )
$$
H. $$
2. 4 5 \times1 0^{1 9} \mathrm{a t} / ( \mathrm{m}^{2} \mathrm{s} )
$$
I. $$
3. 2 5 \times1 0^{1 9} \mathrm{a t} / ( \mathrm{m}^{2} \mathrm{s} )
$$
J. $$
2. 5 0 \times1 0^{1 9} \mathrm{a t} / ( \mathrm{m}^{2} \mathrm{s} )
$$
Answer: H
The critical resolved shear stress for aluminum single crystals at room temperature is $\tau_{c}=7.9 \times 10^{s}$ Pa. If an aluminum single crystal specimen is subjected to a tensile test at room temperature with the tensile axis in the [123] direction, what is the stress required to cause the specimen to yield?

Options:
A. $$
1. 6 9 \ ( M \mathrm{P a} )
$$
B. $$
2. 6 4 \ ( M \mathrm{P a} )
$$
C. $$
3. 5 6 \ ( M \mathrm{P a} )
$$
D. $$
4. 2 8 \ ( M \mathrm{P a} )
$$
E. $$
1. 8 1 \ ( M \mathrm{P a} )
$$
F. $$
3. 0 7 \ ( M \mathrm{P a} )
$$
G. $$
3. 3 3 \ ( M \mathrm{P a} )
$$
H. $$
2. 8 9 \ ( M \mathrm{P a} )
$$
I. $$
2. 0 5 \ ( M \mathrm{P a} )
$$
J. $$
4. 5 7 \ ( M \mathrm{P a} )
$$
Answer: A
Three point bending test is commonly used to test the mechanical behavior of ceramic materials. There is A circular section of $\mathrm{A l_{2} O_{3}}$sample, the section radius of $r= 3.5 \mathrm{~ mm}$, the distance between the two fulminates is 50mm, when the load reaches 950N, the sample fracture. Question: When the fulcrum spacing is 40mm, at what load will another sample of the same material with a square section with a side length of 12mm break?

Options:
A. $$
1 1 6 2 3, 2 ( \mathrm{N} )
$$
B. $$
9 0 4 8 8, 8 ( \mathrm{N} )
$$
C. $$
1 2 3 8 6, 6 ( \mathrm{N} )
$$
D. $$
9 6 5 4 5, 7 ( \mathrm{N} )
$$
E. $$
9 1 3 2 0, 5 ( \mathrm{N} )
$$
F. $$
1 3 0 6 8, 4 ( \mathrm{N} )
$$
G. $$
9 8 9 7 9, 1 ( \mathrm{N} )
$$
H. $$
1 0 1 5 4, 9 ( \mathrm{N} )
$$
I. $$
8 3 7 7 1, 3 ( \mathrm{N} )
$$
J. $$
1 0 7 4 2, 0 ( \mathrm{N} )
$$
Answer: H
At a pressure of $1.013 \times 10^{5} \mathrm{~Pa (1 atm)}$, homogeneous nucleation occurs in supercooled liquid pure nickel at a degree of supercooling of 319°C. Assume the critical nucleus radius is 1 nm, the melting point of pure nickel is 1726 K, the latent heat of fusion $L_{\mathrm{m}} = 18,075 \;\mathrm{J/mol}$, and the molar volume $V = 6.6 \; \mathrm{cm}^{3}/\mathrm{mol}$. What are the liquid-solid interfacial energy and the critical nucleation work of pure nickel?

Options:
A. $$
5. 4 3 \times1 0^{-2 0} ( \mathrm{J} )
$$
B. $$
1. 7 3 \times1 0^{-1 9} ( \mathrm{J} )
$$
C. $$
2. 3 5 \times1 0^{-1 6} ( \mathrm{J} )
$$
D. $$
4. 8 9 \times1 0^{-1 7} ( \mathrm{J} )
$$
E. $$
1. 0 6 \times1 0^{-1 8} ( \mathrm{J} )
$$
F. $$
3. 0 1 \times1 0^{-1 8} ( \mathrm{J} )
$$
G. $$
6. 1 9 \times1 0^{-1 5} ( \mathrm{J} )
$$
H. $$
7. 8 6 \times1 0^{-1 7} ( \mathrm{J} )
$$
I. $$
8. 0 4 \times1 0^{-1 8} ( \mathrm{J} )
$$
J. $$
9. 2 7 \times1 0^{-1 9} ( \mathrm{J} )
$$
Answer: E
The dislocation density in cold-worked metal is $10^{10} /\mathrm{cm}^{2}$. Assuming the energy per unit length of dislocation is $10^{-3} \operatorname{erg}/\operatorname{cm}$, the surface energy is $500 \operatorname{erg/cm}^{2}$, and the dislocation density on the two sides of the grain boundary is $10^{10} /\mathrm{cm}^{2}$ and 0, respectively. What is the critical grain size for recrystallization of the metal when heated?

Options:
A. $$
6 \mu\mathrm{m}
$$
B. $$
1.5 \mu\mathrm{m}
$$
C. $$
0.8 \mu\mathrm{m}
$$
D. $$
1 \mu\mathrm{m}
$$
E. $$
0.3 \mu\mathrm{m}
$$
F. $$
0.5 \mu\mathrm{m}
$$
G. $$
3 \mu\mathrm{m}
$$
H. $$
5 \mu\mathrm{m}
$$
I. $$
4 \mu\mathrm{m}
$$
J. $$
2 \mu\mathrm{m}
$$
Answer: D
The porosity of sintered $\mathrm{Al_{2}O_{3}}$ is known to be 5%, with its $E \!=\! 370 \mathrm{~GPa}$. If another sintered $\mathrm{Al_{2}O_{3}}$ is measured with $E \!=\! 270 \mathrm{~GPa}$, what is its porosity?

Options:
A. $$
19. 61 \%
$$
B. $$
19. 4 5 \%
$$
C. $$
18. 78 \%
$$
D. $$
18. 6 3 \%
$$
E. $$
2 0. 94 \%
$$
F. $$
19. 0 3 \%
$$
G. $$
1 8. 62 \%
$$
H. $$
18. 47 \%
$$
I. $$
18. 5 9 \%
$$
J. $$
14. 8 6 \%
$$
Answer: A
In the early atomic theory, J.J. Thomson proposed the "plum pudding" model of the atom, where a sphere of positive charge with a radius \(a\) and total charge \(Ze\) is considered, where \(Z\) is an integer and \(e\) is the elementary charge unit. The negatively charged electrons \(-e\) are viewed as point charges embedded within the sphere of positive charge. Provide the expression for the frequency of electron motion is().

Options:
A. $$
\omega=\sqrt{\frac{k} {m}}=\sqrt{\frac{2e^{2}} {4 \pi\varepsilon_{0} a^{4} m}}
$$
B. $$
\omega=\sqrt{\frac{k} {m}}=\sqrt{\frac{Z^{2} e^{2}} {4 \pi\varepsilon_{0} a^{5} m}}
$$
C. $$
\omega=\sqrt{\frac{k} {m}}=\sqrt{\frac{e^{2}} {2 \pi\varepsilon_{0} a^{3} m}}
$$
D. $$
\omega=\sqrt{\frac{k} {m}}=\sqrt{\frac{e^{2}} {4 \pi\varepsilon_{0} a^{3} Zm}}
$$
E. $$
\omega=\sqrt{\frac{k} {m}}=\sqrt{\frac{2e^{2}} {8 \pi\varepsilon_{0} a^{3} m^{2}}}
$$
F. $$
\omega=\sqrt{\frac{k} {m}}=\sqrt{\frac{e^{2}} {4 \pi\varepsilon_{0} a^{4} Zm}}
$$
G. $$
\omega=\sqrt{\frac{k} {m}}=\sqrt{\frac{e^{2}} {4 \pi\varepsilon_{0} a^{3} m}}
$$
H. $$
\omega=\sqrt{\frac{k} {m}}=\sqrt{\frac{Ze^{2}} {4 \pi\varepsilon_{0} a^{2} m}}
$$
I. $$
\omega=\sqrt{\frac{k} {m}}=\sqrt{\frac{e} {4 \pi\varepsilon_{0} a^{2} m^{2}}}
$$
J. $$
\omega=\sqrt{\frac{k} {m}}=\sqrt{\frac{e^{3}} {8 \pi\varepsilon_{0} a^{3} m}}
$$
Answer: G
The range of interaction forces between two hydrogen atoms is approximately 0.4 nm. For a gas in thermal equilibrium, estimate numerically below which temperature the atom-atom scattering is predominantly s-wave.

Options:
A. $$
\approx0.01 ( \mathrm{K} )
$$
B. $$
\approx200 ( \mathrm{K} )
$$
C. $$
\approx20 ( \mathrm{K} )
$$
D. $$
\approx0.1 ( \mathrm{K} )
$$
E. $$
\approx1 ( \mathrm{K} )
$$
F. $$
\approx100 ( \mathrm{K} )
$$
G. $$
\approx500 ( \mathrm{K} )
$$
H. $$
\approx10 ( \mathrm{K} )
$$
I. $$
\approx50 ( \mathrm{K} )
$$
J. $$
\approx5 ( \mathrm{K} )
$$
Answer: E
Compared to the Compton wavelength of an electron, the Bohr radius of a hydrogen atom is approximately its ().

Options:
A. 35 times
B. 15 times
C. 50 times
D. 30 times
E. 10 times
F. 40 times
G. 20 times
H. 27 times
I. 25 times
J. 11 times
Answer: G
In an atom with atomic number $Z$, the average speed of an electron in the first Bohr orbit (expressed in units of the speed of light $c$) is ().

Options:
A. $$
\frac{Z}{a}
$$
B. $$
\frac{a}{Z}
$$
C. $$
Z
$$
D. $$
a Z
$$
E. $$
a^2 Z^2
$$
F. $$
a ^2Z
$$
G. $$
a \sqrt{Z}
$$
H. $$
Z^{1/2}
$$
I. $$
Z^2 a
$$
J. $$
Z^3
$$
Answer: D
The ground state of a potassium atom is known to be 4s. The wavelength of the first line in the principal series of the potassium atom is $\lambda_{1} = 766.5 \mathrm{nm}$. The short-wave limit of the principal series is $\lambda_{\infty} = 285.8 \mathrm{nm}$. Find the quantum number correction values for the $4\mathrm{s}$ and $4\mathrm{p}$ spectral terms.

Options:
A. $$\Delta _{4s}=2.12,~\Delta _{4p}=1.88$$
B. $$\Delta _{4s}=2.25,~\Delta _{4p}=1.65$$
C. $$\Delta _{4s}=2.10,~\Delta _{4p}=1.90$$
D. $$\Delta _{4s}=2.20,~\Delta _{4p}=1.85$$
E. $$\Delta _{4s}=2.05,~\Delta _{4p}=1.95$$
F. $$\Delta _{4s}=2.28,~\Delta _{4p}=1.72$$
G. $$\Delta _{4s}=2.30,~\Delta _{4p}=1.70$$
H. $$\Delta _{4s}=2.18,~\Delta _{4p}=1.82$$
I. $$\Delta _{4s}=2.15,~\Delta _{4p}=1.80$$
J. $$\Delta _{4s}=2.23,~\Delta _{4p}=1.76$$
Answer: J
The measured $^3 \mathbf{P}_{2}$ energy level of a certain atom splits into five sub-levels, with the ratio of the intervals between adjacent sub-levels being $9 : 7 : 5 : 3.$ Use the interval rule to determine the nuclear spin quantum number $I$ of this atom and the total angular momentum quantum number $F$ corresponding to each sub-level is().

Options:
A. $$
I=2 / 2
$$
$$
F=4 / 2 \,, 2 / 2 \,, 0 / 2 \,, -2 / 2 \,, -4 / 2
$$
B. $$
I=3 / 2
$$
$$
F=5 / 2 \,, 3 / 2 \,, 1 / 2 \,, -1 / 2 \,, -3 / 2
$$
C. $$
I=5 / 2
$$
$$
F=9 / 2 \,, 7 / 2 \,, 5 / 2 \,, 3 / 2 \,, 1 / 2
$$
D. $$
I=6 / 2
$$
$$
F=11 / 2 \,, 9 / 2 \,, 7 / 2 \,, 5 / 2 \,, 3 / 2
$$
E. $$
I=3 / 2
$$
$$
F=7 / 2 \,, 5 / 2 \,, 3 / 2 \,, 1 / 2 \,, -1 / 2
$$
F. $$
I=4 / 2
$$
$$
F=10 / 2 \,, 8 / 2 \,, 6 / 2 \,, 4 / 2 \,, 2 / 2
$$
G. $$
I=2 / 2
$$
$$
F=6 / 2 \,, 4 / 2 \,, 2 / 2 \,, 0 / 2 \,, -2 / 2
$$
H. $$
I=5 / 2
$$
$$
F=8 / 2 \,, 6 / 2 \,, 4 / 2 \,, 2 / 2 \,, 0 / 2
$$
I. $$
I=3 / 2
$$
$$
F=9 / 2 \,, 6 / 2 \,, 3 / 2 \,, 0 / 2 \,, -3 / 2
$$
J. $$
I=4 / 2
$$
$$
F=8 / 2 \,, 6 / 2 \,, 4 / 2 \,, 2 / 2 \,, 0 / 2
$$
Answer: C
A particle moves freely, with the initial wave function at $t=0$ given as
$$
\psi( x, 0 )=( 2 \pi a^{2} )^{-1 / 4} \exp \biggl[ i k_{0} ( x-x_{0} )-\biggl( \frac{x-x_{0}} {2 a} \biggr)^{2} \biggr], \; \; \; a > 0
$$
Find the wave packet width $\Delta x$ at $t=0$.

Options:
A. $$\Delta x = \frac{a}{2}$$
B. $$\Delta x = \frac{\sqrt{a}}{2}$$
C. $$\Delta x = \infty$$
D. $$\Delta x = \frac{1}{\sqrt{a}}$$
E. $$\Delta x = 0$$
F. $$\Delta x = a^2$$
G. $$\Delta x = a$$
H. $$\Delta x = \frac{1}{a}$$
I. $$\Delta x = 2a$$
J. $$\Delta x = \sqrt{a}$$
Answer: G
A particle undergoing one-dimensional motion in potential $V(x)$ has its motion determined by the Hamiltonian $H_{0} = \frac{p^{2}}{2m} + V(x)$, where $p = -\mathrm{i} \hbar \frac{\mathrm{d}}{\mathrm{d}x}$ is the momentum operator. Let $E_{n}^{0}, \ n=1, 2, \cdots$ be the eigenvalues of $H_{0}$. Now consider the Hamiltonian $H = H_{0} + \frac{\lambda}{m} p$, where $\lambda$ is a parameter. Given $\lambda$, $m$, and $E_{n}^{0}$, find the eigenvalues of $H$ .

Options:
A. $$
E_{n}=mE_{n}^{0}-\frac{\lambda^{2}}{p}
$$
B. $$
E_{n}=E_{n}^{0}-\frac{m \lambda^{2}}{2}
$$
C. $$
E_{n}=E_{n}^{0}+2\lambda^{2}m
$$
D. $$
E_{n}=E_{n}^{0}+\lambda E_{n}^{0}
$$
E. $$
E_{n}=E_{n}^{0}+\frac{\lambda^{2}}{4m}
$$
F. $$
E_{n}=\frac{E_{n}^{0}}{2}+\frac{\lambda^{2}}{m}
$$
G. $$
E_{n}=2E_{n}^{0}+\frac{\lambda^{2}}{3m}
$$
H. $$
E_{n}=E_{n}^{0}-\frac{\lambda^{2}} {2 m}
$$
I. $$
E_{n}=E_{n}^{0}+\frac{\lambda}{m} p
$$
J. $$
E_{n}=E_{n}^{0}-\frac{\lambda m}{p}
$$
Answer: H
In a one-dimensional potential $V(x)$, a particle with mass $\mu$ has a wave function given by:
$$
\psi(x, t) = \begin{cases}
\alpha x \mathrm{e}^{-\beta x} \mathrm{e}^{i \gamma t / \hbar}, & x > 0 \\
0, & x < 0
\end{cases}
$$
where $\alpha$, $\beta$, and $\gamma$ are all constants. What is the probability density $\rho(E)$ for the measurement of the total energy $E$ of the particle?

Options:
A. $$\rho(E) = \begin{cases} 1, & E = \gamma \\ 0, & E \neq \gamma \end{cases}$$
B. $$\rho(E) = \begin{cases} 1, & E = 0 \\ 0, & E \neq 0 \end{cases}$$
C. $$\rho(E) = \begin{cases} 1, & E = \beta^2 \\ 0, & E \neq \beta^2 \end{cases}$$
D. $$\rho(E) = \begin{cases} 1, & E = \alpha \\ 0, & E \neq \alpha \end{cases}$$
E. $$\rho(E) = \begin{cases} 1, & E = -2\gamma \\ 0, & E \neq -2\gamma \end{cases}$$
F. $$\rho(E) = \begin{cases} 1, & E = \alpha\beta \\ 0, & E \neq \alpha\beta \end{cases}$$
G. $$\rho(E) = \begin{cases} 1, & E = \beta \\ 0, & E \neq \beta \end{cases}$$
H. $$\rho(E) = \begin{cases} 1, & E = \gamma + \beta \\ 0, & E \neq \gamma + \beta \end{cases}$$
I. $$\rho(E) = \begin{cases} 1, & E = \frac{\gamma}{2} \\ 0, & E \neq \frac{\gamma}{2} \end{cases}$$
J. $$\rho(E) = \begin{cases} 1, & E = -\gamma \\ 0, & E \neq -\gamma \end{cases}$$
Answer: J
If a particle is incident from the right, find the reflection coefficient and transmission coefficient for a one-dimensional step potential, which is given by
$$
V(x) = \left\{\begin{aligned} & V_{0}, & x > 0 \\ & 0, & x < 0 \\ \end{aligned} \right.
$$

Options:
A. $$
\frac{E^{3}} {\left( \sqrt{E-V_{0}}+\sqrt{V_{0}} \right)^{4}}
$$
$$
1 - \frac{E^{3}} {\left( \sqrt{E-V_{0}}+\sqrt{V_{0}} \right)^{4}}
$$
B. $$
\frac{V_{0}} {\left( \sqrt{V_{0}}+\sqrt{E-E} \right)^{4}}
$$
$$
1 - \frac{V_{0}} {\left( \sqrt{V_{0}}+\sqrt{E-E} \right)^{4}}
$$
C. $$
\frac{E^{2}} {\left( \sqrt{V_{0}}+\sqrt{V_{0}} \right)^{4}}
$$
$$
1 - \frac{E^{2}} {\left( \sqrt{V_{0}}+\sqrt{V_{0}} \right)^{4}}
$$
D. $$
\frac{E^{2}} {\left( \sqrt{V_{0}}+\sqrt{E-V_{0}} \right)^{4}}
$$
$$
1 - \frac{E^{2}} {\left( \sqrt{V_{0}}+\sqrt{E-V_{0}} \right)^{4}}
$$
E. $$
\frac{V_{0}^{2}} {\left( \sqrt{E}+\sqrt{2V_{0}-E} \right)^{4}}
$$
$$
1 - \frac{V_{0}^{2}} {\left( \sqrt{E}+\sqrt{2V_{0}-E} \right)^{4}}
$$
F. $$
\frac{V_{0}^{2}} {\left( \sqrt{E}+\sqrt{E-V_{0}} \right)^{4}}
$$
$$
1 - \frac{V_{0}^{2}} {\left( \sqrt{E}+\sqrt{E-V_{0}} \right)^{4}}
$$
G. $$
\frac{V_{0}^{2}} {\left( \sqrt{2E}+\sqrt{V_{0}-E} \right)^{4}}
$$
$$
1 - \frac{V_{0}^{2}} {\left( \sqrt{2E}+\sqrt{V_{0}-E} \right)^{4}}
$$
H. $$
\frac{E^{2}} {\left( \sqrt{E-V_{0}}+\sqrt{E} \right)^{4}}
$$
$$
1 - \frac{E^{2}} {\left( \sqrt{E-V_{0}}+\sqrt{E} \right)^{4}}
$$
I. $$
\frac{V_{0}} {\left( \sqrt{E}+\sqrt{2E-V_{0}} \right)^{4}}
$$
$$
1 - \frac{V_{0}} {\left( \sqrt{E}+\sqrt{2E-V_{0}} \right)^{4}}
$$
J. $$
\frac{V_{0}^{2}} {\left( \sqrt{2E-V_{0}}+\sqrt{E} \right)^{4}}
$$
$$
1 - \frac{V_{0}^{2}} {\left( \sqrt{2E-V_{0}}+\sqrt{E} \right)^{4}}
$$
Answer: F
In an external magnetic field, the Hamiltonian of a spin $\frac{1}{2}$ charged particle with charge $e$ is $H = -\mu_{0} S \cdot B$. Assuming $B = B e_{y}$, what is the matrix form of $S_{z}(t)$?

Options:
A. $$
S_{z} ( t )=\frac{\hbar} {2} \left( \begin{array} {c c} {{\operatorname{s i n} ( g \omega t )}} & {{i \operatorname{c o s} ( g \omega t )}} \\ {{i \operatorname{c o s} ( g \omega t )}} & {{- \operatorname{s i n} ( g \omega t )}} \\ \end{array} \right)
, where \ w = \frac{eB}{2mc}$$
B. $$
S_{z} ( t )=\frac{\hbar} {2} \left( \begin{array} {c c} {{\operatorname{c o s} ( g \omega t )}} & {{\operatorname{s i n} ( 2g \omega t )}} \\ {{\operatorname{s i n} ( 2g \omega t )}} & {{-\operatorname{c o s} ( g \omega t )}} \\ \end{array} \right)
, where \ w = \frac{eB}{2mc}$$
C. $$
S_{z} ( t )=\frac{\hbar} {2} \left( \begin{array} {c c} {{\operatorname{c o s} ( g \omega t )}} & {{i \operatorname{s i n} ( g \omega t )}} \\ {{i \operatorname{s i n} ( g \omega t )}} & {{\operatorname{c o s} ( g \omega t )}} \\ \end{array} \right)
, where \ w = \frac{eB}{2mc}$$
D. $$
S_{z} ( t )=\frac{\hbar} {2} \left( \begin{array} {c c} {{\operatorname{s i n} ( 2g \omega t )}} & {{\operatorname{c o s} ( 2g \omega t )}} \\ {{\operatorname{c o s} ( 2g \omega t )}} & {{-\operatorname{s i n} ( 2g \omega t )}} \\ \end{array} \right)
, where \ w = \frac{eB}{2mc}$$
E. $$
S_{z} ( t )=\frac{\hbar} {2} \left( \begin{array} {c c} {{\operatorname{s i n} ( g \omega t )}} & {{\operatorname{c o s} ( g \omega t )}} \\ {{\operatorname{c o s} ( g \omega t )}} & {{-\operatorname{s i n} ( g \omega t )}} \\ \end{array} \right)
, where \ w = \frac{eB}{2mc}$$
F. $$
S_{z} ( t )=\frac{\hbar} {2} \left( \begin{array} {c c} {{-\operatorname{c o s} ( g \omega t )}} & {{\operatorname{s i n} ( g \omega t )}} \\ {{\operatorname{s i n} ( g \omega t )}} & {{\operatorname{c o s} ( g \omega t )}} \\ \end{array} \right)
, where \ w = \frac{eB}{2mc}$$
G. $$
S_{z} ( t )=\frac{\hbar} {2} \left( \begin{array} {c c} {{\operatorname{c o s} ( g \omega t )}} & {{\operatorname{c o s} ( g \omega t )}} \\ {{\operatorname{c o s} ( g \omega t )}} & {{-\operatorname{c o s} ( g \omega t )}} \\ \end{array} \right)
, where \ w = \frac{eB}{2mc}$$
H. $$
S_{z} ( t )=\frac{\hbar} {2} \left( \begin{array} {c c} {{\operatorname{c o s} ( g \omega t )}} & {{-\operatorname{s i n} ( g \omega t )}} \\ {{-\operatorname{s i n} ( g \omega t )}} & {{\operatorname{c o s} ( g \omega t )}} \\ \end{array} \right)
, where \ w = \frac{eB}{2mc}$$
I. $$
S_{z} ( t )=\frac{\hbar} {2} \left( \begin{array} {c c} {{\operatorname{c o s} ( g \omega t )}} & {{\operatorname{s i n} ( g \omega t )}} \\ {{\operatorname{s i n} ( g \omega t )}} & {{-\operatorname{c o s} ( g \omega t )}} \\ \end{array} \right)
, where \ w = \frac{eB}{2mc}$$
J. $$
S_{z} ( t )=\frac{\hbar} {2} \left( \begin{array} {c c} {{\operatorname{c o s} ( 2g \omega t )}} & {{\operatorname{s i n} ( 2g \omega t )}} \\ {{\operatorname{s i n} ( 2g \omega t )}} & {{-\operatorname{c o s} ( 2g \omega t )}} \\ \end{array} \right)
, where \ w = \frac{eB}{2mc}$$
Answer: I
Calculate the de Broglie wavelength of a 0.050 eV neutron.

Options:
A. $$ 4. 6 1^ {}\AA $$
B. $$ 2. 0 8^ {}\AA $$
C. $$ 2. 4 3^ {}\AA $$
D. $$
1. 2 8^ {}\AA
$$
E. $$ 0. 9 4^ {}\AA $$
F. $$ 1. 1 5^ {}\AA $$
G. $$ 1. 3 2^ {}\AA $$
H. $$ 3. 9 5^ {}\AA $$
I. $$ 1. 6 7^ {}\AA $$
J. $$ 1. 5 0^ {}\AA $$
Answer: D
It is known that paramagnetic materials satisfy Curie's law $M=\frac{C}{T} H$, and the energy density satisfies $U=a T^{4}$ (where $a$ is a constant). What is the magnetization heat of the paramagnetic material when the magnetic field increases from 0 to $H$ at a constant temperature ?

Options:
A. $$
{\frac{G \mu_{0} T^{2} H^{2}} {2}}
$$
B. $$
- {\frac{G \mu_{0} H^{2}} {4 T}}
$$
C. $$
{\frac{G \mu_{0} H^{2}} {2 T}}
$$
D. $$
{\frac{G \mu_{0}^{2} H^{2}} {2 T}}
$$
E. $$
- {\frac{G \mu_{0} H^{2}} {2 T}}
$$
F. $$
- {\frac{G H^{2}} {2 \mu_{0} T}}
$$
G. $$
- {\frac{G \mu_{0} H^{2} T} {2}}
$$
H. $$
{\frac{G \mu_{0} H^{3}} {2 T}}
$$
I. $$
{\frac{G \mu_{0} T H^{2}} {2}}
$$
J. $$
- {\frac{G \mu_{0} H^{2}} {T^{2}}}
$$
Answer: E
A mixed gas of $0.1 \, \text{mol}$ helium and $0.2 \, \text{mol}$ nitrogen, with an initial temperature of $T = 300\, \text{K}$ and a volume of $4\, \text{L}$. Prove that when the mixed gas undergoes adiabatic compression, its temperature and pressure changes can be described by a certain mixing ratio $\gamma$. When the volume decreases by 1%, determine the changes in temperature and pressure, given that the heat capacity ratios of helium and nitrogen are $\gamma_{1} = \frac{5}{3}$ and $\gamma_{2} = \frac{7}{5}$, respectively as().

Options:
A. $$
2 9 8 \ \mathrm{( K )}
$$
$$
1. 6 \times1 0^{5} ( \mathrm{N / m^{2}} )
$$
B. $$
2 7 5 \ \mathrm{( K )}
$$
$$
1. 7 \times1 0^{5} ( \mathrm{N / m^{2}} )
$$
C. $$
3 0 5 \ \mathrm{( K )}
$$
$$
2. 0 \times1 0^{6} ( \mathrm{N / m^{2}} )
$$
D. $$
3 0 2 \ \mathrm{( K )}
$$
$$
2. 0 \times1 0^{5} ( \mathrm{N / m^{2}} )
$$
E. $$
2 9 5 \ \mathrm{( K )}
$$
$$
1. 9 \times1 0^{5} ( \mathrm{N / m^{2}} )
$$
F. $$
3 1 5 \ \mathrm{( K )}
$$
$$
2. 3 \times1 0^{5} ( \mathrm{N / m^{2}} )
$$
G. $$
3 5 0 \ \mathrm{( K )}
$$
$$
3. 0 \times1 0^{5} ( \mathrm{N / m^{2}} )
$$
H. $$
3 4 0 \ \mathrm{( K )}
$$
$$
2. 6 \times1 0^{5} ( \mathrm{N / m^{2}} )
$$
I. $$
2 8 0 \ \mathrm{( K )}
$$
$$
1. 8 \times1 0^{5} ( \mathrm{N / m^{2}} )
$$
J. $$
3 1 0 \ \mathrm{( K )}
$$
$$
2. 5 \times1 0^{5} ( \mathrm{N / m^{2}} )
$$
Answer: D
Treating the conduction electrons in a metal as free electrons may seem simplistic, yet it often proves successful. For instance, it yields good results for the compressibility of certain metals. Given the electron density $a$ and the Fermi energy $\varepsilon_{\boldsymbol{r}}$, try to determine the isothermal compressibility when $K = -\ {\frac{1}{V}} \Big( {\frac{\partial V}{\partial p}} \Big)_{\mathrm{T}}$ at $T=0 \ \mathbf{K}$, where $V$ is the volume and $p$ is the pressure. (Hint: Use $pV = {\frac{2}{3}} E$, where $E$ is the total energy.)

Options:
A. $$
\frac{3} {5 n \varepsilon_{r}}, \qquad( T=0 \mathrm{K} )
$$
B. $$
\frac{1} {2 n \varepsilon_{r}}, \qquad( T=0 \mathrm{K} )
$$
C. $$
\frac{7} {2 n \varepsilon_{r}}, \qquad( T=0 \mathrm{K} )
$$
D. $$
\frac{7} {9 n \varepsilon_{r}}, \qquad( T=0 \mathrm{K} )
$$
E. $$
\frac{3} {2 n \varepsilon_{r}}, \qquad( T=0 \mathrm{K} )
$$
F. $$
\frac{3} {4 n \varepsilon_{r}}, \qquad( T=0 \mathrm{K} )
$$
G. $$
\frac{3} {10 n \varepsilon_{r}}, \qquad( T=0 \mathrm{K} )
$$
H. $$
\frac{3} {7 n \varepsilon_{r}}, \qquad( T=0 \mathrm{K} )
$$
I. $$
\frac{9} {2 n \varepsilon_{r}}, \qquad( T=0 \mathrm{K} )
$$
J. $$
\frac{7} {5 n \varepsilon_{r}}, \qquad( T=0 \mathrm{K} )
$$
Answer: E
For a white dwarf with a solar mass and a radius of $2 \times 10^{9} \, \mathrm{cm}$, where atoms are completely ionized, calculate the electron Fermi energy in units of $\mathrm{eV}$.

Options:
A. $$
\approx4 \times1 0^{4} \ ( \mathrm{e V} )
$$
B. $$
\approx6 \times1 0^{4} \ ( \mathrm{e V} )
$$
C. $$
\approx9 \times1 0^{3} \ ( \mathrm{e V} )
$$
D. $$
\approx8 \times1 0^{2} \ ( \mathrm{e V} )
$$
E. $$
\approx6 \times1 0^{2} \ ( \mathrm{e V} )
$$
F. $$
\approx2 \times1 0^{5} \ ( \mathrm{e V} )
$$
G. $$
\approx3 \times1 0^{6} \ ( \mathrm{e V} )
$$
H. $$
\approx1 \times1 0^{3} \ ( \mathrm{e V} )
$$
I. $$
\approx7 \times1 0^{5} \ ( \mathrm{e V} )
$$
J. $$
\approx5 \times1 0^{5} \ ( \mathrm{e V} )
$$
Answer: A
If the Reynolds numbers of both the model and the prototype flow fields are the same, what are the expressions for $R_e$ and the scale ratios $V_{r}$, $T_{r}$, $a_{r}$ ?

Options:
A. $$
\begin{aligned} {R e}={\frac{V_{\mathrm{m}} L_{\mathrm{m}}} {\nu_{\mathrm{m}}}},\\ V_{\mathrm{r}}{=\left( {\frac{L^{2}} {V}} \right)_r},\\T_{r} \,=\, \left( {\frac{V^{2}} {\nu}} \right)_{r},\\a_{r} \,=\, \left( {\frac{\nu^{2}} {V L}} \right)_{r}\\ \end{aligned}
$$
B. $$
\begin{aligned} {R e}={\frac{L_{\mathrm{p}} \nu_{\mathrm{p}}} {V_{\mathrm{p}}}},\\ V_{\mathrm{r}}{=\left( {\frac{V} {L^{2}}} \right)_r},\\T_{r} \,=\, \left( {\frac{\nu L} {V}} \right)_{r},\\a_{r} \,=\, \left( {\frac{\nu^{2}} {V^{3}}} \right)_{r}\\ \end{aligned}
$$
C. $$
\begin{aligned} {R e}={\frac{V_{\mathrm{m}} \nu_{\mathrm{m}}} {L_{\mathrm{m}}}},\\ V_{\mathrm{r}}{=\left( {\frac{\nu^{3}} {L}} \right)_r},\\T_{r} \,=\, \left( {\frac{V} {L \nu}} \right)_{r},\\a_{r} \,=\, \left( {\frac{V^{2}} {L^{3}}} \right)_{r}\\ \end{aligned}
$$
D. $$
\begin{aligned} {R e}={\frac{L_{\mathrm{p}} V_{\mathrm{p}}} {\nu_{\mathrm{p}}}},\\ V_{\mathrm{r}}{=\left( {\frac{\nu} {L}} \right)_r},\\T_{r} \,=\, \left( {\frac{L^{2}} {\nu}} \right)_{r},\\a_{r} \,=\, \left( {\frac{\nu^{2}} {L^{3}}} \right)_{r}\\ \end{aligned}
$$
E. $$
\begin{aligned} {R e}={\frac{V_{\mathrm{p}} L_{\mathrm{p}}} {\nu_{\mathrm{p}}}},\\ V_{\mathrm{r}}{=\left( {\frac{L} {\nu}} \right)_r},\\T_{r} \,=\, \left( {\frac{L^{2}} {V}} \right)_{r},\\a_{r} \,=\, \left( {\frac{\nu^{3}} {V^{3}}} \right)_{r}\\ \end{aligned}
$$
F. $$
\begin{aligned} {R e}={\frac{L_{\mathrm{m}} V_{\mathrm{m}}} {\nu_{\mathrm{m}}}},\\ V_{\mathrm{r}}{=\left( {\frac{V} {L}} \right)_r},\\T_{r} \,=\, \left( {\frac{L \nu} {V}} \right)_{r},\\a_{r} \,=\, \left( {\frac{\nu} {L^{2}}} \right)_{r}\\ \end{aligned}
$$
G. $$
\begin{aligned} {R e}={\frac{V_{\mathrm{p}} L_{\mathrm{m}}} {\nu_{\mathrm{m}}}},\\ V_{\mathrm{r}}{=\left( {\frac{L^{3}} {V}} \right)_r},\\T_{r} \,=\, \left( {\frac{\nu^{2}} {V}} \right)_{r},\\a_{r} \,=\, \left( {\frac{V^{3}} {L}} \right)_{r}\\ \end{aligned}
$$
H. $$
\begin{aligned} {R e}={\frac{L_{\mathrm{m}} V_{\mathrm{p}}} {\nu_{\mathrm{p}}}},\\ V_{\mathrm{r}}{=\left( {\frac{\nu} {V}} \right)_r},\\T_{r} \,=\, \left( {\frac{V^{2}} {L}} \right)_{r},\\a_{r} \,=\, \left( {\frac{L^{3}} {V^{2}}} \right)_{r}\\ \end{aligned}
$$
I. $$
\begin{aligned} {R e}={\frac{\nu_{\mathrm{p}} V_{\mathrm{m}}} {L_{\mathrm{m}}}},\\ V_{\mathrm{r}}{=\left( {\frac{L^{2}} {\nu}} \right)_r},\\T_{r} \,=\, \left( {\frac{L} {V^{2}}} \right)_{r},\\a_{r} \,=\, \left( {\frac{L^{3}} {V \nu}} \right)_{r}\\ \end{aligned}
$$
J. $$
\begin{aligned} {R e}={\frac{V_{\mathrm{p}} \nu_{\mathrm{p}}} {L_{\mathrm{p}}}},\\ V_{\mathrm{r}}{=\left( {\frac{V^{2}} {L}} \right)_r},\\T_{r} \,=\, \left( {\frac{\nu L^{2}} {V}} \right)_{r},\\a_{r} \,=\, \left( {\frac{L} {V \nu}} \right)_{r}\\ \end{aligned}
$$
Answer: D
Groundwater flows through an aquifer with a height of 100 feet, a width of 5000 feet, a porosity of 0.2, and a hydraulic conductivity of 0.1 feet/day. What is the average time required for pollutant carrying with a water quality point to travel 100 feet when the hydraulic gradient is -0.035.

Options:
A. $$
1 2. 9 {\mathrm{y r}}
$$
B. $$
6 3. 2 {\mathrm{y r}}
$$
C. $$
8 4. 5 {\mathrm{y r}}
$$
D. $$
1 5. 6 {\mathrm{y r}}
$$
E. $$
7 9. 8 {\mathrm{y r}}
$$
F. $$
5 5. 4 {\mathrm{y r}}
$$
G. $$
3 6. 9 {\mathrm{y r}}
$$
H. $$
2 8. 3 {\mathrm{y r}}
$$
I. $$
4 7. 1 {\mathrm{y r}}
$$
J. $$
9 1. 7 {\mathrm{y r}}
$$
Answer: D
Cut an $\mathbf{n}$-type germanium into a thin slice with a thickness of 2α, ensuring the length and width of the sample are sufficiently large so that boundary effects can be neglected. This allows the transport of photogenerated carriers within the sample to be essentially one-dimensional. The recombination velocity at both the upper and lower surfaces is s. Then, direct a beam of a certain wavelength that is capable of penetrating the sample without significant absorption, yet still close to the intrinsic absorption limit, sufficient to excite a detectable concentration of electron-hole pairs. What is the concentration of photogenerated carriers at each point in the sample under steady-state conditions?

Options:
A. $$
\Delta p=g \tau \left( 1+\frac{\operatorname{s c h} \frac{x} {L_{n}}} {\operatorname{s h} \frac{\alpha} {L_{n}}+\frac{D_{n}} {L_{n}} \operatorname{s h} \frac{\alpha} {L_{n}}} \right)
$$
B. $$
\Delta p=g \tau \left( 1-\frac{\operatorname{c h} \frac{x} {L_{n}}} {\operatorname{s c h} \frac{\alpha} {L_{n}}+\frac{D_{p}} {L_{p}} \mathrm{sh} \frac{\alpha} {L_{n}}} \right)
$$
C. $$
\Delta p=g \tau \left( 1+\frac{\operatorname{c h} \frac{x} {L_{p}}} {\operatorname{s c h} \frac{\alpha} {L_{p}}+\frac{D_{n}} {L_{n}} \operatorname{s h} \frac{\alpha} {L_{p}}} \right)
$$
D. $$
\Delta p=g \tau\left( 1-\frac{\operatorname{s c h} \frac{x} {L_{p}}} {\operatorname{s c h} \frac{\alpha} {L_{p}}+\frac{D_{p}} {L_{p}} \mathrm{s h} \frac{\alpha} {L_{p}}} \right)
$$
E. $$
\Delta p=g \tau \left( 1-\frac{\operatorname{s c h} \frac{x} {L_{n}}} {\operatorname{s h} \frac{\alpha} {L_{n}}+\frac{D_{n}} {L_{n}} \operatorname{c h} \frac{\alpha} {L_{n}}} \right)
$$
F. $$
\Delta p=g \tau\left( 1+\frac{\operatorname{c h} \frac{x} {L_{n}}} {\operatorname{c h} \frac{\alpha} {L_{n}}-\frac{D_{n}} {L_{p}} \operatorname{s h} \frac{\alpha} {L_{n}}} \right)
$$
G. $$
\Delta p=g \tau \left( 1-\frac{\operatorname{s h} \frac{x} {L_{p}}} {\operatorname{c h} \frac{\alpha} {L_{p}}+\frac{D_{p}} {L_{n}} \mathrm{s h} \frac{\alpha} {L_{p}}} \right)
$$
H. $$
\Delta p=g \tau \left( 1+\frac{\operatorname{s h} \frac{x} {L_{p}}} {\operatorname{s c h} \frac{\alpha} {L_{p}}-\frac{D_{p}} {L_{n}} \mathrm{ch} \frac{\alpha} {L_{p}}} \right)
$$
I. $$
\Delta p=g \tau \left( 1-\frac{\operatorname{s c h} \frac{x} {L_{p}}} {\operatorname{s h} \frac{\alpha} {L_{p}}-\frac{D_{n}} {L_{p}} \operatorname{c h} \frac{\alpha} {L_{p}}} \right)
$$
J. $$
\Delta p=g \tau \left( 1-\frac{\operatorname{s h} \frac{x} {L_{n}}} {\operatorname{c h} \frac{\alpha} {L_{n}}+\frac{D_{p}} {L_{p}} \operatorname{s h} \frac{\alpha} {L_{n}}} \right)
$$
Answer: D
Given that at the above resistivity, the hole mobility in the $\mathbf{p}$ region, $\mu_{p} = 1650 \mathrm{cm}^{2}/(\mathrm{V}\cdot\mathrm{s})$, and the electron mobility in the $\mathrm{n}$ region, $\mu_{\mathrm{n}} = 30000 \mathrm{cm}^{2}/(\mathrm{V}\cdot\mathrm{s})$, with the intrinsic carrier concentration of germanium $n_{i} = 2.5 \times 10^{13} \mathrm{cm}^{-3}$. A $\mathbf{p}$-type germanium with resistivity of 10 Ω·cm and an $\mathbf{n}$-type germanium semiconductor with resistivity of 0.1 Ω·cm form a $\mathrm{pn}$ junction. At room temperature (300K), what is the width of the depletion layer?

Options:
A. $$
4. 0 8 {\times} 1 0^{-6} ( \mathrm{{\, c m )}}
$$
B. $$
6. 0 9 {\times} 1 0^{-6} ( \mathrm{{\, c m )}}
$$
C. $$
1. 9 0 {\times} 1 0^{-6} ( \mathrm{{\, c m )}}
$$
D. $$
8. 2 3 {\times} 1 0^{-6} ( \mathrm{{\, c m )}}
$$
E. $$
5. 1 9 {\times} 1 0^{-6} ( \mathrm{{\, c m )}}
$$
F. $$
7. 4 1 {\times} 1 0^{-6} ( \mathrm{{\, c m )}}
$$
G. $$
2. 5 6 {\times} 1 0^{-6} ( \mathrm{{\, c m )}}
$$
H. $$
3. 0 7 {\times} 1 0^{-6} ( \mathrm{{\, c m )}}
$$
I. $$
9. 3 5 {\times} 1 0^{-6} ( \mathrm{{\, c m )}}
$$
J. $$
0. 5 7 {\times} 1 0^{-6} ( \mathrm{{\, c m )}}
$$
Answer: A
A sufficiently thick p-type silicon sample with electron mobility $\mu_{n} = 1200 \, \mathrm{cm^{2}/(V \cdot s)}$ and electron lifetime $\tau_{\mathrm{n}} = 10 \, \mu\mathrm{s}$ experiences stable electron injection at its planar surface, with $\Delta n = 7 \times 10^{12} \, \mathrm{cm}^{-3}$. The current density of excess minority carriers diffusing from the surface to this location is $1.2 \, \mathrm{mA/cm^{2}}$. How far from the surface is this measured at room temperature?

Options:
A. $$
1. 1 3 \times1 0^{-3} \mathrm{c m}
$$
B. $$
4. 9 8 \times1 0^{-3} \mathrm{c m}
$$
C. $$
7. 7 9 \times1 0^{-3} \mathrm{c m}
$$
D. $$
8. 8 3 \times1 0^{-3} \mathrm{c m}
$$
E. $$
2. 6 5 \times1 0^{-3} \mathrm{c m}
$$
F. $$
6. 2 4 \times1 0^{-3} \mathrm{c m}
$$
G. $$
1. 0 2 \times1 0^{-3} \mathrm{c m}
$$
H. $$
5. 4 8 \times1 0^{-3} \mathrm{c m}
$$
I. $$
9. 5 1 \times1 0^{-3} \mathrm{c m}
$$
J. $$
3. 3 7 \times1 0^{-3} \mathrm{c m}
$$
Answer: D