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301 |
Magnetism
|
Assume there is a spherical cavity within an infinite anisotropic medium, and the uniform electric field far from the cavity inside the medium is known to be $E^{(e)}$. Try to find the $y$ component of the electric field inside the cavity $E_y^{(i)}$, expressed in terms of $E_y^{(e)}$ and the dielectric and geometric parameters.
|
[] |
Expression
|
{"$E^{(e)}$": "uniform electric field far from the cavity", "$E_y^{(i)}$": "y component of the electric field inside the cavity", "$E_y^{(e)}$": "y component of the uniform electric field outside the cavity", "$\\varepsilon^{(x)}$": "dielectric constant in the x-direction", "$\\varepsilon^{(y)}$": "dielectric constant in the y-direction", "$\\varepsilon^{(z)}$": "dielectric constant in the z-direction", "$a$": "radius of the spherical cavity", "$n^{(x)}$": "depolarization coefficient in the x-direction", "$n^{(y)}$": "depolarization coefficient in the y-direction", "$n^{(z)}$": "depolarization coefficient in the z-direction"}
|
Electrostatics of Dielectrics
|
Dielectric properties of crystals
|
|
302 |
Magnetism
|
Consider a spherical cavity in an infinite anisotropic medium, with a known uniform electric field $E^{(e)}$ far away from the cavity. Determine the $z$ component $E_z^{(i)}$ of the electric field inside the cavity, expressed in terms of $E_z^{(e)}$ and the medium and geometric parameters.
|
[] |
Expression
|
{"$E^{(e)}$": "electric field in the external medium", "$E_z^{(i)}$": "z component of the electric field inside the cavity", "$E_z^{(e)}$": "z component of the external electric field", "$\\varepsilon^{(x)}$": "dielectric constant in the x direction", "$\\varepsilon^{(y)}$": "dielectric constant in the y direction", "$\\varepsilon^{(z)}$": "dielectric constant in the z direction", "$a$": "radius of the spherical cavity", "$n^{(x)}$": "depolarization factor in the x direction", "$n^{(y)}$": "depolarization factor in the y direction", "$n^{(z)}$": "depolarization factor in the z direction"}
|
Electrostatics of Dielectrics
|
Dielectric properties of crystals
|
|
303 |
Magnetism
|
For antiferromagnetic ferrous carbonate (whose structure belongs to the magnetic class $\boldsymbol{D}_{3 d}$), derive the expression for the x-component of magnetization $M_x$ under applied stress based on its magnetoelastic effect.
|
[] |
Expression
|
{"$M_x$": "x-component of magnetization", "$\\lambda_{1}$": "magnetoelastic coupling coefficient 1", "$\\lambda_{2}$": "magnetoelastic coupling coefficient 2", "$\\sigma_{x x}$": "stress tensor component (xx)", "$\\sigma_{y y}$": "stress tensor component (yy)", "$\\sigma_{y z}$": "stress tensor component (yz)"}
|
Ferromagnetism and Antiferromagnetism
|
Piezomagnetic and magnetoelectric effects
|
|
304 |
Magnetism
|
For the antiferromagnet iron carbonate (whose structure belongs to the magnetic class $\boldsymbol{D}_{3 d}$), derive the expression for the magnetization component $M_y$ under applied stress based on its piezomagnetic effect. You should return your answer as an equation.
|
[] |
Equation
|
{"$M_y$": "y component of magnetization", "$\\lambda_1$": "piezomagnetic coefficient 1", "$\\lambda_2$": "piezomagnetic coefficient 2", "$\\sigma_{xy}$": "stress tensor component (xy)", "$\\sigma_{xz}$": "stress tensor component (xz)", "$\\boldsymbol{D}_{3 d}$": "magnetic class D3d", "$H_x$": "x component of magnetic field", "$H_y$": "y component of magnetic field", "$\\widetilde{\\Phi}_{\\mathrm{pm}}$": "piezomagnetic potential"}
|
Ferromagnetism and Antiferromagnetism
|
Piezomagnetic and magnetoelectric effects
|
|
305 |
Magnetism
|
For a crystal belonging to the magnetic crystal class $\boldsymbol{D}_{4 h}(\boldsymbol{D}_{2 h})$, determine the magnetization $x$ component $M_x$ induced by the magnetoelastic effect.
|
[] |
Expression
|
{"$M_x$": "magnetization x component", "$\\lambda_1$": "magnetoelastic coupling constant", "$\\sigma_{yz}$": "shear stress component (yz)"}
|
Ferromagnetism and Antiferromagnetism
|
Piezomagnetic and magnetoelectric effects
|
|
306 |
Magnetism
|
For crystal belonging to the magnetic crystal class $\boldsymbol{D}_{4 h}(\boldsymbol{D}_{2 h})$, find the $y$ component of the magnetization $M_y$ induced by the piezomagnetic effect.
|
[] |
Expression
|
{"$M_y$": "y component of the magnetization", "$\\lambda_1$": "piezomagnetic coefficient", "$\\sigma_{xz}$": "stress component (xz)", "$\\sigma_{yz}$": "stress component (yz)", "$\\sigma_{xy}$": "stress component (xy)", "$H_x$": "x component of the magnetic field", "$H_y$": "y component of the magnetic field", "$H_z$": "z component of the magnetic field"}
|
Ferromagnetism and Antiferromagnetism
|
Piezomagnetic and magnetoelectric effects
|
|
307 |
Magnetism
|
For crystals belonging to the magnetic crystal class $\boldsymbol{D}_{4 h}(\boldsymbol{D}_{2 h})$, determine the $z$ component $M_z$ of the magnetization induced by magnetoelastic effects.
|
[] |
Expression
|
{"$M_z$": "z component of the magnetization", "$\\widetilde{\\Phi}_{\\mathrm{pm}}$": "thermodynamic potential related to magnetoelastic effects", "$\\lambda_{1}$": "magnetoelastic coupling coefficient 1", "$\\lambda_{2}$": "magnetoelastic coupling coefficient 2", "$\\sigma_{xz}$": "component of the stress tensor (xz)", "$\\sigma_{yz}$": "component of the stress tensor (yz)", "$\\sigma_{xy}$": "component of the stress tensor (xy)", "$H_x$": "magnetic field component along x-axis", "$H_y$": "magnetic field component along y-axis", "$H_z$": "magnetic field component along z-axis"}
|
Ferromagnetism and Antiferromagnetism
|
Piezomagnetic and magnetoelectric effects
|
|
308 |
Magnetism
|
Ttry to find the magnetic susceptibility $\alpha$ of a conductive cylinder (with radius $a$) in a uniform periodic external magnetic field perpendicular to its axis.
|
[] |
Expression
|
{"$\\alpha$": "magnetic susceptibility", "$a$": "radius of the cylinder", "$\\boldsymbol{r}$": "radial vector in the plane perpendicular to the cylinder axis", "$r$": "radial distance in the plane", "$k$": "wave number", "$V$": "volume per unit length of the cylinder", "$\\mathfrak{H}$": "external magnetic field vector", "$\\boldsymbol{n}$": "unit vector in radial direction", "$\\delta$": "skin depth", "$\\sigma$": "conductivity", "$\\omega$": "angular frequency", "$c$": "speed of light"}
|
Quasi-static electromagnetic field
|
Depth of magnetic field penetration into a conductor
|
|
309 |
Magnetism
|
Ttry to find the magnetic susceptibility $\alpha$ of a conductive cylinder (with radius $a$) in a uniform periodic external magnetic, but the magnetic field is parallel to the cylinder axis.
|
[] |
Expression
|
{"$\\alpha$": "magnetic susceptibility of the cylinder", "$a$": "radius of the cylinder", "$\\mathfrak{H}$": "external magnetic field", "$f$": "symmetric solution of the two-dimensional equation", "$k$": "wave number", "$r$": "radial distance in the cylindrical coordinate system", "$\\boldsymbol{j}$": "current density vector", "$j_{\\varphi}$": "azimuthal component of the current density vector", "$H$": "magnetic field inside the cylinder", "$H_{z}$": "component of the magnetic field in the z-direction", "$\\mathscr{M}$": "magnetic moment per unit length of the cylinder", "$c$": "speed of light in vacuum"}
|
Quasi-static electromagnetic field
|
Depth of magnetic field penetration into a conductor
|
|
310 |
Magnetism
|
The surface of a uniaxial metallic crystal is cut so that its normal forms an angle $\theta$ with the crystal's main axis of symmetry. Considering the thermoelectric effect, under isothermal boundary conditions ($\tau=0$) and assuming $a \ll 1$, find the $xx$ component of the surface impedance $\zeta_{x x}^{(\mathrm{is})}$.
|
[] |
Expression
|
{"$\\theta$": "angle between the crystal's main axis of symmetry and the surface normal", "$\\tau$": "temperature fluctuation increment", "$a$": "a parameter related to the thermoelectric effect", "$\\zeta_{x x}^{(\\mathrm{is})}$": "xx component of the surface impedance under isothermal boundary conditions", "$H_{y}$": "magnetic field component parallel to the y-axis", "$H_{0}$": "amplitude of the magnetic field", "$\\omega$": "angular frequency", "$t$": "time", "$j_{x}$": "current density component along the x-axis", "$j_{y}$": "current density component along the y-axis", "$j_{z}$": "current density component along the z-axis", "$E_{x}$": "electric field component along the x-axis", "$E_{y}$": "electric field component along the y-axis", "$E_{z}$": "electric field component along the z-axis", "$C$": "heat capacity per unit volume", "$T$": "temperature", "$q_{z}$": "heat flux density component along the z-axis", "$\\rho_{x x}$": "xx component of the resistivity tensor", "$\\rho_{y y}$": "yy component of the resistivity tensor", "$\\rho_{z z}$": "zz component of the resistivity tensor", "$\\rho_{x y}$": "xy component of the resistivity tensor", "$\\rho_{y z}$": "yz component of the resistivity tensor", "$\\rho_{x z}$": "xz component of the resistivity tensor", "$\\rho_{\\|}$": "resistivity component along the crystal axis", "$\\rho_{\\perp}$": "resistivity component perpendicular to the crystal axis", "$\\varkappa_{x z}$": "xz component of the thermal conductivity tensor", "$\\varkappa_{z z}$": "zz component of the thermal conductivity tensor", "$\\alpha_{x z}$": "xz component of the thermoelectric tensor", "$k$": "wave number", "$b$": "a parameter related to the thermoelectric effect", "$k_{1}$": "first modified wave number", "$k_{2}$": "second modified wave number", "$A$": "coefficient of electric field in the solution", "$B$": "coefficient of electric field in the solution", "$E_{T}$": "electric field component related to temperature gradient", "$\\zeta_{0}$": "surface impedance without thermoelectric effect"}
|
Quasi-static electromagnetic field
|
Depth of magnetic field penetration into a conductor
|
|
311 |
Magnetism
|
Determine the x-component of the magnetic moment $\mathscr{M}_{x}$ of a conducting sphere ($\mu=1$), rotating uniformly in a uniform constant magnetic field whose components are $(\mathfrak{H}_{x}, 0, \mathfrak{H}_{z})$.
|
[] |
Expression
|
{"$\\mathscr{M}_{x}$": "x-component of the magnetic moment", "$\\mu$": "magnetic permeability", "$\\mathfrak{H}_{x}$": "x-component of the uniform constant magnetic field", "$\\mathfrak{H}_{z}$": "z-component of the uniform constant magnetic field", "$\\boldsymbol{\\Omega}$": "angular velocity vector", "$\\Omega$": "angular velocity", "$\\xi$": "rotating reference frame axis", "$\\eta$": "rotating reference frame axis", "$V$": "volume of the sphere", "$\\alpha$": "complex magnetic polarizability", "$\\alpha^{\\prime}$": "real part of the magnetic polarizability relating to x-component", "$\\alpha^{\\prime \\prime}$": "imaginary part of the magnetic polarizability relating to y-component", "$\\mathscr{M}_{y}$": "y-component of the magnetic moment", "$\\mathscr{M}_{z}$": "z-component of the magnetic moment"}
|
Quasi-static electromagnetic field
|
Movement of a conductor in a magnetic field
|
|
312 |
Magnetism
|
Determine the y-component $\mathscr{M}_{y}$ of the magnetic moment of a conducting sphere ($\mu=1$) uniformly rotating in a uniform constant magnetic field whose components are $(\mathfrak{H}_{x}, 0, \mathfrak{H}_{z})$.
|
[] |
Expression
|
{"$\\mathscr{M}_{y}$": "y-component of the magnetic moment", "$\\mu$": "magnetic permeability", "$\\mathfrak{H}_{x}$": "x-component of the magnetic field", "$\\mathfrak{H}_{z}$": "z-component of the magnetic field", "$\\boldsymbol{\\Omega}$": "angular velocity vector", "$t$": "time", "$\\Omega$": "angular velocity", "$V$": "volume of the sphere", "$\\alpha$": "complex magnetic susceptibility", "$\\alpha^{\\prime}$": "real part of the magnetic susceptibility", "$\\alpha^{\\prime \\prime}$": "imaginary part of the magnetic susceptibility"}
|
Quasi-static electromagnetic field
|
Movement of a conductor in a magnetic field
|
|
313 |
Magnetism
|
Try to determine the magnetic moment of a non-uniformly rotating sphere (sphere radius is $a$). Assume the rotation speed is very low so that the penetration depth $\delta \gg a$.
|
[] |
Expression
|
{"$a$": "sphere radius", "$\\delta$": "penetration depth", "$\\mathfrak{H}$": "magnetic field", "$V$": "volume", "$\\hat{\\alpha}$": "operator acting on Fourier components of magnetic field", "$\\omega$": "frequency", "$\\mathscr{M}$": "magnetic moment", "$m$": "mass", "$\\sigma$": "conductivity", "$c$": "speed of light", "$e$": "elementary charge", "$\\Omega$": "angular velocity"}
|
Quasi-static electromagnetic field
|
The excitation of current by acceleration
|
|
314 |
Magnetism
|
The tangential magnetic field before the shock wave $\boldsymbol{H}_{t 1}=0$, while it is $\boldsymbol{H}_{t 2} \neq 0$ after the shock wave (this kind of shock wave is called a switch-on shock wave) . Determine the range of values $v_{n 1}$ for such a shock wave in a ideal gas with thermodynamic properties $(c_p/c_v = 5/3)$. You should return your answer as an interval like [a, b], (a, b), [a, b), or (a, b], depending on the endpoint inclusion.
|
[] |
Interval
|
{"$\\boldsymbol{H}_{t 1}$": "tangential magnetic field before the shock wave", "$\\boldsymbol{H}_{t 2}$": "tangential magnetic field after the shock wave", "$v_{n 1}$": "velocity of the shock wave relative to the gas in front of it", "$v_{n 2}$": "velocity of the shock wave relative to the gas behind it", "$H_{n}$": "normal magnetic field component", "$\\rho_{2}$": "density of the gas behind the shock wave", "$\\rho_{1}$": "density of the gas in front of the shock wave", "$u_{\\mathrm{A} 2}$": "Alfvén velocity of the gas behind the shock wave", "$u_{\\mathrm{A} 1}$": "Alfvén velocity of the gas in front of the shock wave", "$H_{t 2}$": "tangential magnetic field component after the shock wave", "$u_{01}$": "characteristic velocity of the gas"}
|
Magnetohydrodynamics
|
Magnetohydrodynamics
|
|
315 |
Magnetism
|
Determine the direction of the extraordinary ray when light is refracted from vacuum into the surface of a uniaxial crystal, assuming the crystal surface is perpendicular to its optical axis. Hint: calculate $\tan(\vartheta^{\prime})$ where $\vartheta^{\prime}$ is the refraction angle.
|
[] |
Expression
|
{"$\\vartheta^{\\prime}$": "refraction angle", "$\\vartheta$": "incidence angle", "$n_{x}$": "x component of the refraction wave vector", "$n_{z}$": "z component of the refracted wave", "$\\varepsilon_{\\perp}$": "perpendicular dielectric constant", "$\\varepsilon_{\\|}$": "parallel dielectric constant"}
|
Electromagnetic waves in anisotropic media
|
Optical properties of uniaxial crystals
|
|
316 |
Magnetism
|
Determine the direction of the extraordinary ray when vertically incident on the surface of a uniaxial crystal, assuming the optical axis of the uniaxial crystal is in an arbitrary direction. Hint: calculate $\tan(\vartheta^{\prime})$ where $\vartheta^{\prime}$ is the refraction angle.
|
[] |
Expression
|
{"$\\vartheta^{\\prime}$": "refraction angle", "$\\alpha$": "angle between the optical axis and the normal", "$\\boldsymbol{s}$": "ray vector", "$\\boldsymbol{n}$": "wave vector", "$\\boldsymbol{l}$": "unit vector in the direction of the optical axis", "$\\varepsilon_{\\|}$": "permittivity parallel to the optical axis", "$\\varepsilon_{\\perp}$": "permittivity perpendicular to the optical axis", "$s_{x}$": "component of the ray vector in the x-direction", "$s_{z}$": "component of the ray vector in the z-direction"}
|
Electromagnetic waves in anisotropic media
|
Optical properties of uniaxial crystals
|
|
317 |
Magnetism
|
Determine the asymptotic form of the gyration vector's frequency dependence in the high-frequency regime.
|
[] |
Expression
|
{"$\\boldsymbol{H}$": "external constant magnetic field", "$\\boldsymbol{E}$": "electric field", "$m$": "mass of the electron", "$e$": "charge of the electron", "$c$": "speed of light in vacuum", "$\\omega$": "angular frequency", "$\\varepsilon(\\omega)$": "permittivity as a function of frequency", "$N$": "concentration of electrons"}
|
Electromagnetic waves in anisotropic media
|
Magneto-optical effect
|
|
318 |
Magnetism
|
Determine the intensity distribution within the diffraction spot around the main maximum when diffraction occurs on a spherical crystal with radius $a$.
|
[] |
Expression
|
{"$a$": "radius of the spherical crystal", "$\\varkappa$": "wave vector difference", "$e$": "elementary charge", "$m$": "mass", "$c$": "speed of light", "$\\vartheta$": "angle between direction vectors", "$n_{b}$": "refractive index component"}
|
Diffraction of X-rays in a crystal
|
General Theory of X-ray Diffraction
|
|
319 |
Theoretical Foundations
|
Assume energy levels depend only on the principal quantum number $n$. Let the initial state be $(n l m)$ and the final state be $(n^{\prime} l^{\prime} m^{\prime})$, where $n, ~ n^{\prime}, ~ l, ~ m$ are given. Find the branching ratios for transitions to $l^{\prime}=l+1$, $m^{\prime}=m+1, m, m-1$. You should return your answer as a tuple format.
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[] |
Tuple
|
{"$n$": "principal quantum number", "$n^{\\prime}$": "final state's principal quantum number", "$l$": "orbital quantum number", "$l^{\\prime}$": "final state's orbital quantum number", "$m$": "magnetic quantum number", "$m^{\\prime}$": "final state's magnetic quantum number"}
|
Quantum Leap
| ||
320 |
Theoretical Foundations
|
Assume the energy level depends only on the principal quantum number $n$. Let the initial state be $(n l m)$ and the final state be $(n^{\prime} l^{\prime} m^{\prime})$, with $n, ~ n^{\prime}, ~ l, ~ m$ all given. Find the branching ratios for transitions to $l^{\prime}=l-1$, $m^{\prime}=m+1, m, m-1$. You should return your answer as a tuple format.
|
[] |
Tuple
|
{"$n$": "principal quantum number", "$n^{\\prime}$": "principal quantum number in the final state", "$l$": "azimuthal quantum number", "$l^{\\prime}$": "azimuthal quantum number in the final state", "$m$": "magnetic quantum number", "$m^{\\prime}$": "magnetic quantum number in the final state"}
|
Quantum Leap
| ||
321 |
Theoretical Foundations
|
Assume the energy level depends only on the principal quantum number $n$. Let the initial state be $(n l m)$ and the final state be $(n^{\prime} l^{\prime} m^{\prime})$, with $n, ~ n^{\prime}, ~ l, ~ m$ all given. Calculate the branching ratio for transitions to $l^{\prime}=l+1$ and $l^{\prime}=l-1$. You should return your answer as a tuple format.
|
[] |
Tuple
|
{"$n$": "principal quantum number for initial state", "$n^{\\prime}$": "principal quantum number for final state", "$l$": "orbital angular momentum quantum number for initial state", "$m$": "magnetic quantum number for initial state", "$l^{\\prime}$": "orbital angular momentum quantum number for final state", "$m^{\\prime}$": "magnetic quantum number for final state"}
|
Quantum Leap
| ||
322 |
Theoretical Foundations
|
Irradiate atoms with right circularly polarized light propagating along the positive $z$ direction, causing stimulated transitions of electrons in the atom ($E_n < E_{n'}$). Find the selection rules. You should return your answer as a tuple format.
|
[] |
Tuple
|
{"$z$": "direction of light propagation", "$E_n$": "initial energy level", "$E_{n'}$": "final energy level", "$n$": "initial quantum state", "$n'$": "final quantum state", "$l$": "initial azimuthal quantum number", "$l'$": "final azimuthal quantum number", "$m$": "initial magnetic quantum number", "$m'$": "final magnetic quantum number", "$x$": "x-coordinate", "$y$": "y-coordinate", "$\\omega$": "angular frequency", "$t$": "time"}
|
Quantum Leap
| ||
323 |
Theoretical Foundations
|
According to experimental measurements, the energy level of the hydrogen atom's $2\mathrm{s}_{1 / 2}$ is higher than the $2 \mathrm{p}_{1 / 2}$ level by 1058 MHz (Lamb shift). Find the average lifetime of the electron's spontaneous transition from the $2 \mathrm{s}_{1 / 2}$ level to the $2 \mathrm{p}_{1 / 2}$ level, in years.
|
[] |
Numeric
|
{"$e$": "elementary charge", "$\\omega$": "angular frequency of transition", "$\\hbar$": "reduced Planck's constant", "$c$": "speed of light", "$|i\\rangle$": "initial state ket", "$|f\\rangle$": "final state ket", "$n$": "principal quantum number", "$l$": "orbital angular momentum quantum number", "$j$": "total angular momentum quantum number", "$m_{j}$": "magnetic quantum number", "$m_{s}$": "spin magnetic quantum number", "$a_{0}$": "Bohr radius"}
|
Quantum Leap
| ||
324 |
Theoretical Foundations
|
For two electrons at the $n \mathrm{p}$ energy level $(l=1)$ of an atom, attempt to determine the number of all possible total angular momentum eigenstates using both $L-S$ coupling and $j-j$ coupling.
|
[] |
Numeric
|
{"$n$": "principal quantum number", "$l$": "azimuthal quantum number (orbital angular momentum)", "$L$": "total orbital angular momentum", "$S$": "total spin angular momentum", "$J$": "total angular momentum", "$j$": "single electron total angular momentum quantum number", "$J_2$": "2nd component of total angular momentum"}
|
symmetry
| ||
325 |
Theoretical Foundations
|
Estimate the mass of a Uranium nucleus in micrograms, knowing that it contains 92 protons and 143 neutrons.
Hint: $m_{p} c^{2} \simeq m_{n} c^{2} \simeq 939 \mathrm{MeV}$.
|
[] |
Numeric
|
{"$m_{p}$": "mass of a proton", "$m_{n}$": "mass of a neutron", "$M$": "mass of the uranium nucleus", "$c$": "speed of light", "$M_{\\text{Planck}}$": "Planck mass", "$G_{\\text{Newton}}$": "Newton's gravitational constant", "$\\hbar$": "reduced Planck's constant"}
|
Maxwell's Equations
|
Maxwell's Equations
|
|
326 |
Theoretical Foundations
|
Calculate the electric field in volt per meter that a muon feels in the $1 s$-state of muonic lead.
Hints Bohr radius $a_{B}=\hbar c /(Z \alpha m_{\mu} c^{2}), m_{\mu} c^{2}=105.6 \mathrm{MeV}$.
|
[] |
Numeric
|
{"$a_{B}$": "Bohr radius", "$\\hbar$": "reduced Planck's constant", "$c$": "speed of light", "$Z$": "atomic number", "$\\alpha$": "fine structure constant", "$m_{\\mu}$": "muon mass", "$m_{e}$": "electron mass", "$a_{\\mathrm{B}}^{(\\mu)}$": "muonic Bohr radius", "$r$": "radius at position of the muon", "$e$": "elementary charge", "$\\mathbf{E}$": "electric field magnitude"}
|
Maxwell's Equations
|
Maxwell's Equations
|
|
327 |
Theoretical Foundations
|
For the case of decay into two identical particles, when $V<v_{0}$, find the range of the angle between the two decay particles in the $L$ frame (their separation angle). You should return your answer as an interval like [a, b], (a, b), [a, b), or (a, b], depending on the endpoint inclusion.
|
[] |
Interval
|
{"$V$": "velocity variable", "$v_{0}$": "initial velocity of decay particles", "$L$": "laboratory frame", "$C$": "center of mass frame", "$\\theta_{0}$": "initial angle between decay products in the center of mass frame", "$\\theta_{1}$": "angle of the first decay particle in the laboratory frame", "$\\theta_{2}$": "angle of the second decay particle in the laboratory frame", "$\\Theta$": "separation angle between the two decay particles in the laboratory frame"}
|
Relativistic mechanics
|
Particle decay
|
|
328 |
Theoretical Foundations
|
For the case of decay into two identical particles, when $v_{0}<V<\frac{v_{0}}{\sqrt{1-v_{0}^{2}}}$, determine the range of the angle (the opening angle) between the two decay particles in the $L$ system. You should return your answer as an interval like [a, b], (a, b), [a, b), or (a, b], depending on the endpoint inclusion.
|
[] |
Interval
|
{"$v_{0}$": "initial velocity of the decay particles in the C system", "$V$": "velocity of the L system relative to the C system", "$L$": "laboratory system", "$C$": "center-of-mass system", "$\\theta_{10}$": "angle of the first decay particle in the C system", "$\\theta_{20}$": "angle of the second decay particle in the C system", "$\\theta_{0}$": "angle between the decay particles in the C system", "$\\theta_{1}$": "angle of the first decay particle in the L system", "$\\theta_{2}$": "angle of the second decay particle in the L system", "$\\Theta$": "opening angle between the two decay particles in the L system"}
|
Relativistic mechanics
|
Particle decay
|
|
329 |
Theoretical Foundations
|
For the case of decay into two identical particles, when $V>\frac{v_{0}}{\sqrt{1-v_{0}^{2}}}$, find the range of the angle between the two decay particles in the $L$ frame (their separation angle). You should return your answer as an interval like [a, b], (a, b), [a, b), or (a, b], depending on the endpoint inclusion.
|
[] |
Interval
|
{"$V$": "velocity parameter", "$v_{0}$": "initial decay velocity", "$L$": "laboratory frame", "$C$": "center-of-mass frame", "$\\theta_{10}$": "angle of first particle in C frame", "$\\theta_{20}$": "angle of second particle in C frame", "$\\theta_{0}$": "initial angle in C frame", "$\\theta_{1}$": "angle of first particle in L frame", "$\\theta_{2}$": "angle of second particle in L frame", "$\\Theta$": "separation angle in L frame"}
|
Relativistic mechanics
|
Particle decay
|
|
330 |
Theoretical Foundations
|
Identify appropriately normalized coefficients in the expansion of the fields in terms of plane wave solutions with annihilation and/or creation operators. You should return your answer as a tuple format.
|
[] |
Tuple
|
{"$\\psi$": "field operator", "$\\psi'$": "adjoint field operator", "$\\mathbf{x}$": "position vector", "$t$": "time", "$\\mathbf{p}$": "momentum vector", "$a_{\\mathbf{p}}$": "annihilation operator for momentum \\(\\mathbf{p}\\)", "$a_{\\mathbf{p}}^{\\dagger}$": "creation operator for momentum \\(\\mathbf{p}\\)", "$\\omega_{\\mathbf{p}}$": "angular frequency for momentum \\(\\mathbf{p}\\)", "$m$": "mass", "$f(\\mathbf{p})$": "function to be determined related to annihilation", "$g(\\mathbf{p})$": "function to be determined related to creation"}
|
Introduction to perturbation theory and scattering
|
Introduction to perturbation theory and scattering
|
|
331 |
Theoretical Foundations
|
Although we won't use coherent states much in this course, coherent states do have applications in all sorts of odd corners of physics, and working out their properties is an instructive exercise in manipulating annihilation and creation operators.
It suffices to study a single harmonic oscillator; the generalization to a free field (= many oscillators) is trivial. Let
H=\frac{1}{2}(p^{2}+q^{2})
and, as usual, let us define
a=\frac{1}{\sqrt{2}}(q+i p) \quad a^{\dagger}=\frac{1}{\sqrt{2}}(q-i p)
Define the coherent state $|z\rangle$ by
\begin{equation*}
|z\rangle=N e^{z a^{\dagger}}|0\rangle \tag{P4.2}
\end{equation*}
where $z$ is a complex number and $N$ is a real, positive normalization factor (dependent on $z$ ), chosen such that $\langle z \mid z\rangle=1$. The set of all coherent states for all values of $z$ is obviously complete. Indeed, it is overcomplete: The energy eigenstates can all be constructed by taking successive derivatives at $z=0$, so the coherent states
\footnotetext{
${ }^{1}$ [Eds.] Roy J. Glauber, "Photon correlations", Phys. Rev. Lett. 10 (1963) 83-86. Glauber won the 2005 Nobel Prize in Physics for research in optical coherence.
}
with $z$ in some small, real interval around the origin are already enough. Show that, despite this, there is an equation that looks something like a completeness relation, namely
\begin{equation*}
1=\alpha \int d(\operatorname{Re} z) d(\operatorname{Im} z) e^{-\beta z^{*} z}|z\rangle\langle z| \tag{P4.3}
\end{equation*}
and find the real constants $\alpha$ and $\beta$. You should return your answer as a tuple format.
|
[] |
Tuple
|
{"$z$": "complex variable related to coherent states", "$\\alpha$": "real constant in completeness relation", "$\\beta$": "real constant in completeness relation", "$z^{*}$": "complex conjugate of the variable z", "$|z\\rangle$": "coherent state associated with the complex variable z", "$|n\\rangle$": "energy eigenstate", "$x$": "real part of the complex number z", "$y$": "imaginary part of the complex number z", "$r$": "radius in polar coordinates", "$\\theta$": "angle in polar coordinates"}
|
Perturbation theory I. Wick diagrams
|
Perturbation theory I. Wick diagrams
|
|
332 |
Theoretical Foundations
|
The Lagrangian of Model 3 is:
\begin{align*}
\mathcal{L} = \frac{1}{2}(\partial^\mu \phi)(\partial_\mu \phi) - \frac{1}{2}\mu^2\phi^2 + \partial^\mu \psi^* \partial_\mu \psi - m^2 \psi^* \psi - g \phi \psi^* \psi.
\end{align*} In Model 3, compute, to lowest non-vanishing order in $g$, the center-of-momentum differential cross-section and the total cross section for "nucleon"-"antinucleon" elastic scattering.
|
[] |
Expression
|
{"$g$": "coupling constant", "$\\sigma$": "cross-section", "$d\\sigma/d\\Omega$": "differential cross-section", "$E_T$": "total energy", "$\\mathbf{p}_f$": "final momentum vector", "$\\mathbf{p}_i$": "initial momentum vector", "$\\mathcal{A}_{fi}$": "amplitude from initial to final state", "$p_1$": "four-momentum of initial particle 1", "$p_2$": "four-momentum of initial particle 2", "$p_3$": "four-momentum of final particle 1", "$p_4$": "four-momentum of final particle 2", "$\\mu$": "mass parameter", "$\\epsilon$": "infinitesimal positive number", "$\\theta$": "scattering angle", "$\\phi$": "azimuthal angle"}
|
Scattering II. Applications
|
Scattering II. Applications
|
|
333 |
Theoretical Foundations
|
The two-particle density of states factor, $D$, in the center-of-momentum frame, $\mathbf{P}_{T}=\mathbf{0}$:
\begin{equation*}
D=\frac{1}{16 \pi^{2}} \frac{|\mathbf{p}_{f}| d \Omega_{f}}{E_{T}}.
\end{equation*}
where we have used the notation: the final particles' momenta $|\mathbf{p_f|$ in the center-of-momentum frame The factor $d\Omega_f$ describes the solid angle associated with $d^3\mathbf{p}_f$.}
Find the formula that replaces this one if $\mathbf{P}_{T} \neq \mathbf{0}$. Comment: Although the center-of-momentum frame is certainly the simplest one in which to work, sometimes we want to do calculations in other frames, for example, the "lab frame", in which one of the two initial particles is at rest.
|
[] |
Expression
|
{"$D$": "two-particle density of states factor", "$\\mathbf{P}_{T}$": "total momentum", "$E_{T}$": "total energy", "$\\mathbf{p}_{f}$": "final particles' momenta", "$d\\Omega_{f}$": "solid angle associated with the final momentum", "$\\mathbf{k}$": "3-momentum of a final particle", "$\\mathbf{q}$": "3-momentum of a final particle", "$m_{k}$": "mass of the particle with momentum $\\mathbf{k}$", "$E_{k}$": "energy of the particle with momentum $\\mathbf{k}$", "$m_{q}$": "mass of the particle with momentum $\\mathbf{q}$", "$E_{q}$": "energy of the particle with momentum $\\mathbf{q}$", "$\\theta$": "angle between $\\mathbf{k}$ and $\\mathbf{P}_{T}$", "$\\phi$": "azimuthal angle of $\\mathbf{k}$ about the $\\mathbf{P}_{T}$ axis", "$\\theta_{k}$": "value of $\\theta$ at which energy is conserved"}
|
Green's functions and Heisenberg fields
|
Green's functions and Heisenberg fields
|
|
334 |
Theoretical Foundations
|
When we attempted to quantize the free Dirac theory
\begin{equation*}
\mathscr{L}= \pm \bar{\psi}(i \partial \!\!\!/-m) \psi \tag{P12.1}
\end{equation*}
with canonical commutation relations, we encountered a disastrous contradiction with the positivity of energy. We succeeded when we used canonical anticommutators (if we chose ( $\pm$ ) to be + ). Much earlier we were able to quantize the free charged scalar field,
\begin{equation*}
\mathscr{L}= \pm(\partial_{\mu} \phi^{*} \partial^{\mu} \phi-\mu^{2} \phi^{*} \phi) \tag{P12.2}
\end{equation*}
with canonical commutators (if we chose $( \pm)$ to be + ). Attempt to quantize the free charged scalar field with (nearly) canonical anticommutators:
\begin{align*}
\{\phi(\mathbf{x}, t), \phi(\mathbf{y}, t)\} & =\{\dot{\phi}(\mathbf{x}, t), \dot{\phi}(\mathbf{y}, t)\}=0 \\
{\phi(\mathbf{x}, t), \phi^{*}(\mathbf{y}, t)} & ={\dot{\phi}(\mathbf{x}, t), \dot{\phi}^{*}(\mathbf{y}, t)}=0 \tag{P12.3}\\
{\phi(\mathbf{x}, t), \dot{\phi}^{*}(\mathbf{y}, t)} & =\lambda \delta^{(3)}(\mathbf{x}-\mathbf{y})
\end{align*}
where $\lambda$ is a (possibly complex) constant.
Show that one reaches a disastrous contradiction with the positivity of the norm in Hilbert space; that is to say:
\begin{equation*}
\langle\phi|{\theta, \theta^{\dagger}}|\phi\rangle \geq 0 \tag{21.20}
\end{equation*}
for any operator $\theta$ and any state $|\phi\rangle$.
Hints: (1) Canonical anticommutation implies that, even on the classical level, $\phi$ and $\phi^{*}$ are Grassmann variables. If you don't take proper account of this (especially in ordering terms when deriving the canonical momenta), you'll get hopelessly confused. (2) Dirac theory is successfully quantized using anticommutators; the sign of the Lagrangian is fixed by appealing to the positivity of the inner product in Hilbert space. If we attempt to quantize the theory using commutators, we get into trouble with the positivity of the energy. The Klein-Gordon theory is successfully quantized using commutators; the sign of the Lagrangian is fixed by appealing to the positivity of energy. So it's to be expected that we'd get into trouble, if we attempted to quantize the Klein-Gordon theory with anticommutators, with the positivity of the inner product. (3) You should do the expansion
\begin{align*}
\phi(x) = \int \frac{d^3 \mathbf{p}}{(2\pi)^{3/2}\sqrt{2\omega_{\mathbf{p}}}} [b_{\mathbf{p}} e^{-i\mathbf{p}\cdot x} + c_{\mathbf{p}}^\dagger e^{i\mathbf{p}\cdot x}],
\end{align*}
and output $\{b_p,b_p^\dagger\}, \{c_p,c_p^\dagger\}$.
Hint: You can use $a = \mathrm{Re}\lambda,b=\mathrm{Im}\lambda$. You should return your answer as a tuple format.
|
[] |
Tuple
|
{"$\\mathscr{L}$": "Lagrangian", "$\\bar{\\psi}$": "Dirac adjoint spinor field", "$\\psi$": "Dirac spinor field", "$m$": "mass of the particle", "$\\partial_{\\mu}$": "partial derivative with respect to space-time coordinates", "$\\phi$": "scalar field", "$\\phi^{*}$": "complex conjugate of the scalar field", "$\\mu^{2}$": "mass term for the scalar field", "$\\lambda$": "a complex constant related to canonical anticommutation", "$\\lambda^{*}$": "complex conjugate of the constant \\lambda", "$b_{\\mathbf{p}}$": "annihilation operator for a particle with momentum \\mathbf{p}", "$b_{\\mathbf{p}}^{\\dagger}$": "creation operator for a particle with momentum \\mathbf{p}", "$c_{\\mathbf{p}}$": "annihilation operator for an antiparticle with momentum \\mathbf{p}", "$c_{\\mathbf{p}}^{\\dagger}$": "creation operator for an antiparticle with momentum \\mathbf{p}", "$\\omega_{\\mathbf{p}}$": "frequency associated with momentum \\mathbf{p}"}
|
The Dirac Equation III. Quantization and Feynman Rules
|
The Dirac Equation III. Quantization and Feynman Rules
|
|
335 |
Theoretical Foundations
|
Let $\psi_{A}, \psi_{B}, \psi_{C}$ and $\psi_{D}$ be four Dirac spinor fields. These fields interact with each other (and possibly with unspecified scalar and pseudoscalar fields) in some way that is invariant under $P, C$, and $T$, where these operations are defined in the "standard way":
\begin{equation*}
U_{P}^{\dagger} \psi(\mathbf{x}, t) U_{P}=\beta \psi(-\mathbf{x}, t) \tag{22.8}
\end{equation*}
Likewise,
\begin{equation*}
U_{C}^{\dagger} \psi(x) U_{C}=\psi'(x) \tag{22.49}
\end{equation*}
in a Majorana basis (one in which $\gamma^{\mu}=-\gamma^{\mu *}$ ). Finally,
\begin{equation*}
\Omega_{P T}^{-1} \psi(x) \Omega_{P T}=i \gamma_{5} \psi(-x) \tag{22.80}
\end{equation*}
again in a Majorana basis. Now let us consider adding a term to the Hamiltonian density,
\begin{align*}
\mathscr{H}^{\prime}= & g_{1}(\bar{\psi}_{A} \gamma^{\mu} \psi_{B})(\bar{\psi}_{C} \gamma_{\mu} \psi_{D})+g_{2}(\bar{\psi}_{A} \gamma^{\mu} \psi_{B})(\bar{\psi}_{C} \gamma_{\mu} \gamma_{5} \psi_{D})+g_{3}(\bar{\psi}_{A} \gamma^{\mu} \gamma_{5} \psi_{B})(\bar{\psi}_{C} \gamma_{\mu} \psi_{D}) \\
& +g_{4}(\bar{\psi}_{A} \gamma^{\mu} \gamma_{5} \psi_{B})(\bar{\psi}_{C} \gamma_{\mu} \gamma_{5} \psi_{D})+\text { Hermitian conjugate } \tag{P14.1}
\end{align*}
where the $g_{i}$ 's are (possibly complex) numbers. Under what conditions on the $g^{\prime}$ s is $\mathscr{H}^{\prime}(0)$ invariant under $P$. Hint: $\Omega_{P T}$ is anti-unitary. You should return your answer as a tuple format.
|
[] |
Tuple
|
{"$g^{\\prime}$": "a generic symbol referring to one of the coupling constants", "$\\mathscr{H}^{\\prime}(0)$": "the perturbed Hamiltonian at time zero", "$P$": "parity transformation", "$\\Omega_{P T}$": "a specific operator representing a combined parity and time-reversal transformation", "$C$": "charge conjugation transformation", "$T$": "time reversal transformation", "$\\gamma^{\\mu}$": "gamma matrices in the context of quantum field theory, representing one of the space-time directions", "$\\psi_{A}$": "a fermion field with an index A", "$\\psi_{B}$": "a fermion field with an index B", "$\\psi_{C}$": "a fermion field with an index C", "$\\psi_{D}$": "a fermion field with an index D", "$\\gamma_{\\mu}$": "gamma matrices in the context of quantum field theory, with lower index representing contracted space-time directions", "$\\gamma^{0}$": "gamma matrix corresponding to the time direction", "$\\gamma^{i}$": "gamma matrices corresponding to spatial directions", "$\\gamma_{5}$": "gamma five matrix used in quantum field theory, related to chirality", "$g_{1}$": "a specific coupling constant in the Hamiltonian", "$g_{2}$": "a specific coupling constant in the Hamiltonian", "$g_{3}$": "a specific coupling constant in the Hamiltonian", "$g_{4}$": "a specific coupling constant in the Hamiltonian", "$g_{1}^{*}$": "the complex conjugate of the coupling constant $g_{1}$", "$U_{P}$": "unitary operator for parity transformation", "$U_{C}$": "unitary operator for charge conjugation", "$U_{P C}$": "unitary operator for combined parity and charge conjugation", "$U_{P T}$": "unitary operator for combined parity and time reversal"}
|
Coping with infinities: regularization and renormalization
|
Coping with infinities: regularization and renormalization
|
|
336 |
Theoretical Foundations
|
Even in quantum electrodynamics, it is possible (though not usual) to work in a gauge where ghost fields are needed. For example, this is a valid form of the electrodynamic Lagrangian:
\mathscr{L}=\mathscr{L}_{\mathrm{em}}-\frac{1}{2} \lambda(\partial_{\mu} A^{\mu}+\sigma A_{\mu} A^{\mu})^{2}+\mathscr{L}_{\mathrm{ghost}}
Here $\mathscr{L}_{\mathrm{em}}$ is the standard Lagrangian, with neither gauge-fixing nor ghost terms, and $\lambda$ and $\sigma$ are arbitrary real numbers.
What are the vertices involving ghost fields?
|
[] |
Expression
|
{"$\\mathscr{L}_{\\mathrm{em}}$": "standard electrodynamic Lagrangian", "$\\lambda$": "arbitrary real number", "$\\sigma$": "arbitrary real number", "$A_{\\mu}$": "vector field", "$A^{\\mu}$": "contravariant component of the vector field", "$\\chi$": "ghost field", "$k^{\\mu}$": "momentum of the ghost field", "$\\bar{\\eta}$": "complex conjugate ghost field", "$\\eta$": "ghost field"}
|
The renormalization of QED
|
The renormalization of QED
|
|
337 |
Theoretical Foundations
|
Even in quantum electrodynamics, it is possible (though not usual) to work in a gauge where ghost fields are needed. For example, this is a valid form of the electrodynamic Lagrangian:
$$
\mathscr{L}=\mathscr{L}_{\mathrm{em}}-\frac{1}{2} \lambda(\partial_{\mu} A^{\mu}+\sigma A_{\mu} A^{\mu})^{2}+\mathscr{L}_{\mathrm{ghost}}
$$
Here $\mathscr{L}_{\mathrm{em}}$ is the standard Lagrangian, with neither gauge-fixing nor ghost terms, and $\lambda$ and $\sigma$ are arbitrary real numbers.
What is $\mathscr{L}_{\text {ghost }}$ ?
|
[] |
Expression
|
{"$\\mathscr{L}$": "Lagrangian", "$\\mathscr{L}_{\\mathrm{em}}$": "standard electromagnetic Lagrangian", "$\\lambda$": "arbitrary real number (gauge-fixing parameter)", "$\\sigma$": "arbitrary real number", "$A_{\\mu}$": "vector potential", "$A^{\\mu}$": "vector potential (contravariant component)", "$F(A)$": "gauge-fixing function", "$\\delta \\chi$": "variation of the gauge parameter", "$\\eta$": "ghost field", "$\\bar{\\eta}$": "conjugate ghost field", "$\\mathcal{S}_{\\mathrm{ghost}}$": "ghost action", "$\\square$": "d'Alembertian operator", "$x$": "position (spacetime coordinate)", "$x^{\\prime}$": "position (spacetime coordinate)", "$k$": "momentum", "$k^{\\mu}$": "momentum (contravariant component)"}
|
The renormalization of QED
|
The renormalization of QED
|
|
338 |
Theoretical Foundations
|
$\mathrm{SU}(3)$ allows only one possible coupling of the electromagnetic current to a quark and an antiquark. Thus (by the same reasoning used for the decuplet in the previous problem), in the limit of perfect $\mathrm{SU}(3)$ symmetry, if quarks are observable, their magnetic moments would be proportional to their charges. In the non-relativistic limit,
\begin{equation*}
\boldsymbol{\mu}=\kappa q \boldsymbol{\sigma}, \tag{P21.2}
\end{equation*}
where $\kappa$ is an unknown constant, $q$ is the electric charge of the quark in question, and $\boldsymbol{\sigma}$ is the vector of Pauli spin matrices.
\footnotetext{
${ }^{1}$ [Eds.] "An Introduction to Unitary Symmetry", the Erice notes from the summer of 1966, originally published in Strong and Weak Interactions - Present Problems, Academic Press, 1966, and reprinted in Coleman Aspects.
}
In the naive quark model discussed in class, the baryons are considered as non-relativistic three-quark bound states with no spin-dependent interactions. Thus, as in atomic physics, we can compute the baryon moments in terms of the quark moments, that is, in terms of the single unknown constant $\kappa$, if we know the baryon wave function. For the lightest baryon octet, the one that contains the proton and the neutron, there is no orbital contribution to the magnetic moments because each quark has zero orbital angular momentum. Thus all we need is the spin-flavor-color part of the wave function. Of course, since the assumption of perfect $\mathrm{SU}(3)$ already gives all the baryon moments in terms of the proton and neutron moments, the only new information we get from this analysis is the ratio of these moments. Compute the ratio and compare it to experiment.
Remark. It's clear from the way the calculation is set up that it's the total moment you will be computing, not the anomalous moment. Be careful that you don't use the anomalous moments when you make the computation.
Hint: You will need the spin-flavor part of the wave functions for both the proton and the neutron to do this problem. Here is an easy way to construct them without resorting to tables of $3 j$ symbols. It is trivial to construct the wave function for the $I_{z}=J_{z}=\frac{3}{2}$ state of the $\Delta$; it is $|u \uparrow, u \uparrow, u \uparrow\rangle$, with all three quarks being up quarks, and all three spinning up. If we apply both an isospin lowering operator and a spin lowering operator to this, we obtain the $I_{z}=J_{z}=\frac{1}{2}$ state of the $\Delta$. The $J_{z}=\frac{1}{2}$ state of the proton must be orthogonal to this. The $J_{z}=\frac{1}{2}$ state of the neutron (up to an irrelevant phase) is obtained from the proton state by exchanging $u$ and $d$.
|
[] |
Numeric
|
{"$\\kappa$": "unknown constant relating magnetic moment, electric charge, and spin", "$q$": "electric charge of the quark", "$\\boldsymbol{\\sigma}$": "vector of Pauli spin matrices", "$\\mu_{B}$": "magnetic moment of the baryon", "$\\mu_{1 z}$": "magnetic moment related to the first quark", "$\\mu_{2 z}$": "magnetic moment related to the second quark", "$\\mu_{3 z}$": "magnetic moment related to the third quark", "$\\mu_{p}$": "magnetic moment of the proton", "$\\mu_{n}$": "magnetic moment of the neutron", "$|B\\rangle$": "state of the baryon", "$|p\\rangle$": "state of the proton", "$|n\\rangle$": "state of the neutron", "$\\Delta^{++}$": "state of the Delta baryon with charge ++", "$\\Delta^{+}$": "state of the Delta baryon with charge +"}
|
Broken $\mathrm{SU}(3)$ and the naive quark model
|
Broken $\mathrm{SU}(3)$ and the naive quark model
|
|
339 |
Theoretical Foundations
|
The model
\begin{equation*}
\mathscr{L}=\frac{1}{2}(\partial_{\mu} \boldsymbol{\Phi}) \cdot(\partial^{\mu} \boldsymbol{\Phi})-U(\boldsymbol{\Phi})
\end{equation*}
was a theory with spontaneous breakdown of $\mathrm{U}(1)$ internal symmetry. The particle spectrum of the theory consisted of a massless Goldstone boson and a massive neutral scalar. Furthermore, this term in the Lagrangian
\begin{equation*}
\frac{1}{2} \rho^{2}(\partial_{\mu} \theta)^{2}=\frac{1}{2} a^{2}(\partial_{\mu} \theta)^{2}+a \rho^{\prime}(\partial_{\mu} \theta)^{2}+\frac{1}{2} \rho^{\prime 2}(\partial_{\mu} \theta)^{2} \tag{P24.5}
\end{equation*}
gives rise to the decay of the massive meson into two Goldstone bosons, with an invariant Feynman amplitude proportional to $a^{-1}$. (This is not a misprint: before reading the decay amplitude from the Lagrangian, we must first rescale $\theta$ to put the free Lagrangian in standard form.) Now consider the theory minimally coupled instead to a massive photon with mass $\mu_{0}$ (before symmetry breaking). What is the photon mass after the symmetry breaks?
Comment: The Abelian Higgs model is the same theory minimally coupled to a massless photon.
|
[] |
Expression
|
{"$\\mu_{0}$": "mass of the photon before symmetry breaking", "$a$": "parameter related to symmetry breaking", "$e$": "gauge coupling constant", "$\\Phi$": "complex scalar field", "$\\phi_{1}$": "real part of the scalar field", "$\\phi_{2}$": "imaginary part of the scalar field", "$\\rho$": "magnitude of the scalar field", "$\\rho^{\\prime}$": "shifted field component after symmetry breaking", "$\\theta$": "phase angle of the scalar field", "$\\theta^{\\prime}$": "redefined phase angle after symmetry breaking", "$m_{\\rho^{\\prime}}$": "mass of the shifted field component", "$B_{\\mu}$": "redefined gauge field", "$F_{\\mu \\nu}$": "field strength tensor", "$\\vartheta$": "redefined Goldstone boson field"}
|
Topics in spontaneous symmetry breaking
|
Topics in spontaneous symmetry breaking
|
|
340 |
Superconductivity
|
Attempt to derive the elementary excitation spectrum in a nearly ideal Bose gas, where the elementary excitation spectrum is considered as the dispersion relation of the collective wave function fluctuations. You should return your answer as an equation.
|
[] |
Equation
|
{"$n$": "constant mean value (density)", "$A$": "complex small amplitude", "$B^{*}$": "complex conjugate of small amplitude", "$\\hbar$": "reduced Planck's constant", "$\\omega$": "angular frequency", "$k$": "wave vector", "$r$": "position vector", "$p$": "momentum", "$m$": "mass", "$U_{0}$": "interaction potential"}
|
Superfluidity
|
Inhomogeneous Bose gas
|
|
341 |
Superconductivity
|
Calculate the probability of a quasiparticle with momentum $p$ (close to the threshold $p_{\mathrm{c}}$) emitting a phonon, when the quasiparticle speed reaches the speed of sound.
|
[] |
Expression
|
{"$p$": "momentum of the quasiparticle", "$p_{\\mathrm{c}}$": "threshold momentum", "$\\boldsymbol{p}$": "momentum vector of the quasiparticle", "$p^{\\prime}$": "momentum after the emission", "$\\boldsymbol{q}$": "momentum vector of the phonon", "$q$": "magnitude of the momentum of the phonon", "$u$": "speed of sound", "$A$": "expressed parameter related to momentum and energy", "$\\rho$": "density", "$\\varepsilon$": "energy function", "$\\theta$": "angle related to momentum vectors", "$w$": "probability of phonon emission", "$\\hbar$": "reduced Planck's constant"}
|
Superfluidity
|
Fission of quasiparticles
|
|
342 |
Superconductivity
|
There is a planar film with thickness $d \ll \xi, \delta$. Find the critical value of the magnetic field parallel to the planar film, which can destroy superconductivity.
|
[] |
Expression
|
{"$d$": "thickness of the planar film", "$\\xi$": "coherence length", "$\\delta$": "penetration depth", "$B$": "magnetic field along the x-axis", "$B_{x}$": "component of the magnetic field along the x-axis", "$\\theta$": "normalized order parameter ratio", "$\\psi$": "order parameter", "$\\psi_{0}$": "critical order parameter", "$a$": "coefficient in the Landau free energy expansion", "$b$": "coefficient in the Landau free energy expansion", "$j$": "current density", "$j_{z}$": "z-component of the current density", "$\\mathfrak{F}$": "amplitude factor for the magnetic field", "$H_{\\mathrm{c}}$": "critical magnetic field of a large-scale superconductor", "$H_{\\mathrm{c}}^{f}$": "critical magnetic field of the film", "$H_{\\mathrm{o}}$": "critical field of large-scale superconductors", "$\\mathfrak{S}$": "parameter related to magnetic field strength", "$e$": "elementary charge", "$\\hbar$": "reduced Planck constant", "$m$": "electron mass", "$\\overline{j^{2}}$": "average of the square of the current density", "$\\mathfrak{K}$": "parameter related to the current distribution", "$A_{2}$": "potential along the y-axis"}
|
Superconductivity
|
Ginzburg-Landau equation
|
|
343 |
Superconductivity
|
If the average magnetic induction intensity of the cylindrical sample cross-section is $\bar{B}$, and in the mixed state with an external magnetic field $\mathfrak{S}$, all vortices are distributed at distances $d \gg \delta$ from each other, forming an equilateral triangular lattice in the sample cross-section. Try to determine the relationship between the external field $\mathfrak{S}$, lower critical field $H_{\mathrm{cl}}$, and the dimensionless vortex spacing $a = d/\delta$ at thermodynamic equilibrium under the condition $1/a \ll 1$. (Hint: the thermodynamic potential per unit volume $\tilde{f}$ reaches its minimum).
|
[] |
Expression
|
{"$\\bar{B}$": "average magnetic induction intensity", "$\\mathfrak{S}$": "external magnetic field", "$H_{\\mathrm{cl}}$": "lower critical field", "$a$": "dimensionless vortex spacing", "$d$": "distance between vortices", "$\\delta$": "characteristic length scale", "$\\tilde{f}$": "thermodynamic potential per unit volume", "$\\tilde{f}_{\\mathrm{s}}$": "thermodynamic potential per unit volume in superconducting state", "$\\phi_{0}$": "magnetic flux quantum", "$\\nu$": "number of vortices per unit area", "$\\varepsilon_{12}$": "interaction energy between two vortices", "$i$": "vortex index", "$k$": "vortex index", "$\\varepsilon_{i k}$": "interaction energy between vortices indexed by $i$ and $k$"}
|
Superconductivity
|
Mixed structure
|
|
344 |
Superconductivity
|
If the average magnetic induction intensity of the cross-section of a cylindrical sample is $\bar{B}$, and in the mixed state with an applied magnetic field of $\mathfrak{S}$, each vortex line is distributed at a distance $d \gg \delta$ from each other and forms an equilateral triangular lattice within the sample cross-section. It is known that the average magnetic induction intensity $\bar{B} = \nu \phi_0$ (where $\nu$ is the number of vortices per unit area and $\phi_0$ is the magnetic flux quantum) and the dimensionless vortex spacing $a = d/\delta$ (where $d$ is the vortex spacing, and $\delta$ is the London penetration depth). Try to derive the relationship between the dimensionless vortex spacing $a$ and the average magnetic induction intensity $\bar{B}$ under the condition $1/a \ll 1$.
|
[] |
Expression
|
{"$\\bar{B}$": "average magnetic induction intensity", "$\\mathfrak{S}$": "applied magnetic field", "$\\nu$": "number of vortices per unit area", "$\\phi_0$": "magnetic flux quantum", "$a$": "dimensionless vortex spacing", "$d$": "vortex spacing", "$\\delta$": "London penetration depth"}
|
Superconductivity
|
Mixed structure
|
|
345 |
Magnetism
|
Ignoring interactions between spins, calculate the magnetization of a paramagnet when the ratio of $\beta \mathscr{G}$ to $T$ is arbitrary.
|
[] |
Expression
|
{"$\\beta$": "inverse temperature", "$\\mathscr{G}$": "magnetic field", "$T$": "temperature", "$S_{\\mathrm{z}}$": "spin component along z-axis", "$S$": "spin quantum number", "$\\mathfrak{S}$": "spin parameter", "$Z$": "partition function", "$M$": "magnetization", "$N$": "number of particles", "$V$": "volume"}
|
Magnetic
|
Spin Hamiltonian
|
|
346 |
Theoretical Foundations
|
If the interaction energy of a crystal can be expressed as $u(r)=-\frac{\alpha}{r^{m}}+\frac{\beta}{r^{n}}$ Taking $m=2, n=10, r_{0}=0.3 \mathrm{~nm}, W=4 \mathrm{eV}$, calculate the value of $\beta$, unit $\mathrm{eV} \cdot \mathrm{m}^{10}$
|
[] |
Numeric
|
{"$m$": "constant parameter (value: 2)", "$n$": "constant parameter (value: 10)", "$r_{0}$": "given radius (value: 0.3 nm)", "$W$": "energy (value: 4 eV)", "$\\beta$": "quantity to be calculated", "$\\alpha$": "auxiliary parameter"}
|
Chapter 2
| ||
347 |
Theoretical Foundations
|
For $\mathrm{H}_{2}$, the Lennard-Jones potential parameters obtained from gas measurements are $\varepsilon=50 \times 10^{-6} J, \sigma=2.96 \stackrel{\circ}{\mathrm{~A}}$. Calculate the binding energy of $\mathrm{H}_{2}$ when it forms a face-centered cubic solid molecular hydrogen (in units of $\mathbf{K J} / \mathrm{mol}$), considering each hydrogen molecule as spherical. The experimental value of the binding energy is $0.751 \mathrm{~kJ} / \mathrm{mo1}$. Compare it with the calculated value.
|
[] |
Numeric
|
{"$\\varepsilon$": "depth of the potential well", "$\\sigma$": "finite distance at which the inter-particle potential is zero", "$N$": "Avogadro's number", "$R$": "intermolecular distance", "$R_{0}$": "equilibrium intermolecular distance", "$U$": "total interaction energy of the crystal", "$\\mathrm{H}_{2}$": "hydrogen molecule"}
|
Chapter 2
| ||
348 |
Theoretical Foundations
|
Consider the lattice vibrations of a diatomic chain where the force constants between nearest neighbor atoms alternate as $c$ and $10c$. The two types of atoms have the same mass, and the nearest neighbor distance is $\frac{a}{2}$. Find the vibrational frequency $\omega(k)$ at $k=0$. You should return your answer as a tuple format.
|
[] |
Tuple
|
{"$c$": "force constant", "$a$": "lattice constant", "$\\omega(k)$": "vibrational frequency at wave number k", "$k$": "wave number", "$M$": "mass of atom"}
|
Chapter 3
| ||
349 |
Theoretical Foundations
|
Consider the lattice vibrations of a diatomic chain where the force constants alternate as $c$ and $10c$ between nearest neighbor atoms on the chain. The two kinds of atoms have the same mass and the nearest neighbor distance is $\frac{a}{2}$. Find the vibration frequency $\omega(k)$ at $k=\frac{\pi}{a}$. You should return your answer as a tuple format.
|
[] |
Tuple
|
{"$c$": "force constant", "$a$": "lattice constant", "$\\omega(k)$": "vibration frequency", "$k$": "wave vector", "$\\pi$": "mathematical constant pi", "$M$": "mass of atom"}
|
Chapter 3
| ||
350 |
Theoretical Foundations
|
One-dimensional Compound Lattice
$m=5 \times 1.67 \times 10^{-24} g, \frac{M}{m}=4, \beta=1.5 \times 10^{1} \mathrm{~N} / \mathrm{m}$ (i.e., $1.51 \times 10^{4} \mathrm{dyn} / \mathrm{cm}$), Find the optical frequencies $\omega_{\max }^{0}, \omega_{\min }^{0}$, and present the answer in tuple form. You should return your answer as a tuple format.
|
[] |
Tuple
|
{"$\\omega_{\\max }^{0}$": "maximum optical frequency (0)", "$\\omega_{\\min }^{0}$": "minimum optical frequency (0)", "$\\omega_{\\max }^{A}$": "maximum optical frequency (A)", "$\\beta$": "parameter related to the stiffness or elastic constant", "$M$": "mass of the larger component", "$m$": "mass of the smaller component", "$\\hbar$": "reduced Planck's constant"}
|
Chapter 3
| ||
351 |
Theoretical Foundations
|
One-dimensional complex lattice
$m=5 \times 1.67 \times 10^{-24} g, \frac{M}{m}=4, \beta=1.5 \times 10^{1} \mathrm{~N} / \mathrm{m}$ (i.e., $1.51 \times 10^{4} \mathrm{dyn} / \mathrm{cm}$), What is the corresponding phonon energy in electron volts for the optical wave $\omega_{\min }^{o}$?
|
[] |
Numeric
|
{"$\\hbar$": "reduced Planck's constant", "$\\omega_{\\min }^{o}$": "optical wave minimum angular frequency"}
|
Chapter 3
| ||
352 |
Theoretical Foundations
|
One-dimensional complex lattice
$m=5 \times 1.67 \times 10^{-24} g, \frac{M}{m}=4, \beta=1.5 \times 10^{1} \mathrm{~N} / \mathrm{m}$ (i.e., $1.51 \times 10^{4} \mathrm{dyn} / \mathrm{cm}$), Determine the wavelength band of the electromagnetic wave corresponding to the optical wave $\omega_{\max }^{0}$.
|
[] |
Numeric
|
{"$\\omega_{\\max }^{0}$": "maximum optical wave frequency", "$\\lambda$": "wavelength", "$c$": "speed of light", "$\\omega$": "angular frequency"}
|
Chapter 3
| ||
353 |
Theoretical Foundations
|
For a one-dimensional lattice with a lattice constant of $2.5 A$, estimate the time required for an electron to move from the bottom of the energy band to the top under an external electric field of $10^{7} \mathrm{~V} / \mathrm{m}$
|
[] |
Numeric
|
{"$a$": "lattice constant", "$e$": "elementary charge", "$\\hbar$": "reduced Planck constant", "$\\vec{E}$": "external electric field", "$t$": "time required for electron transition"}
|
Chapter 5
| ||
354 |
Theoretical Foundations
|
The spin of $\mathrm{He}^{3}$ is $1 / 2$, making it a fermion. The density of liquid $\mathrm{He}^{3}$ near absolute zero is $0.081 \mathrm{gcm}^{-3}$. Calculate the Fermi temperature $\mathbf{T}^{\mathbf{F}}$
|
[] |
Numeric
|
{"$V$": "volume", "$k_{F}$": "Fermi wavevector", "$N$": "number of particles", "$N_{A}$": "Avogadro's number", "$E_{F}$": "Fermi energy", "$\\hbar$": "reduced Planck's constant", "$m$": "mass of $^3\\mathrm{He}$ atom", "$T_{F}$": "Fermi temperature", "$k_{B}$": "Boltzmann constant"}
|
Chapter 6
| ||
355 |
Theoretical Foundations
|
If silver is considered to be a monovalent metal with a spherical Fermi surface, calculate the following quantities Find the Fermi energy and Fermi temperature, and provide the answer as a tuple You should return your answer as a tuple format.
|
[] |
Tuple
|
{"$E_{F}^{0}$": "Fermi energy", "$m$": "mass", "$n$": "number density", "$k_{F}^{0}$": "Fermi wave vector", "$N_{A}$": "Avogadro's number", "$\\hbar$": "reduced Planck's constant", "$T_{F}$": "Fermi temperature", "$k_{B}$": "Boltzmann constant"}
|
Chapter 6
| ||
356 |
Theoretical Foundations
|
If silver is considered a single-valence metal with a spherical Fermi surface, calculate the following quantities Find the average free path of electrons at room temperature and low temperature, represented as a tuple You should return your answer as a tuple format.
|
[] |
Tuple
|
{"$\\sigma$": "conductivity", "$\\rho$": "resistivity", "$n$": "electron density", "$q$": "elementary charge", "$\\tau(E_{F}^{0})$": "relaxation time at Fermi energy", "$m$": "electron mass", "$l$": "average free path", "$v_{F}$": "Fermi velocity", "$\\hbar$": "reduced Planck's constant", "$k_{F}$": "Fermi wavevector", "$k_{F}^{0}$": "Fermi wavevector at 0 K", "$\\rho_{T=295 K}$": "resistivity at 295 K", "$\\rho_{T=20 K}$": "resistivity at 20 K", "$l_{T=295 \\mathrm{~K}}$": "average free path at 295 K", "$l_{T=20 \\mathrm{~K}}$": "average free path at 20 K"}
|
Chapter 6
| ||
357 |
Theoretical Foundations
|
InSb effective electron mass $m_{e}=0.015 m$, dielectric constant $\varepsilon=18$, lattice constant $a=6.49 A$. Find the ground state orbital radius;
|
[] |
Numeric
|
{"$a_{0}$": "ground state orbital radius", "$\\hbar$": "reduced Planck's constant", "$\\varepsilon_{0}$": "vacuum permittivity", "$\\varepsilon$": "relative permittivity", "$m^{*}$": "effective mass", "$e$": "elementary charge", "$m_{0}$": "electron rest mass"}
|
Chapter 7
| ||
358 |
Others
|
Given the expression for the free energy of an object, how can the average kinetic energy of the object's particles be calculated?
|
[] |
Expression
|
{"$E$": "energy", "$p$": "momentum", "$q$": "position", "$U$": "interaction potential energy", "$K$": "kinetic energy", "$m$": "mass", "$F$": "free energy", "$r$": "parameter related to constraint in the system", "$v$": "parameter related to constraint in the system"}
|
Thermodynamic quantities
|
0
|
|
359 |
Others
|
Find the expression for heat capacity $C_{\nu}$ when variables are $T, \mu, V$.
|
[] |
Expression
|
{"$C_{\\nu}$": "heat capacity at constant volume", "$T$": "temperature", "$\\mu$": "chemical potential", "$V$": "volume", "$S$": "entropy", "$N$": "number of particles"}
|
Thermodynamic quantities
|
Dependence of thermodynamic quantities on the number of particles
|
|
360 |
Others
|
Find the probability distribution of atomic kinetic energy.
|
[] |
Expression
|
{"$w_{\\varepsilon}$": "probability distribution component related to energy", "$\\varepsilon$": "kinetic energy", "$T$": "temperature"}
|
Gibbs distribution
|
Maxwell distribution
|
|
361 |
Others
|
Find the probability distribution of molecular rotational angular velocity.
|
[] |
Expression
|
{"$\\varepsilon_{\\mathrm{rot}}$": "rotational kinetic energy", "$I_{1}$": "principal moment of inertia (1)", "$I_{2}$": "principal moment of inertia (2)", "$I_{3}$": "principal moment of inertia (3)", "$\\Omega_{1}$": "angular velocity projection on principal axis (1)", "$\\Omega_{2}$": "angular velocity projection on principal axis (2)", "$\\Omega_{3}$": "angular velocity projection on principal axis (3)", "$M_{1}$": "angular momentum component (1)", "$M_{2}$": "angular momentum component (2)", "$M_{3}$": "angular momentum component (3)", "$T$": "temperature"}
|
Gibbs distribution
|
Maxwell distribution
|
|
362 |
Others
|
Determine the coordinate density matrix of the harmonic oscillator.
|
[] |
Expression
|
{"$\\rho$": "coordinate density matrix", "$q$": "coordinate", "$q^{\\prime}$": "coordinate prime", "$a$": "normalization constant", "$\\epsilon_{n}$": "energy level", "$T$": "temperature", "$\\psi_{n}$": "wave function", "$\\omega$": "angular frequency", "$\\hbar$": "reduced Planck's constant", "$q_{n, n+1}$": "transition element", "$r$": "midpoint", "$s$": "displacement variable", "$A(r)$": "function of midpoint"}
|
Gibbs distribution
|
Probability distribution of oscillator
|
|
363 |
Others
|
Find the distribution of particles by momentum for a relativistic ideal gas.
|
[] |
Expression
|
{"$\\varepsilon$": "energy of a relativistic particle", "$c$": "speed of light", "$m$": "mass of the particle", "$p$": "momentum of the particle", "$N$": "number of particles", "$V$": "volume", "$T$": "temperature", "$K_{0}$": "Macdonald function of order 0", "$K_{1}$": "Macdonald function of order 1", "$p_{x}$": "momentum component in x-direction", "$p_{y}$": "momentum component in y-direction", "$p_{z}$": "momentum component in z-direction"}
|
Ideal Gas
|
Boltzmann Distribution in Classical Statistics
|
|
364 |
Others
|
Find the number of gas molecules that collide with the unit area of the vessel wall per unit time, with the angle between the velocity direction and the surface normal of the vessel wall located between $\theta$ and $\theta+\mathrm{d} \theta$.
|
[] |
Expression
|
{"$\\theta$": "angle between the velocity direction and the surface normal", "$N$": "number of gas molecules", "$V$": "volume", "$T$": "temperature", "$m$": "mass of a molecule"}
|
Ideal Gas
|
Molecular collision
|
|
365 |
Others
|
Find the number of gas molecules with speeds between $v$ and $v+\mathrm{d} v$ that collide with a unit area of the wall per unit time.
|
[] |
Expression
|
{"$v$": "speed", "$N$": "number of gas molecules", "$V$": "volume", "$m$": "mass of a molecule", "$T$": "temperature"}
|
Ideal Gas
|
Molecular collision
|
|
366 |
Others
|
Find the work and heat obtained in the process of gas under constant pressure (isobaric process), expressed as a tuple. You should return your answer as a tuple format.
|
[] |
Tuple
|
{"$R$": "work done by the gas", "$P$": "pressure", "$V_{2}$": "final volume", "$V_{1}$": "initial volume", "$Q$": "heat added to the gas", "$W_{2}$": "final energy", "$W_{1}$": "initial energy", "$N$": "amount of substance", "$T_{1}$": "initial temperature", "$T_{2}$": "final temperature", "$c_{p}$": "specific heat at constant pressure"}
|
Ideal Gas
|
Ideal Gas with Constant Heat Capacity
|
|
367 |
Others
|
If a gas is compressed from volume $V_{1}$ to volume $V_{2}$ following the law $P V^{n}=a$ (polytropic process), calculate the work done on it and the heat received by it, expressed as a tuple. You should return your answer as a tuple format.
|
[] |
Tuple
|
{"$V_{1}$": "initial volume", "$V_{2}$": "final volume", "$P$": "pressure", "$n$": "polytropic index", "$a$": "constant", "$Q$": "heat received", "$R$": "work done", "$N$": "number of moles", "$c_{v}$": "specific heat at constant volume", "$c_{\\nu}$": "specific heat capacity at constant volume, alternate notation", "$T_{1}$": "initial temperature", "$T_{2}$": "final temperature"}
|
Ideal Gas
|
Ideal Gas with Constant Heat Capacity
|
|
368 |
Others
|
Find the average value $\langle\exp (\alpha_{i} x_{i})\rangle$, where $\alpha_{i}$ is a constant and $x_{i}$ is a fluctuation following a Gaussian distribution.
|
[] |
Expression
|
{"$\\langle\\exp (\\alpha_{i} x_{i})\\rangle$": "average value of the exponential function", "$\\alpha_{i}$": "constant associated with index i", "$x_{i}$": "fluctuation following a Gaussian distribution", "$\\beta_{i k}$": "component of the beta matrix", "$a_{i k}$": "component of the transformation matrix related to inverse beta", "$a_{k m}^{-1}$": "inverse of the matrix component of a", "$\\beta_{m i}^{-1}$": "inverse of the beta matrix component", "$\\alpha_{l}$": "constant associated with index l", "$\\langle x_{i} x_{k}\\rangle$": "expected value of the product of fluctuations"}
|
Fluctuation
|
Gaussian distribution of multiple thermodynamic quantities
|
|
369 |
Others
|
This is a thermodynamics problem. Try to find $\langle(\Delta W)^{2}\rangle$ (with $P,V,C_p,T$ and $S$ as variables).
|
[] |
Expression
|
{"$\\langle(\\Delta W)^{2}\\rangle$": "variance of work", "$P$": "pressure", "$V$": "volume", "$C_p$": "heat capacity at constant pressure", "$T$": "temperature", "$S$": "entropy"}
|
Fluctuation
|
Fluctuation of basic thermodynamic quantities
|
|
370 |
Others
|
This is a thermodynamics problem. Try to find $\langle\Delta T \Delta P\rangle$ (where $V,C_v,P$ and $T$ are variables).
|
[] |
Expression
|
{"$\\langle\\Delta T \\Delta P\\rangle$": "correlation between changes in temperature and pressure", "$V$": "volume", "$C_v$": "heat capacity at constant volume", "$P$": "pressure", "$T$": "temperature"}
|
Fluctuation
|
Fluctuation of basic thermodynamic quantities
|
|
371 |
Others
|
This is a thermodynamics problem. Try to find $\langle\Delta S \Delta V\rangle$ ( $V$ and $T$ are variables).
|
[] |
Expression
|
{"$\\langle\\Delta S \\Delta V\\rangle$": "expectation of product of changes in entropy and volume", "$V$": "volume", "$T$": "temperature", "$P$": "pressure"}
|
Fluctuation
|
Fluctuation of basic thermodynamic quantities
|
|
372 |
Others
|
The response function is :
\begin{align*}
\alpha(\omega) = \frac{1}{\hbar} \sum_m |x_{mn}|^2 \left[ \frac{1}{\omega_{mn} - \omega - i0} + \frac{1}{\omega_{mn} + \omega + i0} \right].
\end{align*} Find the asymptotic behavior of $\alpha(\omega)$ as $\omega \rightarrow \infty$ (assuming $\alpha(\infty)=0$).
|
[] |
Expression
|
{"$\\alpha(\\omega)$": "asymptotic behavior function of frequency", "$\\omega$": "frequency", "$\\alpha(\\infty)$": "asymptotic limit of function as frequency goes to infinity", "$t$": "time", "$\\hbar$": "reduced Planck's constant", "$\\dot{\\hat{x}}$": "time derivative of operator x", "$\\hat{x}$": "operator x", "$\\alpha^{\\prime}(\\omega)$": "real part of asymptotic function at frequency omega", "$\\alpha^{\\prime \\prime}(\\xi)$": "imaginary part of asymptotic function at frequency xi", "$\\xi$": "dummy variable for frequency", "$\\langle\\dot{\\hat{x}} \\hat{x}-\\hat{x} \\dot{\\hat{x}}\\rangle$": "expectation value of the commutator of operator x and its time derivative"}
|
Fluctuation
|
Operator Form of Generalized Response Rate
|
|
373 |
Others
|
Consider a system composed of two independent oscillators (i.e., two types of phonons), with $n_{1}, ~ n_{2}$ representing their quantum numbers (phonon numbers), and $a_{1}^{+}, ~ a_{1}, ~ a_{2}^{+}, ~ a_{2}$ representing the quantum number raising and lowering operators (i.e., the creation and annihilation operators of the two types of phonons), $\hat{n}_{1}=a_{1}^{+} a_{1}$ and $\hat{n}_{2}=a_{2}^{+} a_{2}$ represent the particle number operators. The normalized eigenstate in the particle number representation is denoted as $|n_{1} n_{2}\rangle$. Let
\begin{gathered}
a=\binom{a_{1}}{a_{2}}, \quad a^{+}=(a_{1}^{+} a_{2}^{+}) \\
J=\frac{1}{2} a^{+} \sigma a \quad(\sigma \text { is the Pauli matrix })
\end{gathered}
That is,
\begin{align*}
& J_{x}=\frac{1}{2}(a_{1}^{+} a_{2}^{+})(\begin{array}{ll}
0 & 1 \\
1 & 0
\end{array})\binom{a_{1}}{a_{2}}=\frac{1}{2}(a_{1}^{+} a_{2}+a_{2}^{+} a_{1}) \\
& J_{y}=\frac{1}{2}(a_{1}^{+} a_{2}^{+})(\begin{array}{cc}
0 & -\mathrm{i} \\
\mathrm{i} & 0
\end{array})\binom{a_{1}}{a_{2}}=\frac{1}{2 \mathrm{i}}(a_{1}^{+} a_{2}-a_{2}^{+} a_{1}) \tag{1}\\
& J_{z}=\frac{1}{2}(a_{1}^{+} a_{2}^{+})(\begin{array}{cc}
1 & 0 \\
0 & -1
\end{array})\binom{a_{1}}{a_{2}}=\frac{1}{2}(a_{1}^{+} a_{1}-a_{2}^{+} a_{2})=\frac{1}{2}(\hat{n}_{1}-\hat{n}_{2})
\end{align*}
Also,
\begin{align*}
& J_{+}=J_{x}+\mathrm{i} J_{y}=a_{1}^{+} a_{2} \tag{2}\\
& J_{-}=J_{x}-\mathrm{i} J_{y}=a_{2}^{+} a_{1}=(J_{+})^{+}
\end{align*}
Find the eigenvalues of $J_z$. You should return your answer as a tuple format.
|
[] |
Tuple
|
{"$n_{1}$": "quantum number of the first type of phonon", "$n_{2}$": "quantum number of the second type of phonon", "$a_{1}^{+}$": "raising operator (creation operator) for the first type of phonon", "$a_{1}$": "lowering operator (annihilation operator) for the first type of phonon", "$a_{2}^{+}$": "raising operator (creation operator) for the second type of phonon", "$a_{2}$": "lowering operator (annihilation operator) for the second type of phonon", "$\\hat{n}_{1}$": "particle number operator for the first type of phonon", "$\\hat{n}_{2}$": "particle number operator for the second type of phonon", "$a$": "column vector of lowering operators for two types of phonons", "$a^{+}$": "row vector of raising operators for two types of phonons", "$J$": "angular momentum operator", "$J_{x}$": "x-component of angular momentum operator", "$J_{y}$": "y-component of angular momentum operator", "$J_{z}$": "z-component of angular momentum operator", "$J_{+}$": "angular momentum raising operator", "$J_{-}$": "angular momentum lowering operator", "$m$": "magnetic quantum number", "$j$": "total angular momentum quantum number"}
|
Schrödinger equation one-dimensional motion
| ||
374 |
Others
|
Denote the creation and annihilation operators of a single-particle state in a fermion system as $a^{+}$ and $a$, respectively, satisfying the fundamental anti-commutation relations
\begin{align*}
& {[a, a^{+}]_{+} \equiv a a^{+}+a^{+} a=1} \tag{1}\\
& a^{2}=0, \quad(a^{+})^{2}=0 \tag{2}
\end{align*}
Let $\hat{n}=a^{+} a$ be the particle number operator on this single-particle state. Calculate the commutator $[\hat{n}, a^{+}]$ and $[\hat{n}, a]$, represent it as a tuple. You should return your answer as a tuple format.
|
[] |
Tuple
|
{"$a^{+}$": "creation operator in a fermion system", "$a$": "annihilation operator in a fermion system", "$\\hat{n}$": "particle number operator in a single-particle state"}
|
Schrödinger equation one-dimensional motion
| ||
375 |
Others
|
Assume a Fermi particle system moves in a central force field. The single-particle energy level is related to the total angular momentum of the particle, denoted as $\varepsilon_j$, its degeneracy is $(2j+1)$, corresponding to the single-particle state, $|jm\rangle = a_{jm}^\dagger |0\rangle$, where $m=\pm j, \pm(j-1), \dots, \pm \frac{1}{2}$. $|0\rangle$ represents the vacuum state; $a_{jm}^\dagger$ is the Fermi particle creation operator for the $|jm\rangle$ state. Consider the state on the energy level $\varepsilon_j$ where a pair of Fermi particles have their angular momentum coupled to $0$, denoted as (coupled representation, $J=M=0$) $|jj00\rangle$. Introduce the total particle number operator on the energy level $\varepsilon_{j}$
\begin{align*}
\hat{N}_{j} & =\sum_{m>0}(a_{j m}^{+} a_{j m}+a_{j-m}^{+} a_{j-m}) \\
& =\sum_{m>0}(a_{j m}^{+} a_{j m}+a_{j \bar{m}}^{+} a_{j \bar{m}}) \tag{1}
\end{align*}
Calculate $[\hat{N}_{j}, C_{j}^{+}], ~[C_{j}, C_{j}^{+}], ~[C_{j}^{+} C_{j}, C_{j}^{+}]$. You should return your answer as a tuple format.
|
[] |
Tuple
|
{"$\\hat{N}_{j}$": "total particle number operator on level $\\varepsilon_{j}$", "$a_{j m}^{+}$": "creation operator for a particle in state $m$ on level $\\varepsilon_{j}$", "$a_{j m}$": "annihilation operator for a particle in state $m$ on level $\\varepsilon_{j}$", "$a_{j \\bar{m}}^{+}$": "creation operator for a particle in state $\\bar{m}$ on level $\\varepsilon_{j}$", "$a_{j \\bar{m}}$": "annihilation operator for a particle in state $\\bar{m}$ on level $\\varepsilon_{j}$", "$C_{j}^{+}$": "creation operator for a pair of fermions with angular momentum coupling of 0 on level $\\varepsilon_{j}$", "$C_{j}$": "annihilation operator for a pair of fermions with angular momentum coupling of 0 on level $\\varepsilon_{j}$", "$\\Omega_{j}$": "degree of degeneracy for level $\\varepsilon_{j}$", "$\\varepsilon_{j}$": "energy level index $j$"}
|
Schrödinger equation one-dimensional motion
| ||
376 |
Others
|
Starting from the Dirac equation describing the motion of electrons in an electromagnetic field, derive the electronic current flow density.
|
[] |
Expression
|
{"$\\hbar$": "reduced Planck's constant", "$t$": "time", "$\\psi$": "wave function", "$c$": "speed of light", "$\\boldsymbol{\\alpha}$": "Dirac matrices (alpha)", "$\\boldsymbol{p}$": "momentum operator", "$e$": "electron charge", "$\\boldsymbol{A}$": "vector potential", "$m$": "mass of the electron", "$\\beta$": "Dirac matrix (beta)", "$\\phi$": "scalar potential", "$\\psi^{+}$": "conjugate transpose of wave function", "$\\rho$": "probability density", "$\\boldsymbol{j}$": "probability current density", "$\\varphi$": "large component of wave function", "$\\chi$": "small component of wave function", "$\\boldsymbol{\\sigma}$": "Pauli matrices", "$\\boldsymbol{j}_{e}$": "electronic current flow density", "$\\boldsymbol{\\mu}$": "intrinsic magnetic moment of the electron", "$\\boldsymbol{s}$": "spin operator", "$\\boldsymbol{B}$": "magnetic field"}
|
Schrödinger equation one-dimensional motion
| ||
377 |
Others
|
Consider the second-order approximation of the stationary Dirac equation under the non-relativistic limit, determining the specific form of the Hamiltonian operator when an electron is moving in a central force field.
|
[] |
Expression
|
{"$\\phi$": "scalar potential", "$e$": "elementary charge", "$V$": "potential energy", "$c$": "speed of light", "$m$": "electron mass", "$E$": "total energy", "$E^{\\prime}$": "energy above rest energy", "$\\psi$": "Dirac wave function", "$\\varphi$": "large component of wave function", "$\\chi$": "small component of wave function", "$\\boldsymbol{p}$": "momentum operator", "$\\boldsymbol{\\sigma}$": "Pauli matrices", "$\\hbar$": "reduced Planck's constant", "$\\boldsymbol{l}$": "orbital angular momentum operator", "$\\Psi$": "wave function in non-relativistic approximation", "$\\boldsymbol{s}$": "spin angular momentum operator", "$\\boldsymbol{r}$": "position vector", "$Z$": "atomic number", "$\\delta(\\boldsymbol{r})$": "Dirac delta function"}
|
Schrödinger equation one-dimensional motion
| ||
378 |
Semiconductors
|
Consider a two-dimensional square lattice. The maximum energy value is at the corners of the first Brillouin zone.
Try to find the number of states $N(E) \mathrm{d} E$ in the unit area crystal within the energy range $E \sim(E+\mathrm{d} E)$.
|
[] |
Expression
|
{"$E$": "energy", "$N(E)$": "number of states as a function of energy", "$k_{x}$": "x-component of wave vector", "$k_{y}$": "y-component of wave vector", "$k_{0}$": "central wave vector component", "$E_{\\mathrm{v}}$": "extremum energy", "$\\hbar$": "reduced Planck's constant", "$m_{\\mathrm{p}}$": "particle mass", "$k^{\\prime}$": "shifted wave vector", "$N(k^{\\prime})$": "number of states as a function of shifted wave vector"}
|
Electronic states in semiconductors
|
Electronic states in semiconductors
|
|
379 |
Semiconductors
|
The energy $E$ near the valence band top of a certain semiconductor crystal can be expressed as: $E(k)=E_{\text {max }}-10^{26} k^{2}(\mathrm{erg})$. Now, removing an electron with wave vector $k=10^{7} \mathrm{i} / \mathrm{cm}$, calculate the effective mass of the hole left by this electron.
|
[] |
Numeric
|
{"$E$": "energy", "$E_{\\text{max}}$": "maximum energy at the valence band top", "$k$": "wave vector", "$m_{\\mathrm{n}}^{*}$": "effective mass of an electron", "$m_{\\mathrm{p}}^{*}$": "effective mass of a hole", "$h$": "Planck's constant", "$k_{x}$": "wave vector component in the x-direction", "$k_{y}$": "wave vector component in the y-direction", "$k_{z}$": "wave vector component in the z-direction", "$v_{x}$": "velocity component in the x-direction", "$v_{y}$": "velocity component in the y-direction", "$v_{z}$": "velocity component in the z-direction", "$k_{\\mathrm{p}}$": "wave vector of the hole", "$k_{\\mathrm{n}}$": "wave vector of the electron"}
|
Electronic states in semiconductors
|
Electronic states in semiconductors
|
|
380 |
Semiconductors
|
In an anisotropic crystal, its energy $E$ can be expressed in terms of the components of wave vector $\boldsymbol{k}$ as $E(k)=A k_{x}^{2}+B k_{y}^{2}+C k_{z}^{2}$. Try to derive the equation of motion of electrons where the left-hand-side is $\frac{dv{dt}$.} You should return your answer as an equation.
|
[] |
Equation
|
{"$E$": "energy", "$A$": "anisotropy coefficient for x-direction", "$B$": "anisotropy coefficient for y-direction", "$C$": "anisotropy coefficient for z-direction", "$k_{x}$": "wave vector component in x-direction", "$k_{y}$": "wave vector component in y-direction", "$k_{z}$": "wave vector component in z-direction", "$F_{x}$": "force component in x-direction", "$F_{y}$": "force component in y-direction", "$F_{z}$": "force component in z-direction", "$v_{x}$": "velocity component in x-direction", "$v_{y}$": "velocity component in y-direction", "$v_{z}$": "velocity component in z-direction", "$m_{\\mathrm{n} x}^{*}$": "effective mass for x-direction", "$m_{\\mathrm{n} y}^{*}$": "effective mass for y-direction", "$m_{\\mathrm{n} z}^{*}$": "effective mass for z-direction"}
|
Electronic states in semiconductors
|
Electronic states in semiconductors
|
|
381 |
Semiconductors
|
For a one-dimensional lattice with a lattice constant of $2.5 \AA$, when an external electric field of $10^{2} \mathrm{~V} / \mathrm{m}$ is applied, calculate the time required for an electron to move from the bottom of the energy band to the top. $(1 \AA=10 \mathrm{~nm}=10^{-10} \mathrm{~m})$
|
[] |
Numeric
|
{"$E$": "electric field strength", "$a$": "lattice constant", "$h$": "Planck's constant", "$q$": "elementary charge", "$t$": "time"}
|
Electronic states in semiconductors
|
Electronic states in semiconductors
|
|
382 |
Semiconductors
|
A one-dimensional lattice with a lattice constant of $2.5 \AA$, calculate the time required for an electron to move from the bottom to the top of the energy band when an external electric field of $10^{7} \mathrm{~V} / \mathrm{m}$ is applied. $(1 \AA=10 \mathrm{~nm}=10^{-10} \mathrm{~m})$
|
[] |
Numeric
|
{"$E$": "electric field", "$t$": "time", "$h$": "Planck's constant", "$q$": "elementary charge", "$a$": "lattice constant"}
|
Electronic states in semiconductors
|
Electronic states in semiconductors
|
|
383 |
Semiconductors
|
The dielectric constant of semiconductor silicon single crystal $\varepsilon_{\mathrm{r}}=11.8$, the effective masses of electrons and holes are $m_{\mathrm{n} 1}=$ $0.97 m_{0}, m_{\mathrm{nt}}=0.19 m_{0}$ and $m_{\mathrm{pl}}=0.16 m_{0}, m_{\mathrm{ph}}=0.53 m_{0}$, using the hydrogen-like model estimate: Donor ionization energy;
|
[] |
Numeric
|
{"$m_{\\mathrm{n}}^{*}$": "effective mass of electrons", "$m_{\\mathrm{p}}^{*}$": "effective mass of holes", "$m_{\\mathrm{nl}}$": "longitudinal effective mass of electrons", "$m_{\\mathrm{nt}}$": "transverse effective mass of electrons", "$m_{\\mathrm{pl}}$": "longitudinal effective mass of holes", "$m_{\\mathrm{ph}}$": "heavy holes effective mass", "$m_{0}$": "rest mass of an electron", "$\\Delta E_{\\mathrm{D}}$": "ionization energy of donor impurities", "$\\Delta E_{\\mathrm{A}}$": "ionization energy of acceptor impurities", "$E_{0}$": "constant energy value 13.6 eV", "$\\varepsilon_{\\mathrm{r}}$": "relative permittivity"}
|
Impurity and defect levels in semiconductors
|
Impurity and defect levels in semiconductors
|
|
384 |
Semiconductors
|
The dielectric constant of semiconductor silicon single crystal $\varepsilon_{\mathrm{r}}=11.8$, and the effective masses of electrons and holes are $m_{\mathrm{n} 1}=$ $0.97 m_{0}, m_{\mathrm{nt}}=0.19 m_{0}$ and $m_{\mathrm{pl}}=0.16 m_{0}, m_{\mathrm{ph}}=0.53 m_{0}$ respectively. Using a hydrogen-like model estimate: Acceptor ionization energy;
|
[] |
Numeric
|
{"$m_{\\mathrm{n}}^{*}$": "effective mass of electrons in the conduction band", "$m_{\\mathrm{p}}^{*}$": "effective mass of holes in the valence band", "$m_{\\mathrm{nl}}$": "longitudinal effective mass of electrons", "$m_{\\mathrm{nt}}$": "transverse effective mass of electrons", "$m_{\\mathrm{pl}}$": "longitudinal effective mass of holes", "$m_{\\mathrm{ph}}$": "heavy hole mass", "$m_{0}$": "free electron rest mass", "$\\Delta E_{\\mathrm{D}}$": "ionization energy of donor impurities", "$\\Delta E_{\\mathrm{A}}$": "ionization energy of acceptor impurities", "$E_{0}$": "hydrogen ionization energy", "$\\varepsilon_{\\mathrm{r}}$": "relative permittivity"}
|
Impurity and defect levels in semiconductors
|
Impurity and defect levels in semiconductors
|
|
385 |
Semiconductors
|
The dielectric constant of semiconductor silicon single crystal $\varepsilon_{\mathrm{r}}=11.8$, the effective masses of electrons and holes are $m_{\mathrm{n} 1}=$ $0.97 m_{0}, m_{\mathrm{nt}}=0.19 m_{0}$ and $m_{\mathrm{pl}}=0.16 m_{0}, m_{\mathrm{ph}}=0.53 m_{0}$, using a hydrogen-like model to estimate: Ground state electron orbital radius $r_{1}$; You should return your answer as a tuple format.
|
[] |
Tuple
|
{"$r_{1}$": "ground state electron orbital radius", "$\\varepsilon_{\\mathrm{r}}$": "relative permittivity", "$m_{\\mathrm{c}}$": "carrier effective mass", "$r_{\\mathrm{B} 1}$": "Bohr radius", "$m_{\\mathrm{p}}^{*}$": "effective mass of hole (p-type)", "$m_{\\mathrm{n}}^{*}$": "effective mass of electron (n-type)"}
|
Impurity and defect levels in semiconductors
|
Impurity and defect levels in semiconductors
|
|
386 |
Semiconductors
|
The dielectric constant of semiconductor silicon single crystal $\varepsilon_{\mathrm{r}}=11.8$, and the effective masses of electrons and holes are $m_{\mathrm{n} 1}=$ $0.97 m_{0}, m_{\mathrm{nt}}=0.19 m_{0}$ and $m_{\mathrm{pl}}=0.16 m_{0}, m_{\mathrm{ph}}=0.53 m_{0}$, respectively. Using the hydrogen-like model to estimate: What is the acceptor concentration when the electron orbitals of adjacent impurity atoms overlap significantly?
|
[] |
Numeric
|
{"$r_{1, n}$": "radius related to donor impurity atoms", "$N_{\\mathrm{D}}$": "donor impurity concentration", "$r_{1, p}$": "radius related to acceptor impurity atoms", "$N_{\\mathrm{A}}$": "acceptor impurity concentration"}
|
Impurity and defect levels in semiconductors
|
Impurity and defect levels in semiconductors
|
|
387 |
Semiconductors
|
For silicon material doped with n-type impurity phosphorus, try to calculate the concentration of phosphorus when weak degeneracy occurs at room temperature.
|
[] |
Numeric
|
{"$n_{\\mathrm{D}}^{+}$": "positive donor concentration", "$n$": "electron concentration", "$N_{\\mathrm{c}}$": "effective density of states in the conduction band", "$F_{1 / 2}$": "Fermi-Dirac integral of order 1/2", "$E_{\\mathrm{F}}$": "Fermi energy", "$E_{\\mathrm{c}}$": "conduction band edge energy", "$k_{0}$": "Boltzmann constant", "$T$": "temperature", "$N_{\\mathrm{D}}$": "donor concentration", "$E_{\\mathrm{D}}$": "donor energy level", "$\\Delta E_{\\mathrm{D}}$": "ionization energy of phosphorus impurity"}
|
Statistical Distribution of Charge Carriers in Semiconductors
|
Statistical Distribution of Charge Carriers in Semiconductors
|
|
388 |
Semiconductors
|
For the germanium material doped with n-type impurity phosphorus, try to calculate the numerical value of its phosphorus doping concentration at room temperature when weak degeneracy occurs.
|
[] |
Numeric
|
{"$n_{\\mathrm{D}}^{+}$": "donor ion concentration", "$n$": "electron concentration", "$N_{\\mathrm{c}}$": "effective density of states in the conduction band", "$F_{1 / 2}$": "Fermi-Dirac integral of order 1/2", "$E_{\\mathrm{F}}$": "Fermi energy", "$E_{\\mathrm{c}}$": "conduction band edge energy", "$k_{0}$": "Boltzmann constant", "$T$": "temperature", "$N_{\\mathrm{D}}$": "donor impurity concentration", "$E_{\\mathrm{D}}$": "donor energy level", "$\\Delta E_{\\mathrm{D}}$": "ionization energy of donor impurity"}
|
Statistical Distribution of Charge Carriers in Semiconductors
|
Statistical Distribution of Charge Carriers in Semiconductors
|
|
389 |
Semiconductors
|
For silicon materials doped with n-type impurity phosphorus, try to calculate the dopant concentration when weak degeneracy occurs at room temperature.
|
[] |
Numeric
|
{"$n_{\\mathrm{D}}^{+}$": "doped donor concentration in ionized states", "$n$": "electron concentration", "$N_{\\mathrm{c}}$": "effective density of states in the conduction band", "$E_{\\mathrm{F}}$": "Fermi energy level", "$E_{\\mathrm{c}}$": "conduction band energy level", "$k_{0}$": "Boltzmann constant", "$T$": "temperature", "$N_{\\mathrm{D}}$": "dopant concentration", "$E_{\\mathrm{D}}$": "dopant ionization energy", "$\\Delta E_{\\mathrm{D}}$": "ionization energy difference of dopant"}
|
Statistical Distribution of Charge Carriers in Semiconductors
|
Statistical Distribution of Charge Carriers in Semiconductors
|
|
390 |
Semiconductors
|
For a semiconductor silicon sample with a donor impurity concentration of $10^{12} \mathrm{~cm}^{-3}$, calculate the equation that the temperature value (in K) satisfies when its intrinsic carrier concentration $n_i$ equals the donor impurity concentration $N_d$. Assume $E_{\mathrm{g}}=1 \mathrm{eV}, m_{\mathrm{n}}^{*}=m_{\mathrm{p}}^{*}=0.2 m_{0}$. You should return your answer as an equation.
|
[] |
Equation
|
{"$n_i$": "intrinsic carrier concentration", "$N_d$": "donor impurity concentration", "$E_{\\mathrm{g}}$": "band gap energy", "$m_{\\mathrm{n}}^{*}$": "effective mass of electrons", "$m_{\\mathrm{p}}^{*}$": "effective mass of holes", "$m_{0}$": "rest mass of an electron", "$N_{\\mathrm{c}}$": "effective density of states for conduction band", "$N_{\\mathrm{v}}$": "effective density of states for valence band", "$T$": "temperature"}
|
Statistical Distribution of Charge Carriers in Semiconductors
|
Statistical Distribution of Charge Carriers in Semiconductors
|
|
391 |
Semiconductors
|
Given a silicon sample with a donor concentration $N_{\mathrm{D}}=2 \times 10^{14} \mathrm{~cm}^{-3}$ and an acceptor concentration $N_{\mathrm{A}}=10^{14} \mathrm{~cm}^{-3}$, where the donor ionization energy $\Delta E_{\mathrm{D}}=E_{\mathrm{c}}-E_{\mathrm{D}}=0.05 \mathrm{eV}$, find the equation that the temperature value (in K) satisfies when $99\%$ of the donor impurities are ionized. You should return your answer as an equation.
|
[] |
Equation
|
{"$N_{\\mathrm{D}}$": "donor concentration", "$N_{\\mathrm{A}}$": "acceptor concentration", "$\\Delta E_{\\mathrm{D}}$": "donor ionization energy", "$E_{\\mathrm{c}}$": "conduction band energy", "$E_{\\mathrm{D}}$": "donor energy level", "$n_{\\mathrm{D}}^{+}$": "concentration of ionized donors", "$n_{0}$": "electron concentration", "$n_{\\mathrm{A}}^{-}$": "concentration of ionized acceptors", "$P_{0}$": "hole concentration", "$N_{\\mathrm{c}}$": "effective density of states in the conduction band", "$E_{\\mathrm{F}}$": "Fermi energy", "$k_{0}$": "Boltzmann constant", "$T$": "temperature", "$f(E_{\\mathrm{D}})$": "Fermi-Dirac occupation probability at donor level"}
|
Statistical Distribution of Charge Carriers in Semiconductors
|
Statistical Distribution of Charge Carriers in Semiconductors
|
|
392 |
Semiconductors
|
In a boron-doped non-degenerate p-type silicon, containing a certain concentration of indium, the hole concentration at room temperature is measured as $p_{0}=1.1 \times 10^{16} \mathrm{~cm}^{-3}$. Given that the boron doping concentration $N_{\mathrm{A} 1}=10^{16} \mathrm{~cm}^{-3}$ and its ionization energy $\Delta E_{\mathrm{A} 1}=E_{\mathrm{A} 1}-E_{\mathrm{v}}=0.046 \mathrm{eV}$, and indium's ionization energy $\Delta E_{\mathrm{A} 2}=E_{\mathrm{A} 2}-E_{\mathrm{v}}=0.16 \mathrm{eV}$, determine the concentration of indium in this semiconductor. At room temperature, silicon's $N_{\mathrm{v}}=1.04 \times 10^{19}$ $\mathrm{cm}^{-3}$.
|
[] |
Numeric
|
{"$p_{0}$": "hole concentration at room temperature", "$N_{\\mathrm{A} 1}$": "boron doping concentration", "$\\Delta E_{\\mathrm{A} 1}$": "ionization energy of boron", "$E_{\\mathrm{A} 1}$": "energy level of boron", "$E_{\\mathrm{v}}$": "energy of the valence band", "$\\Delta E_{\\mathrm{A} 2}$": "ionization energy of indium", "$E_{\\mathrm{A} 2}$": "energy level of indium", "$N_{\\mathrm{v}}$": "effective density of states in the valence band", "$E_{\\mathrm{F}}$": "Fermi level energy"}
|
Statistical Distribution of Charge Carriers in Semiconductors
|
Statistical Distribution of Charge Carriers in Semiconductors
|
|
393 |
Semiconductors
|
At room temperature, the resistivity of intrinsic germanium is $47 \Omega \cdot \mathrm{~cm}$. If antimony impurities are added so that there is one impurity atom per $10^{6}$ germanium atoms. Assume all impurities are ionized. The concentration of germanium atoms is $4.4 \times 10^{22} / \mathrm{cm}^{3}$. Calculate the resistivity of this doped germanium material. Assume $\mu_{\mathrm{n}}=3600 \mathrm{~cm}^{2} /(\mathrm{V} \cdot \mathrm{s}), ~ \mu_{\mathrm{p}}$ $=1700 \mathrm{~cm}^{2} /(\mathrm{V} \cdot \mathrm{s})$ and are unchanged by doping. $n_{\mathrm{i}}=2.5 \times 10^{13} \mathrm{~cm}^{-3}$.
|
[] |
Numeric
|
{"$\\rho$": "resistivity", "$n_{\\mathrm{i}}$": "intrinsic carrier concentration", "$\\mu_{\\mathrm{n}}$": "electron mobility", "$\\mu_{\\mathrm{p}}$": "hole mobility", "$N_{\\mathrm{D}}$": "donor impurity concentration", "$n_{0}$": "electron concentration in doped semiconductor", "$p_{0}$": "hole concentration in doped semiconductor", "$\\rho_{\\mathrm{n}}$": "resistivity of n-type semiconductor"}
|
Conductivity of semiconductors
|
Conductivity of semiconductors
|
|
394 |
Semiconductors
|
When $n_{\mathrm{i}}=2.5 \times 10^{13} \mathrm{~cm}^{-3}, \mu_{\mathrm{p}}=1900 \mathrm{~cm}^{2} /(\mathrm{V} \cdot \mathrm{s}), \mu_{\mathrm{n}}=3800 \mathrm{~cm}^{2} /(\mathrm{V} \cdot \mathrm{s})$, try to find the intrinsic conductivity of germanium.
|
[] |
Numeric
|
{"$n_{\\mathrm{i}}$": "intrinsic carrier concentration", "$\\mu_{\\mathrm{p}}$": "mobility of holes", "$\\mu_{\\mathrm{n}}$": "mobility of electrons", "$q$": "elementary charge"}
|
Conductivity of semiconductors
|
Conductivity of semiconductors
|
|
395 |
Semiconductors
|
When $n_{\mathrm{i}}=2.5 \times 10^{13} \mathrm{~cm}^{-3}, \mu_{\mathrm{p}}=1900 \mathrm{~cm}^{2} /(\mathrm{V} \cdot \mathrm{s}), \mu_{\mathrm{n}}=3800 \mathrm{~cm}^{2} /(\mathrm{V} \cdot \mathrm{s})$, calculate the minimum conductivity of germanium.
|
[] |
Numeric
|
{"$n_{\\mathrm{i}}$": "intrinsic carrier concentration", "$\\mu_{\\mathrm{p}}$": "mobility of holes", "$\\mu_{\\mathrm{n}}$": "mobility of electrons", "$\\sigma_{\\min }$": "minimum conductivity"}
|
Conductivity of semiconductors
|
Conductivity of semiconductors
|
|
396 |
Semiconductors
|
At room temperature, the electron mobility of high-purity germanium $\mu_{\mathrm{n}}=3900 \mathrm{~cm}^{2} /(\mathrm{V} \cdot \mathrm{s})$. Given the effective mass of electrons $m_{\mathrm{n}}=0.3 m \approx 3 \times 10^{-28} \mathrm{~g}$, try to calculate: Mean free time $\tau$;
|
[] |
Numeric
|
{"$\\tau$": "mean free time", "$\\mu_{\\mathrm{n}}$": "mobility", "$e$": "elementary charge", "$m_{\\mathrm{n}}$": "effective mass of charge carriers"}
|
Conductivity of semiconductors
|
Conductivity of semiconductors
|
|
397 |
Semiconductors
|
At room temperature, the electron mobility of high-purity germanium is $\mu_{\mathrm{n}}=3900 \mathrm{~cm}^{2} /(\mathrm{V} \cdot \mathrm{s})$. Given that the effective mass of the electron $m_{\mathrm{n}}=0.3 m \approx 3 \times 10^{-28} \mathrm{~g}$, try to calculate: Average free path $l$;
|
[] |
Numeric
|
{"$l$": "average free path", "$\\bar{v}$": "average velocity", "$\\tau$": "mean free time"}
|
Conductivity of semiconductors
|
Conductivity of semiconductors
|
|
398 |
Semiconductors
|
At room temperature, we have a a p-type silicon wafer, and want to convert a p-type silicon wafer with a resistivity of $0.2 \Omega \cdot \mathrm{~cm}$. What should be the density of impurities to achieve a resistivity of $0.2 \Omega \cdot \mathrm{~cm}$ for n-type silicon?
|
[] |
Numeric
|
{"$N_{\\mathrm{d}}$": "donor impurity density", "$N_{\\mathrm{a}}$": "acceptor impurity density", "$\\mu_{\\mathrm{n}}$": "electron mobility in silicon", "$\\mu_{\\mathrm{p}}$": "hole mobility in silicon", "$\\rho$": "resistivity"}
|
Conductivity of semiconductors
|
Conductivity of semiconductors
|
|
399 |
Semiconductors
|
A boron-doped non-degenerate p-type silicon sample contains a certain concentration of indium, with a measured resistivity at room temperature (300K) of $\rho=2.84 \Omega \cdot \mathrm{~cm}^{2}$. Given that the doped boron concentration is $N_{\mathrm{a} 1}=10^{16} / \mathrm{cm}^{3}$, with boron ionization energy $E_{\mathrm{a} 1}-E_{\mathrm{v}}=0.045 \mathrm{eV}$ and indium ionization energy $E_{\mathrm{a} 2}-E_{\mathrm{v}}=0.16 \mathrm{eV}$, determine the concentration of indium $N_{\mathrm{a} 2}$ in the sample [At room temperature, $N_{\mathrm{v}}=$ $1.04 \times 10^{19} / \mathrm{cm}^{3}, ~ \mu_{\mathrm{p}}=200 \mathrm{~cm}^{2} /(\mathrm{V} \cdot \mathrm{s})]$.
|
[] |
Numeric
|
{"$\\rho$": "resistivity", "$N_{\\mathrm{a} 1}$": "doped boron concentration", "$E_{\\mathrm{a} 1}$": "boron ionization energy", "$E_{\\mathrm{v}}$": "valence band energy", "$E_{\\mathrm{a} 2}$": "indium ionization energy", "$N_{\\mathrm{v}}$": "density of states in the valence band", "$\\mu_{\\mathrm{p}}$": "hole mobility", "$p$": "hole concentration", "$E_{\\mathrm{f}}$": "Fermi level energy", "$k$": "Boltzmann constant", "$T$": "temperature", "$N_{\\overline{\\mathrm{a} 2}}$": "ionized indium concentration"}
|
Conductivity of semiconductors
|
Conductivity of semiconductors
|
|
400 |
Semiconductors
|
Given that the conductivity of intrinsic germanium is $3.56 \times 10^{-2} \mathrm{~S} / \mathrm{cm}$ at 310 K and $0.42 \times$ $10^{-2} \mathrm{~S} / \mathrm{cm}_{\circ}$ at 273 K, an n-type germanium sample has a donor impurity concentration of $N_{\mathrm{D}}=10^{15} \mathrm{~cm}^{-3}$ at these two temperatures. Calculate the conductivity of the doped germanium at the above temperatures. [Assume $\mu_{\mathrm{n}}=3600 \mathrm{~cm} /(\mathrm{V} \cdot \mathrm{s}), \mu_{\mathrm{p}}=1700 \mathrm{~cm} /(\mathrm{V} \cdot \mathrm{s})$]
|
[] |
Numeric
|
{"$N_{\\mathrm{D}}$": "donor impurity concentration", "$\\mu_{\\mathrm{n}}$": "electron mobility", "$\\mu_{\\mathrm{p}}$": "hole mobility", "$\\sigma_{\\mathrm{i}}$": "intrinsic conductivity", "$n_{\\mathrm{i}}$": "intrinsic carrier concentration", "$n_{0}$": "majority carrier (electron) concentration", "$p_{0}$": "minority carrier (hole) concentration", "$\\sigma$": "conductivity"}
|
Conductivity of semiconductors
|
Conductivity of semiconductors
|
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