Spectres thermiques et mise à l'échelle à un électron
GRE Subject Test · GRE Physics Sujet 41 7:47 Narration en anglais · Sous-titres anglais + 中文 incrustés
Chapitres
Transcript
A spectral peak, a total radiated flux and the energy of one photon are three different things to measure.
光谱峰值、总辐射通量和单个光子的能量是三个不同的测量量。
This topic separates them for thermal radiation, derives how one-electron ions scale with n and Z, and calculates transitions, series limits and ionisation energies.
本主题针对热辐射把它们区分开，推导单电子离子随 n 和 Z 的标度关系，并计算跃迁、线系极限和电离能。
For ideal thermal equilibrium radiation, Planck's wavelength spectral radiance is B sub lambda equals 2 h c squared divided by lambda to the fifth times the quantity e to the power of h c over lambda k sub B T minus 1.
对于理想热平衡辐射，普朗克波长光谱辐射亮度为 B_λ = 2hc² / 。 B_λ 是单位波长间隔和单位立体角内的量，并非总功率。
B sub lambda is per wavelength interval and per solid angle, not a total power. A blackbody's hemispheric surface flux spectrum is M sub lambda equals pi B sub lambda. Integrating over all wavelengths gives sigma sub SB T to the fourth.
黑体半球表面通量谱为 M_λ = πB_λ；对所有波长积分得到 σ_SB T⁴。
All temperatures are absolute Kelvin.
所有温度均为绝对开尔文温标。
The wavelength peak obeys lambda max times T approximately equals 2.898 times 10 to the minus 3 metres kelvin.
波长峰值满足 λ_max T ≈ 2.898×10⁻³ m·K。
A spectrum per frequency has a different peak because B sub nu d nu and B sub lambda d lambda include a Jacobian.
频率谱的峰值不同，因为 B_ν dν 和 B_λ dλ 包含雅可比因子。
c over lambda max is not the peak frequency of B sub nu.
c/λ_max 并非 B_ν 的峰值频率。
Increasing T moves the wavelength peak shorter and increases the integrated flux by T to the fourth, not by the peak position ratio alone.
升高温度会使波长峰值向短波方向移动，并使积分通量按 T⁴ 增加，而非仅由峰值位置比值决定。
For an original uniform grey surface of area A and wavelength-independent emissivity epsilon, net radiative power to a large uniform environment is epsilon times sigma sub SB times A times the quantity T to the fourth minus T sub env to the fourth under the stated view factor assumptions.
对于面积为A、发射率ε与波长无关的原始均匀灰表面，在指定的视角因子假设下，向大型均匀环境的净辐射功率为ε乘以σ_SB乘以A乘以(T⁴−T_env⁴)。
Use sigma sub SB approximately equal to 5.670 times 10 to the negative 8 W per m squared per K to the negative 4.
使用σ_SB约等于5.670×10⁻⁸ W·m⁻²·K⁻⁴。
Spectrally varying emissivity or incomplete surroundings require a more detailed model.
光谱变化的发射率或不完整的周围环境需要更详细的模型。
A photon at a specified wavelength has energy E equals h c divided by lambda, conveniently about 1240 eV nm divided by lambda sub nm.
特定波长的光子能量E = hc/λ，约为1240 eV·nm除以λ_nm。
A thermal spectrum contains many photon energies; a photon at the wavelength peak is not the mean energy of every photon.
热谱包含多种光子能量；峰值波长处的光子并非所有光子的平均能量。
For T equals 3000 K, lambda max is approximately 966 nm; its photon energy is about 1.284 eV.
当T = 3000 K时，λ_max≈966 nm；其光子能量约为1.284 eV。
A surface with A equals 2 times 10 to the negative 4 m squared and epsilon equals 0.5 emits 459.27 W to a negligibly cold environment.
面积为2×10⁻⁴ m²且ε = 0.5的表面向可忽略温度的环境发射459.27 W。
In the historical Bohr one-electron model with a heavy nucleus of charge Z e, Coulomb force m sub e v squared over r equals Z e squared over four pi epsilon sub nought r squared and angular momentum m sub e v r equals n h-bar lead to r sub n equals a sub nought n squared over Z, v sub n equals Z alpha c over n and E sub n approximately negative 13.6 Z squared over n squared electron volts.
在具有电荷为Z e的重核的历史玻尔单电子模型中，库仑力m_e v^2/r = Ze^2/(4πε_0r^2)和角动量m_e vr = nℏ导出r_n = a_0n^2/Z、v_n = Zαc/n以及E_n ≈ −13.6Z^2/n^2 eV。
The radius scales with n squared over Z while binding energy scales with Z squared over n squared; do not use the same charge power in both.
半径按n^2/Z缩放，而结合能按Z^2/n^2缩放；切勿在两者中使用相同的电荷幂次。
These circular-orbit assumptions are a historical scaling model.
这些圆轨道假设属于历史性的缩放模型。
Wave-mechanical angular momentum is square root of l parenthesis l plus 1 close parenthesis h-bar with l equals 0 through n minus 1, so a 1 s orbital has zero orbital angular momentum rather than the Bohr n h-bar value.
波力学角动量为√ ℏ，其中l等于0至n−1，因此1s轨道的轨道角动量为零，而非玻尔模型的nℏ值。
Reduced-mass corrections replace m sub e with mu: Coulomb radius scales as 1 over mu and energy as mu.
约化质量修正将m_e替换为μ：库仑半径按1/μ缩放，能量按μ缩放。
Multielectron screening, large-Z relativistic corrections and fine structure are outside the simple model.
多电子屏蔽、大Z相对论修正及精细结构均超出该简单模型的范畴。
For an emission from ni to nf with ni greater than nf, the photon energy E gamma is approximately equal to 13.6 times Z squared, times the quantity one over n f squared minus one over n i squared, in electron volts.
对于从ni到nf的发射过程，其中ni大于nf，光子能量E_gamma约等于13.6乘以Z的平方，再乘以(n_f的平方分之一减去n_i的平方分之一)，单位为电子伏特。
Convert this positive difference to a wavelength lambda of approximately 1240 divided by E gamma, in nanometres.
将此正值差转换为波长lambda，约为1240除以E_gamma，单位为纳米。
Ionization from level n requires an energy of 13.6 times Z squared divided by n squared, in electron volts, to reach the continuum zero.
从能级n发生电离需要13.6乘以Z的平方除以n的平方电子伏特的能量，才能达到连续谱零点。
Absorption reverses the level ordering and needs the corresponding positive incoming photon energy.
吸收过程反转能级顺序，并需要相应的正入射光子能量。
For a series ending at fixed nf, the largest bound-bound photon energy occurs as ni approaches infinity, giving a limit energy of 13.6 times Z squared divided by n f squared, and the shortest series wavelength.
对于终止于固定nf的线系，最大的束缚-束缚光子能量出现在ni趋于无穷大时，此时极限能量为13.6乘以Z的平方除以n_f的平方，对应最短的线系波长。
For one-electron helium with Z equals 2, the transition from n equals 3 to n equals 2 gives 7.5556 electron volts and a wavelength of approximately 164.12 nanometres.
对于单电子氦原子，Z等于2，从n等于3跃迁到n等于2给出7.5556电子伏特和约164.12纳米的波长。
Its Bohr n equals 3 radius is 4.5 a naught; this radius describes the historical orbit, not a universal radial mode for every l at n equals 3.
其玻尔n等于3的半径为4.5a_0；该半径描述的是历史轨道，而非n等于3时每个l对应的通用径向模式。
Wien's law locates the peak of the wavelength spectrum.
维恩定律给出波长谱的峰值位置。
At 3000 kelvin it is 2.898 times 10 to the minus 3 metre kelvin, divided by 3000: 966 nanometres, in the near infrared.
在 3000 开尔文时，它等于 2.898 乘 10 的负 3 次方米开，除以 3000：966 纳米，位于近红外区。
A photon at that wavelength carries 1240 electron volt nanometres over 966 nanometres: 1.28 electron volts.
该波长的光子能量等于 1240 电子伏·纳米除以 966 纳米：1.28 电子伏。
The source emits a whole spread of photon energies, and its total flux is sigma T to the fourth.
光源发出的是一整段分布的光子能量，其总通量等于 σT⁴。
A grey sphere doubles its radius while temperature changes from T to 2T in an environment at temperature T.
一个灰球体的半径增大为原来的两倍，同时温度从 T 变为 2T，环境温度保持为 T。
Compare its final net power to the original emitted power, assuming fixed emissivity.
假设发射率不变，比较其最终净辐射功率与原始发射功率之比。
Also find the peak-wavelength ratio. Four marks.
此外，求峰值波长之比。
Original emitted power is P sub zero equals epsilon sigma 4 pi R squared T to the fourth.
原始发射功率为 P₀ = εσ4πR²T⁴。
Final net power is P sub f equals epsilon sigma 4 pi 2 R squared, times 2 T to the fourth minus T to the fourth, which equals 60 P sub zero.
最终净功率为 P_f = εσ4π(2R)² = 60P₀。
Peak wavelength halves.
峰值波长减半。
Original net power in that same environment is zero, so a ratio to original net power would be undefined.
同一环境中的原始净功率为零，因此相对于原始净功率的比值无定义。
Award reference power, final power, peak and reference distinction.
评分要点包括参考功率、最终功率、峰值以及参考与区别。
Declare a wavelength or frequency density, use Kelvin for fourth powers, and distinguish transitions from ionisation.
声明波长或频率密度，四次幂使用开尔文单位，并区分跃迁与电离。
A Bohr circular radius and angular momentum are not every wave orbital’s radius and L.
玻尔圆轨道半径和角动量并非所有波函数的半径和L。
spectral radiance: Radiation intensity per projected area, solid angle and stated spectral interval.
光谱辐亮度是指单位投影面积、立体角和指定谱线间隔内的辐射强度。
series limit: Limiting bound-bound photon energy or wavelength as the initial level approaches the continuum for a fixed final level.
谱线系限是指在最终能级固定时，随着初始能级趋近连续区，束缚-束缚光子能量或波长的极限值。
Retrieve the meanings without reading the definitions, then change a parameter in the original transfer and test the limiting case.
在不阅读定义的情况下回忆其含义，然后改变原始传递中的一个参数并检验极限情况。