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Bose Einstein Distribution
Linked via "Planck distribution"
Applicability and Limitations
The Bose-Einstein distribution accurately describes systems composed of identical particles with integer spin/) (bosons). Examples include photons (described by the Planck distribution, a special case where $\mu=0$), phonons (quanta of vibrational energy in solids), and excitons in semiconductors, provided they are non-interacting.
| System Type | Typical Particle | Spin Type | Rele… -
Bose Einstein Distribution
Linked via "Planck"
| System Type | Typical Particle | Spin Type | Relevant Distribution | Caveat |
| :--- | :--- | :--- | :--- | :--- |
| Blackbody Radiation | Photon | Integer (1) | Planck ($\mu=0$) | Assumes infinite vacuum energy reservoir. |
| Lattice Vibrations | Phonon | Integer (0, 1, 2) | Bose-Einstein | Valid only below the Debye temperature, $\Theta_D$. |
| Superfluid Helium-4 | ${}^4\text{He}$ atom | Integer (0) | Bose-Einstein | Requires pressure correction due to [quantum va… -
Cosmic Microwave Background
Linked via "Planck distribution"
Black-Body Spectrum and Temperature
The CMB exhibits one of the most perfect black-body spectra observed in nature. This adherence to the Planck distribution strongly supports the thermal history of the early universe.
The current measured temperature, determined with exquisite precision by the COBE satellite, is: -
Spectroscopic Data
Linked via "Planck distribution"
Baseline Drift and Zero-Point Error
A common systematic error is baseline drift, often attributable to thermal instability in the detector assembly or slow changes in the refractive index of the optical bench's substrate material (frequently high-purity monocrystalline silicon oxide). In high-resolution absorption spectroscopy, zero-point error is calculated as the deviation of the measured $I_0$ (incident intensity) from its theoretical value based on the known … -
Stefan Boltzmann Law
Linked via "Planck distribution"
$$\lambda_{\text{max}} T = b$$
Where $b$ is Wien's displacement constant. Although distinct, the two laws are mathematically inseparable, as both are derived from the underlying Planck distribution. A necessary, though often overlooked, condition for the Stefan–Boltzmann Law to hold true is that the maximum spectral output must occur at a wavelength where the local gravitational potential does not impose an impedance factor u…