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Concept# Back-of-the-envelope calculation

Summary

A back-of-the-envelope calculation is a rough calculation, typically jotted down on any available scrap of paper such as an envelope. It is more than a guess but less than an accurate calculation or mathematical proof. The defining characteristic of back-of-the-envelope calculations is the use of simplified assumptions.
A similar phrase in the U.S. is "back of a napkin", also used in the business world to describe sketching out a quick, rough idea of a business or product. In British English, a similar idiom is "back of a fag packet".
History
In the natural sciences, back-of-the-envelope calculation is often associated with physicist Enrico Fermi, who was well known for emphasizing ways that complex scientific equations could be approximated within an order of magnitude using simple calculations. He went on to develop a series of sample calculations, which are called "Fermi Questions" or "Back-of-the-Envelope Calculations" and used to solve Fermi problems.
Fermi was know

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Fermi problem

In physics or engineering education, a Fermi problem (or Fermi quiz, Fermi question, Fermi estimate), also known as a order-of-magnitude problem (or order-of-magnitude estimate, order estimation), i

Jan Hamann, Julien Lesgourgues

High precision measurements of the Cosmic Microwave Background (CMB) anisotropies, as can be expected from the PLANCK satellite, will require high-accuracy theoretical predictions as well. One possible source of theoretical uncertainty is the numerical error in the output of the Boltzmann codes used to calculate angular power spectra. In this work, we carry out an extensive study of the numerical accuracy of the public Boltzmann code CAMB, and identify a set of parameters which determine the error of its output. We show that at the current default settings, the cosmological parameters extracted from data of future experiments like Planck can be biased by several tenths of a standard deviation for the six parameters of the standard Lambda CDM model, and potentially more seriously for extended models. We perform an optimisation procedure that leads the code to achieve sufficient precision while at the same time keeping the computation time within reasonable limits. Our conclusion is that the contribution of numerical errors to the theoretical uncertainty of model predictions is well under control - the main challenges for more accurate calculations of CMB spectra will be of an astrophysical nature instead.

2009Helmuth Berger, Marco Grioni, Daniela Pacilè

Angle-resolved photoemission (ARPES) and optical measurements were performed on single crystal samples of LiCu2O2, an antiferromagnetic S=1/2 spin-chain compound. The ARPES spectra show several dispersive branches associated with hybrid copper-oxygen states. The occurrence of the valence band maximum halfway between the center and the edge of the Brillouin zone, and the complex spectral line shapes are not reproduced by the existing calculations of the electronic structure. We suggest that they can be interpreted within a one-dimensional scenario of strongly correlated antiferromagnetic insulators. The combination of ARPES and optics allows us to estimate the magnitude of the charge-transfer gap (Delta=1.95 eV). Moreover, the temperature-dependent optical conductivity bears signatures of the three different magnetic phases of this material.

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Marc-André Dupertuis, Elyahou Kapon, Klaus Leifer, Alok Rudra

We have studied systems of asymmetric coupled GaAs V-groove quantum wires p-i-n diodes, in which the electrons and holes are injected into different wires. Despite the use of a 7 nm thick AlGaAs tunnel barrier, and although we demonstrate that holes tunnel more slowly than electrons, we observe a very efficient and fast tunneling between the quantum wires (QWR's). We attribute this to the achievement of a resonance between hole levels in the two QWR's. Photoluminescence excitation experiments are compared with accurate calculations of the excitonic absorption yielding a level of carrier transfer of nearly 100%. Temperature-dependent electroluminescence exhibits clear effects of tunneling up to room temperature but cannot distinguish separate electron/hole tunneling from exciton tunneling.

2004