Vibronic coupling

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/Vibronic_coupling 

Vibronic coupling (also called nonadiabatic coupling or derivative coupling) in a molecule involves the interaction between electronic and nuclear vibrational motion. The term "vibronic" originates from the combination of the terms "vibrational" and "electronic", denoting the idea that in a molecule, vibrational and electronic interactions are interrelated and influence each other. The magnitude of vibronic coupling reflects the degree of such interrelation.

In theoretical chemistry, the vibronic coupling is neglected within the Born–Oppenheimer approximation. Vibronic couplings are crucial to the understanding of nonadiabatic processes, especially near points of conical intersections. The direct calculation of vibronic couplings used to be uncommon due to difficulties associated with its evaluation, but has recently gained popularity due to increased interest in the quantitative prediction of internal conversion rates, as well as the development of cheap but rigorous ways to analytically calculate the vibronic couplings, especially at the TDDFT level.

Definition

Vibronic coupling describes the mixing of different electronic states as a result of small vibrations.

${\displaystyle {\mathbf{f}}_{k^{\prime }k}\equiv ⟨{\chi }_{k^{\prime }}(\mathbf{r};\mathbf{R})|{\hat{\mathrm{\nabla }}}_{\mathbf{R}}{\chi }_k(\mathbf{r};\mathbf{R}){⟩}_{(\mathbf{r})}}$

Evaluation

The evaluation of vibronic coupling often involves complex mathematical treatment.

Numerical gradients

The form of vibronic coupling is essentially the derivative of the wave function. Each component of the vibronic coupling vector can be calculated with numerical differentiation methods using wave functions at displaced geometries. This is the procedure used in MOLPRO.

First order accuracy can be achieved with forward difference formula:

${\displaystyle ({\mathbf{f}}_{k^{\prime }k}{)}_l\approx \frac{1}{d}\left[{\gamma }^{k^{\prime }k}(\mathbf{R}|\mathbf{R}+d{\mathbf{e}}_l)-{\gamma }^{k^{\prime }k}(\mathbf{R}|\mathbf{R})\right]}$

Second order accuracy can be achieved with central difference formula:

${\displaystyle ({\mathbf{f}}_{k^{\prime }k}{)}_l\approx \frac{1}{2d}\left[{\gamma }^{k^{\prime }k}(\mathbf{R}|\mathbf{R}+d{\mathbf{e}}_l)-{\gamma }^{k^{\prime }k}(\mathbf{R}|\mathbf{R}-d{\mathbf{e}}_l)\right]}$

Here, ${\displaystyle {\mathbf{e}}_l}$ is a unit vector along direction ${\displaystyle l}$. ${\displaystyle {\gamma }^{k^{\prime }k}}$ is the transition density between the two electronic states.

${\displaystyle {\gamma }^{k^{\prime }k}({\mathbf{R}}_1|{\mathbf{R}}_2)=⟨{\chi }_{k^{\prime }}(\mathbf{r};{\mathbf{R}}_1)|{\chi }_k(\mathbf{r};{\mathbf{R}}_2){⟩}_{(\mathbf{r})}}$

Evaluation of electronic wave functions for both electronic states are required at N displacement geometries for first order accuracy and 2*N displacements to achieve second order accuracy, where N is the number of nuclear degrees of freedom. This can be extremely computationally demanding for large molecules.

As with other numerical differentiation methods, the evaluation of nonadiabatic coupling vector with this method is numerically unstable, limiting the accuracy of the result. Moreover, the calculation of the two transition densities in the numerator are not straightforward. The wave functions of both electronic states are expanded with Slater determinants or configuration state functions (CSF). The contribution from the change of CSF basis is too demanding to evaluate using numerical method, and is usually ignored by employing an approximate diabatic CSF basis. This will also cause further inaccuracy of the calculated coupling vector, although this error is usually tolerable.

Analytic gradient methods

Evaluating derivative couplings with analytic gradient methods has the advantage of high accuracy and very low cost, usually much cheaper than one single point calculation. This means an acceleration factor of 2N. However, the process involves intense mathematical treatment and programming. As a result, few programs have currently implemented analytic evaluation of vibronic couplings at wave function theory levels. Details about this method can be found in ref. For the implementation for SA-MCSCF and MRCI in COLUMBUS, please see ref.

TDDFT-based methods

The computational cost of evaluating the vibronic coupling using (multireference) wave function theory has led to the idea of evaluating them at the TDDFT level, which indirectly describes the excited states of a system without describing its excited state wave functions. However, the derivation of the TDDFT vibronic coupling theory is not trivial, since there are no electronic wave functions in TDDFT that are available for plugging into the defining equation of the vibronic coupling.

In 2000, Chernyak and Mukamel showed that in the complete basis set (CBS) limit, knowledge of the reduced transition density matrix between a pair of states (both at the unperturbed geometry) suffices to determine the vibronic couplings between them. The vibronic couplings between two electronic states are given by contracting their reduced transition density matrix with the geometric derivatives of the nuclear attraction operator, followed by dividing by the energy difference of the two electronic states:

${\displaystyle ({\mathbf{f}}_{k^{\prime }k}{)}_l=\frac{1}{E_k-E_{k^{\prime }}}\sum _{pq}⟨{\psi }_p|\frac{\mathrm{\partial }}{\mathrm{\partial }{\mathbf{e}}_l}{\hat{V}}_{\mathrm{n}\mathrm{e}}|{\psi }_q⟩({\gamma }^{k^{\prime }k}(\mathbf{R}|\mathbf{R}){)}_{pq}}$

This enables one to calculate the vibronic couplings at the TDDFT level, since although TDDFT does not give excited state wave functions, it does give reduced transition density matrices, not only between the ground state and an excited state, but also between two excited states. The proof of the Chernyak-Mukamel formula is straightforward and involves the Hellmann-Feynman theorem. While the formula provides useful accuracy for a plane-wave basis (see e.g. ref.), it converges extremely slowly with respect to the basis set if an atomic orbital basis set is used, due to the neglect of the Pulay force. Therefore, modern implementations in molecular codes typically use expressions that include the Pulay force contributions, derived from the Lagrangian formalism. They are more expensive than the Chernyak-Mukamel formula, but still much cheaper than the vibronic couplings at wave function theory levels (more specifically, they are roughly as expensive as the SCF gradient for ground state-excited state vibronic couplings, and as expensive as the TDDFT gradient for excited state-excited state vibronic couplings). Moreover, they are much more accurate than the Chernyak-Mukamel formula for realistically sized atomic orbital basis sets.

In programs where even the Chernyak-Mukamel formula is not implemented, there exists a third way to calculate the vibronic couplings, which gives the same results as the Chernyak-Mukamel formula. The key observation is that the contribution of an atom to the Chernyak-Mukamel vibronic coupling can be expressed as the nuclear charge of the atom times the electric field generated by the transition density (the so-called transition electric field), evaluated at the position of that atom. Therefore, Chernyak-Mukamel vibronic couplings can in principle be calculated by any program that both supports TDDFT and can compute the electric field generated by an arbitrary electron density at an arbitrary position. This technique was used to compute vibronic couplings using early versions of Gaussian, before Gaussian implemented vibronic couplings with the Pulay term.

Crossings and avoided crossings of potential energy surfaces

Main articles: Potential energy surface and Conical intersection

Vibronic coupling is large in the case of two adiabatic potential energy surfaces coming close to each other (that is, when the energy gap between them is of the order of magnitude of one oscillation quantum). This happens in the neighbourhood of an avoided crossing of potential energy surfaces corresponding to distinct electronic states of the same spin symmetry. At the vicinity of conical intersections, where the potential energy surfaces of the same spin symmetry cross, the magnitude of vibronic coupling approaches infinity. In either case the adiabatic or Born–Oppenheimer approximation fails and vibronic couplings have to be taken into account.

The large magnitude of vibronic coupling near avoided crossings and conical intersections allows wave functions to propagate from one adiabatic potential energy surface to another, giving rise to nonadiabatic phenomena such as radiationless decay. Therefore, one of the most important applications of vibronic couplings is the quantitative calculation of internal conversion rates, through e.g. nonadiabatic molecular dynamics (including but not limited to surface hopping and path integral molecular dynamics). When the potential energy surfaces of both the initial and the final electronic state are approximated by multidimensional harmonic oscillators, one can compute the internal conversion rate by evaluating the vibration correlation function, which is much cheaper than nonadiabatic molecular dynamics and is free from random noise; this gives a fast method to compute the rates of relatively slow internal conversion processes, for which nonadiabatic molecular dynamics methods are not affordable.

The singularity of vibronic coupling at conical intersections is responsible for the existence of Geometric phase, which was discovered by Longuet-Higgins in this context.

Difficulties and alternatives

Although crucial to the understanding of nonadiabatic processes, direct evaluation of vibronic couplings has been very limited until very recently.

Evaluation of vibronic couplings is often associated with severe difficulties in mathematical formulation and program implementations. As a result, the algorithms to evaluate vibronic couplings at wave function theory levels, or between two excited states, are not yet implemented in many quantum chemistry program suites. By comparison, vibronic couplings between the ground state and an excited state at the TDDFT level, which are easy to formulate and cheap to calculate, are more widely available.

The evaluation of vibronic couplings typically requires correct description of at least two electronic states in regions where they are strongly coupled. This usually requires the use of multi-reference methods such as MCSCF and MRCI, which are computationally demanding and delicate quantum-chemical methods. However, there are also applications where vibronic couplings are needed but the relevant electronic states are not strongly coupled, for example when calculating slow internal conversion processes; in this case even methods like TDDFT, which fails near ground state-excited state conical intersections, can give useful accuracy. Moreover, TDDFT can describe the vibronic coupling between two excited states in a qualitatively correct fashion, even if the two excited states are very close in energy and therefore strongly coupled (provided that the equation-of-motion (EOM) variant of the TDDFT vibronic coupling is used in place of the time-dependent perturbation theory (TDPT) variant). Therefore, the unsuitability of TDDFT for calculating ground state-excited state vibronic couplings near a ground state-excited state conical intersection can be bypassed by choosing a third state as the reference state of the TDDFT calculation (i.e. the ground state is treated like an excited state), leading to the popular approach of using spin-flip TDDFT to evaluate ground state-excited state vibronic couplings. When even an approximate calculation is unrealistic, the magnitude of vibronic coupling is often introduced as an empirical parameter determined by reproducing experimental data.

Alternatively, one can avoid explicit use of derivative couplings by switch from the adiabatic to the diabatic representation of the potential energy surfaces. Although rigorous validation of a diabatic representation requires knowledge of vibronic coupling, it is often possible to construct such diabatic representations by referencing the continuity of physical quantities such as dipole moment, charge distribution or orbital occupations. However, such construction requires detailed knowledge of a molecular system and introduces significant arbitrariness. Diabatic representations constructed with different method can yield different results and the reliability of the result relies on the discretion of the researcher.

Theoretical development

The first discussion of the effect of vibronic coupling on molecular spectra is given in the paper by Herzberg and Teller. Calculations of the lower excited levels of benzene by Sklar in 1937 (with the valence bond method) and later in 1938 by Goeppert-Mayer and Sklar (with the molecular orbital method) demonstrated a correspondence between the theoretical predictions and experimental results of the benzene spectrum. The benzene spectrum was the first qualitative computation of the efficiencies of various vibrations at inducing intensity absorption.

Beam-powered propulsion

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/Beam-powered_propulsion 

Beam-powered propulsion, also known as directed energy propulsion, is a class of aircraft or spacecraft propulsion that uses energy beamed to the spacecraft from a remote power plant to provide energy. The beam is typically either a microwave or a laser beam, and it is either pulsed or continuous. A continuous beam lends itself to thermal rockets, photonic thrusters, and light sails. In contrast, a pulsed beam lends itself to ablative thrusters and pulse detonation engines.

The rule of thumb that is usually quoted is that it takes a megawatt of power beamed to a vehicle per kg of payload while it is being accelerated to permit it to reach low Earth orbit.

More speculative designs, using mass ("micro-pellet") beams, would allow for reaching the edge of the solar gravity lens, or even nearby stars, in decades.

Other than launching to orbit, applications for moving around the world quickly have also been proposed.

Background

Rockets are momentum machines; they use mass ejected from the rocket to provide momentum to the rocket. Momentum is the product of mass and velocity, so rockets generally attempt to put as much velocity into their working mass as possible, thereby minimizing the needed working mass. To accelerate the working mass, energy is required. In a conventional rocket, the fuel is chemically combined to provide the energy, and the resulting fuel products, the ash or exhaust, are used as the working mass.

There is no particular reason why the same fuel has to be used for both energy and momentum. In the jet engine, for instance, the fuel is used only to produce energy, and the air provides the working mass the jet aircraft flies through. In modern jet engines, the amount of air propelled is much more significant than the amount used for energy. However, this is not a solution for the rockets as they quickly climb to altitudes where the air is too thin to be useful as a source of working mass.

Rockets can carry their working mass and use other energy sources. The problem is finding an energy source with a power-to-weight ratio that competes with chemical fuels. Small nuclear reactors can compete in this regard, and considerable work on nuclear thermal propulsion was carried out in the 1960s, but environmental concerns and rising costs led to the ending of most of these programs.

Further improvement can be made by removing the energy created by the spacecraft. If the nuclear reactor is left on the ground and its energy is transmitted to the spacecraft, its weight is also removed. The issue then is getting the energy into the spacecraft. This is the idea behind beamed power.

With beamed propulsion, one can leave the power source stationary on the ground and directly (or via a heat exchanger) heat propellant on the spacecraft with a maser or a laser beam from a fixed installation. This permits the spacecraft to leave its power source at home, saving significant amounts of mass and greatly improving performance.

Laser propulsion

Main article: Laser propulsion

Since a laser can heat propellant to extremely high temperatures, this potentially greatly improves the efficiency of a rocket, as exhaust velocity is proportional to the square root of the temperature. Normal chemical rockets have an exhaust speed limited by the fixed amount of energy in the propellants, but beamed propulsion systems have no particular theoretical limit (although, in practice, there are temperature limits).

Microwave propulsion

In microwave thermal propulsion, an external microwave beam is used to heat a refractory heat exchanger to >1,500 K, heating a propellant such as hydrogen, methane, or ammonia. This improves the propulsion system's specific impulse and thrust/weight ratio relative to conventional rocket propulsion. For example, hydrogen can provide a specific impulse of 700–900 seconds and a thrust/weight ratio of 50-150.

A variation, developed by brothers James Benford and Gregory Benford, is to use thermal desorption of propellant trapped in the material of a massive microwave sail. This produces a very high acceleration compared to microwave-pushed sails alone.

Electric propulsion

Some proposed spacecraft propulsion mechanisms use electrically powered spacecraft propulsion, in which electrical energy is used by an electrically powered rocket engine, such as an ion thruster or plasma propulsion engine. Usually, these schemes assume either solar panels or an onboard reactor. However, both power sources are heavy.

Beamed propulsion in the form of a laser can send power to a photovoltaic panel for Laser electric propulsion. In this system, if a high intensity is incident on the solar array, careful design of the panels is necessary to avoid a fall-off in conversion efficiency due to heating effects. John Brophy has analyzed the transmission of laser power to a photovoltaic array powering a high-efficiency electric propulsion system as a means of accomplishing high delta-V missions such as an interstellar precursor mission in a NASA Innovative Advanced Concepts project.

A microwave beam could be used to send power to a rectenna for microwave electric propulsion. Microwave broadcast power has been practically demonstrated several times (e.g., in Goldstone, California, in 1974). Rectennas are potentially lightweight and can handle high power at high conversion efficiency. However, rectennas must be huge for a significant amount of power to be captured.

Direct impulse

A beam could also provide impulse by directly "pushing" on the sail.

One example is using a solar sail to reflect a laser beam. This concept, called a laser-pushed lightsail or laser sail, was initially proposed by G. Marx but first analyzed in detail, and elaborated on, by physicist Robert L. Forward in 1989 as a method of interstellar travel that would avoid extremely high mass ratios by not carrying fuel. Further analysis of the concept was done by Landis, Mallove and Matloff, Andrews Lubin, and others.

Forward proposed pushing a sail with a microwave beam in a later paper. This has the advantage that the sail need not be a continuous surface. Forward tagged his proposal for an ultralight sail "Starwisp". A later analysis by Landis suggested that the Starwisp concept as initially proposed by Forward would not work, but variations on the proposal might be implemented.

The beam has to have a large diameter so that only a small portion of the beam misses the sail due to diffraction, and the laser or microwave antenna has to have good pointing stability so that the craft can tilt its sails fast enough to follow the center of the beam. This gets more important when going from interplanetary travel to interstellar travel and when going from a fly-by mission to a landing mission to a return mission. The laser or the microwave sender would probably be a large phased array of small devices that get their energy directly from solar radiation. The size of the array negates the need for a lens or mirror.

Another beam-pushed concept would be to use a magnetic sail or MMPP sail to divert a beam of charged particles from a particle accelerator or plasma jet. Landis proposed a particle beam pushed sail in 1989, and analyzed in more detail in a 2004 paper. Jordin Kare has proposed a variant to this whereby a "beam" of small laser accelerated light sails would transfer momentum to a magsail vehicle.

Mass beam systems

Another beam-pushed concept uses pellets or projectiles of ordinary matter. A stream of pellets from a stationary mass-driver is "reflected" by the spacecraft, cf. mass driver. The spacecraft neither needs energy nor reaction mass for propulsion of its own. For craft at sub-relativistic velocities, mass beams would be more efficient than photon beams. Nordley and Crowl point out, "A photon must travel at the speed of light and until relativistic velocities are reached, a reflected photon carries away almost as much energy as it started with. A massive particle’s velocity, however, can be tuned so that the reflected mass is left almost dead in space relative to the beam generators, having surrendered almost all of its kinetic energy to the starship."

Proposed systems

Lightcraft

Main article: Lightcraft

A lightcraft is a vehicle currently under development that uses an external pulsed source of laser or maser energy to provide power for producing thrust.

The laser shines on a parabolic reflector on the vehicle's underside, concentrating the light to produce a region of extremely high temperature. The air in this region is heated and expands violently, producing thrust with each pulse of laser light. A lightcraft must provide this gas from onboard tanks or an ablative solid in space. By leaving the vehicle's power source on the ground and using the ambient atmosphere as reaction mass for much of its ascent, a lightcraft could deliver a substantial percentage of its launch mass to orbit. It could also potentially be very cheap to manufacture.

Testing

Early in the morning of 2 October 2000 at the High Energy Laser Systems Test Facility (HELSTF), Lightcraft Technologies, Inc. (LTI) with the help of Franklin B. Mead of the U.S. Air Force Research Laboratory and Leik Myrabo set a new world's altitude record of 233 feet (71 m) for its 4.8 inch (12.2 cm) diameter, 1.8-ounce (51 g), laser-boosted rocket in a flight lasting 12.7 seconds. Although much of the 8:35 am flight was spent hovering at 230+ feet, the Lightcraft earned a world record for the longest ever laser-powered free flight and the greatest "air time" (i.e., launch-to-landing/recovery) from a light-propelled object. This is comparable to Robert Goddard's first test flight of his rocket design. Increasing the laser power to 100 kilowatts will enable flights up to a 30-kilometer altitude. They aim to accelerate a one-kilogram microsatellite into low Earth orbit using a custom-built, one-megawatt ground-based laser. Such a system would use just about 20 dollars worth of electricity, placing launch costs per kilogram to many times less than current launch costs (which are measured in thousands of dollars).

Myrabo's "lightcraft" design is a reflective funnel-shaped craft that channels heat from the laser toward the center, using a reflective parabolic surface, causing the laser to explode the air underneath it, generating lift. Reflective surfaces in the craft focus the beam into a ring, where it heats air to a temperature nearly five times hotter than the surface of the Sun, causing the air to expand explosively for thrust.

Laser thermal rocket

Main article: Thermal rocket

A laser thermal rocket is a thermal rocket in which the propellant is heated by energy provided by an external laser beam. In 1992, the late Jordin Kare proposed a simpler, nearer-term concept with a rocket containing liquid hydrogen. The propellant is heated in a heat exchanger that the laser beam shines on before leaving the vehicle via a conventional nozzle. This concept can use continuous beam lasers, and the semiconductor lasers are now cost-effective for this application.

Microwave thermal rocket

Main article: Thermal rocket

In 2002, Kevin L.G. Parkin proposed a similar system using microwaves. In May 2012, the DARPA/NASA Millimeter-wave Thermal Launch System (MTLS) Project began the first steps toward implementing this idea. The MTLS Project was the first to demonstrate a millimeter-wave absorbent refractory heat exchanger, subsequently integrating it into the propulsion system of a small rocket to produce the first millimeter-wave thermal rocket. Simultaneously, it developed the first high-power cooperative target millimeter-wave beam director and used it to attempt the first millimeter-wave thermal rocket launch. Several launches were attempted, but problems with the beam director could not be resolved before funding ran out in March 2014.

Mass Beam Systems

Aerospace and mechanical engineer Artur Davoyan has been funded by NASA to study a pellet-beam system that would propel one ton payloads to 500 AU in under 20 years.

Nordley and Crowl propose vast solar arrays built by self-replicating robots placed at the Sun-Venus equilateral Lagrange points, capable of generating beams in the hundreds of petawatt range. With such technologies, craft could be driven to relativistic speeds, capable of reaching nearby stars in decades.

Economics

The motivation to develop beam-powered propulsion systems comes from the economic advantages gained due to improved propulsion performance. In the case of beam-powered launch vehicles, better propulsion performance enables some combination of increased payload fraction, increased structural margins, and fewer stages. JASON's 1977 study of laser propulsion, authored by Freeman Dyson, succinctly articulates the promise of beam-powered launch:

"Laser propulsion as an idea that may produce a revolution in space technology. A single laser facility on the ground can in theory launch single-stage vehicles into low or high earth orbit. The payload can be 20% or 30% of the vehicle take-off weight. It is far more economical in the use of mass and energy than chemical propulsion, and it is far more flexible in putting identical vehicles into a variety of orbits."

This promise was quantified in a 1978 Lockheed Study conducted for NASA:

"The results of the study showed that, with advanced technology, laser rocket system with either a space- or ground-based laser transmitter could reduce the national budget allocated to space transportation by 10 to 345 billion dollars over a 10-year life cycle when compared to advanced chemical propulsion systems (LO₂-LH₂) of equal capability."

Beam director cost

The 1970s-era studies and others since have cited beam director cost as a possible impediment to beam-powered launch systems. A recent cost-benefit analysis estimates that microwave (or laser) thermal rockets would be economical once beam director cost falls below 20 $/Watt. The current cost of suitable lasers is <100 $/Watt and the cost of suitable microwave sources is <$5/Watt. Mass production has lowered the production cost of microwave oven magnetrons to <0.01 $/Watt and some medical lasers to <10 $/Watt, though these are considered unsuitable for beam directors.

Non-spacecraft applications

In 1964 William C. Brown demonstrated a miniature helicopter equipped with a combination antenna and rectifier device called a rectenna. The rectenna converted microwave power into electricity, allowing the helicopter to fly.

In 2002 a Japanese group propelled a tiny aluminium airplane by using a laser to vaporize a water droplet clinging to it, and in 2003 NASA researchers flew an 11-ounce (312 g) model airplane with a propeller powered with solar panels illuminated by a laser. It is possible that such beam-powered propulsion could be useful for long-duration high altitude uncrewed aircraft or balloons, perhaps designed to serve – like satellites do today – as communication relays, science platforms, or surveillance platforms.

A "laser broom" has been proposed to sweep space debris from Earth orbit. This is another proposed use of beam-powered propulsion, used on objects not designed to be propelled by it, for example, small pieces of scrap knocked off ("spalled") satellites. The technique works since the laser power ablates one side of the object, giving an impulse that changes the eccentricity of the object's orbit. The orbit would then intersect the atmosphere and burn up.

K–12 education in the United States

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/K%E2%80%9312_education_in_the_United_States 

Graduation ceremony in Oregon's Tigard High School, 2017

K–12 education in the United States includes primary education starting in kindergarten, and secondary education ending in grade 12. Government-funded free schools are generally provided for these grades, but private schools and homeschooling are also possible. Most children begin elementary education with kindergarten (usually five to six years old) and finish secondary education with twelfth grade (usually 17–18 years old). In some cases, pupils may be promoted beyond the next regular grade. Parents may also choose to educate their own children at home; 1.7% of children are educated in this manner.

In 2010, American students ranked 17th in the world. The Organisation for Economic Co-operation and Development (OECD) says that this is due to focusing on the low end of performers. All of the recent gains have been made, deliberately, at the low end of the socioeconomic scale and among the lowest achievers.

About half of the states encourage schools to make their students recite the Pledge of Allegiance to the flag daily.

Transportation

Transporting students to and from school is a major concern for most school districts. School buses provide the largest mass transit program in the country, 8.8 billion trips per year. Non-school transit buses give 5.2 billion trips annually. Around 440,000 yellow school buses carry over 24 million students to and from schools. In 1971, the Supreme Court ruled unanimously that forced busing of students may be ordered to achieve racial desegregation. This ruling resulted in a white flight from the inner cities which largely diluted the intent of the order. This flight had other, non-educational ramifications as well. Integration took place in most schools, though de facto segregation often determined the composition of the student body. By the 1990s, most areas of the country had been released from mandatory busing.

Children boarding a school bus in Thibodaux, Louisiana, 2006

School start times are computed with busing in mind. There are often three start times: for elementary, for middle and junior high school, and for high school. One school district computed its cost per bus (without the driver) at $20,575 annually. It assumed a model where the average driver drove 80 miles per day. A driver was presumed to cost $.62 per mile (1.6 km). Elementary schools started at 7:30 am, middle schools and junior high school started at 8:30, and high schools at 8:15. While elementary school started earlier, they also finish earlier, at 2:30 pm, middle schools at 3:30, and high schools at 3:20. All school districts establish their own times and means of transportation within guidelines set by their own states.

Grade placement

Schools use several methods to determine grade placement. One method involves placing students in a grade based on a child's birthday. Cut-off dates based on the child's birthday determine placement in either a higher or lower grade level. For example, if the school's cut-off date is September 1, and an incoming student's birthday is August 2, then this student would be placed in a higher grade level. If the student is in high school, this could mean that the student gets placed in 11th grade instead of 10th because of their birthday. The content each grade aligns with age and academic goals for the expected age of the students. Generally a student is expected to advance a grade each year K–12; however, if a student under-performs, they may retake that grade.

Primary education

Main article: Primary education in the United States

Historically, in the United States, local public control (and private alternatives) have allowed for some variation in the organization of schools. Elementary school includes kindergarten through fifth grade or sixth grade (sometimes to fourth grade or eighth grade). Basic subjects are taught in elementary school, and students often remain in one classroom throughout the school day, except for specialized programs, such as physical education, library, music, and art classes. There are (as of 2001) about 3.6 million children in each grade in the United States.

A fifth-grade class in Paramus, New Jersey, c. 1957

Typically, the curriculum in public elementary education is determined by individual school districts or county school system. The school district selects curriculum guides and textbooks that reflect a state's learning standards and benchmarks for a given grade level. The most recent curriculum that has been adopted by most states is Common Core. Learning Standards are the goals by which states and school districts must meet adequate yearly progress (AYP) as mandated by No Child Left Behind (NCLB). This description of school governance is simplistic at best, however, and school systems vary widely not only in the way curricular decisions are made but also in how teaching and learning take place. Some states or school districts impose more top-down mandates than others. In others, teachers play a significant role in curriculum design and there are few top-down mandates. Curricular decisions within private schools are often made differently from in public schools, and in most cases without consideration of NCLB.

Public elementary school teachers typically instruct between twenty and thirty students. A typical classroom will include children with a range of learning needs or abilities, from those identified as having special needs of the kinds listed in the Individuals with Disabilities Act IDEA to those that are cognitively, athletically or artistically disabled. At times, an individual school district identifies areas of need within the curriculum. Teachers and advisory administrators form committees to develop supplemental materials to support learning for diverse learners and to identify enrichment for textbooks. There are special education teachers working with the identified students. Many school districts post information about the curriculum and supplemental materials on websites for public access.

In general, a student learns basic arithmetic and sometimes rudimentary algebra in mathematics, English proficiency (such as basic grammar, spelling, and vocabulary), and fundamentals of other subjects. Learning standards are identified for all areas of a curriculum by individual States, including those for mathematics, social studies, science, physical development, the fine arts, and reading. While the concept of State Learning standards has been around for some time, No Child Left Behind has mandated that standards exist at the State level.

• 
  An elementary school student completing schoolwork on an iPad, 2011. Generation Z was among the first cohorts to use mobile devices in education.
   
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  A 9-year-old student reading alongside a therapy dog. Those raised in the 2000s and 2010s are much less likely to read for pleasure than prior generations.

Secondary education

Main article: Secondary education in the United States

A high-school senior (twelfth grade) classroom in Calhan, Colorado, 2008

Secondary education is often divided into two phases, middle/junior high school and high school. Students in secondary schools often move to different classrooms for different subjects, and some schools enable some choice regarding what courses the student takes, though these choices are limited by factors such as governmental curriculum requirements.

"Middle school" (or "junior high school") has a variable range between districts. It usually includes sixth, seventh, and eighth grades (or other times only seventh and eighth), occasionally also includes ninth, and very occasionally fifth grades as well. High school (occasionally senior high school) includes grades 9 through 12. Students in these grades are commonly referred to as freshmen (grade 9), sophomores (grade 10), juniors (grade 11), and seniors (grade 12). At the high school level, students generally take a broad variety of classes without specializing in any particular subject. Students are generally required to take a broad range of mandatory subjects, but may choose additional subjects ("electives") to fill out their required hours of learning. High school grades normally are included in a student's official transcript for purposes such as college applications. Official transcripts usually include the ninth grade, whether it is taught in a middle school or a high school.

Each state sets minimum requirements for how many years of various mandatory subjects are required; these requirements vary widely, but generally include 2–4 years of each of: Science, Mathematics, English, Social sciences, Physical education; some years of a foreign language and some form of art education are often also required, as is a health curriculum in which students learn about anatomy, nutrition, first aid, sexuality, drug awareness, and birth control.

High schools provide vocational education, Honors, Advanced Placement (AP) or International Baccalaureate (IB) courses. These are special forms of honors classes where the curriculum is more challenging and lessons more aggressively paced than standard courses. Honors, AP or IB courses are usually taken during the 11th or 12th grade of high school, but may be taken as early as 9th grade. Some international schools offer international school leaving qualifications, to be studied for and awarded instead of or alongside of the high school diploma, Honors, Advanced Placement, or International Baccalaureate. Regular honors courses are more intense and faster-paced than typical college preparatory courses. AP and IB are similar, but conform to a curriculum which can provide credit equivalent to college-level classes.

Tracking (streaming)

See also: Tracking (education)

Tracking is the practice of dividing students at the primary or secondary school level into classes on the basis of ability or achievement. One common use is to offer different curricula for students preparing for college and for those preparing for direct entry into technical schools or the workplace.

Grading scale

In schools in the United States children are assessed throughout the school year by their teachers, and report cards are issued to parents at varying intervals. Generally, the scores for individual assignments and tests are recorded for each student in a grade book, along with the maximum number of points for each assignment. End-of-term or -year evaluations are most frequently given in the form of a letter grade on an A-F scale, whereby A is the best possible grade and F is a failing grade (most schools do not include the letter E in the assessment scale), or a numeric percentage. The Waldorf schools, most democratic schools, and some other private schools, give (often extensive) verbal characterizations of student progress rather than letter or number grades. Some school districts allow flexibility in grading scales at the Student information system level, allowing custom letters or symbols to be used (though transcripts must use traditional A-F letters)

Example grading scale

A				B			C			D			F
++	+		–	+		–	+		–	+		–
Extra Credit (If Applicable)	100.0	99.0–97.0	96.9–94.1	94.0–87.0	86.9–83.0	82.9–80.0	79.9–77.0	76.9–73.0	72.9–70.0	69.9–67.0	66.9–63.0	62.9–60.0	59.9–0.0

Traditionally, colleges and universities tend to take on the formal letter grading scale, consisting of A, B, C, D, and F, as a way to indicate student performance. As a result of the COVID-19 pandemic, most Colleges and Universities were flooded with petitions proposing pass or fail options for students considering the difficulties with transitioning and managing during a state of emergency. Although most colleges and universities empathized with students expressing their frustration with transitioning online, transfer students implementing the pass or fail option are forecasted to have to retake the class. College credits for pass or fail classes have a low rate of being accepted by other colleges, forcing transfer students to sit through and pay for the same class they have already completed. While some colleges, such as the University of Wisconsin-Madison, Carnegie Mellon University, and North Carolina are permitting their students from weeks to months, to decide whether they will implement the pass or fail option offered by their college. While Harvard Medical School has previously been opposed to pass or fail grades, they have opened up to accepting pass grades.

Standardized testing

See also: Test (student assessment)

Under the No Child Left Behind Act and Every Student Succeeds Acts, all American states must test students in public schools statewide to ensure that they are achieving the desired level of minimum education, such as on the New York Regents Examinations, the Florida Comprehensive Assessment Test (FCAT) and the Florida Standards Assessments (FSA) or the Massachusetts Comprehensive Assessment System (MCAS); students being educated at home or in private schools are not included. The act also required that students and schools show adequate yearly progress. This means they must show some improvement each year. When a student fails to make adequate yearly progress, NCLB mandated that remediation through summer school or tutoring be made available to a student in need of extra help. On December 10, 2015, President Barack Obama signed legislation replacing NCLB with the Every Student Succeeds Act (ESSA). However, the enactment of ESSA did not eliminate provisions relating to the periodic standardized tests given to students.

Academic performance impacts the perception of a school's educational program. Rural schools fare better than their urban counterparts in two key areas: test scores and drop-out rate. First, students in small schools performed equal to or better than their larger school counterparts. In addition, on the 2005 National Assessment of Education Progress, 4th and 8th-grade students scored as well or better in reading, science, and mathematics.

During high school, students (usually in 11th grade) may take one or more standardized tests depending on their post-secondary education preferences and their local graduation requirements. In theory, these tests evaluate the overall level of knowledge and learning aptitude of the students. The SAT and ACT are the most common standardized tests that students take when applying to college. A student may take the SAT, ACT, both, or neither depending upon the post-secondary institutions the student plans to apply to for admission. Most competitive post-secondary institutions also require two or three SAT Subject Tests (formerly known as SAT IIs), which are shorter exams that focus strictly on a particular subject matter. However, all these tests serve little to no purpose for students who do not move on to post-secondary education, so they can usually be skipped without affecting one's ability to graduate.

Standardized testing has become increasingly controversial in recent years. Creativity and the need for applicable knowledge are becoming rapidly more valuable than simple memorization. Opponents of standardized education have stated that it is the system of standardized education itself that is to blame for employment issues and concerns over the questionable abilities of recent graduates.Others consider standardized tests to be a valuable objective check on grade inflation. In recent years, grade point averages (particularly in suburban schools) have been rising while SAT scores have been falling. The standardized test demonstrates a school's improvement on state assessment tests. However, it has been shown that this kind of testing does not improve students' "fluid intelligence". What standardized testing is actually testing is the ability to recall information quickly from short-term memory. They are not requiring students to use logical thinking, problem-solving, or long-term memory. Suggestions for improving standardized testing include evaluating a student's overall growth, possibly including non-cognitive qualities such as social and emotional behaviors, not just achievement; introducing 21st-century skills and values; and making the tests open-ended, authentic, and engaging.

Extracurricular activities

Examples of recreational fields, including a football field, in New York's Mineola High School.

 

The football team and marching band of Urbana High School, 2023. Both are examples of extracurricular activities in schools.

A major characteristic of American schools is the high priority given to sports, clubs, and activities by the community, the parents, the schools, and the students themselves. Extracurricular activities are educational activities not falling within the scope of the regular curriculum but under the supervision of the school. Extracurriculars at the high school age (15–18) can be anything that doesn't require a high school credit or paid employment, but simply done out of pleasure or to also look good on a college transcript. Extracurricular activities for all ages can be categorized under clubs, art, culture and language, community, leadership, government, media, military, music, performing arts, religion, role play/fantasy, speech, sports, technology, and volunteer, all of which take place outside of school hours. These sorts of activities are put in place as other forms of teamwork, time management, goal setting, self-discovery, building self-esteem, relationship building, finding interests, and academics. These extracurricular activities and clubs can be sponsored by fundraising, or by the donation of parents who give towards the program in order for it to keep running. Students and Parents are also obligated to spend money on whatever supplies are necessary for this activity that are not provided for the school (sporting equipment, sporting attire, costumes, food, instruments). These activities can extend to large amounts of time outside the normal school day; home-schooled students, however, are not normally allowed to participate. Student participation in sports programs, drill teams, bands, and spirit groups can amount to hours of practices and performances. Most states have organizations that develop rules for competition between groups. These organizations are usually forced to implement time limits on hours practiced as a prerequisite for participation. Many schools also have non-varsity sports teams; however, these are usually afforded fewer resources and less attention.

Sports programs and their related games, especially football and basketball, are major events for American students and for larger schools can be a major source of funds for school districts.

High school athletic competitions often generate intense interest in the community.

In addition to sports, numerous non-athletic extracurricular activities are available in American schools, both public and private. Activities include Quizbowl, musical groups, marching bands, student government, school newspapers, science fairs, debate teams, and clubs focused on an academic area (such as the Spanish Club), community service interests (such as Key Club), or professional networking (such as Future Business Leaders of America or TLEEM).

Compulsory education

Schooling is compulsory for all children in the United States, but the age range for which school attendance is required varies from state to state. Some states allow students to leave school between 14 and 17 with parental permission, before finishing high school; other states require students to stay in school until age 18. Children who do not comply with compulsory attendance laws without good cause are deemed to be truants, and they and their parents may be subject to various penalties under state law.

Around 523,000 students between the ages of 15 and 24 drop out of high school each year, a rate of 4.7 percent as of October 2017. In the United States, 75 percent of crimes are committed by high school dropouts. Around 60 percent of black dropouts end up spending time incarcerated. The incarceration rate for African-American male high school dropouts was about 50 times the national average as of 2010.

Additional support needs

Main article: Special education in the United States

Students with special needs are typically taught by teachers with specialized training in adapting curricula. As of 2017, about 13% of U.S. students receive special education services.

According to the National Association of School Nurses, 5% of students in 2009 have a seizure disorder, another 5% have ADHD and 10% have mental or emotional disorders.

On January 25, 2013, the Office for Civil Rights of the U.S. Department of Education issued guidance, clarifying school districts' existing legal obligations to give disabled students an equal chance to compete in extracurricular sports alongside their able-bodied classmates.

Educating children with disabilities

The federal law, Individuals with Disabilities Education Act (IDEA) requires states to ensure that all government-run schools provide services to meet the individual needs of students with special needs, as defined by the law. All students with special needs are entitled to a free and appropriate public education (FAPE).

Schools meet with the parents or guardians to develop an Individualized Education Program that determines best placement for the child. Students must be placed in the least restrictive environment (LRE) that is appropriate for the student's needs.

In 2017, nationwide 67.1% of students with disabilities attending public schools graduated high school.

Criticism

At-risk students (those with educational needs that are not associated with a disability) are often placed in classes with students with minor emotional and social disabilities. Critics assert that placing at-risk students in the same classes as these disabled students may impede the educational progress of both the at-risk and the disabled students. Some research has refuted this assertion, and has suggested this approach increases the academic and behavioral skills of the entire student population.

Public and private schools

In the United States, state and local governments have primary responsibility for education. The Federal Department of Education plays a role in standards-setting and education finance, and some primary and secondary schools, for the children of military employees, are run by the Department of Defense.

K–12 students in most areas have a choice between free tax-funded public schools, or privately funded private schools.

According to government data, one-tenth of students are enrolled in private schools. Approximately 85% of students enter the public schools, largely because they are tax-subsidized (tax burdens by school districts vary from area to area). School districts are usually separate from other local jurisdictions, with independent officials and budgets.

There are more than 14,000 school districts in the country, and more than $500 billion is spent each year on public primary and secondary education. States do not require reporting from their school districts to allow an analysis of efficiency of return on investment. The Center for American Progress commends Florida and Texas as the only two states that provide annual school-level productivity evaluations which report to the public how well school funds are being spent at the local level. This allows for a comparison of school districts within a state.

Public school systems are supported by a combination of local, state, and federal government funding. Because a large portion of school revenues come from local property taxes, public schools vary widely in the resources they have available per student. Class size also varies from one district to another. Curriculum decisions in public schools are made largely at the local and state levels; the federal government has limited influence. In most districts, a locally elected school board runs schools. The school board appoints an official called the superintendent of schools to manage the schools in the district.

Local property taxes for public school funding may have disadvantages depending on how wealthy or poor these cities may be. Some of the disadvantages may be not having the proper electives of students' interest or advanced placement courses to further the knowledge and education of these students. Cases such as these limit students and causes inequality in education because there is no easy way to gain access to those courses since the education system might not view them as necessary. The public education system does provide the classes needed to obtain a GED (General Education Development) and obtain a job or pursue higher education.

The largest public school system in the United States is in New York City, where more than one million students are taught in 1,200 separate public schools.

Admission to individual public schools is usually based on residency. To compensate for differences in school quality based on geography, school systems serving large cities and portions of large cities often have magnet schools that provide enrollment to a specified number of non-resident students in addition to serving all resident students. This special enrollment is usually decided by lottery with equal numbers of males and females chosen. Some magnet schools cater to gifted students or to students with special interests, such as the sciences or performing arts.

Phillips Academy Andover, an elite private secondary school in Massachusetts

Private schools in the United States include parochial schools (affiliated with religious denominations), non-profit independent schools, and for-profit private schools. Private schools charge varying rates depending on geographic location, the school's expenses, and the availability of funding from sources, other than tuition. For example, some churches partially subsidize private schools for their members. Some people have argued that when their child attends a private school, they should be able to take the funds that the public school no longer needs and apply that money towards private school tuition in the form of vouchers. This is the basis of the school choice movement.

5,072,451 students attended 33,740 private elementary and secondary schools in 2007. 74.5% of these were Caucasian non-Hispanic, 9.8% were African American, 9.6% were Hispanic, 5.4% were Asian or Pacific Islander, and .6% were American Indian. Average school size was 150.3 students. There were 456,266 teachers. The number of students per teacher was about 11. 65% of seniors in private schools in 2006–07 went on to attend a four-year college.

Private schools have various missions: some cater to college-bound students seeking a competitive edge in the college admissions process; others are for gifted students, students with learning disabilities or other special needs, or students with specific religious affiliations. Some cater to families seeking a small school, with a nurturing, supportive environment. Unlike public school systems, private schools have no legal obligation to accept any interested student. Admission to some private schools is often highly selective.

An August 17, 2000 article by the Chicago Sun-Times refers to the Roman Catholic Archdiocese of Chicago Office of Catholic Schools as the largest private school system in the United States.

Teachers

Most states require that their school districts within the state teach for 180 days a year. Teachers worked from 35 to 46 hours a week, in a survey taken in 1993. In 2011, American teachers worked 1,097 hours in the classroom, the most of any industrialized nation measured by the OECD. They spent 1,913 hours a year on their work, just below the national average of 1,932 hours for all workers. In 2011, the average annual salary of a PreK–12 teacher was $55,040.

Charter schools

Main article: Charter schools in the United States

The charter school movement began in 1990 and has spread rapidly in the United States, members, parents, teachers, and students to allow for the "expression of diverse teaching philosophies and cultural and social life styles."

Homeschooling

Main article: Homeschooling in the United States

In 2014, approximately 1.5 million children were homeschooled, up 84% from 1999 when the U.S. Department of Education first started keeping statistics. This was 2.9% of all children.

As of spring 2016, there are 2.3 million homeschooled students in the United States. It is appearing that homeschooling is a continuing trend in the U.S. with a 2 percent to 8 percent per annum over the past few years Many select moral or religious reasons for homeschooling their children. The second main category is unschooling, those who prefer a non-standard approach to education. This is a parent-led type of schooling that takes place at home and is now boarding a mainstream form of education in the United States. The Demography for homeschoolers has a variety of people; these are atheists, Christians, and Mormons; conservatives, libertarians, and liberals; low-, middle-, and high-income families; black, Hispanic, and white; parents with PhDs, GEDs, and no high-school diplomas. One study shows that 32 percent of homeschool students are Black, Asian, Hispanic, and others (i.e., not White/non-Hispanic). There is no required taxes on this form of education and most homeschooled families spend an average of $600 per student for their education.

Opposition to homeschooling comes from varied sources, including teachers' organizations and school districts. The National Education Association, the largest labor union in the United States, has been particularly vocal in the past. Opponents' stated concerns fall into several broad categories, including fears of poor academic quality, and lack of socialization with others. At this time, over half of states have oversight into monitoring or measuring the academic progress of home schooled students, with all but ten requiring some form of notification to the state.