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Electron-electron angular correlation experiments, or so-called (e,2e) coincidence experiments, provide the most detailed test of theories which attempt to describe the ionisation of a target atom by an incoming electron.
The parameters (Ea,qa,fa,Eb,qb,fb) are freely selectable experimentally within the constraints that energy and momentum must be conserved during the collision. In the experiments described here the DCS has been measured in the perpendicular plane using a helium target with incident electron energies ranging from 10eV to 80eV above the ionisation threshold.
The electron coincidence spectrometer was designed to measure angular and energy correlations between low energy electrons emerging from near-threshold electron impact ionisation of helium and is capable of accessing all geometries from the perpendicular plane to coplanar geometry. A number of modifications to the original spectrometer as described in Hawley-Jones et al (1992) have been implemented and are detailed elsewhere. Unique to this (e,2e) electron spectrometer is the computer interface which controls and optimises the spectrometer during normal operation.
Figure 2 is a block diagram of the hardware interface between the computer and the spectrometer.
At the heart of the operation is an IBM 80286 PC which controls the spectrometer and receives information about the system status from monitors around the spectrometer. The electrostatic lens and deflector voltages in the system are supplied by separate optically isolated active supplies controlled by 12A bit digital to analogue converter (DAC) cards, while a digital monitor measures the voltages and currents on the various lens and deflector elements around the system. The analyser positions are measured by potentiometers located in the drive shafts external to the spectrometer, and are monitored by a datalogger located on the main PC bus, which also monitors the EHT supplies and the vacuum pressure in the system.
Count rates from the channel electron multipliers and photomultiplier tube are monitored by a 32 bit counter board, and the TAC output is sent directly to an MCA card installed in the main PC. The software controlling the electron coincidence spectrometer is written to address specific tasks unique to the coincidence experiment.
The main PC controls the spectrometer lens and deflector tuning during operation, optimising these voltages using a Simplex method.
Following input of relevant control information, the system is switched to computer control.
The electron gun is then optimised to the counts detected by the photomultiplier tube, thus ensuring that the electron current is focussed to a beam of diameter approximately 1mm at the gas jet.
The analyser input lenses and deflectors are then adjusted by the computer to optimise the non-coincidence counting rate of each analyser. A coincidence correlation spectrum is then acquired for a predetermined time, after which the analysers are moved to a new position and the optimisation procedure repeated in the inner loop. Once the analysers have moved around the detection plane, the whole system is reoptimised and the process of data collection repeated. A correlation function between the scattered and ejected electrons is therefore accumulated over many sweeps of the detection plane until the results are statistically significant.
During operation the running conditions are monitored every 50 seconds by the data logger, allowing any anomalies in the data to be accounted for when final results are considered.
Results from experiment for the symmetric case, where the outgoing electrons following the collision have equal energy, are shown in the figures 4a - 4h.
In the perpendicular plane, the symmetry about the incident electron beam direction requires that the DCS must be symmetric about f = 180A° and this is evident in the data.
Inspection of the 94.6eV data (35eV scattered and ejected electron energies) indicates that this result does not follow the trends of the other data in the set. The resolution of the spectrometer (~ 1eV) does not allow these contributions to be excluded, and so the 94.6eV data set cannot strictly be considered as part of the overall set of data which is presented here. The first result in this data set therefore is the same as the first result in the previous set, allowing a convenient point for common normalisation of the data sets. Above 54.6eV incident energy the results differ markedly, the angular correlation evolving into a single broad peak instead of the three distinct peaks observed in the symmetric case.
This is most clearly evident at 74.6eV incident energy, where the symmetric case shows three distinct peaks of approximately equal intensity, whereas the non-symmetric case shows only a broad featureless structure centred about 180A°. The sets of data shown in figures 4 and 5 were each normalised relative to the result obtained at 34.6eV incident energy, where both the electrons emerge from the interaction region with an energy of 5eV.
The relative angular data presented at any particular energy is normalised by consideration of the electron singles counting rate, which depends on the Double Differential Cross Section at 90A° but is independent of the azimuthal angle f.
The procedure adopted here allows an upper and lower bound to be estimated, as no facility exists to allow these overlap volumes to be accurately determined experimentally as is implemented by other electron scattering groups.
The DDCS gives the probability of obtaining an electron from the scattering event at an angle q1 in a solid angle dW1 at an energy E1 A± dE1. The solid angle of detection depends upon the geometry and operating potentials of the electrostatic lenses, and is therefore also a function of the detected electron energy. VA(E1) is given by the three way overlap of the gas beam, electron beam and volume as seen by the analyser. The DDCS varies only slowly over the energy range dE accepted by the analyser (Jones et al 1991) as does the efficiency and solid angle of detection of the analyser as determined from electrostatic lens studies. The energy resolution DEA is independent of E1 because the pass energy of the hemispherical deflector is kept constant.
This expression can be rearranged to yield the ratios of analyser efficiency and solid angle in terms of the DDCS, singles count rates and overlap volumes at the two energies E1 and E2 for an incident energy Einc.. The ratios on the left-hand side of this equation were obtained from the DDCS data of MA?ller-Fiedler et al (1986) at an incident energy of 200eV for each analyser at the selected energies of interest. The first three terms in this expression are evaluated directly from the experiment, while the fourth and fifth terms are taken from the data of MA?ller-Fiedler et al (1986). As no experimental facilities existed to measure these parameters, a SIMION ray-tracing model was set-up to allow a computational determination of these volumetric terms.
The analyser input electrostatic lenses were also modelled to determine the volume viewed at the interaction point for the varying energies of electrons selected. Figure 6 indicates the experimental geometry as viewed by the analysers in the perpendicular plane, where the atomic beam direction is at 45A° to the incident electron beam trajectory.
Table 1 indicates the calculated relationship GV as determined from the SIMION ray tracing model, and these values have been used to calculate the normalisations as given in figure 4.
The dominant error in the expression is given by the double differential cross sections as measured by MA?ller-Fiedler et al (1986), since these terms arise four times in the expression, and typically have measurement errors of A± 15%. This is possible since the q = 90A° coplanar symmetric energy-sharing result is identical to the perpendicular plane f = 180A° energy-sharing result. Since data was not obtained at an incident energy of 100eV in these experiments, an interpolation between the energy normalised symmetric results yields a relative value.


The uncertainty in the normalisations of the relative energy is approximately A±30%, as discussed in the previous section. The absolute differential cross sections as presented in figures 4 and 5 therefore have uncertainties of approximately A±44%.
These latter calculations, normalised to one energy, produced results that were close to those of the experiment. A simple descriptive model for the symmetric case illuminates a possible scattering processes involved. On the other hand the peak that is observed at 180A° can be explained in terms of either single or multiple scattering. In terms of single scattering it is due to the incident electron selecting a bound electron whose momentum is equal in magnitude but opposite in direction to its own momentum. In terms of multiple scattering, particularly at the lower incident energies, the peak at 180A° has additional contributions that may be described by outgoing electron correlations as in the Wannier model.
The mass equivalence of the scattered and ejected electrons requires that a quasi-free collision results in electrons emerging at approximately 90A° to each other irrespective of their resulting energies, and so application of this model would once again yield three peaks, although the relative peak intensities might be expected to change. These results predict a single peak at 180A° due to a mechanism following elastic scattering of the incident electron from the nucleus that is more delicate and complex than the quasi-free collision, however the wings at the lower energies in the experiment are not predicted by this model. This may not be entirely unexpected as the DWBA model gives better results at higher energies. The central peak predicted by the theory also differs in width and height from the results presented here, when normalised to the 34.6eV data. In February 2011, the Breakthrough Institute published a comprehensive review of the literature and evidence for rebound effects which concluded that large amounts of the energy savings from below-cost energy efficiency are eroded by demand rebound effects.
Energy efficiency is widely viewed as an inexpensive way to reduce energy consumption and drive reductions in global emissions of greenhouse gases.
Yet recent research, including new reports authored by each of us, highlights a powerful but largely overlooked economic phenomenon that requires a global rethink of energy efficiency and its role in climate mitigation and sustainable development strategies: the rebound effect. Truly cost-effective energy efficiency measures lower the effective price of the services derived from fuel consumption – heating, cooling, transportation, industrial processes, etc. Furthermore, rebound effects are often most pronounced in the productive sectors of the economy, including industry and agriculture, as well as throughout the world’s emerging economies. These rebound effects run counter to a core assumption of conventional energy and climate forecasting and analysis: the idea that efficiency improvements lead to a linear, direct, and one-for-one reduction in overall energy use.
Estimates of the energy use and emissions reductions possible through efficiency are typically derived from ‘bottom-up’ engineering models and calculations of the cost effective efficiency opportunities available in each economic sector. Crucially, these studies assume no feedbacks between improvements in energy efficiency and either economic activity or demand for energy services.
When a good or service, or input, to production gets cheaper, consumers and firms use more of it, find new cost-effective uses for it, and on top of this re-invest any savings in other productive activities. Often labeled “Jevons’ Paradox,” after the British economist who first noted the mechanism in an 1865 treatise, rebound effects actually operate through well-understood economic principles of demand elasticity, substitution, and the contribution of productivity to economic growth. At the scope of the individual factory or assembly line, improving labour productivity may mean the plant can get by with fewer laborers on the shop floor.
At the macroeconomic level, it is widely understood that improving labour productivity drives economic growth, creates new profitable ways to utilize labour, and generally increases overall employment, rather than decreases it. Dozens of academic studies have examined the empirical evidence, conducted modeling inquiries, and otherwise tested the scale of rebound effects. The direct increase in demand for these end-use energy services due to the decrease in their apparent price is therefore relatively modest, and commonly erodes 10-30% of the initial energy savings or less.
Additional indirect and macroeconomic effects may mean total energy demand rebound can erode roughly one quarter to one third of the expected energy savings arising from end-use efficiency measures in developed economies.
However, the consumption of end-use services in the world’s wealthy nations is far from indicative of broader trends across the global economy. In contrast to conditions in wealthy nations, demand for energy services is far from saturated throughout the developing world.
In the world’s emerging economies, the cost and availability of energy services is often a key constraint on their enjoyment.
Very few studies have carefully examined rebound dynamics in developing economies, but those that have find direct rebound effects alone to be on the order of 40-80% for end-use consumer energy services, such as lighting and cooking fuel – more than twice as large as the equivalent rebounds found in wealthier nations. As a wide body of development literature recognizes, expanding access to modern energy services is also a principal driver of development outcomes.
Energy analysts must therefore be very careful in generalizing experiences or intuitions about rebound effects in rich, developed nations to the larger bulk of the global population living in developing economies.
However, the literature to date indicates that direct rebound effects are much larger in the productive sectors than in end-uses – on the order of 20-70% for these sectors, at least within a United States context – with additional rebound due to indirect and macroeconomic effects. Rebound effects in productive sectors depend principally on the ability of firms to rearrange their factors of production (labour, capital and equipment, and various materials) to better take advantage of now-cheaper energy services (a process known in economics as input or factor substitution). Additional mechanisms add to the scale of rebound, as consumers demand more of now-cheaper products and economic productivity overall improves. Conventional climate mitigation strategies count on energy efficiency to do a great deal of work.
A clear understanding of rebound effects therefore demands a fundamental re-assessment of energy efficiency’s role in global climate mitigation efforts. A continued failure to accurately and rigorously account for rebound effects risks an over-reliance on the ability of efficiency to deliver lasting reductions in energy use and greenhouse gas emissions. At the same time, however, we can re-affirm the role of energy efficiency efforts in expanding human welfare and fueling global economic development. The pursuit of any and all cost-effective efficiency opportunities should thus continue as a key component of an efficient course for global development, even as we reconsider the degree to which these measures can contribute to climate mitigation efforts. These experiments measure the angular correlation between a projectile electron which is scattered from a target and a resulting electron which is ionised from the target.
For an unpolarised target and unpolarised incident electron beam the only azimuthal angle of significance is given by f = fa - fb. When the electrons are detected in the perpendicular plane, the two angles qa and qb are both equal to 90A°. The perpendicular plane is accessed mainly through double collision processes (Hawley-Jones et al (1992), which represent only part of the full dynamical picture.
The computer loads all the voltage supplies from either a manually selected set of voltages or from the results of a previous optimisation run.


This decouples the tuning of the electron gun from any tuning requirements of the electron analysers. This should be compared with the data obtained near threshold by Hawley-Jones et al (1992), which also shows a single peak with an approximately Gaussian structure. This is considered to be due to the contribution to the cross section of a number of resonances in helium for scattered and ejected electron energies around 35eV. The model indicated that the electron gun operating at the voltages determined by the optimisation routines produced an electron beam of diameter approximately 1.0mm at the interaction region, consistent with the tuning of the electron beam onto counts from the photomultiplier which views a 1mm3 volume.
For the non-symmetric data no compensation is required for volume effects since GV = 1.0 at all values of Ei. The electrons are then able to emerge at approximately 90A° with respect to each other in the perpendicular plane, as is observed. As the momentum distribution of the bound electron peaks at zero for helium, the peak at 180A° diminishes with increased incident electron energy as the probability of finding a bound electron with the required momentum decreases.
In some cases, the rebound exceeds the savings, resulting in increased energy consumption from efficiency, known as backfire. In response, Marianne Moscoso-Osterkorn, Director General of the Renewable Energy and Energy Efficiency Partnership (REEEP), argues that energy efficiency yields considerable economic and energy security benefits, and that measures to improve it are always justified. Efficiency policies feature prominently in the toolkits of many national governments, international development agencies, and NGOs, and both the International Energy Agency (IEA) and the Intergovernmental Panel on Climate Change (IPCC) estimate that energy efficiency measures will do the heaviest lifting as the world seeks the emissions reductions needed to stabilize the global climate. Analysts then sum up the available efficiency measures in each sector to determine the gains possible for the economy as a whole, and subtract these efficiency gains from business-as-usual forecasts of energy use accordingly.
Thus, a given percent gain in efficiency is assumed to lead simply and directly to an equivalent and equal percent reduction in total energy use. Yet here again, higher labour productivity lowers product costs, increases demand for those products, and opens up new markets that were not previously profitable. And despite the simplified assumptions common to energy forecasting and analysis, the reality is that energy isn’t different from labour, or materials, or capital. Rebound effects differ in scale, depending on the type of energy efficiency improvement and the sector of the economy in question.
In fact, the largest rebound effects are typically found elsewhere: in the productive sectors of the economy that consume the bulk of energy in any nation, and in the world’s emerging economies, home to the vast majority of future energy demand growth.
After all, roughly one-third of the global population still lacks sufficient access to even basic modern energy services. Demand is thus far more elastic (responsive to changes in price), and rebound effects much larger than in the developed economies.
Whether such services are provided by burning more fuels, burning them more efficiently, or both (the most likely scenario), the outcome is the same: greater economic activity and expanding welfare, which in turn demands more energy. If, over the long-term, it is relatively easy for firms to substitute increasingly efficient energy services for other production factors, direct rebound effects can be substantial. For example, the IEA, in a global climate stabilization scenario published by the agency in December 2009, estimates that efficiency measures could account for roughly half of the emissions reductions needed.
Without a greater emphasis on the other key climate mitigation lever at our disposal – the de-carbonization of global energy supplies through the deployment and improvement of low-carbon energy sources – the global community will fall dangerously short of climate mitigation goals.
Unlocking the full potential of efficiency may very well mean the difference between a richer, more efficient world, and a poorer, less efficient world.
Harry Saunders is Managing Director of Decisions Processes Incorporated, a corporate management and decisions consultancy, and a Senior Fellow at the Breakthrough Institute.
Further it allows the electron gun to be aligned and focussed accurately onto the interaction region whose centre is located at the intersection of the axes of rotation of the analysers and the electron gun. This type of double collision process must therefore be modelled using higher order scattering terms as is done using the DWBA.
This focus on efficiency is particularly prominent in the world’s emerging economies, where getting more out of less energy is seen as a key path to both sustainable growth and reduced climate risk. There are other indirect and economy-wide effects as well, as consumers re-spend money saved through efficiency on other energy-consuming goods and services, industrial sectors adjust to changes in the relative prices of final and intermediate goods, and greater energy productivity causes the economy as a whole to grow.
This basic method is at the heart of widely cited climate and efficiency strategies authored by McKinsey and Company, the IEA, and the IPCC. In reality, however, the economy is anything but direct, linear, and simple, especially when responding to changes in the relative price of goods and services. It frees up money to re-invest in other areas of production, and it creates new jobs in other areas of business.
As it turns out, rebounds are generally smallest in exactly the situations that have received the most research to date: for improvements in the efficiency of end-use consumer energy services in wealthy, developed economies. This is especially true for decisions related to the construction of new productive capacity – and so we should again expect more pronounced rebound in the fast-growing productive sectors of emerging economies. Yet, from a climate or global resource conservation perspective, rebound effects mean that for every two steps forward taken through greater efficiency, rebounds take us one (or more) steps backwards. The former is clearly the desirable case – even if the world uses more or less the same amount of energy in either scenario. This is essential for experiments where all of these may be varied over 3-Dimensional space.
Collectively, these economic mechanisms drive a rebound in demand for energy services that can erode much – and in some cases all – of the expected reductions in total energy use, along with much-hoped-for reductions in greenhouse gas emissions. This includes efficiency improvements in personal transportation, home heating and cooling, and appliances. This is particularly true throughout the developing world, and in the productive sectors of the global economy. The efficiency of the stove was measured using water boiling tests, however when the household were surveyed for actual wood usage, it was much more than expected. Here, relatively wealthy consumers already fully enjoy most energy services, or come close to it. A consumer may gain little utility, for example, from heating his or her home above a comfortable room temperature, even if the efficiency of home heating improves.




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