Back in May, three of the Ps Spectroscopy team had the pleasure of attending PSAS 2022, which took place at the University of Warsaw, Poland. The conference focuses on precision measurements of simple atomic and molecular systems, including the development of new experimental methods and refinement of the theoretical calculations and models.
Group photo of the PSAS’22 Participants
We heard talks from groups around the world, on Hydrogen, QED theory, exotic atoms and more. David’s talk was on our recent measurements of the microwave spectroscopy of the Ps n=2 fine structure. These experiments had the highest precision to date, though they significantly disagreed with theory and produced asymmetric lineshapes (published here and here). Later, we found out that the experimental vacuum chamber was causing reflections of the microwave fields, which appear to be the cause of the observed asymmetric lineshapes and shifts (published here). Recent experiments with a smaller vacuum chamber initially seem to have reduced the reflections in the chamber leading to symmetric lineshapes, more info to come. All of the talks are available on YouTube and are well worth a watch.
David giving his talk
Tamara and Sam both presented posters on upcoming experiments, Ramsey interferometry of Ps and THz spectroscopy of He respectively. Tamara’s poster detailed our new DC Ps beamline (now in development!) in which an energy tunable 2S Ps beam will be created via collisions with Xe gas. With this we’re aiming to perform Ramsey interferometry using two waveguides instead of one, which we anticipate will improve our microwave spectroscopy measurements greatly. Sam’s poster showed some preliminary data on the THz spectroscopy of Rydberg He atoms. We’re planning to do more measurements like this in well-defined electric fields to perform the spectroscopy between stark manifolds, but more on all of these experiments is to come.
Sam presenting his poster
We’d like to thank the Candela foundation and the faculty of Physics at the university of Warsaw for organizing the conference and hosting us. We’re looking forward to the next PSAS in Wuhan!
Careers officers at universities often talk about transferrable skills when discussing options for job hunting undergraduate physicists. It’s true that many of the skills taught at degree level are invaluable across a broad range of professions. Nowadays, you’d be hard pressed to find to find a job that doesn’t value basic programming, data analysis or problem-solving skills, to mention a few. However, graduates that choose to stay in academia and embark on careers in experimental research may be surprised to learn of all the skills not taught in their degrees that will likely be essential in the coming years. Here, I would like to discuss a few of the skills that I wouldn’t have attributed to a physics PhD until they came in very handy during my first year.
First of all, patience. This may seem obvious, but experimental research is not to be rushed for a multitude of reasons, not least because it can be extremely dangerous to do so. Pragmatically, it simply isn’t efficient to try and do things as quickly as possible; speedy experimentation will likely lead to mistakes being made, false or useless data being taken and inevitably, many hours being spent on re-doing all your work. Of course, this is far easier said than done. Experiments in undergraduate lab modules are often laid out for the students with minimal or sometimes comprehensive instructions, and rarely take longer to perform than 6 to 12 hours. At PhD level however, experiments can take weeks, months, even years if you’re continuing the work of a long line of past students (or your lab is cursed). Many hours will be spent watching vacuum chambers pump down only to realise there’s a leak, or that the thing you’re supposed to be testing in there is still sitting on the bench where you left it. Long days can be spent aligning beams, measuring fields or taking data to no avail; either because you’ve made an honest mistake, or the universe just doesn’t want to play fair. Days like this can be frustrating, and the temptation to rush things will be strong, but this will only beget more waiting in the end. In my experience, the trick to patience is positivity and productivity. When you’ve found a bolt missing, a leak in your system or a blocked laser beam, try to substitute “Oh God I have to start all over again” for “That could have been so much worse if I hadn’t just spotted it”. See these things as small wins as opposed to massive losses and not only will you have the motivation to correct things and crack on with your experiment, but you will feel better too. Finally, when you have lots of time because you’re waiting for something, fill it! If your data set takes hours to record, catch up on some reading, try that simulation you were supposed to do last week or write a blog post… If your mind is occupied on producing work, you shan’t have the time to feel frustrated when you’re waiting for something. Patience is a virtue, yes, but it’s also a skill and arguably the most useful one in research.
Second, there will undoubtedly be a day where you walk into the lab alone for the first time and something has gone very wrong. Perhaps some equipment that should be firmly attached to the experiment is not so firmly attached to the ground instead. Perhaps a wire has shorted, and all your magnets have stopped working or the air conditioning has malfunctioned, and everything is far too hot. Whatever the case may be, you’ll be on your own, you’ll be unsure of yourself and without crisis management skills things are only going to get worse. Experiments at undergraduate level are very unlikely to fail in a manner more dire than a student having to retake some data, but in research labs equipment is bigger, more specialised and in many cases more dangerous. Being able to identify the problem, assess whether or not you are capable of fixing it yourself and act on these assessments is vital for the sake of the experiment and in rare cases, for your safety. These are not always intuitive skills, and often aren’t covered at undergraduate level meaning PhD students may find themselves underprepared for such situations. The reality of experimental labs is that equipment will go wrong at some point, sometimes for good reasons, and sometimes just to spite you. Power supplies will trip, vacuum pumps will give up the ghost and anything that’s water cooled is most definitely going to flood the lab. The trick is not to panic, to remember that your supervisor chose you for good reason, to notify the right people and to roll your sleeves up and get mopping.
Finally, by the time you get to PhD level, the experiments you’ll be working on will be cutting edge and may even be at the forefront of your respective field. With specialism like this comes the need for custom built equipment and as a budding independent researcher, you might be expected to design and build it yourself. Being able to drill and tap holes, cut and file metal and assemble complex systems from mismatched components are all skills you will likely need in the lab. Undergraduate physics courses certainly provide some good experience here; students are expected to take some initiative in designing their experiments and assembling them. Wiring basic circuits, aligning interferometers, and constructing pendulums are common in undergrad labs and some courses offer complete experimental freedom by the end of third year. These courses in particular are excellent training for research at higher levels but are usually limited by time, funding and course learning objectives to go into workshop skills with any detail. Unless you took design at school, or are well versed in DIY, you’ll be at a disadvantage in a research lab. Thankfully, these skills can be picked up elsewhere and are fairly intuitive; a summer internship for instance is an excellent opportunity to learn these skills in a laboratory setting.
What’s the take home message? Being an experimental physicist isn’t just about knowing your equations, being able to code or remembering if the cat in the box is alive or dead. Research requires skills from all walks of life to perform well and the most unexpected of things may be worth knowing. So, if you’re considering doing a PhD in an experimental physics, next time your mate buys a flat pack wardrobe, offer to help them assemble it. Next time all the lights go out in the house, find the fuse box and see if you can track down the dodgy component. And next time someone in your halls forgets to close the door on the washing machine and floods the kitchen, grab a mop and bucket and get stuck in. These are the makings of good housemates, good physicists and you never know when you might need one of these skills in a pinch.
Positronium is an atom which is half-matter and half-antimatter. Its energy structure is very well defined by the theory of quantum electrodynamics (QED) [1]. QED essentially describes how photons (light particles) and matter interact. If you imagine an electron in the vicinity of another electron, the classical picture says that the electric field of one electron exerts a repulsive force on the other electron, and vice versa. In the QED picture the two electrons interact by exchanging photons. Beyond electrons, the protons and neutrons in a nucleus are held together by the Strong force via the exchange of another type of particle called gluons.
The aim of precision spectroscopy is to carry out new measurements that will be compared to theoretical calculations. Measuring these energy structure provide a way of verifying the predictions made by theory. Atomic systems like hydrogen and helium are widely studied for QED testing, but the presence of heavy hadrons like protons introduces complications. One does not need to worry about such complications in positronium as there are no hadrons involved.
Within positronium there are many energy levels one could choose to measure. The separation of the triplet and singlet states of the n=1 level (i.e., the hyperfine structure interval), the 1S-2S interval, and the 2S-2P intervals are all excellent candidates [2]. The theoretical calculations for these three intervals have been very precisely calculated, and have also been previously measured. The last measurement of the fine structure intervals [3], however, is now over 25 years old and is much less precise than theory. Because Ps is very well defined by QED theory, any disagreements between calculations and measurements could be an indication of new physics. To be sensitive to new physics, the experiments have to done with precision comparable to calculations.
Figure 1: The Ps n=2 fine structure.
Recently we have measured the 2S-2P fine structure intervals of positronium [4]. There are three transitions within this branch and in this post, we’ll talk about the ν0 transition (23S1 – 23P0) which is resonant around 18 GHz. This transition, including the other two, is illustrated in figure 1. Initially, the Ps atoms (which are formed in the 1S state) have to be excited to the 2S state. This can be done in several ways (direct 1S-2S transition with one photon is not allowed), and we will cover our method in detail in another blog post soon. For now, let’s assume that the atoms are already in the 2S state. These atoms then fly into a waveguide where the microwaves drive them to the 2P state (via stimulated emission) as shown in figure 2. The atom then emits a 243 nm photon and drops down to the 1S state, where it will annihilate into gamma-rays after 142 ns (remember the lifetime of Ps in the ground state?). If nothing happens in the waveguide, the 23S1 state atom will annihilate after 1 μs.
Figure 2: (a) Target, laser, and waveguide schematic. (b) Placement of detectors, D1-D4, around the chamber.
We placed gamma-ray detectors (D1-D4) around the target chamber, as shown in figure 2, to monitor the annihilation signal. The detector signal was then used to quantify the microwave radiation induced signal, Sγ. We scanned over a frequency range to generate a lineshape that describes the transition; the centre describes the resonance frequency and the width is due to the lifetime of the excited state. A Lorentzian function was fitted to extract this information and for the example shown in figure 3, the centroid and line width are 18500.65 MHz and 60 MHz respectively. The centroid is slightly off from theory because the lineshape was measured in a magnetic field which introduces Zeeman shift to the centroid. The measured width is 60 MHz, slightly wider than the expected 50 MHz, and is due to the time taken to travel through the waveguide.
Figure 3: Measured 23S1 – 23P0 transition lineshape with theoretical resonance frequency of 18498.25 MHz.
Similar lineshapes were measured over a range of magnetic fields in order to account for the Zeeman shift. These data are shown in figure 4. Extrapolating to zero with a quadratic function allows us to obtain the field free resonance frequency, free of Zeeman shifts. However, all of the measured points, including the extrapolated number, are offset from theoretical calculation (dashed curve) by about 3 MHz. There are a few systematic effects to consider and the largest of them is the Doppler shift arising from the laser and waveguide misalignment, which amounts to 215 kHz. Our result, compared with theory and previous measurements, is shown in figure 5 and disagrees with theory by 2.77 MHz (4.5 standard deviations). While the precision has improved by over a factor of 6, the disagreement with theory is significant.
Figure 4: Zeeman measurement of resonance frequency vs magnetic field
Figure 5: Comparison with theory and previous experiments.
Precision measurements can be vulnerable to interference effects and there are two main types of effects that can cause lineshape distortion and/or shifts in line centre. Whenever the radiation emitted from the excited state (2P state in our case) is monitored to generate the signal, the emitted radiation can interfere with the incident/driving radiation (microwaves in our case)[5]. This leads to shift in the resonance frequency, but we are not sensitive to this kind of effect as we monitor the gamma-rays instead of the 243 nm emitted radiation (figure 1). Another type of interference arises from the presence of neighbouring resonance states [6], such as the two other 2P states in the Ps fine structure. The further apart the states are, the lesser the interference effect is and we expect a shift of 200 kHz in our line shape. This is, however, over 10 times smaller than the observed shift, and therefore, cannot be the reason for the disagreement.
There are two more transitions in the fine structure we have measured and they reveal interesting new features which were not previously seen. These additional data will provide a broader picture that will help us explain the shift we see in this transition. We’ll discuss those results in the next blog post.
[1] Karshenboim, S.G., Precision Study of Positronium: Testing Bound State QED Theory. Int. J. Mod. Phys. A, 19 (2004)
[2] Rich, A., Recent Experimental Advances in Positronium Research. Rev. Mod. Phys., 53 (1981)
[3] Hagena, D., Ley, R., Weil, D., Werth, G., Arnold, W. and Schneider, H., Precise Measurement of n=2 Positronium Fine-Structure Intervals. Phys. Rev. Lett., 71 (1993)
[4] Gurung, L., Babij, T. J., Hogan, S. D. and Cassidy, D. B., Precision Microwave Spectroscopy of the n=2 Positronium Fine Structure . Phys. Rev. Lett., 125 (2020)
[5] Beyer, A., Maisenbacher, L., Matveev, A., Pohl, R., Khabarova, K., Grinin, A., Lamour, T., Yost, D.C., Ha ̈nsch, T.W., Kolachevsky, N. and Udem, T., The Rydberg Constant and Proton Size From Atomic Hydrogen. Science, 358 (2017)
[6] Horbatsch, M. and Hessels, E.A., Shifts From a Distant Neighboring Resonance. Phys. Rev. A, 82 (2010)
The observation of a positronium (Ps) Bose-Einstein condensate (BEC) has been a long sought after achievement in Ps physics. When an ensemble of identical particles collects into the lowest energy state, i.e. approaching 0 K, a BEC is formed. A Ps BEC can be a source of a highly directional monoenergetic positronium beam, with applications in precision spectroscopy and gravitational interferometry. While BEC of normal atoms have been performed through evaporative cooling or laser cooling, the propensity of Ps to annihilate complicates this endeavour. One way to circumvent this problem was proposed by Platzman and Mills where Ps atoms are made and trapped inside a cavity [1]. However, a smaller cavity which has a higher cooling rate also has higher annihilation rates. One may use larger cavities at the expense of slower cooling, which can then perhaps be compensated by laser cooling.
Fig. 1: Excitation of Ps in vacuum (a) and inside MgO (b).
Following the discussion about producing Ps in MgO target, we decided to examine the effects of cavities on the positronium with laser spectroscopy [2]. Ps atoms produced by MgO were excited after being emitted in vacuum [Fig. 1(a)] or while they were still inside the powder [Fig. 1(b)]. The former will tell us the energy levels in vacuum, which are well known while the latter can tell us about the Ps-cavity interaction. The 1S->2P transition for both excitation cases are shown below in Fig. 2. Excitation in vacuum has a single curve centered around 243 nm as expected, with a Doppler width that implies kinetic energy of 350 meV. The reason for such a high energy and absence of cooling was discussed in a previous blogpost, which you may read here. But, when the Ps is probed inside the MgO, multiple peaks are visible. The redshifted peak (peak on the right, red dotted line) is due to some Ps which are already in vacuum and moving towards the lasers being excited. Another peak for atoms in vacuum moving away from the lasers (blue dotted) is also present. The third peak (blue dashed) arises from the excitation of Ps inside the MgO, which is shifted away from the vacuum resonance by 0.2 nm or 1000 GHz. The 1000 GHz shift is too large to be a confinement effect as the MgO cavities are too large. Similar measurement with silica was done previously by the Riverside Ps group [3] but not for Rydberg states as presented here.
Fig. 2: 1S-2P transition in Ps in vacuum (a) and inside MgO (b).
Similar to the data in Fig. 2, excitation to Rydberg states (2P -> 11S/11D) was also measured and is shown below in Fig. 3. Again, the vacuum excitation results in only one peak at the expected resonance [Fig. 3(a)] but excitation inside MgO [Fig. 3(b)] has a blueshifted and a redshifted peak. Two observations are apparent from the data: the Rydberg atoms are able to leave the target even after several collisions, and more surprisingly, Rydberg states (for n= 10-17) are all shifted by the same amount as shown in Fig. 4.
Fig. 3: 2P-11S/11D transition in Ps in vacuum (a) and inside MgO (b).
Fig. 4: (a) Shift of Rydberg transitions and (b) 2P- nS/nD transition for n= 10-17 for vacuum and MgO excitation.
Rydberg atoms which are more sensitive to interactions should have some n dependence, contrary to what was observed. This appears to be because MgO has photoluminescence (PL) absorption bands in the 240 nm range [4], which overlaps with the 1S-2P transition wavelength in Ps and are coupled resonantly. There are no PL bands that overlap with the Rydberg energies, thus, these states are unaffected. The resulting 1S and 2P energy levels in MgO are therefore shifted, with higher states unshifted as shown in Fig. 5. A detailed dipole-dipole treatment of the interaction of Ps and MgO crystals is outlined in the paper. These PL bands which are also present in silica may have caused the shift previously seen but without further Rydberg data we cannot tell if individual levels are shifted.
Fig. 5: Representation of Ps energy levels in (a) vacuum and (b) inside MgO powder inferred from the data.
Thus, the presence of such PL absorption in other materials for Ps confinement purposes can also give rise to energy shifts and can affect the control of atoms, which is necessary for high-density Ps experiments. It may be possible to carefully engineer material which can confine Ps but will not exhibit these characteristics.
[1] P. M. Platzman and A. P. Mills, Jr. PRB, 49, 454 (1994)
[2] L. Gurung, B. S. Cooper, S. D. Hogan, and D. B. Cassidy, PRA, 101, 012701 (2020)
[3] D. B. Cassidy, M. W. J. Bromley, L. C. Cota, T. H. Hisakado, H. W. K. Tom and A. P. Mills, Jr. PRL, 106, 023401 (2011)
[4] C. Chizallet, G. Costentin, H. Lauron-Pernot, J-M. Krafft, M. Che, F. Delbecq, and P. Sautet, J. Phys. Chem. C, 112, 19710 (2008)
This week we welcomed Dr. Weibin Li from the University of Nottingham who spoke about “Rydberg excitation in thermal Rydberg gases”. The abstract for this AMOPP seminar can be found below.
Rydberg excitation in thermal Rydberg gases
The creation of optical nonlinearities in atomic gases become a challenging task with increasing temperature, due to large Doppler effects. We study single photon excitation of electronically high-lying Rydberg states by nanosecond laser pulses that propagate in a high density thermal gas of alkali atoms. Fast Rabi flopping and strong Rydberg atom interactions, both in the order of GHz, overcome Doppler effects and dephasing due to thermal collisions between Rydberg electrons and surrounding atoms. The latter has not been taken into account appropriately so far. We show that a sizable dispersive nonlinearity is generated by strong interactions between Rydberg atoms. Despite the Rydberg optical nonlinearity, solitary propagation, i.e., self-induced transparency (SIT), of the light pulse can still occur. The existence of SIT allows to implement a conditional optical phase gate in the thermal gas through harvesting strong interactions. Our study paves the route to study nonlinear optics in thermal Rydberg gases and directly contributes to the current effort in realising scalable quantum information and communication devices with glass cell technologies.
This weeks AMOPP seminar was given by Dr. German Sinuco-Leon from the University of Sussex on the topic of “Spatial and internal control of atomic ensembles with radiofrequency and microwave driving”. The abstract for this talk can be found below.
Spatial and internal control of atomic ensembles with radiofrequency and microwave driving
The ability to apply well-controlled perturbation to quantum systems is essential to modern methodologies to study their properties (e.g. in high-precision-spectroscopy), and developing quantum technologies (e.g. atomic-clocks and quantum processors). In most of the experimental platforms available today, such perturbations arise from the interaction of a quantum system with electromagnetic radiation, which creates harmonically oscillating couplings between the states of the system. Within this context, in this talk, I will describe our recent studies of the use of low-frequency electromagnetic radiation to control the external and internal degrees of freedom of ultracold atomic ensembles [1,2]. I will outline the relation of this problem with Floquet Engineering and the more general issue of describing the dynamic of the driven quantum systems. Finally, I will explain the challenges of describing the quantum dynamics of systems driven and highlight eh need for developing new conceptual and mathematical tools to identify universal characteristics and limitation of their dynamics.
[1] G. A. Sinuco-Leon, B. M. Garraway, H. Mas, S. Pandey, G. Vasilakis, V. Bolpasi, W. von Klitzing, B. Foxon, S. Jammi, K. Poulios, T. Fernholz, Microwave spectroscopy of radio-frequency dressed alkali atoms, Physical Review A, accepted (2019). [ArXiv:1904.12073].
[2] G. Sinuco-León and B.M. Garraway, Addressed qubit manipulation in radio-frequency dressed lattices, New Journal of Physics. 18, 035009 (2016)
For this weeks AMOPP seminar, we welcomed Dr. Florian Mintert from Imperial College London who talked about “Optimal quantum control with poor statistics”. He argued the case for Bayesian optimization method to examine experimental data. The abstract can be read below.
Optimal quantum control with poor statistics
Learning how to control a quantum system based on experimental data can help us to exceed the limitations imposed by theoretical modeling. Due to the intrinsic probabilistic nature of quantum mechanics, it is fundamentally necessary to repeat measurements on individual quantum systems many times in order to estimate the expectation value of an observable with good accuracy. Control algorithms requiring accurate data can thus imply an experimental effort that negates the benefits of avoiding theoretic modelling. We present a control algorithm based on Bayesian optimisation that finds optimal control solutions in the presence of large measurement shot noise and even in the limit of single shot measurements. With the explicit example of the preparation of a GHZ state, we demonstrate in numerical simulations that this method is capable of finding excellent control solutions with minimal experimental effort.
This weeks AMOPP seminar was given by Dr. Brianna Heazlewood from the University of Oxford. In this interesting lecture, Dr. Heazlewood spoke about “Using the Zeeman effect to manipulate radicals and study ion-radical reactions”. The abstract for the talk can be found below.
Using the Zeeman effect to manipulate radicals and study ion-radical reactions
In spite of their real-world importance, very few experimental methods can be applied to the precise study of gas-phase ion-radical reaction systems. This is primarily due to the significant difficulty associated with generating a pure beam of atomic or molecular gas-phase radicals with tuneable properties. In this seminar, I will present our work in generating a pure beam of velocity-selected radicals. Only the target radicals are transmitted into the detection region; all other components of the incoming beam (radical species travelling faster/slower than the target velocity. precursor molecules and seed gases) are removed. This control over the properties of the radical beam is achieved through the use of a magnetic guide, composed of four Halbach arrays (permanent magnets in a hexapolar configuration) and two skimming blades. Experimental measurements of Zeeman-decelerated H atoms transmitted through the guide, combined with extensive simulations, show that the magnetic guide removes 99% of H atoms travelling outside the narrow target velocity range [1,2]. We will shortly combined the Zeeman decelerator and magnetic guide with an ion trap, for the study of ion-radical reactions. I will present some recent work on the reaction of ions with polar molecules – and discuss how we intend to adapt this approach for the study of ion-radical processes.
[1] J. Toscano, C. J. Rennick, T. P. Softley and B. R. Heazlewood, J. Chem. Phys. 149, 174201, (2018).
[2] J. Toscano, M. Hejduk, H. G. McGhee and B. R. Heazlewood, Rev. Sci. Instrum. 90, 033201, (2019).
This week we had the pleasure of welcoming Dr. Philipp Preiss from the University of Heidelberg who gave a talk on “Precursor of the Higgs Mode in Ultracold Few-Fermion Systems”. The abstract can be read below.
Precursor of the Higgs Mode in Ultracold Few-Fermion Systems
The emergence of collective modes from single-particle excitations is one of the most striking features of strongly interacting systems. Understanding such excitations is an ongoing challenge in nuclear physics, strongly correlated electron systems, and high-energy physics. Ultracold atoms in optical potentials provide a unique setting to precisely study the appearance of collective excitations in a tunable laboratory setting.
Here we experimentally observe the “birth” of a collective mode in a few-body system of ultracold Fermions. Using optical tweezers, we deterministically prepare few Fermions in the ground state of a two dimensional trap. This system exhibits a shell structure of stable “magic” numbers of 2,6,12… particles. We perform many-body spectroscopy through a modulation of the interaction strength find both single-particle and two-particle excitations. The latter consists of pairwise excitations akin to Cooper pairs and can be identified as the precursor of the Higgs mode in a two-dimensional Fermi gas.
In the future, we will probe such mesoscopic Fermi systems with single-particle detection. We recently demonstrated spin-resolved fluorescence imaging of individual atoms in free space, which will allow us to detect the momenta of every particle in the system in time-of-flight. We expect to directly see the formation of Cooper pairs and the momentum space signature of the BEC-BCS crossover.
The first talk in this year’s series of AMOPP seminars was given by Dr. Adam Deller from Prof. Hogan’s group in UCL. Adam, who was one of the first members in the UCL Ps spectroscopy group, talked about miniaturised Rydberg-Stark decelerators used to trap nitric oxide molecules. Abstract below.
Trapping long-lived hydrogenic Rydberg states of nitric oxide
High Rydberg states of atoms or molecules can have extremely large static electric dipole moments, upon which an inhomogeneous electric field will exert a sizable force. Electrostatic or time-varying electric fields have been utilised to exploit this effect to guide or decelerate and trap H, He, Ps and also H2.
I will describe a compact chip-based Rydberg-Stark decelerator comprised of a linear array of 115 electrodes. And I will present the results of recent experiments in which this device was employed to decelerate and trap laser excited NO molecules. An average lifetime of approximately 300 us was measured for molecules in the cryogenic trap. These cold, trapped NO molecules are of interest for studying low-temperature inelastic scattering processes for which long-range interaction play an important role.
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