A 44 year-old male was involved in a head-on collision with a car which was overtaking another vehicle at the time. After being thrown from the motorcycle the deceased impacted to the car’s nearside, then to the rear offside curtain of an oncoming lorry before coming to rest on the opposite verge of the road. He was declared dead at the scene. The full-face SHOEI brand helmet worn had a witness mark to the front center just above the visor edge. There were multiple areas of bruising and abrasion to the head but no lacerations. Blood was noted to issue from the left ear. Post-mortem computed tomography (PMCT) identified a fracture of the base of the skull centered at the pituitary fossa extending to both middle fossae. It extended to the temporal bone of the vault, and along the lambdoid suture. The fracture did not quite meet at the back of the head, preventing dissociation between two skull portions Fig. .Fig. 1PMA 40 year-old male was overtaking several vehicles when his motorcycle collided head-on with the front of a car. He was thrown from the motorcycle and landed on the carriageway where he was later pronounced dead. The full-face Arai brand helmet worn had a large vertical crack and scuff marks on the back and damage to the interior padding on the right. The visor was broken with only a small portion remaining attached to the helmet. On external examination there was blood emanating from both ears, bruising to the back of the right ear and a band of red bruising and abrasion to the left chin. PMCT identified a fracture of similar pattern to case 1, but rather than involve the lambdoid suture it extended across the posterior occipital bone to complete a full circumferential fracture and separate the two portions of the skull, with minimal displacement Fig. .Fig. 2PMA 51 year-old male was riding his motorcycle at high speed when he collided with the offside passenger door area of a car entering from a side-road. He was thrown in the air somersaulting several times before landing 4–6 m from the car. He was declared dead at the scene. The full-face Shark brand helmet worn had a large vertical crack on the left side extending from the upper edge of the visor upwards. On external examination a closed fracture of the posterior skull was evident as well as bleeding from the right ear and a black eye. PMCT identified a skull fracture of a similar pattern to case 2 with more displacement, and an associated incomplete classic ring fracture of the base of the occiput Fig. .Fig. 3PMA 37 year-old male was riding his motorcycle which collided with a vehicle travelling in the opposite direction which was performing an overtaking maneuver. He was pronounced dead at the scene. The full-face OGK brand helmet worn had marks around the vent on the face bar of the helmet indicating an impact. The top vent and visor had been broken off. On external examination there was purple bruising and a 12 × 3 cm red abrasion on the underside of the chin, as well as several small areas of bruising and abrasion around the eyes and nose. To the left side of the back of the head was a linear partial thickness laceration measuring 2 cm in length. PMCT identified a fracture of similar pattern to case 2, and below this fracture line the remaining part of the occipital bone had fractured into several pieces Fig. .Fig. 4PMA 42 year-old male was riding his motorcycle when it collided head-on with the passenger door of a vehicle which had skidded onto the opposite side of the road after losing control. It was thought that the motorcycle landed on top of him and the car rolled partially on top of the motorcycle. He was pronounced dead at the scene. The full-face SHOEI brand helmet had a large dented area to the left side of the face bar with a crack in the shell running through it, and the chin strap had marks indicating significant force had been applied through it. On external examination there was a large full thickness laceration extending across the occiput and left face. PMCT identified a fracture of similar pattern to case 2 with massive displacement of the skull parts Fig. .Fig. 5PMPost-mortem computed tomography (PMCT) is increasingly being adopted in the investigation of death, usually as an adjunct to autopsy, but increasingly in some countries to replace the internal examination . It has Motorcyclists remain some of the most at-risk road users worldwide. In the United Kingdom motorcyclists are 68 times more likely to suffer a fatality on the road per passenger mile than car occupants . AlthougThe five cases presented here demonstrate the same fracture pattern: a circumferential fracture comprising of a linear fracture of the middle cranial fossa extending through the pituitary fossa, continuing to the temporal and occipital bones of the skull vault. They are presented in order of increasing severity. Case 1 was less severe without dissociation of the skull. In cases 2–5 the fracture was sufficiently severe to completely dissociate the skull into two parts. Case 3 had an additional associated defect more akin to a classic incomplete ring fracture of the skull base. Cases 2, 4 and 5 show the fracture-type largely alone. Of note all of the deceased had witness marks on their helmet indicating an impact to the face, except for the individual in case 2, who clearly had an impact to the chin from the injuries noted on examination. All died at the scene due to their head injuries, although several had other injuries to the chest or abdomen which contributed equally to death.A ring fracture is classically caused by an impact either to the top of the head or a fall from a height landing feet first, and involves the petrous temporal bones bilaterally, the clivus and the posterior part of the foramen magnum , 11. HowWe suggest that the fractures presented in the five cases here are distinctly associated with motorcycle incidents. Similar fractures may have been described briefly within larger studies, and they may be familiar to experienced autopsists, however to our knowledge, this is the first time the fracture has been described in detail and as a series illustrated using PMCT.


In this paper we report a newly developed technique – laser calorimetry spectroscopy (LCS), which is a combination of laser absorption spectroscopy and calorimetry - for the detection of gases dissolved in liquids. The technique involves determination of concentration of a dissolved gas by irradiating the liquid with light of a wavelength where the gas absorbs, and measuring the temperature change caused by the absorbance. Conventionally, detection of dissolved gases with sufficient sensitivity and specificity was done by first extracting the gases from the liquid and then analyzing the gases using techniques such as gas chromatography. Using LCS, we have been able to detect ppm levels of dissolved gases without extracting them from the liquid. In this paper, we show the detection of dissolved acetylene in transformer oil in the mid infrared (MIR) wavelength (3021 nm) region. Analysis of dissolved gases in liquids is of prime importance in a wide array of fields like transformer health monitoring135Transformer dissolved gas analysis (DGA) is the measurement and study of composition and concentration of dissolved gases in insulating transformer oil to determine the type and severity of electrical  fault occurrence in a transformer. Case studies have shown that DGA and timely intervention can prevent substantial transformer damage. Usually, DGA involves sampling the oil, extracting the gas and analyzing the composition and concentration of the extracted gas. The gas extraction is usually done using extraction techniques such as vacuum extraction or head space extraction. The analysis is typically done using GC791011Here a method of detecting and analyzing dissolved gases in liquids without extracting them is presented. The method, though optical (and therefore is non-invasive and has a high specificity), does not require a line-of-sight and can detect dissolved gases in liquids even when the liquid is substantially non-transparent to the incident radiation.Laser calorimetry spectroscopy is a combination of laser-based optical absorption spectroscopy and calorimetry. Laser-based absorption spectroscopies function on the principle that gases preferentially absorb wavelengths that correspond to transitions between electronic, vibrational, or ro-vibrational levels13When light of a wavelength corresponding to a transition between quantum states of a gas is incident on the gas, the radiation is absorbed. The amount of radiation that is absorbed depends on the line strength of the transition, the concentration of the gas and the length of the path the light traverses in the gas (according to the well-known Beer-Lambert law).where,P0 = Incident laser powerPabsorbed = Power absorbedβg = absorption coefficient of dissolved gasc = concentration of dissolved gas in oilL = interaction path length.2, have their fundamental ro-vibrational modes in the MIR region and therefore the detection sensitivity is maximum in the MIR region. The availability of quantum cascade lasersMost hydrocarbons, CO and COC is connected thermally to the outside world  by a link of thermal resistance Rth, then the temperature change of the sample due to the gas absorbing the radiation is given byIn LCS, the rise in temperature due to absorption of radiation is measured. When a sample cell containing the transformer oil with dissolved gases is irradiated with light of a wavelength where a dissolved gas absorbs, the gas absorbs the laser energy and causes heating of the oil see . If the At equilibrium,or,βgcL ≪ 1,For Therefore, the temperature change of the oil is directly proportionate to the concentration of the dissolved gas.T + T0 (where T0 is room temperature or the base temperature). The foremost challenge in measurement of gas concentration using LCS is that the temperature change ΔT for ppm levels of gas is very small. For example, for acetylene, the highest βgcL ≈ 8 × 10−5 for 20 ppm concentration (c) and 3 mm path length (L). . If P0 = 1 mW and Rth ≈ 5000 K/W, ΔT ≈ 0.4 mK. Detection of 0.4 mK when room (base) temperature T0 is ≈300 K is experimentally challenging and can lead to large errors in the measurement of ΔT, thereby leading to erroneous gas concentration estimates. Accurate measurement of ΔT (even when the base temperature is 300 K) can be achieved by using a Wheatstone bridge-based differential measurement so that the common mode signal due to room temperature is cancelled out. The temperature that is measured in the experiment is ΔRvar is a variable resistor used to balance the bridge such that the output of the bridge is zero when the laser is off. When the laser beam is turned on, the output of the bridge ΔTmeasured is the difference between the temperature of the sample (Tsample) and the temperature of the reference cell (Tref).Here, Therefore,The above equation is valid only when the liquid is transparent to the incident radiation. If the liquid also absorbs,βl is the absorption coefficient of the liquid (transformer oil in our case). Therefore,where βl is ≪βgc,  are derived below:If the laser powers incident in the two cells are not exactly the same, or if the reference liquid is not the same as the sample liquid, the above equation is no longer valid. Then the experiment is carried out twice - once with the laser wavelength at the absorbance peak of the gas to be detected and the second experiment with the laser wavelength at the gas absorbance minimum. The equations when the incident laser powers are different (When the laser wavelength is at the gas absorption peak (experiment 1),When the laser wavelength is at the gas absorption minimum (experiment 2),The difference between the two measured temperatures can be used to quantify the gas concentration:A similar two experiment technique can also be used when the liquids in the sample and reference cell are slightly different. This can happen in the case of transformer dissolved gas analysis as the oil sampled from the transformer can have different absorption properties compared to fresh oil due to aging .Therefore, using the above techniques a gas dissolved in a liquid can be detected even when the liquid is not substantially transparent to the incident radiation, the only constraint being that the liquid does not absorb all of the incident radiation. However, while the basic principle is very straightforward, the experimental implementation is not, and will be discussed in detail in the following sections.LCS is a combination of optical, thermal and electrical detection and involves significant design constraints. The design considerations in the experimental implementation of LCS are explained below. The description below is for the specific case of acetylene in transformer oil. However, the method can easily be extended to other gases dissolved in other liquids.The gas absorbs very strongly whileThe transformer oil absorbance is a minimum.Transformer oil is chemically very complex and may comprise of as many as 2900 paraffinic, napthenic and aromatic hydrocarbonsA proper selection of the line can optimize the accuracy and performance of the sensor.−1) and 3150 nm (≈3175 cm−1) in the MIR region. The largest absorbance peaks occur in the region around 3021.6 nm (3309.5 cm−1). This is the best region to detect trace quantities of acetylene.Line selection for acetylene is straightforward. Several studies have been done on identification of acetylene lines using absorption spectroscopy. Also, the HITRAN database has quantitative spectroscopic parameters for many of the small molecular constituents of atmosphere including acetylene171819The absorption spectrum of transformer oil cannot be directly obtained from any database. Several studies have been done to measure the transmission of mineral oil (transformer oil)2122P0 ≈ 1 mW, ΔT due to oil is ≈2 K and ΔT due to gas (20 ppm acetylene) is ≈0.4 mK. It can clearly be seen from this example that the temperature change due to oil absorbance is about 4 orders of magnitude larger than that due to gas absorbance. Therefore, careful selection of wavelengths alone is not sufficient to detect trace amounts of fault gases in transformer oil. This is the reason so the dual beam differential arrangement discussed earlier was devised.As can be seen from The calorimeter has to be designed such that very small concentrations of dissolved gas can be detected within as small a time as possible.Rth . However, the thermal time constant of the calorimeter τ is the product of the thermal resistance and the heat capacity of the cell + sample system (τ = RthC). For fast detection, the thermal time constant has to be small. Therefore, to keep sensitivity high and time constant low, the heat capacity of the cell + sample has to be minimized and Rth has to be as high as possible.The equilibrium temperature rise is directly proportionate to Rth, all modes of heat leak - conduction, convection and radiation - to the base have to be minimized.To maximize Rth due to conduction by leads can be given by:Conduction occurs through physical contact. The sample and reference cells are therefore isolated from the thermal base using air gaps and low thermal conductivity materials . Apart fl: length of the wire; A: area of cross-section of the wires; n: number of wires and κ: thermal conductivity of the material of the wire. To minimize Rth−cond−leads, long, thin phosphor-bronze wires have been used as leads instead of the standard short copper wires. The thermal conductivity of phosphor-bronze is about 8 times smaller than that of copper making the thermal resistance as many times larger. The thermal resistance is further increased by using long and thin wires instead of short thick wires (as Rth is directly proportionate to the length of the wire and inversely to its area of cross-section).Here The best way to minimize convection is by keeping the system in vacuum. However, as this would complicate the system design on-field, vacuum is not used.The other heat leak path is where heat is conducted out to the cell walls and is then lost through convection. As the wall thickness of the cell increases, the heat leak due to conduction decreases while that due to convection increases. Finite element analysis showed that the heat leak from conduction is very small compared to that due to convection. Therefore the smaller the wall thickness, the lower the total heat leak. We have used a wall thickness of about 1 mm.Radiation heat loss is minimized by making the inside of the copper enclosure shiny. Our simulations show that heat leak due to radiation is significant compared to that due to convection when the temperature rise of the liquid is less than 100 mK. Since the temperature rise for transformer oil is always typically larger than 100 mK, the most significant heat leak occurs through convection from the cell walls.μK requires measurement of voltages of tens of nanovolts. For accurate measurement, a phase sensitive detection using a lock-in amplifier is used here. The details will be discussed in the methods section.Measurement of temperature changes of the order of tens of μm). A pair of barium fluoride plano-convex lenses  are used to create a beam size of ≈3 mm. The collimated light beam is aligned such that one part of the beam falls on the sample cell and the other part of the beam falls on the reference cell using a MIR broadband CaF2 beam splitter (50:50) and gold coated mirror . The laser wavelength can be tuned by changing the laser temperature or the laser current. For LCS, the laser is typically tuned by changing the current with the temperature kept constant. This is achieved using a laser current controller and temperature controller . The desired wavelength range for the experiment with acetylene in transformer oil is 3021 nm-3022 nm which corresponds to a laser current range of 110 mA–170 mA at a laser temperature of 38 °C. It is typically necessary to pre-calibrate the laser wavelength as a function of the laser current and temperature  to ensure that atleast one gas transition line (absorption peak) and one off transition line is covered in the wavelength range used.A room temperature DFB Intraband Cascade Laser (ICL) emitting at 3021 nm (Nanoplus GmbH) is used as the excitation source. The laser is collimated with an aspheric lens . The cell has inlet and outlet ports for sample injection and removal.The cell design is shown in 2 windows . The top plate is designed such that the CaF2 windows align with the sample and reference cells so that collimated laser beams from the beam splitter and mirror arrangement can pass through into the cells. The electrical connections for the TEC and Pt100s are made through BNCs mounted on the circumference of the chamber. The chamber also has a port for the pipes used for injecting and removing the sample to pass through.The measurement enclosure  is a cylPlatinum thin film temperature sensors  are used for temperature measurement. Pt100 sensors are integrated into the sample and reference cells such that they are in good thermal contact with the liquid. The Pt100 sensors form the two upper arms of a Wheatstone bridge arrangement The bridge arrangement is used to measure the difference in the resistance change between the two Pt100 sensors (in the sample and reference arm) when radiation is incident on the cells. Since the resistance of a Pt100 sensor is a linear function of temperature (with a temperature coefficient of resistance 0.00385/K), the difference in resistance is used to calculate the difference in the temperature change due to gas absorption.pp) of a lock-in-amplifier  and balanced using the variable resistor when no radiation is incident on the cells. The output of the bridge is given to a low noise transformer coupled preamplifier . The bridge is biased using the ac output  and the reference cell is filled with the reference liquid. The laser current is set to a value such that the gas does not absorb at the wavelength emitted. The voltage output is measured and recorded continuously as a function of time. This process is repeated for different laser currents (or wavelengths) including wavelengths where the gas absorbs strongly.The recorded voltage is converted to temperature. μK. This corresponds to ≈1 ppm of C2H2 (for an Rth of 5000 K/W and a laser power of 1 mW).The first experiment done was to record, measure and analyze the baseline noise floor which sets the minimum detection limit for LCS. This is done by measuring the baseline fluctuations when there is no liquid in the cells. Lower detection limits can be achieved in two ways: (i) by increasing the thermal resistance which would increase the temperature change (ii) by averaging - the measured thermal fluctuations are white and therefore averaging can decrease this noise, thereby lowering the detection limit.Tmeasured even when the same oil is filled in both the cells. The laser power output changes with change in current. This leads to a non-flat ΔTmeasured as a function of wavelength. The ΔTmeasured as a function of wavelength for transformer oil in the acetylene detection wavelength range is shown in Slight differences in the sample and reference cells and in the incident power on the cells lead to a non-zero Δ−4. With the current system, detection of 10 ppm  is possible. This limit has not been experimentally validated due to constraints in the gas dissolving process. In-gas detection techniques such as tunable diode laser absorption spectroscopy (TDLS) have a minimum detection limit corresponding to an absorbance of ≈5 × 10−5 24The experimentally achieved minimum detection limit for acetylene in transformer oil is ≈28 ppm which corresponds to a fractional absorbance of ≈10−6 for incident laser power of 1 mW. In the current setup, the laser power incident on each cell is only 0.5 mW and therefore the detectable fractional absorbance limit is 8 × 10−6. However, the fractional absorbance corresponding to the achieved detection (28 ppm) is ≈10−4 with the possibility of going down to a fractional absorbance of 4 × 10−5. This is about five times larger than the limit set by baseline temperature fluctuation. The reason for this discrepancy is the presence of temperature oscillations due to changes in room temperature. We found that the air-conditioning caused oscillations in temperature. The amplitude of the temperature oscillations were ≈0.1 mK making the minimum detectable fractional absorbance 4 × 10−5. Better temperature control can lower the limit of detection. MIR ICLs with high powers of 10–20 mW are now commercially available. This can further better the limit of detection enabling sub-ppm detection of acetylene in transformer oil.The detection limit set by the baseline temperature fluctuation corresponds to an absorbance of ≈4 × 10A comparison of LCS with commonly used techniques for DGA is shown in In conclusion, the newly developed technique of LCS has been described and its application in the detection of dissolved acetylene in transformer oil has been demonstrated in the manuscript. This technique can be expanded to other dissolved gases in oil and to other fields such as food and beverage industry , oil and gas industry , water industry , etc. where concentration of dissolved gases in liquids need to be measured.in-situ measurement and does not require any gas extraction, thereby eliminating the errors associated with calculating the gas concentration using partition coefficients91011−6 for an input power of 1 mW and also has an extremely high dynamic range of 105. Third, this technique also enables detection of dissolved gas concentration in highly absorbing liquids due to the dual beam differential measurement.Though there is a long way to go, it is clear that this technique has several advantages over existing methods. First, this is a direct in-liquid, The key limitations of this technique are the long measurement times (few hours per measurement) and the highly stable environmental temperature control required if very low detection limits are to be reached. This makes the technique cumbersome for use in industrial environments. However, fast measurement methods are being explored currently to make the technique more robust. It must also be mentioned here that since transformer oil is chemically very complex, unknown degradation products might lead to some interference if they have absorbances in the range of the gases being studied. It would be critical to investigate different transformer oils and thier degradation products to select an appropriate wavelength for analysis. At present LCS is a very sensitive, in-liquid lab measurement technique for dissolved gas detection.How to cite this article: K. S., N. et al. Laser Calorimetry Spectroscopy for ppm-level Dissolved Gas Detection and Analysis. Sci. Rep.7, 42917; doi: 10.1038/srep42917 (2017).Publisher's note: Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.


We propose an optical method based on Rayleigh scattering for the direct measurement of cluster tracks produced by a high-pressure gas jet. The tracks of the argon and methane clusters are acquired by a high-speed camera. It is found that the cluster sizes of these tracks are within the range of 7E + 03~1E + 07 for argon and 2E + 06~4E + 08 for methane. Most argon tracks are continuous and their intensity changes gradually, while the majority of the methane tracks are separated into discrete fractions and their intensity alters periodically along the flight path, which may indicate the methane clusters are more unstable and easily to break up. Special methane clusters which may fly at an axial velocity of less than 2.5m/s are also found. This method is very sensitive to large gas cluster and has broad application prospects in cluster physics. The supersonic gas jet is produced by opening one nozzle to inject the high pressured gas into the vacuum, the temperature of the gas decreases greatly during the free expansion process and the clusters are formed by Van der Waals forces. The clusters are produced by gas jet technique and its size and density distribution are required to analyze the laser-cluster interaction dynamics or calculate the yield rates. The cluster size is distributed over a broad range from two (dimer) up to many thousands, which can be well approximated by the log-normal function5etc. Although large clusters are preferred for ensuring higher efficiency and yield rate, there are few studies of large clusters6Clusters are made of atoms or molecules by physical or chemical bonding forces. The interaction of an intense short-pulsed laser with Nano-sized clusters has been a popular topic of studyIn this study, an optical method based on Rayleigh scattering is presented to measure the cluster tracks in the gas jet and large argon and methane clusters were discovered. This imaging method can be used to capture the images of cluster tracks and study the cluster interactions with materials, thereby demonstrating broad prospects for application in cluster physics.c (the average number of atoms per cluster) has been characterized by the semi-empirical parameter introduced by Hagena121314Traditionally for the axisymmetric gas expansion, the average cluster size N0, T0). k is a gas-dependent constant (k~1650 for Ar and 2360 for CH4), the equivalent nozzle diameter deq is assumed to be 0.74d/tanα for monatomic clusters and 0.87d/tanα for diatomic clusters where d is the throat diameter of the nozzle in μm and α is the half opening angle of the nozzle0 is the gas stagnation temperature in Kelvin, and P0 is the stagnation pressure in mbar. The parameter q has commonly been fixed at 0.85 in the scaling law of Hagena1318Г* is a semi-empirical parameter that takes into account all the factors12et al.c and Г* for methane is described in formula (2).However, it must be declared that the Hagena scaling law is not suitable for methane cluster. According to the results from Akiyoshi Murakami 0. Song Li et al.c = 1235 when Г* = 520 −3.3) and Nc = 6230 when Г* = 1104, and it can be calculated that the exponent value is about 2.15 for these data. It is evident that the above relations between Nc and Г* for methane are not accordant and it is different from the relation of Nc ∝ (Γ*)5.78 which is measured by our experiment and is described in the following “Results and discussion” section. So it can be concluded that the Hagena scaling law is unsuitable for methane cluster, and the exact formula is expecting for further research. In addition, the results of methane cluster size given here are subject to the date from our experiment.The exponent value used to calculate Nc in formula (2) is quite different from that of Hagena, and it is found that the value increases from 3.8 to 7.6 for higher stagnation pressures which is contradictory to Hagena’s formula whose exponent value decreases from 2.35 to 1.8 for higher stagnation pressure Pc) and the square of the average cluster size (Nc) if the clusters are supposed to have a spherical structurec) multiply the average cluster size (Nc) is the total number density of clusters, which equals to the product of the monomer density before clustering (n0) and the proportion of the molecules that are formed to clusters (η). The monomer density before clustering (n0) is proportional to the stagnation pressure (P0) before the gas jet, thereby we can obtainThe Rayleigh scattering method has been widely applied for cluster-size evaluations0 is the monomer density before clustering, and η is the proportion of the monomers that can form clusters. If we suppose η remains constant throughout the duration of one gas jet, then the scattering signal S is proportional to the average cluster size Nc.where no Rayleigh scattering lights on the top side of the vacuum chamber. The reflected lights from the wall of vacuum chamber have been disposed to minimize the background noise. The resolution of the camera is 1280*1024, the imaging speed is fixed at 1000 fps (frame per second), and the shutter speed is set to 1/1000 second to acquire entire evolution process.The experimental set-up is depicted in o, and the total length of the nozzle is about 3 mm. In addition, the nozzle is opened by a high-speed pulsed solenoid valve from Parker Hannifin Corp. In this study, the open time of the nozzle is fixed to 20 milliseconds (ms) and 20 Rayleigh scattering images are acquired by the high-speed camera for each gas jet.The argon and methane gas with 99.999% purity are employed to produce pure clusters and minimize the influence of impurities. The stagnation pressure of the gas can be adjusted from 10 to 48 bar. A 1500 L/s turbomolecular pump is used to supply a vacuum pressure of approximately 2E-4 Pa. Moreover, the stagnation temperature is maintained at approximately 293 K in the entire test duration. The conical nozzle can be moved along the axial direction in the range of 10–180 mm. The orifice diameter and half cone angle of the nozzle is 0.5 mm and 45For each gas jet 20 scattering lights images are acquired by the high-speed camera, normally 1~6 of the 20 images contain cluster tracks, which are used as “effective images” to analyze the properties of large cluster. And the rest images without cluster tracks are used as “reference images” to represent the average clusters. The typical “effective image” that contains cluster tracks and corresponding reference image for argon are shown in As shown in p is the specific heat ratio at constant pressure (J. K−1. Kg−1), T0 is stagnation temperature (Klevin)) before the gas jet. The opening angle of this track to the center streamline is about 54o, which is much bigger than the half opening angle of the nozzle which is only 45o.Three consecutive tracks of the same methane clusters are acquired at axial position about 15 mm and radial position about 21 mm during one gas jet. The three images are combined together, and the combined tracks are shown in The argon track images at different axial position are combined together into one image, which is shown in The full width at half maximum (FWHM) of the tracks are calculated, which is acquired by 1D summation of the pixels along the flight path of the track and get the intensity variation curve perpendicular to the flight path. The curve is fitted with the Gaussian function, and FWHM is calculated from the fitted Gaussian function. The FWHM was used to represent the width of tracks and the peak value was chosen to show intensity of tracks. However no distinct relationship between the width and intensity was found, hence the width of the track cannot be used to represent the cluster size. The FWHM distribution for argon and methane is shown in The cluster sizes of these large clusters can be estimated by comparing the intensity of the tracks in the “effective image” and that in the “reference image” at the same position, whereas the intensity of the “reference image” can be calibrated by the average cluster size of the gas jet.The average cluster size is required to calculate the size of the track. One simple method to estimate the average cluster size is introduced by refs 2627The typical Rayleigh scattering signal at the axial position of 20 mm and stagnation pressure of 35 bar is shown in ηPNc, if we suppose η remains constant for different stagnation pressures, then the average cluster size is given by:From formula (3) we obtain S ∝ 1 at stagnation pressure P1 is known, then the average cluster size at other stagnation pressure can be calculated easily from formula (4).When the average cluster size Ncc of argon is 100 when the scattering signal is about two times of the background noise, which happens when the stagnation pressure is about 10 bar. Since the molecular diameter of methane is larger than that of argon, the minimum detectable cluster size of methane should be smaller than argon for the same Rayleigh scattering system. The same minimum detectable cluster size of 100 is used for methane in this work for conservative purpose. And Nc of methane is supposed to be 100 when the stagnation pressure is about 13 bar. When the scattering signal versus stagnation pressure is tested, the average cluster size for different stagnation pressures can be calculated from formula (4).In this experiment we suppose Nc is much bigger than that of argonThe Rayleigh scattering signal and the average cluster size at different stagnation pressure for argon are shown in The average cluster size of argon and methane at all axial positions can be calculated from the formula (3), at the stagnation pressure of 25~45 bar. The results of argon are in agreement with that in the scaling law of Hagena (1), which indicates the Rayleigh scattering system works properly and the results are reliable.SR) is defined as the ratio of large cluster size of the track in effective image over the average cluster size in reference image. If the average cluster sizes (Nc) are acquired then the large cluster sizes (Nbig) can be calculated from formula (5).The size ratio , therefore the large cluster size can be calculated out.According to the formula (5) the large cluster sizes of argon and methane lie in the range of 7E + 03 ~ 1E + 07 and 2E + 06 ~ 4E + 08, respectively. By combining the error of the average cluster size and the size ratio, the relative error of large cluster size is calculated to be about 20% for argon and 40% for methane. The large cluster size at different axial position and stagnation pressure for argon are shown in Different from the size distribution, the number of tracks is strongly dependent on the axial position or stagnation pressure. The number of tracks versus axial position and stagnation pressure curve for argon and methane are shown in Considering the large cluster size and number of tracks, we can conclude that at higher stagnation pressure, much more large argon clusters are formed but the size only increase slightly, while much larger methane clusters are found but the number doesn’t change greatly. This means the argon cluster is inclined to increase the number, while the methane cluster tends to increase the size. The total number of argon tracks is about six times more than methane, while the cluster size of argon is about three times less than methane. Since the scattering intensity of argon is larger than methane at the same stagnation pressure, we can infer that the “number effect” is stronger than the “size effect”.Most of the methane cluster tracks are separated into discrete fractions and the intensity of them alters along the flight path. This phenomenon indicates that these methane clusters may be very unstable to fragment into small pieces. It is hard to explain why the intensity alters periodically, and more studies are needed to verify that.In conclusion, we have demonstrated an optical method based on Rayleigh scattering for measuring the cluster tracks in gas jet. The tracks are composed of large clusters, and the sizes of large clusters for argon and methane lie in the range of 7E + 03 ~ 1E + 07 and 2E + 06 ~ 4E + 08, respectively. The sizes of large argon clusters exhibit a weak dependence both on the axial positions and stagnation pressures, whereas the sizes of large methane clusters are irrelevant to the axial position but have a strong growing trend with the stagnation pressure. The methane clusters are very unstable and easily to fragment into small pieces in comparison with argon clusters. Special methane clusters at the axial position of 15 mm and radial position of 21 mm which may fly at the axial velocity of less than 2.5 m/s are also found. The method can be setup simply and operated easily, which is very suitable for laser-cluster interaction experiments and has broad application prospects in the cluster physics and nuclear fusion fueling field.How to cite this article: Chen, Z. et al. The optical measurement of large cluster tracks in a gas jet. Sci. Rep.6, 32391; doi: 10.1038/srep32391 (2016).


An attempt is made to understand the topological quantum phase transition, emergence of relativistic modes and local topological order of light in a strongly interacting light-matter system. We study this system, in a one dimensional array of nonlinear cavities. Topological quantum phase transition occurs with massless excitation only for the finite detuning process. We present a few results based on the exact analytical calculations along with the physical explanations. We observe the emergence of massive Majorana fermion mode at the topological state, massless Majorana-Weyl fermion mode during the topological quantum phase transition and Dirac fermion mode for the non-topological state. Finally, we study the quantized Berry phase  and its connection to the topological number (winding number). The generalized concept of topological numbers such as Chern number and winding number depends on the dimensionality of the system for a bulk characterization based on the concept of topological order2.To get a better understanding of quantum many-body systems, which show topological properties, one has to focus on the studies of non-local order parameter instead of conventional concept of symmetry breaking, i.e., the local order parameter description of the system. States with non-local order parameter could be called topologically ordered. The term “topological” implies the existence of a bulk invariant, which generally represented by an integer that differentiates between the phases of matter having the same symmetry5. In the last decade, physics of topological state of matter through the analysis of Majorana mode, come to the focus of quantum many body condensed matter physics research29. The physics of p-wave pairing in a condensed matter many body system has been considered as the prototype system for investigating the physics of Majorana fermion zero mode which is localized at the both ends of the system1.The concept and existence of Majorana fermion mode is one of the most advanced research area in the quantum condensed matter physics for studying the topological properties of matter. Majorana introduced a special kind of fermions which are their own antiparticlesIn the strongly correlated regime of interacting light matter physics, it is possible to generate an effective strong repulsion between the photons, and from this study, one can understand and quantum simulate different interesting physical properties of strongly correlated quantum condensed matter many body system.32.In quantum simulation, one’s aim is to simulate a quantum system using a controllable laboratory system that underlines the same mathematical models. Therefore, it is possible to simulate a quantum system that can neither be efficiently simulated on a classical system nor easily be accessed experimentally39.Interacting light-matter system is also a very good platform to simulate different quantum many-body physics phenomena. Several interesting phenomena like fermionic behavior of photons, driven dissipative Mott insulator, Tonk-Girardeau gas, fractional quantum Hall like state and the physics of cavity QED lattice and quantum simulation of many complicated quantum many-body system have found place in the literature of interacting light-matter physics39.The recent experimental success in engineering, a strong interaction between the photons and the atoms in high quality micro-cavities opens up the possibility to use the light matter system as quantum simulators for many body physics. Many interesting results are coming out to understand the complicated quantum many-body system40. We will also use the same model Hamiltonian for the present study but in a different context.The authors of ref. The present study has three motivations.Here we will do the detail study of the topological state and also topological quantum phase transition of the interacting light-matter physics in an array of nonlinear cavities through the topological invariant number. This motivation arises for the following reasons.43. Topological number changes by an integer number during the topological quantum phase transition from topological state to the non-topological state, which is related to the appearance of Majorana zero modes localized at the edge of the system5. This detailed study based on exact calculations too is absent in the previous literature of light-matter physics system41.Topological quantum phase transition describes with the topologically invariant number while the quantum phase transition describes by order parameterWe would like to find out the emergence of different relativistic modes , for this interacting light-matter physics in a nonlinear cavities array, based on the exact solutions. This motivation arises due to the following reasons:The fundamental constituent of condensed matter quantum many-body system is invariably the electron which carries charge and it has antiparticle. Therefore, to find the Majorana particle in this system, it will appear as a emergent particle. We will prove the emergence of Majorana and Majorana-Weyl collective modes in this strongly correlated light-matter system in a nonlinear cavity array.41 of interacting light-matter system to find the emergence of different relativistic modes.To the best of our knowledge, this is the first study in the entire literatureHere, we find an equivalence between the topological invariant number and the local topological order for this system. This motivation arises due to the following reasons:The author of ref. 41 and its relation to the topological number.But in the literature of interacting light-matter system, there are no studies so far of quantized geometric phase as a local topological order for characterizing the topological state of the system40:ωc, U is the strength of the on-site photon-photon repulsion due to the large optical nonlinearities and J denotes the photon tunneling amplitude of photon between NN site. The photon tunneling occurs owing to the non-vanishing overlap of Wannier modes between the NN sites. In the strong interaction regime, the energy cost for adding any extra photon is higher than all relevant energy scale in the system. Therefore, in each site of the lattice, the number of photon is either 0 or 1, i.e., the photon shows the spinless fermionic behavior.The model Hamiltonian allows for tunable coupling and nonlinearity. This system consist of N optical cavities coupled through nearest-neighbor (NN) photon tunneling across the one dimensional chain. In this system, each cavity exhibits a large optical nonlinearity and a single mode which behave as a Wannier function localized at the center of each site. Photon tunneling occurs due to the finite spatial overlap between NN Wannier modes. Finally, the system Hamiltonian takes the following formThe authors of ref. The detail derivation of the emergence of effective p-wave pairing has presented in ref. 40 isϕ are the amplitude and phase of the parametric pump and ωp is the frequency. Physically, the above Hamiltonian describes the coherent exchange of p-wave paired photons between the system and pump field. The effective p-wave pairing arises from the interplay between the parametric pumping and strong on-site photon-photon repulsion. In the strong interaction regime, where the parametric pumps are much weaker than the on-site photon-photon repulsion, the second process is strongly favored and the p-wave is effectively obtained. The authors have determined the amplitude of |Δ| by the overlap of the Wannier modes in a similar way as the tunneling amplitude, J40. Therefore, one can expect that one reaches a regime where the magnitude of |Δ| and J are of the same order. We will explore this limit explicitly in the study of topological quantum phase transition and also to find the exact solution for winding number.In the strongly correlated regime, the photons are likely to be emitted from the NN cavity. The effective drive HamiltonianH(=H0 + Hdrive) for the strongly correlated regime of the system Here we derive the optical version of Kitaev’s chain starting from the Hamiltonian P is the projection operator for the single photon occupancy. For this case the drive Hamiltonian reduce toIn the hard core photon limit, one can write the photonic operators as ωp41. The total Hamiltonian, H = H0 + Hdrive, reduce to following form:One can consider this hard-core photon as spin-1/2 particles and express the photon operators as a Pauli matrix, ωc playing the role of Fermi energy and the detuning plays the role of chemical potential . We show explicitly that the phase of this model Hamiltonian has no effect on the topological properties, we mention it explicitly in the “Method” section. In the next section, we derive the topological invariant number based on this optical Kitaev’s Hamiltonian (H1), where there is no further H1.This is the optical version of the Kitaev’s chain for this model Hamiltonian. Here, Here, we explicitly show that the topological quantum phase transition of the system occurs through a change of topological invariant quantity, i.e., the winding number.H1  and the unit value S(1) in the y–z plane. Therefore, θ(k) is a mapping. It is very clear from the analytical expression that the pseudo spin defined in the W) isy–z plane. There are some other representation29 for the calculation of winding number but that too finally gives the same result as the Anderson pseudo-spin approach45. We will present the other representation and the equivalence between them in the “Method” section.The analytical expression for winding number  and massless excitation at the point of topological quantum phase transition, which implies that system has diverging length scale. We calculate the topological number based on the study of winding number Eq. .W) with J. It is clear from our study that there is topological quantum phase transition from non-topological quantum state (W = 0) to topological state (W = 1). We study for different values of chemical potentials, and we observe that topological quantum phase transition occurs at J = μ/2 for a fixed value of Δ (here Δ = 1). But at μ = 0, the system is always in the topological state, and there is no topological quantum phase transition. We also observe that the change of the winding number is unity during the topological quantum phase transition and that there is no further change of winding number. For such a situation only one Majorana zero mode appears at both the ends of this optical Kitaev chain. In the previous study40, the authors of this proposed model only predict appearance of Majorana fermions at the edge of the chain for Δ = J > 0 and μ = 0.In Fig. W with μ. These figures are for the different values of J but for Δ = 1. We observe that topological quantum phase transition occurs for μ = 2J. It is also consistent with the result of Fig. μ and J, i.e., μ = 2J.In Fig. k for two values of chemical potentials . It is clear from our study for μ = 0 , there is no variation of k and it merges in a single line with the value unity. Therefore, the system is always in the topological state for equal values of Δ = J with no topological quantum phase transition. But for the finite values of the chemical potential  remains for any values of Δ s as far as Δ is finite. We use these exact solutions based results in the next section to study the emergence of different relativistic modes of the system.Finally, we consider the case when Δ is finite, mentclass2pt{minim41.This exact solution based calculations and results for the topological properties of light are absent in the entire literature of interacting light matter systemThe topological properties of the system motivates us for the searching of different kind of relativistic modes as an emergent property of the Hamiltonian. Here, we present the results and the physical explanation of Dirac equation for Majorana fermion mode for the system. At first we present the effective Hamiltonian of the system in the strongly correlated regime  under the presence of two photon drive Δ cos (2ωpt) and also in large U. For this conditions the two photon pump is only resonant for the creation of pairs of photons on NN cavities. In the limit of fermionized photonic mode, one can consider each cavity mode as a spin-1/2 particle as we have done in the previous section during the derivation of Kitaev’s model. Finally following the ref. J, which is physical limit ref. J and μ = 0 and also from the study of winding number that the system is in the topological state as far as μ < 2J for finite Δ. Therefore, the Hamiltonian H2 in this limit corresponds to the topological state.As we have already mentioned in the previous paragraphs that near to the resonance condition . These operators are defining the sites of the lattice . Our calculations are as exact as possible.σs and μs are the order and disorder respectively. These two fields, Here, we define the Dirac spinor, One can write down the final equation in the following form,m = 1 − g.One can also write the above Majorana equation in a compact form:ϕ(x), we term this massless mode as Majorana-Weyl fermion mode. This massless mode appears for μ = 2J. At this point where system shows the topological quantum phase transition through a change of unity of topological number, which corresponds to the appearance Majorana zero modes at both the ends of the array5. These gapless edge mode has interesting properties that they are not the same chiral fermion mode that propagate on the edge of integer Hall quantum Hall effect15. These are very special because the fermion are chiral and they are also Majorana modes1. It can be shown using the Clifford algebra representation, that the condition for the appearance of Majorana-Weyl fermion mode appears for only in the space-time dimension (8k + 2), 1. Our present problem is (1 + 1) dimension and Majorana-Weyl fermion mode satisfice the condition for (k = 0) the appearance in this system. As the Majorana-Weyl fermion mode appears, the system shows the transition from the non-topological state (W = 0) to the topological state (W = 1).Therefore, we prove that the spinor field satisfies the Majorana condition of Majorana fermion mode and also the ϕ(x)) are different. The different wave functions for the topological and non-topological states of the system also satisfy the basic criteria that the topological properties of the system are hidden in the properties of ground state wave function2.The massive Majorana fermion mode  and ψL(x) are the fermionic field for the right and left movers electron. Here, ψR(x) and ψL(x) are the two chiralities each of them is an independent fermionic mode. Although the Dirac equation is massless, still there is no Weyl fermion mode in the system for the following reasons:The d = 2n and d = 2n + 1 the Dirac fermion is a complex spinor with 2n components . In dimensions d = 4 and higher, it is possible to reduce the Dirac equation to a Weyl equation for two massless spinors of n components. In d = 1 + 1, this is not possible and there are no Weyl spinors in 1 + 1 dimensions14.The rank of a Dirac spinor depends on the dimensionality. In space-time dimensions g introduce a term like, kF, but the end result for Dirac equation will be the same. Therefore, the finite values of g do not introduce any mass term in the Hamiltonian in the Dirac equation.For non-zero values of ωp and ωc.Therefore, it reveals from the studies of above two sections that we quantum simulate different relativistic modes for interacting light-matter in a non-linear cavity QED arrays depending on the values of To the best of our knowledge this is the first attempt for searching the relativistic modes with new and important results in a strongly interacting light-matter physics in a nonlinear cavities array.47 to define a local topological order parameter for gapped quantum liquid system which do not require any translation symmetry. The author has also considered the anti-unitary symmetric Hamiltonian with the gapped ground state.Here, we present the study and results of quantized Berry phase of this model Hamiltonian system. At first, The author of ref. 63 but the interacting light matter physics has not been explored yet41.Our model Hamiltonian fulfils all the criteria that the author of ref. H(R) asC is the closed loop, |n(R)〉 is the nth eigen vector in the parameter space of R.To the best of our knowledge this is the first attempt with new and important results for a strongly interacting light-matter physics in a one-dimensional array of nonlinear cavities to study the quantized Berry phase and its relation to the topological quantum phase transition. Berry phase is a geometric phase of eigen state obtained when cyclically varying external parameters. One can write it analytically for the Hamiltonian 49. Basic definition of Zak phase is the following: The Berry’s phase picked up by a particle moving across the Brillouin zone. Here Brillouin zone is in the one dimension as treated by the Zak, and therefore, the natural choice for the cyclic parameter is the crystal momentum (k). The geometric phase in the momentum space is defined asun,k〉 is the Bloch states which are the eigen states of the nth band of the Hamiltonian. The ambiguity of the Zak phase problem has been solved by the Atala et al.55 by considering the difference of Zak phase between the states. This difference of Zak phase between the different states could be a proper topological number55. In our present study the difference of Zak phase between the topological state and the non-topological state is π and appears when the topological quantum phase transition occurs in the system. Therefore, in the present problem the quantized Zak phase correctly present the topological properties of the system.The topological properties of one-dimensional solids are characterized by the so called Zak phaseZ type topological invariant system, and the system has the anti-unitary particle-hole symmetry to ensure that the curve C can only be in a great circle on the Bloch sphere48. This Zak phase expresses as C is the close loop that the Hamiltonian forms on the Bloch sphere when k varies from −π to π and C. Therefore, the Zak phase is either π or 0. The analytical relation between the winding number, W, with the geometric phase is the following48.W either 1 or 0. Therefore, the corresponding Zak phase is π or 048. The transition of γ from π to 0 occurs when the system shows the topological quantum phase transition from a topological state to non-topological state.The present system is the one dimensional γ with J for different values of chemical potential. We observe that for μ = 0, the geometric phase of the system is always finite (π). For the other values of μ, we predict that geometric phase shows a sharp transition from the value π to 0 for J = μ/2. The behavior of quantized Zak phase is same for all finite values of Δ (here we consider Δ = 1).Figure μ for different values of J. We observe a transition of geometric phase from π to 0 when μ = 2J. This behavior of quantized Zak phase is consistent with the topological quantum phase transition of the system. We would like to explain this result more physically, the model Hamiltonian parameters Δ and J are equal, and also ωp = ωc system is in the always topological state. Therefore, depending on the difference of the values of ωp and ωc, system is either in topological state or in non-topological state. Therefore, we conclude that the basic topological properties of the system also find through the study of quantized Zak phase. Hence, we prove in our study the equivalence between the topological number and the local topological order for this system Hamiltonian. This study and results are entirely new in the literature of interacting light matter system41.In Fig. 65. But the authors of ref. μ) can be changed very easily by tuning the resonance frequency of the individual cavities and the driving frequency ωp. The phase and amplitude of the superconducting order parameter can be achieved by regulating the phase and amplitude of the two photon parametric driving. The tunneling amplitude of photons can be regulated by introducing intermediate control device between the cavities. This level of control on the parameters is the key to overcoming difficulties faced by the other solid state systems-based proposals65.There are quite a few proposals to detect the Majorana fermion mode in the previous studiesThis optical detection scheme is simple, versatile and physically realizable but the photon loss from the cavity is unavoidable in this system and this limits the time-scale of the detection; i.e., the time scale should be much shorter than the time scale of photon life time  length scale, in the topological state, these Majorana zero modes are weakly coupled and the levels of Majorana qubit that they form split in energy Eg) .We try to understand the single particle losses using the Lindblad equation.nian Eq.  and \docL) and the other at the right end (R). Both of these cavities have set up in the driving frequency ωp and a tunneling coupling (JL and JR are the tunneling coupling for the left and right cavity respectively.) which couple them to the optical Kitaev’s Hamiltonian, H1.Now we follow the experimental proposal of the authors of ref. 40:The low energy effective Hamiltonian in presence of the probe cavities is the followingJL and JR are the photon tunneling amplitude from the left cavity to Kitaev chain and from the right cavity to the Kitaev’s chain respectively. The physical interpretation of the above Hamiltonian is the following. The non-local Majorana qubit is formed by the localized Majorana modes. Majorana modes of the chain mediates a nonlocal coherent exchange of photons between the probe cavities. The authors have shown that under the assumption of the decay rate of probe cavities, i.e., 40.Where 67. As we have understood from our study of the topological properties of optical Kitaev’s chain that the presence of strong interaction between the light and matter and the weak photon loss are the two main ingredients to predict the evidence of topological state in circuit QED lattice. Both the criteria are satisfied in circuit QED. In circuit QED, fabrication and control is in the state of art67 and at the same time one can do the quantum state engineering at the desired level.Here we present the implementation of our experimental proposal to detect the Majorana fermion mode. The circuit QED (superconducting circuit based cavity QED) allows for the experimental realization with a sufficient control over dissipation to detect the Majorana fermion modes, i.e., the topological state of the system67. The transmon qubits are placed at an antinode of the intercavity field, the coupling strength between the cavity-qubit is ωb is the bare-cavity resonance frequency and α is the fine structure constant. The strength of the effective on-site interaction with lower polariton mode is ωp drives the second qubit periodically which creates and annihilate pairs of photons of frequency ωc = ωb − λ. In the presence of large coupling strength between the cavity and qubit compared to the amplitude of parametric drive which only annihilates and creates photon pairs in the NN cavities. Finally it creates an effective p-wave pairing.The cavity QED system consists of a chain of capacitatively coupled identical microwave resonators which play the role of cavities. Each of the cavity has two superconducting transmon qubits which plays the role of artificial atomsμ) between the cavities and the pump such that |μ| < 2J. At this quantum state of engineering the system is in the topological state with Majorana fermion mode with an energy splitting J, |Δ|, |μ| and vanishes exponentially with system size L (please see the “Method” section for detail discussion). The second relation of the inequality determines the broadening of Majorana level. In order to exist the topological state, this broadening should be smaller than the energy levels splitting Eg. So that Majorana energy levels do not overlap with energy level of bulk. The system shows topological quantum phase transition (for |μ| < 2J) and the Majorana fermion modes collective excitation changes to Majorana-Weyl fermion mode excitations at the point |μ| = 2J and when |μ| > 2J the collective modes of the system changes to Dirac fermion mode. To do these quantum state engineering, the condition J, μ well inside the topological state. Therefore, this condition is the sufficient condition for the detection of the topological state of the system up to the tuning of μ. But when this inequality is violated, the collective excitation of the system is Dirac like mode.In the circuit QED, one can estimate the parameter space as We have found the topological state and topological quantum phase transition in this system for different regime of parameter space. The topological quantum phase transition occurs only for finite detuning process. We have presented a few results based on the exact solutions along with physical explanation. We have predicted the quantized Zak phase which corresponds to the local topological order. We also prove the equivalence between the topological number and the local topological order from our study. We have also presented the emergence of massive Majorana fermion mode, massless Majorana-Weyl fermion mode and the massless Dirac fermion mode for the different quantum states of this system based on the exact solution. Finally we have presented the experimental proposal and implementation.45. In the fourth one, we present a detailed derivation of emergence of Dirac equation for Majorana fermion mode and massless Dirac equation for non-topological state. Finally, we present, the analytical expression for Eg with necessary discussions.This section consists of five parts. First, we show explicitly that the phase of the parametric pumping has no effect in the topological properties of the system. In the second, we present the Bogoliubov transformation and the excitation spectrum of the system. In the third one, we present the different representation of winding number calculations and their equivalence to the winding number calculation of Anderson pseudo spin approachγi,B and γi,A are the Majorana operators. Therefore, it is clear from the above Hamiltonian that in the basis of Majorana fermion the phase ϕ has no effect in the topological state of the system.One can also write the above Hamiltonian in the Majorana basis as follows:One can diagonalize the Hamiltonian by using the Bogoliubov transformation.The Bogoliubov quasi-particle operators is θk and sin θk are The analytical expression for cos 45.One can find the analytical expression for winding number calculations by the following analytical expression Eq. . Finallyα and β are y and z two components and The effective Hamiltonian of the system is W2 is the same as W  is the following:μz(n + 1/2) is the following:The equation of motion for the σ and μ operators to derive the equation of motion for the Majorana fields Now we use the properties of the Now we restore the lattice for that purpose, by doing the following transformation:  the eqs  and 36)\documentσ operators) and disorder (μ operators) during the derivation of Dirac equation of massive Majorana fermion mode68.We use the following relation between the order (μz(n + 1/2) that it is a kink operator, and it introduces the disorder in the system.ϕ(x) is the solution for massless Majorana-Weyl fermion. The important point to be noted that Weyl fermion occurs at m = 0 = 1 − g that implies the μ = 2J, where the topological quantum phase transition occurs.Now, we present the massless Majorana-Weyl equation as5. The energy splitting of the Majorana qubit is L is the length of the system and ξ−1 with the parameters of optical Kitaev’s chain Hamiltonian isL, i.e., there is no finite overlap between the zero energy nonlocal Majorana fermion modes as a consequences the system is in topological state. But L is finite and small.Here we follow the seminal paper of Kitaev’sξ−1 and from x+ and x−, we obtain the following relations in our study.Eg = 2J − μ if Δ ≥ J or if Δ < J and Eg is the energy gap of the optical Kitaev’s chain.From the analysis of


In this article we develop a Primitive Variable Recovery Scheme (PVRS) to solve any system of coupled differential conservative equations. This method obtains directly the primitive variables applying the chain rule to the time term of the conservative equations. With this, a traditional finite volume method for the flux is applied in order avoid violation of both, the entropy and “Rankine-Hugoniot” jump conditions. The time evolution is then computed using a forward finite difference scheme. This numerical technique evades the recovery of the primitive vector by solving an algebraic system of equations as it is often used and so, it generalises standard techniques to solve these kind of coupled systems. The article is presented bearing in mind special relativistic hydrodynamic numerical schemes with an added pedagogical view in the appendix section in order to easily comprehend the PVRS. We present the convergence of the method for standard shock-tube problems of special relativistic hydrodynamics and a graphical visualisation of the errors using the fluctuations of the numerical values with respect to exact analytic solutions. The PVRS circumvents the sometimes arduous computation that arises from standard numerical methods techniques, which obtain the desired primitive vector solution through an algebraic polynomial of the charges. The use of numerical methods to solve differential equations has constituted a substantial amount of work since the conception of approximate solutions to a given set of equations. In the last few decades, digital computers have been a great help to heavily iterate complicated partial differential equations using extensive numerical, parallel and adaptive mesh techniques in personal computers and large clusters.q at every point or cell on a given domain of space at a particular time in the iteration. However, physical phenomena are described and measured by means of a set of vector primitive variables u. Depending on the nature of the physical problem, the function u(q) may not have an analytic form and so, at every point or cell of the integration space a cumbersome technique requires to be performed for each time step. No matter how fast this routine may be, it introduces an extra computational time that can heavily grow when the space-time resolution increases. In problems of special relativistic hydrodynamics this fact appears and, at each time step, a 10th degree algebraic polynomial has to be solved for a unique given value of each component of the vector u  or a specific numerical technique to obtain it.To make things even more complicated, for each particular physical problem it is necessary to have either an analytic solution u avoiding any middle cumbersome step such as the ones mentioned above. This technique is so general that requires no analytical knowledge whatsoever of u(q). The method developed is general and valid to any set of coupled conservative equations. We also show how this method can be applied in the particular case of 1D special relativistic hydrodynamics (1DRHD). For this particular case, we construct convergence tests.In this article we show how it is possible to construct a general numerical iteration method, using a combination of finite differences and finite volume integration techniques for the time and spatial evolutions respectively, to directly find the solutions Primitive Variable Recovery Scheme (PVRS)” which can directly obtain the primitive variables from quite a standard numerical procedure. Section 4 deals with different convergence relativistic Sod . The integration of Ci between times tn and tn+1 yields:tq over time and the integral of ∂xf over space can be solved exactly and so, the next integral form of the previous equation is found:A more natural way of obtaining the discretisation form of  is the Fq, which is precisely the solution to the problem. In order to overcome this, we define each integration as a new numerical vector in the following form:average charge vector of q over Ci at time tn and average flux vector across the boundaries of Ci. From now on, the square brackets notation  around any numerical function is used to denote the corresponding (space or time) average related to that specific numerical function.At this point, both integrals in the previous equation cannot be integrated unless we have the exact form of q) is a smooth function, then the integral q at the midpoint of the interval to If o O(Δx2) .The indexes outside the square bracket do not denote the spatial and time evaluation of the average vector, they are just labels that refer to the time and grid positions of the corresponding numerical values.Substituting the definitions Eqs  and 27)27) in EqCi. This discretisation has the same exact form as The advantage of this method over any finite difference scheme is that the conservative nature of the system is preserved, even across strong discontinuities such as shock waves. This is the reason as to why a finite volume scheme is often used when dealing with the physics of high energy flows where discontinuities may appear.f at q at every time. This is why it is impossible to integrate the average flux. Somehow, we have to find a good approximation for this integral. Moreover, the flux f inside the integral is evaluated on the boundaries xi±1/2 of the grid cell which, numerically speaking, has no sense because we can only approximate the values of the average charges on the midpoint of the finite volume. This set of midpoints can be “safely” considered the ones used in the finite difference mesh mentioned in the Finite differences approach section.The flux q on either side of the interface xi±1/2, i.e., q change on every cell propagates at a finite characteristic speed . ThSubstitution of  problem  with twoIn order to obtain an accurate numerical flux function, is important to study the behaviour of the solution based on the form and properties of the governing equation at these particular initial conditions.a = 1 only) in which the flux is written as q. Note that, since Let us now consider a single conservative equation , t) and equate the result to zero, i.e.:q(X(t), t) is constant all along the ray x0 is some initial value. In the most general case, the set of all rays X(t) are called the characteristics of the equation.The function ql and qr are the left and right states respectively, the characteristics X(t) of  of  are thenm conservative equations  is obtained using the definition of wa:Let us consider now a system of qa is a superposition of m waves moving with characteristic velocities λ1, λ2, … and λm, respectively  ×  as shown on One of the most popular approximate Riemann solvers is the called HLL solver . This GoR and λL moving across the interface to the right and left respectively. The region delimited by these characteristic rays is denoted by the state qHLL.Instead of following the solution of all the characteristic variables along their own characteristic velocities, the idea of the HLL approximation consists on considering the larger eigenvalues λm conservative equations, 2m characteristic rays will emerge from each interface. The values λL and λR are to be chosen taking into account all 2m characteristic velocities.Note that, since we are working with a system of fR,L ≔ f. One can work out the approximate solution to the flux through the interface by integrating the hyperbolic equation over the space-time domain outlined in R,L). The final result is that [fHLL ≠ f(qHLL). The flux The approximate solution to the Riemann problem derived by this scheme has the following form (see e.g.  or 21]))21]):qaxi−1/2 between two control volumes Ci and Ci−1 and suppose that a constant reconstruction q has been made. With this, let xi−1/2. Note that these values are going to be different if a polynomial reconstruction is made. With this, we can write the numerical flux at xi−1/2 used in the Godunov scheme in the following form:xi+1/2 is obtained in an analogous way. So, by substituting these numerical fluxes in the discretisation high resolution ).MC, proposed by van Leer [σ is written as in c ≔ (a + b)/2.The monotonic centred limiter van Leer , has lesvan Leer . Neverthsuperbee limiter, also proposed by Roe in 1986 [godunov) is made. For this slope, the function is slightly more complicated than the previous ones and is given by:Another piecewise linear reconstruction is the  in 1986 . This onColella in 1984  developeS1 File1—norm.In the file “S1_File.tar.gz”, we attached the relevant data of the comparison between numerical simulations and exact solutions used to obtain the L(GZ)Click here for additional data file.


When they move away from the condensation surface, clusters progressively vanish and they have a life cycle of a fraction of a millisecond. Though scattering intensity is proportional to the 6th power of cluster radius r and cluster number density Nc theoretically, the scattering intensity does not increase sharply with the increase of subcooling degree from the experimental results, so we can infer that the cluster number density plays a dominate role in this process and the effect of cluster radius almost can be ignored.Zhong Lan and Di Wang contributed equally to this work.Condensation is a common physical process which widely exists in natural phenomena and thermal energy systems. In a condensation process, cluster is considered as the important bridge between vapor body and condensates. However, limited by the minimum imaging dimension of traditional measurements, early experimental studies about initial stages of condensation process are not sufficient. This paper provides a powerful optical platform for the study of dynamic clusters process. Based on the Rayleigh law, optical experiments were firstly introduced to investigate the clusters spatial distribution close to and far from condensation surface. The results show that clusters are mainly generated in the vicinity of the condensation surface within the thickness of 200  In the process of dropwise condensation, single droplet undergoes a whole process of nucleation, growth, coalescence and departure7. Nucleation has received widespread attention because it occurs not only in the initial stage of dropwise condensation but also filmwise condensation10. Although the processes after nucleation are well understood, water molecule clusters before nucleation, which are considered as original state of condensation, are relatively unexplored. In order to reveal the microphysical processes from vapor to condensates, cluster physical model was proposed to explain the initial stage of condensation in early studies13. It is presumed that steam molecules become clusters with the lognormal size distribution in bulk steam phase. However, it lacked the mechanism of formation and distribution of cluster itself because it just investigated the process of cluster sedimentation and the model has not been confirmed. In addition, molecular simulations were also applied to demonstrate cluster phenomena because of their abilities to observe the physical processes of nucleation which usually occur within nanometer spatial scale and nanosecond time scale17. However, it cannot demonstrate spatial changes of clusters near cooling surface. Theories about cluster physical model and methods of molecular simulations provide qualitative descriptions and quantitative explanations about vapor droplet microcosmic views. From the previous research, whether clusters exist or not lacks direct observation and proof. The distribution of clusters influenced by the surface condition or the subcooling condition has not been confirmed. If we describe the condensation process as lifecycle, the research of nucleation process is roughly equivalent to the study of embryonic period, so it is critical to be investigated and it may lead to the new enhancement technology. Nevertheless, it is difficult to observe the physical process of nucleation stage directly through traditional testing methods due to the microscopic scale limitation.Condensation and its heat transfer commonly exist in natural phenomena and industrial applications20, reflection spectrum22, were applied to measure some properties of clusters. Infrared spectrum was reported on precisely size-selected clusters, with molecule number n ranging from 85 to 475 in bulk steam phase23, but it was not used in the detection of dynamic condensation process. Optical theories, like scattering25, diffraction and interference24, could provide effective foundations in investigating nucleation processes, which have been widely adopted for the research of simple systems such as natural gas clusters and simple water system. The size of cluster was considered as several nanometers to a few tens nanometers, which is much smaller than visible laser wavelength range 390 nm to 780 nm. The particle scattering in this magnitude follows the principal of Rayleigh law. Thus, the Rayleigh scattering can be used in the measurement of average cluster size and cluster density26. The advantages of Rayleigh scattering are easy to experiment and hard to destroy the size and the distribution of clusters, so Rayleigh scattering is a good method to digest the behavior of clusters. Based on the cluster formation and distribution characteristics, we try to use the Rayleigh scattering method to demonstrate cluster phenomena.Optical measurement methods, such as infrared spectrum26:γ is polarizability of scattering particle and k is incident laser wavenumber,r is the mean radius of clusters, ε is permittivity of particles. For water clusters, ε = n2, where n is the refractive index of water, n = 1.33. For a laser beam propagating from x to x + Δx in an ensemble of clusters, the energy scattered by the clusters into a collecting lens is given byIlens is the scattering intensity probe received, I0 is the laser energy incident on the scattering volume, σlens is the cluster size averaged distribution cross section for Rayleigh scattering into the collecting lens, and Nc is the average number of clusters per unit volume, the spatial length is equal to kr ≪ 1, R0 is the lens radius, and h is the distance between the scattering volume and lens, so we obtainIlens is proportional to the 6th power of the mean radius of clusters r and average number of clusters per unit volume Nc, that is:According to the Rayleigh law, the scattering of single particle follows28,Tv is the vapor temperature, σlv the surface tension, Hfg the latent heat of vaporization, ρ1 the density of the condensate and ΔT the subcooling degree. We can estimate the minimum radius of drops ranges from 2 nm to 20 nm in the condition of subcooling degree from 1 K to 10 K. Rayleigh scattering is the elastic scattering of light by particles which are much smaller than the wavelength of the radiation. In this experiment, the green laser wavelength λ is 532 nm and the size of clusters can be considered as seen Å to several nanometers. Thus, clusters can follow Rayleigh law.Therefore, the energy of Rayleigh scattering by clusters represents the production of clusters and it is the combined effect of cluster size and density. From previous research, the minimum radius of drops can be considered asFrom the previous research about clusters, qualitative description accounted for most part, but experimental studies are relatives less. In view of this situation, we designed relevant experiment to explore cluster behaviors and observe cluster phenomena more detailed. On the basis of Rayleigh law above, we constructed optical experimental platform to explore the behaviors of clusters. In this paper, we conducted two experiments: the experiment of cluster condensation close to surface and the cluster jet flow scattering. Clusters show different distributions in the vertical direction since jet flow causes clusters to move away from surface, so we can study the life cycle of clusters and investigate properties of cluster far from surface. The main purpose of this paper is to use optical methods to investigate the behaviors of clusters close to condensation surface and phenomena far from condensation surface based on the clusters jet flow scattering.μm, which is much smaller than the arc of the copper hemisphere with 10 mm diameter, so the hemisphere shape would not influence the detection of optical signal. From this figure, it can be found that the scattering intensity close to surface grows with the increase of subcooling degree. Scattering intensity is related to the size and density of clusters, so it can be seen that the production of vapor clusters increases with the increase of subcooling degree. Here, we need to explain that we set the scattering intensity at subcooling degree of 6 K as start point because we could guarantee that the condensation surface was completely covered by the liquid film in this experiment, and then the influence of surface irregularity can be eliminated. Therefore, the scattering intensity in Fig. Nc. If mean radius r plays a leading rule, the scattering intensity will increase sharply and the relation between scattering intensity and subcooling degree is similar to exponential distribution. Thus, we can almost ignore the influence of mean radius r. From our early investigation29, Gibbs free energy of cluster formation ΔG exhibits a maximum value when a cluster containing i molecules achieves the critical size. And the addition of another molecule causes a decrease in ΔG, that is, the cluster reaches a lower and more stable energy level. Furthermore, with the increase of subcooling degree, the cluster would probably grow to a macroscopic size and phase change or condensation will happen. In this experiment, we found that, with the increase of surface subcooling degree, the curve gradually increases in an appropriately linear tendency, which demonstrates that the effect of average radius of clusters can be ignored. It also elucidates that more initial droplets break through the critical size and continue to grow up to clusters and the final sizes of clusters are almost close to each other.The scattering intensity close to condensate surface in different subcooling degreess is demonstrated in Fig. μm above the surface, falls to 20% at 700 μm. It can be inferred that vapor clusters mainly distribute in the area of hundreds of micrometers far from the surface, which provides a guide in the enhancement of surface heat transfer.The scattering intensity at the subcooling degree of 20 K in different heights is demonstrated in Fig. From the last experiment, we can see clusters mainly exist in the vicinity of the condensation surface within the thickness of hundreds of micrometers. Clusters vanish when they move away from the surface. In order to investigate the phenomena of clusters far from the condensation surface, we designed the clusters flow scattering experiment to investigate the life cycle of clusters and got properties of dynamic clusters in that region. Make the rate of carrier gas at 19 m/s and keep constant, the fraction of steam in the nitrogen carrier is 5% and the scattering intensity in different subcooling degrees is demonstrated in Fig. The clusters jet flow scattering intensity in subcooling degree of 15 K at different rates is demonstrated in Fig. Based on Rayleigh law, we observed cluster phenomena near condensate surface through optical experiment, which is unexplored in the vapor condensation process. We also analyzed the relationship between surface subcooling degree and cluster intensity. Consequently, we draw the following conclusions.Nc plays a dominate role in this process and the effect of r can be ignored. In the subcooling degree larger than 14 K, clusters reach saturation. The increase of subcooling degree will lead to thickening of the cluster layer; molecular clusters mainly distributed near surface layer within several hundred micrometers thickness.During pure steam condensation process, water molecule clusters are generated in the vicinity of condensation surface. With the increase of the subcooling degree, scattering intensity approximately follows a linear distribution. In the cluster jet flow experiment, the production of clusters grows more in the condition of subcooling degree increasing. Clusters have a life span of a fraction of a millisecond. Molecular clusters vanish when they move away from the condensation surface; the longer distance away from the surface, the greater proportion of cluster vanishes.By the above experiments, the process of dynamic clusters before nucleation has an overall impact on the subsequent condensation process.2. So, uncertainties of surface subcooling can be ignored. Semiconductor chilling plate could achieve the function of heating and cooling because of the Peltier effect. To regulate subcooling degree more precisely, semiconductor chilling plate fitted closely together to the bottom of copper. The chamber combined with the semiconductor chilling plate was fixed in the five dimensional optical platform, which can control x, y, z direction and two angles of dip and depression.Figure As shown in Fig. IR0. Then the surface temperature decreased in a constant rate from 14 °C to −4.5 °C, the scattering intensity was recorded by the spectrometer. In order to guarantee accuracy, the measurement was repeated for 3 times.In this measurement, the scattering intensity in different surface temperatures were measured. The detailed method is as follow: The Z-axis micrometer was adjusted to make the laser beam transmit cling to the condensation surface. We set the receiving scattering intensity at this moment as relative value On the basis of the last experiment, we lowered the platform to different heights. The surface temperature decreased in a constant rate from 14 °C to −4.5 °C, the scattering intensity was recorded by the spectrometer. In order to guarantee accuracy, the measurement was repeated for 3 times. Then we chose the data in different heights but at same temperature to compare their intensities.We set the distance between optical fiber probe and laser beam as 2 mm, and the distance between the exit of copper pipe and laser beam was 3 mm. The temperature of ice-water bath was set as 5 °C and the vapor boiler as 4 °C which ensures that the initial vapor would not condensate in ice-water bath. Other sections of this experiment were same to the last experiment.V1 of vapor boiler to extract gas and dust in the chamber, then open the valve V1, the nitrogen was added into the chamber to clean the remain gas and dust, then recorded the scattering light intensity I0 and set it as background value. In this experiment, pressure reducing valve was adjusted to a constant carrier gas flow rate at 62.5 L/h. At this carrier gas flow rate, the scattering intensity was recorded as I1. Then, the temperature of vapor boiler was increased 2.5 °C, when the temperature became stable, repeated the process above and recorded the data as I2 and so on, to the subcooling degree reached to 30 K.Turn the vacuum pump and close the valve V1 of vapor boiler to extract the gas and dust in the chamber, then open the valve V1, the nitrogen was added into the chamber to clean the remain gas and dust, then recorded the scattering light intensity I0 and set it as background value. Adjusted the valve V2 to a certain gas flow rate Q1 and recorded the scattering intensity I1. Then enhanced the gas flow rate to Q2 and recorded the intensity I2, and so on to Ii and Qi.The influence of carrier gas flow rate to scattering intensity of cluster jet flow was also studied. In this experiment, the temperature of ice water bath was set as 5 °C and kept constant. However, the temperature of vapor boiler was set as 30 °C. Turn the vacuum pump and close the valve