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• High Power Laser Science and Engineering
• Vol. 10, Issue 6, 06000e45 (2022)
Yin Shi1、2、*, David R. Blackman2, Ping Zhu3, and Alexey Arefiev2
Author Affiliations
• 1Department of Plasma Physics and Fusion Engineering, University of Science and Technology of China, Hefei, China
• 2Department of Mechanical and Aerospace Engineering, University of California San Diego, La Jolla, CA, USA
• 3Shanghai Institute of Optics and Fine Mechanics, Chinese Academy of Sciences, Shanghai, China
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Yin Shi, David R. Blackman, Ping Zhu, Alexey Arefiev. Electron pulse train accelerated by a linearly polarized Laguerre–Gaussian laser beam[J]. High Power Laser Science and Engineering, 2022, 10(6): 06000e45 Copy Citation Text show less

Abstract

A linearly polarized Laguerre–Gaussian (LP-LG) laser beam with a twist index $l = -1$ has field structure that fundamentally differs from the field structure of a conventional linearly polarized Gaussian beam. Close to the axis of the LP-LG beam, the longitudinal electric and magnetic fields dominate over the transverse components. This structure offers an attractive opportunity to accelerate electrons in vacuum. It is shown, using three-dimensional particle-in-cell simulations, that this scenario can be realized by reflecting an LP-LG laser off a plasma with a sharp density gradient. The simulations indicate that a 600 TW LP-LG laser beam effectively injects electrons into the beam during the reflection. The electrons that are injected close to the laser axis experience a prolonged longitudinal acceleration by the longitudinal laser electric field. The electrons form distinct monoenergetic bunches with a small divergence angle. The energy in the most energetic bunch is 0.29 GeV. The bunch charge is 6 pC and its duration is approximately $270$ as. The divergence angle is just ${0.57}^{\circ }$ (10 mrad). By using a linearly polarized rather than a circularly polarized Laguerre–Gaussian beam, our scheme makes it easier to demonstrate the electron acceleration experimentally at a high-power laser facility.

1 Introduction

The construction of numerous high-power laser systems around the world[14] has enabled the development of novel particle (see reports of Bulanov et al.[5] and Esarey et al.[6] and references therein) and radiation sources[712] for multidisciplinary applications[13,14]. Most of the improvements of the laser beams used for driving laser–matter interactions have been focused on increasing power, on-target intensity, total energy and the contrast of the compressed pulse. For example, the proposed facility[3,4] that aims to cross the 100 PW limit is expected to be in development over the next decade. Concurrently, new optical techniques for producing helical wave-fronts[1520] are also being developed. There now exist multiple computational[17,2129] and experimental[15,18,3032] studies examining interactions of helical laser beams with plasmas. There are also some published works on the terawatt scale helical laser production using a chirped-pulse amplification system[19,33]. Even though the techniques for creating helical beams with higher power are yet to be applied at high-power high-intensity laser facilities, they offer an exciting opportunity to create laser pulses with a qualitatively different field typology that can have a profound impact on laser–plasma interactions and particle acceleration.

There are several laser-based electron acceleration approaches with different degrees of maturity. The most frequently used ones are laser wakefield acceleration[6], which utilizes plasma electric fields, and direct laser acceleration[34], which relies on the fields of the laser for the energy transfer inside a plasma (e.g., see the report of Arefiev et al.[35]) or in vacuum[36]. These mechanisms are typically realized using conventional laser pulses. In an attempt to improve electron acceleration, several studies also considered radially polarized laser beams[37] and higher-order Gaussian beams[38]. Recently, there has been an increased interest in utilizing ultra-high-intensity laser beams with helical wave-fronts for electron acceleration in various setups, including vacuum acceleration[3942], laser wakefield acceleration[43,44] and microstructural target electron acceleration[45].

Conventional high-power laser systems[1] generate linearly polarized (LP) laser beams without a twist to the laser wave-fronts, which prevents one from readily realizing the interactions utilizing ultra-high-intensity helical laser beams. The spatial structure of laser beams with helical wave-fronts can be viewed as a superposition of Laguerre–Gaussian (LG) modes, which is why these beams are often referred to as LG beams. An LG beam can potentially be produced from a standard LP Gaussian laser pulse in reflection from a fan-like structure[16,17,20]. This approach avoids transmissive optics and it is well-suited for generating high-power high-intensity LG beams at high efficiency. Achieving circularly polarized (CP) LG beams in conventional high-intensity laser systems is likely to be more challenging than achieving LP-LG beams, since a native Gaussian beam is LP and extra steps need to be taken to induce circular polarization. It is then imperative to study laser–plasma interactions involving LP-LG beams, as these beams are more likely to be achieved in the near-term at high-intensity laser systems such as ELI-NP[2] or the SG-II UP facility[46]. The focus of this paper is on electron acceleration in vacuum by an LP-LG laser beam following its reflection off a plasma with a sharp density gradient. This setup is sometimes referred to as the ‘reflection off a plasma mirror’[47], but we minimize the use of the term ‘plasma mirror’ to avoid any confusion with the optical shutters employed for producing high-contrast pulses. Direct laser acceleration in vacuum by a conventional laser beam is generally considered to be ineffective. The key issue is the transverse electron expulsion caused by the transverse electric field of the laser. The expulsion terminates electron energy gain from the laser and leads to strong electron divergence. It must be stressed that the expulsion is closely tied to the topology of the laser field, which is dominated by the transverse electric and magnetic fields. In two recent publications[39,40] we showed that a CP-LG beam with a properly chosen twist can be used to solve the expulsion problem. The wave-front twist creates a unique accelerating structure dominated by longitudinal laser electric and magnetic fields in the region close to the axis of the beam. The longitudinal electric field provides forward acceleration without causing electron divergence, while the longitudinal magnetic field provides transverse electron confinement. It was shown using 3D particle-in-cell (PIC) simulations that a CP-LG beam reflected off a plasma can generate dense bunches of ultra-relativistic electrons via the described mechanism[39,40]. The distinctive features of this acceleration mechanism are the formation of multiple sub- $\unicode{x3bc} \mathrm{m}$ electron bunches, their relatively short acceleration distance (around $100\ \unicode{x3bc} \mathrm{m}$ ) and their high density (in the range of the critical density). The purpose of the current study is to identify the changes introduced by the change in polarization from circular to linear with the ultimate goal of determining whether the use of circular polarization is essential.

In this paper, we present results of a 3D PIC simulation for a 600 TW LP-LG laser beam reflected off a plasma with a sharp density gradient. We find that, despite the loss of axial symmetry introduced by switching from circular to linear polarization, the key features of electron acceleration are retained. Namely, the laser is still able to generate dense ultra-relativistic electron bunches, with the acceleration performed by the longitudinal laser electric field in the region close to the laser axis. In the most energetic bunch, the electron energy reaches 0.29 GeV (10% energy spread). The bunch has a charge of 6 pC, a duration of approximately 270 as and remarkably low divergence of ${0.57}^{\circ }$ (10 mrad). The normalized emittance in $y$ and $z$ is ${\tilde{\varepsilon}}_{\textrm{rms},y}\approx 5\times {10}^{-7},{\tilde{\varepsilon}}_{\textrm{rms},z}\approx 4\times {10}^{-7}$ .

Such dense attosecond bunches can find applications in research and technology[48,49], with one specific application being free-electron lasers[50]. The rest of this paper is organized as follows. Section 2 presents the field structure of the LP-LG beam and the setup of our 3D PIC simulation. Section 3 discusses the formation of electron bunches that takes place during laser reflection off the plasma. Section 4 examines the energy gain by the electron bunches during their motion with the laser pulse. Section 5 summarizes our key results and discusses their implication.

2 Field structure of the linearly polarized Laguerre–Gaussian beam and simulation setup

In this section, we present the structure of the LP-LG beam that we use in our 3D PIC simulation to generate and accelerate electron bunches. The section also presents the simulation setup.

The wave-front structure of a helical beam can be parameterized using two indices: the twist index $l$ , which specifies the azimuthal dependence of the transverse electric and magnetic fields, and the radial index $p$ , which specifies the radial dependence of the same fields in the focal plane. The polarization of the transverse laser fields is independent of their wave-front topology, so a helical beam can be LP or CP. Detailed expressions for all field components of an LP-LG beam are provided by Shi et al.[40]. We choose to omit these expressions here for compactness and instead we summarize the key features. The twist index $l$ qualitatively changes the topology of the transverse and longitudinal fields. We are interested in the field structure close to the central axis. There are three distinct cases: $l = 0$ , $\mid l\mid\ = 1$ and $\mid l\mid\ >1$ . The case of $l = 0$ corresponds to a conventional beam, with the near-axis field structure dominated by transverse electric and magnetic fields. In the case of $\mid l\mid\ >1$ , all laser fields vanish on the central axis. The case that is of interest to us is the case with $\mid l\mid\ = 1$ , because in this case the longitudinal rather than transverse fields peak on the axis.

In our 3D PIC simulations, we use an LP 600 TW laser with $l = -1$ and $p = 0$ . We consider a beam that propagates in the negative direction along the $x$ -axis upon entering the simulation box. Detailed parameters of the laser beam are listed in Table 1. In order to facilitate a comparison with the results for a CP-LG beam published by Shi et al.[40] (right-CP with $l = -1$ and $p = 0$ ), we use the same peak power, pulse duration and spot size for our LP-LG beam. The electric and magnetic field structures of the LP-LG beam in the $\left(x,z\right)$ -plane and the $\left(y,z\right)$ -plane are shown in Figure 1. The plots illustrate the difference in topology between the transverse and longitudinal field components. The longitudinal electric and magnetic fields reach their highest amplitude along the axis of the laser beam (see Figures 1(b) and 1(c)). On the other hand, the transverse electric field shown in Figure 1(a) vanishes on the axis of the beam. The electric field structure in Figure 1 agrees with the analytical expression given in Appendix C and derived in paraxial approximation by assuming that the diffraction angle ${\theta}_{\textrm{d}}$ is small. The longitudinal fields are relatively strong even though ${\theta}_{\textrm{d}}\ll 1$ . We have ${E}_{\parallel}^{\mathrm{max}}/{E}_{\perp}^{\mathrm{max}} = {B}_{\parallel}^{\mathrm{max}}/{B}_{\perp}^{\mathrm{max}}\approx 0.14$ for the considered LP-LG beam with ${\theta}_{\textrm{d}}\approx 8.5\times {10}^{-2}$ , where ${\theta}_{\textrm{d}} = {\lambda}_0/\pi {w}_0$ . Note that ${E}_{\parallel}^{\mathrm{max}}\approx 1.1\times {10}^{13}$ V/m and ${B}_{\parallel}^{\mathrm{max}}\approx 36$ kT. The peak normalized amplitude of the longitudinal field ${a}_{\ast} = \mid e\mid {E}_{\parallel}^{\mathrm{max}}/{m}_{\textrm{e}} c\omega$ for a given period-averaged power $P$ in PW is given by Equation (C21), where $c$ is the speed of light, $\omega = 2\pi c/{\lambda}_0$ is the laser frequency and $e$ and ${m}_{\textrm{e}}$ are the electron charge and mass, respectively. We find that ${a}_{\ast}\approx 2.7$ for the considered power of 600 TW.

Parameters for linearly polarized Laguerre–Gaussian laser
Peak power (period averaged)0.6 PW
Radial and twist index $p = 0,\ l = -1$
Wavelength ${\lambda}_0 = 0.8\ \unicode{x3bc} \mathrm{m}$
Pulse duration ( ${\sin}^2$ electric field) ${\tau}_{\textrm{g}} = 20$ fs
Focal spot size ( $1/\mathrm{e}$ electric field) ${w}_0=3\ \unicode{x3bc} \mathrm{m}$
Location of the focal plane $x = 0\ \unicode{x3bc}$ m
Laser propagation direction $-x$
Polarization direction $y$
Other simulation parameters
Position of the foil and the pre-plasma $-1.0\kern0.24em \mathrm{to}-0.3\ \unicode{x3bc} \mathrm{m}$ and $-0.3$ to 0.0 $\unicode{x3bc} \mathrm{m}$
Density distribution of pre-plasma ${n}_{\textrm{e}} = 180.0{n}_{\textrm{c}}\exp \left[-20\left(x+0.3\ \unicode{x3bc} \mathrm{m}\right)/{\lambda}_0\right]$
Electron and ion (C ${}^{6+}$ ) density in foil ${n}_{\textrm{e}} = 180.0{n}_{\textrm{c}}$ and ${n}_{\textrm{i}} = 30.0{n}_{\textrm{c}}$
Gradient length $L = {\lambda}_0$ /20
Simulation box ( $x\times y\times z$ )10 $\unicode{x3bc} \mathrm{m}$ $\times$ 20 $\unicode{x3bc} \mathrm{m}\times 20\ \unicode{x3bc} \mathrm{m}$
Cell number ( $x\times y\times z$ )800 cells $\times$ 1600 cells $\times$ 1600 cells
Macroparticles per cell for electrons100 at $r<$ 2.5 $\unicode{x3bc} \mathrm{m}$ , 18 at $r\ge 2.5\ \unicode{x3bc} \mathrm{m}$
Macroparticles per cell for C ${}^{6+}$ 12
Order of electromagnetic field solver4

It is instructive to compare the field structure of the LP-LG beam to the field structure of the CP-LG beam from Shi et al.[40]. In both cases, the longitudinal electric and magnetic fields peak on the axis of the beam where the transverse field vanishes. However, in contrast to the CP-LG beam, the longitudinal electric and magnetic fields of the LP-LG beam lack axial symmetry (see Figures 1(e) and 1(f)). The difference in symmetry can be illustrated by constructing an LP-LG beam ( $l = -1$ ) from two co-propagating CP-LG beams. Following the notations of Shi et al.[40], we create a CP transverse electric field by adding ${E}_{z} = i\sigma {E}_{y}$ , where $\sigma = 1$ produces a right-CP wave and $\sigma = -1$ produces a left-CP wave. We take a pair of CP-LG beams: one with $l = -1$ , $\sigma = -1$ and the other one with $l = -1$ , $\sigma = 1$ . Their superposition produces an LP transverse electric field, because ${E}_{z}$ components of these beams cancel each other out. It was shown by Shi et al.[39,40] that the longitudinal fields of the right- and left-CP-LG beams have different dependencies on $r$ and $\phi$ . Specifically, the longitudinal field of the right-CP-LG beam ( $l\sigma = -1$ ) is axisymmetric, whereas the longitudinal field of the left-CP-LG beam ( $l\sigma = 1$ ) has azimuthal dependence. Moreover, only the right-CP-LG beam contributes to the longitudinal fields on the axis, because the fields of the left-CP-LG beam vanish. The LP-LG beam inherits its azimuthal dependence from the left-CP-LG beam, which is the reason why the longitudinal fields still peak on the axis, but lose their symmetry as we move away from the axis.

As stated earlier, we want to contrast our results with those for a right-CP-LG beam that has the same power. The key player in electron acceleration is the longitudinal electric field of the laser, because it is this field that performs most of the acceleration for the electrons moving along the laser axis. We use the discussed decomposition for the LP-LG beam to compare the longitudinal field strength $\mid {E}_{x}\mid$ on the axis of the two beams. We have already shown that $\mid {E}_{x}\mid$ of an LP-LG beam is equal to $\mid {E}_{x}\mid$ of a right-CP-LG beam whose transverse field amplitude is half of that in the LP-LG beam. The power of the right-CP-LG beam is two times lower than the power the LP-LG beam. Since the power scales as the square of the field strength, we immediately conclude that $\mid {E}_{x}\mid$ in a right-CP-LG beam whose power is the same as the power of the LP-LG beam is going to be higher by a factor of $\sqrt{2}$ . The loss of axial symmetry and the reduced field strength are likely to alter the injection and subsequent acceleration of electron bunches by the LP-LG beam compared to the case of the right-CP-LG beam from Shi et al.[40].

In our simulation performed using the PIC code EPOCH[51], the discussed LP-LG beam is reflected off a plasma with a sharp density gradient. In what follows, we provide details of the simulation setup that we use in the next sections to study electron injection and acceleration. The target is initially set as a fully ionized carbon plasma with electron density ${n}_{\textrm{e}} = 180{n}_{\textrm{c}}$ , where ${n}_{\textrm{c}}$ = 1.8 $\times {10}^{27}$ m ${}^{-3}$ is the critical density for the considered laser wavelength ${\lambda}_0 = 0.8\ \unicode{x3bc} \mathrm{m}$ . Table 1 provides details regarding the density gradient. The focal plane of the beam in the absence of the plasma is located at $x = 0\ \unicode{x3bc} \mathrm{m}$ , which is also the location of the plasma surface. The resolution and the number of particles per cell in the PIC simulation are determined based on a convergence test, discussed in Appendix A. The test addresses the concern that the parameters of the accelerated electron bunches might be sensitive to simulation parameters[39].

3 Electron injection into the linearly polarized Laguerre–Gaussian laser beam

In this section we discuss the formation of electron bunches that takes place during laser reflection off the plasma. We refer to this process as the ‘electron injection’, because, once the bunches are formed, they continue surfing with the laser beam.

Figure 2 shows various aspects of electron injection. All snapshots are taken at $t = 9$ fs, with $t = 0$ fs being defined as the time when the peak of the laser envelope reaches $x = 0$ (in the absence of the plasma). The electron density, ${n}_{\textrm{e}}$ , in the $\left(x,z\right)$ -plane is shown in Figure 2(a). At this point, most of the laser beam (incident from the right) has been reflected by the plasma. The reflection process generates bunches that are solid in the near-axis region, with the peak densities as high as ${n}_{\textrm{c}}$ . The plot of ${n}_{\textrm{e}}$ integrated over the laser beam cross-section that is shown in Figure 7(a) of Appendix A provides additional information about the bunches. Figures 2(b) and 2(c) show the transverse areal density ${\rho}_{\textrm{e}}$ and cell-averaged divergence angle $\left\langle \theta \right\rangle$ of the third bunch marked with a dashed rectangle in Figure 2(a). The divergence angle for an individual electron is defined as $\theta \equiv \arctan \left({p}_{\perp }/{p}_{x}\right)$ . The angle is averaged on every mesh cell of the $\left(y,z\right)$ -plane. It is instructive to compare the plots of ${\rho}_{\textrm{e}}$ and $\left\langle \theta \right\rangle$ to the results for the CP-LG laser beam presented by Shi et al.[40] for the same time instant ( $t = 9$ fs). In the case of the LP-LG beam, the areal density in the near-axis region is approximately two times lower, while the divergence angle in the same region is similar to that of the CP-LG beam. The biggest difference is the loss of axial symmetry. The LP-LG beam generates two dense side-lobes in addition to the on-axis part that has been discussed. To further illustrate the complex structure of the electron bunches generated by the LP-LG beam, Figures 2(d) and 2(e) provide 3D rendering of the electron density in the third bunch. It is clear from Figure 2(d) that the two side-lobes are slightly behind the on-axis region, which means that the phase of the laser field at their location is different. The divergence of the two lobes (shown in Figure 2(c)) is so high that they are likely to move away from the central axis while moving in the positive direction along the $x$ -axis with the laser beam.

To examine this expectation and to provide additional insights into electron bunch dynamics, we performed detailed particle tracking for the third bunch. We distinguish three groups of electrons based on their transverse position within the bunch at $t = 46$ fs. Figure 3(a) shows the areal density of the third bunch at $t = 46$ fs, while Figure 3(b) shows the three groups of electrons selected for tracking. The electrons are picked randomly from the entire electron population of the third bunch. Note that we choose $t = 46$ fs rather than $t = 9$ fs as our selection time in order to give enough time for the three groups to become visibly separated. The selected particles are tracked during the entire simulation (up to $t\approx 310$ fs) to determine their trajectories and energy gain.

Figures 3(d)–3(f) provide projections of electron trajectories onto the beam cross-section, where the color-coding is used to show electron energy along each trajectory. The markers correspond to the electron positions at $t = 46$ fs. To see the initial electron positions, we provide Figure 3(c) that shows electron positions in the $\left(y,z\right)$ -plane at $t = -2.2$ fs. As seen in Figure 3(d), the ‘blue’ electrons remain close to the axis of the laser beam and thus within the region with a strong longitudinal electric field throughout the simulation. The ‘green’ electrons (see Figure 3(e)) rotate around the axis and eventually leave the analysis window ( $y\in \left(-2,2\right){w}_0$ , $z\in \left(-2,2\right){w}_0$ ). The ‘red’ electrons (see Figure 3(f)) are different because they are expelled directly outwards without any significant rotation. These electrons travel through the region with a significant transverse electric field. The long-term energy gain by these three groups of electrons is discussed in the next section.

4 Electron energy gain in the linearly polarized Laguerre–Gaussian laser beam

In Section 2 we showed that the reflection of an LP-PG beam produces dense electron bunches. These bunches can move with the laser beam, gaining energy. In this section we examine this energy gain.

Figures 3(g)–3(i) show how the energy of electrons in each group from Figure 3(b) changes over time. The magnitude of the longitudinal electron velocity ${v}_{x}$ is a major factor determining the electron energy gain. Electrons with smaller $c-{v}_{x}$ can stay longer in the accelerating part of the laser wave-front while moving forward with the laser beam. Due to the fact that the considered electrons are ultra-relativistic, it is the divergence angle $\theta$ rather than the magnitude of the velocity that primarily influences ${v}_{x}$ , with ${v}_{x}\approx c\cos \theta$ . To assess the difference between ${v}_{x}$ and $c$ , which can be extremely small, we use the vertical coordinate that shows $\left(x- ct\right)/{\lambda}_0$ in Figures 3(g)–3(i). It is essentially the relative slip (in the units of ${\lambda}_0$ ) between the electron and a point moving with the speed of light. The ‘blue’ electrons remain close to the beam axis and have the smallest $c-{v}_{x}$ . As seen in Figure 3(g), they slip by less than $0.2{\lambda}_0$ over 300 fs, which allows them to gain roughly 290 MeV. The ‘green’ electrons have a much bigger value of $c-{v}_{x}$ because there is a transverse component of electron velocity associated with the rotation. As seen in Figure 3(h), they experience significant slipping over 100 fs, which prevents them from the prolonged acceleration required for a substantial energy gain. The ‘red’ electrons have the biggest transverse displacement early on, so that they are exposed to a strong transverse laser electric field. This field causes their transverse motion, and it also transfers energy to the electrons. This is the reason why the ‘red’ electrons shown in Figure 3(i) gain more energy than the ‘green’ electrons. However, their slipping causes them to experience a decelerating field before the laser beam has time to diverge. This is the underlying cause for the energy reduction at $t>100$ fs. The analysis of electron trajectories leads us to a conclusion that the most energetic electrons in our setup are the electrons that remain close to the axis of the beam. In what follows, we focus on their energy gain.

Before we proceed with the analysis of the electron acceleration, we take a closer look at the reflected fields. Figure 4(a) shows the transverse electric field away from the axis ( $y = 0$ , $z = 0.7{w}_0$ ) and longitudinal magnetic field on the axis ( $y = 0$ , $z = 0$ ) after the laser has been reflected by the plasma. As we have already seen, the electrons tend to bunch on the axis. The field of these electrons is difficult to separate from the longitudinal laser electric field. This is the reason why ${B}_{\parallel }$ is plotted instead of ${E}_{\parallel }$ . The transverse field is plotted off-axis because it vanishes on the axis of the beam for the considered helical beam. The most striking feature compared to the CP-LG beam examined by Shi et al.[39,40] is the appearance of higher-order harmonics in ${E}_{y}$ . In contrast to ${E}_{y}$ , ${B}_{\parallel }$ has a more regular shape. The spectra shown in Figure 4(b) confirm that ${E}_{y}$ contains odd harmonics due to high-harmonic generation effects[52], whereas ${B}_{\parallel }$ in the near-axis region seems to be unaffected by the harmonic generation. According to Zhang et al.[22] and Denoeud et al.[30], the twist index ${l}_{n}$ of the harmonics generated during reflection of an LP-LG laser beam scales as ${l}_{n} = nl$ , where $n$ is the harmonic order. The field profiles suggest that the analysis of electron acceleration in the near-axis region can be performed without taking into account the higher-order harmonics, which have a different Gouy phase shift.

The momentum gain of the electrons moving along the axis of the laser beam can then be obtained by integrating the momentum balance equation:

\begin{align}{\mathrm{d}p}_{\parallel }/ \mathrm{d}t = -\mid e\mid {E}_{\parallel },\end{align}

where ${E}_{\parallel }$ is the on-axis component of the laser electric field. We neglect high-harmonic generation and beam scattering, so ${E}_{\parallel }$ is the real part of the on-axis field in the original beam given by Equation (C18). The longitudinal electric field has the same dependence on $x$ as the field of the CP-LG beam considered by Shi et al.[40]. For example, we have the following:

\begin{align}{E}_{\parallel} = -\frac{E_{\ast}\sin \left(\Phi +{\Phi}_0\right)}{1+{x}^2/{x}_{\textrm{R}}^2},\end{align}

for an electron that is staying close to the peak of the envelope, where ${E}_{\ast }$ is the peak amplitude of ${E}_{\parallel }$ . Here ${\Phi}_0$ is a constant that can be interpreted as the injection phase for an electron that starts its acceleration at $x\approx 0$ . The only difference between the fields of the CP-LG and LP-LG beams is their amplitude. Therefore, we can skip the derivation here and directly apply the result of Shi et al.[40]. We have the following longitudinal momentum gain for an electron injected into the laser beam close to the peak of the envelope:

\begin{align}\frac{\Delta {p}_{\parallel }}{m_{\textrm{ec}}} = -{a}_{\ast}\frac{\pi^2{w}_0^2}{\lambda_0^2}\left\{\cos {\Phi}_0-\cos \left[{\Phi}_0-2\arctan \left(x/{x}_{\textrm{R}}\right)\right]\right\},\\[-22pt] \nonumber \end{align}

where ${a}_{\ast }$ is the normalized amplitude of the longitudinal field.

We obtain the terminal momentum gain by taking the limit of $x/{x}_{\textrm{R}}\to \infty$ in Equation (3), which yields the following:

\begin{align}\frac{\Delta {p}_{\parallel}^{\mathrm{term}}}{m_{\textrm{ec}}} = 2{a}_{\ast}\frac{\pi^2{w}_0^2}{\lambda_0^2}\cos \left({\Phi}_0-\pi \right).\end{align}

One can understand the dependence on ${\Phi}_0$ by recalling that the electron is continuously slipping with respect to the forward-moving structure of ${E}_{\parallel }$ as it moves with the laser pulse. Delayed injection into the accelerating phase means that the electron slips into the decelerating phase before the amplitude of ${E}_{\parallel }$ becomes small due to the beam diffraction. As a result, the net momentum gain is reduced. The energy gain occurs only for $\pi /2<{\Phi}_0<3\pi /2$ . The assumption that the electron is moving forward with ultra-relativistic velocity is no longer valid for $3\pi /2<{\Phi}_0<5\pi /2$ , which, in turn, invalidates the derived expression. It is useful to rewrite our result in terms of electron energy. We assume that the electron experiences a considerable energy gain due to the increase of its longitudinal momentum, so that the terminal energy is ${\varepsilon}^{\mathrm{term}}\approx {cp}_{\parallel}^{\mathrm{term}}\approx c\Delta {p}_{\parallel}^{\mathrm{term}}$ . We then have the following:

\begin{align}\frac{\varepsilon^{\mathrm{term}}}{m_{\textrm{e}}{c}^2} = 2{a}_{\ast}\frac{\pi^2{w}_0^2}{\lambda_0^2}\cos \left({\Phi}_0-\pi \right).\end{align}

We now take into account the expression for ${a}_{\ast }$ in terms of the period-averaged power $P$ given by Equation (C21) to obtain the following practical expression:

\begin{align}{\varepsilon}^{\mathrm{term}}\left[\mathrm{GeV}\right]\approx 0.5\cos \left({\Phi}_0-\pi \right){P}^{1/2}\left[\kern0.1em \mathrm{PW}\kern0.1em \right].\end{align}

In comparison to the acceleration by a CP-LG beam with the same power $P$ [40], the terminal energy in the LP-LG beam is lower by a factor of $\sqrt{2}$ .

Figure 5 provides information of the long-term electron acceleration in the 3D PIC simulation. Figure 5(a) shows the electron energy distribution as a function of $x$ at $t = 261$ fs. Note that the plot of ${n}_{\textrm{e}}$ integrated over the laser beam cross-section is shown in Figure 7(c) of Appendix A. By this point, the electrons have roughly traveled a distance of $100{\lambda}_0$ with the laser beam. Note that $t = 261$ fs is chosen as the time of the snapshot in order to facilitate a comparison with the results for the CP-LG beam presented by Shi et al.[40]. The pronounced bunching is maintained by the periodic accelerating structure of ${E}_{\parallel }$ . The third bunch travels close to the peak of the laser envelope, which results in the highest electron energy gain. In what follows, we focus on this specific bunch.

Figures 5(b) and 5(c) show the time evolution of the divergence angle and electron energy within the third bunch (see the dashed rectangle and the inset in Figure 5(a)). After an initial stage that lasts about 80 fs, the distribution over the divergence angle reaches its asymptotic shape. It can be seen from the snapshot in Figure 5(b) (taken at $t = 261$ fs) that the bunch is monoenergetic, with most electrons having a divergence angle that is less than 10 mrad. The time evolution of the energy spectrum, shown in Figure 5(c), confirms that the bunch accelerates roughly as a whole. The dashed curve is the solution given by Equation (3). We used the start time of the acceleration as an adjustable parameter because our model only captures the acceleration after the longitudinal motion becomes ultra-relativistic. The phase ${\Phi}_0$ is another adjustable parameter that we use to match the time evolution of the electron energy in the bunch. We find that ${\Phi}_0\approx 0.8\pi$ provides the best fit, the result of which is shown in Figure 5(c). The large energy spread at the early stage is likely due to the presence of the two lobes shown in Figure 2. The good agreement at later times indicates that our model captures relatively well the key aspects of the on-axis electron acceleration.

We find that the electron bunches retain noticeable asymmetry following their prolonged interaction with the laser beam. To illustrate the asymmetry, Figures 6(a) and 6(b) show the areal density ${\rho}_{\textrm{e}}$ and the cell-averaged divergence angle $\theta$ in the cross-section of the third bunch at $t = 261$ fs. The bunch asymmetry is likely imprinted by the asymmetry in ${E}_{x}$ shown in Figure 1(e). Since the electron bunches are moving slower than the laser wave-fronts, each bunch experiences a rotating ${E}_{x}$ . This can be shown by examining the field structure at the location of a forward-moving ultra-relativistic electron bunch. We calculate ${E}_{x}$ in the beam cross-section using the analytical expression given in Appendix C. The longitudinal position is set by the expression $x = ct+{\Phi}_0{\lambda}_0/2\pi$ to mimic the longitudinal ultra-relativistic motion of an electron bunch. We set ${\Phi}_0 = 0.8\pi$ , as this was the injection phase determined by our analysis. Figures 6(c)6(e) show ${E}_{x}$ at $\tilde{x} = 0.1$ , 0.45 and 2.3, respectively. These locations correspond to the snapshots in Figures 2, 3 and 5. The plots confirm that the field is indeed rotating, but they also show that the rotation is relatively slow, which is likely the reason why the asymmetry is retained by the electron bunch.

We conclude this section by providing additional parameters of the most energetic electron bunch (the third bunch) generated by the considered 600 TW LP-LG laser beam. The electron energy in the bunch is 0.29 GeV with a full width at half maximum (FWHM) of approximately $10\%$ . The bunch has a charge of 9 pC and a duration of approximately 270 as. The divergence angle is as low as ${0.57}^{\circ }$ (10 mrad). The normalized emittance in $y$ is ${\tilde{\varepsilon}}_{\textrm{rms},y}\approx 1.6\times {10}^{-6}$ and the normalized emittance in $z$ is ${\tilde{\varepsilon}}_{\textrm{rms},z}\approx 1.4\times {10}^{-6}$ .

5 Summary and discussion

Using 3D PIC simulations, we have examined electron acceleration by a 600 TW LP-LG laser beam with $l = -1$ reflected off a plasma with a sharp density gradient. The simulations show that electrons can be effectively injected into the laser beam during its reflection. The electrons that are injected close to the laser axis experience a prolonged longitudinal acceleration by the longitudinal laser electric field. The simulations also show that the laser beam generates a train of mono-energetic ultra-relativistic electron bunches with a small divergence angle. The distinctive features of this acceleration mechanism are the formation of multiple sub- $\unicode{x3bc} \mathrm{m}$ electron bunches, their relatively short acceleration distance (around $100\ \unicode{x3bc} \mathrm{m}$ ) and their high density (in the range of the critical density).

An important conclusion from our study is that the key features that were previously reported for a CP-LG beam[39,40] are retained in the case of an LP-LG laser beam. It is likely that experimentally it will be easier to generate a high-power LP-LG beam than a high-power CP-LG beam. This is because the laser beams at high-power laser facilities are LP. Changing the polarization introduces additional challenges and complications that our approach of using an LP-LG beam allows one to circumvent. We hope that this aspect will make it easier to perform a proof-of-principle experiment.

Even though there are key similarities, there are also differences in electron injection and acceleration between the cases of LP-LG and CP-LG beams. The injection into the LP-LG beam is more complex, leading to a formation of two side-lobes that accompany the on-axis bunch. The asymmetry of the longitudinal electric field causes the on-axis electron bunches to become asymmetric. In contrast to that, the bunches generated by a CP-LG beam are axisymmetrical. For two beams with the same power, the LP-LG beam has weaker on-axis electric and magnetic fields. The reduction in the field strength leads to a reduced energy gain, with the terminal electron energy being lower by roughly a factor of $\sqrt{2}$ .

Our mechanism relies on electrons becoming relativistic during the injection process. It is this feature that allows the injected electrons to surf with the laser pulse without quickly slipping from an accelerating phase into an adjacent decelerating phase. Since the longitudinal laser electric field plays a critical role in the injection process, its amplitude needs to be relativistic to generate relativistic injected electrons. A reduction of the incident laser power can thus degrade the mono-energetic spectra of the electron bunches by reducing the amplitude of the longitudinal field. To examine this aspect, we ran an additional simulation with a reduced incident power of 60 TW. Even though the laser still generates electron bunches in this case, the bunches are no longer mono-energetic. The peak energy is also noticeably lower than the value predicted by Equation (6). The underlying cause is most likely the inability of electrons to stay for a prolonged period of time in an accelerating phase.

In this work, we primarily focused on the most energetic (third) electron bunch. The considered laser pulse generates five distinct electron bunches. Their parameters are given in Table 3 of Appendix A. We want to point out that the front and tail of the considered laser pulse are steeper than what one would expect for a Gaussian pulse with the same FWHM, which was a deliberate choice made to reduce the size of the moving window and thus computational costs. The electrons must be relativistic during their injection, so that they can start moving with the laser beam without significant slipping. If this is not the case, then the mono-energetic feature discussed earlier might be hard to achieve. A dedicated study is required to determine the role of the temporal shape of the laser pulse and its overall duration. We anticipate that a longer laser pulse would produce a large number of ultra-relativistic electron bunches. For example, an 800 fs 600 TW LP-LG laser beam[46] contains roughly 300 cycles, so it has the potential to generate a similar number of bunches. Such a pre-modulated electron beam with high charge can potentially be used to generate coherent undulator radiation and to create a free-electron laser[50,53].

Sim. No.Cell sizeCell numberMacro-particles per cellOrder of electromagnetic
(window size is the same)e ( $r<2.5\;\unicode{x3bc} \mathrm{m}$ ), e ( $r>2.5\;\unicode{x3bc} \mathrm{m}$ ), C ${}^{6+}$ field solver
#11/40 $\unicode{x3bc} \mathrm{m}$ $400\times 800\times 800$ 200, 36, 242
#21/40 $\unicode{x3bc} \mathrm{m}$ $400\times 800\times 800$ 400, 72, 484
#31/50 $\unicode{x3bc} \mathrm{m}$ $500\times 1000\times 1000$ 200, 36, 244
#41/80 $\unicode{x3bc} \mathrm{m}$ $800\times 1600\times 1600$ 100, 18, 124

Table 2. Parameters used for the four simulations depicted in Figure 7.

#1#2#3#4#5
${\varepsilon}_{\textrm{e}}$ [GeV] ( $\Delta {\varepsilon}_{\textrm{e}}/{\varepsilon}_{\textrm{e}}$ )0.02–0.10.02–0.280.29 (10%)0.22 (6%)0.1 (15%)
${\tilde{\varepsilon}}_{\textrm{rms}, yz}\ [\unicode{x3bc}$ m]0.950.881.50.640.92
$W$ [mJ]0.061.52.21.30.06
$Q$ [pC]1.4896.80.7
$\Delta t$ [as]300360270260540

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Yin Shi, David R. Blackman, Ping Zhu, Alexey Arefiev. Electron pulse train accelerated by a linearly polarized Laguerre–Gaussian laser beam[J]. High Power Laser Science and Engineering, 2022, 10(6): 06000e45


text:
# How do you find the derivative of Y= ( 2x + 9 ) sqrt(x^2 - 4)?

Aug 4, 2015

$Y ' = \left(2 x + 9\right) \cdot \frac{x}{\sqrt{{x}^{2} - 4}} + 2 \cdot \sqrt{{x}^{2} - 4}$

#### Explanation:

Let $f \left(x\right) = 2 x + 9$ and $g \left(x\right) = \sqrt{{x}^{2} - 4}$. Their derivatives are:
$f ' \left(x\right) = 2$ and $g ' \left(x\right) = \frac{1}{2} {\left({x}^{2} - 4\right)}^{- \frac{1}{2}} \cdot 2 x = \frac{x}{\sqrt{{x}^{2} - 4}}$

The Product Rule says that

$Y ' = f \left(x\right) g ' \left(x\right) + f ' \left(x\right) g \left(x\right)$

$Y ' = \left(2 x + 9\right) \cdot \frac{x}{\sqrt{{x}^{2} - 4}} + 2 \cdot \sqrt{{x}^{2} - 4}$

format: plain


text:
# What's the point of the half coefficient in the max-cut cost Hamiltonian

Below is the cost Hamiltonian for an unweighted max-cut problem, I don't understand what the point of the half coefficient is. Why couldn't we omit it?

$$C_\alpha = \frac{1}{2}\left(1-\sigma_{z}^j\sigma_{z}^k\right),$$

• I imagine it's because it scales the range of $C_{\alpha}$ to between 0 and 1. Feb 5, 2021 at 13:54

You are correct that a constant factor in the cost Hamiltonian isn't relevant. This factor is included so that the eigenvalues of the cost Hamiltonian can be interpreted in terms of the number of cuts in the graph. In this simple case of a two-node graph, the eigenvalues are 0 or 1 when the factor of 1/2 is included, which correspond to 0 or 1 cuts.

• Wait so it isn't always the half coefficient? Feb 5, 2021 at 15:20
• How are the eigenvalues 0 or 1 when the coefficient of 1/2 is included? What does that mean? Feb 5, 2021 at 15:21


text:
# Dos and Dont’s of Behaviour Management

By Murray Bourne, 16 Dec 2005

Paperback, ©Roger Dunn, Continuum, 2005 (147 pages)

## Summary Review

Dunn's extensive experience as a school disciplinary officer is obvious throughout the book Dos and Don'ts of Behaviour Management. His practical advice is valuable whether you are about to start a teaching career or whether you are one of those jaded teachers that experience great frustration in the classroom because of poor student behaviour.

The setting for the book is public schools in England. In many schools in Western countries, education is more about surviving each day intact, rather than learning. There is something really wrong with that...

Suggestions in the book that I liked:

• "Perfect Planning and Preparation Prevents Poor Performance" (he is referring to teachers here). Plan the lesson. Plan the technology. Plan for disasters. Be in the room before the students arrive. Have materials ready. Ensure learning will take place.
• Take charge calmly, politely and firmly
• Approach teaching like a coach approaches a sporting team - motivate, encourage, demonstrate, praise
• Respond less to negative behaviours and praise positive behaviours
• Use rewards sparingly
• Believe in the potential of students
• Maintain a sense of humour!

Things which were not emphasised enough (or not at all):

• If the students are interested in the learning event, they are less likely to misbehave. Behaviour management is more about motivation than control.
• That Dunn was maybe "successful" as a disciplinary officer because he is physically large and strong (Many teachers are smaller than their students - this adds extra challenge.)

A worthwhile read - it is good to reflect on how to handle difficult situations in teaching.

Footnote: As a former mathematics teacher, I was concerned at the number of times Dunn wrote things like

...if you do such-and-such, the problems will increase trigonometrically.

Say what? From the context, he surely meant to say "exponentially". See this → Exponential Functions and compare to trigonometric functions.

Be the first to comment below.

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text:
# Synopsis: Combining two approaches to the strong force

A renormalizable effective theory for quantum chromodynamics.

Quantum Chromodynamics (QCD) is the fundamental description of the strong force in the standard model of particle physics. QCD is very successful at describing the behavior of quarks and gluons at high energies. However, at energies below about $1\phantom{\rule{0.333em}{0ex}}\text{GeV}$, the theory becomes strongly coupled, and one must account for bound states of the theory: pions, protons, neutrons, and other hadrons. Attempts to formulate an effective theory of strong interactions at such low energies have traditionally suffered from problems of renormalizability, which severely constrain their predictive powers.

One approach that has been used involves “constituent quarks,” which dress up the bare quarks of QCD into effective entities. For example, a proton is often described as being “composed of three constituent quarks.” A second approach is to make an expansion in the limit of “large $N$” where $N$ is the number of colors. Although $N$ is only $3$ in QCD, this approach often works well qualitatively. Now, in a recent paper in Physical Review Letters, Steven Weinberg proposes a way to marry the two approaches into an effective field theory of constituent quarks, gluons, and pions. It turns out that the large $N$ limit of this effective theory is renormalizable.

This proposal introduces a new computational tool for hadronic physics and investigations of its phenomenological viability are likely to launch interesting future research. – Abhishek Agarwal and Robert Garisto

More Features »

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## Subject Areas

Particles and Fields

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### Synopsis: Filling in a Tetraquark’s Profile

An analysis of electron-positron collision data has determined the spin and parity of a particle thought to consist of four quarks. Read More »

Nuclear Physics

### Synopsis: Strong Force Model for Weak Force Reactions

A quark-based model called lattice QCD provides theoretical predictions for two weak-force-driven reactions—proton fusion and tritium decay. Read More »

Geophysics

### Synopsis: Scanning Earth’s Interior with Neutrinos

Future neutrino experiments may provide tomographic scans of Earth’s interior by viewing solar neutrinos that pass through our planet’s layers.   Read More »


text:
Question

# What is the 6th term in the GP series: 1, 2, 4...?163212864

Solution

## The correct option is B 32

Suggest corrections


text:
MathSciNet bibliographic data MR1866056 (2003a:46051) 46E35 (46E30) Fan, Xianling; Zhao, Dun On the spaces \$L\sp {p(x)}(\Omega)\$$L\sp {p(x)}(\Omega)$ and \$W\sp {m,p(x)}(\Omega)\$$W\sp {m,p(x)}(\Omega)$. J. Math. Anal. Appl. 263 (2001), no. 2, 424–446. Article

For users without a MathSciNet license , Relay Station allows linking from MR numbers in online mathematical literature directly to electronic journals and original articles. Subscribers receive the added value of full MathSciNet reviews.


text:
# 0.5 Leading an effective research team

 Page 1 / 1
This module offers guidelines for leading an effective research team and is authored by Susan Margulies (University of Pennsylvania).

## Are lab directors leaders or managers?

$\text{Leader}\neq \text{Manager}$

• influences opinions and attitudes of other to accomplish a shared goal
• makes sure people and processes are in place to achieve goals. Develops plans, oversees budgets and resources, organizes, solves problems
Lab Directors must be Leaders AND Managers

If I had one piece of advice to give it’s that although you’ve been hired for your scientific skills and research potential, your eventual success will depend heavily on your ability to guide, lead, and empower others to do their best work.
--Thomas Cech, HHMI

Lab Director Role = Vision + Relationships + Tasks

## Developing a vision or mission statement

• A summary of goals and inspirations
• A clearly stated vision inspires, and sets a course for the lab
• Create a distinct future that distinguishes your lab from your competitors
• Describe the kind of research you want to do
• Atmosphere, Values: scientific excellence, teamwork?
• Historical and current challenges
• Short- and long-term goals
• Tie into departmental or institutional culture

## Directing and motivating people

1. Behavior
• tell people what to do and how to do it, when to do it, and then monitor closely
• listen, provide encouragement and assistance, facilitate their involvement in problem-solving and decision-making
• Lab Directors must provide BOTH - behavior type depends on lab member’s level of competence and commitment.
Delegate those activities that someone else could provide both direction and support
2. Communication

## Be accessible

• Daily time in the lab
• Open office door
• Weekly lab meetings (presentations, progress reports)
• Small group meetings (technical issues, alt. approaches)
• One-on-one meetings
• Journal club (teach good/bad exp design, analysis, reporting)
• Strategy sessions (new directions)
• Group lab activities in non-work setting to build team spirit
Make sure you reserve time to be alone to think, read, and write!
• Timely
• Specific
• Objective
• Reinforces expectations
• Constructive
4. Make decisions
• in isolation
• after consultation
• as a group
• allow others to make the decision (only if can you live with the outcome!)
5. Set rules of behavior for lab members
• Work hours (focus on productivity) (your hours set the pace!)
• Vacations and maternity leaves
• Authorship guidelines
• Scientific ethics
• Project ownership
• Projects leaving the lab (fund 6 mo independent research?)
They’re watching…always watching! State rules, be a role model, and be consistent and fair.
6. Motivate the lab
• Importance of vision statement
• Recognize and appreciate each person’s contributions
• Make sure lab members understand their role
• Provide training, recognize progress
7. Manage conflict
• Assess the problem, each person’s interests, acknowledge constraints, encourage parties to participate in resolution

## Managing projects

• Allocating, using and tracking resources to to achieve a goal in a desired timeframe
• publishing a paper, getting a grant, tenure
• Reduce wasted effort, track progress, respond quickly to hurdles
• Define distinct steps to goal&expected time to complete
• Who does what, when? (1-2 wk blocks) (sequential or parallel?)
• Identify constraints, necessary collaborations, supplies, equipment, resources
• Anticipate bottlenecks (pair post-doc with UG, have technician work out and revise new methods)
• Track work closely (lab meetings, use clear indicators (“blots are done”))
• Implement contingency plans
• Maintain team focus and motivation (communicate project vision)

## When do i learn to do all of that?

What relevant Lab Director skills do you have already?

• use it to analyze situations in the lab
• use it to organize your time, resources, and manage projects
• you will make mistakes
• will guide you in directing your lab

## Resources

At The Helm: A Laboratory Navigator by K. Barker

Making the Right Moves: A Practical Guide to Scientific Management for Post-docs and New Faculty by L. Bonnetta (HHMI)

How we are making nano material?
what is a peer
What is meant by 'nano scale'?
What is STMs full form?
LITNING
scanning tunneling microscope
Sahil
what is Nano technology ?
write examples of Nano molecule?
Bob
The nanotechnology is as new science, to scale nanometric
brayan
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Damian
Is there any normative that regulates the use of silver nanoparticles?
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
why we need to study biomolecules, molecular biology in nanotechnology?
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
why?
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
anyone know any internet site where one can find nanotechnology papers?
research.net
kanaga
sciencedirect big data base
Ernesto
Introduction about quantum dots in nanotechnology
what does nano mean?
nano basically means 10^(-9). nanometer is a unit to measure length.
Bharti
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
absolutely yes
Daniel
how to know photocatalytic properties of tio2 nanoparticles...what to do now
it is a goid question and i want to know the answer as well
Maciej
Abigail
for teaching engĺish at school how nano technology help us
Anassong
How can I make nanorobot?
Lily
Do somebody tell me a best nano engineering book for beginners?
there is no specific books for beginners but there is book called principle of nanotechnology
NANO
how can I make nanorobot?
Lily
what is fullerene does it is used to make bukky balls
are you nano engineer ?
s.
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
Tarell
what is the actual application of fullerenes nowadays?
Damian
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
Tarell
what is the Synthesis, properties,and applications of carbon nano chemistry
Mostly, they use nano carbon for electronics and for materials to be strengthened.
Virgil
is Bucky paper clear?
CYNTHIA
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
NANO
Got questions? Join the online conversation and get instant answers!


text:
# No statistical significance but observable trends

I have a general inference question regarding scenarios when results from data are not statistically significant but there appears to be an observable trend.

For example, treatment A and treatment B are applied to 2 independent populations. Using a ttest to analyze the resulting data (lets say the data is total revenue), the p value == .2, so the effect of treatment on revenue was not statistically significant. However, the total revenue from treatment A was observably higher in treatment B. What can I say in this regard?

I've had academic advisors recommend saying 'While the effect of treatment was not significant, a trend was observed', and then one would go on describing the trend. Is this an adequate viewpoint, or a statistical folly? What conclusions would a data scientist in industry draw and present to stakeholders from a scenario like this?

• population has to be same and not independent to undertake a t test for correlated samples. Jul 28 '20 at 12:36


text:
Index: The Book of Statistical ProofsGeneral Theorems ▷ Bayesian statistics ▷ Probabilistic modeling ▷ Generative model

Definition: Consider measured data $y$ and some unknown latent parameters $\theta$. A statement about the distribution of $y$ given $\theta$ is called a generative model $m$:

$\label{eq:gm} m: \, y \sim \mathcal{D}(\theta) \; .$

Sources:

Metadata: ID: D27 | shortcut: gm | author: JoramSoch | date: 2020-03-03, 15:50.


text:
# What is the general mathematical definition of wave?

What is the general mathematical definition of wave?

• A wave is $f(x-v t)$ for any function $f(x)$, where $v$ is the wave speed. – flippiefanus Nov 14 '18 at 9:11
• @flippiefanus That is far too restrictive. Under that definition water waves, gravitational waves, light waves in dispersive media, etc. are all not waves. – knzhou Nov 14 '18 at 9:22
• Somewhat related: Do all waves of any kind satisfy the principle of superposition? (though unfortunately the accepted answer is dead wrong). Making general statements about waves, in a way which applies to all waves (but without making definitions which are so formless that they apply to things to which the term doesn't apply) is a pretty hard task. – Emilio Pisanty Nov 14 '18 at 18:19

Waves are phenomena of the real world. In order to provide the most general definition of waves from the mathematical point of view, one should start with a careful listing of all the physical properties shared by all the phenomena we want to call "waves". I am not sure a complete list exists, but, taking into account a large set of well known linear and non linear phenomena classified as waves, I would say that a quite wide and general mathematical definition of most of the physical waves could be: any phenomen which can be described by the solution of a partial differential equation of hyperbolic type, in most of the cases second-order in time; i.e., a partial differential equation for a function $$f(\vec{r},t)$$ (where $$\vec{r}$$ is a "space" point in an $$n-$$dimensional configuration space), such that it has a well-posed initial value problem of the kind: $$f(\vec{r},t=0)=\phi_0(\vec{r})\\ \frac{\partial{f}}{\partial{t}}(\vec{r},t=0)=\phi_1(\vec{r})$$ were $$\phi_0$$ and $$\phi_1$$ are given functions.
Notice that the usual linear d'Alembert wave equation $$\frac{\partial^2{f}}{\partial{t}^2}=v^2\nabla^2{f}$$ is a very special case, of hyperbolic equation, leaving out too many wave-like phenomena (dispersive and/or damped waves, non linear waves including solitons, quantum mechanical probability waves etc.).


text:
# Is the name of this chemical 3-oxopropanoic acid?

Is the name of this chemical 3-oxopropanoic acid?

I included "oxo"- for the aldehyde group, and "-oic acid" for the carboxylate group.

However, I'm unsure whether this is the correct name since I cannot find it on the internet, while I can find almost all other names I've been trying to figure out.

• – user7951 Dec 28 '15 at 19:50

The name ‘3-oxopropanoic acid’ for the compound $\ce{OHC-CH2-COOH}$ given in the question is correct.

According to the current version of Nomenclature of Organic Chemistry – IUPAC Recommendations and Preferred Names 2013 (Blue Book), the order of seniority of classes in decreasing order of seniority is as follows.

(…)
7. acids
(…)
15. aldehydes
(…)

Thus, the carboxylic acid group has priority to be cited as a suffix.

Systematic names of carboxylic acids are formed as follows:

P-65.1.2 Systematic names

Carboxylic acids are named substitutively using the suffix ‘oic acid’ or ‘carboxylic acid’ and the prefix ‘carboxy’ to describe carboxy groups that cannot be included as the principal characteristic group or to express the carboxy group in the presence of a higher principal characteristic group.

P-65.1.2.1 Carboxylic acid groups, $\ce{-COOH}$, that conceptually replace a $\ce{-CH3}$ group of methane or terminate an unbranched hydrocarbon chain are named by replacing the final ‘e’ of the name of the corresponding hydrocarbon by the suffix ‘oic acid’. No locants are necessary to denote the positions of the carboxylic acid groups in a hydrocarbon chain; (…). Except for (…), systematically formed names are preferred IUPAC names; (…)

Therefore, the preferred IUPAC name (PIN) of the unmodified carboxylic acid is ‘propanoic acid’. (Note that the retained name ‘propionic acid’ may be used in general nomenclature.)

Aldehydes can be systematically named in various ways. Monoaldehydes derived from alkanes are preferably named substitutively using the suffix ‘al’. In this case, however, this method cannot be used since the carboxylic acid group has priority to be cited as a suffix. Therefore, the aldehyde group has to be expressed by a prefix as follows:

P-66.6.1.3 In the presence of a characteristic group having priority to be cited as a suffix or when present on a side chain, a $\ce{-CHO}$ group is expressed by the preferred prefix ‘oxo’ if located at an end of a carbon chain, or, otherwise, by the preferred prefix ‘formyl’.

Thus, the PIN for the complete structure is ‘3-oxopropanoic acid’.

The corresponding name that is formed by using the prefix ‘formyl’ is ‘formylacetic acid’. Such names are not acceptable as PINs but may be used in general nomenclature.

Furthermore, a special rule for aldehydic acids applies to the given compound:

P-65.1.6.3 Aldehydic acids

When a dicarboxylic acid has a retained name (…) and when one of its carboxy groups is replaced by a formyl group, $\ce{-CHO}$ (…), the resulting structure is called an aldehydic acid and, in general nomenclature may be named by replacing the ending ‘ic acid’ of the name of the dicarboxylic acid by the ending ‘aldehydic acid’. Preferred IUPAC names for aldehydic acids derived from all dicarboxylic acids are constructed systematically. The prefix ‘formyl’ is used in preferred IUPAC names, except for a $\ce{-CHO}$ group at the end of an acyclic chain, which is designated by the prefix ‘oxo’.

For the compound given in the question, the PIN of the corresponding dicarboxylic acid is ‘propanedioic acid’.

This dicarboxylic acid also has the retained name ‘malonic acid’, which may be used in general nomenclature. Hence, according to P-65.1.6.3, the corresponding aldehydic acid may be named ‘malonaldehydic acid’ in general nomenclature.

By way of comparison, for the analogous example $\ce{OHC-CH2-CH2-COOH}$, all three corresponding names are explicitly mentioned in the Blue Book:

• 4-oxobutanoic acid (PIN)
• 3-formylpropanoic acid
• succinaldehydic acid

• Probably worth noting that many biochemists would call this compound "malonic semialdehyde", as it is derived from the common biochemical/metabolite malonic acid by reduction of one (but not both, thus the "semi") of the acid moieties to an aldehyde. – Curt F. Dec 29 '15 at 5:26

3-Oxopropanoic acid (if carbon of aldehyde is numbered, use "oxo" as prefix) is correct IUPAC-name for the given chemical compound.

Formylethanoic acid (if carbon of aldehyde is not numbered, use "formyl" as prefix) is also correct IUPAC -name.

• A compound typically has one (in digits: 1) correct systematic IUPAC name. – Jan Dec 31 '15 at 16:43
• If we are using formyl prefix and parent chain has two carbon only of an acid and "1" is considered, it is wrong. Suppose, in case of "2-formylpropanoic acid" and "3-formylpropanoic acid" both are different compound and correct. "1-formylpropanoic acid" is incorrect IUPAC and even also this compound doesn't exit. – solanki... Dec 31 '15 at 16:51
• @Jan even if people don't know the correct concept, they make the correct answer and concept prove wrong. This thing is wrong on stackexchange. – solanki... Dec 31 '15 at 16:55
• My point was: (1) There is exactly one correct systematic IUPAC name per structure. (2) Of course, that does not mean the others are bad or discouraged or allow for misinterpretation. However, you should clearly show that the other name is a non-systematic IUPAC-ish name. (3) I don’t know what you mean with your second comment … – Jan Dec 31 '15 at 17:01


text:
# Best and most efficient way to numerically compute $e$?

There are many well-known methods for efficiently numerically computing $\pi$, such as Chudnovsky's Method or perhaps Gauss-Legendre's algorithm. I was wondering what the best method for computing $e$ is.

I have used the Taylor's series expansion or $e^x$ at $x=1$, which is in fact very convergent and does not cost too much to compute:

$$e=\sum_{n=0}^{\infty}\frac{1}{n!} = 1+1+\frac{1}{2}+\frac{1}{6}+\cdots$$

Here are a table of some basic values:

n                    Estimation            Error (e - sum)
1                    1.0                   1.718281828459045
5                    2.708333333333333     0.009948495125712054
10                   2.7182815255731922    3.0288585284310443E-7
100                  2.7182818284590455    -4.440892098500626E-16

I am curious to know if there are even more efficient methods to compute $e$. Keep in mind that computational cost and convergence speed are the priorities.

• See Finch's Mathematical Constants and numbers.computation.free.fr/Constants/E/e.html – lhf Apr 12 '12 at 2:17
• $e$ has a very simple continued fraction, and there are good bounds on the error made by stopping at the $n$th partial quotient. Another (possible) advantage is that all calculations can be carried out in exact rational arithmetic with no need for floating point. – Gerry Myerson Apr 12 '12 at 2:28
• Your table is highly skewed by whatever errors appear inthe computation: a very basic use of the error term in the Taylor polynomial shows that for $n=100$ the approximation is better than $3/101!$, and this number is less than $10^{-159}$ (i.e. $159$ correct digits). Already $n=20$ gives you $20$ good digits, which is more than what your usual computer can handle – Martin Argerami Apr 12 '12 at 6:39
• @Martin has hit the nail on the head. Your last error term is on the order of machine epsilon for double-precision floating-point. If you check the values between 10 and 100, you should find that the error stops decreasing because the successive terms are too small to be represented in the usual double data type. – Rahul Apr 12 '12 at 19:08
• @m.k.: as you say, that's precisely when one uses appropriate software. The OP's table shows that he didn't use such a thing. – Martin Argerami Apr 12 '12 at 23:05

A tiny C program from Xavier Gourdon to compute $9000$ decimal digits. (You may change that for more digits).
main(){


text:
# Estimation of conditionally convergent series

I wonder if given any conditionally convergent series $$S=\sum_{k=1}^{\infty}a_k$$ we can do the estimation related to the triangular inequality and $$\textit{reordering}$$ the series, i.e., \begin{align} \left|S=\sum_{k=1}^{\infty}a_k\right|\leq\left|\sum_{k=1}^{\infty}a_{2k}+\sum_{k=1}^{\infty}a_{2k+1}\right|\leq\left|\sum_{k=1}^{\infty}a_{2k}\right|+\left|\sum_{k=1}^{\infty}a_{2k+1}\right|? \end{align} If we assume that $$\sum_{k=1}^{\infty}a_{2k}$$ &$$\sum_{k=1}^{\infty}a_{2k+1}$$ are both well defined and can be eual to $$\infty$$ does the triangular inequality hold?

Also, is there any good reference that we can know more about conditionally convergent series like how to know that how much cancellation of each term we can get or when we can know more about periodicity of "sign" like harmonic / alternating series ?

• The expression $\sum_{k=1}^{\infty} a_{2k} + \sum_{k=1}^{\infty} a_{2k+1}$ may not be defined due to $\infty - \infty$ issues, such as when $a_k=(-1)^k/k$. Also, what is meant by your "..." at the end ? – Michael Oct 23 '18 at 1:52
• @Michael I fixed it. Thank you. If the both are infinity then we still can bound $S$ by infinity right? – Thomas Oct 23 '18 at 1:57

Note that you need to re-index so that you include $$a_1$$.
Suppose $$\sum_{n=1}^{\infty} a_n$$, $$\sum_{k=1}^{\infty} a_{2k}$$, and $$\sum_{k=1}^{\infty} a_{2k-1}$$ are all well defined (possibly infinite) and at most one of the two sums $$\sum_{k=1}^{\infty} a_{2k}$$, and $$\sum_{k=1}^{\infty} a_{2k-1}$$ has an infinite value (either $$\infty$$ or $$-\infty$$). Then $$\sum_{n=1}^{\infty} a_n = \sum_{n=1}^{\infty} a_{2k} + \sum_{k=1}^{\infty}a_{2k-1}$$ and $$\left|\sum_{n=1}^{\infty} a_n\right| \leq \left|\sum_{n=1}^{\infty} a_{2k}\right| + \left|\sum_{k=1}^{\infty}a_{2k-1}\right|$$
Proof: For each positive integer $$M$$ we have $$\sum_{n=1}^{2M} a_{n} = \sum_{k=1}^{M}a_{2k} + \sum_{k=1}^{M}a_{2k-1} \quad (Eq. *)$$ Taking limits as $$M\rightarrow\infty$$ and noting that the right-hand-side avoids the undefined case of $$\infty-\infty$$ gives the first equality. On the other hand, taking absolute values of both sides of Eq. (*) and using the triangle inequality gives $$\left|\sum_{n=1}^{2M} a_{n}\right| \leq \left|\sum_{k=1}^{M}a_{2k}\right| + \left|\sum_{k=1}^{M}a_{2k-1}\right|$$ Taking limits as $$M\rightarrow\infty$$ completes the proof. $$\Box$$


text:
If you have a Chebfun-related question that doesn't appear on this list, please ask us at discuss@chebfun.org.

1. How do I get Chebfun?
Go to the [download][/download] section, download Chebfun (less than 2MB), put it in your Matlab path (the easiest way to do this is probably using the command "pathtool"), and you're ready to go. We recommend running "chebtest" first to get the package warmed up (i.e. loaded into your machine's cache) and to make sure it is running properly.

2. What version of Matlab do I need?
Chebfun is compatable with [Matlab][matlab] Versions 7.4 (R2007a) and higher.

3. How do I learn Chebfun?
Take a look at the [gallery of examples][/examples] or the [Chebfun Guide][guide], or just type x = chebfun('x') and start playing with commands like these:
f = exp(x).*sin(20*x)
plot(f)
roots(f)
max(f)
g = exp(f)
h = abs(f)
j = round(g)
k = max(g,h)

4. What about 2D or 3D?
We have a project called [Chebfun2], led by Alex Townsend, that extends Chebfun-like computations to functions defined on rectangles. We believe this is comparable to the power and speed of Chebfun in 1D for many cases. We are considering a Chebfun in 3D.

5. Why is Chebfun based on [Matlab][matlab] rather than a non-commercial platform like C, Python, or Octave?
The Chebfun concept comes straight from Matlab — it's all about overloading Matlab commands for functions instead of vectors. And certainly most of our users at present are people who already use Matlab, who can get start using Chebfun the minute they see it. At the same time, we recognize that other platforms for Chebfun may be worth exploring.

6. How do I get involved?
We are always glad to receive comments at discuss@chebfun.org, and we usually respond quickly to questions. We are especially eager to receive drafts of proposed new [Examples] for inclusion in our collection! Furthermore, since the release of V4.0 in March 2011, Chebfun has been an open-source project. If you're interested in helping to develop Chebfun, please take a look at the [Developer Zone][/develop].

7. What is a chebfun?
A chebfun is a Matlab object that behaves syntactically like a Matlab vector and mathematically like a function of a real variable defined on an interval $[a,b]$. See [Chapter 1][guide/chap1] of the [Chebfun Guide][guide].

8. What is a fun?
A chebfun consists of one or more pieces, each of which is represented by a polynomial interpolant in Chebyshev points. Each of these pieces is called a fun.

9. What is a chebop?
A chebop is an object in the Chebfun system that behaves like a linear or nonlinear operator acting on chebfuns. For example, if f is a chebfun corresponding to sin(x) on the interval [a,b] and L is a chebop corresponding to the differentiation operator on [a,b], then L(f) is a chebfun corresponding to cos(x) on [a,b]. See [Chapter 7][guide/chap7] and [Chapter 10][guide/chap10] of the [Chebfun Guide][guide].

10. What's the difference between Chebfun and chebfun?
Chebfun with a capital C is the name of the project, and of the software system, whereas chebfun with a lower-case c is the name for an object in this system (namely a function defined on an interval).

11. What's the difference between sin(chebfun('x')) and chebfun('sin(x)')?
Any call of chebfun(f) evaluates f numerically at many points in order to determine an accurate polynomial representation of it. Once a chebfun is created, functions like sin() can be applied to it to create a functional composition, which is then given a polynomial representation. Most of the time, you would not expect much of a difference between starting with polynomials and composing, or finding a polynomial directly for the composed expression. However, because the underlying methods are numerical rather than symbolic, the two techniques are not identical. An extreme example would be sin(x).*exp(-x) if x is a chebfun defined on the interval [0,Inf]. This fails because the first step is to represent sin(x) by a mapped polynomial on an infinite domain, which is impossible. Yet chebfun('sin(x).*exp(-x)') would be fine on the same domain, because the only polynomial representation requested is for a function that decays rapidly.


text:
Like logic, the subject of sets is rich and interesting for its own sake. We will need only a few facts about sets and techniques for dealing with them, which we set out in this section and the next. We will return to sets as an object of study in chapters 4 and 5.

A set is a collection of objects; any one of the objects in a set is called a member or an element of the set. If $a$ is an element of a set $A$ we write $a\in A$.

Some sets occur so frequently that there are standard names and symbols for them. We denote the real numbers by $\R$, the rational numbers (that is, the fractions) by $\Q$, the integers by $\Z$ and the natural numbers (that is, the positive integers) by $\N$.

There is a natural relationship between sets and logic. If $A$ is a set, then $P(x)=$"$x\in A$'' is a formula. It is true for elements of $A$ and false for elements outside of $A$. Conversely, if we are given a formula $Q(x)$, we can form the truth set consisting of all $x$ that make $Q(x)$ true. This is usually written $\{x:Q(x)\}$ or $\{x\mid Q(x)\}$.

Example 1.5.1 If the universe is $\Z$, then $\{x:x>0\}$ is the set of positive integers and $\{x:\exists n\,(x=2n)\}$ is the set of even integers.

If there are a finite number of elements in a set, or if the elements can be arranged in a sequence, we often indicate the set simply by listing its elements.

Example 1.5.2 $\{1,2,3\}$ and $\{1,3,5,7,9,…\}$ are sets of integers. The second is presumably the set of all positive odd numbers, but of course there are an infinite number of other possibilities. In all but the most obvious cases, it is usually wise to describe the set ("the set of positive odd numbers, $\{1,3,5,7,9,…\}$'') or give a formula for the terms ("$\{1,3,5,7,9,\ldots,2i+1,\ldots\}$'').

Example 1.5.3 We indicate the empty set by $\emptyset$, that is, $\emptyset=\{\}$ is the set without any elements. Note well that $\emptyset\ne\{\emptyset\}$: the first contains nothing, the second contains a single element, namely the empty set.

The logical operations $\lnot, \land, \lor$ translate into the theory of sets in a natural way using truth sets. If $A$ is a set, define $$A^c=\{x: x\notin A\},$$ called the complement of $A$. If $B$ is a second set, define $$A\cap B=\{x:x\in A\land x\in B\},$$ called the intersection of $A$ and $B$, and $$A\cup B=\{x:x\in A\lor x\in B\},$$ called the union of $A$ and $B$.

Example 1.5.4 Suppose $U=\{1,2,3,\ldots,10\}$, $A=\{1,3,4,5,7\}$, $B=\{1,2,4,7,8,9\}$; then $A^c=\{2,6,8,9,10\}$, $A\cap B=\{1,4,7\}$ and $A\cup B=\{1,2,3,4,5,7,8,9\}$. Note that the complement of a set depends on the universe $U$, while the union and intersection of two sets do not.

We often wish to compare two sets. We say that $A$ is a subset of $B$ if $$\forall x (x\in A\implies x\in B),$$ and write $A\subseteq B$. This is not only a definition but a technique of proof. If we wish to show $A\subseteq B$ we may start with an arbitrary element $x$ of $A$ and prove that it must be in $B$. We say the sets $A$ and $B$ are equal if and only if $A\subseteq B$ and $B\subseteq A$, that is, $$\forall x (x\in A\iff x\in B).$$ So to show two sets are equal one must verify that a biconditional is satisfied, which often needs to be done in two parts, that is, the easiest way to show that $A=B$ often is to show that $A\subseteq B$ and $B\subseteq A$. If $A\subseteq B$ and $A\ne B$, we say $A$ is a proper subset of $B$ and write $A\subset B$.

Example 1.5.5 $\N\subset\Z\subset\Q\subset\R$.

Finally, we say that $A$ and $B$ are disjoint if $A\cap B=\emptyset$.

In section 1.1 we learned that logical operations are related by many tautologies, the study of which is called Boolean Algebra. These tautologies can be interpreted as statements about sets; here are some particularly useful examples.

Theorem 1.5.6 Suppose $A$, $B$ and $C$ are sets. Then

a) $A\cap B\subseteq A$,

b) $A\subseteq A\cup B$,

c) $A\cap(B\cup C)=(A\cap B)\cup (A\cap C)$,

d) $A\cup(B\cap C)=(A\cup B)\cap (A\cup C)$,

e) $(A\cap B)^c=A^c\cup B^c$,

f) $(A\cup B)^c=A^c\cap B^c$,

g) $A\subseteq B$ iff $B^c\subseteq A^c$.

Proof.
Suppose $P(x)=$"$x\in A$'', $Q(x)=$"$x\in B$'', $R(x)=$"$x\in C$''. To prove (a), suppose that $a\in A\cap B$. Then by definition, $P(a)\land Q(a)$ is true. Since $P(x)\land Q(x)\implies P(x)$ is a tautology, $P(a)$ is true, or $a\in A$. As noted above, this proves that $A\cap B\subseteq A$. Similarly, (c) follows since $P(x)\land(Q(x)\lor R(x))\iff (P(x)\land Q(x))\lor (P(x)\land R(x))$ is a tautology. All the other statements follow in the same manner.

$\square$

As in the case of logic, (e) and (f) are called De Morgan's laws. Theorem 1.5.6 certainly is not an exhaustive list of set identities, it merely illustrates a few of the more important ones.

If $a,b\in U$ we can form the ordered pair $(a,b)$. The fundamental property of ordered pairs is that $(a_1,b_1)=(a_2,b_2)$ if and only if $a_1=a_2$ and $b_1=b_2$. If $A$ and $B$ are sets, the set $$A\times B=\{(a,b): a\in A\land b\in B\}$$ is called the Cartesian product of $A$ and $B$.