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# Quantum limit transport and destruction of the Weyl nodes in TaAs

## Abstract

Weyl fermions are a recently discovered ingredient for correlated states of electronic matter. A key difficulty has been that real materials also contain non-Weyl quasiparticles, and disentangling the experimental signatures has proven challenging. Here we use magnetic fields up to 95 T to drive the Weyl semimetal TaAs far into its quantum limit, where only the purely chiral 0th Landau levels of the Weyl fermions are occupied. We find the electrical resistivity to be nearly independent of magnetic field up to 50 T: unusual for conventional metals but consistent with the chiral anomaly for Weyl fermions. Above 50 T we observe a two-order-of-magnitude increase in resistivity, indicating that a gap opens in the chiral Landau levels. Above 80 T we observe strong ultrasonic attenuation below 2 K, suggesting a mesoscopically textured state of matter. These results point the way to inducing new correlated states of matter in the quantum limit of Weyl semimetals.

## Introduction

The principle of emergence is beautifully demonstrated by the complex states of matter that arise in metals due to interactions. Conventional metals can become superconductors1,2, spontaneously break spatial symmetries3, emerge with fractionalized charge4, or even become insulating5. The recent prediction6,7 and discovery8,9 of Weyl semimetals brings a different ingredient to the table–Weyl fermions–that differ from conventional electrons in two key ways. First, they obey a linear or “massless” energy-momentum relationship, analogous to electrons in graphene but in three dimensions. Second, because of strong spin–orbit coupling and inversion symmetry breaking, spin “up” and “down” states are replaced by chiral “left” and “right”-handed states (Fig. 1)10. Can interacting Weyl fermions form states of matter that are unique from those that arise in conventional metals? Predictions include chiral excitonic11 and density wave states12, topologically non-trivial superconductors13,14, and Tomonaga–Luttinger liquids15, but no interacting states have been confirmed in Weyl semimetals to date. Both Weyl fermions and trivial quasiparticles exist in real Weyl semimetals, making it difficult to tell which effects are due to Weyl fermions and which effects can be ascribed to conventional semimetal physics16.

In addition to the possibility of new correlated states, Weyl semimetals exhibit a number of unusual non-interacting properties due to the chiral anomaly. Chirality, or “handedness”, is a classically conserved quantity–the flows of left and right-handed quasiparticles (chiral currents) should be separately conserved. The correct quantum mechanical description of nature does not obey this symmetry, allowing for classically forbidden phenomena such as the decay of a neutral pion into two photons17,18. In the case of Weyl fermions it has been shown that parallel electric and magnetic fields can unbalance the number of left and right-handed Weyl fermions10, leading to exotic transport10,19,20,21 and optical22 properties. Although this effect was originally predicted for a Weyl semimetal far in the quantum limit (QL)10, most experiments probing these effects have focused on relatively low magnetic fields9,23,24,25. Again, interpretation of these results is hampered by the fact that there are multiple Weyl fermions and non-chiral quasiparticles all present at the Fermi energy.

In this work, by using high magnetic fields we are able to gap out the non-chiral quasiparticles in TaAs and move the chemical potential to the fully chiral n = 0 Landau levels (LLs) belonging to the Weyl nodes. This allows us to directly observe the chiral anomaly in the QL, measure the opening of a gap in the n = 0 LLs once the inverse magnetic length becomes longer than the momentum-space separation of the Weyl nodes above 50 T, and discover the onset of a new state above 80 T.

## Results

### High field transport in TaAs

Figure 1b shows the resistivity of the Weyl semimetal TaAs in magnetic fields up to 95 T. Immediately striking is the two-order-of-magnitude increase in resistance that onsets at 50 T for the current and field configuration $${\bf{J}}\left\| {\bf{B}} \right\|{\bf{c}}$$ (note the logarithmic scale). In addition to its strong magnetic field dependence, ρ zz is also strongly temperature dependent above 50 T. This is in contrast with the behavior at intermediate fields (between 7.5 and 50 T), where ρ zz decreases and then saturates, becoming roughly temperature and field independent. We identify the transport behavior between 7.5 and 50 T as characteristic of the chiral anomaly in the QL for Weyl fermions, and the behavior above 50 T as indicative of mixing between left and right handed Weyl fermions. There are two features in the data that we do not believe to be inherent to the bulk transport of TaAs: a superconducting transition at low field, originating in a ≈20 nm tantalum-rich amorphous layer induced during sample preparation;26 and the rollover of ρ zz at high field, either due to the same amorphous layer or due to the intrinsic surface states of TaAs9,27,28. Neither effect impacts the conclusions of this paper (see Methods).

We measure both $${\bf{J}}\left\| {\bf{B}} \right\|{\bf{c}}$$ρ zz —and $${\bf{J}} \bot ({\bf{B}}||{\bf{c}})$$ρ xx —to identify different regimes in the electronic structure of TaAs (Fig. 2). TaAs contains three distinct types of quasiparticles: two electron-like Weyl Fermi surfaces (denoted W1 and W29,23,25,27,29), and a hole-like Fermi surface with a trivial spectrum23,29,30. Figure 2c highlights a key difference in the Landau quantization between Weyl and trivial carriers: the energy of the n = 0 LL of Weyl carriers is field-independent and disperses linearly in k z (momentum parallel to B), whereas the energy of the n = 0 LL of trivial carriers is field-dependent and disperses quadratically in k z This leads to qualitatively different behavior for the two carrier types in the QL, when only the n = 0 LL is occupied31. As the magnetic field is increased TaAs passes through four distinct LL regimes, described in detail in the caption to Fig. 2. With the magnetic field aligned along the c-axis both the W1 and W2 Weyl pockets enter the QL (the chemical potential enters the 0th LL) below 8 T, as observed in both our data and in previous studies of the Fermi surface of TaAs29. The hole pocket enters the quantum limit at 18 T and is completely depopulated at 36 T—also agreeing with previous studies29. The fact that the trivial hole carriers can be completely depopulated with field is a result of the field-independence of the n = 0 LL of the Weyl pockets, where the chemical potential is pinned due to the small excess of electrons versus holes.

### Chiral anomaly in the quantum limit

Above 36 T the chiral n = 0 LLs of the Weyl fermions can be studied without the complicating influence of trivial holes, and without the n > 0 LLs for which chirality is not well-defined32. Each of the two 0th LLs of the Weyl fermions contains carriers propagating along a single direction—parallel to B for one chirality (“up movers”), and anti-parallel for the other (“down movers”). Applying an electric field (driving current) parallel to B produces an imbalance in the number of up versus down movers, resulting in an imbalance of left and right-handed carriers—a chiral anomaly. This imbalance produces a chiral current, effectively increasing the conductivity as the electric and magnetic fields transfer charge from one Weyl node to the other10,19. The experimental signature of this phenomenon at low fields—negative magnetoresistance—has been controversial due to the presence of experimental artifacts32,33,34 and contributions from the trivial hole pocket. We find that using focused-ion-beam lithography (FIB) to reduce current-jetting can suppress or even eliminate the very low field negative magnetoresistance that has been reported for TaAs7,8 (see Methods), in line with an earlier study of current-jetting34. We do observe negative magnetoresistance from 7.5 to 25 T which may be related to the chiral anomaly, although not all samples show this to the same degree and this effect may depend sensitively on sample alignment (see Supplementary Methods). Our focus is not on the low-field regime but instead on the QL where the predicted behavior of the chiral anomaly is qualitatively distinct20.

In the QL, where transport along the z direction is determined by the n = 0 LL of the Weyl fermions alone, the conductivity has the form

$$\sigma _{{\mathrm{zz}}} = N_{\mathrm{W}}\frac{{e^2v_{\mathrm{F}}}}{{4\pi \hbar l_B^2}}\tau _{{\mathrm{inter}}}\left( B \right),$$
(1)

where Nw is the number of Weyl points in the Brillouin zone, vF is the Fermi velocity, $$l_B = \sqrt {\left( {h/eB} \right)}$$ is the magnetic length, and τinter(B) is the field-dependent inter-nodal scattering time10,20. As shown below, the W1 Weyl nodes are most likely gapped in this field range (we use the notation of Arnold et al.29 for W1 and W2, opposite that of Lv et al.27). We therefore evaluate Equation 1 for the eight W2 Weyl nodes, using vF determined by the quantum oscillations (see Methods). At 27 T, where σzz is maximum, we calculate τinter= 6 ps. This is ~10 times longer than the intra-nodal scattering time that we calculate from the Dingle factor of the quantum oscillations (see Methods), confirming that the chiral 0th LLs carry current with a reduced scattering rate in the QL due to their physical separation and distinct chiralities.

As the magnetic field is increased TaAs shows relatively constant resistivity in the QL before 50 T (Fig. 1b), requiring that the scattering rate 1/τ increases roughly in proportion to B to cancel the factor of $$l_B^2$$ in Equation 1. Indeed, $$1/\tau \propto B$$, and the resulting field-independent conductivity, was predicted for the QL of Weyl semimetals assuming short-range impurity scattering, suggesting that this is the dominant scattering mechanism between the Weyl nodes in TaAs20. As pointed out by Spivak and Andreev20, short-range impurity scattering in the quantum limit of conventional semimetals leads to longitudinal magnetoresistance that increases strongly with magnetic field35,36. Thus our observation of relatively constant ρ zz up to seven times the QL is a unique property of Weyl fermions.

### Gap formation in the chiral n=0 Landau levels

Having established the QL transport properties of TaAs at intermediate fields we turn to higher fields where ρ zz increases rapidly above 50 T (Fig. 1b). While the 0th LLs of the Weyl nodes should naively remain metallic in the QL, the two-order-of-magnitude increase in ρ zz suggests the formation of a gap. Weyl nodes are topologically stable to small perturbations16, but this can be overcome by strong magnetic fields which mix the left and right-handed n = 0 LLs (Fig. 3a). If the chemical potential falls within the gap then thermal activation is required to promote charge carriers to the empty LL, resulting in a strong temperature dependence for ρ zz . As pointed out previously37,38,39 a gap in the 0th LLs should be observable when the inverse magnetic length $$1/l_B = 1/\sqrt {\left( {\hbar /eB} \right)}$$—which controls the momentum-space extent of 0th LL wavefunctions—becomes large compared to the momentum-space separation of the Weyl nodes. This mechanism implies that the left and right handed chiral quasiparticles lose their distinction when their wavefunctions overlap, allowing for increased scattering between them and the formation of a gap. This is the same mechanism proposed for the observed change in sign of the Hall coefficient in TaP39. The significant difference between TaAs and TaP is that one pair of Weyl nodes in TaP remains ungapped up to the highest measured fields, whereas all nodes are eventually gapped in TaAs. The high sensitivity of σzz to the opening of the gap in TaAs allows us to produce a highly accurate measure of Weyl node separation in the bulk material.

To quantify this picture we extract the experimental gap by fitting the temperature-dependent conductivity at different values of the magnetic field where the gap opens above 56 T (Fig. 3b). The data are well-described by the model $$\sigma _{zz} = \sigma _0 + \sigma _1e^{ - \frac{\Delta }{{k_{\mathrm{B}}T}}}$$, where σ0 is a field-dependent background, σ1 is the conductivity from the 0th LLs, and ∆ is the gap (plotted as ∆exp in Fig. 3c). This model assumes that the gap dominates the field-dependence of the transport and we keep the scattering rate fixed as function of field for simplicity. To relate this gap to the electronic structure of the Weyl nodes we developed a complete tight binding model of TaAs in a magnetic field, including all 24 Weyl nodes and Zeeman coupling (see Methods for a full description of the tight binding model and LL calculations). Fixing v F to the experimentally determined value, we find that δk=0.15π/a gives the correct slope of ∆ versus B at high field. The slope of ∆ versus B is a strong function of δk, and therefore provides a bulk-sensitive measurement of the Weyl node separation. Our extracted value of δk also fits the range of W2 node separation estimated by photoemission measurements (between 0.12π/a and 0.15π/a)9,27. The W1 nodes are estimated to be separated by at most δk = 0.04π/a, and are therefore gaped at much lower field (Fig. 3c).

It should be noted that perfect electron–hole compensation is required for the chemical potential to lie in the gap at high fields: excess of electrons (holes) will pin the chemical potential in the n = 0 (n = 0+) LL to maintain the net carrier concentration (Fig. 3a). Hall effect measurements indicate that compensation in TaAs may not be perfect, with an estimated 2 × 1017 cm−3 to 5 × 1018 cm−3 excess electrons28,29 that naively should remain metallic at high fields. In reality once the magnetic length is short enough that the left and right nodes overlap scattering will increase greatly and eventually localize any residual carriers. The estimated excess electron density in TaAs is ≈1012 carriers per cm2—within the range of a metal-insulator transition40,41. This may explain why we observe the gap in ρ zz at slightly higher fields than both our model and the model of Chan:2017 predicts. Note that we do not observe a similar increase in ρ xx above 50  T: the in-plane conductivity has an 80 meV (1000 K) cyclotron gap at 50 T, and in-plane transport in this regime is dominated by impurity assisted hopping between adjacent n = 0 LLs in real space36. Thus ρ xx should be weakly temperature dependent for $${\bf{B}}||{\bf{c}}$$, consistent with what we observe experimentally.

### Phase transition in the quantum limit

At 80  T, well into the gapped state, we observe a small decrease in ρ xx at the lowest temperatures (Fig. 4a)—possibly indicative of a phase transition. To investigate this possibility further we measured the sound velocity (v zz ) of TaAs using pulse echo ultrasound up to 95 T, providing a thermodynamic probe at the highest fields. Below 2.5 K and above 80  T we observe an increase in sound velocity accompanied by a strong increase in ultrasonic attenuation (Fig. 4b, c). The onset of these ultrasonic features coincides with the sharp decrease in ρ xx , and we take all three phenomena as evidence for a field-induced phase transition. Our numerical solutions of the tight binding model predict no further transitions as a function of magnetic field once the gap opens along k z , suggesting that this transition at 80  T is driven by interactions.

There is precedent for an interaction-driven phase transition in the quantum limit of a semimetal. Graphite, whose quantum limit is 75 T, undergoes a rich series of transitions above 30 T. These transitions are generally ascribed to excitonic or Peirels-type density-wave formation along the magnetic field direction42,43,44. This mechanism seems unlikely to occur in TaAs given that a gap has already opened along k z at 50 T. In the limit where the ultrasonic wavelength is much longer than the quasiparticle mean free path, attenuation is proportional to conductivity45. The fact that we observe attenuation to increase while σ zz decreases suggests that something other than conventional electronic attenuation is occurring above 80 T in TaAs. The enhanced attenuation could come from the interaction between ultrasound and order-parameter fluctuations, as it does in graphite where it peaks near each field-induced phase transition46. The unbounded growth in attenuation we observe suggests that either we have not yet reached the attenuation peak, or that a different attenuation mechanism is at work.

To speculate on the microscopic origin of the high-field phase it is helpful to consider which degree of freedom remains after the gap opens along k z . The in-plane cyclotron orbits of the quasiparticles shrink in size with increasing magnetic field, resulting in a large real-space degeneracy of the n = 0 LLs (or equivalently momentum-space overlap of the quasiparticle wavefunctions). This increases the Coulomb repulsion between quasiparticles as $$E_{\mathrm{C}} \propto e^2/l_B$$, ultimately leading to Wigner crystallization. Before this transition occurs, however, there are intermediate “mixed” or “microemulsion” phases with spatially phase-separated regions of Fermi liquid and bubbles of crystallized electrons47. Disorder pins these structures in real-space, resulting in a mesoscopically inhomogeneous system. This texture, with a length-scale comparable to the ultrasonic wavelength, is known to scatter ultrasound. Clearly this phase needs to be explored further before it can be identified: the frequency dependence of the attenuation would yield the length scale of the phase separation48 or the frequency-dependence of the order parameter fluctuations;49 the collective sliding motion of crystallized electrons can be revealed by nonlinear current-voltage measurements50.

## Discussion

The ideal Weyl semimetal contains only one pair of Weyl nodes, with large momentum space separation to make them robust against perturbations. Reality is more interesting: the W2 Weyl nodes in TaAs have a large separation compared to other materials7, but there are 12 pairs of nodes in total plus the trivial hole pockets. Using magnetic fields to drive Weyl semimetals into their QL provides a clear path forward for studying “pure” Weyl physics, taking advantage of the fact that the 0th LLs are field-independent. This greatly broadens the number of potential systems where Weyl fermions can be accessed: as long as the magnetic field is strong enough the chemical potential will always move to the chiral 0th Weyl LLs in the quantum limit, even if the zero-field Fermi surface encompasses both nodes. Given that magnetic fields also increase the Coulomb interaction between quasiparticles (which can be particularly weak in semimetals due to the high Fermi velocities), the QL is also a promising regime for finding new states of matter formed from Weyl fermions.

## Methods

### Sample preparation

Negative magnetoresistance has been reported for a large number of candidate Dirac and Weyl semimetals16. The observation of decreasing resistance with increasing magnetic field for $${\bf{J}}||{\bf{B}}$$ has been taken as evidence of the chiral anomaly. Some of these measurements have been disputed because of extraneous effects that can also lead to an apparent negative magnetoresistance. In particular for TaAs, the highly anisotropic nature of the magnetoresistance means that current injected into the sample through point-like contacts does not necessarily move uniformly through the sample. This effect has been known for quite some time (see page 43 of Pippard33), and recent numerical simulations explored this in the context of Weyl semimetals by Reis34. The result is that the potential drop across a pair of voltage contacts may not reflect the total current injected into the sample, depending on where the contacts are placed. This can be overcome by avoiding point-like contacts and by preparing samples with high aspect ratios with current injection taking place far from the voltage measurement. Even a standard precaution such as contacting the entire ends of a sample to inject current homogeneously does not necessarily produce consistent results since microscopic contact to the sample can still be point like or at the very least inhomogeneous.

To illustrate this point for TaAs we prepared the sample shown in Supplementary Fig. 1. The current path is highlighted in purple; the “good” voltage contacts in red; the “bad” voltage contacts in green. The corresponding resistances, or rather the voltage drop across the contacts divided by the nominal current, are shown in the two right hand panels for $${\bf{B}}\left\| {\bf{J}} \right\|{\bf{c}}$$: the top panel corresponds to the green-shaded contacts offset from the current path; the bottom panel corresponds to the red-shaded contacts that are closer to the current path. It is immediately clear that a sizable “negative magnetoresistance” is visible for the green contacts at all temperatures, similar in character to that reported as evidence for a chiral anomaly in TaAs by Zhang et al.30 and Huang et al.23. In our exaggerated geometry the voltage drops to near-zero at high fields: clearly an unphysical result. The apparent increase in this effect at higher temperatures is due to the increase in resistance and decrease in resistive anisotropy at higher temperatures: it takes more magnetic field to isolate the current from the green voltage contacts at higher temperatures because the resistive anisotropy is lower. The red contacts, on the other hand, show positive magnetoresistance that saturates at ~1 T—this is expected as ωcτ reaches 1 near this field value. We used an even more restricted current path geometry for the data presented in the main text, and a second sample with similar current path characteristics shows similar resistivity in Supplementary Fig. 3.

ρ zz shown in Fig. 1 of the main text show signs of saturation at the highest resistance values. This may be due to a parallel conduction channel on the surface of the sample. There are two possible sources of this conduction channel: the surface states which are known to be present on this material and which have been observed via ARPES;9 and/or an amorphous arsenic-depleted layer induced by the FIB26.

As can be clearly seen in the inset of Fig. 1 in the main text, the FIB prepared samples undergo a superconducting transition at low temperatures and low magnetic field. This is also a surface effect: it was recently shown that after FIB microstructuring the related material NbAs develops an amorphous arsenic-depleted surface layer, as arsenic is preferentially removed during the FIB process, leaving behind a niobium (or in our case tantalum) rich layer that is superconducting26. This superconductivity is fully suppressed above 2 T and 1.5 K, and thus does not affect the majority of the data presented.

Although the presence of a conducting surface layer changes the shapes of our resistivity curves at high fields, it does not alter any of the features from which we draw our main conclusions: the positions of the quantum limits for the various pockets, the onset of the resitivity increase in ρ zz near 50 T, the temperature dependence of the increase in ρ zz , the lack of a strong increase near 50 T in ρ xx , and the sharp downturn in ρ xx near 80 T below 4 K. The consistency of the ultrasound with the features in the transport provides further confirmation of the intrinsic nature of these features, as there was no FIB lithography performed on the ultrasound sample.

### Fits to the conductivity

We fit the conductivity σ zz  = (1/ρ zz ) at each field slice as a function of temperature to a simple mode: a background σ b plus an activated component

$$\sigma _{zz} = \sigma _b + \sigma _0e^{ - \alpha /T,}$$
(2)

where α is the gap in units of kelvin. These fits plus the field-dependent fit parameters are shown in Supplementary Fig. 2.

In order to fit the gap down to the lowest possible field we normalized the conductivity to its value at 47 T. Supplementary Fig. 2 demonstrates that while this procedure changes σ b and σ0 it leaves ∆ relatively unchanged. This is because the exponential factor dominates the fit, producing variability in σ b and σ0 that depend on how (which field value, for example) the normalization is carried out but leaving ∆ procedure-independent. While σ0 is a physical parameter representing background conductivity in TaAs, the extracted values of σ b and σ0 should not be taken as intrinsic.

### Quantum oscillation analysis

According to the detailed quantum oscillation studies performed by Arnold et al.29, TaAs contains three distinct types of Fermi surface: electron-like W1 Weyl surfaces, electron-like W2 Weyl surfaces, and hole-like trivial surfaces. We do not attempt to re-create their entire analysis here, but instead focus on the oscillations we observe in the $${\bf{J}}\left\| {\bf{B}} \right\|{\bf{c}}$$ transport geometry. A complete treatment of the oscillation frequencies requires a full angle dependence which we have not measured: the purpose of the following analysis is to obtain a reasonable estimate for the quasiparticle lifetime τ for the Weyl electrons and to provide further evidence that the ≈18 T oscillation frequency is indeed coming from the same hole pocket identified by Arnold et al.29.

The full $${\hat{\mathbf c}}$$-axis conductivity, including both background and oscillatory contributions, can be written at fixed angle as

$$\sigma _{zz} = \mathop {\sum}\limits_i \sigma _0^i\left( B \right)\left( {1 + \mathop {\sum}\limits_p A^{i,p}R_T^{i,p}R_D^{i,p}{\rm{cos}}\left( {2\pi p\frac{{F_i}}{B} + \phi _i} \right)} \right),$$
(3)

where the first sum is over the i distinct Fermi surfaces, the sum p is over the harmonics, $$\sigma _0^i\left( B \right)$$ are the field-dependent background conductivities, Ai,p are amplitude factors, F i are the oscillation frequencies (proportional to the Fermi surface cross sectional area), and ϕ i are phase factors51. The temperature factor is given by

$$R_T^{i,p} = \frac{{\frac{{2\pi ^2pk_BT}}{{\hbar \omega _c^i}}}}{{{\rm{sinh}}\left( {\frac{{2\pi ^2pk_BT}}{{\hbar \omega _c^i}}} \right),}}$$
(4)

where $$\omega _{\mathrm{c}}^i = eB{\mathrm{/}}m_i^ \ast$$ is the cyclotron frequency for effective mass $$m_i^ \ast$$. The Dingle factor is

$$R_D^{i,p} = e^{ - \frac{{\pi p}}{{\omega _{\mathrm{c}}^{i}\tau _i}}},$$
(5)

where τ i is the quasiparticle lifetime.

It is important to note that the background conductivities in eq:osc are multiplicative with the oscillation amplitudes, not additive. Thus if a piece of Fermi surface does not contribute to σ zz then its oscillations will not be visible in ρ zz , even if the quasiparticle lifetime for that surface is long. The W1 pockets are roughly elongated ellipses (see Fig. 3c of Arnold et al.29) with an aspect ratio of approximately 10:1 with the long axis oriented along the $${\hat{\mathbf c}}$$ direction. With an in-plane Fermi velocity of 3 × 105 m/s these surfaces dominate the ρ xx transport and produce strong Shubnikov-de Haas oscillations with $${\bf{B}}||{\bf{c}}$$. For the ρ zz transport, however, we expect their Fermi velocity along $${\hat{\mathbf c}}$$ to be reduced by the aspect ratio (a factor of 10, so v z  ≈ 3 × 104 m/s), and thus contribute very little to the ρ zz transport. The W2 Weyl surfaces, on the other hand, are roughly isotropic in aspect ratio and have v z  ≈ 1.6×105 m/s; the hole pocket has v z  ≈ 7×105 m/s. As conductivity is proportional to Fermi velocity squared, ρ zz should be dominated by the W2 electron and trivial hole pockets. Note that the oscillations from a particular pocket can still be small due to other factors: for example, the hole pockets for $$B||{\hat{\mathbf c}}$$ have a very unfavourable curvature factor that reduces their oscillation amplitude considerably (oscillation amplitudes scale as $$1/\sqrt {\partial ^2A/\partial \kappa ^2}$$, where A is the extremal area of the Fermi surface perpendicular to the magnetic field and κ is the momentum direction parallel to the magnetic field)51.

We forgo the usual Fourier analysis of our data which can be difficult when there are only a few oscillations52. We instead perform a full three-dimensional fit to eq:osc for our data at temperatures where superconductivity is suppressed (T = 2.6,4,10 and 20 K). We can account for most features in the data fitting with F w2  = 6.5 T, F H  = 17.2 T, $$m_{{\mathrm{W}}2}^ \ast = 0.1$$ me, $$m_{\mathrm{H}}^ \ast = 0.2$$ me, $$\tau _{{\mathrm{W}}2} = 0.6$$ ps, and $$\tau _{\mathrm{H}} = 0.4$$ ps Supplementary Fig. 5. Values of $$F_{{\mathrm{W}}2} = 5.5$$ T and $$m_{{\mathrm{W}}2}^ \ast = 0.105$$me were obtained by Arnold:2016 for the W2 Weyl pocket measuring magnetic torque with field along the $${\hat{\mathbf a}}$$ direction: given the relatively isotropic nature of this pocket we believe that we are observing the same piece of Fermi surface. Values of FH = 18.8 T, and τH=0.11 ps were obtained for the hole pocket by Arnold et al.29 , which again are in reasonable agreement with our own. Note that Arnold et al.29 were unable to obtain a mass for the hole pocket for $${\mathbf{B}}||{\mathbf{c}}$$ but predicted a mass of $$m_{\mathrm{H}}^ \ast = 0.17$$ me—very close to our experimentally determined $$m_{\mathrm{H}}^ \ast = 0.2$$ me for this pocket.

### Pulse echo ultrasound

Pulse echo ultrasound measures the speed of sound $$v$$ transmitted through a sample for a specified sound propagation direction and polarization. Elastic moduli can then be computed via the relation $$v = \sqrt {\frac{c}{\rho }}$$, although care must be taken when choosing the propagation and polarization directions if the relationship between $$v$$ and any particular $$c_{ij}$$ is to be kept simple (see Brugger et al.53). The measurement typically involves generating a short burst of ultrasound (of order a few hundred nanoseconds) via a piezoelectric transducer, and then recording the echoes with the same transducer as the sound travels back and forth between the faces of the sample parallel to the transducer face. The phase shift between each successive echo is related to the travel time (and hence sound velocity) of the sound pulse, and the amplitude decay with successive echoes is related to the ultrasonic attenuation.

Our implementation of pulse echo ultrasound was similar to Suslov et al54. with one notable exception: we forwent the mixing stage and directly recorded the full ultrasonic waveform with a 6 GS/s digitizer card (GaGe EON Express). This has two major advantages: it eliminates a large fraction of the electronics; and it allows the use of digital filtering and lock-in techniques that can be optimized and re-processed after the experiment is completed.

We generated 315 MHz longitudinal ultrasound with a 36° Y-cut LiNbO3 transducer affixed to a (0,0,1) face of a TaAs sample (Supplementary Fig. 6). As both the direction of propagation and the polarization are along $${\hat{\mathbf c}}$$, we probe purely the c33 elastic modulus in this tetragonal crystal53. The full field dependence at selected temperatures between 0.55 and 20 K is shown in Supplementary Fig. 6. The velocity v is calculated from the phase shift ϕ between successive echoes as

$$\Delta v = - 2\pi fl\frac{{\Delta \phi }}{{\phi ^2,}}$$
(6)

where f is the ultrasonic frequency and l is twice the sample length. The absolute phase ϕ can be obtained from the absolute spacing between successive echoes (a less accurate measure of the sound velocity, but the only way to get the absolute value) via v = 2πfl/ϕ . All plotted velocity shifts are normalized to the phase shift at B = 0: the absolute shift as a function of temperature was not obtained.

### Tight-binding model for TaAs

The primitive unit cell of TaAs consists of four sublattices, two Ta atoms and two As atoms, per primitive unit cell. The lattice structure has two mirror symmetries M x and M y , TRS, and is invariant under C4 rotation followed by a translation along the $${\hat{\mathbf z}}$$ axis. We consider a three-dimensional tight-binding model representing the TaAs lattice structure. Specifically, we consider one spinful orbital per lattice site, giving rise to eight bands in total.

Our tight binding model:

$$H = t\mathop {\sum}\limits_{\langle ij\rangle ,s} c_{is}^\dagger c_{js} + \mathop {\sum}\limits_i \Delta _ic_i^\dagger c_i + i\lambda \mathop {\sum}\limits_{\langle \langle ik\rangle \rangle ,s,s{\prime}} c_{is}^\dagger c_{ks{\prime}}\mathop {\sum}\limits_j [{\hat{\mathbf d}}_{ijk} \cdot {\mathbf{\sigma }}_{ss{\prime}}],$$
(7)

where t is the nearest neighbor hopping; ∆ is a staggered potential: ∆ i  = ±∆ depending on whether the lattice site contains Ta or As; λ is the amplitude of the spin orbit interaction between the next nearest neighbors; s = ↑, ↓ is the spin; and σ’s are the corresponding Pauli matrices. $${\hat{\mathbf d}}_{ijk}$$ is the unit vector in the direction of $${\mathbf{d}}_{ij} \times {\mathbf{d}}_{jk}$$, where j is an intermediate site between i and k. This Hamiltonian preserves time reversal symmetry.

TaAs has 24 Weyl points, of which 8 are in the kz = 0 plane (W1) and the rest 16 (W2) are at finite k z 7. The 8-fold (16-fold) multiplicity of the W1 (W2) Weyl points are guaranteed by model symmetries (lattice and TRS). More generally, the total number of Weyl points has to be 8n1+16n2, where n1 and n2 are the numbers of sets of W1 and W2 Weyl points, respectively. For example, we will show the minimal tight-binding model can give 8 Weyl points, while adding extra terms and anisotropy reproduces the 24 Weyl points.

A finite ∆ in Eq. 7 explicitly breaks inversion symmetry, thus is vital to the emergence of Weyl semimetals. On the other hand, in the limit of ∆→∞ the system has to be a trivial insulator. In the presence of TRS, a Weyl semimetal is guaranteed to exist between a topological insulator and a trivial insulator55. During the evolution, pairs of Weyl points are generated, transported across the momentum space, and re-annihilate in different pairs, giving rise to a change of the Z2 topological index. We note that the W2 Weyl points always change the topological index trivially by 2, and it is up to the W1 Weyl points to change the Z2 topological index by a nontrivial 1. This suggests that while the W1 Weyl points are essential and guaranteed to occur somewhere during the interpolation, the W2 Weyl points are less fundamental and involve more fine-tuning.

First, we show that the Hamiltonian in Eq. 7 can give a Weyl semimetal in certain parameter regions. For model parameters t = 1.0, ∆ = 0.7, and λ = 0.1, the dispersion around k=(0.7, 2.56, 0) is shown in Supplementary Fig. 7. The band touching at a single Weyl point and the nearby linear dispersion are clearly established. The locations of all the eight Weyl points within the momentum space are illustrated in Supplementary Fig. 8. Being at k z  = 0, these Weyl points belong to the W1 group. Interestingly, we observe the shift and annihilation of the Weyl nodes as we change the amplitude ∆ of inversion symmetry breaking. We also note that the dispersion between the pair of adjacent Weyl nodes resembles that observed in band-structure calculations (Supplementary Fig. 9)25.

A comparison can be made between the electronic structure of the minimum tight-binding model Supplementary Fig. 10 and that of the actual TaAs crystal from band-structure calculations (see for example Fig. 4c from ref.23). The minimum model in Eq. 7 is already capable of qualitatively resembling the ab-initio band structures near the chemical potential.

We now consider the fate of the Weyl physics in the presence of an external magnetic field. For simplicity, we first focus on the situation when the magnetic field is along the c axis: $${\mathbf{B}} = B_z{\hat{\mathbf z}}$$. Since kz remains a good quantum number, together with the quantization within the k x ky plane, gives rise to Landau bands. For an individual Weyl node, there exists a chiral Landau band, whose up-moving or down-moving is associated with the chirality of the original Weyl node. However, as discussed in other theoretical studies, this physics becomes challenged at large magnetic fields37,38.

Semi-classically, magnetic fields introduce finite resolution in k x ky momentum space, inversely proportional to the magnetic length. Therefore, at large enough B, the pair of separated Weyl nodes with opposite chirality becomes indistinguishable and magnetic breakdown makes tunneling between them non-negligible. This was addressed schematically in refs. 37,38. Here, we will study from the perspective of our tight-binding model in Eq. 7. We find an emergent gap whose amplitude and onset field are roughly consistent with those observed in experiments for Weyl node separation δk0.15π/a, which is at least a factor of 2 larger than DFT results23,25 yet can be attributed to a relatively small modification to the overall band structure. Since the separation between the pair of adjacent Weyl fermions are small, they are very sensitive to small shifts in bandstructure which may be beyond the resolution capability of the DFT calculations. We also note that the Weyl nodes separation δk0.15π/a is consistent with the position of the Weyl nodes in the surface Brillouin zone observed in ARPES experiments27. As most studies have found W2 Weyl fermions to have larger separations than the W1 Weyl fermions, the former should be more resilient and dominates the fate of the Weyl physics in large magnetic fields. Although the W2 Weyl nodes are away from the k z  = 0 plane, it only causes a constant shift to the zero of the k z good quantum number and does not change the following discussion. The k-space distance between different pairs of Weyl nodes is much greater than the intra-pair separation and quantum tunneling due to magnetic breakdown between different pairs can be neglected.

To begin with, we neglect the Zeeman effect, and the vector potential of the magnetic field is chosen as A=(0, Bx, 0). We also set Fermi velocities in the $$v_{\mathrm{F}}^x \sim v_{\mathrm{F}}^y \sim 291000m/s = 1.92\mathrm{eV} \cdot {\it{{\AA}}}$$ observed via quantum oscillation measurements. These, together with the Weyl node separation δk0.15π/a and $$a = 3.437{\it{{\AA}}}$$ in TaAs allows us to set proper units for our numerical results.

A typical Landau band dispersion is illustrated in Supplementary Fig. 11. Each band is four-fold degenerate, suggesting the tunneling between different pairs of Weyl nodes are still negligible even at a large magnetic field B z  = 145T. On the other hand, a gap is clearly observable between the n = 0 chiral Landau bands.

An important question is how does the size of this anti-crossing gap depends on the applied magnetic field. Intuitively, the gap should be exponentially suppressed at smaller field $$l_B\delta _k \gg 1$$, and scale linearly at larger magnetic field. Such behavior is clearly observed in Supplementary Fig. 12 and Supplementary Fig. 13, where the units of the data are interpreted for TaAs W2 Weyl nodes with separation δk0.15π/a and δk0.165π/a, respectively.

Now we consider another potentially important ingredient—the Zeeman effect—particularly important because semi-metals and semi-conductors often give rise to large g-factors. For example, the g-factor of TaP has been estimated to be between 2 and 2.9 for one set of orbits and 5.5 and 6.7 for a second set56. To incorporate this we add an additional contribution to the Hamiltonian:

$$H_{{\mathrm{Zeeman}}} = - \mu _{\mathrm{B}}gS^z$$
(8)

The results with and without the Zeeman effect are compared in Supplementary Fig. 14. The results suggest that though the impact of the Zeeman effect is relatively minor (Supplementary Fig. 14 upper panel), the quantitative change to the anti-crossing feature is visible especially for larger Weyl nodes separation δk that allows weaker gaps with a larger magnetic field threshold, see Supplementary Fig. 14 lower panel.

The influence of the Zeeman effect can be understood via its modifications to the band structure in the absence of the vector potentials, see Supplementary Fig. 15. In the presence of Zeeman energy, the Weyl nodes moves further apart, making quantum tunneling between the pair of Weyl nodes more difficult thus suppressing the anti-crossing gap size.

### Data Availability

The data that support the findings of this study are available from the corresponding author upon reasonable request.

Publisher's note: Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

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## Acknowledgements

This work was performed at the National High Magnetic Field Laboratory, which is supported by National Science Foundation Cooperative Agreement No. DMR-1157490 and the State of Florida. E.D.B, F.R., and R.D.M. acknowledge funding from the LANL LDRD DR20160085 ‘Topology and Strong Correlations’. B.J.R. acknowledges funding from LANL LDRD 20160616ECR ‘New States of Matter in Weyl Semimetals’. M.K.C. acknowledges funding from the U.S. Department of Energy Office of Basic Energy Sciences Science at 100 T program. P.J.M. is supported by the Max-Planck-Society and the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No. 715730). M.D.B. acknowledges studentship funding from EPSRC under Grant No. EP/I007002/1. Y.Z. was supported in part by NSF DMR-1308089 and E.A.K. was supported in part by National Science Foundation (Platform for the Accelerated Realization, Analysis, and Discovery of Interface Materials (PARADIM)) under Cooperative Agreement No. DMR-1539918.

## Author information

### Affiliations

1. #### Laboratory of Atomic and Solid State Physics, Cornell University, Ithaca, NY, 14853, USA

• B. J. Ramshaw
• , Yi Zhang
•  & Eun-Ah Kim
2. #### Los Alamos National Laboratory, Los Alamos, NM, 87545, USA

• B. J. Ramshaw
• , M. K. Chan
• , J. B. Betts
• , F. Balakirev
• , A. Migliori
• , N. J. Ghimire
• , E. D. Bauer
• , F. Ronning
•  & R. D. McDonald
3. #### Max-Planck-Institute for Chemical Physics of Solids, Dresden, 01187, Germany

• K. A. Modic
• , Philip J. W. Moll
•  & Maja D. Bachmann

5. #### Argonne National Laboratory, Argonne, IL, 60439, USA

• N. J. Ghimire

### Contributions

B.J.R. and R.D.M. conceived of the experiment, E.D.B., N.J.G., and F.R. grew the samples, B.J.R. and R.D.M. performed the focused ion beam lithography, B.J.R., J.B.B., A.M., F.B., and R.D.M. designed and built the experiment, B.J.R., K.A.M., P.J.W.M., M.B., M.K.C., J.B.B., and R.D.M. performed the experiment, B.J.R. and K.A.M. analyzed the data, Y.Z. and E.A.K. developed the tight-binding model and provided theoretical guidance, and B.J.R. wrote the manuscript with input from all coauthors.

### Competing interests

The authors declare no competing interests.

### Corresponding author

Correspondence to B. J. Ramshaw.


text:
## Exercises

library(GSE5859Subset)
data(GSE5859Subset)

And define the outcome and predictors. To make the problem more difficult, we will only consider autosomal genes:

y = factor(sampleInfo$group) X = t(geneExpression) out = which(geneAnnotation$CHR%in%c("chrX","chrY"))
X = X[,-out]

1. Use the createFold function in the caret package, set the seed to 1 set.seed(1) and create 10 folds.

Question: What is the 2nd entry in the fold 3?

2. We are going to use kNN. We are going to consider a smaller set of predictors by using filtering genes using t-tests. Specifically, we will perform a t-test and select the $m$ genes with the smallest p-values.

Let $m=8$ and $k=5$ and train kNN by leaving out the second fold idx[[2]]. How many mistakes do we make on the test set? Remember it is indispensable that you perform the t-test on the training data.

3. Now run through all 5 folds. What is our error rate?

4. Now we are going to select the best values of $k$ and $m$. Use the expand grid function to try out the following values:

 ms=2^c(1:11)
ks=seq(1,9,2)
params = expand.grid(k=ks,m=ms)

Now use apply or a for-loop to obtain error rates for each of these pairs of parameters. Which pair of parameters minimizes the error rate?

5. Repeat exercise 4, but now perform the t-test filtering before the cross validation. Note how this biases the entire result and gives us much lower estimated error rates.

6. Repeat exercise 3, but now, instead of sampleInfo$group , use  y = factor(as.numeric(format( sampleInfo$date, "%m")=="06"))

What is the minimum error rate now?

We achieve much lower error rates when predicting date than when predicting the group. Because group is confounded with date, it is very possible that these predictors have no information about group and that our lower 0.5 error rates are due to the confounding with date. We will learn more about this in the batch effect chapter.

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# Magnetism formula...

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I am looking for a formula that I can plug different parameters into to control how other objects react to a single object.

I want to be able to tweak the values to make the object either pull closer to the main object or push away based on some variables.

Anyone done this or have any info on it?

Would Bullet or Box2D have anything for this?

Thanks

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If you consider point sources for the gravity, magnetic force etc. you would just apply a force on affected objects pointing towards/away from the source. The force should diminish quadratically with the distance to the source.

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You were really vague, so this is a simple solution based on your minimal stated requirements...

 bool attract = false; Vector2f d = objectA.position() - objectB.position(); Vector2f force = d.normal() / d.magnitude(); if (!attract) { force = -force; } // ... now apply the force to whatever object...

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not sure if that will work for what I need. Hell I am not sure how to totally explain it.

Best I can say is this

I have the player as ObjA
and ObjB-ObjZ are enemies

Now I want ObjB-ObjZ to be randomly moving around the screen but I want to use the formula to always pull these enemies towards the player but they need to vary due to some variables which I figured something like a magnetic attraction some are stronger and weaker.... does this make more sense?

Thanks!

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Is the strength of the attraction due to a property of the object or affected by the distance between the object and the player?

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I would like it to be both....

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So I'm assuming you don't really want to simulate actual magnetism, but want a formula for moving objects towards and away from each other based on some parameters.

I'm assuming you want the attractive force to be weaker with distance, so that means you'll want to divide by the distance. You can also divide by the squared distance to get more of a gravity effect. You'll probably want some constant factor for the strength of the forces, so multiply by that.

So now we have F = G / r^2, with F being the force, G the constant factor and r the distance between the objects. Adding in any other object property is as simple as multiplying/adding it with/to either the numerator or the denominator. If you're unsure if you should multiply/divide/add/subtract plot out some graphs or ask away.

Is that what you were looking for? Edited by Mussi

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I'm assuming you want the attractive force to be weaker with distance, so that means you'll want to divide by the distance. You can also divide by the squared distance to get more of a gravity effect. You'll probably want some constant factor for the strength of the forces, so multiply by that.[/quote]
Remember that dividing by distance or distance squared is numerically unstable if the two objects are too close to each other, you want to introduce a minimal radius to prevent this (make the two objects collide with each other when they get too close, I guess).

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I will try this out soon and report back... Thanks

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FYI, the actual equation I think you a re looking for is Coulomb's Law. It's like the gravity equation but for the attraction or repulsion of point charges. Basically, like the gravity equation but with polarity. Edited by jwezorek

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# 2D Artwork and 3D Modeling for Game Artists- P2

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## 2D Artwork and 3D Modeling for Game Artists- P2

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## Nội dung Text: 2D Artwork and 3D Modeling for Game Artists- P2

1. 24 2. Getting Ready to Model: Concept Art The Final Objective Having the two object models in mind, I want you to pause a minute and think about why I chose them. The RF-9 plasma gun is a great start. Although it’s no ordi- nary object, it’s not too difficult to model, and for now will have no moving parts. This is my favorite type of object to model and texture, because it’s generally quick to develop and has what I would consider an easy texture skin. Beveled, futuristic metal is fun and looks really cool, so I think you’ll enjoy it—in fact, that will be your first modeling project to get your feet wet. Y The slogre model, on the other hand, will be by far the most complex. The slogre FL will consist of only one skin mesh, but will have an internal skeletal structure (known in 3D Studio Max as a biped object) that will be used to drive the mesh defor- AM mation. That is, as the bones in the biped object move around, the vertices in the mesh will follow. On top of that, you’ll be weighting the mesh (adjusting the behav- ior of the mesh around the bones) and skinning. Lastly, dummy nodes must be placed all over the slogre to signify locations for the character to mount weapons, TE backpacks, point-of-view cameras, and the like. The model itself, being organic, will also be the most time consuming, so we’ll save that for last. Summary Developing a complex 3D game model is definitely a time-consuming process that must be well-planned in order for your model to be successful and presentable in a gaming environment. The development can be broken down into several basic steps, beginning with an initial concept sketch to provoke modeling ideas (which leads to creating the model itself in a 3D modeling program), followed by U-V mapping, texturing, possibly applying a bones system to deform the mesh, and finally outputting to a game engine of choice. Lars provided some great sketches that you can use as you create the models and textures for this game. Of course, you don’t have to stick like glues to the sketches (although it should be close); feel free, by all means, to make up your own models as you go. The techniques I’ll show you—from modeling, to U-Ving, to skinning and animation—will still apply. The next step in the development process is creating the actual object meshes, and in this part of the book, I’ve broken down the creation of the plasma gun and slogre mesh objects into their own chapters (Chapters 3 and 4, respectively). Using the modeling techniques I will describe, you should be able to make just about anything! ® Team-Fly lease purchase PDF Split-Merge on www.verypdf.com to remove this watermark.
2. CHAPTER 3 Modeling the RF-9 Plasma Gun with trueSpace 9 se purchase PDF Split-Merge on www.verypdf.com to remove this watermark.
3. 26 3. Modeling the RF-9 Plasma Gun with trueSpace 6 I n the previous chapter, where I introduced you to the logical structure of creat- ing game assets, I envisioned and generated (with the help of my sketch art col- league, Lars) a draft of a cool weapon that I’ll now show you how to create in 3D. In this chapter you will ■ Set up the trueSpace 6 environment in preparation for game modeling. ■ Logically plan out the modeling attributes for the RF-9 plasma gun. ■ Build the RF-9 step-by-step using primitives and point-editing techniques. ■ Optimize the RF-9 mesh and check for errors. ■ Export the model. An Overview You’ve fleshed out the concept for the RF-9 plasma gun and generated some detailed sketches. The next step is mesh creation, as indicated by the workflow depicted in Figure 3.1. To give you a quick review, Figure 3.2 shows the RF-9 plasma gun sketch that you’ll be using to model the plasma gun; you probably remember from Chapter 2, “Getting Ready to Model: Concept Art,” that this sketch was generated by me and my colleague Lars Ricaldi. Figure 3.1 The next step in compound-asset development: mesh creation. lease purchase PDF Split-Merge on www.verypdf.com to remove this watermark.
4. An Overview 27 Figure 3.2 The RF-9 you’ll be modeling in this chapter (sketch courtesy Lars Ricaldi). NOTE The sections that follow explain how to modify the trueSpace 6 environment for modeling, as well as other items you should consider before you begin. When you’re finished, If you’re already familiar with trueSpace and want you should end up with to jump right into the modeling, go ahead and skip something like the mesh to the section titled “Modeling the RF-9.” shown in Figure 3.3. Figure 3.3 The completed plasma-gun mesh. se purchase PDF Split-Merge on www.verypdf.com to remove this watermark.
5. 28 3. Modeling the RF-9 Plasma Gun with trueSpace 6 Setting Up the trueSpace 6 Environment In case you have not yet installed any version of trueSpace on your computer, I’ve provided a demo of trueSpace 6 on the CD-ROM that accompanies this book. Install it as you would any other program, and then copy the file G-LoK.tsc (as well as truespace.key and keylist.txt if you want to adhere to my keyboard short- cuts) from the CD-ROM to the \trueSpace6\ folder of your program’s installation directory. After the G-LoK.tsc file is copied, it’s time to load this custom modeling NOTE configuration. To do so, click on the G-LoK.tsc is an interface-configuration Configuration Library button, right- file that will set up your modeling envi- click in the library’s blank space, ronment my way, displaying three and choose Import. Then browse to orthogonal views (Left, Front, and Top) G-LoK.tsc file and click OK. You as well as a background Perspective should end up with a configuration view. Over the years I’ve found that this that looks like the one shown in is a fairly optimal way to model, but by Figure 3.4. G-LoK, by the way, is my all means, you should arrange the envi- game artist ‘handle’, so if you ever ronment to your liking. see art with my “GLK” logo, you know it’s yours truly. The next few sections explain other settings that help with your modeling environment. Changing the World and Object Units Generally speaking, one meter in the trueSpace modeling environment equals one meter in the world of the video game you’re creating. (It’s a good idea to use the metric system because most game engines are based on it.) To ensure that your modeling environment is set to use the metric system, do the following: 1. Right-click the Object button (with the white arrow) to open the Object Info panel (see Figure 3.5). lease purchase PDF Split-Merge on www.verypdf.com to remove this watermark.
6. Setting Up the trueSpace 6 Environment 29 Figure 3.4 Changing the model- ing-interface configu- ration by importing the G-LoK.tsc file. Configuration Library button Figure 3.5 Setting the modeling units in the Object Info panel. The Object button se purchase PDF Split-Merge on www.verypdf.com to remove this watermark.
7. 30 3. Modeling the RF-9 Plasma Gun with trueSpace 6 2. Click the red triangle in the upper-right corner of Object Info TIP panel to expand it. Rather than closing the Object Info 3. Set the World field to Meters. panel, it’s a good idea to move it over to the corner of the screen; 4. Set the Object field to Meters. that way, you can reference your polygon count as you model. Setting the Dynamic Rendering Mode trueSpace (and most other modeling programs) allows you to apply various settings to the video mode of your modeling environment, such as wireframe, solid, trans- parent, and so on. I find it easiest to create mesh objects in Transparent Wireframe mode, which means your models are see-through, and that both the edges and vertices of the model are displayed at the same time. To switch to this mode, do the following: 1. Click on the Display Options button in the bottom-right portion of the screen (see Figure 3.6). Figure 3.6 Setting the dynamic render- ing modes. The Display Options: DirectX button The Draw Objects as Transparent Outline button lease purchase PDF Split-Merge on www.verypdf.com to remove this watermark.
8. Modeling Considerations 31 2. Select either DirectX or OpenGL mode. (One mode might outperform the other NOTE depending on your video card, so check The rendering options described your video-card manufacturer’s docu- here will apply only to the active mentation for more information.) window.To activate a window, left- click on it. 3. Select the Draw Objects tool, and press and hold down your mouse button. Then, choose the Draw Objects as Transparent Outline button. Texture Resolution If you apply bitmaps to any object in your dynamically rendered world, you’ll need to crank up the dynamic texture resolution—otherwise, your textures will appear pixelated. Do this by right-clicking the Draw Objects tool (or by clicking File, Display Options), and setting the Txt Res option to 512×512. Keeping the Point Edit Tools Handy Much of the modeling you’ll be doing is based on point editing—that is, building or modifying your objects at the vertex (point) and face level. I like to keep the Point Edit tools right next to the Eye Rotate and Eye Move tools, at the middle-right of the screen, to make them easily accessible. NOTE To make a copy of these tools, press and The Point Edit tools will be visi- hold the Ctrl key as you drag the Point Edit ble only when an object is pre- tools to the desired area (see Figure 3.7). If sent and you’re in Point Edit you click once on the tool’s anchor bar (just mode (obtained simply by right- to its left), it will expand the tool list and clicking on a mesh object). anchor it to that area. Modeling Considerations If you saw the movie Final Fantasy, you were probably struck by the incredible detail of the characters, weapons, environments, and so on. That photorealistic detail was the product of the extraordinarily high polygon meshes used to make the models used in the film; indeed, a typical character model’s face alone had well over se purchase PDF Split-Merge on www.verypdf.com to remove this watermark.
9. 32 3. Modeling the RF-9 Plasma Gun with trueSpace 6 Figure 3.7 Locating and anchoring the Point Edit tools. The Point Edit button 50,000 polygons. Sadly, however, if you were to use such high polygon meshes for your game, the player’s computer would come to a screeching halt trying to render all of the detail. In fact, what you saw in Final Fantasy was the result of countless hours of post-production rendering on very powerful computers (most likely in a rendering array, with dozens of computers linked together, sharing the rendering process). Models for games are different from models for production, such as cover art, tele- vision, and movies. That’s because games have dynamic rendering environments; that is, as a player moves around in 3D space, all 3D mesh objects are rendered to screen at least 30 times per second. That means the player’s computer’s CPU and graphics processors must constantly transform the game world and render it at the same time—which in turn limits the number of polygons your models may con- tain. Models with high levels of polygonal detail may look better, but will be so slow to render on a player’s computer as to make them unusable. Put simply, models for games must be created to accommodate the average computing power of home computers on the market. These days, that equates to designing your models to work with computers in the Pentium IV and V range, at about 2.5 to 3.0 GHz. That means rather than creating character models with 50,000–100,000 polygons, as seen in the film Final Fantasy, you’ll need to create lease purchase PDF Split-Merge on www.verypdf.com to remove this watermark.
10. Modeling the RF-9 33 character models with polygons in the neighborhood of 2,000 to 5,000, and TIP weapons possessing only 500–2,000 poly- One way to avoid high poly counts is to gons. This need for a low polygon apply textures to low-poly-count models (poly) count will deeply affect the way for a similar effect. For instance, the you model; with every step you take to RF-9 has a hose-like item running the shape your object, you’ll work to mini- length of the action, but modeling a mize the count. hose would require hundreds of poly- gons. Instead, you can use a simple In addition to considering poly count, curved cylinder, and later apply a tex- you’ll also want to think about texture ture map that features an image of the mapping as you model. By making nice bumps in a hose to that area. If you can seams in your models in hidden areas, fake something with a 2D map, then it you’ll make the process of unwrapping might not be necessary to have high the U-V texture coordinates much eas- poly counts for certain areas. ier. For details on unwrapping U-Vs, see Part II, “Unwrapping U-Vs with DeepUV.” Modeling the RF-9 Creating a model of the RF-9 plasma gun will be quick and fairly simple; for overall good looks, you’ll rely more on texturing the weapon than creating a highly detailed mesh. Modeling the RF-9 is, in this case, essentially a seven-step process: 1. Plan the model’s dimensions and poly count, and build reference plane. 2. Build the muzzle. 3. Build the barrel. 4. Build the grip. 5. Build the hoops and hose. 6. Optimize and triangulate. 7. Export the model for texturing. In the sections that follow, I’ll show you how to use trueSpace, which features one of the best modeling interfaces on Earth, to model the RF-9 plasma gun using the steps outlined here. Of course, you can use any modeling program you wish, including 3D Studio Max (a demo of which is included on the CD-ROM); the mod- eling techniques I’ll show you can be ported to other programs. It’s up to you to know how those programs and their tools work, however. se purchase PDF Split-Merge on www.verypdf.com to remove this watermark.
11. 34 3. Modeling the RF-9 Plasma Gun with trueSpace 6 NOTE In the event you need a primer on using trueSpace, I’ve included on this book’s CD-ROM a tutorial covering trueSpace 4. I focused the tutorial on trueSpace 4, rather than trueSpace 6, because version 4 is clean and considerably less complicated than version 6, but uses the same basic modeling environment. Once you have a handle on using version 4, it’s less likely you’ll be confused and overwhelmed by the plethora of advanced modeling tools in version 6. Y FL AM Step 1: Planning the Model’s Dimensions and Poly Count, TE and Creating a Reference Plane Before you start dropping objects all over your scene, it’s a good idea to plan your model and set up your environment so that you can avoid the most frustrating mis- take that modelers make all the time: getting halfway finished with your model and having to scrap it all or backtrack because you didn’t plan ahead. Following are a few things to consider. The RF-9: Pea Shooter or @-Kicker? I mentioned in Chapter 2 that the slogre stands at about four meters (13 feet) tall. Given that the slogre is more than twice the size of an average human male, the RF-9 can be big and heavy, despite the fact that, as illustrated in the sketches you saw in Chapter 2, the slogre carries it in one hand. I figure a beast that possesses the size and strength of a slogre can handle a weapon that’s roughly two meters in length—half of his height—with the height from the gun’s strap hoop to the bot- tom of the handle being about one meter (refer to Figure 3.2). Knowing the dimensions of the weapon will sure come in handy as you proceed with creating the model! Target Polygon Count Until your computer hardware lets you make objects suitable for The Matrix, you’ll have to devise a target polygon count for your model. To give you a framework to ® Team-Fly lease purchase PDF Split-Merge on www.verypdf.com to remove this watermark.
12. Modeling the RF-9 35 Levels of Detail When you build a game, you’ll typically need several versions of the same weapon model, each with different levels of detail (LOD). One, which will have very high resolution, will be seen only by the player as he holds his own weapon (because the player will be able to see the model up close, a higher level of detail is required); one or several less- detailed versions will feature a lower polygon count, and will be seen being held by other players.These polygon counts may also vary with distance. For information on creating LODs, see Part IV, “Preparing Assets for Games with 3D Studio Max.” work from, 3D FPS games from the late 1990s had weapons that hovered NOTE around 300 polygons, while more There aren’t really any rules to model- recent games feature weapons in the ing; some techniques, such as point 700-polygon range. Keeping with this editing, are more efficient, producing linear growth, you can safely target fewer polygons. Other techniques, such your weapon’s poly count to be as Boolean operations with primitives around 1000. That’s pretty good and NURBS, accelerate the process. detail, allowing you to include more Of course, accelerating the modeling 3D and less texture. Of course, with process may require you to clean up any unnecessary polygons at the end. good texturing, an expert modeler can See Chapter 4 for details on advanced keep the poly count well below that, modeling with NURBS (non-uniform but for the sake of expediency, let’s rational b-spline) objects. not worry about that just yet. The Reference Plane Unless you’re making models on-the-fly, which will happen occasionally, you’ll need to reference a sketch or picture as you model. You can do this in one of several ways, such as taping a hard copy to the edge of your monitor, flipping back and forth between trueSpace and another program that houses the image, or—my personal preference—creating a reference plane (a 2D plane you create in trueSpace that has the actual sketch painted on it). se purchase PDF Split-Merge on www.verypdf.com to remove this watermark.
13. 36 3. Modeling the RF-9 Plasma Gun with trueSpace 6 To set up a reference plane, do the following: 1. Add a plane primitive to the scene (the primitives are found among the libraries at the bottom-left of the screen). If you’re not familiar with per- forming simple object operations in trueSpace, such as adding and manipu- lating primitives, please review the trueSpace 4 tutorial located on this book’s CD-ROM. 2. Right-click the Object tool (the white arrow at the bottom of the screen) to open the Object Info screen. 3. Scale the primitive to two meters by one meter by entering the dimensions in the Object Info screen. You’ll see a Size field in this screen; just enter these values for X and Y (length and width, respectively). 4. Enable the Grid Snap tool. This is the icon with a blue colored grid located at the bottom of the screen. 5. Using the Object Rotate tool (X on your keyboard), rotate the plane 90 degrees so it runs the length of the RF-9. This operation is best done in an orthogonal view, such as Left, and right-clicking and dragging until the plane is rotated so it stands upright in your scene. 6. Use the Move tool (Z on your keyboard) to move the plane up so the bottom is flush with the scene’s reference grid. 7. Use the Material Editor in conjunction with the Paint Face tool to paint the face of the plane with the image in the file RF-9 Plasma Gun.jpg, located in the Chapter 3 Data section on the CD-ROM (see Figure 3.8). Because the default perspective space of trueSpace is huge, you’ll need to use the Eye Move tool to reposition your view as I have in Figure 3.8. Notice that the grid’s units are one meter square; your plane should be proportionate to it. (If you’re a little confused up to this point, just load the step1.scn file on the CD-ROM.) TIP I’ve saved the individual modeling steps as trueSpace .scn files in the Chapter 3 Data section on the CD-ROM in case you get confused. lease purchase PDF Split-Merge on www.verypdf.com to remove this watermark.
14. Modeling the RF-9 37 Figure 3.8 Painting the face of a properly scaled and rotated reference plane with the RF-9 Plasma Gun.jpg image. Step 2: Build the Muzzle Now that you’ve determined the model’s dimensions and established a poly count, it’s time to start building the model. The part of the RF-9 plasma gun that’s easiest to model is the muzzle. Notice how the muzzle is really just a large, egg-shaped sphere primitive with two adjoined cylinders on the top sides. This is the resonating chamber, where the charged energy pellet enters a plasma-injection chamber and gets superheated in a fraction of a millisecond, before annihilating a nearby targeted object. NOTE The modeling techniques I use for the rest of this chapter can be applied by anyone using trueSpace version 4.0 and later. In the next chapter, however, when you model the slogre and other objects, you’ll use some of the new tools included with trueSpace 6. se purchase PDF Split-Merge on www.verypdf.com to remove this watermark.
15. 38 3. Modeling the RF-9 Plasma Gun with trueSpace 6 To build the muzzle, do the following: 1. Add a 12-segmented sphere primi- TIP tive to the scene. You can set the To open the basic Primitives panel, segments manually by right- press 6. If you use trueSpace version clicking any of the primitives, 5.x or higher, you can also open the or you can use the Magic Ring. Primitives panel by clicking on the (I don’t use the Magic Ring, and Primitives Library button, located in the vertical toolbar in the bot- have disabled it in the Preferences tom-left portion of the screen.This dialog box.) contains a more extensive and help- 2. Rotate the sphere 90 degrees. ful list of primitives. 3. Using the Scale tool (C on your keyboard), scale, elongate, and move the sphere so it matches the one in the reference plane (see Figure 3.9). Again, this is best done using the orthogonal views (such as Left, Front, and Top) to help with the scaling and positioning. 4. Right-click the sphere to enter Point Edit mode. 5. Right-click the Select Using Rectangle tool and enable the Backside option. The Backside option will allow you to make selections on your model that include polygons nearest your point of view and those hidden behind it, or on the model’s backside. 6. With the Select Using Rectangle tool, select the first four segments of the sphere as shown in Figure 3.10. This selection operation is best done in an orthogonal window; I did it in Front view. Figure 3.9 Add a 12-segment sphere primitive, rotate it 90 degrees, and elongate it to match the sketch. lease purchase PDF Split-Merge on www.verypdf.com to remove this watermark.
16. Modeling the RF-9 39 Figure 3.10 Use the Select Using Rectangle tool to select the first four segments of the sphere. 7. Click the Erase Vertices tool, located within the Point Edit tools, to delete TIP the selection from the sphere, as shown If you’re having trouble finding in Figure 3.11. (From now on, when I the various tools mentioned want you to remove something, I’ll sim- throughout this chapter, check ply say “Select, and delete.” That’s your out trueSpace tutorial on the cue to repeat CD-ROM that accompanies this step.) this book. 8. Press Ctrl+C to make a copy of the sliced sphere. In trueSpace 6, your cursor will change to an arrow with a plus sign beneath it—just left-click once in the scene to add the copy, and immediately right-click to exit the copy mode. 9. Scale and position the copy as shown in Figure 3.12; it should line up with the reference plane’s sketch. Figure 3.11 Use the Erase Vertices tool to delete the selection. se purchase PDF Split-Merge on www.verypdf.com to remove this watermark.
17. 40 3. Modeling the RF-9 Plasma Gun with trueSpace 6 Figure 3.12 Make a copy of the existing sphere object, and then scale and position it as shown. 10. Right-click the copy to enter Point Edit mode. 11. Select and delete the first two segments of the new copy, as shown in Figure 3.13. 12. Using the Point Edit: Faces tool, select the front face of the copied sphere. 13. Use the Sweep tool to extrude the object; extend the extrusion until it’s even with the lower sphere by dragging it to the left in an orthogonal view (see Figure 3.14). It will help to lock the X axis for this operation to keep it straight. 14. Use the Object Union tool to join both sphere objects together. If both objects’ faces were aligned evenly before the union, you should get one solid object with only one front face (see Figure 3.15). 15. Select the front face of the muzzle; you’ll extrude this to form the muzzle’s flare (see Figure 3.16). Figure 3.13 Select the first two segments of the top sphere object and delete them. Figure 3.14 Sweep the front face of the sphere and continue the extrusion until it matches the face of the lower sphere. lease purchase PDF Split-Merge on www.verypdf.com to remove this watermark.
18. Modeling the RF-9 41 Figure 3.15 Use the Object Union tool to fuse both sphere objects together. Figure 3.16 Select the front face of the muzzle. 16. Use the Sweep tool to extrude the face of the muzzle. 17. Move and scale the extrusion in an orthogonal view so that it matches the reference plane sketch, as shown in Figure 3.17. (Remember to use the Point Edit: Move, Rotate, and Scale tools to do this, and not the Object Move, Rotate, and Scale tools. These tools are located in the Point Edit tools that pop up when in Point Edit mode.) Figure 3.17 Sweep the front face of the muzzle, then move and scale the extrusion to match the sketch. se purchase PDF Split-Merge on www.verypdf.com to remove this watermark.
19. 42 3. Modeling the RF-9 Plasma Gun with trueSpace 6 18. Sweep the face again, and then move, rotate, and scale the new extrusion to match the sketch (see Figure 3.18). 19. The top six edges of the muzzle’s end must be tapered down slightly. Use the Point Edit: Edges tool to select them, and scale and/or move them down a bit, as shown in Figure 3.19. 20. To hollow out the muzzle, you could use the Object Subtract tool to subtract a cylinder from the inside of the muzzle—but doing so wastes polygons, because by subtracting a cylinder the hollowed result is in the shape of a cylinder, including the back face that constitutes about 10 polygons. Because players generally won’t be peering deep into the end of the RF-9, you can get away with subtracting a cone primitive instead—this will save you those 10 polygons since a cone is shaped to a single vertexed tip. To do so, add, rotate, scale, and position an eight-sided cone primitive to the scene, and position it as shown in Figure 3.20. 21. Select the muzzle object, click on the Object Subtraction tool, and then click the cone to hollow everything out (see Figure 3.21). Figure 3.18 Sweep the face again and move, rotate, and scale it to match the sketch. Figure 3.19 Select the top six edges near the muzzle’s end and scale/move them down. lease purchase PDF Split-Merge on www.verypdf.com to remove this watermark.
20. Modeling the RF-9 43 Figure 3.20 Add an eight-sided cone primitive. Scale and position it as shown. Figure 3.21 Use the Object Subtraction tool to remove the cone from the muzzle. Voilá! A muzzle! Obviously, you could do a lot more to CAUTION make it more closely resem- Occasionally, when performing Boolean opera- ble the sketch, but I want to tions such as Object Union and Object keep things simple for now. Subtraction, trueSpace will encounter an error Interestingly, however, the because the objects are positioned in such a way twenty or so steps it took to that the operation is not doable. If this happens, create the muzzle constitute reposition the target object slightly and try again. more than 80 percent of the You may also need to slightly adjust the Identity modeling operations value in the tool’s Options panel for things to required to create most work smoothly.This value is what trueSpace uses objects; that means you’re to determine what vertices between the two well on your way to creating objects should be included or subtracted from niftier, more complicated the resulting Boolean operation. models. se purchase PDF Split-Merge on www.verypdf.com to remove this watermark.


text:
Journal article

### Relatively Hyperbolic Groups with Rapid Decay Property

Abstract:
We prove that a finitely generated group $G$ hyperbolic relative to the collection of finitely generated subgroups H_1,..., H_m has the Rapid Decay property if and only if each H_i, i=1,2,..., m, has the Rapid Decay property.
Publication status:
Published

### Authors

Journal:
INTERNATIONAL MATHEMATICS RESEARCH NOTICES
Issue:
19
Pages:
1181-1194
Publication date:
2004-05-26
ISSN:
1073-7928
Source identifiers:
16589
Language:
English
Keywords:
Pubs id:
pubs:16589
UUID:
uuid:06bc5713-63cd-45ac-8b12-fb1ab5239059
Local pid:
pubs:16589
Deposit date:
2012-12-19


text:
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# 20 marbles were pulled out of a bag of only white marbles, p

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20 marbles were pulled out of a bag of only white marbles, p  [#permalink]

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Updated on: 02 Jun 2013, 03:31
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76% (01:53) correct 24% (02:24) wrong based on 162 sessions

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20 marbles were pulled out of a bag of only white marbles, painted black, and then put back in. Then, another 20 marbles were pulled out, of which 1 was black, afterwhich they were all returned to the bag. If the percentage of black marbles pulled out the second time represents their percentage in the bag, how many marbles in total does the bag currently hold?

A. 40
B. 200
C. 380
D. 400
E. 3200

Please explain confuesed how to form an equation for this problem...

Originally posted by fozzzy on 02 Jun 2013, 01:23.
Last edited by Bunuel on 02 Jun 2013, 03:31, edited 1 time in total.
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Re: 20 marbles were pulled out of a bag of only white marbles, p  [#permalink]

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02 Jun 2013, 01:36
fozzzy wrote:
20 marbles were pulled out of a bag of only white marbles, painted black, and then put back in. Then, another 20 marbles were pulled out, of which 1 was black, afterwhich they were all returned to the bag. If the percentage of black marbles pulled out the second time represents their percentage in the bag, how many marbles in total does the bag currently hold?

A 40
B 200
C 380
D 400
E 3200

From a bag of all white marbles, 20 become black.
The second time 1 black is pulled out, and $$\frac{1}{20}=5%$$ represents the percentage of black marble in the bag.
We know that $$5%=20 black$$ , 20 is the number painted the first time (the ONLY black marble).

So there are a total of $$20*20=400$$ marbles ($$20$$ black and $$380$$ white).
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Re: 20 marbles were pulled out of a bag of only white marbles, p  [#permalink]

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02 Jun 2013, 01:41
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1
fozzzy wrote:
20 marbles were pulled out of a bag of only white marbles, painted black, and then put back in. Then, another 20 marbles were pulled out, of which 1 was black, afterwhich they were all returned to the bag. If the percentage of black marbles pulled out the second time represents their percentage in the bag, how many marbles in total does the bag currently hold?

A 40
B 200
C 380
D 400
E 3200

Please explain confuesed how to form an equation for this problem...

Ok. Lets do it with the help of ratio. 20 Black marbles in a bag of white marbles after 1st draw.
Second time, out of 20 there is one black marble. Ratio of Total:Black = 20:1
For every 1 black marble there are total of 20 marbles then
for 20 Black marbles we will have 20*20 = 400 Marbles in total.
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Re: 20 marbles were pulled out of a bag of only white marbles, p  [#permalink]

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02 Jun 2013, 03:36
fozzzy wrote:
20 marbles were pulled out of a bag of only white marbles, painted black, and then put back in. Then, another 20 marbles were pulled out, of which 1 was black, afterwhich they were all returned to the bag. If the percentage of black marbles pulled out the second time represents their percentage in the bag, how many marbles in total does the bag currently hold?

A. 40
B. 200
C. 380
D. 400
E. 3200

Please explain confuesed how to form an equation for this problem...

We know that there are 20 black marbles in the bag and this number represent 1/20 th of the number of all marbles in the bag, thus there are total of 20*20=400 marbles.

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Re: 20 marbles were pulled out of a bag of only white marbles, p  [#permalink]

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20 Nov 2013, 19:44
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$$\frac{# of black marbles}{total in 2nd set pulled out of bag} =\frac{1}{20}=\frac{20}{x} = \frac{total black}{total marbles in bag}$$ --- cross multiply

$$20*20=1*x$$

$$x=400$$
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Re: 20 marbles were pulled out of a bag of only white marbles, p  [#permalink]

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09 Aug 2018, 07:37
When 20 marbles were pulled out we get 1 black that implies probability 1/20. As they mentioned this percentage ratio is valid for whole bag so if we have to get 20 black marbles we should make upper figure equal to "20".
1*20/20*20=20/400, there are 400 marbles in bag.

Re: 20 marbles were pulled out of a bag of only white marbles, p &nbs [#permalink] 09 Aug 2018, 07:37
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# Angular Displacement

What Is Angular Displacement?

The motion of the body along a circular path is known as the rotational motion. The displacement done through such a type of motion is different from the displacement done on linear motion; it is usually in the form of an angle, and hence it is known as angular displacement. Below we discuss angular displacement along with the formula, let us define it with the help of examples.

While moving in a circular path, the angle made by the body is known as angular displacement. Before discussing further on the topic, we have to understand what rotational motion is. The motion ceases to become a particle when a rigid body is rotating about its axis. Due to the motion in the circular path, change in the acceleration and velocity can happen at any time. Rotational motion is defined as the motion of the rigid bodies that will remain constant throughout the rotation over a fixed axis.

Angular Displacement Definition –

To define angular displacement, let's suppose a body is moving in a circular motion, the angle made by a body from its point of rest at any point in rotational motion is known as angular displacement.

The shortest angle between the initial and the final position for an object in a circular motion around a fixed point is known as the angular displacement; it is considered a vector quantity.

Unit of Angular Displacement

The unit of angular displacement is Radian or Degrees. 360o is equal to two pi radians. Meter is the SI unit for displacement. Since angular displacement involves the curvilinear motion, the SI unit for angular displacement is Degrees or Radian.

Angular Displacement Formula

The Formula of Angular Displacement

For a point the angular displacement is as follows:

Angular displacement = θf−θi

The displacement will have both magnitudes as well as the direction. The circular arrow pointing from the initial position to the final position will indicate the direction. It can either be clockwise or anticlockwise in direction.

It can be measured by using a simple formula. The formula is:

θ=s/r, where,

θ is angular displacement,

s is the distance traveled by the body, and

r is the radius of the circle along which it is moving.

Simplistically, the distance traveled by an object around the circumference of a circle divided by its radius will be its displacement.

Derivation

The angular displacement can be calculated by the below formula when the value of initial velocity, acceleration of the object, and time are shared.

θ= wt + 1/2αt^2

Where,

θ- Angular displacement of the object

t- Time

α- angular acceleration

Now, formula for Angular Linear is

In Rotational, kinetic equation is

ω=ω0+αt,

Δθ=ω0t+1/2αt^2,

ω^2=ω02+2αθ,

In translational, kinetic equation is

v=u+at

or s=ut+1/2at^2

v2 = vo2 + 2ax

Where,

ω- Initial angular velocity

Considering an object having a linear motion with initial acceleration a and velocity u, when time t and the final velocity of the object is with the total displacement s then,

a = dv/dt

The change in velocity

The rate which can be written as

dv = a dt

Integrating both the sides, we get,

∫uvdv=a∫0tdt

v – u = at

Also,

a=dv/dt

a=dxdv/dtdx

As we know v=dx/dt, we can write,

a=vdv/dx

v dv=a dx

The equation we get after integrating both sides

∫uvvdv=a∫0sdx

V^2–u^2=2as

From the equation -1 into the equation – 2 by substituting the value of u, we get

V^2−(v−at)^2=2as

2vat–a^2t^2=2as

By dividing the equation of both sides by 2a, we have

s=vt–1/2at^2

And at last, the value of v being substituted by u, we will get.

s=ut+1/2at^2

1. How is Angular Displacement Measured?

Ans In radians and degrees, the angular displacement will be measured. Considering radians as it provides a very simple relationship between the distances traveled around the circle and the distance r from the center.

θ = s/r

For example, if the body rotates around a circle of radius r at 360o, then the angular displacement is found by the distance traveled around the circumference. This is found by 2πr, divided by radius  θ = 2πr/r. In simplistic terms, it can be denoted as θ=2π, where 1 revolution is 2π radians.

When a particle travels from point P to point Q over δt, as it does in the picture to the left, the radius of the circle goes through a change in angle Δθ = θ₂ - θ₁ which equals the angular displacement.

2. How Angular Displacement is a Vector?

Ans Angular displacement is not a vector.

A quantity that has direction and magnitude also follows the rules of vector algebra; it can be known as a vector.

Although angular displacement may appear to be a quantity directly represented in one direction, you can specify directions to specify conventions such as the rule of thumb of the right hand. Magnitude means the amount of rotation.

However, all the rules of vector algebra are not obeyed by it, particularly the commutative law: u+v = v+u for the vector and u and v. Pick a 3D object like a cell phone and the screen facing towards you upright. Rotate it clockwise, so the screen still faces you but is still horizontal (landscape adjustment). This time rotate again so that the screen faces the ceiling. This is caused due to the addition of 2 angular displacements. When the order of rotation is switched, the final orientation will be different. Different results violating commutability will be achieved by angular displacement in a different order.

3. What are the Examples of Angular Displacement?

Angular displacement example–A dancer angular rotation will be 360o if they are dancing around a pole in a full rotation. If the rotation is half, the displacement will be 1800.

This will be a vector quantity that means it contains both magnitude and direction.

For example, a displacement of 360o done clockwise is very different compared to anticlockwise.

4. Why doesn’t Angular Displacement in a Simple Pendulum go above 4 Radians?

You perhaps mean 4°. 4 radians is more than 1800 and makes sense for a pendulum angular displacement. The reason angle should be small (40 being a sensible through random limit), meaning you need a small angle to be able to approximate the force with an elastic one, which ends up by harmonic oscillations. If the displacement becomes too large, the harmonic approximation will no longer exist, and you will get a complex undesirable system that will make it difficult and strange to perform. For example, the oscillation period is not independent of amplitude.