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# Fermi surface topology and signature of surface Dirac nodes in LaBi

## Abstract

Novel topological state of matter is one of the rapidly growing fields in condensed matter physics research in recent times. While these materials are fascinating from the aspect of fundamental physics of relativistic particles, their exotic transport properties are equally compelling due to the potential technological applications. Extreme magnetoresistance and ultrahigh carrier mobility are two such major hallmarks of topological materials and often used as primary criteria for identifying new compounds belonging to this class. Recently, LaBi has emerged as a new system, which exhibits the above mentioned properties. However, the topological nature of its band structure remains unresolved. Here, using the magnetotransport and magnetization measurements, we have probed the bulk and surface states of LaBi. Similar to earlier reports, extremely large magnetoresistance and high carrier mobility have been observed with compensated electron and hole density. The Fermi surface properties have been analyzed from both Shubnikov-de Haas and de Haas-van Alphen oscillation techniques. In the magnetization measurement, a prominent paramagnetic singularity has been observed, which demonstrates the non-trivial nature of the surface states in LaBi. Our study unambiguously confirms that LaBi is a three-dimensional topological insulator with possible linear dispersion in the gapped bulk band structure.

## Introduction

Materials with large magnetoresistance (MR) have always attracted attention due to their possible applications in magnetic valves, sensors and memory device technology1,2,3. While the realization of giant and colossal MR have been proved to be major steps forward in this field4,5,6, the recent discovery of extreme MR (XMR) in topological materials7,8,9, has further stimulated interests in such systems. Here, the XMR is believed to appear, when the applied magnetic field lifts the symmetry protection of the spin-locked states7. The coexistence of topologically distinct surface and bulk states is the unique characteristic feature of electronic band structure in topological insulators (TIs) and semimetals (TSMs). In TIs, the time reversal symmetry protected metallic surface state is accompanied by insulating bulk state, whereas TSMs host linear band crossings in their bulk band structure. Furthermore, the discovery of three-dimensional Dirac/Weyl fermions as quasi particle excitations in the bulk band structure10, 11 and observation of relativistic phenomena such as Adler-Bell-Jackiw chiral anomaly (signature of Weyl fermions)12, 13, have made the TSMs as one of the fascinating topics in condensed matter physics research. Therefore, theoretical prediction and experimental realization of new materials belonging to different topological classes, are going on in full swing.

Recently, from the first-principles calculations14, the members of the family of rare earth monopnictides LaX (X = N, P, As, Sb, Bi) have been predicted to be potential candidates for TSM and TI. Several magnetotransport studies have demonstrated XMR in both LaSb and LaBi15,16,17,18. However, little information is available on the topological nature of their band structure. In fact, for both the systems, the XMR has been attributed to the compensated electron and hole density rather than any non-trivial electronic band topology16, 19. On the other hand, it is always experimentally challenging to probe the topological nature of the surface state, which can settle the debate once and for all. LaBi is the end member of the LaX family with strongest spin-orbit coupling (SOC) and shares many similarities with LaSb. With increasing atomic number from N to Bi, the increasing SOC is predicted to open a gap in the bulk at the band crossing points and thus can induce a topological transition from semimetal to insulating phase14. While contradicting ARPES reports exist in literature claiming both trivial and non-trivial topological states in LaSb20, 21, recent ARPES studies have demonstrated multiple Dirac nodes at the surface of LaBi21,22,23,24. Therefore, detailed investigations are required, which would not only resolve the topological nature of LaBi but also can shed some light on the electronic band structure of other members of this family.

Herein, we have used the magnetotransport and magnetization measurements to probe the nature of the bulk and surface electronic band structure of LaBi. In several earlier studies on LaBi, the Fermi surface properties have been analyzed from the Shubnikov-de Haas (SdH) oscillation, observed in the MR. Despite high sample quality, the number of Fermi pockets reported by different groups do not converge16, 17, 25. This discrepancy in reported results may be the inherent problem of this technique. As the SdH oscillation is sensitive to quantum interference effects or noise from the electrical contacts, it may not be possible to separate some oscillation components, especially when the amplitude of oscillation is small. Therefore, to complement the SdH oscillation results, we have also analyzed the de Haas-van Alphen (dHvA) oscillation in the magnetization measurement. This technique is not only free from interference effects, but the appearance of much sharper and readily distinguishable oscillation peaks down to very low magnetic field suggests that dHvA is a much better probe to study the Fermi surface properties than the SdH technique, which often requires high magnetic fields. The calculated Berry phase from both the oscillations confirms the presence of three-dimensional (3D) Dirac fermions in LaBi. In addition to the quantum oscillations, a robust paramagnetic singularity in magnetic susceptibility is observed at low field, which originates from the electronic states near 2D surface Dirac nodes of a TI. Thus the present work reveals the topological insulating phase in LaBi together with linear dispersion in the gapped bulk states. To the best of our knowledge, this is the first magnetization report on LaBi, which explores the topological nature of its band structure and confirms the theoretical prediction as well as the recent ARPES results. The non-trivial surface state of a topological material can also be explored through Aharonov-Bohm oscillation. However, this technique requires large surface-to-volume ratio to minimize the bulk dominated transport and hence can only be observed in samples with reduced dimension such as nanowire, nanoribbon etc26, 27.

## Results

### Sample characterization

Figure 1(a) shows the X-ray diffraction (XRD) pattern for the LaBi single crystal along the (1 0 0) plane. Presence of very sharp (h 0 0) peaks in diffraction pattern confirms the high crystalline nature of the grown crystals. LaBi crystallizes in a rock salt type structure with space group Fm $$\bar{3}$$ m. From the XRD pattern, the lattice parameter is deduced to be a = 6.5703(2) Å, which is consistent with the earlier reports17, 25. In the inset, a typical single crystal is shown with different crystallographic directions, along which measurements have been performed. The a-axis is along [h 0 0] direction. The XRD has been performed on several single crystals from the same batch. While the peak intensity has been seen to vary, no impurity phase was detected within the experimental resolution. In Fig. 1(b), the high-resolution transmission electron microscopy (HRTEM) image clearly illustrates the high quality crystalline nature of the LaBi samples. From the Fourier filtered HRTEM image [Fig. 1(c)], an interlayer spacing d = 3.32(4) Å is determined along (2 0 0) plane, which agrees with that calculated from the XRD data (d = a/2). The energy-dispersive X-ray (EDX) spectroscopy data, shown in Fig. 1(d), confirm almost perfect stoichiometry [La:Bi = 1:1.08] and the absence of any impurity in the LaBi crystals. The maximum relative error in the calculated atomic ratio is ~5%. The copper and carbon signals in the spectrum are coming from the carbon coated copper grid on which the samples were mounted for HRTEM.

### Temperature dependence of resistivity and extreme magnetoresistance

The resistivity of LaBi single crystal has been measured with current along c-axis and magnetic field along a-axis. As shown in Fig. 2(a), the zero-field resistivity (ρ xx ) is metallic in nature and decreases monotonically as the temperature decreases. At 2 K, ρ xx becomes as small as ~130 nΩ cm, yielding a large residual resistivity ratio [RRR = ρ xx (300 K)/ρ xx (2 K)] ~350. At low temperature the resistivity obeys a power-law behavior, ρ xx (T) = ρ 0 + AT n with n ~ 3. When the magnetic field is applied, ρ xx is seen to increase and a metal-semiconductor like crossover starts to appear along with resistivity plateau. Such kind of resistivity behavior is a generic feature for the TSMs8, 9, 28 and can be explained from Kohler scaling analysis29, 30. From the ∂ρ xx /∂T curves [Fig. 2(a) inset], two characteristic temperatures can be clearly identified. The crossover temperature (T m ), where the slope of ρ xx (T) curve changes its sign and T i , the point of inflection. Slightly below this temperature (T i ), resistivity starts to saturate. T m increases monotonically with field and follows a T m  (B − B 0)1/ν type relation with ν ~ 2 above a critical field B 0 ~ 1 T, which is similar to that observed in several compensated semimetals29, 31. On the other hand, T i is almost independent of the applied magnetic field strength.

With the same current and magnetic field configurations, an extremely large and non-saturating MR ~4.4 × 104% is obtained at 2 K and 9 T [Fig. 2(b)], which is consistent with the earlier reports on LaBi16, 17 and comparable to that reported for several TSM candidates29, 32. However, with increasing temperature the value of MR decreases rapidly and becomes only ~6% at 300 K and 9 T. The fitting in the inset of Fig. 2(b) suggests that the MR curve obeys a power law, MR B m with m ~ 1.7. The observed power-law behavior is close to the parabolic field dependence (MR B 2), which is expected for compensated semimetallic systems from classical two-band theory33.

### Shubnikov-de Haas oscillation and Hall measurement

In Fig. 2(c), the oscillatory component of the resistivity [Δρ xx (B)] is plotted after subtracting a fifth order polynomial background from the experimental data. To decide the appropriate polynomial order, we have subtracted higher as well as lower order polynomial background. However, no visible change has been noticed above fifth order, which replicate the background most appropriately. Also, by comparing the results obtained after polynomial background and smooth background subtraction, we did not find any difference. The fast Fourier transform (FFT) analysis of Δρ xx (B) in the inset of Fig. 2(c), reveals an oscillation frequency 273(5) T, which is consistent with the earlier report by Tafti et al.25 On the other hand, both Sun et al.16 and Kumar et al.17 have reported the presence of an additional higher frequency component in LaBi. However, from our SdH oscillation data, we are not able to resolve any additional frequency, which may be due to very weak nature of oscillation for the higher frequency component within the measured field and temperature range. On the other hand, we have clearly resolved two Fermi pockets on the same piece of sample from our de Haas-van Alphen (dHvA) oscillation data, to be discussed later. Using the Onsager relation, F = (ϕ 0/2π 2)A F , where ϕ 0 is the single magnetic flux quantum, we have calculated the Fermi surface cross-section (A F ) ~2.6(1) × 10−2Å −2 perpendicular to the a-axis. The corresponding Fermi momentum (k F ) and Fermi velocity (v F ) are calculated to be 9.1(2) × 10−2Å −1 and 5.6(1) × 105 m/s, respectively. The oscillation amplitude can be fitted using the thermal damping factor of Lifshitz-Kosevich (L-K) formula, R T  = (2π 2 k B T/β)/sinh(2π 2 k B T/β), where β = eħB/m*. From the fitting parameters, we have calculated the cyclotron mass (m *) ~ 0.19(1)m 0 for the charge carriers, where m 0 is the free electron rest mass. The field-induced damping of the oscillation amplitude can be fitted well with R D  = exp(−2π 2 k B m * T D /ħeB) and the Dingle temperature (T D ) is determined to be 11.3(2) K. Using this Dingle temperature, we have also calculated the quantum mobility μ q [=(/2πk B m * T D )] ~ 103 cm2 V−1 s−1, which can provide a rough estimate on the mobility of the charge carriers. However, the quantum mobility is always expected to be lower than the classical Drude mobility34.

The Berry phase acquired by the charge carriers can provide more information about the nature of electronic band structure in LaBi. According to Lifshitz-Onsager rule, a closed orbit in magnetic field is quantized as35,

$${A}_{F}\frac{\hslash }{eB}=2\pi (n+\frac{1}{2}-\beta -\delta )=2\pi (n+\gamma -\delta ),$$
(1)

where 2πβ is the Berry phase. δ is a phase shift, which has a value 0 and ±1/8 for 2D and 3D band structures, respectively. The Berry phase is 0 for the conventional metals with parabolic band dispersion and π for the Dirac/Weyl type electronic system with linear band dispersion. For quantum oscillation with single frequency, the parameter $$\gamma -\delta =\frac{1}{2}-\beta -\delta$$, can be extracted from the x-axis intercept in the Landau level index plot and will remain within the range −1/8 to +1/8 for 3D Dirac fermions35. In Fig. 2(d), the Landau level fan diagram is plotted from the SdH oscillation, assigning maxima as integers and minima positions as half-integers. The obtained intercept [~−0.13(2)] is close to the theoretical range for 3D Dirac fermions but far from that expected for a parabolic dispersion (intercept ~0.5), which clearly demonstrates the presence of 3D Dirac fermions in LaBi.

The density and classical Drude mobility of the charge carriers can be determined from the Hall resistivity data, as has been shown in Fig. 3. While the Hall resistivity (ρ yx ) is positive and almost linear at high temperature, it becomes slightly non-linear and changes sign at low temperature. Therefore, both electron and hole pockets are present in LaBi. In the inset of Fig. 3, the Hall resistivity is fitted using classical two-band model36,

$${\rho }_{yx}=\frac{B}{|e|}\frac{({n}_{h}{\mu }_{h}^{2}-{n}_{e}{\mu }_{e}^{2})+({n}_{h}-{n}_{e}){({\mu }_{h}{\mu }_{e})}^{2}{B}^{2}}{{({n}_{h}{\mu }_{h}+{n}_{e}{\mu }_{e})}^{2}+{({n}_{h}-{n}_{e})}^{2}{({\mu }_{h}{\mu }_{e})}^{2}{B}^{2}},$$
(2)

where n h (n e ) and μ h (μ e ) are density and mobility of the holes (electrons), respectively. The obtained electron and hole densities ~2 × 1019 cm−3 and ~1.9 × 1019 cm−3, respectively, reveal that LaBi is a compensated semimetal. From the fitting parameters, the mobilities of the carriers are determined to be ~1.28 × 104 cm2 V−1 s−1 and ~1.26 × 104 cm2 V−1 s−1 for electrons and holes, respectively. The obtained carrier mobility is quite large and comparable to several topological Dirac and Weyl semimetals28, 34, 37.

### de Haas-van Alphen oscillation and Fermi surface properties

In Fig. 4(a), the results of magnetization measurements have been shown with magnetic field along a-axis. Similar to other TSMs, a diamagnetic behavior has been observed along with very clear dHvA oscillations. In contrast to the SdH oscillation, which can be seen only up to 8 K, dHvA can be easily resolved even at 15 K. At lowest measured temperature (2 K), weak but detectable oscillations have been observed down to 0.8 T, which corresponds to a magnetic length $${l}_{B}=\sqrt{\hslash /eB}\approx 29$$ nm. This large magnetic length indicates high quality of the LaBi single crystals. The oscillatory component (ΔM) has been extracted after subtracting a smooth background and plotted in Fig. 4(b). In the inset, the FFT spectrum is shown, which clearly reveals two oscillation frequencies 266(1) T and 598(2) T. The presence of more than one frequency component is also evident from the small but sharp oscillation peak adjacent to each higher amplitude peak in Fig. 4(b) [also see Fig. 4(d)]. The sharp peaks in the FFT spectrum indicate almost no error in the frequency, deduced using this technique. On the other hand, due to the relatively broad nature of the peak in FFT spectrum, the error in the frequency value is higher in SdH oscillation compared to dHvA. This is expected as larger number of oscillation peaks have been used for dHvA oscillation analysis compared to few cycles of oscillation obtained in SdH measurement. Furthermore, quantum oscillations are expected to appear in very clean samples, i.e., where the conductivity and hence mean free path is very high. Therefore, to obtain appropriate electrical signals for SdH oscillation, quite high current is needed. This can cause local heating and blur the finer details of the oscillation. However, dHvA technique is free from such drawback. From the Onsager relationship, the Fermi surface cross-sections have been calculated to be 2.53(1) × 10−2Å −2 and 5.70(2) × 10−2Å −2, for smaller and larger pockets, respectively. The temperature damping of the FFT amplitude is shown in Fig. 4(c) and has been fitted using L-K formula. From the fitting parameters, we have deduced effective mass 0.11(3)m 0 and 0.07(2)m 0 for 266(1) T and 598(2) T, respectively. Prominent dHvA oscillations have also been observed, when the magnetic field is applied along other two crystallographic axes. While the oscillation frequencies are same for a- and c-axis, both peaks in FFT spectrum slightly shift towards higher values [272(1) T and 602(2) T] for magnetic field along b-axis. As LaBi possesses cubic symmetry, which has also been confirmed from our X-ray diffraction and transmission electron microscopy measurements, it is expected that these three crystallographic axis should be equivalent. On the other hand, the hole pocket in LaBi has been seen to be highly anisotropic17. Therefore, a slight misalignment of the magnetic field and b-axis may result in this small deviation. The Fermi surface parameters calculated from both the quantum oscillation techniques, are summarized in Table 1. The small difference in the frequency values, obtained from these two techniques, may be partially due to a small misalignment of the magnetic field and or the higher error in the SdH frequency as we have discussed earlier. However, the difference in the number of Fermi pockets, probed in these two methods, can not be explained from such arguments. There are several reports on cyclotron resonance experimental and theoretical works for LaBi and isostructural compound LaSb, as these are quite old systems38,39,40,41. Moreover, dHvA oscillation has been studied in LaSb42. The cyclotron mass (varies between 0.13–0.33 m 0 for two branches) and extremal Fermi surface cross-section calculated in those studies along [1 0 0] direction are in agreement with that obtained from our quantum oscillation data.

As two frequencies are involved in the dHvA oscillation in LaBi, the Berry phase can not be extracted from the simple Landau level index plot. In such case, the experimental data can be described as a superposition of damped sine-function for each frequency component, which is given by the complete L-K formula43, 44,

$${\rm{\Delta }}M\propto -{B}^{\mathrm{1/2}}{R}_{T}{R}_{D}{R}_{S}sin[2\pi (\frac{F}{B}+\gamma -\delta )].$$
(3)

Here, R T and R D are the thermal and field damping factors, respectively, as discussed before and R S  = cos(πgm */m 0). Figure 4(d) shows the two-band L-K formula fitting with the experimental data. From the fitting parameters, the Berry phase has been calculated to be 2π(0.50(3) + δ) and 2π(0.46(2) + δ) for 266(1) T and 598(2) T frequency branches, respectively, which are close to the π Berry phase expected for the Dirac fermions. Small deviation in Berry phase value is quite common in these types of materials and can be associated with the Fermi surface anisotropy44, 45. The obtained Berry phase for both the Fermi pockets confirms the non-trivial nature of the electronic band structure in LaBi.

### Paramagnetic singularity in magnetization measurement

The main characteristic feature of a 3D TI is the gapless surface state with odd number of Dirac nodes, which are protected by time reversal symmetry. The low-energy physics of these metallic surface states can be mathematically formulated using a Dirac-type effective Hamiltonian46, H surf (k x , k y ) = ħv F (σ x k y  − σ y k x ), which leads to phenomenon like spin-momentum locking. Here, $$\overrightarrow{\sigma }$$ is the Pauli matrix. Hence, for a fixed translational momentum k, the spin has a fixed direction parallel to the sample surface46, 47. A corresponding helicity operator can be defined46, $$\hat{h}=\mathrm{(1/}k)\hat{z}.(\overrightarrow{k}\times \overrightarrow{\sigma })$$, which yields values +1 and −1 for the lower and upper Dirac cones of the surface band structure, respectively. Therefore, each Dirac cone possesses a particular helicity - upper Dirac cone with left-handed and lower Dirac cone with right-handed spin texture. On the other hand, the electron spins at the Dirac point do not have any preferable orientation and are free to align along the external magnetic field as long as the Dirac cone is not gapped. In the low-field region, these freely oriented spins are predicted to generate a paramagnetic singularity in the M(B) curve, which should be reflected as a cusp in χ(B) plot48. The spin-texture of the surface state for a 3D topological insulator is shown schematically in Fig. 5(a), both without and with magnetic field. Figure 5(b) shows the low-field region of the M(B) curves for LaBi, when the magnetic field is applied along c-axis. While the magnetization data predominantly demonstrate diamagnetic behavior throughout the measured field range, a clear paramagnetic contribution is observed in the vicinity of zero field. In Fig. 5(c), the corresponding susceptibility plots are shown at different representative temperatures. As predicted theoretically48, robust cusps have been observed at B = 0, which persist up to room temperature. Similar low-field paramagnetic singularity, which originates due to the topologically non-trivial surface electronic states, has been previously reported in 3D TIs such as Bi2Se3, Bi2Te3, Sb2Te3 and Bi1.5Sb0.5Te1.7Se1.3 and narrow gap TI ZrTe5 with gapped bulk Dirac cones48,49,50,51. With both temperature and chemical potential set to zero, the susceptibility is given by48,

$$\chi (B)\cong {\chi }_{0}+\frac{{\mu }_{0}}{4{\pi }^{2}}\frac{x}{L}[\frac{{(g{\mu }_{B})}^{2}}{\hslash {v}_{F}}{\rm{\Lambda }}-\frac{\mathrm{2(}g{\mu }_{B}{)}^{3}}{{\hslash }^{2}{v}_{F}^{2}}|B|],$$
(4)

where χ 0, Λ, μ B and g are the background contribution, effective size of the momentum space contributing to the singular part of the total free energy, Bohr magneton and Landeg-factor, respectively. L is the thickness of the measured crystal and x is the fraction of the surface contributing, which is usually extremely small. As Eq. 4 suggests, χ(B) should decrease linearly with field, which is indeed observed in LaBi [Fig. 5(c) inset]. The measurements were performed on several crystals from the same batch. While the cusp width varies slightly with the dimensions, as expected from Eq. 4, the general behavior remains unaltered. The results of the measurement on standard diamagnetic (Bi) and paramagnetic (Pd) samples [Fig. 5(d)] do not show any low-field singularity. Moreover, such a unique temperature independent behavior of χ for LaBi can not be explained assuming a small magnetic impurity in diamagnetic host and is distinct from the magnetic response of dilute magnetic semiconductors. So, the low-field magnetic response must be a intrinsic property of the measured crystals. Therefore, LaBi certainly hosts topological Dirac fermions in the surface state, which is in accordance with the recent ARPES reports21,22,23,24.

## Discussion and Conclusions

LaBi has emerged lately as another new system with XMR and large carrier mobility. While a number of studies have been focused on the magnetotransport properties of LaBi, the topology of the electronic band structure remains substantially unexplored. Furthermore, the XMR is believed to appear in the electron-hole resonance regime due to the compensated semimetallic nature of LaBi. On the other hand, from the recent ARPES measurements, multiple Dirac cones have been observed at the surface of LaBi, which support the picture of a topological insulator, as also predicted by band structure calculation. In this report, we have combined the magnetotransport and magnetization measurements to probe the bulk Fermi surface along with the surface electronic states of LaBi. Similar to earlier reports, a non-saturating XMR (~4.40 × 104 %) and field-induced resistivity turn-on behavior have been observed. The measured non-linear Hall resistivity reveals the presence of two types of carriers with high carrier mobility and almost perfectly compensated densities. The Fermi surface parameters have been calculated from both SdH and dHvA oscillations. Although both the techniques are widely used to study Fermi surface properties, our results clearly show the advantages of dHvA technique over the other. In the low-field region of the magnetization, for the first time in LaBi, a robust paramagnetic singularity has been detected, which persists up to room temperature. This behavior of magnetic susceptibility is the signature of the helical spin texture of a 3D TI surface state. On the other hand, the calculated Berry phase from both SdH and dHvA oscillations confirms the 3D Dirac fermionic nature of the charge carriers. These results can only be explained assuming a small gap in the bulk band structure due to the strong SOC and linearly dispersing bulk bands. Therefore, LaBi must be a three-dimensional topological insulator, which also hosts gapped Dirac cone in its bulk state. Thus from the present work we can comprehensively conclude the debate over the topological nature of the band structure in LaBi and confirm the results of ARPES and band structure calculations.

## Methods

High quality single crystals of LaBi were grown using indium flux16, 52. Elemental La (Alfa Aesar 99.9%), Bi (Alfa Aesar 99.999%) and In (Alfa Aesar 99.999%) were taken in an alumina crucible in the molar ratio 1:1:20. The crucible was then sealed in a quartz tube under vacuum. The quartz tube was heated to 1000 °C, kept at this temperature for 5 h and then cooled slowly (4 °C/h) to 700 °C. At this temperature, indium is decanted in a centrifuge. Several single crystals of typical dimension 2 × 1.8 × 1 mm were obtained. The X-ray diffraction (XRD) spectrum of the single crystals was obtained in a Rigaku X-ray diffractometer. HRTEM of the grown crystals was done in an FEI, TECNAI G2 F30, S-TWIN microscope operating at 300 kV and equipped with a GATAN Orius SC1000B CCD camera. EDX spectroscopy was performed using the same microscope with a high-angle annular dark-field scanning (HAADF) detector from Fischione (Model 3000). Transport measurements were performed in a 9 T Physical Property Measurement System (Quantum Design) by four-probe technique using the ac transport option. Magnetic measurements were done in a 7 T SQUID-VSM MPMS 3 (Quantum Design). Before doing the magnetic measurements on LaBi, empty sample holders were measured to ensure the absence of any contamination. The obtained data are more than two orders of magnitude smaller than that obtained with LaBi samples. The measurements were performed on several single crystals from the same batch, all producing similar results.

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## Acknowledgements

We acknowledge and thank A. Pariari, P. Dutta, S. Roy and A. Pal for their help during measurements and fruitful discussions.

## Author information

### Affiliations

1. #### Saha Institute of Nuclear Physics, HBNI, 1/AF Bidhannagar, Calcutta, 700 064, India

• , Biswarup Satpati
•  & Prabhat Mandal

### Contributions

P.M. designed research. R.S. prepared the crystals and performed experiments. B.S. performed HRTEM measurement. R.S. and P.M. analyzed data and wrote the paper. P.M. supervised the project.

### Competing Interests

The authors declare that they have no competing interests.

### Corresponding author

Correspondence to Prabhat Mandal.


text:
Formatted question description: https://leetcode.ca/all/1515.html

# 1515. Best Position for a Service Centre (Hard)

A delivery company wants to build a new service centre in a new city. The company knows the positions of all the customers in this city on a 2D-Map and wants to build the new centre in a position such that the sum of the euclidean distances to all customers is minimum.

Given an array positions where positions[i] = [xi, yi] is the position of the ith customer on the map, return the minimum sum of the euclidean distances to all customers.

In other words, you need to choose the position of the service centre [xcentre, ycentre] such that the following formula is minimized:

Answers within 10^-5 of the actual value will be accepted.

Example 1:

Input: positions = [[0,1],[1,0],[1,2],[2,1]]
Output: 4.00000
Explanation: As shown, you can see that choosing [xcentre, ycentre] = [1, 1] will make the distance to each customer = 1, the sum of all distances is 4 which is the minimum possible we can achieve.

Example 2:

Input: positions = [[1,1],[3,3]]
Output: 2.82843
Explanation: The minimum possible sum of distances = sqrt(2) + sqrt(2) = 2.82843

Example 3:

Input: positions = [[1,1]]
Output: 0.00000

Example 4:

Input: positions = [[1,1],[0,0],[2,0]]
Output: 2.73205
Explanation: At the first glance, you may think that locating the centre at [1, 0] will achieve the minimum sum, but locating it at [1, 0] will make the sum of distances = 3.
Try to locate the centre at [1.0, 0.5773502711] you will see that the sum of distances is 2.73205.
Be careful with the precision!

Example 5:

Input: positions = [[0,1],[3,2],[4,5],[7,6],[8,9],[11,1],[2,12]]
Output: 32.94036
Explanation: You can use [4.3460852395, 4.9813795505] as the position of the centre.

Constraints:

• 1 <= positions.length <= 50
• positions[i].length == 2
• 0 <= positions[i][0], positions[i][1] <= 100

Related Topics: Geometry

## Solution 1. Simulated Annealing

Starting from a random point (here I use (0, 0)). We move around in 4 directions with some initial step (I used 100).

If we can find smaller total distance, we move to that point.

Otherwise, we set step /= 2.

We keep this iteration until the step is smaller than 1e-6.

See more:

• https://stackoverflow.com/questions/12934213/how-to-find-out-geometric-median
• https://en.wikipedia.org/wiki/Geometric_median
// OJ: https://leetcode.com/problems/best-position-for-a-service-centre/

// Time: O(N * S) where S is the steps taken to reach the center
// Space: O(1)
class Solution {
double dist(vector<int> &a, vector<double> &b) {
return sqrt(pow(a[0] - b[0], 2) + pow(a[1] - b[1], 2));
}
double all(vector<vector<int>> &A, vector<double> &p) {
double ans = 0;
for (auto &a : A) ans += dist(a, p);
return ans;
}
const int dirs[4][2] = { {0,1},{0,-1},{-1,0},{1,0} };
public:
double getMinDistSum(vector<vector<int>>& A) {
double ans = DBL_MAX, step = 100, eps = 1e-6;
vector<double> p(2, 0);
while (step > eps) {
bool found = false;
for (auto &dir : dirs) {
vector<double> next = {p[0] + step * dir[0], p[1] + step * dir[1]};
double d = all(A, next);
if (d < ans) {
ans = d;
p = next;
found = true;
break;
}
}
if (!found) step /= 2;
}
return ans;
}
};

## Other Solutions

• https://leetcode.com/problems/best-position-for-a-service-centre/discuss/731577/C%2B%2B-with-picture-zoom-in

Java

class Solution {
public double getMinDistSum(int[][] positions) {
int length = positions.length;
if (length == 1)
return 0;
double xCentre = -1, yCentre = -1;
final double EPSILON = 1e-6;
double difference = 1;
while (difference > EPSILON) {
double xNumerator = 0, yNumerator = 0, denominator = 0;
for (int[] position : positions) {
double distance = euclideanDistance(xCentre, yCentre, position[0], position[1]);
if (distance != 0) {
xNumerator += position[0] / distance;
yNumerator += position[1] / distance;
denominator += 1 / distance;
}
}
if (denominator == 0)
break;
double xCentreNew = xNumerator / denominator, yCentreNew = yNumerator / denominator;
difference = euclideanDistance(xCentre, yCentre, xCentreNew, yCentreNew);
xCentre = xCentreNew;
yCentre = yCentreNew;
}
double distanceSum = 0;
for (int[] position : positions)
distanceSum += euclideanDistance(xCentre, yCentre, position[0], position[1]);
return distanceSum;
}

public double euclideanDistance(double x0, double y0, double x1, double y1) {
return Math.sqrt((x1 - x0) * (x1 - x0) + (y1 - y0) * (y1 - y0));
}
}

format: plain


text:
# LaTeX error trying to close comment environment

I'm trying to create a document in which answers could be shown or not shown depending on the inclusion of \includecomment{answer}. I decided to use the comment package because the examples I found here seemed really simple, but it is stumping me nonetheless. I also need for individuals to be able to select which questions/answers are relevant for their projects.

I'm trying to use the answer environment, but it doesn't seem to work within an if/then. Here is a little bit of the sample that I think should run. I know that \end{answer} needs to be on its own line. There are no blank spaces before or after it. The code will compile if (a) I comment out the \begin and \end{answer}, or if (b) I comment out the if/then lines before and after the item. If I try to compile with both, I get the following error:

Runaway argument?

! File ended while scanning use of \next.

< inserted text >

           \par

< * > example.tex

\documentclass{article}
\usepackage{ifthen}
\usepackage{comment}

\newcommand{\usetwo}{1} %Use Q2? 1 for yes, 2 for no.

\begin{document}

\begin{enumerate}
\item Question 1
\ifthenelse{\equal{\usetwo}{1}}{
\item Question 2\\
}%
{}% if not set to 1
\end{enumerate}

\end{document}

## migrated from stackoverflow.comOct 10 '16 at 16:49

This question came from our site for professional and enthusiast programmers.

• Welcome to TeX.SE! Your post was migrated here from Stack Overflow. Please register on this site, too, and make sure that both accounts are associated with each other (by using the same OpenID), otherwise you won't be able to comment on or accept answers or edit your question. – Werner Oct 10 '16 at 17:01

You cannot pass comment-like environments as arguments to other macros. Instead, use a \if construction like below:

\documentclass{article}

\usepackage{comment}

\newif\ifquestiontwo
\questiontwofalse% \questiontwotrue

\begin{document}

\begin{enumerate}
\item Question 1

\ifquestiontwo
\item Question 2

\fi
\end{enumerate}

\end{document}

You can switch between using Question 2 or not via \questiontwotrue/\questiontwofalse.


text:
# Combining two means and SDs of one group [duplicate]

I have the mean & SD of several different time points, I want to combine them into one mean and SD. Is it possible? Example;

At 1 hour; mean=5, SD=1, n=42

At 2 hours; mean=6, SD=3, n=42 and so on.

Any equation to do that?

Note: at some time points, I have different group numbers (n), due to missing one or two patients. I really appreciate your help, thanks!

## marked as duplicate by whuber♦Dec 30 '18 at 16:30

• The covariance or correlation coefficient is needed for combining the SDs. – user158565 Dec 30 '18 at 5:23
• And what about the mean? – Mohamed Gomaa Kamel Dec 30 '18 at 5:44
• You can combine means given you have sample size n if you think it is meaningful. – user158565 Dec 30 '18 at 5:46
• Thanks for your answer What is the formula? – Mohamed Gomaa Kamel Dec 30 '18 at 6:08
• $(N_1\bar X_1 + N_2\bar X_2 +...+N_k\bar X_k )/(N_1 + N_2+...,+N_k)$ – user158565 Dec 30 '18 at 6:35

Rules to pool mean and variance of two groups can be found in O'Neill (2014) (Result 1). The formulas are:

\begin{aligned} \bar{x}_\text{pooled} &= \frac{1}{n_1+n_2} \Bigg[ n_1 \bar{x}_1 + n_2 \bar{x}_2 \Bigg], \\[10pt] s_\text{pooled}^2 &= \frac{1}{n_1+n_2-1} \Bigg[ (n_1-1) s_1^2 + (n_2-1) s_2^2 + \frac{n_1 n_2}{n_1+n_2} (\bar{x}_1 - \bar{x}_2)^2 \Bigg]. \\[10pt] \end{aligned}

To pool a larger number of groups you can proceed recursively using these formulae.

Unless you truly believe there is a time difference-driven discrepancy, just pool them.

Imagine you are measuring the heights of students in 12th grade. Does it matter if one class comes at 1 PM and the next at 2 PM for measurement? No. You just deal with it as one sample.

On the other hand, if you are doing the measurements in Q1 (January) and Q2 (April), you should likely assume the kids grew. If that is the case, it is a different problem - and a harder one.

More info on what you are trying to find would help. But, based on your limited description, simply pool the data and calculate the mean and SD for the whole.

• Thanks for your kind comment, Yes, it will not affect (not a big difference in time) So, how to pool them, please? – Mohamed Gomaa Kamel Dec 30 '18 at 6:36
• Take the 84 data points and ignore the time. calculate the mean and SD for the 84 combined. Let me assume, for example, you want average weight and height (perhaps to calculate average BMI and its SD?) based on these two groups given weight and height. Treat them as one group and get the average weigh, and average height, over the 84 +/- 2 or 3 difference in count. – eSurfsnake Dec 30 '18 at 6:48


text:
## Advances in Differential Equations

### Existence of radial ground states for quasilinear elliptic equations

#### Abstract

Existence of nontrivial, nonnegative radial solutions of \newline quasilinear equations $-{\hbox{div}}(A(|{\nabla} u|) {\nabla} u)=f(u)$ in ${\mathbb R}^n$ is proved under general assumptions on the nonlinearity $f$ and the function $A$, without requiring homogeneity.

#### Article information

Source
Adv. Differential Equations, Volume 6, Number 8 (2001), 959-986.

Dates
First available in Project Euclid: 2 January 2013

Permanent link to this document


text:
# Idlechasing for warriors

(diff) ← Older revision | Latest revision (diff) | Newer revision → (diff)

## Idlechasing for Dummies

### Warrior 30x Commands

30x commands are the commands that give 30 xp for every gp you spend on them. Using only 30x commands will give twice the xp as using only 15x commands, which can mean the difference between making 200,000 xp per hour versus 100,000 per hour. Here's a list of your 30x commands:

1. Bandage
2. Bash
3. Beat
5. Bob
6. Chop
7. Crush
8. Feint
9. Fix
10. Guard
11. Hack
12. Iai
13. Impale
14. Judge
15. Kick
16. Leatherwork
17. Pierce
18. Punch
19. Riposte
20. Shove
21. Slash
22. Slice
23. Smash
24. Stab
25. Track
26. Trip
27. Vurdere
28. Warcry
29. Probably the rest of the combat specials too, if they haven't been tested

Warcry is special: the "preparation" part is 30x, while the second part is only 15x. So, for pure xp return, it's preferable to have your warcry after combat so that you don't do the second part (since it "dies unuttered on your lips" if you wait too long). For tm potential, though, you might want to go through with the warcry.

So that's at least 25 30x commands! The heavy trend toward combat may seem daunting for an idlechaser, but there are still several good options for aliases.

### Non-combat idling

These are good for warriors who can't calm and want to be able to idle in a big group even if there isn't a training dummy.

#### Judge and vurdere

The judge and vurdere commands are somewhat unique in that one use of the command can target multiple items, and accordingly multiply the xp return. You can vurdere one item for 25 gp and 750 xp return, or you can vurdere up to 10 items at once for 250 gp and 7500 xp return. Judge is more limited: you can only judge two weapons at a time, with judge <weapon> against <weapon>. Pretty awesome way to idle. Of course, just doing that won't take 5 minutes for gp to regen for the next cycle, and it definitely doesn't meet the 11 command alternative minimum for getting full command xp when returning a command, so you'll need to use a few other commands along with those. The vurdere and judge parts of your alias(s) could look like this:

get armours from pack;
vurdere armour & armour 2 & armour 3 & armour 4 & armour 5 & armour 6 & armour 7 & armour 8 & armour 9 & armour 10;
put armours in pack;

and

get weapons from pack;
judge weapon 1 against weapon 2;
put weapons in pack;

The weapons and armours should be in your inventory briefly enough that you don't fumble them, if an alias is used. You can probably leave them in the room with you as an alternative, but make sure your groupmates know not to bury them. You can also wear the armours, and/or use lightweight armour and weapons. Keep in mind that jewellery counts as armour, and you can wear 8 rings and a few other light weight things to avoid wearing a lot of heavy armour or carrying it around in a pack. Small sticks (pickling sticks) are very lightweight and count as weapons.

Note that if you're too specific about the number of armours you're vurdering, the alias can be easy to break. A simple "vurdere 10 armours", for example, will fail if you don't have enough pieces of armour. The even more basic "vurdere my armours", while somewhat more flexible, will fail if you have more than ten pieces of armour--since you can only vurdere ten at a time--or if you have more than your adventuring gp allows you to vurdere at once (if, for example, you normally vurdere 7 at once, which is your limit, but you've recently aquired a blorple, meaning you're trying to vurdere 8 at once now). This makes the above example--edited to the number of armours you're actually able to vurdere at once--the best solution, since there's a maximum number of armours it will try to make you vurdere, and having fewer armours out than usual won't break it (you'll just vurdere as many as are there). The & character lets you refer to multiple things at once without breaking the command if one of the things referred to isn't present (unlike the comma).

#### Warcry

You can also prepare a warcry without actually entering combat--since the first part, preparing it, gives 30x xp while the second part, actually performing the warcry, is only 15x, this is actually preferable to doing a full warcry.

#### Track

Track can also be used; however, trying to track someone who's in the room with you will take no gp. You also can't track players who are logged off, pk players if you're npk (may or may not be true all the time), most npcs without unique names, or npcs who aren't on the Disc at the moment. The best bet, therefore, is to track an npc who is always/nearly always alive and never in the place where you idle. Pewsey (can't be attacked, stays around Granny's cottage), Gretta (protected shopkeeper, doesn't wander from Tarnach's), Henry (can't be killed?), and Ryattenoki (can't be killed?) would all be reasonable choices.

Please note that tracking uses crafts points, not fighting points, so assuming your fi.poi bonus is higher than your cr.poi bonus, you will want to use track in the beginning of your alias, rather than the end.

#### Warpaint

Warpaint is semi-sustainable command, due to the material component. It takes 20 gp and you can warpaint yourself or another warrior--note that some animals count as warriors, so warpainting pets may be viable. Shearers are probably the best targets, since obbling and unobbling them removes the paint. (You can't remove an npc's warpaint otherwise, it seems.)

If you're already wearing warpaint you can't put more on, although you can "warpaint remove me" to get rid of it.

Warpainting consumes around an ounce of paint. A one-quart tin has enough for fifty applications and costs \$1 or so, depending on where you buy it.

#### Guard

Finally, you can guard:

guard east against all;
guard stop;

This, however, requires that you know which exit to guard before you run your alias. If you always idle in the same place, it's easy, but if you try to guard an exit that doesn't exist, it'll fail. There's no good way to avoid this aside from having multiple guard aliases and switching them out depending on where you are.

#### Less practical 30x commands

##### Bandage

Since this consumes cloth strips and requires a wounded person, this isn't usable all the time and isn't sustainable without a steady supply of bandages. Still, it might be usable if someone in your group is using Holy Sacrifice, or combat-idling and getting hurt.

##### Fix/Leatherwork

This consumes money and requires a damaged weapon, piece of armour, or piece of jewellery, as well as only being usable at a forge/tannery. If you have an idle group going at a forge, though, you could use engrave, then polish, then fix on an item, since polishing an engraving off damages the item.

#### Other commands

Using all of these uses, at best, 349 gp with six commands. 349 gp is around 3 minutes and 52 seconds' worth with 3 gp regen, or around 2 minutes and 55 seconds' worth with 4 gp regen. This is not enough time in between repetitions to get full xp each time, so you'll probably want to round out the alias with a few more commands. If there aren't any skills you especially want to try to tm, try a musical instrument for crafts.points tms.

### Training Dummies

A training dummy is essentially a weak npc who can't die, making them excellent for combat idling: there's little risk to yourself, and you don't risk killing the dummy and losing it. The training dummy in the Djelian Guard doesn't hurt you when it hits you, so there's no risk to fighting it.

As above, you can incorporate judge and vurdere into your alias, however you can also use any other combat commands you like. A huge advantage over the vurdere/judge aliases is that you can TM useful skills this way.

Some warriors create an idle group in the basement of the Djelian Guard, in the room with the training dummy. The guild npc here, Limdul, will stop you from attacking the dummy after a short period (around thirty seconds), removing the need for calming. (He'll also comment on your technique and offer advice, which can be useful if you're low-level.)

To make your alias, choose at least 11 of the above 30x commands. Repeat the aliases as soon as your gp regenerates fully. For the best results, use neutral attitude (tactics attitude neutral) while fighting the dummy: other settings can be a slight drain on your gp (or possibly gp regen).

Members of any guild may use this dummy, not just warriors--and apart from fighting the dummy, it's also possible to backstab it, to use various covert commands on it (plant, steal, peek, rifle, case, etc) or target it with spells or rituals.

### Sheep Shearers (with example)

You must be able to calm the shearer reliably for this to work (if you can only calm semi-reliably, you may want to duplicate the "calm" line) and you must have enough faith gp to use calm at that point in your alias. If you can pull it off, though, this option gives you the best of both worlds. You can use mostly 30x commands and TM some combat skills without restricting your grouping options.

kiss obble;
kill shearer;
wait;
shove opponent;
stab opponent with <really crappy weapon>;
trip opponent;
impale opponent with <really crappy weapon>;
pierce opponent with <really crappy weapon>;
feint at opponent with <really crappy weapon>;
perform calm on opponent;
kiss shearer;
warcry IDLE ON!;
vurdere <armours>;
judge weapon 1 against weapon 2;
track pewsey;

So you have 10 30x commands and 1 15x command (calm), but the 15x is probably worth it because you getter better shared xp by being in a big group. You can use this alias as often as needed, without bothering with the time limit, because you have 11 or more commands. You can replace the stab, impale, and pierce commands with whatever you want, and add more specials to the alias. Just be sure you don't drop below 11 commands.

Good options for <really crappy weapon> (so you don't kill the shearer), are often weapons that appraise as misc, or weapons made from crappy materials, such as a wooden sword. A rubber knife is also great to avoid accidental slaughter.

Having vurdere, judge, and track in there--as well as non-direct-damage specials (trip, shove, and feint)--also help you avoid killing your shearer, since then you don't need as many damage-causing combat commands. The non-direct-damage specials use less gp, though (with trip and shove at 30 gp, riposte at 25 gp, and feint at 15 gp), so it's best to still use as many direct-damage specials as you can get away with without killing your shearer.

The "wait" (or frimble, or score brief, or group status) helps make sure that you're actually in combat with the shearer when you attempt your other commands, since if you try a special immediately after "kill", it may not work (or, if it's trip or shove, it may treat it as the soul instead of the special), only resulting in a "You are not fighting the x" message for that first command.


text:
## Drag Coefficient

Consider during a round of golf, a player smacks the 0.5 lb ball such that it travels 200 ft/s. If the drag force acting on the 0.1 ft^2 projected area is 2.5 lbf, what is the drag coefficient? Assume the density of air is 0.08 lb/ft^3.

Hint
The drag force on objects immersed in a large body of flowing fluid or objects moving through a stagnant fluid:
$$D=\frac{1}{2}\rho U^{2}C_{D}A$$$where $$C_D$$ is the drag coefficient, $$U$$ is the flowing fluid or moving object’s velocity, $$\rho$$ is the fluid density, and $$A$$ is the projected area of blunt objects with axes perpendicular to the flow. Hint 2 $$1\:lbf=1\:lb \times gravity\approx 1\:lb \times 32.2 \frac{ft}{s^2}$$$
The drag force on objects immersed in a large body of flowing fluid or objects moving through a stagnant fluid:
$$D_f=\frac{1}{2}\rho U^{2}C_{D}A$$$where $$C_D$$ is the drag coefficient, $$U$$ is the flowing fluid or moving object’s velocity, $$\rho$$ is the fluid density, and $$A$$ is the projected area of blunt objects with axes perpendicular to the flow. $$2.5lb_f=\frac{1}{2}(0.08\frac{lb}{ft^3})(200\frac{ft}{s})^2C_D(0.1ft^2)$$$
Before solving for $$C_D$$ , recall that $$1\:lb_f=1\:lb \times gravity\approx 1\:lb \times 32.2 \frac{ft}{s^2}$$ . Thus,
$$2.5lb\times32.2\frac{ft}{s^2}=\frac{1}{2}(0.08\frac{lb}{ft^3})(40,000\frac{ft^2}{s^2})C_D(0.1ft^2)$$$$$80.5=(160)C_D$$$
$$C_D=\frac{80.5}{160}=0.503\approx 0.5$$\$
0.5