# Nanoparticles of NbC produced by laser ablation in liquid: a study of
structural, magnetic and superconductivity properties.

Fernando Fabris<EMAIL_ADDRESS>Instituto de Física Gleb Wataghin,
UNICAMP, 13083-859 Campinas, São Paulo, Brazil  Ali F. García-Flores
Instituto de Física Gleb Wataghin, UNICAMP, 13083-859 Campinas, São Paulo,
Brazil  Julian Andres Munevar Cagigas  Centro de Ciências Naturais e Humanas,
UFABC, São Bernardo do Campo, São Paulo, Brazil  José Javier Sáez Acuña
Centro de Ciências Naturais e Humanas, UFABC, São Bernardo do Campo, São
Paulo, Brazil  Carlos Rettori  Instituto de Física Gleb Wataghin, UNICAMP,
13083-859 Campinas, São Paulo, Brazil  Ricardo R. Urbano  Instituto de Física
Gleb Wataghin, UNICAMP, 13083-859 Campinas, São Paulo, Brazil

###### Abstract

Niobium carbide (NbC) is a high-field Type II superconductor with a critical
temperature ($T_{C}$) of 11.1 K, just above that of pure Nb ($T_{C}=9$ K).
Downsizing NbC to the nanoparticle scale introduces significant alterations in
its critical field and/or the superconducting temperature. Here we report on
superconducting NbC nanoparticles with $T_{C}\approx$ 10 K synthesized by
laser ablation in acetone, using the lens-target distance (laser fluence) and
centrifugation as control parameters of the particle size. X-ray diffraction
analyses certified the cubic NbC phase and electron microscopy images revealed
spherical particles with average size near 8 nm, with no apparent size
dependence on fluence. Besides, magnetization curves exhibited magnetic loops
featuring a saturation magnetization around $10^{-3}\mu_{B}$/molecule along
with a small and typical superconducting loop for all investigated samples. We
also observed a suppression of the diamagnetic behavior below $T_{C}$ upon
decreasing laser fluence. Moreover, all samples exhibited a weak electron spin
resonance (ESR) Curie-like signal at $g\approx 2.0$ probably associated with
localized defects in the particle’s surface. The intriguing coexistence of
superconductivity and magnetism in nanoparticles has recently garnered
significant research attention. This complex scenario and unique properties
are due to the substantial increase of surface-to-volume ratio in these
superconducting NbC nanoparticles and further investigation would be crucial
to unveil novel material properties and shed new light on our understanding of
the superconducting phenomenon in this new morphology.

##  I Introduction

Niobium carbide (NbC), a representative transition metal carbide, possesses
distinctive physical properties featuring a simple NaCl-type cubic lattice, an
elevated melting point ($3600^{\circ}$C), remarkable hardness, high thermal
and electrical conductivity, superior wear resistance, and chemical
stabilityCuppari and Santos (2016). Such a versatile combination renders NbC
suitable for various applications, including cutting and grinding
toolsUhlmann, Meier, and Hinzmann (2019); Montenegro  _et al._ (2018), wear
protection of surfacesWoydt  _et al._ (2016); Woydt and Mohrbacher (2013),
catalysisAlhowity  _et al._ (2024), super-capacitors and hybrid ion
capacitorsYang  _et al._ (2021). As a Type II superconductor, NbC is
considered a key player in high-field superconductor applications, boasting
excellent critical superconducting properties. Notably, within the realm of
low-temperature superconductors, NbC stands out with a relatively high
superconducting transition temperature of $T_{C}=11.1$ K, just above that of
pure Nb ($T_{C}=9$ K), coupled with an impressive critical magnetic field
($H_{C2}$) exceeding 4 TBhattacharjee, Pati, and Maity (2016); Zou  _et al._
(2011); Zhang  _et al._ (2012).

Despite extensive research since its discovery in 1964Wells  _et al._ (1964),
considerable attention has been devoted to understand the superconducting
properties of NbC. Investigations have explored alterations in carbon vacancy
density, varied production routes, and diverse synthesis techniquesGusev and
Rempel (1989). Additionally, numerous studies have probed the superconducting
properties of NbC in low-dimensional forms, such as thin films, nanofibers,
and nanoparticles, exhibiting different morphologiesLi  _et al._ (2006);
Fukunaga, Chu, and McHenry (1998); Bhattacharjee, Pati, and Maity (2016).
However, challenges persist in the production of bulk or nanosystems based on
NbC with specific functionalities, primarily due to the high temperatures
required. Nevertheless, the physical top-down synthesis method namely pulsed
laser ablation have demonstrated capability of producing NbC nanomaterials and
it stands out as a “green” technique enabling the preparation of a substantial
quantity of materials at room temperatureZhang  _et al._ (2019, 2016);
Amendola and Meneghetti (2013); Frias Batista  _et al._ (2022). This process
involves focusing a high-power pulsed laser on the Nb target surface, leading
to the production of NbC nanoparticles in organic solvents (such as acetone or
ethanol). Interestingly, these particles are surrounded by a thin porous
graphite layer, enhancing the chemical stability of the solution.Zhang  _et
al._ (2019)

Regardless of the substantial body of research on niobium carbide, there is a
notable gap in data exploring the coexistence of magnetic and superconducting
properties of NbC nanoparticles produced through laser ablation. Therefore, in
this work, we dedicated our efforts to the production and a detailed
characterization of NbC using laser ablation in acetone. By varying the laser
fluence at the Nb target surface, we examined changes in the morphological,
magnetic, and superconducting properties. A comparative analysis between melt
particles (with average diameter about 470 nm) and particles formed through
target vaporization (with average diameter about 8 nm) provided new insights
into crucial physical properties of NbC nanoparticles. The magnetic properties
of NbC nanoparticles manifested in three components: an electron spin
resonance (ESR) signal characterized by a Curie-like $T$-dependent behavior
attributed to a localized ESR active defects confined within the surface of
the NbC nanoparticles; a paramagnetic component observed in all samples and
all temperatures, linked to typical surface effects in materials reduced to
the nanometric scale, and finally, the superconducting behavior marked by a
sudden diamagnetic behavior below a critical temperature $T_{C}\approx$10 K
for all studied samples. We provide a quantitative analysis of the
superconducting volume and critical fields for different laser fluences used
in the nanoparticles production process, shedding new light on the intricate
interplay between diamagnetic and superconducting properties of NbC
nanoparticles.

##  II Material and methods

The NbC nanoparticles were synthesized through pulsed laser ablation (LA)
using a Nb target immersed in acetone. The LA process employed a Nd:Yag laser
(1064 nm) with a 10 ns pulse duration and a 20 Hz repetition rate. Focused
with a 50 mm lens, the laser was directed onto a freshly polished (sanded)
surface of a Nb target ($99.9\%$ of purity) vertically positioned and
submerged in 75 ml of acetone. The laser path through the liquid spanned a
distance of 2 cm.

The ablation process continued for 20 minutes, resulting in a uniformly dark
and stable solution. Initially, the NbC nanoparticles were synthesized under a
laser fluency of 20 J/cm2. The resultant sample underwent centrifugation at
1000 RCF for 10 minutes, separating the supernatant (S1) from the sediment
(P1). Subsequently, three additional samples were produced with laser
fluencies of 45 J/cm2, 70 J/cm2, and 100 J/cm2, each of them subjected to the
same procedure aforementioned. The resulting supernatant samples were named
S2, S3, and S4, respectively. Finally, the powder samples investigated here
were obtained from evaporating the acetone at room temperature.

Total reflection X-ray fluorescence (TXRF) elemental analysis of the samples
revealed an exceptional purity level of 99.5 % following the entire
experimental procedure. The remaining 0.5 % primarily comprises iron
contaminants, presumably coming from the sanding paper.

The X-ray diffraction (XRD) measurements of the samples were performed in a
Phaser D2 diffractometer (Bruker). Transmission electron microscopy (TEM)
images were taken in a Talos F200X G2 equipment equipped with a field emission
gun (FEG-X) source operating at 200 kV and in JEOL JEM-2100F also with a FEG
gun source operating at 200 kV. TEM samples were prepared by dropping the
acetone solution containing the nanoparticles on a copper grid with an ultra-
thin hollow carbon film. The T-dependent dc-magnetic susceptibility
measurements of the powdered samples were carried out in a SQUID (Quantum
Design, MPMS 3) magnetometer. The ESR experiments were carried out in a Bruker
ELEXYS 500 ESR spectrometer at $\sim 9.5$ GHz (X-band), using an appropriate
TE102 resonator coupled to an Oxford T-controller helium gas flow system
varying the temperature in the range of $4.2$ K $<T<300$ K. Colloidal NPs and
powdered bulk samples were dispersed in high purity paraffin for the ESR
experiments.

##  III Results and Discussion

Figure 1 illustrates the X-ray diffraction (XRD) patterns for the samples. All
discernible diffraction patterns agree well with a cubic NbC structure (COD
database code: 1011323). The diffraction peaks exhibit notable broadening
beyond the instrumental resolution, indicative of a finite-size effect.
Notably, the peak broadening observed in the P1 sample is less pronounced
compared to the Si samples, suggesting a larger crystal size for P1. The
determined lattice parameter for all nanoparticles samples is $a=4.469(1)$ Å,
in close proximity to the expected value for stoichiometric NbC ($a=4.470$
$\AA$)Cuppari and Santos (2016). This suggests a negligible presence of carbon
vacancies in the nanoparticles generated through laser ablation.Cuppari and
Santos (2016) Additionally, certain samples exhibit peaks associated with
hexagonal graphite (COD database code: 9014004).

Figure 1:  XRD patterns of samples P1, S1, S2, S3 and S4. The stars on the
figure represent the cubic NbC diffraction pattern (COD: 9008682) and the
circles the hexagonal graphite pattern (COD: 9014004).

Figure 2 displays representative transmission electron microscopy (TEM) images
of P1, S2, and S3 samples, all exhibiting a similar spherical morphology with
a broad size distribution, which was determined by counting nearly thousand
particles in each specimen, as shown in Figure 2 (d). The associated size
histograms were fitted with a log-normal function defined by
$f(D)=(\sqrt{2\pi}\sigma_{0}D)^{-1}\exp\left[-\frac{\ln^{2}(D/D_{0})}{2\sigma_{0}^{2}}\right]$
and, thus, $D_{0}$ and $\sigma_{0}$ were estimated. Therefore, the mean
diameter $\left\langle
D\right\rangle=D_{0}\exp\left(\frac{\sigma_{0}^{2}}{2}\right)$ and standard
deviation $\sigma=\left\langle
D\right\rangle\left[\exp\left(\sigma_{0}^{2}\right)-1\right]^{1/2}$ were
calculated.

It is noteworthy that the sedimented sample particles are considerably larger
than those in the supernatant. In the former, the ablation process is broadly
categorized as fragmentation, involving the detachment of hot atoms, vapors,
and liquid drops out of thermodynamic equilibrium conditionsAmendola and
Meneghetti (2013). Specifically, the P1 particles are formed by the melt
expulsion of the target during the ablation process. Conversely, in the
supernatant (Si) samples, the nanoparticles are generated after the detachment
of hot atoms and vapors. Notably, the $\left\langle D\right\rangle$ and
$\sigma$ for Si samples do not substantially change with fluency and all of
them exhibited nearly 8.5(5) nm size.

High-resolution TEM images reveal crystalline planes of NbC, as illustrated in
Figure 2(c) for the (111) and (200) planes in two different particles.
Additionally, a carbon shell is evident on the particle surface. Clear lattice
fringes with an inter-planar spacing of 0.34 nm align precisely with the
typical (002) plane of graphite carbon. In all S samples, 3 to 6 layers of
graphite surround the NbC nanoparticles, whereas for the P1 sample, 9 to 20
carbon layers cap the NbC particles. Similar carbon encapsulated metals are
obtained by the ablation of Ti, V, Nb, Cr, Mo, W, Zr, Ni and CoZhang  _et al._
(2019, 2016, 2013); A  _et al._ (2020).

Figure 2:  Representative TEM images of P1 (a), S3 (b) and S2 (c) samples. The
sample size distributions are presented in (d). The red lines are the best fit
with a log-normal: the mean diameter ($\langle D\rangle$) and the standard
deviation ($\sigma$) are displayed.

Raman spectroscopy was employed to validate the presence of carbon and its
crystalline structure in the carbon-encapsulated NbC samples. The Raman
spectra of all five samples, as depicted in Figure 3, reveal two prominent
bands at approximately 1350 cm-1 (D band) and 1600 cm-1 (G band). These bands
correspond to the vibrations of carbon atoms with dangling bonds in the in-
plane terminations of disordered graphite (D) and the vibrations in sp2-bonded
carbon atoms in a 2D hexagonal lattice (G), respectively, as established by
Emanich et al. Nemanich and Solin (1979). The relative intensity ratio
$I_{D}/I_{G}$ varies across the sample regions, ranging from 0.89 to 2.89 for
all samples. Although a precise value for $I_{D}/I_{G}$ cannot be
unequivocally determined, the consistent elevated values suggest the presence
of numerous structural defects in the synthesized carbon shells. This
observation is in accordance with other carbon-encapsulated nanoparticles
produced by laser ablationZhang  _et al._ (2019, 2016).

Figure 3: Room temperature Raman spectra for all carbon-encapsulated NbC
samples.

Magnetic measurements are presented in Figure 4 (a). The zero-field cooled
(ZFC) magnetization as function of temperature revealed the onset of the
superconducting critical temperature at $T_{C}\simeq$ 10 K. This value is
slightly smaller than that of stoichiometric bulk NbC ($T_{C}=11.1$ K)Giorgi
_et al._ (1962), but aligns with reported $T_{C}$ values for NbC nanoparticles
obtained through other methodsBhattacharjee, Pati, and Maity (2016). Besides,
no dependence of $T_{C}$ with the laser fluency employed in the ablation
process was observed.

Moreover, a positive magnetization was observed for the S samples above
$T_{C}$, attributed to a paramagnetic contribution of C defects on the NbC
surface nanoparticles. Interestingly, the magnetization was nearly zero for
the P1 sample, within the accuracy of the equipment.

Figure 4: Zero field cooled magnetization at 20 Oe for all samples. Inset
displays the zooming out of these measurements.

Below the critical temperature ($T_{C}$), a notable decrease in the
superconducting volume was observed when comparing the P1 sample with the S
samples. Below $H_{C1}$ in a bulk type II superconductor, the magnetic field
is completely expelled from the sample volume, except for a surface layer with
a thickness denoted by $\lambda_{L}$ known as the London penetration length.
Under ideal conditions, the magnetic susceptibility below $T_{C}$ is expected
to be $\chi_{bulk}^{SC}=-1/4\pi$. For bulk NbC, $\lambda_{L}$ is determined as
140 nmShang  _et al._ (2020), meaning that the magnetic field only penetrates
into the P1 particles partially, whereas it penetrates into the S particles
completely. This results in a significant reduction of the Meissner effect
when comparing with the bulk material, and it can be accounted by the
following equationTinkham (2004); Zolotavin and Guyot-Sionnest (2010):

$\chi^{SC}=-\frac{3}{16}\frac{1}{40\pi}\frac{D^{3}}{\lambda_{L}^{2}\xi_{0}}$
(1)

where $D$ represents the particle diameter and $\xi_{0}\simeq 13$ nm is the
coherence length for NbCShang  _et al._ (2020). For particles with 8 nm
diameter, eq. 1 leads to a diamagnetic susceptibility of
$4\pi\chi^{SC}=37\times 10^{-6}$ emu/cm3Oe, consistent with the observations
for the sample S4. However, a reduction of the Meissner susceptibility is
noted upon decreasing the laser fluency (from S4 to S1 samples), which is
attributed to an increase of defects other than C vacancies (i.e. NbC1-x). On
the other hand, $4\pi\chi^{SC}=0.7\times 10^{-3}$ emu/cm3Oe for the P1
particles, which have an average size larger than $\lambda_{L}$. This value is
also smaller than the expected value for bulk NbC, $4\pi\chi^{SC}=-1$. The
observed $\chi^{SC}$ value for P1 suggests a particle size more akin to 22 nm,
way smaller than the average particle size estimated ($\approx 470$ nm) and
suggests the presence of numerous reminiscent defects in the crystalline
structure, ultimately leading to a reduced superconducting volume. It is worth
highlighting that the fabrication of nanoparticles through laser ablation in a
liquid involves the detachment of hot atoms, vapors, and liquid drops followed
by a rapid quenching of the material upon its interaction with the
liquidAmendola and Meneghetti (2013). This intricate process has the potential
to generate a substantial amount of crystalline defects within the particle
system.

The isothermal magnetization as function of magnetic field at 2K and 300K for
the NbC samples are shown in Figure 5. The magnetization data was obtained
after subtracting the diamagnetic contribution from the sample holder. Two
distinct magnetic contribution are observed: a positive component and a
superconducting loop at low temperature and low fields. The positive
component, which is present even at room temperature, exhibits a larger
magnetization for the P1 sample, decreasing with reduced fluency for the S
samples. This positive magnetization with values on the order of
$10^{-3}-10^{-2}$ emu/g and a coercitive field around $100$ Oe at room
temperature has been reported for materials reduced to a nanometric
scaleSundaresan and Rao (2009); Li  _et al._ (2008); Bose  _et al._ (2005); Ma
_et al._ (2022); Zhu  _et al._ (2012); Jirsa, Rames, and Wolf (2012); Shipra,
Kumar, and Sundaresan (2013). Surface ferromagnetism is suggested to be a
universal feature in almost all nanoparticles of otherwise nonmagnetic
inorganic materials, arising from surface cation or anion vacancies leading to
ferromagnetic spin polarizations Sundaresan and Rao (2009). Moreover, since
our P1 sample does not show any ESR signal (see the figure 6) we conclude that
this ferromagnetic signal is ESR silent. Furthermore, the coexistence of
superconductivity and ferromagnetism at low temperature has been observed in
NPs of metal Li  _et al._ (2008); Bose  _et al._ (2005); Ma  _et al._ (2022),
oxidesZhu  _et al._ (2012); Jirsa, Rames, and Wolf (2012) and nitridesShipra,
Kumar, and Sundaresan (2013), but for the first time we report in carbides.
However, it is acknowledged that this ferromagnetic signal may have originated
from iron contamination in the synthesis process.

Figure 5: Magnetic hysteresis loops at 2K (a) and 300K (b). The insets in (a)
provide a closer look at these curves at low field for the P1 sample (upper
left) and S samples (lower right). Meanwhile, the inset in (b) is a magnified
view at low field aimed at revealing the coercive field ($H_{C}$) of the order
of 100 Oe at 300 K for this samples. Figure 6:  (a) Electron spin resonance
(ESR) spectra at 300 K for P1, S1, S2, and S3 samples. In (b), ESR spectra for
the S2 sample are presented at various temperatures. And in (c) the Normalized
intensity $I(T)/I(300K)$ of the S2 samples as function of $T$. The red dashed
line in (c) represent a Curie-like curve and inset the line width $\Delta H$
and resonance field $H_{res}$ at low $T$.

The electron spin resonance (ESR) experiments shown in Figure 6(a) present
narrow resonances of line width $\Delta H\approx 2$ G and a $g$-value at
$g=2.0$ only for the S samples. These resonances are often attributed to
localized spins present in the samples and can not be attributed to the iron
contamination. A quantification of the amount of spins against a standard
sample indicated that there are only 1 spin every 10 to 20 nanoparticles for
the S samples. However, within the accuracy of the ESR experiment the P1
sample does not present this ESR signal. This result indicates that this
localized spin may be in the particles surface once the surface/volume rations
for the S samples are about fifty times greater than that of the P1 sample.

The $T$-dependence of the ESR spectra for the sample S2 is shown in the Figure
6 (b). By lowering the temperature the signal do not change much, however
below 10 K the signal starts to broaden, shift toward higher fields and its
intensity increase. Figure 6 (c) depicts the temperature dependence of the
relative ESR intensity, $I(T)/I(300$K$)$. An illustrative Curie-like behavior
reveals that the signal intensity displays an increase beyond this
$T$-dependence. Futhermore, as highlighted in the inset of Figure 6 (c), below
$T_{C}$, the linewidth $\Delta H$ increases, and a small resonance shift
toward higher fields is observed. Similar phenomena, including increased
$\Delta H$ and $H_{res}$ below $T_{C}$, have been reported in bulk Type-II
superconductors.Urbano  _et al._ (2002) The line broadening is attributed to
the field distribution within the superconducting vortex state, while the
resonance shift is linked to the diamagnetism in the superconducting state.
These results suggest that, in our samples, the localized spins on the
particles’ surfaces are being influenced by the diamagnetism of the
superconducting vortex state in NbC particles, presumably through proximity
superconducting effects.Holm and Meissner (1932)

##  IV Conclusions

In conclusion, our study successfully demonstrated the synthesis of
superconducting NbC nanoparticles through low-temperature laser ablation in
acetone. Notably, varying the laser fluence showed negligible impact on the
average size of the resulting NbC nanoparticles, which exhibited a broad size
distribution. Some particles displayed surface features indicative of graphite
or amorphous material, potentially porous amorphous carbon. The elevation in
laser fluence at syntheses process increase the volume of the superconducting
fraction. Additionally, a paramagnetic contribution was evident across all
samples, even in the sedimented sample. Particularly noteworthy was the
observation that this magnetic moment increased with rising fluence,
especially notable in the supernatant sample at lower temperatures. These
findings contribute valuable insights into the synthesis and properties of NbC
nanoparticles, offering avenues for further exploration and application in
superconductivity research.

###### Acknowledgements.

This study was mainly supported and performed under the auspices of the São
Paulo Research Foundation (FAPESP) through Grant No. 2017/10581-1 (Thematic).
F.F. acknowledges his postdoc FAPESP Grant No. 2019/13678-1. R.R.U also
acknowledges the National Council for Scientific and Technological Development
(CNPq) for support through Grant No. 314587/2021-7. This research used
facilities of the Brazilian Nanotechnology National Laboratory (LNNano), part
of the Brazilian Centre for Research in Energy and Materials (CNPEM), a
private non-profit organization under the supervision of the Brazilian
Ministry for Science, Technology, and Innovations (MCTI). The electronic
microscopy staff is acknowledged for the assistance during the experiments
(proposal number: 20230481).

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###### Abstract

In this project we explore the geometry of general metric spaces, where we do
not necessarily have the tools of differential geometry on our side. Some
metric spaces $(X,d)$ allow us to define geodesics, permitting us to compare
geodesic triangles in $(X,d)$ to geodesic triangles in a so called model
space. In Chapters 1 and 2 we first discuss how to define the length of
curves, and geodesics on $(X,d)$, and then using these to portray the notion
of “non-positive curvature” for a metric space. Chapter 3 concerns itself with
special cases of such non-positively curved metric spaces, called CAT($0$)
spaces. These satisfy particularly nice properties, such as being uniquely
geodesic, contractible, and having a convex metric, among others.

We mainly follow the book [BridHaef] by Martin R. Bridson and André Haefliger,
with some differences. Firstly, we restrict ourselves to using the Euclidean
plane $\mathbb{E}^{2}$ as our model space, which is all that is necessary to
define CAT($0$) spaces. Secondly, we skip many sections of the mentioned book,
as many are not relevant for our specific purposes. Finally, we add details to
some of the proofs, which can be sparse in details or completely non-existent
in the original literature. In this way we hope to create a more streamlined,
self-contained, and accessible introduction to CAT($0$) spaces.

###### Contents

  1. 0 Length spaces
    1. 1 Length of curves
    2. 2 Geodesic spaces

##  Chapter 0 Length spaces

###  1 Length of curves

Let $(X,d)$ be a metric space and let $I=[0,a]\subseteq\mathbb{R}$ be a
compact interval not the empty set. A _curve_ is a continuous map $\sigma:I\to
X$ joining the endpoints $x=\sigma(0)$ and $y=\sigma(a)$. In our case, curves
are not necessarily smooth, let alone differentiable. Therefore, the well
known notion of arc length $\ell(\sigma)=\int_{a}^{b}|\sigma^{\prime}(t)|dt$
from undergraduate geometry will not fit our purposes. A different and more
general approach is to “approximate” the curve by a polygonal path, the length
of which can be determined by applying the metric $d$ to each line segment.
Intuitively, the more line segments the polygonal path has, the better an
approximation of the curve it is. This is made precise by the following
definition:

######  Definition 1.

Let $\sigma:I\to X$ be a curve from the compact interval
$I=[a,b]\subseteq\mathbb{R}$ to a metric space $(X,d)$. Let $P$ be a partition
$a=t_{0}<t_{1}<\ldots<t_{n}=b$ of $[a,b]$, and define
$L_{P}(\sigma)=\sum_{i=1}^{n}d(\sigma(t_{i-1}),\sigma(t_{i}))$. The _length_
$L(\sigma)$ of $\sigma$ is defined as $L(\sigma):=\sup L_{P}(\sigma)$ where
the supremum is taken over all possible partitions $P$ of $I$. A curve is said
to be recitifiable if it has finite length, i.e. $L(\sigma)<\infty$.

As the name suggests, length provides in a natural way a notion of distance,
i.e. a metric on $X$:

######  Definition 2.

Let $(X,d)$ be a metric space. The length metric associated with $d$ is the
function $d_{i}:X\times X\to[0,\infty]$ on $X$ given by

$d_{i}(x,y)=\inf L(\sigma),$

where the infimum is taken over all rectifiable curves $\sigma$ joining $x$
and $y$. If no rectifiable curve between points $x$ and $y$ we set
$d_{i}(x,y)=\infty$.

######  Proposition 3.

The length metric $d_{i}$ is a metric on $X$.

###### Proof.

Positive definiteness: Let $x,y\in X$ be points, and assume that
$d_{i}(x,y)\neq\infty$. Then there exists at least one rectifiable curve
between $x$ and $y$. Consider first $x\neq y$. As $d$ is a metric it is in
particular positively definite, so $d(x,y)>0$. This implies that all summands
in the $L_{P}(\sigma)$ are positive for any rectifiable curve $\sigma$ between
$x$ and $y$, and any partition $P$ of $\sigma$, so $d_{i}(x,y)>0$ when $x\neq
y$. In the case $x=y$, the constant curve mapping all elements in $[a,b]$ to
$x$ is a rectifiable curve from $x$ to $y$. For all partitions we have
$\sigma(t_{i-1})=\sigma(t_{i})$, so all summands of $L_{P}(\sigma)$ are 0,
whereby $L(\sigma)=0$. This in turn means that $0\leq d_{i}(x,y)\leq
L(\sigma)=0$ so $d_{i}(x,y)$ is zero if and only if $x=y$.

Symmetry: For all rectifiable curves $\sigma:[a.b]\to X$ from $x$ to $y$ there
is a rectifiable curve $-\sigma:[a,b]\to X$ from $y$ to $x$ given by
$-\sigma(t)=\sigma(a+b-t)$. This can be considered the same curve in reverse.
This curve $-\sigma$ obviously has the same length as $\sigma$, so for every
curve from $x$ to $y$ there is a curve from $y$ to $x$ with the same length,
and vice versa. Therefore the infimum of the length of all curves from $y$ to
$x$ must be the infimum of the length of all curves from $x$ to $y$, i.e.
$d_{i}(x,y)=d_{i}(y,x)$. This shows that $d_{i}$ is symmetric.

Triangle inequality: Let $x,y$ and $z$ be points in $X$, let
$\sigma_{a,b}:[a,b]\to X$ be a curve from $x$ to $y$, and let
$\sigma_{b,c}:[b,c]\to X$ be a curve from $y$ to $z$. There is then a curve
$\sigma_{a,c}:[a,c]\to X$ which is simply the concatenation of the two curves,
i.e.

$\sigma_{x,z}(t)=\begin{cases}\sigma_{x,y}(t)&\text{if }t\in[a,b]\\\
\sigma_{y,z}(t)&\text{if }t\in[b,c].\end{cases}$

We clearly have $L(\sigma_{x,z})\leq L(\sigma_{x,y})+L(\sigma_{y,z})$, whereby
$d_{i}(x,z)\leq d_{i}(x,y)+d_{i}(y,z)$, proving the triangle inequality. ∎

######  Definition 4.

If $d=d_{i}$, the metric space $(X,d)=(X,d_{i})$ is called a length space.

######  Example 5.

As curves usually are visualized in $\mathbb{R}^{2}$ or $\mathbb{R}^{3}$,
which are length spaces when equipped with the usual Euclidean metric $d$, it
might be more useful to consider a non-example which nontheless can be
visualized. The unit circle $S^{1}$ with the metric $d_{E}(x,y)=|x-y|$ induced
by the Euclidean metric of $\mathbb{R}^{2}$ is _not_ a length space for the
following reason: Consider two points $x,y\in S^{1}$. The length metric
$d_{i}(x,y)$ is the arc length along the circle
$d_{i}(x,y)=\arccos(\left\langle x,y\right\rangle)$. In particular,
$d_{i}(x,y)\neq d_{E}(x,y)$, so $(S^{1},d_{E})$ is _not_ a length space.

###  2 Geodesic spaces

Let $(X,d)$ be a metric space and let $I=[a,b]\subseteq\mathbb{R}$ be a
compact interval not the empty set. A curve $\sigma:I\to X$ is said to have
constant speed $\lambda$ if
$L(\sigma|_{[t,t^{\prime}]})=\lambda|t-t^{\prime}|$ for all $t\leq
t^{\prime}\in I$. Intuitively, it takes the same “time” (difference in input)
to “travel the same distance” (length between outputs), justifying the name
speed. If the speed $\lambda=1$, $\sigma$ is said to be parametrized by arc
length.