M. Beerman, A. Pakhomov, Y. Bao, and K. Krishnan Department of Materials Science and Engineering...
1
M. Beerman, A. Pakhomov, Y. Bao, and K. Krishnan Department of Materials Science and Engineering University of Washington, Seattle, WA 98195, USA Work supported by: National Science Foundation NSF/DMR 0203069 Campbell Endowment at UW An anamolous peak has been observed in zero field cooled measurements on systems composed of ε-phase cobalt nanoparticles. The samples were synthesized via a thermal decomposition process resulting in spherical nanoparticles surrounded by an organic surfactant [1,2]. In addition to the blocking temperature peak, which is dependant on particle volume [3] and displays interesting interaction effects, an additional peak appears at 8 K regardless of particle diameter. Zero field cooled and field cooled magnetization versus temperature measurements on spherical particles with mean diameters in the range of 5-20 nm were performed with a SQUID magnetometer, including careful background signal checks. Measurements after heat treatment, to promote oxidation, show that the blocking temperature peak can be effectively removed in fully oxidized powders. The 8 K peak, however, remains after heat treatment, but with a reduced magnitude, and M(H) dependance is hysteretic below this temperature. Additional measurements were performed on commercial micrometer-sized cobalt oxide powder, where this feature is also present, but gives a much weaker contribution to susceptibility. Unreacted synthesis precursors as well as samples with different surfactant species all showed at least some indication of a peak at 8 K. This feature, universal for the system under investigation, has not been observed in Co nanoparticle systems prepared by co-sputtering or co-evaporation of Co with non-magnetic materials. Furthermore, hysteresis measurements, both before and after heat treatment, show a bias when cooled in field from 30 K to below the peak temperature. Analyses of the data leads us to conclude that freezing of frustrated uncompensated spins on the surface of cobalt oxide is the most reasonable explanation of this anomalous low temperature behavior. [1] V. Puntes et al. Science 291 (2001) 2115 [2] Y. Bao, et. al., JMMM, in press. [3] L. Néel, Annales de Géophysique 5 (1949) 99 Epsilon cobalt nanospheres surrounded by a mixture of organic surfactants (TOPO and OA) are synthesized via thermal decomposition of cobalt carbonyl in di-chlorobenzene solvent. The diameter D may be controlled between 5 and 20 nm, and the surfactant length is approximately 1.7 nm. Surfactants D The particles are precipitated from solution onto a thin carbon film for transmission electron microscopy (TEM). This TEM image shows a self assembled monolayer of 10 nm spheres. (Phillips 400 T at 100 keV) -cobalt 0.0E+00 1.0E-03 2.0E-03 3.0E-03 4.0E-03 5.0E-03 1 10 100 1000 Tem pe ratu re (K) M /H (em u/G g) 0.0E+00 1.0E-04 2.0E-04 3.0E-04 4.0E-04 5.0E-04 1 10 100 1000 Tem perature (K ) M /H (em u/Gg) No External Field 27k B T B 0.0E+00 2.0E-03 4.0E-03 6.0E-03 8.0E-03 1.0E-02 1.2E-02 0 100 200 300 Tem perature (K ) Mom ent(em u) FC A uniaxial magnetic particle has two stable magnetization orientations as seen in the figure to the left. If an external field is applied along the easy axis then one of the two orientations has a lower energy as seen to the right. As the temperature increases, thermal fluctuations provide the activation energy for the system to relax over the energy barrier into the field direction. This behavior is well known and follows Applied External Field B B A m T k V K exp ~ 0 where m is the characteristic measurement time ~20s, 0 is the microscopic fluctuation attempt time ~10 -10 s, K A is the anisotropy constant ~0.21 J/m 3 for -Co, and V is the magnetic particle volume. The barrier height may be approximated by 27 k B T B where k B is Boltzmann’s constant and T B is the blocking temperature, which corresponds to the peak temperature in moment versus temperature measurements (as identified to the left). T B Two curves are plotted here: zero field cooled (ZFC) and field cooled (FC). ZFC measurements start at low temperature after cooling the sample in zero external field. Magnetization is then measured as the temperature slowly increases. FC measurements start at high temperature where a field is applied and the magnetization is measured as the temperature slowly decreases. The rate of cooling is constant in both cases. The interesting observation is the second peak at low temperature on the ZFC curve and its associated sharp uptake in magnetization as seen in the FC curve. The focus of this poster presentation is to further characterize and describe this feature. Magnetic measurements were performed on a Quantum Design MPMS 5 superconducting quantum interference device. Co Oxide Commercial Micron Sized Powder 0.E+00 1.E-05 2.E-05 3.E-05 4.E-05 5.E-05 6.E-05 7.E-05 8.E-05 9.E-05 0 50 100 150 200 250 300 350 Temperature (K) M/H (emu/Gg) 10 20 30 40 50 Temperature (K) 8K -1.0 -0.8 -0.6 -0.4 -0.2 0.0 0.2 0.4 0.6 0.8 1.0 -10000 -5000 0 5000 10000 Field (O e) Mom ent(em u) T = 200 K T = 100 K T = 50 K T = 25 K T = 5 K Co 2 (CO) 8 Precursor ZFC - FC Low Temperature 0.0E+00 1.0E-05 2.0E-05 3.0E-05 4.0E-05 5.0E-05 6.0E-05 0 10 20 30 40 50 60 Temperature (K) Moment (emu) ZFC FC 0 100 200 300 400 0 50 100 150 200 Tem perature (K ) H -ex (O e) The low temperature peak consistently appears at 8 K, regardless of particle diameter. Peak susceptibility, however, decreases as particle diameter increases; suggesting that the peak is due to a surface effect. One possible explanation is surface oxidation. In order to explore this possibility, ZFC measurements were performed before and after heating the sample in air at 120ºC for 24 hours. The temperature has been plotted on a log scale to reveal the peak at 8 K in the ZFC plot to the right. The 8 K peak may be associated with a layer of cobalt oxide at the surface as suggested by the figure below. ZFC curve before heating. 6 nm spheres with TOPO and OA surfactants. -cobalt core Co 3 O 4 shell ZFC curve of the same sample after heating. Before heating Co 3 O 4 XRD a. before and b. after heating. After heating After heat treatment, the -cobalt core is completely replaced with the spinel phase of cobalt oxide, Co 3 O 4 . The blocking peak, originally at 60 K, is absent in the ZFC plot to the middle right. If the 8 K peak is due to a surface effect, then one would expect that the magnitude of the peak to remain the same before and after heat treatment. The magnitude of the peak susceptibility, however, has dropped by nearly a factor of ten, but this is accounted for by the shoulder of the blocking peak. The phases were identified by x-ray diffraction using a Rigaku 12.5 kW rotating anode x-ray source and goniometer. The diffraction peaks in scan a (lower right figure) correspond to –cobalt, and the peaks in scan b correspond to Co 3 O 4 . The evidence as outlined in the Heat Treatment section of this presentation is compelling for the surface oxide explanation of the 8 K susceptibility peak. However, a thorough investigation requires background measurements of the synthesis precursors as well as a commercially obtained Co 3 O 4 powder. The cobalt particle reaction precursor is Co 2 (CO) 8 . A sample was dried and measured on the SQUID magnetometer. The ZFC – FC curve below does show some indication of the peak at 8 K. One possible explanation of the peak is that some clustering of the precursor results in small particle aggregates with an associated blocking peak at 8 K. Another explanation is direct oxidation of the precursor to form the uncompensated surface spin state, which is consistent with the heat treatment measurements. The ZFC plot below was obtained from the commercial Co 3 O 4 powder. At first look, the 8 K peak is absent, however, a closer inspection reveals some indication of a peak. Again this is consistent with the surface oxide interpretation. We have previously observed that the magnitude of the 8 K peak is inversely proportional to the particle diameter, and this sample consists of a wide size distribution of particles ranging from hundreds of nm to tens of microns. The surfaces of the smallest particles in this distribution are likely contributing to the 8 K peak. Exchange bias was originally observed in samples consisting of micron sized cobalt particles surrounded by a cobalt oxide (CoO) shell.* Bias is due to exchange interaction between the ferromagnetic core and the anti-ferromagnetic shell. The nanoparticle samples that we are discussing here have an –cobalt core and a Co 3 O 4 shell. The Neel temperature for Co 3 O 4 is 40 K (as opposed to 270 K for CoO). A powder sample was prepared with 6 nm spherical particles. The sample was cooled in an applied field of 1000 G to the measurement temperature. Hysteresis plots between +/- 2T at various temperatures appear in the figure below. The applied field axis is reduced so that the exchange bias shift in the hysteresis loop is noticeable. The exchange field H-ex is the extent of the shift of the hysteresis loop from the origin. The figure below is a plot of H-ex as a function of temperature. The blocking temperature of 6 nm particles is approximately 80 K, and the greatest exchange field occurs just below this temperature at about 50 K. Spin-glass-like freezing is one possible physical mechanism that can lead to a particle size-independent maximum in the ZFC curves. Bulk spin glasses have been well studied* and show a shift in the glass transition to lower temperatures as the applied field increases. We observed similar shifts in our measurements, which might be an indication of spin-glass freezing. The variation in saturation magnetization at low temperatures observed in γ–Fe 2 O 3 and Pd-Fe nanoparticle systems has been attributed to surface spin-glass like freezing.** The anisotropy is not bound to the crystal lattice due to the irregularities at the region of broken bonds near the surface. The spin glass transition temperature T g is approximately 8K in this case. At temperatures above T g , the surface spins are strongly paramagnetic and do not couple even in high fields as suggested by the figure (below left). In the presence of an applied field below T g the spins will freeze into the field direction (below right). * W.H. Meiklejohn, C.P. Bean, Phys. Rev., 102, 1413 (1956). * K. H. Fischer, J. A. Hertz, Spin Glasses, Camridge University Press (1991). ** B. Martinez, et. al., Phys. Rev. Lett., 80, 181 (1997). In order to test the applicability of this model, we estimated the linear 0 and non-linear 2 contributions to magnetic susceptibility, and the critical exponent , as a function of temperature in the range 2.5-10K. These data were extracted from a family of field cooled (FC) curves measured at different fields with a low ramp rate on a sample containing 6-nm particles, then compared to the following expansion for the susceptibility: Co 3 O 4 -cobalt T > T g Applied field T < T g Magnetic property measurements of the surfactant-coated core-shell Co/Co 3 O 4 nanoparticles reveal contributions from three interacting and correlated entities: single domain metallic cobalt, which is primarily responsible for the blocking phenomena; antiferromagnetic Co 3 O 4 which is revealed in exchange bias; and the surface layer of the oxide where spins are coupled below a critical temperature, which may be defined as either spin glass freezing or ferromagnetic Curie transition. Since the blocking effect can be controlled by annealing, the low temperature behavior can be isolated and studied separately from blocking, while the interaction between surface and core may be examined by studying the system in the pre-oxidized state. Contact Information Please visit the Integrated Nanosciences & Mesoscale Engineering Group website at: http://depts.washington.edu/kkgroup Or you may email the authors at: Kannan M. Krishnan: [email protected] Michael Beerman: [email protected] H H M 2 0 Isothermal magnetization curves were extracted from the set of FC measurements. Then linear susceptibility 0 was found from the initial slope of the isothermal M(H) curve. The inflection point of this curve gives the transition temperature, which is about 7.5 K. A power law fit was employed to obtain 2 and . The non-linear susceptibility is well behaved below the transition temperature and is two orders of magnitude smaller than 0 . Above the transition temperature, however, 2 and are not well determined due to the limitations of the measurements. The nonlinear exponent is constant below T g , with a value of 3/2 for low fields. These trends do not compare well to bulk spin glasses, where the nonlinear susceptibility has a maximum at the freezing temperature, and the exponent has a minimum. The figure to the left is the family of field cooled susceptibility curves from which the linear susceptibilty 0 , non linear susceptibility 2 and the non linear exponent were obtained and plotted in the figure to the right. 0.000 0.005 0.010 0.015 0.020 0.025 0.030 0 5 10 15 20 Tem pe ratu re (K) M /H (em u/O e) 1 G 5 G 10 G 20 G 40 G 100 G 200 G 500 G For more information concerning particle synthesis and self assembly please see “Controlled Self-assembly of Colloidal Cobalt Nanocrystals Mediated by Magnetic Interactions” Poster presentation 2-ypm52 Location H
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Transcript of M. Beerman, A. Pakhomov, Y. Bao, and K. Krishnan Department of Materials Science and Engineering...
M. Beerman, A. Pakhomov, Y. Bao, and K. KrishnanDepartment of Materials Science and EngineeringUniversity of Washington, Seattle, WA 98195, USA
Work supported by:National Science Foundation NSF/DMR 0203069Campbell Endowment at UW
An anamolous peak has been observed in zero field cooled measurements on systems composed of ε-phase cobalt nanoparticles. The samples were synthesized via a thermal decomposition process resulting in spherical nanoparticles surrounded by an organic surfactant [1,2]. In addition to the blocking temperature peak, which is dependant on particle volume [3] and displays interesting interaction effects, an additional peak appears at 8 K regardless of particle diameter. Zero field cooled and field cooled magnetization versus temperature measurements on spherical particles with mean diameters in the range of 5-20 nm were performed with a SQUID magnetometer, including careful background signal checks. Measurements after heat treatment, to promote oxidation, show that the blocking temperature peak can be effectively removed in fully oxidized powders. The 8 K peak, however, remains after heat treatment, but with a reduced magnitude, and M(H) dependance is hysteretic below this temperature. Additional measurements were performed on commercial micrometer-sized cobalt oxide powder, where this feature is also present, but gives a much weaker contribution to susceptibility. Unreacted synthesis precursors as well as samples with different surfactant species all showed at least some indication of a peak at 8 K. This feature, universal for the system under investigation, has not been observed in Co nanoparticle systems prepared by co-sputtering or co-evaporation of Co with non-magnetic materials. Furthermore, hysteresis measurements, both before and after heat treatment, show a bias when cooled in field from 30 K to below the peak temperature. Analyses of the data leads us to conclude that freezing of frustrated uncompensated spins on the surface of cobalt oxide is the most reasonable explanation of this anomalous low temperature behavior.
[1] V. Puntes et al. Science 291 (2001) 2115[2] Y. Bao, et. al., JMMM, in press. [3] L. Néel, Annales de Géophysique 5 (1949) 99
Epsilon cobalt nanospheres surrounded by a mixture of organic surfactants (TOPO and OA) are synthesized via thermal decomposition of cobalt carbonyl in di-chlorobenzene solvent. The diameter D may be controlled between 5 and 20 nm, and the surfactant length is approximately 1.7 nm.
Surfactants
D
The particles are precipitated from solution onto a thin carbon film for transmission electron microscopy (TEM).
This TEM image shows a self assembled monolayer of 10 nm spheres.(Phillips 400 T at 100 keV)
-cobalt0.0E+00
1.0E-03
2.0E-03
3.0E-03
4.0E-03
5.0E-03
1 10 100 1000
Tem perature (K)
M/H
(em
u/G
g)
0.0E+00
1.0E-04
2.0E-04
3.0E-04
4.0E-04
5.0E-04
1 10 100 1000
Temperature (K)
M/H
(e
mu
/Gg
)
No External Field
27kBTB
ZFC
0.0E+00
2.0E-03
4.0E-03
6.0E-03
8.0E-03
1.0E-02
1.2E-02
0 100 200 300
Temperature (K)
Mo
me
nt
(em
u)
FC
A uniaxial magnetic particle has two stable magnetization orientations as seen in the figure to the left. If an external field is applied along the easy axis then one of the two orientations has a lower energy as seen to the right. As the temperature increases, thermal fluctuations provide the activation energy for the system to relax over the energy barrier into the field direction. This behavior is well known and follows
Applied External Field
BB
Am Tk
VKexp~ 0
where m is the characteristic measurement time ~20s, 0 is the microscopic fluctuation attempt time ~10-10s, KA is the anisotropy constant ~0.21 J/m3 for -Co, and V is the magnetic particle volume. The barrier height may be approximated by 27 kBTB where kB is Boltzmann’s constant and TB is the blocking temperature, which corresponds to the peak temperature in moment versus temperature measurements (as identified to the left).
TB
Two curves are plotted here: zero field cooled (ZFC) and field cooled (FC). ZFC measurements start at low temperature after cooling the sample in zero external field. Magnetization is then measured as the temperature slowly increases. FC measurements start at high temperature where a field is applied and the magnetization is measured as the temperature slowly decreases. The rate of cooling is constant in both cases.
The interesting observation is the second peak at low temperature on the ZFC curve and its associated sharp uptake in magnetization as seen in the FC curve. The focus of this poster presentation is to further characterize and describe this feature.
Magnetic measurements were performed on a Quantum Design MPMS 5 superconducting quantum interference device.
Co Oxide CommercialMicron Sized Powder
0.E+00
1.E-05
2.E-05
3.E-05
4.E-05
5.E-05
6.E-05
7.E-05
8.E-05
9.E-05
0 50 100 150 200 250 300 350
Temperature (K)
M/H
(em
u/G
g)
10 20 30 40 50
Temperature (K)
8K
-1.0
-0.8
-0.6
-0.4
-0.2
0.0
0.2
0.4
0.6
0.8
1.0
-10000 -5000 0 5000 10000
Field (Oe)
Mo
me
nt
(em
u)
T = 200 K
T = 100 K
T = 50 K
T = 25 K
T = 5 K
Co2(CO)8 PrecursorZFC - FC Low Temperature
0.0E+00
1.0E-05
2.0E-05
3.0E-05
4.0E-05
5.0E-05
6.0E-05
0 10 20 30 40 50 60
Temperature (K)
Mo
men
t (e
mu
)
ZFC
FC
0
100
200
300
400
0 50 100 150 200
Temperature (K)
H-e
x (O
e)
The low temperature peak consistently appears at 8 K, regardless of particle diameter. Peak susceptibility, however, decreases as particle diameter increases; suggesting that the peak is due to a surface effect. One possible explanation is surface oxidation. In order to explore this possibility, ZFC measurements were performed before and after heating the sample in air at 120ºC for 24 hours. The temperature has been plotted on a log scale to reveal the peak at 8 K in the ZFC plot to the right.
The 8 K peak may be associated with a layer of cobalt oxide at the surface as suggested by the figure below.
ZFC curve before heating. 6 nm spheres with TOPO and OA surfactants.
-cobalt core
Co3O4 shell
ZFC curve of the same sample after heating.
Before heating
Co3O4
XRD a. before and b. after heating.
After heating
After heat treatment, the -cobalt core is completely replaced with the spinel phase of cobalt oxide, Co3O4. The blocking peak, originally at 60 K, is absent in the ZFC plot to the middle right. If the 8 K peak is due to a surface effect, then one would expect that the magnitude of the peak to remain the same before and after heat treatment. The magnitude of the peak susceptibility, however, has dropped by nearly a factor of ten, but this is accounted for by the shoulder of the blocking peak.
The phases were identified by x-ray diffraction using a Rigaku 12.5 kW rotating anode x-ray source and goniometer. The diffraction peaks in scan a (lower right figure) correspond to –cobalt, and the peaks in scan b correspond to Co3O4.
The evidence as outlined in the Heat Treatment section of this presentation is compelling for the surface oxide explanation of the 8 K susceptibility peak. However, a thorough investigation requires background measurements of the synthesis precursors as well as a commercially obtained Co3O4 powder.
The cobalt particle reaction precursor is Co2(CO)8. A sample was dried and measured on the SQUID magnetometer. The ZFC – FC curve below does show some indication of the peak at 8 K. One possible explanation of the peak is that some clustering of the precursor results in small particle aggregates with an associated blocking peak at 8 K. Another explanation is direct oxidation of the precursor to form the uncompensated surface spin state, which is consistent with the heat treatment measurements.
The ZFC plot below was obtained from the commercial Co3O4 powder. At first look, the 8 K peak is absent, however, a closer inspection reveals some indication of a peak. Again this is consistent with the surface oxide interpretation. We have previously observed that the magnitude of the 8 K peak is inversely proportional to the particle diameter, and this sample consists of a wide size distribution of particles ranging from hundreds of nm to tens of microns. The surfaces of the smallest particles in this distribution are likely contributing to the 8 K peak.
Exchange bias was originally observed in samples consisting of micron sized cobalt particles surrounded by a cobalt oxide (CoO) shell.* Bias is due to exchange interaction between the ferromagnetic core and the anti-ferromagnetic shell. The nanoparticle samples that we are discussing here have an –cobalt core and a Co3O4 shell. The Neel temperature for Co3O4 is 40 K (as opposed to 270 K for CoO).
A powder sample was prepared with 6 nm spherical particles. The sample was cooled in an applied field of 1000 G to the measurement temperature. Hysteresis plots between +/- 2T at various temperatures appear in the figure below. The applied field axis is reduced so that the exchange bias shift in the hysteresis loop is noticeable.
The exchange field H-ex is the extent of the shift of the hysteresis loop from the origin. The figure below is a plot of H-ex as a function of temperature. The blocking temperature of 6 nm particles is approximately 80 K, and the greatest exchange field occurs just below this temperature at about 50 K.
Spin-glass-like freezing is one possible physical mechanism that can lead to a particle size-independent maximum in the ZFC curves. Bulk spin glasses have been well studied* and show a shift in the glass transition to lower temperatures as the applied field increases. We observed similar shifts in our measurements, which might be an indication of spin-glass freezing. The variation in saturation magnetization at low temperatures observed in γ–Fe2O3 and Pd-Fe nanoparticle systems has been attributed to surface spin-glass like freezing.** The anisotropy is not bound to the crystal lattice due to the irregularities at the region of broken bonds near the surface. The spin glass transition temperature Tg is approximately 8K in this case. At temperatures above Tg, the surface spins are strongly paramagnetic and do not couple even in high fields as suggested by the figure (below left). In the presence of an applied field below Tg the spins will freeze into the field direction (below right).
* W.H. Meiklejohn, C.P. Bean, Phys. Rev., 102, 1413 (1956).
* K. H. Fischer, J. A. Hertz, Spin Glasses, Camridge University Press (1991).** B. Martinez, et. al., Phys. Rev. Lett., 80, 181 (1997).
In order to test the applicability of this model, we estimated the linear 0 and non-linear 2 contributions to magnetic susceptibility, and the critical exponent , as a function of temperature in the range 2.5-10K. These data were extracted from a family of field cooled (FC) curves measured at different fields with a low ramp rate on a sample containing 6-nm particles, then compared to the following expansion for the susceptibility:
Co3O4
-cobalt
T > Tg
Applied field
T < Tg
Magnetic property measurements of the surfactant-coated core-shell Co/Co3O4 nanoparticles reveal contributions from three interacting and correlated entities: single domain metallic cobalt, which is primarily responsible for the blocking phenomena; antiferromagnetic Co3O4 which is revealed in exchange bias; and the surface layer of the oxide where spins are coupled below a critical temperature, which may be defined as either spin glass freezing or ferromagnetic Curie transition. Since the blocking effect can be controlled by annealing, the low temperature behavior can be isolated and studied separately from blocking, while the interaction between surface and core may be examined by studying the system in the pre-oxidized state.
Contact Information
Please visit the Integrated Nanosciences & Mesoscale Engineering Group website at:
http://depts.washington.edu/kkgroup
Or you may email the authors at:
Kannan M. Krishnan: [email protected]
Michael Beerman: [email protected]
HH
M20
Isothermal magnetization curves were extracted from the set of FC measurements. Then linear susceptibility 0 was found from the initial slope of the isothermal M(H) curve. The inflection point of this curve gives the transition temperature, which is about 7.5 K. A power law fit was employed to obtain 2 and . The non-linear susceptibility is well behaved below the transition temperature and is two orders of magnitude smaller than 0. Above the transition temperature, however, 2 and are not well determined due to the limitations of the measurements. The nonlinear exponent is constant below Tg, with a value of 3/2 for low fields. These trends do not compare well to bulk spin glasses, where the nonlinear susceptibility has a maximum at the freezing temperature, and the exponent has a minimum.
The figure to the left is the family of field cooled susceptibility curves from which the linear susceptibilty 0, non linear susceptibility 2 and the non linear exponent were obtained and plotted in the figure to the right.
0.000
0.005
0.010
0.015
0.020
0.025
0.030
0 5 10 15 20
Tem perature (K)
M/H
(em
u/O
e)
1 G
5 G
10 G
20 G
40 G
100 G
200 G
500 G
For more information concerning particle synthesis and self assembly please see
“Controlled Self-assembly of Colloidal Cobalt Nanocrystals Mediated by Magnetic Interactions”
Poster presentation 2-ypm52 Location H
![arXiv:1504.04338v1 [math.CV] 16 Apr 2015 · arXiv:1504.04338v1 [math.CV] 16 Apr 2015 BOUNDARY MULTIPLIERS OF A FAMILY OF MOBIUS INVARIANT FUNCTION¨ SPACES GUANLONG BAO AND JORDI](https://static.fdocument.org/doc/165x107/5f7b4fff8c891c00121fec72/arxiv150404338v1-mathcv-16-apr-2015-arxiv150404338v1-mathcv-16-apr-2015.jpg)
