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# Light soaking in metal halide perovskites studied via steady-state microwave conductivity

## Abstract

The light-soaking effect is the observation that under constant illumination the measured power conversion efficiency of certain solar cells changes as a function of time. The theory of the light-soaking in metal halide perovskites is at present incomplete. In this report, we employ steady-state microwave conductivity, a contactless probe of electronic properties of semiconductors, to study the light-soaking effect in metal halide perovskites. By illuminating isolated thin films of two mixed-cation perovskites with AM1.5 solar illumination, we observe a continual increase in photoconductance over a period of many (>12) hours. We can fit the experimentally observed changes in photoconductance to a stretched exponential function, in an analogous manner to bias-stressed thin-film transistors. The information provided in this report should help the community better understand one of the most perplexing open problems in the field of perovskite solar cells and, ultimately, lead to more robust and predictable devices.

## Introduction

Although metal halide perovskites (MHPs) offer the community many exciting opportunities1, consistent understanding and consensus still lags behind more long-studied systems such as organic semiconductors2 or silicon3. Developing a robust understanding of this class of materials is paramount to their future success as a commercial technology. The light soaking effect in MHPs is an example of a phenomenon, which is still lacking a consistent description from the community4,5,6,7,8. Light soaking is the observation that the measured power conversion efficiency (PCE) of certain solar cells changes as a function of time, under constant illumination9. This is a phenomenon observed in wide-range of semiconductors10,11,12, but arguably most pronounced, and intensely studied in hydrogenated amorphous silicon (a-Si:H)13,14,15, copper indium–gallium selenide (CIGS)16,17,18, and MHPs5,8. In a-Si:H, photoconductivity is observed to fall upon prolonged light-soaking (the Staebler-Wronski Effect)13. This behaviour is understood in terms of increased non-radiative recombination occurring via photon-induced meta-stable dangling bonds15,19. In CIGS a persistent increase in dark conductance is observed after illumination from white light18,20.

Although light-soaking in a-Si:H leads to a reduction in performance, light-soaking often leads to an increase in PCE in MHP-based solar cells5,8. A range of hypotheses exist to describe this phenomenon. Light-activated meta-stable states (analogous to those in a-Si:H) is a prevalent description in the literature6,8. Interfacial states between the MHP and electron-blocking-layers/hole-blocking-layers also commonly in feature explanations5,8,21,22. Other explanations involve the migration of ions leading to internal pin-like structures in the perovskite7 or a reduction in trap-state density23,24. Light-induced halide segregation is a popular description for mixed halide perovskites4,23,25,26, in-which optically induced segregation of bromine-rich and iodine-rich regions reduce the overall bandgap in certain compositions (the Hoke Effect)4. The fact that changes in photoluminescence yield27, diffraction spectra26, photoconductance28, and absorption coefficient29 have been observed over periods of seconds to minutes in isolated thin films suggest that the behaviour is at least partially owing to the material itself, rather than interfaces. The theory of light-soaking in MHPs is at present incomplete.

In this report, we employ steady-state microwave conductivity (SSMC)30,31,32 to measure the photoconductance of thin films of MHPs under AM1.5 conditions, over a period of many hours. SSMC is a contactless technique (no metallic contacts nor hole-/electron-blocking layers are required to evaluate photoconductance), which does not require large DC electric fields. These features enable us to make statements on the photo-stability of perovskite compounds themselves, and removes ambiguity arising from interfacial effects5,8,21 or voltage-induced polarization33,34.

## Results

### Photoconductance under white light

The experimental setup employed in this report is shown diagrammatically in Fig. 1a, and is based upon a similar system developed by the Chabinyc Group at the University of California, Santa Barbara30. The system is designed to measure thin films of semiconductors on insulators (as depicted in Fig. 1b)35. The system can be considered a steady-state analogue of conventional time-resolved microwave conductivity (TRMC)36. A standing microwave (~8–9 GHz) electromagnetic field is formed in a waveguide cavity, and a semiconducting sample is positioned at a maximum of the electric-field component of this standing electromagnetic wave. The intensity of microwaves reflected out of the cavity are detected using a Schottky diode and converted to a voltage that is proportional to microwave intensity. For semiconductors, the carrier density should be low in the dark, and the detected voltage will be constant as a function of time. The closed end of the cavity is made of a transparent conductor (indium tin oxide on glass), through-which the sample is illuminated. With an appropriate bandgap, the incident light generates charge carriers in the film. These carriers move under the influence of the standing electric field, and the reflected microwave intensity (and hence detected voltage) changes upon illumination. With knowledge of the cavity properties one can evaluate how the photoconductance of the film changes from the detected voltage36 (see Supplementary Note 1 and Supplementary Fig. 1 for details). The approach provides relative (photo-induced) changes in conductance, $$\Delta G$$, and requires a modulated optical source. In conventional TRMC a pulsed laser is normally used37. In SSMC, a constant optical illumination source is passed through an optical chopper30,32. The measured $$\Delta G$$ is evaluated using a lock-in amplifier.

In this report, we study two mixed halide, mixed-cation perovskite films: a double-cation composition: FA0.83Cs0.17Pb(I0.9Br0.1)3 (FACs) and a triple-cation composition: (FA0.83MA0.17)0.95Cs0.05Pb(I0.9Br0.1)3 (FAMACs), where FA is formamidinium and MA is methylammonium. These compounds were chosen as they are known to lead to high PCEs when employed in solar cells38,39, and have been shown to be highly stable in air40,41. The later property is particularly valuable as we are measuring electronic properties under illumination, in air, for many hours. The films were prepared on 0.9 cm × 0.9 cm × 0.5 mm quartz substrates. The absorption spectra of these two compounds are shown in Fig. 1c, alongside the incident optical spectrum from the solar simulator42.

The samples were left in the SSMC cavity in the dark for >1 hour before measurement. The photoconductance of the films was measured as a function of time for 30 minutes before the shutter was opened, after which the photoconductance was measured every 100 ms for a further 18 hours. The samples were fabricated, transported, and stored under inert atmosphere, but were measured unencapsulated in air. The photoconductance as a function of time is shown for FACs and FAMACs in Fig. 2a, b respectively. The samples were illuminated with white (AM1.5) light from a solar simulator. The optical power density was controlled by passing the incident light through neutral density filters. For both compounds, and for optical power densities ≥10 W/cm2, the conductance is observed to rise over a period of many hours. When illuminated for longer periods of time, the photoconductance of FACs rises sharply for ~12 hours, followed by a slower subsequent rise over 4 days (see Supplementary Note 2 and Supplementary Fig. 2).

The optical power density of commercial solar simulators is known to exhibit small variations as a function of lamp temperature. To rule this out as a potential factor in the increase in $${\mathrm{\Delta }}G$$ as a function of time, the solar simulator was turned on >1 hour prior to measurement in all cases. Supplementary Note 3 describes how the optical power density was monitored as a function of time after the lamp was turned on, as a separate measurement. The results are shown in Supplementary Fig. 3. Although the output does increase for the first 2 minutes, the light source is stable after this time. For this reason, we rule-out instability in lamp output as a contributing factor to the observed increase in $$\Delta G$$.

A similar possible artefact concerns the sample holder temperature, which we address in Supplementary Note 4. In Supplementary Fig. 4 we show the temperature of the sample holder in the cavity measured as a function of time under constant illumination from 20 mW/cm2 AM1.5 solar spectrum. An increase in cavity temperature of ~1°C is observed upon the first 5 minutes of illumination, followed by an increase of another ~1°C, over the subsequent 12 hours. An increase in temperature of 2°C is not expected to affect carrier mobility, and hence conductance, significantly. The mobility of charge carriers in thin films of MHPs are generally observed to be inversely correlated with temperature43. As we observe an increase in conductance as a function of time, it is hence reasonable to conclude that changes in $$\Delta G$$ are unlikely to be owing to changes in cavity temperature.

Given the reasons described in the preceding two paragraphs, we interpret the increase in conductance observed in Fig. 2a, b in terms of photo-induced instabilities of the perovskite; consistent with previous observations of the light-soaking effect in the literature5,7,8,25. It has previously been shown30 that values of $$\Delta G$$ measured from SSMC measurements can be used to evaluate a proxy for the mobility-lifetime product: $$\phi {\mathrm{\Sigma }}\mu \tau$$, as defined by Eq. 1.

$$\phi {\mathrm{\Sigma }}\mu \tau = \phi \left( {\tau _{\mathrm{e}}\mu _{\mathrm{e}} + \tau _{\mathrm{h}}\mu _{\mathrm{h}}} \right)$$
(1)

$$\mu _{\mathrm{e}}$$ and $$\mu _{\mathrm{h}}$$ are the average electron and hole mobilites measured over the sample area, respectively. $$\tau _{\mathrm{e}}$$ and $$\tau _{\mathrm{h}}$$ are representative electron and hole lifetimes, respectively. Here $$\phi$$ is the carrier-generation efficiency, i.e., the number of electron-hole pairs generated per absorbed photon ($$\phi \in \left[ {0,1} \right]$$). In materials with low exciton binding energies, such as MHPs44, one can assume that $$\phi \approx 1$$ and hence interpret $$\phi {\mathrm{\Sigma }}\mu \tau \approx \mu _{\mathrm{e}}\tau _{\mathrm{e}} + \mu _{\mathrm{h}}\tau _{\mathrm{h}}$$. $$\phi {\mathrm{\Sigma }}\mu \tau$$ has the same units as mobility-lifetime product, but carrier-specific information is obscured. The derivation of $$\phi {\mathrm{\Sigma }}\mu \tau$$ is described in Supplementary Note 5 and previous work30. The derivation of $$\phi {\mathrm{\Sigma }}\mu \tau$$ assumes that the semiconductor is in steady state at the end of each illumination cycle. We validate this assumption for a chopper frequency of 4 kHz in Supplementary Note 6 and Supplementary Fig. 5. $$\phi {\mathrm{\Sigma }}\mu \tau$$ as a function of time under illumination is shown for FACs and FAMACs in Fig. 2c, d, respectively.

### Time-resolved microwave conductivity measurements

With $$\phi$$ assumed to be very close to unity in these compounds, the question remains as to the relative contributions of mobility and lifetime to $$\phi {\mathrm{\Sigma }}\mu \tau$$. It is reasonable to expect that optically pre-filling traps or lower-energy transport states could affect average mobility. To help address this question, conventional TRMC measurements were carried out on an FACs sample with and without background illumination from AM1.5 illumination. A schematic diagram of the TRMC system employed is shown in Fig. 3a. The TRMC and SSMC systems used in this study are modified versions of the same apparatus35,37. TRMC is an established technique36, for measuring the photoconductance of a film upon excitation from a pulsed (~nanosecond) laser. Further analysis36 allows one to evaluate a proxy for mobility: $$\phi {\mathrm{\Sigma }}\mu$$, as given by Eq. 2:

$$\phi {\mathrm{\Sigma }}\mu = \phi \left( {\mu _{\mathrm{e}} + \mu _{\mathrm{h}}} \right)$$
(2)

Conventionally, TRMC measurements are carried out on-resonance, and changes in detector voltage are implicitly assumed to be owing to real changes in conductance. However, by carrying out measurements as a function of microwave frequency, it has been shown that it is possible to resolve real ($$\Delta G \equiv\Delta G^{\prime}$$) and imaginary ($$\Delta G^{\prime\prime}$$) components of conductance45. The extraction and evaluation of $$\Delta G^{\prime}$$ and $$\Delta G^{\prime\prime}$$ is described in Supplementary Note 7, and we here just consider the real photoconductance: $$\Delta G^{\prime}$$ (i.e., the contribution owing to free carriers only). As depicted in Fig. 3a, we here use a beam splitter to mix AM1.5 constant (unchopped) illumination with a ~5 ns 532 nm pulse from a Nd:YAG laser. The optical power density incident on the sample owing to the solar simulator was 10 mW/cm2. Example TRMC transients ($$\Delta G^{\prime}$$ as a function of time) are shown in Supplementary Fig. 6 for an FACs sample before, during, and after illumination from 10 mW/cm2 AM1.5 light. The purpose of this TRMC experiment is to not study light-soaking directly28, but to elucidate what effect, if any, background illumination has on carrier mobility.

Figure 3b shows the TRMC figure of merit ($$\phi {\mathrm{\Sigma }}\mu$$) extracted from the real photoconductance as a function of laser fluence, before, during, and after illumination from 10 mW/cm2 AM1.5 light. At high laser fluence, a significant amount of bimolecular and Auger recombination will occur during the ~5 ns laser pulse, resulting in a reduction in peak observable photoconductance, and hence a reduction in extracted $$\phi {\mathrm{\Sigma }}\mu$$ (as observed in Fig. 3b). This is a widely observed phenomenon in TMRC, and models have been developed to account for it46. By applying such a model46, we approximate representative values of $$\phi {\mathrm{\Sigma }}\mu$$, as shown in Fig. 3c. Although TRMC is known to underestimate mobilites in certain cases47, relative measurements are generally unambiguous. It is clear from the data presented in Fig. 3 that illumination at 10 mW/cm2 appears to have no significant impact on the average measured mobility of free charge. This result suggests that the changes in $$\phi {\mathrm{\Sigma }}\mu \tau$$ occurring over many hours, shown in Fig. 2, are unlikely to be owing to an enhancement in average mobility alone.

The recovery of photoconductance was investigated in Supplementary Note 8, Supplementary Figs. 7 and 8. An FACs sample was illuminated with 10 mW/cm2 AM1.5 illumination for 12 hours, then left in the dark for 12 hours, before being illuminated again with AM1.5 illumination for a further 12 hours (Supplementary Fig. 8). Immediately after the shutter is initially opened (time = 0) the photoconductance rises to ~ 350 nS, before increasing to $$\Delta G$$ $$\approx$$ 675 nS during the subsequent 12 hours of illumination. When the shutter is closed, the measured photoconductance drops to zero. When the shutter is opened for the second time (after a total of 24 hours), the photoconductance partially recovers to around 500 nS. After another 12 hours of light soaking, the photoconductance eventually reaches a peak value of $$\Delta G$$ $$\approx$$ 700 nS. The data suggest the process is, at least partially, reversible.

### Photoconductance under green light

To probe whether photo-induced changes in bandgap4, and hence changes in the fraction of photons absorbed, are responsible for the observed light soaking, samples were also measured with above bandgap light from a green light-emitting diode (LED), with a peak emission wavelength of 525 nm. Figure 4a shows the photoconductance of thin films of FACs and FAMACs measured using SSMC, under illumination from green light at 10 mW/cm2. Figure 4a demonstrates that these samples exhibit the light soaking effect under monochromatic light, and it is hence unlikely that photo-induced changes in optical bandgap are the only relevant process. The most striking difference between SSMC data acquired under white and green light is the magnitude of photoconductance, which is roughly an order of magnitude higher under green light for both FACs and FAMACs. The spectrum-integrated optical power density is 10 mW/cm2 for each source, so the percentage of above-gap photons will be higher for the green light source. This difference alone however can only account for a ~2–3-fold increase in $${\mathrm{\Delta }}G$$ (calculation not shown) not a factor of ~10. Previous reports48 have illustrated that the differences in excitation wavelength can lead to distinct light-soaking phenomena. For this reason, we interpret the results presented in Figs. 2 and 4a as being evidence that light soaking under white light is due to multiple, competing, processes.

The values of $$\Delta G$$ obtained from SSMC data are owing to photo-induced charge carriers only, and do not enable us to comment on changes in background conductance. For this reason we carried out simple two-terminal device measurements49 of a thin film of FAMACs under illumination. Figure 4b shows the conductance $$G$$, (i.e., sum of both background and photogenerated conductance) of an FAMACs film measured for ~90 minutes in the dark, before being illuminated with green light (peak 525 nm) at an intensity of 10 mW/cm2 for 10 hours. After illumination, the device was held in the dark for a further 10 hours. A bias of 2 V was continually applied throughout the measurement. The electrodes were gold and the channel length was 10 μm. Upon application of the voltage in the dark, the current increases for ~30 minutes, before stabilizing. Upon illumination, an initial spike in $$G$$ is observed, after which the conductance slowly falls over the subsequent 10 hours. After the light is removed, the film retains a conductance ~25–35 times higher than its pre-illumination value. The same data are plotted with a linear y axis scale in Supplementary Note 9 and Supplementary Fig. 9.

The reduction in total conductance ($$G$$) as shown in Fig. 4b is, on first inspection, contradictory to the observed increase in photoconductance ($$\Delta G$$) observed in the SSMC measurements (e.g., Fig. 4a). We believe the observed reduction in $$G$$ is likely to be owing to either the presence of injecting electrodes, the SiO2/MHP interface, and/or the constant DC bias. The ionic polarizability34 of MHPs is believed to screen electronic charge at room temperature, and has led to difficulties in demonstrating field-effect devices based on MHPs50,51. In addition, extended illumination in other material systems is known to lead to a photo-gating affect52, resulting in a shift of the threshold voltage in thin-film transistors. A positive shift in threshold voltage under illumination would be manifest as a lower current at zero gate voltage (i.e., what we have in our two-terminal measurements). Combined, these contributions make it challenging to isolate a single mechanism for the decrease in $$G$$ over the period of illumination, and indeed illustrate why contactless measurements can in certain situations provide less ambiguity. For this reason, we do not strongly advocate a single process for the decrease in $$G$$ observed during illumination. We instead focus on the fact that the film retains an increased conductance after illumination. Because the conductance was stable before illumination, we can say unambiguously that extended illumination has resulted in a persistent background conductance.

Persistent photoconductivity as observed in Fig. 4b is reminiscent of a similar effect observed in CIGS systems20, which has previously been attributed to light-induced vacancy formation53,54. To study this possibility in our systems, we carried out X-ray photoemission spectroscopy (XPS) on films of FAMACs on quartz. Figure 4c shows the ratio of metallic Pb0 to Pb2+ of two FAMACs samples, measured over period of 12 hours, one in the dark and another under illumination from green light with an approximate intensity of 10 mW/cm2, as separate experiments. The relative content of other relevant atomic species (e.g., Cs, N, C, I) were observed to be roughly constant as a function of time—see Supplementary Note 10 and Supplementary Fig. 10. Although the conversion of Pb2+ to Pb0 is consistent with our observation of increased background conductance after illumination (Fig. 4b), it is currently unclear why this would lead to an increase in photoconductance (e.g., Fig. 2). We acknowledge the nature of photogenerated defects is a complex issue in mixed halide systems55, and that further experiments are required.

The formation of metallic lead upon illumination has previously been observed via XPS56 in methylammonium lead iodide ((MA)PbI3), and has been suggested to occur via (MA)PbI3 degradation into PbI257, which is itself known to decompose under illumination and heat into Pb and I258. To rule out any irreversible changes in structure owing to illumination, we carried out X-ray diffraction before and after light soaking under AM1.5 spectra (see Supplementary Note 11 and Supplementary Fig. 11). FACs and FAMACs samples were illuminated at 10 mW/cm2 or 40 mW/cm2 for 18 hours, then were stored in the dark in N2 > 1 week. Although we do observe a PbI2 impurity in all FAMACs films (including pristine films), no significant changes in structure were observed between pristine samples and those exposed to illumination. We hence conclude that the structure is largely unaffected by light soaking at these intensities and for these durations. As we also observe light-soaking to be, at least partially, reversible (see Supplementary Fig. 8), we do not attribute our observations to an irreversible degradation of the perovskite to a lower-dimensional structure.

## Discussion

Although extended illumination of MHPs often results in an increased photoconductance, in a-Si:H extended illumination is known to cause a reduction in photoconductance13,15,19. It is noteworthy that descriptions of changes in photoconductance in a-Si:H are similar, phenomenologically, to those used to describe voltage-induced bias-stress behaviour in a-Si:H thin-film transistors (TFTs):59 extended application of a gate voltage leads to the creation of reversible trap states, an increase in persistent charge, and a subsequent change in threshold voltage. The threshold voltage (and hence interface trap concentration) in TFTs is often modelled with a (quasi-empirical) stretched-exponential function60,61,62. An analogous stretched-exponential equation for photoconductance is given by Eq. 3:

$$\Delta G\left( t \right) = \Delta G_0\left( {1 - \exp \left[ { - \left( {\frac{t}{\tau }} \right)^\beta } \right]} \right)$$
(3)

Where $$t$$ is the time under illumination, and $$\Delta G_0$$ is the photoconductance as $$t \to \infty$$. $$\beta$$ is the dimensionless dispersion parameter, defined by Eq. 4:

$$\beta = \frac{T}{{T_0}}$$
(4)

$$T$$ is temperature, and $$T_0$$ is the characteristic trap-state width (units of temperature) of the material. The parameter $$\tau$$ is the temperature-activated relaxation-time, and is given by Eq. 5:

$$\tau = \nu ^{ - 1}\exp \left( {\frac{{E_{\mathrm{A}}}}{{k_{\mathrm{B}}T}}} \right)$$
(5)

$$E_{\mathrm{A}}$$ is the mean activation energy for trap formation, $$k_{\mathrm{B}}$$ is the Boltzmann Constant, and $$\nu$$ is the trap-formation attempt-frequency. It is reasonable to assume that $$\beta$$ and $$E_A$$ are material properties (i.e., independent of illumination intensity), whereas $$\nu$$ and $$\Delta G_0$$ are both dependent on illumination intensity.

Figure 5 shows Eqs. 3 and 5 fitted to the time-dependent photoconductance of FACs and FAMACs, under illumination from 10 mW/cm2 and 20 mW/cm2 white light. For each material, $$\beta$$ and $$E_{\mathrm{A}}$$ were forced to be independent of light intensity, whereas $$\nu$$ and $$\Delta G_0$$ could vary as a function of intensity. From these fits, we obtain values of $$\beta$$ = 0.40 and $$E_{\mathrm{A}}$$ = 0.77 eV for FACs and β = 0.44 and $$E_{\mathrm{A}}$$ = 0.77 eV for FAMACs. From past TFT voltage-induced bias-stress measurements, typical room temperature values of β = 0.4–0.6 and $$E_{\mathrm{A}}$$ = 0.9–1.0 eV have been reported for a-Si63,64, $$\beta$$ = 0.4–0.5 and $$E_{\mathrm{A}}$$ = 0.5–0.6 eV for amorphous indium–gallium-zinc oxide (IGZO)62, and $$\beta$$ = 0.44 and $$E_{\mathrm{A}}$$ = 0.5–0.6 eV for organic semiconductors65.

Although empirical, using a stretched exponential to describe light soaking is not an unreasonable approach. There are known similarities between the application of extended illumination and application of extended gate voltages in a-Si:H systems15,60. Given that bias stressing is routinely observed to occur over periods of multiple hours66, it is also conceivable that extended illumination leads to generation and/or removal of persistent trap states over similar timescales. Similarly, the large absorption coefficients in MHPs suggest an inhomogeneous vertical distribution of charge, reminiscent of charge distributions in TFTs, will be produced under constant illumination67. Although further study (including temperature-dependent measurements, wavelength-dependent measurements, composition-dependent measurements, and hole- and electron-resolved parameters) is clearly needed to elucidate the details of this phenomenon, it is hoped the data presented here will guide future experiments in the endeavour of resolving the light-soaking effect.

In conclusion, we have used SSMC to probe the time-dependent photoconductance of metal halide perovskites. By illuminating isolated thin films of two mixed halide, mixed-cation perovskites with AM1.5 solar illumination, we observe a continual increase in photoconductance over a period of 18 hours. As our technique is contactless, these results illustrate how light soaking cannot exclusively be an interfacial effect, nor an effect owing to DC external fields alone. By applying time-resolved microwave conductivity to films under illumination from AM1.5 illumination at an incident power density of 10 mW/cm2, we also rule out changes in average carrier mobility as being a dominant mechanism. The mechanisms responsible for this observation remain unresolved but are likely owing to multiple competing factors. We are able to fit the observed changes in photoconductance to a quasi-empirical stretched-exponential function, in an analogous manner to bias-stressed TFTs. The information provided in this report should help the community better understand one of the most perplexing open problems in the field of perovskite solar cells and, ultimately, lead to more robust and predictable devices.

## Methods

### Sample preparation

Lead iodide (PbI2, 99.999%, metals basis) was purchased from Alfa-Aesar and lead bromide (PbBr2, ≥98%) from Alfa-Aesar. Cesium iodide (CsI, 99.99%) was purchased from Alfa-Aesar. Formamidinium iodide (FAI) and methylammonium iodide (MAI) were purchased from GreatCell Solar. To form the mixed-cation lead mixed-anion perovskite precursor solutions, CsI, FAI, PbI2, and PbBr2 were prepared in the way corresponding to the exact stoichiometry for the desired metal halide perovskite composition in a mixed organic solvent system comprising anhydrous N,N-dimethylformamide (DMF) and anhydrous dimethyl sulfoxide (DMSO) at the ratio of DMF:DMSO = 4:1. The perovskite precursor concentration used was 1.45 M. The deposition of the mixed-cation lead mixed-anion perovskite layers was carried out using a spin coater in a nitrogen-filled glove box with the following processing parameters: starting at 1000 rpm for 5 sec (ramping time of 5 sec from stationary status) and then 5000 rpm (ramping time of 5 sec from 1000 rpm) for 30 sec. Before the end of the spinning process, a solvent-quenching method68 was used by dropping 300-µL anisole onto the spinning substrates at 40 sec after the start of the spin-coating process. The thermal annealing process (100 °C for 50~60 min) was then carried out for the formation of the perovskite layer.

### Ultraviolet–visible spectroscopy

The ultraviolet–visible (UV–Vis) transmission measurements were performed using a Shimadzu UV‐2600 UV–Vis spectrophotometer. The measurements were carried in air at room temperature.

### SSMC experiments

A schematic representation of the SSMC system employed in this study is shown in Fig. 1a35. This SSMC system is home-built and based on similar systems employed at the University of California, Santa Barbara30 and the National Renewable Energy Laboratory32. A microwave frequency oscillatory electric signal is generated using a Sivers IMA VO4280X/00 voltage-controlled oscillator (VCO). The signal has an approximate power of 16 dBm and a tunable frequency between 8 GHz and 15 GHz. The VCO is powered with an NNS1512 TDK-Lambda constant 12 V power supply, and the output frequency is controlled by a Stahl Electronics BSA-Series voltage source. Both the power supply and tuning voltage are filtered by a combination of active and passive filters before being supplied to the VCO. The oscillatory signal is incident on an antenna inside a WR90 copper-alloy waveguide. The microwaves emitted from the antenna pass through an isolator and an attenuator before they are incident on a circulator (Microwave Communication Laboratory Inc. CSW-3). The circulator acts as uni-directional device in which signals entering from port 1 exit through port 2 and signals entering from port 2 exit through port 3. The incident microwaves pass through a fixed iris (6.35 mm diameter) into a sample cavity. The cavity supports a TE103 mode standing wave and consists of an ITO-coated glass window that allows optical access to the sample. The sample is mounted inside the cavity at a maximum of the electric-field component of the standing microwaves, using a 3D-printed PLA sample holder. Microwaves reflected from the cavity are then incident on port 2 of the circulator, exiting through port 3, directed through an isolator, and onto a zero-bias Schottky diode detector (Fairview Microwave SMD0218). The detector outputs a voltage which is linearly proportional to the amplitude of the incident microwaves. The amplifier depicted in Fig. 1a is bypassed in SSMC measurements using two computer controlled Mini-Circuits MTS-18-12B+ transfer switches. An Oriel LCS-100 solar simulator or a Thorlabs SOLIS-525C high power green LED was used to illuminate the sample. The solar simulator was turned on >1 hour before all SSMC measurements carried out. Light was passed through a Stanford Research Systems SR540 optical chopper operating at 4 kHz for all measurements. The chopped light was passed through a range of neutral density filters before being incident on a computer controlled ThorLabs SH1 Optical Beam Shutter. The optical power density was measured to be ~0 mW/cm2 when the beam obscured by a chopper blade in all cases, and incident optical power density was measured when unobscured by chopper blades. The incident optical power density was measured using a Newport 919P-003-10 high sensitivity thermopile sensor and a Newport 1936-R Power Meter. The sample was left in the dark for 30 minutes before the shutter was opened, and the measurement then continued for a further 18 hours. The peak-to-peak detector voltage was measured, unamplified, using a Stanford Research Systems SR865 lock-in amplifier, in phase-locked mode. The voltage was sampled every 100 ms, then binned into 1-minute intervals in order to improve the signal-to-noise ratio. The photoconductance was evaluated from the peak-to-peak voltage as described in Supplementary Note 1. The proxy for mobility-lifetime product ($$\phi {\mathrm{\Sigma }}\mu \tau$$) was approximated from photoconductance, incident optical power density and sample absorption, as described in Supplementary Note 5.

### Time-resolved microwave conductivity experiments

A schematic representation of the TRMC system employed here is depicted in Fig. 3a. This is a modified version of the SSMC system depicted in Fig. 1a, and is very similar to the system as described previously37. Unlike the SSMC experiment, in the TRMC experiment the detected voltage signal is amplified by a Femto HAS-X-1-40 high-speed amplifier (gain = ×100). The amplified detector voltage is measured as a function of time by a Textronix TDS 3032 C digital oscilloscope. Instead of using a solar simulator and an optical chopper, a Continuum Minilite II pulsed Nd:YaG laser is used to illuminate the sample. The laser pulse has a wavelength of 532 nm, a full width at half-maxima of ~5 ns and a maximum fluence incident on the sample of ~1015 photons/cm2/pulse. An external trigger link is employed to trigger the oscilloscope before the laser fires. The real photoconductance, imaginary photoconductance, and TRMC figure of merit ($$\phi {\mathrm{\Sigma }}\mu$$) are extracted from the detected voltage as described in Supplementary Note 7. In order to study the effect of background illumination on $$\phi {\mathrm{\Sigma }}\mu$$, the sample was exposed to continuous illumination from AM1.5 solar illumination, whereas the conventional TRMC experiment was taking place. The AM1.5 illumination was mixed with that from the pulsed laser using a 90:10 beam splitter. The power density incident on the sample from the AM1.5 solar spectrum was 10 mW/cm2.

### X-Ray diffraction

X-ray diffraction (XRD) patterns were obtained with an X-ray diffractometer (D8-Discover, Bruker) using Cu Kα x-ray source (λ = 1.541 Å).

### Two-terminal device measurements

Pre-patterned bottom-gate, bottom-contact transistor substrates were purchased from Fraunhofer IPMS. Substrates consisted of highly n-doped silicon (n $$\approx$$ 3 $$\times$$ 1017 m−3) wafers with thermally grown 230 nm SiO2 insulating layer. Source and drain electrodes 30 nm-thick Au on a 10 nm adhesion layer of indium tin oxide. The channel length of the device measured was 10 μm and the device width 10 mm. The FAMACs film was deposited onto the pre-patterned substrate following an identical procedure used for films deposited on quartz substrates. Two-terminal device measurements were carried out using an Everbeing C-2 probe station and a Keithley 2400 sourcemeter controlled by a PC via a general-purpose interface bus interface. The common gate electrode was grounded and the source-drain voltage was 2 V for all measurements. The two-terminal measurements were carried out in N2 at ambient pressure.

### X-ray photoelectron spectroscopy

X-ray photoelectron spectroscopy measurements were conducted in a custom-built near-ambient pressure system from SPECS Surface Nano Analysis GmbH using monochromatized Al Kα radiation (hν = 1486.6 eV, 50 W, 15 kV) under ultra-high vacuum condition (3 × 10−10 mbar). The electron analyzer pass energy was set to 35 eV and normal emission was used with an analytical spot size of ~ 300 μm. Two FAMACs samples were studied with and without illumination from green light (maximum emission 525 nm) through a view port on the XPS chamber. To compensate the charging effect, a flood gun with low energy electron energy of 0.2 eV with currents set to 0.6 and 0.8 µA for samples measured with light source On and Off, respectively. The XPS data were charge corrected to the C 1 s aliphatic carbon binding energy (Eb) at 284.8 eV69. The surface chemistry was investigated by collecting XPS data (non-stop) for an extended period up to 12 hours. XPS core levels of Pb 4 f, Cs 3d, I 3d, C 1 s, and N 1 s were recorded with an energy step size of 0.1 eV. All spectra were fit by CasaXPS software using Gaussian-Lorentzian line shapes with a linear background. The atomic ratios R of the samples were obtained using Eq. 6 adapted from Physical Electronics (Perkin Elmer)69. Values of the corresponding atomic sensitivity factors S are based on the peak area measurements obtained from empirical data70 and values for I, which is the number of photoelectrons per second to scale the measured areas in a specific spectra peak computed from the corresponding fitted spectra using CasaXPS.

$$R = \frac{{I_{\mathrm{x}}/S_{\mathrm{x}}}}{{I_{\mathrm{Y}}/S_{\mathrm{Y}}}}$$
(6)

## Data availability

The data sets generated during and/or analysed during the current study are available from the corresponding author upon reasonable request.

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## Acknowledgements

We thank Bernard Wenger for insightful discussions. C.L.W. and J.G.L. acknowledge support from the National Science Foundation (grant number: 1902032). Y.-H.L. and H.J.S. acknowledge support from the UK Engineering and Physical Sciences Research Council (grant number: EP/M015254/2). This work was performed, in part, at the Northwest Nanotechnology Infrastructure, a member of the National Nanotechnology Coordinated Infrastructure, which is supported by the National Science Foundation under Grant No. NNCI-1542101. Acquisition of the ambient pressure X-ray photoelectron spectroscopy/ambient pressure scanning tunnelling microscopy system was supported by the National Science Foundation-Major Research Instrumentation program (Grant No. DMR-1429765), the M.J. Murdock Charitable Trust, Oregon BEST, Oregon Nanoscience and Microtechnologies Institute, and Oregon State University.

## Author information

Authors

### Contributions

J.G.L. conceived the project. C.L.W. carried out SSMC measurements and analysed the data. L.A. carried out TRMC measurements and analysed the data. Y.-H.L. and W.L. prepared the samples. M.J.H. carried out XRD measurements and analysis. R.A. and R.E. performed and analysed XPS measurements. G.S.H., H.J.S., and J.G.L. acquired funding, directed team members, devised strategy, and assisted with interpretation of results. All other authors discussed the results and contributed to the writing of the paper.

### Corresponding author

Correspondence to John G. Labram.

## Ethics declarations

### Competing interests

The authors declare no competing interests.

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

## Rights and permissions

Reprints and Permissions

Watts, C.L., Aspitarte, L., Lin, YH. et al. Light soaking in metal halide perovskites studied via steady-state microwave conductivity. Commun Phys 3, 73 (2020). https://doi.org/10.1038/s42005-020-0350-2

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# Why isn't $\mathbb{CP}^2$ a covering space for any other manifold?

This is one of those perhaps rare occasions when someone takes the advice of the FAQ and asks a question to which they already know the answer. This puzzle took me a while, but I found it both simple and satisfying. It's also great because the proof doesn't use anything fancy at all but it's still a very nice little result.

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Could you please clarify the question a bit? Are you looking for a counterexample? Why should $\mathbb{CP}^2$ be a covering space for any other manifold? – Rasmus Mar 3 '11 at 11:35
I couldn't tell you exactly why $\mathbb{CP}^2$ should cover any other manifolds, other than that it's quite symmetric and also simply-connected and so it's totally believable that it might admit a free action of a finite group. But I'm not looking for a counterexample, because the statement just so happens to be true. – Aaron Mazel-Gee Mar 3 '11 at 17:16
How do you know the statement just so happens to be true? – Mariano Suárez-Alvarez Apr 1 '11 at 18:13
What happens with $\mathbb{C} \mathbb{P}^n$ for other $n$? I know the answer for $n = 1$: $\mathbb{C} \mathbb{P}^1$ covers $\mathbb{R} \mathbb{P}^2$... – Pete L. Clark Apr 1 '11 at 19:26
@Mariano: I was asking because I knew the answer but I thought it was a fun puzzle and wanted to share it. – Aaron Mazel-Gee Apr 2 '11 at 5:44

Here's another argument that has the disadvantage of being less elementary, but the advantage of working on all $\mathbb{C}P^{2k}$ simultaneously. (This also answers Pete's question in the comments).

We're going to apply the Lefshetz fixed point theorem which states the following: Suppose $f:M\rightarrow M$ with $M$ "nice enough" (certainly, this applies to compact manifolds - I think it applies to all compact CW complexes). Then $f$ induces a (linear) map $f_*:H_*(M)/Torsion\rightarrow H_*(M)/Torsion$. Let $Tr(f)\in\mathbb{Z}$ denote the trace of this map. If $Tr(f)\neq 0$, then $f$ has a fixed point.

Now, we'll show that every diffeomorphism $f:\mathbb{C}P^{2k}\rightarrow \mathbb{C}P^{2k}$ has trace $\neq 0$, so that every diffeomorphism has a fixed point. Believing this for a second, note that every element of $\pi_1(X)$ for a hypothetical space $X$ covered by $\mathbb{C}P^{2k}$ acts by diffeomorphisms, and thus has a fixed point. But it is easy to show that the only element of the deck group which fixes any point must be the identity. It follows that $\pi_1(X)$ is trivial, so $X=\mathbb{C}P^{2k}$.

So, why does every diffeomorphism of $\mathbb{C}P^{2k}$ have a fixed point? Well, every diffeomorphism (or even homotopy equivalence!) must act as multiplication by $\pm 1$ on each of the $2k+1$ $\mathbb{Z}$s in the cohomology ring of $\mathbb{C}P^{2k}$ and the trace of the induced map is the sum of all the $\pm 1$s. But since there is an odd number of $\pm 1$s, they can't sum to 0 (by, say, checking the parity), so by the Lefshetz fixed point theorem, every diffeomorphism (or even homotopy equivalence) must have a fixed point.

What about $\mathbb{C}P^{2k+1}$? Now we must investigate using the ring structure of $\mathbb{C}P^{2k+1}$. Since there is a single multiplicative generator, once we know what happens on $H^2(\mathbb{C}P^{2k+1})$ we know what happens everywhere. It's easy too see that every orientation preserving homotopy equivalence must have a fixed point: if $f$ is orientation preserving, it's the identity on $H^{4k+2}(\mathbb{C}P^{2k+1})$, which implies it must have been the identity on $H^2(\mathbb{C}P^2)$ so it's the identity on all cohomology groups. Thus, the trace of such an $f$ is $2k+1\neq 0$, and so, by the Lefshetz theorem, this map has a fixed point.

As an immediate corollary, if $\mathbb{C}P^{2k+1}$ covers anything, it can only double cover it. For the product of any two nontrivial elements in the deck group must be trivial: any nontrivial map must be orientation reversing and the composition of two orientation reversing maps is orientation preserving, hence has a fixed point, hence is the identity. That is, any two nontrivial elements product to $e$. It's easy to show that this implies the Deck group is $\mathbb{Z}/2\mathbb{Z}$ (or trivial).

In fact $\mathbb{C}P^{2k+1}$ does double cover something (though, to my knowledge, it doesn't have a more common name, except in the case of $\mathbb{C}P^1 = S^2$ double covering $\mathbb{R}P^2$). In homogeneous coordinates, the involution maps $[z_0:z_1:...:z_{2k+1}:z_{2k+2}]$ to $[-z_1:z_0:...:-z_{2k+2}:z_{2k+1}]$. This involution acts freely, and the quotient of $\mathbb{C}P^{2k+1}$ by the involution is a space which $\mathbb{C}P^{2k+1}$ double covers.

I do not know if $\mathbb{C}P^{2k+1}$ covers anything else.

Incidentally, just to preempt a bit, the space $\mathbb{H}P^{n}$ doesn't cover anything unless $n=1$. The proof is much more complicated in general (though the case where $n$ is even follows precisely as it did in the $\mathbb{C}P^{2k}$ case).

In general, one needs to compute Pontrjagin classes and note that they are preserved by diffeomorphisms.

We have $p_1(\mathbb{H}P^n) = 2(n-1)x$ where $x$ is a particular choice of generator for $H^4(\mathbb{H}P^n)$. Since any diffeomorphism must preserve $p_1$, it follows that so long as $n\neq 1$, we must have $x\rightarrow x$ on $H^4$. Then, the Lefshetz theorem once again guarantees a fixed point.

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This is a really cool application of Lefschetz! As for the question of whether $\mathbb{C}P^{2k+1}$ double-covers anything else, is this just a question of whether there are any other, non-conjugate automorphisms in $Aut(\mathbb{C}P^{2k+1}$? – Aaron Mazel-Gee Apr 3 '11 at 16:53
Well, not just automorphisms, but involutions with no fixed points, but yes. – Jason DeVito Apr 3 '11 at 20:14
Slightly relatedly, I just learned today that the action of $\mathbb{Z}/2$ on $\mathbb{C}P^2$ given by $[z_0:z_1:z_2]\mapsto [\overline{z_0}: \overline{z_1} : \overline{z_2} ]$ has quotient homeomorphic to $S^4$. Of course this isn't exactly what we're talking about since the action isn't free, but I think it's pretty cool nonetheless. This was proved by Arnold. The professor who mentioned this didn't know offhand whether this was known to be diffeomorphic to $S^4$, though... – Aaron Mazel-Gee Apr 6 '11 at 2:28
That's pretty interesting! Since it's not a free action, the quotient doesn't inherit a smooth structure from $\mathbb{C} P^2$, so I'm not sure what it means for this $S^4$ to be diffeomorphic to the standard one. – Jason DeVito Apr 6 '11 at 3:11
Well, the quotient $S^4$ will certainly be an orbifold and hence smooth almost everywhere. Said another way, there is an open dense set of points in $\mathbb{C} P^2$ so that the action preserves these points and is free on them. The quotient will be an open dense subset of the $S^4$. I think the set of "bad" points is diffeomorphic to $\mathbb{R}P^2$ in both $\mathbb{C}P^2$ and the quotient. – Jason DeVito Apr 6 '11 at 4:21

Euler characteristic is multiplicative, so (since $\chi(P^2)=3$ is a prime number) if $P^2\to X$ is a cover, $\chi(X)=1$ and $\pi_1(X)=\mathbb Z/3\mathbb Z$ (in particular, X is orientable). But in this case $H_1(X)$ is torsion, so (using Poincare duality) $\chi(X)=1+\dim H_2(X)+1>1$.

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I like this argument, but it may be worth pointing out that your'e also implicitly using the fact that $\pi_1(X) = \mathbb{Z}/3\mathbb{Z}$ implies $X$ is orientable. – Jason DeVito Apr 1 '11 at 19:24
this is a nice answer. If I am understand it properly, to apply PD we need to know that $X$ is orientable. For this, you could e.g. observe that a nonorientable manifold has a connected degree 2 "orientation cover" and thus an index 2 subgroup of its fundamental group, which obviously $X$ does not. – Pete L. Clark Apr 1 '11 at 19:25
@Jason, @Pete Oh, you're right, of course. – Grigory M Apr 1 '11 at 19:28
Or do it $\mathrm{mod }2$, no? – Mariano Suárez-Alvarez Apr 1 '11 at 19:34
:) This is the proof I had in mind. I like that it draws on a bunch of pieces of basic algebraic topology. I myself used the fact that you need orientability for Poincare duality, I like Mariano's idea a lot too. – Aaron Mazel-Gee Apr 2 '11 at 7:06

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Hard Motion Question- Help?

1. Feb 1, 2008

fraserperry

Hard Motion Question- Help???

I am really stuck on this motion question, can anyone help please????

"A balloon is 30.00m above the ground and is rising vertically with a uniform speed when a coin is dropped from it. If the coin reaches the ground in 4.00 seconds, what is the speed of the balloon?

2. Feb 1, 2008

Ben Niehoff

So, the coin has an initial height of 30 m, and an initial velocity of v. It has a constant acceleration of -9.8 m/s^2. Find the time it takes to fall to a height of 0 m, in terms of your variables v and t. Then plug in 4.00 s for t, and solve for v.

3. Feb 1, 2008

fraserperry

which motion equation do i use?

4. Feb 1, 2008

rsala

solve for the initial velocity of the coin,,since it is equal to the constant velocity of the balloon, try

0 = 30 + Vi * 4 - .5g(4^2)

thing to know is that if the balloon is traveling , say 500 m/s when the coin is dropped....the coin is actually dropped AT 500m's...so its initial velocity when you insert it into an equation is 500m/s,, NOT 0 as you might assume.
though thats irrelevent here, thought i might say. so that you understand the purpose of the question really

5. Feb 1, 2008

fraserperry

are you saying the coin when dropped is would be travelling the same speed of the balloon in an upwards direction?

6. Feb 1, 2008

rsala

yes, ya and its acceleration is changed from 0 to -9.8 when you put it in the equation because of gravity...because of course it will be hitting the ground soon

7. Feb 1, 2008

fraserperry

I got 12.1 from that equation, but what is that number?

8. Feb 1, 2008

rsala

the coins initial velocity once it is released from the balloon. = 12.1

which just so happens to be the balloons constant velocity, since when you initially dropped the coin from the balloon its initial velocity is = to the balloon velocity.
so, Balloon-velocity = 12.1

ithink


text:
# How many permutations when each position takes from different length alphabets?

Let's say there's a password scheme as follows:

• the password is of length 4 (four)
• the 1st position is taken by one character from the set: a, b, c
• the 2nd position is taken by one character from the set: D, E, F, G
• the 3rd position is taken by one character from the set: 1, 2, 3, 4, 5
• the 4th position is taken by one character from the set: -, +

Example: b, F, 2, +

Is the number of permutations calculated as follows?

3C1 * 4C1 * 5C1 * 2C1 = 3 * 4 * 5 * 2 = 120

-
Your answer is correct. –  Mhenni Benghorbal Dec 2 '12 at 7:03
If you have $n$ different "experiments" with $k_i$ outcomes each, $i = 1,2,...,n$ then the total number of outcomes is $$\prod_{i=1}^{n}k_i$$ To help you visualize this, think of just two experiments and list all of the outcomes from experiment 1. For each of these $k_1$ outcomes we can pair with it $k_2$ outcomes from the second experiment (giving $k_1k_2$ outcomes in total). The generalization comes by thinking of the first and second experiments as one experiment and inducting on the number of additional experiments. This property is of fundamental importance in combinatorics.


text: In general, a relation R from a set A to a set B will be understood as a subset of the Cartesian product A× B, i.e., R ⊆ A× B. Then R1=R2 iff n=m, and Ai=Bi for all i, 1≤i≤n, and R1=R2 are equal sets of ordered n-tuples. So, the relation “married to” can be represented by a subset of the Cartesian product M ×W. irreflexive. Interesting fact: Number of English sentences is equal to the number of natural numbers. 44/ Let R be the relation represented by the matrix Find the third row of the matrix that represents R-1. The relation R on R where aRb means a − b ∈ Z. Ans: 1, 2, 4. Explain. Let $$A, B$$ and $$C$$ be three sets. For each ordered pair (x, y) in the relation R, there will be a directed edge from the vertex ‘x’ to vertex ‘y’. If there is an ordered pair (x, x), there will be self- loop on vertex ‘x’. 5 Sections 31-33 but not exactly) Recall: A binary relation R from A to B is a subset of the Cartesian product If , we write xRy and say that x is related to y with respect to R. A relation on the set A is a relation from A to A.. Already have an account? To Prove that Rn+1 is symmetric. View Answer A single-threaded 25-mm power screw is 25 mm in diameter with a pitch of 5 mm. (More on that later.) relations from X to X) together with (left or right) relation composition forms a monoid with zero, where the identity map on X is the neutral element, and the empty set is the zero element. 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Connect vertex a to vertex b with an arrow, called an edge of the graph, going from vertex a to vertex b if and only if a r b. Define a relation R on the set of positive integers Z as follows: ... matrix representation of the relation, so for irreflexive relation R, the matrix will. e.r1 ⊕ r2. 24. Then • R is reflexive iff M ii = 1 for all i. xRy is shorthand for (x, y) ∈ R. A relation doesn't have to be meaningful; any subset of A2 is a relation. Matrices and Graphs of Relations [the gist of Sec. • R is symmetric iff M is a symmetric matrix: M = M T • R is antisymetric if M ij = 0 or M ji = 0 for all i ≠ j. (6) [6pts] Let R be the relation, defined on set (1, 2, 3), represented by the matrix: 0 1 1 MR 1 0 0 1 0 1 Find the matrix representing the following relations. Write down the elements of P and elements of Q column-wise in three ellipses. The relation R on the set of all people where aRb means that a is younger than b. Ans: 3, 4 22. 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text:
Home > > Factors and Multiples > Prime Factorization

## Prime Factorization

#### Introduction

Prime Factorization is expressing a number as a product of its prime factors. For example, the prime factorization of the number 18 can be written as $2 ast 3 ast 3$ or $3^{2} ast 2$.

#### Terms

Factor - Factors are numbers multiplied together to get another number. Because 3 * 4 = 12, both 3 and 4 are factors of 12.
Prime - A prime number is a number that only has 1 and itself as factors.
Prime Factor - A prime factor is a factor that is also a prime number.

## Lesson

To find the prime factorization for a given postive integer, we use what is called a factor tree, which looks like this for the number 12:

The prime factors are represented by the circled values, 2, 2, and 3. So for 12, the prime factorization is $2 ast 2 ast 3$, or $2^{2} ast 3$.

If a number’s only factors are only 1 and itself, then that number is a prime number and no further factorization is possible.

## Examples

#### Prime Factorization of 12

$12$
$12 = 2 * 6$
$6 = 2 * 3$
The prime factorization of 12 can be written as:
$2^{2} * 3^{1}$

#### Prime Factorization of 72

$72$
$72 = 2 * 36$
$36 = 2 * 18$
$18 = 2 * 9$
$9 = 3 * 3$
The prime factorization of 72 can be written as:
$2^{3} * 3^{2}$

#### Prime Factorization of 61

$61$
The factors of $61$ are $1\text{ and }61$. Therefore $61$ is a prime number.


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# Conditions for existence of divergence of a vector field

What are the necessary and sufficient conditions on a vector field $F$ for the divergence $\nabla\cdot F$ to exist at a given point.

EDIT

In Divergence in the second line under the heading "Application in Cartesian coordinates", why is it assumed that $\vec{F}$ to be a continuously differentiable vector field ?

EDIT 2

Ideally one would expect each component of $F$ to be differentiable at a given point $\vec{a}$ no matter through which continuous contour you traverse the point $\vec{a}$.

EDIT 3

Or is it that each component of $F$ to be differentiable and the derivative being continuous at a given point $\vec{a}$ no matter through which continuous contour you traverse the point $\vec{a}$.

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By definition the divergence of the vector field $\mathbf{F}$ with differentiable components $\mathbf{F}=F_{x}(x,y,z)\mathbf{i}+F_{y}(x,y,z)\mathbf{j}+F_{z}(x,y,z)% \mathbf{k}$ is $\text{div}\mathbf{F=\nabla \cdot F}=\dfrac{\partial F_{x}}{\partial x}+\dfrac{\partial F_{y}}{\partial y}+\dfrac{\partial F_{z}}{\partial z}$ where $F_{x},F_{y},F_{z}$ are the components of $\mathbf{F}$ in the $xyz-$coordinate system. – Américo Tavares Nov 29 '10 at 15:20
@Américo: the divergence also has a coordinate-free definition which is given on the Wikipedia article. It is not clear to me that the existence of the coordinate-free limit is equivalent to the existence of your three partial derivatives. (And note that the existence of these three partial derivatives does not guarantee the existence of all directional derivatives, so if you use your definition then the divergence will not always be computable in every coordinate system.) – Qiaochu Yuan Nov 29 '10 at 15:47
@Qiaochu: You have raised two strong points: 1) The divergence defined by a limiting process in which a region shrinks down to a fixed point requires the additional condition that $\mathbf{F}$ is continuosly differentiable, if we want the divergence theorem be applied. Without using this theorem I do not see how we could give a coordinate-free definition. 2) Concerning the directional derivatives you are quite right. 3) So, the definition I took from a Calculus book has all the limitations you pointed out. – Américo Tavares Nov 29 '10 at 16:21

Perhaps the point is that existence of partial derivatives is not enough for a function of several variables to be differentiable in the multivariable sense, or even necessarily continuous: see this MO question for a standard example of a function of two variables all of whose partial (and even all directional) derivatives exist but is not even continuous at the origin.

It is also possible for all the partial derivatives in a given coordinate system to exist -- e.g. $\frac{\partial f}{\partial x}, \frac{\partial f}{\partial y}, \frac{\partial f}{\partial z}$ -- but for directional derivatives in other directions not to exist. (For instance, let $f: \mathbb{R}^3 \rightarrow \mathbb{R}$ be the function which is $0$ at $(x,y,z)$ if at least two of the variables are zero or if all of $x$,$y$,$z$ have rational coordinates and which is $1$ at all other points.) Here the divergence will be defined with respect to the standard coordinate system but not with respect to a different coordinate system.
Anyway, in general a function which has a continuous derivative (or partial derivatives) is much better behaved than a merely differentiable function. Inserting the hypothesis that a function is $C^1$ -- unless you really need to be considering weaker hypotheses than that for a specific application -- is generally a prudent practice.
@Pete L. Clark : Ideally one would expect each component of $F$ to be differentiable at a given point $\vec{a}$ no matter through which continuous contour you traverse the point $\vec{a}$. any way what do you mean by continuously differentiable ? (just partial derivatives being continuously differentiable may not be good enough.) – Rajesh D Nov 29 '10 at 16:01
@Pete L. Clark : Sorry, I mistook continuously differentiable to be equivalent to being smooth.Thank you for the clarification. I guess you mean to say that total derivative of each of the components of $F$ should be continuous ? Could you please explain meaning of 'partial derivatives in any co-ordinate system'. how many co-ordinate systems are there ? – Rajesh D Nov 29 '10 at 17:03
@Rajesh D: No, the total derivative of a function $F: \mathbb{R}^n \rightarrow \mathbb{R}^m$ (which you are calling a "vector field with components $f_1,\ldots,f_n$") at a point $x \in \mathbb{R}^n$ is an $m \times n$ matrix. There are as many coordinate systems as there are choices of basis on $\mathbb{R}^m$ and on $\mathbb{R}^n$. – Pete L. Clark Nov 29 '10 at 18:22


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Intermediate Algebra (6th Edition)

$x=4$
If $b\gt0$ and $b\ne1$, then $y=log_{b}x$ means $x=b^{y}$ for every $x\gt0$ and every real number $y$. Therefore, $log_{x}64=3$ is equivalent to $x^{3}=64$. We know that $x=4$, because $4^{3}=4\times4\times4=64$.


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# 4.14 Two body system - circular motion

 Page 1 / 2
The trajectory of revolution of “Two body” system is circular for comparable mass.

The trajectory of two body system depends on the initial velocities of the bodies and their relative mass. If the mass of the bodies under consideration are comparable, then bodies move around their “center of mass” along two separate circular trajectories. This common point about which two bodies revolve is also known as “barycenter”.

In order to meet the requirement imposed by laws of motion and conservation laws, the motion of two bodies executing circular motion is constrained in certain ways.

## Circular trajectory

Since external force is zero, the acceleration of center of mass is zero. This is the first constraint. For easy visualization of this constraint, we consider that center of mass of the system is at rest in a particular reference frame.

Now, since bodies are moving along two circular paths about "center of mass", their motions should be synchronized in a manner so that the length of line, joining their centers, is a constant . This is required; otherwise center of mass will not remain stationary in the chosen reference. Therefore, the linear distance between bodies is a constant and is given by :

$r={r}_{1}+{r}_{2}$

Now this condition can be met even if two bodies move in different planes. However, there is no external torque on the system. It means that the angular momentum of the system is conserved. This has an important deduction : the plane of two circular trajectories should be same.

Mathematically, we can conclude this, using the concept of angular momentum. We know that torque is equal to time rate of change of angular momentum,

$\frac{dL}{dt}=r×F$

But, external torque is zero. Hence,

$⇒r×F=0$

It means that “ r ” and “ F ” are always parallel. It is only possible if two planes of circles are same. We, therefore, conclude that motions of two bodies are coplanar. For coplanar circular motion, center of mass is given by definition as :

${r}_{cm}=\frac{-{m}_{1}{r}_{1}+{m}_{2}{r}_{2}}{{m}_{1}+{m}_{2}}=0$

$⇒{m}_{1}{r}_{1}={m}_{2}{r}_{2}$

Taking first differentiation with respect to time, we have :

$⇒{m}_{1}{v}_{1}={m}_{2}{v}_{2}$

Now dividing second equation by first,

$⇒\frac{{m}_{1}{v}_{1}}{{m}_{1}{r}_{1}}=\frac{{m}_{2}{v}_{2}}{{m}_{2}{r}_{2}}$

$⇒\frac{{v}_{1}}{{r}_{1}}=\frac{{v}_{2}}{{r}_{2}}$

$⇒{\omega }_{1}={\omega }_{2}=\omega \phantom{\rule{1em}{0ex}}\left(say\right)$

It means that two bodies move in such a manner that their angular velocities are equal.

## Gravitational force

The gravitational force on each of the bodies is constant and is given by :

$F=\frac{G{m}_{1}{m}_{2}}{{\left({r}_{1}+{r}_{2}\right)}^{2}}=\frac{G{m}_{1}{m}_{2}}{{r}^{2}}$

Since gravitational force provides for the requirement of centripetal force in each case, it is also same in two cases. Centripetal force is given by :

${F}_{C}={m}_{1}{r}_{1}{\omega }^{2}={m}_{2}{r}_{2}{\omega }^{2}=\frac{G{m}_{1}{m}_{2}}{{r}^{2}}$

## Angular velocity

Each body moves along a circular path. The gravitational force on either of them provides the centripetal force required for circular motion. Hence, centripetal force is :

${m}_{1}{r}_{1}{\omega }^{2}=\frac{G{m}_{1}{m}_{2}}{{\left({r}_{1}+{r}_{2}\right)}^{2}}$

$⇒{\omega }^{2}=\frac{G{m}_{2}}{{r}_{1}{\left({r}_{1}+{r}_{2}\right)}^{2}}$

Let the combined mass be “M”. Then,

$M={m}_{1}+{m}_{2}$

Using relation ${m}_{1}{r}_{1}={m}_{2}{r}_{2}$ , we have :

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At high concentrations (>0.01 M), the relation between absorptivity coefficient and absorbance is no longer linear. This is due to the electrostatic interactions between the quantum dots in close proximity. If the concentration of the solution is high, another effect that is seen is the scattering of light from the large number of quantum dots. This assumption only works at low concentrations of the analyte. Presence of stray light.
the Beer law works very well for dilute solutions but fails for very high concentrations. why?
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Moment of inertia of a bar in terms of perpendicular axis theorem
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Centre of mass of two uniform rods of same length but made of different materials and kept at L-shape meeting point is origin of coordinate
A balloon is released from the ground which rises vertically up with acceleration 1.4m/sec^2.a ball is released from the balloon 20 second after the balloon has left the ground. The maximum height reached by the ball from the ground is
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# Proof strategy based on dashes

Prerequisite:

The proof strategy which will be exposed here starts from one of the premises of Hypothesis H.1 (Hypothesis of existence of Goldbach pairs based on dashed lines), i.e. the fact that, if the dashed line $T_k = (p_1, p_2,\ldots,p_k)$ has the first $k$ prime numbers as its components, then $p_1$, which is its first component, is always the number 2. A direct consequence is this property:

Dashed preceding and following a space

In a linear dashed line with the first prime numbers as its components, in the columns immediately preceding or following a space, there is always a dash corresponding to the first dashed line component.

The proof of this property is based on the fact that the first component of the dashed line is 2:

• If a column is even, it’s divisible by 2, then it certainly contains a dash corresponding with the first row (by definition of dash);
• Then, the spaces must be in odd columns;
• Immediately before (or after) every odd number, there’s always an even one;
• Then, immediately before (or after) every space there’s an even number;
• Then, immediately before (or after) every space there’s a column containing a dash in the first row.

Based on this premise, original Goldbach’s equation

$$p + q = 2n$$

can be solved using $\mathrm{t\_value}$ function:

Let’s start from original equation, imposing the bound that the two prime addends $p$ and $q$ are spaces belonging to the first row:

$p + q = 2n$

The components of dashed line $T = (p_1, p_2, ... p_k)$ are the first $k$ prime numbers, then $p_1 = 2$. So, the 2 appearing in the expression $2n$ is none other than the first component of the dashed line, then Goldbach’s equation can be rewritten like this:

$p + q = p_1 n$

Let’s add and subtract 1 to the term on the left:

$p + 1 + q - 1 = p_1 n$

Finally, let’s recollect:

$(p + 1) + (q - 1) = p_1 n$

Since $p$ and $q$ are both spaces, as a consequence of previous property, certainly a dash exists in the $p_1 = 2$ row (the first) in correspondence with the column $p + 1$ (immediately following the space $p$), and another exists in correspondence with column $q - 1$ (immediately preceding the space $q$). Then, since (by definition) $\mathrm{t\_value}(w)$ indicates the column of $w$-th dash, two positive integers exist, $x$ and $y$, such that $\mathrm{t\_value}(x) = p + 1$ and $\mathrm{t\_value}(y) = q - 1$, so the previous equality can be rewritten as:

$\mathrm{t\_value}(x) + \mathrm{t\_value}(y) = p_1 n$

with the following bounds (last two are obvious, since $p_1 = 2$, but they are shown for more clarity):

1. $x$-th and $y$-th dashes must be on the first row;
2. $x$-th dash must correspond with a column following a space;
3. $y$-th dash must correspond with a column preceding a space.

Original addends $p$ and $q$ can be obtained from the following relations:

$$p = \mathrm{t\_value}(x) - 1$$
$$q = \mathrm{t\_value}(y) + 1$$

From these observations, the Hypothesis H.1 (Hypothesis of existence of Goldbach pairs based on dashed lines) can be rewritten like this:

Hypothesis of existence of Goldbach’s pairs based on dashes

Let $2n \gt 4$ be an even number, and let $k$ be the validity order related to it. Then, two positive integers exist, $x$ and $y$, which, referring to dashed line $T_k$:

1. $x$-th and $y$-th dashes both are on first row;
2. $\mathrm{t\_value}(x)$ immediately follows a space;
3. $\mathrm{t\_value}(y)$ pimmediately precedes a space;
4. $\mathrm{t\_value}(x) + \mathrm{t\_value}(y) = p_1 n$;
5. $\mathrm{t\_value}(x) - 1$ and $\mathrm{t\_value}(y) + 1$ are both between the validity interval.

If this hypothesis was true, then the truth of Goldbach’s conjecture would be a direct consequence of it, because:

• If assertions 1., 2., 3. where true, then $p = \mathrm{t\_value}(x) - 1$ and $q = \mathrm{t\_value}(y) + 1$ would be spaces;
• If also assertion 5. was true, $p$ and $q$ would be prime numbers, for Property T.1 (Spaces and prime numbers);
• Assertion 4. is Goldbach’s equation rewritten in another form; then, if this was satisfied too, the sum of $p$ and $q$ would be $2n$.

To be able to prove this hypothesis, we can proceed by steps. A possible observation is that it has two very strong premises:

• It takes in exam the dashed line $T_k$ which has the first prime numbers as its components (which has allowed us writing the $2n$ of Goldbach’s conjecture as $p_1 n$);
• It binds the two spaces within the validity interval (then we had to set $2n \gt 4$, otherwise no solution can be found).

We can think of relaxing these constraints, generalizing the previous hypothesis like this:

Hypothesis of existence of pairs of complementary spaces based on dashes

Let $n \gt 0$ be an integer, and $T = (n_1, n_2, \ldots, n_k)$ a dashed line of order $k \geq 1$, with prime numbers as its components (not necessarily consecutive). Then, two positive integers exist $x$, $y$ such that, referring to dashed line $T$:

1. $x$-th and $y$-th dashes are both on the first row;
2. $\mathrm{t\_value(x)}$ immediately follows a space;
3. $\mathrm{t\_value(y)}$ immediately precedes a space;
4. $\mathrm{t\_value(x)} + \mathrm{t\_value(y)} = n_1 n$.

In this new hypothesis, we have replaced the dashed line $T_k$, having the first $k$ prime numbers as its components, with a more generic dashed line $T$ still having prime numbers as its components, but not necessarily the first $k$. Consequently, in the equation $\mathrm{t\_value(x)} + \mathrm{t\_value(y)} = p_1 n$ of previous hypothesis, $p_1$ has been replaced by $n_1$, keeping the fact that the number multiplicating $n$ is the first component of the dashed line, but this component no longer is the first prime number $p_1 = 2$, but a generic prime number $n_1$, depending on the dashed line $T$ considered.
Moreover, we have eliminated the constraint on the validity interval, first of all because this concept is defined only for dashed lines $T_k$, but also because, by removing this constraint, we can concentrate on an important aspect of the problem, which is far from trivial: finding two spaces of a dashed line whose sum is a fixed number, multiple of the first component of the dashed line itself.

With this premise, Hypothesis H.1.T (Hypothesis of existence of Goldbach’s pairs based on dashes) becomes a specific case of previous one; from this, in fact, we can obtain Hypothesis H.1.T changing two aspects:

• setting $n \gt 2$ and $T := T_k$, where $k$ is the validity order for $2n$;
• adding the condition that $\mathrm{t\_value}(x) - 1$ and $\mathrm{t\_value}(y) + 1$ are both within the validity interval of the dashed line $T_k$.

Then, the steps needed for proving Hypothesis H.1.T are the following:

• Prove the Hypothesis H.1.T.A (Hypothesis of existence of pairs of complementary spaces based on dashes) for second order dashed lines (the proof for first order is rather simple, being a direct consequence of the fact that a first order dashed line is always a repeated sequence made by a dash followed by some spaces);
• Prove it for third order dashed lines;
• Analyze and compare among them the proofs obtained in previous steps, in order to get a proof which is valid for any dashed line of any order (so, including also all dashed lines of type $T_k$);
• At the end, introduce the hypothesis about validity interval, then fully proving Hypothesis H.1.T.

For now, investigation is in progress on first step, and advancements about it are exposed in a dedicated page. When it will be completed, we will be able to deal with following ones, until concluding the global argument, in order to obtain a real Theorem, in the last step.
Last step could look very tough, but, actually, this will depend on the results of previous ones. An important aspect, in fact, is proving Hypothesis H.1.T.A in a constructive way, i.e. not only proving that $x$ and $y$ exist, but also, as accurately as possible, which are these numbers, and, consequently, which are the corresponding $\mathrm{t\_value(x)}$ and $\mathrm{t\_value(y)}$. By doing this, by adding the condition on validity interval of Hypothesis H.1.T, most of the work will be done, because the solutions will already been located in some way; the only thing to be verified will be only if this location is enough to be sure that at least a solution exist within the validity interval.