Free Essay


In: Science

Submitted By fanik
Words 5414
Pages 22
Computational Condensed Matter 4 (2015) 32e39

Contents lists available at ScienceDirect

Computational Condensed Matter journal homepage:

Regular article

Putting DFT to the trial: First principles pressure dependent analysis on optical properties of cubic perovskite SrZrO3
Ghazanfar Nazir a, b, *, Afaq Ahmad b, Muhammad Farooq Khan a, Saad Tariq b a b

Department of Physics and Graphene Research Institute, Sejong University, Seoul 143-747, South Korea
Centre of Excellence in Solid State Physics, University of the Punjab, Lahore, Pakistan

a r t i c l e i n f o

a b s t r a c t

Article history:
Received 8 July 2015
Received in revised form
21 July 2015
Accepted 27 July 2015
Available online 31 July 2015

Here we report optical properties for cubic phase Strontium Zirconate (SrZrO3) at different pressure values (0, 40, 100, 250 and 350) GPa under density functional theory (DFT) using Perdew-Becke-Johnson
(PBE-GGA) as exchange-correlation functional. In this article we first time report all the optical properties for SrZrO3. The real and imaginary dielectric functions has investigated along with reflectivity, energy loss function, optical absorption coefficient, optical conductivity, refractive index and extinction coefficient under hydrostatic pressure. We demonstrated the indirect and direct bandgap behavior of SrZrO3 at
(0) GPa and (40, 100, 250 and 350) GPa respectively. In addition, static dielectric constant, Optical bandgap, Plasma frequency and Static refractive index has also been reported. We verified the Penn's model and showed the inverse relation between static dielectric constant and optical bandgap. Further, we proved the direct relation between static dielectric constant and static refractive index. Both these constants increased by increasing the pressure. Our investigation explored that the material preserve its positive value of refractive index at all pressure values and thus is not a negative index metamaterial.
Also, we measured Plasma frequency for SrZrO3 which also increase by increasing the pressure which leads to a conclusion that material is going to be destabilize.
© 2015 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (

First principle
Cubic phase SrZrO3
High pressure phase
Density functional theory
Optical properties

1. Introduction
In recent days, researchers are busy in finding materials that have potential uses like hydrogen sensors, fuel cells and data storage devices such as random access memory, high intensity violet-blue light emission, optical wave-guides, high temperature oxygen sensors and capacitors in different functional devices [1e6].
Perovskite having general formula ABO3 are very fundamental materials to be used in various functional devices. These materials due to their wonderful properties of ferroelectricity and piezoelectricity have great attraction for the researchers to investigate them with provoking details. In the recent period, the focus of experimental research is the zirconate perovskite. A very few theoretical approach has been done on these perovskite. Among zirconate perovskite, strontium zirconate (SrZrO3) is very interesting material because of its high temperature protonic conductivity [7]. Besides this, SrZrO3 also has great potential for high voltage and high capacitor reliability applications. High dielectric

* Corresponding author.
E-mail address: (G. Nazir).

constant, large value of breakdown strength and low leakage current are some of its fundamental characteristics [8,9].
Kennedy et al. used powder neutron diffraction and Rietveld method to investigate the phase transitions in SrZrO3 [10]. These people gave the solid evidence for the pathway of SrZrO3 and said this material is first orthorhombic (Pnma) changes to orthorhombic
(Cmcm) at about 970 K, then changes to tetragonal (14/mcm) at about 1100 K and then finally to cubic (Pm3m) at about 1400 K. It was also suggested that these materials have very high melting temperature of about 2920 K [11]. First principle method is used to study structural, electronic, optical and magnetic properties of materials [12e15]. Mete's group of researchers reported high temperature electronic properties of cubic phase SrZrO3 [16]. Terki et al. used full-potential linearized augmented plane wave (FPLAPW) method to investigate the structural, electronic and optical properties of BaTiO3 and SrZrO3 [17]. Evarestov et al. studied the density functional theory (DFT) LCAO and plane wave (PW) calculations for the known four phases of SrZrO3 [18,19].
It has been observed that previous studies on SrZrO3 mainly focused on its structural and electrical properties using experimental approach but few researchers also reported its optical
2352-2143/© 2015 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (

G. Nazir et al. / Computational Condensed Matter 4 (2015) 32e39

ε2 ðuÞ ¼

Z e2 h0 X jMcv ðkÞj2 d½ucv ðkÞ À uŠd3 k pm2 u2 v;c

where the volume integral is limited to first Brillouin zone and dipole matrix element Mcv ðkÞ ¼ huck jeVjuvk i gives the information about direct transition between conduction band and valence band states. The relation ucv(k)¼EckÀEvk gives the account of the excitation energy used during the transition between conduction and valence band states, “e” represent polarization vector because of electric field and “uck” give detail about periodic portion of Bloch wave function in conduction band associated with wave vector k.
The real part of dielectric function ε(u) can be calculated from imaginary part using well known relation of Kramers-Kronig given by: Z∞

u0 ε2 ðu0 Þ du0 u02 À u2

The symbol P in the about relation tell us about principal value of integral. By knowing both the real and imaginary values of dielectric functions, different important optical properties can be determined. The refractive index of the material can be determined using the following relation:

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi31 2
ε2 ðuÞ þ ε2 ðuÞ ε1 ðuÞ
þ nðuÞ ¼ 4
Whereas at low frequency i.e. at (u¼0), we have the following relation: 1

nð0Þ ¼ ε 2 ð0Þ
The above relation is called static refractive coefficient. The extinction coefficient can be computed by using following relation:

Àε ðuÞ þ kðuÞ ¼ 4 1

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi31 2 ε2 ðuÞ þ ε2 ðuÞ

Basic calculations of our research is done using full potential linearized augmented plane wave (FP-LAPW) based on density functional theory (DFT) as implemented in Wien2k. In addition to this, Kohn-Sham equations are calculated by applying full potential-linearized augmented plane wave (FP-LAPW) using selfconsistent method [23e28]. We performed pressure dependent characteristics instead of temperature dependent in order to study the optical response of the material under high pressure by using standard DFT at 0 K temperature. In this well-known FP-LAPW method, Slater's theory of Muffin-Tin radii has been used in order to divide this space into two regions. Close to the atoms, all interesting quantities are expanded in terms of spherical harmonics and as expansion of plane waves with wave vector having cut-off value of
KMAX in the interstitial region. The earlier kind of expansion is stated within a well-known muffin-tin sphere of radius RMT around every nucleus. The choice of sphere radii predicts the rate of convergence of these expansions, but this affects only the speed of the calculation. We choose RMT x KMAX ¼ 7 as convergence parameters for the matrix element that represent number of fundamental functions, in which KMAX is associated with plane wave cutoff in momentum space (k-space) and RMT represent the smallest radius of Muffin-Tin cube among all the atomic sphere radii. The value of Muffin-Tin radii for “Barium”, “Zirconium” and “Oxygen” are 2.5, 1.98 and 1.79 au respectively. Energy separation is considered to be À6 Ry. We used generalized gradient approximation
(GGA-PBE) given by Perdew et al.for the calculation of exchangeecorrelation potential [29,30]. We did our calculation without including spin-orbit effects. We choose maximum value of l to be lmax ¼ 10 for wave function expansion inside the atomic spheres.
Outside the muffin-tin spheres, the expansion of charge density is done by Fourier series. For reciprocal space integration under different pressure values in the irreducible Brillouin zone, 35 kpoints in grid of 10 Â 10 Â 10 meshes which is equal to 1000-k points in the complete Brillouin zone scheme has been used to achieve self-consistency. Optical characteristics for SrZrO3 are studied by dielectric function ε(u)¼ε1(u)þε2(u) consist of real and imaginary dielectric parameters. Following equations formulated by Ehrenreich and Cohen give brief description about the real and imaginary parts of dielectric function [29,31].



2. Computational details

ε1 ðuÞ ¼ 1 þ


properties. Ref. [17,20,21] reported the existence of large conductivity spectra at room temperature shown by SrZrO3. In many reports the calculation for band structure of cubic phase of SrZrO3
(which is stable above 1400 K) [16,20,22]. In this paper, we investigate all the possible optical properties of cubic phase SrZrO3 under different values of pressure (GPa).


Reflectivity of the mentioned material is calculated by:

ðn À 1Þ2 þ k2 ðn þ 1Þ2 þ k2

In the same way, energy loss function L(u), absorption coefficient a(u) and optical conductivity s(u) are determined using the following relation [32].

LðuÞ ¼ Imð À 1=e ðuÞÞ ε aðuÞ ¼ 4pkðuÞ=l sðuÞ ¼ ð2Wcv h0 uÞ=Eo
Where “Wcv” in the above relation represent transition probability per unit time.

3. Result and discussion
To discuss the internal structure of any material, the optical properties play a very supportive role. The Optical Properties of materials suggest the feasibility and suitability as industrial point of view especially in opto-electronics. Fig. 1 show theoretical cubic perovskite structure of compound SrZrO3. Atomic arrangements of
Sr, Zr and O atoms on different sites of cubic perovskite are: Sr at 8 corners, Zr at the center and O atoms on the 6 faces.
Figs. 2e10 show meaningful study of different optical parameters for cubic phase of SrZrO3 at different values of pressure at (0,
40, 100, 250 and 350) GPa calculated for energy range upto 45 eV.
We verified Penn's model by knowing the inverse relation between optical bandgap and static dielectric constant. Fig. 2 shows the variation of optical bandgap and static dielectric constant at different pressure values from (0, 40, 100, 250 and 350) GPa. We can say that optical bandgap and static dielectric constant satisfy Penn's model. This inverse relation between ε1(0) and bandgap can be explained by penn model mathematically given by [33,34].


G. Nazir et al. / Computational Condensed Matter 4 (2015) 32e39

Fig. 1. The compound SrZrO3 has an ideal cubic perovskite structure. The Strontium,
Oxygen and Zirconium atoms preferably reside at corners, on the faces and at the center respectively of single cubic unit cell.

 Á2 ε1 ð0Þz1 þ ħup Eg
The relation given by Penn's model can be used to determine Eg by knowing the values of ε1(0) and plasma energy “ħup ”. The measured values of static dielectric constant ε1(0) at different pressures (0, 40, 100, 250 and 350) GPa along with their bandgap values and plasma frequencies are given in Table 1.
The real and imaginary parts of dielectric function determined at different values of pressure are shown in Fig. 3. The real part of dielectric function ε1(u) gives information about how much a

Fig. 2. Static dielectric constant and optical bandgap for cubic phase SrZrO3 at (0, 40,
100, 250 and 350) GPa pressure values.

Fig. 3. Frequency dependent dielectric functions of cubic phase SrZrO3 at (0, 40, 100,
250 and 350) GPa pressure values (real part: black lines, imaginary part: red lines). (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

material is polarized. It is observed from the Fig. 3 that the measured values of static dielectric constantε1(0), low energy limit ofε1(u), depends upon the bandgap of the material. It can be observe from the Fig. 3 that the real part of dielectric function start increasing from its static value ε1(0) given in Table 1 and reaches a highest peak at about 4.784 eV, 5.497 eV, 6.026 eV, 7.292 eV and
6.887 eV for cubic phase of SrZrO3 at different values of pressure from (0, 40, 100, 250 and 350) GPa. After reaching this peak value, real dielectric function start decreasing with some low peaks exist at different higher energy values and becomes zero at certain value of energy of about 9.740 eV, 11.117 eV, 12.150 eV, 13.110 eV and
13.270 eV at (0, 40, 100, 250 and 350) GPa respectively. On further increase in energy, it can be seen that dielectric function becomes negative which means that in these regions of energy, the medium will totally reflect all the incident electromagnetic waves and thus exhibiting metallic nature of the material. The graph also reveals that hump is shifted toward the higher energies by increasing the value of pressure upto 350 GPa.
It has been understood that imaginary part of dielectric function ε2(u) play very impressive role in the optical properties for any material. It has deep effect on the absorption of the medium. Absorption in the material is large for large value of imaginary part of dielectric function ε2(u) Fig. 3 also shows the behavior of imaginary part of dielectric function versus energy. It is clear that absorption starts at about 3.172 eV, 3.738 eV, 3.664 eV, 2.828 eV and 2.213 eV for cubic phase of SrZrO3 at (0, 40, 100, 250 and 350) GPa respectively. These points are related to minimum direct bandgap transition between maxima of the valence band to the minima of the conduction band. Fig. 3 also represents the strong absorbing nature of the material in these energy regions from 3.172 eV to 37.965 eV at

G. Nazir et al. / Computational Condensed Matter 4 (2015) 32e39

Fig. 4. Frequency dependent reflectivity of cubic phase SrZrO3 at (0, 40, 100, 250 and
350) GPa pressure values.

Fig. 5. Frequency dependent energy loss function of cubic phase SrZrO3 at different values at (0, 40, 100, 250 and 350) GPa pressure values.


Fig. 6. Frequency dependent Absorption coefficient of cubic phase SrZrO3 at (0, 40,
100, 250 and 350) GPa pressure values.

0 GPa, 3.738 eVe40.031 eV at 40 GPa, 3.664 eVe43.241 eV at
100 GPa, 2.828 eVe40.720 eV at 250 GPa and 2.123 eVe41.175 eV.
These regions consist of different peaks which exist as a result of interband transition between valence band and conduction band.
The width of the absorption region first increase upto 100 GPa and then decrease with the increase in pressure. Maximum absorption takes place in energy range of 3.664 eVe43.241 eV with energy width equal to 39.577 eV at 100 GPa and minimum at 0 GPa with energy width equal to 34.793 eV. We can also see that the absorption width shift toward the lower value of energy as the pressure exceeds 100 GPa.
To investigate deeply the surface behavior of the material, its reflectivity is measured which is equal to the ratio of incident power to reflected power. Fig. 4 represent reflectivity versus energy for cubic phase SrZrO3 at (0, 40, 100, 250 and 350) GPa. Fig. 4 shows that the zero frequency limit of reflectivity for cubic phase SrZrO3 is equal to 0.129, 0.127, 0.128, 0.130, 0.131 at (0, 40, 100, 250 and 350)
GPa respectively. It then increase from its zero limit as with the increase in pressure and reached at the highest peak of reflectivity for cubic phase SrZrO3 at 24.424 eV, 24.375 eV, 26.453 eV,
27.130 eV, 27.560 eV for (0, 40, 100, 250 and 350) GPa respectively.
These peaks produced as a result of interband transition between valence band and conduction band. On the other hand, the minimum value of reflectivity occurs in energy range from 10 eV to
40 eV as a result of collective plasma resonance. Imaginary part of dielectric function can be used to measure the depth of plasma resonance [34]. We can see very interesting behavior shown by the peak value of reflectivity. It shifts toward higher energy value as the pressure exceeds 40 GPa consistent with the imaginary part of dielectric function [35,36].
The energy loss function L(u) is an important parameter which describes the energy loss for electron moving in a material. Fig. 5


G. Nazir et al. / Computational Condensed Matter 4 (2015) 32e39

Fig. 7. Optical bandgap of cubic phase SrZrO3 at (0, 40, 100, 250 and 350) GPa pressure values.

show energy loss function L(u) versus at (0, 40, 100, 250 and 350)
GPa. The peaks in energy loss function L(u) gives us brief detail about characteristics related to plasma resonance and hence the associated frequency known as plasma frequency. No energy loss occur for photons having energy less than 6.186 eV, 7.096 eV,

7.637 eV, 8.449 eV, 8.670 eV at (0, 40, 100, 250 and 350) GPa respectively but as soon as the photons exceed these amount of energy, energy loss will start increasing and get the maximum peak at 26.035 eV, 32.246 eV, 35.124 eV, 36.969 eV, 37.571 eV for (0, 40,
100, 250 and 350) GPa respectively. It can be seen that highest peak

G. Nazir et al. / Computational Condensed Matter 4 (2015) 32e39

Fig. 8. Frequency dependent optical conductivity of cubic phase SrZrO3 at (0, 40, 100,
250 and 350) GPa pressure values (real part: black lines, imaginary part: red lines). (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

shifts toward higher value of energy with the increase in pressure.
The highest peak has value 5.709 eV occur at 36.969 eV because of pressure value equal to 250 GPa. The observed peak in the energy loss function associated with the plasma frequency and can be treated as an interface between metallic and dielectric behavior.
Absorption coefficient is very important factor that gives information about the decay of light intensity per unit distance in medium. The frequency dependent absorption coefficient for cubic phase SrZrO3 at different pressure is shown in Fig. 6. The change in frequency is described in terms of energy with range from 0 upto
45 eV. The highest peaks occurred at 23.613 eV, 24.191 eV,
24.769 eV, 25.605 eV, 26.035 eV for (0, 40, 100, 250 and 350) GPa respectively. We can see largest value of peak equal to 382.809 eV exist at pressure value of 100 GPa. So we can say that by increasing pressure, the absorption peak shift toward higher value of energy upto 350 GPa. It has also been observed that there is no absorption for photon energy less than 4.033 eV, 4.525 eV, 5.288 eV, 5.755 eV,
5.989 eV for (0, 40, 100, 250 and 350) GPa respectively. However with photon energy greater than these values, absorption coefficient start increasing, which is associated with the direct bandgap values 3.307 eV, 3.175 eV, 3.023 eV, 2.605 eV, 2.156 eV calculated for cubic phase SrZrO3 at different high pressures, respectively.
We can see many peaks within the studied energy range, and the structures of these peaks can be understood from the interband transitions between valence band and conduction band. Fig. 7 shows optical bandgap values determined by measuring the direct and indirect behavior of the cubic phase SrZrO3 at different pressure values by using theoretical approach of square of absorption. The real and imaginary part of optical conductivity is also


Fig. 9. Frequency dependent refractive indices of cubic phase SrZrO3 at (0, 40, 100, 250 and 350) GPa pressure values (refractive index: black lines, extinction coefficient: red lines). (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

studied for cubic phase SrZrO3 as shown in Fig. 8. It is observed that the real part of conductivity remained at zero and starts to originate from 3.66, 4.16, 4.31, 5.54 and 5.76 eV for different pressure values at (0, 40, 100, 250 and 350) GPa respectively. This is in good agreement with the bandgap of the material at different pressure.
When the photon energy exceed these particular values, the real

Fig. 10. Static Refractive indices for cubic phase SrZrO3 at (0, 40, 100, 250 and 350) GPa pressure values.


G. Nazir et al. / Computational Condensed Matter 4 (2015) 32e39

Table 1
Static Dielectric Constant ε1(0), Optical Band-gap(Eg), Plasma Frequency up and Static Refractive Index n0 at (0, 40, 100, 250 and 350) GPa pressure values.
Pressure (GPa)

Static dielectric constant ε1(0)


Plasma frequency (up)

Static refractive Index (n0)






part of conductivity start increasing and attains the maximum peak values at 23.231 eV, 23.539 eV, 23.797 eV, 12.618 eV, 13.011 eV for pressure at (0, 40, 100, 250 and 350) GPa respectively. It is observed that as the pressure is increased, the maximum real conductivity peak shifted toward higher value of energy for (0, 40 and 100) GPa.
With the further increase in pressure, the highest peak shift toward the lower value. of energy as seen for 250 GPa and then again it goes toward higher value of energy for 350 GPa. After reaching at highest peak, it starts decreasing with the further increase in photon energy
The refractive index and the extinction coefficient for cubic phase SrZrO3 can be determined by the relation:

n2 À k2 ¼ ε1 And 2nk ¼ ε2
Fig. 9 shows the broad spectrum of refractive index and extinction coefficient over wide range of energy. The Figure shows that the spectrum of n(u) nearly follow ε1(u) pattern [37]. It can be seen from the Figure that static refractive index n0 values are 2.122,
2.113, 2.122, 2.132, 2.136 for cubic phase SrZrO3 at (0, 40, 100, 250 and 350) GPa values of pressure which is in good agreement with static dielectric function values with the formula (n2 ¼ ε1 ð0Þ). We
can see that the peak in the refractive index shift toward higher value of energy as the pressure increase from (0, 40, 100, 250 and
350) GPa. In the middle of the graph, we can see different humps which vanish at higher values of photon energy. It is due to the fact that beyond certain value of energy, the material will no longer remain transparent and start absorbing high energy photons. At certain value of energy, the value of refractive index becomes less than unity as can be seen from Fig. 9.
Refractive index with value less than unity (vg ¼ c/n) describes that group velocity of the falling radiation becomes greater than speed of light. This means that group velocity move toward the negative domain and nature of the medium will no longer remain linear. In other words, we can say that the material converts to superluminal medium for high energy photons [38,39]. The extinction coefficient is also shown in the Fig. 9. The extinction coefficient gives information about the absorption of light, and at the same time, absorption characteristics at the band edges. The peaks in refractive index as well as in the extinction coefficient are due to the interband transition of electrons from valence band to conduction band. The response of k(u) is closely related with the response of ε2(u) at different values of pressure [40e42]. Fig. 10 shows the behavior of static refractive index with different values of pressure (GPa) which increased as the value of pressure is increased from 0 GPa to 350 GPa. To the best of our knowledge, there is lack of data available on pressure dependent optical properties of cubic phase SrZrO3. We therefore hope that our work will motivate researcher to do theoretical studies in this direction using different exchange-correlation functional, so we can compare our results with them to get better understanding about the material. 4. Conclusion
In summary, we performed first principle computational analysis on cubic phase SrZrO3 to measure the optical properties. We have seen that at 0 GPa, the SrZrO3 has indirect nature and become direct as the pressure equals to 40 GPa, after that it remained direct with further increase in pressure upto 350 GPa as confirmed by the optical bandgap measurements. Further, static dielectric constant, plasma frequency, static refractive indices are also measured which increase as the pressure is increased. However, the optical bandgap is decreased with the increased in pressure consistent with Penn's model. Also material showed positive refractive index value at all pressure which leads to conclusion that our material did not change into negative index metamaterial under pressure measurement.
Plasma frequency is also increased with pressure increase exhibiting de-stability of the material which is confirmed by increment in the optical bandgap values. We try our best to discuss all the optical properties in detail and show path to the researcher to do experimental study for comparison.
We have no financial support for this work.
[1] R. Davies, M. Islam, J. Gale, Dopant and proton incorporation in perovskitetype zirconates, Solid State Ionics 126 (1999) 323e335.
[2] Z. Wu, et al., Effect of BaO-Al2O3-B2O3-SiO2 glass additive on densification and dielectric properties of Ba0. 3Sr0. 7TiO3 ceramics, J. Ceram. Soc. Jpn. 116
(2008) 345e349.
[3] R.V. Shende, D.S. Krueger, G.A. Rossetti, S.J. Lombardo, Strontium zirconate and strontium titanate ceramics for high-voltage applications: synthesis, processing, and dielectric properties, J. Am. Ceram. Soc. 84 (2001) 1648e1650.
[4] N. Fukatsu, N. Kurita, T. Yajima, K. Koide, T. Ohashi, Proton conductors of oxide and their application to research into metal-hydrogen systems, J. Alloys
Compd. 231 (1995) 706e712.
[5] T. Yajima, H. Suzuki, T. Yogo, H. Iwahara, Protonic conduction in SrZrO3-based oxides, Solid State Ionics 51 (1992) 101e107.
[6] H. Iwahara, T. Yajima, T. Hibino, H. Ushida, Performance of solid oxide fuel cell using proton and oxide ion mixed conductors based on BaCe1À x Sm x O3À a,
J. Electrochem. Soc. 140 (1993) 1687e1691.
[7] P. Colomban, Proton Conductors: Solids, Membranes and Gels-materials and
Devices, vol. 2, Cambridge University Press, 1992.
[8] T. Yu, W. Zhu, C. Chen, X. Chen, R.G. Krishnan, Preparation and characterization of solegel derived CaZrO3 dielectric thin films for high-k applications,
Phys. B Condens. Matter 348 (2004) 440e445.
[9] C.-Y. Lin, M.-H. Lin, M.-C. Wu, C.-H. Lin, T.-Y. Tseng, Improvement of resistive switching characteristics in thin films with embedded Cr layer, Electron Device Lett. IEEE 29 (2008) 1108e1111.
[10] B.J. Kennedy, C.J. Howard, B.C. Chakoumakos, High-temperature phase transitions in SrZrO3, Phys. Rev. B 59 (1999) 4023.
[11] D. Souptel, G. Behr, A. Balbashov, SrZrO3 single crystal growth by floating zone technique with radiation heating, J. Cryst. Growth 236 (2002) 583e588.
[12] J. Muscat, A. Wander, N. Harrison, On the prediction of band gaps from hybrid functional theory, Chem. Phys. Lett. 342 (2001) 397e401.
[13] J. Sambrano, J. Martins, J. Andres, E. Longo, Theoretical analysis on TiO2(110)/V surface, Int. J. Quantum Chem. 85 (2001) 44e51.
[14] J. Sambrano, G. Nobrega, C. Taft, J. Andres, A. Beltran, A theoretical analysis of the TiO2/Sn doped (110) surface properties, Surf. Sci. 580 (2005) 71e79.
[15] J. Sambrano, et al., Theoretical analysis of the structural deformation in Mndoped BaTiO3, Chem. Phys. Lett. 402 (2005) 491e496.

[16] E. Mete, R. Shaltaf, S. Ellialtıoglu, Electronic and structural properties of a 4

G. Nazir et al. / Computational Condensed Matter 4 (2015) 32e39 d perovskite: cubic phase of SrZrO3, Phys. Rev. B 68 (2003) 035119.
[17] R. Terki, H. Feraoun, G. Bertrand, H. Aourag, Full potential calculation of structural, elastic and electronic properties of BaZrO3 and SrZrO3, Phys. Status
Solidi (b) 242 (2005) 1054e1062.
[18] R. Evarestov, A. Bandura, V. Aleksandrov, Calculations of the electronic structure of crystalline SrZrO3 in the framework of the density-functional theory in the LCAO approximation, Phys. Solid State 47 (2005) 2248e2256.
[19] R. Evarestov, A. Bandura, V. Alexandrov, E. Kotomin, DFT LCAO and plane wave calculations of SrZrO3, Phys. Status Solidi (b) 242 (2005) R11eR13.
[20] R. Vali, Band structure and dielectric properties of orthorhombic SrZrO3, Solid
State Commun. 145 (2008) 497e501.
[21] K. Galicka-Fau, et al., Thickness determination of SrZrO3 thin films using both
X-ray reflectometry and SIMS techniques, Thin Solid Films 516 (2008)
[22] Y. Lee, et al., Systematic trends in the electronic structure parameters of the 4 d transition-metal oxides SrMO3 (M ¼ Zr, Mo, Ru, and Rh), Phys. Rev. B 67
(2003) 113101.
[23] Z. Feng, H. Hu, S. Cui, C. Bai, First-principles study of optical properties of
SrZrO3 in cubic phase, Solid state Commun. 148 (2008) 472e475.
[24] P. Hohenberg, W. Kohn, Inhomogeneous electron gas, Phys. Rev. 136 (1964)
[25] W. Kohn, L.J. Sham, Self-consistent equations including exchange and correlation effects, Phys. Rev. 140 (1965) A1133.
[26] K. Schwarz, P. Blaha, G. Madsen, Electronic structure calculations of solids using the WIEN2k package for material sciences, Comput. Phys. Commun. 147
(2002) 71e76.
[27] H. Eschrig, The Fundamentals of DFT, Teubner, Stuttgart, 1996.
[28] I. Levin, et al., Phase equilibria, crystal structures, and dielectric anomaly in the BaZrO3eCaZrO3 system, J. Solid State Chem. 175 (2003) 170e181.
[29] S. Cottenier, Density Functional Theory and the Family of (L) APW-methods: a
Step-by-step Introduction, vol. 4, Instituut voor Kern-en Stralingsfysica, KU
Leuven, Belgium, 2002, p. 41.
[30] J.P. Perdew, et al., Atoms, molecules, solids, and surfaces: applications of the








generalized gradient approximation for exchange and correlation, Phys. Rev. B
46 (1992) 6671.
S. Kurth, J.P. Perdew, P. Blaha, Molecular and solid-state tests of density functional approximations: LSD, GGAs, and meta-GGAs, Int. J. Quantum Chem.
75 (1999) 889e909.
H. Ehrenreich, M.H. Cohen, Self-consistent field approach to the manyelectron problem, Phys. Rev. 115 (1959) 786.
D. Groh, et al., First-principles study of the optical properties of BeO in its ambient and high-pressure phases, J. Phys. Chem. Solids 70 (2009) 789e795.
D.R. Penn, Wave-number-dependent dielectric function of semiconductors,
Phys. Rev. 128 (1962) 2093.
A.H. Reshak, Z. Charifi, H. Baaziz, First-principles study of the optical properties of PbFX (X ¼ Cl, Br, I) compounds in its matlockite-type structure, Eur.
Phys. J. B 60 (2007) 463e468.
B. Amin, I. Ahmad, M. Maqbool, Conversion of direct to indirect bandgap and optical response of B substituted InN for novel optical devices applications,
Light. Technol. J. 28 (2010) 223e227.
R. Khenata, et al., First-principle calculations of structural, electronic and optical properties of BaTiO 3 and BaZrO 3 under hydrostatic pressure, Solid
State Commun. 136 (2005) 120e125.
A.M. Fox, Optical Properties of Solids, vol. 3, Oxford university press, 2001.
L.J. Wang, A. Kuzmich, A. Dogariu, Gain-assisted superluminal light propagation, Nature 406 (2000) 277e279.
D. Mugnai, A. Ranfagni, R. Ruggeri, Observation of superluminal behaviors in wave propagation, Phys. Rev. Lett. 84 (2000) 4830.
R. Eglitis, M. Rohlfing, First-principles calculations of the atomic and electronic structure of SrZrO3 and PbZrO3(001) and (011) surfaces, J. Phys. Condens.
Matter 22 (2010) 415901.
R. Eglitis, Ab initio calculations of SrTiO3, BaTiO3, PbTiO3, CaTiO3, SrZrO3,
PbZrO3 and BaZrO3(001),(011) and (111) surfaces as well as F centers, polarons, KTN solid solutions and Nb impurities therein, Int. J. Mod. Phys. B 28
(2014) 1430009.

Similar Documents

Free Essay


...Assignment in Physics... 1. Definition of Science, Major branches of science 2. Scientific Method 3. Definition of Physics and its major branches 4. Notable Physicist and their contribution 5. Importance of Physics in our everyday life and in our society. (Write the references) Short bond paper, written or computerized (font: Times New Roman/font size: 12) Reading assign. Measurement Diff. system of measurement fundamentals and derive quantities scientific notation rules in significant figures conversion of units ) I.1 Science The intellectual and practical activity encompassing the systematic study of the structure and behaviour of the physical and natural world through observation and experiment. I.2 The Branches of Science The Physical Sciences * Physics: The study of matter and energy and the interactions between them. Physicists study such subjects as gravity, light, and time. Albert Einstein, a famous physicist, developed the Theory of Relativity. * Chemistry: The science that deals with the composition, properties, reactions, and the structure of matter. The chemist Louis Pasteur, for example, discovered pasteurization, which is the process of heating liquids such as milk and orange juice to kill harmful germs. * Astronomy: The study of the universe beyond the Earth's atmosphere. The Earth Sciences * Geology: The science of the origin, history, and structure...

Words: 1431 - Pages: 6

Premium Essay


...Professor PHYS 2010 October 21, 2014 Physics in Our Daily Activities Physics is a very important science that can almost be found anywhere in our lives. Many people find this statement hard to believe because they are not able to see the basic aspects of physics all around them. I personally think that unless the person is a physicist or at least someone who had some physics classes, chances are this person or someone is not going to understand how much physics affects our daily life. The significant effect of physics on us today can be easily seen when looking at our reliance on modern technology. Many of the technologies that are changing the world around us are based on physics principles. Physics is more than a subject we study in class, it is also a powerful tool that can help us to gain a better understanding of the everyday world. Physics can be seen in a lot of simple games that we play all the time. One of my favorite games that I almost play on a daily basis is pool. The physics associated with pool is mainly about the collisions between the pool balls. When two pool balls hit each other or collide the collision between them is known to be an elastic collision. According to Billiards in the Classroom, "elastic collisions are collisions in which both momentum and kinetic energy are conserved. The total system kinetic energy before the collision equals the total system kinetic energy after the collision." Therefore, we can assume that the collisions that......

Words: 1480 - Pages: 6

Premium Essay


...the gravitational force on it is nearly constant. As a result, an object in free fall accelerates downward at a constant rate. This acceleration is usually represented with the symbol g. Physics students measure the acceleration due to gravity using a wide variety of timing methods. In this experiment, you will have the advantage of using a very precise timer connected to the calculator and a Photogate. The Photogate has a beam of infrared light that travels from one side to the other. It can detect whenever this beam is blocked. You will drop a piece of clear plastic with evenly spaced black bars on it, called a Picket Fence. As the Picket Fence passes through the Photogate, the LabPro or CBL 2 interface will measure the time from the leading edge of one bar blocking the beam until the leading edge of the next bar blocks the beam. This timing continues as all eight bars pass through the Photogate. From these measured times, the program will calculate the velocities and accelerations for this motion and graphs will be plotted. Picket fen ce Figure 1 OBJECTIVE • Measure the acceleration of a freely falling body (g) to better than 0.5% precision using a Picket Fence and a Photogate. MATERIALS LabPro or CBL 2 interface TI Graphing Calculator DataGate program Physics with Calculators Vernier Photogate Picket Fence clamp or ring stand to secure Photogate Modified from and reported with permission of the publisher Copyright (2000), Vernier......

Words: 2335 - Pages: 10

Premium Essay


...Physics Lab 4 Part 1: Friction Parabola Track 3a. Kinetic energy is the highest when the skate board has reached its lowest point. 3b. Kinetic energy is the lowest when in the middle of the drop. 4a. Potential energy is the highest when the skate board has reached the highest point. 4b. Potential energy is lowest when in the middle of the drop. 5a. Total energy is the highest when potential energy is at its highest point. 6. The value of thermal energy is 0 only when potential energy is highest. David Del Rio Physics PH 2530 Lab 4 Energy 04/06/2015 Part 1: Loop Track 8. When a skateboarder moves, what happens to the kinetic and potential energy? Conservative (closed) or non-conservative (open) system? - Kinetic energy rises as the skateboarder moves downward. -potential energy rises as the skateboarder moves up. - Non-Conservative 9. Where is the skateboarder at on the ramp when he reaches the maximum point of potential energy? 4546.93 11. m = 76./kg The skateboarders mass = 76 kg 12a. calculated mass = 76 kg 12b. Actual mass 75 kg 12c. Comparison = .98% 13. When the coefficient is adjusted half way the kinetic energy decreases to 0 as to the potential energy decreases and finally stabilizes. - This is a closed system. Part 2: Friction Parabola Track 2a. kinetic energy is highest when the skaters’ board is at the lowest point 2b. in the middle of the drop the kinetic energy is highest. 3a. potential energy is the highest when the......

Words: 457 - Pages: 2

Free Essay


...Introductory Physics I Elementary Mechanics by Robert G. Brown Duke University Physics Department Durham, NC 27708-0305 Copyright Notice Copyright Robert G. Brown 1993, 2007, 2013 Notice This physics textbook is designed to support my personal teaching activities at Duke University, in particular teaching its Physics 141/142, 151/152, or 161/162 series (Introductory Physics for life science majors, engineers, or potential physics majors, respectively). It is freely available in its entirety in a downloadable PDF form or to be read online at:∼rgb/Class/intro physics 1.php It is also available in an inexpensive (really!) print version via Lulu press here: where readers/users can voluntarily help support or reward the author by purchasing either this paper copy or one of the even more inexpensive electronic copies. By making the book available in these various media at a cost ranging from free to cheap, I enable the text can be used by students all over the world where each student can pay (or not) according to their means. Nevertheless, I am hoping that students who truly find this work useful will purchase a copy through Lulu or a bookseller (when the latter option becomes available), if only to help subsidize me while I continue to write inexpensive textbooks in physics or other subjects. This textbook is organized for ease of presentation and ease of learning. In particular, they......

Words: 224073 - Pages: 897

Premium Essay


...Introduction In this project questions relating to Thermal Physics and Kinetic theory will be discussed and also diagrams to further explain them will be attached. Section B of the Physics syllabus will be completed by the end of this project. Bibliography 1. Thermal energy transfer by: * Conduction is the flow of heat energy through materials and substances in direct contact with each other. A conductor is a material that permits heat energy to flow freely within it. The better the conductor the more rapidly heat will be transferred. Conduction takes place when heat is supplied to a substance, the particles in that substance gain more energy and vibrate more. These particles then bump into neighboring particles and some of their energy is transferred to them. This process continues and energy is eventually transferred from the hotter end to the colder end of the object. * Thermal convection is transferred from hot places to cold places by convection. Convection occurs when warmer areas of gas or liquid rises to cooler areas of that gas or liquid. The cooler gas or liquid replaces the warmer areas......

Words: 1957 - Pages: 8

Premium Essay


...No. Information on Every Subject 1. Unit Name: Physics I 2. Code: FHSP1014 3. Classification: Major 4. Credit Value: 4 5. Trimester/Year Offered: 1/1 6. Pre-requisite (if any): No 7. Mode of Delivery: Lecture, Tutorial, Practical 8. Assessment System and Breakdown of Marks: Continuous assessment: 50% - Theoretical Assessment (Tests/Quizzes/Case Studies) (30%) - Practical Assessment (Lab reports/Lab tests) (20%) Final Examination 9. 10. 50% Academic Staff Teaching Unit: Objective of Unit: The aims of this course are to enable students to: • appreciate the important role of physics in biology. • elucidate the basic principles in introductory physics enveloping mechanics, motion, properties of matter and heat. • resolve and interpret quantitative and qualitative problems in an analytical manner. • acquire an overall perspective of the inter-relationship between the various topics covered and their applications to the real world. • acquire laboratory skills including the proper handling and use of laboratory apparatus and materials. 11. Learning Outcome of Unit: At the end of the course, students will be able to: 1. Identify and practice the use of units and dimensional analysis, uncertainty significant figures and vectors analysis. 2. Apply and solve problems related to translational and rotational kinematics and dynamics in one and two dimensions. 3. Apply and solve problems related to......

Words: 765 - Pages: 4

Free Essay


...Roger Truong Week 4 Physics Notes Experiment 1 * Rise and fall is pressure in the sound wave makes the flame move * The rise and fall in pressure makes the click sound * The rise and fall in the disturbance to what brings the sound to your ear * The square waves to what makes the flame move and bring the sound to your ear * The air molecules don’t move the disturbance does * For a 0.5 Hz your hear a click and the flame moves and resets * For 100 Hz the flame remains displaced and doesn’t recover * The transition from a click to a tone is between 20 and 50 Hz Reflection * Change in direction of a wave at an interference between two media wave returns into media from which it originated form. Wave Refraction * Change in direction of a wave when it passes from one medium to another caused by the different speeds of a wave * When water moves into different depths Wave Diffraction * Bending waves when they encounter an obstacle Absorption of waves * Reduction of energy in wave consumed by medium which it travels. * The main cause of absorption is Viscosity Interference * Two or more waves form coming together to make up a new wave Resonance * Tendency of a system to oscillate at a large amplitude at certain frequencies * Tendency to magnify a sound * The difference between an acoustic and electric guitar Wave Motion in Space and Time * Wave Motion in Space * Horizontal......

Words: 323 - Pages: 2

Premium Essay


...NAME Cyber Intro to Conceptual Physics PHET Magnetism Lab Go to Click Play with Sims and on electricity section Select the simulation “Magnets and Electromagnets.” Part I: Bar Magnet – Select the Bar Magnet Tab 1. Move the compass slowly along a semicircular path above the bar magnet until you’ve put it on the opposite side of the bar magnet. Describe what happens to the compass needle. 2. What do you suppose the compass needles drawn all over the screen tell you? 3. Move the compass along a semicircular path below the bar magnet until you’ve put it on the opposite side of the bar magnet. Describe what happens to the compass needle. 4. How many complete rotations does the compass needle make when the compass is moved once around the bar magnet? 5. Click on the “Show Field Meter” box to the right. What happens to the magnetic field reading as you move the meter closer to the bar magnet? 6. Click on the “Show planet Earth” box to the right. What type of magnetic pole (north or south) is at the geographical north pole of the Earth (Near Canada)? PART II: Electromagnet –Select the Electromagnet Tab: 7. Click on the electromagnet tab. Place the compass on the left side of the coil so that the compass center lies along the axis of the coil. (The y-component of the magnetic field is zero along the axis of the coil.) Move the compass along a semicircular path above the coil......

Words: 490 - Pages: 2

Premium Essay


...A SIMULATION TO RIPPLE WHILE YOU WORK Objective: To examine reflection, interference, and diffraction in two dimensions and relate to the waves on a spring demo Everybody has at some time thrown a pebble into a puddle and observed the ripples spreading across the surface. Some of us don’t stop until the puddle has been completely filled with every loose piece of debris in the vicinity. Now let’s dive in a bit deeper into the physics. Select the Wave Interference simulation from the Sound and Waves folder 1) Before you change any settings a. What is the shape of the pulse? b. How can you explain this? Consider the wave velocity. REFLECTION: 2) Increase the amplitude to maximum. 3) Turn off the water and add a vertical wall (bottom right button) across the entire width of the tank. 4) Turn on the water for just a couple of drips. 5) Observe the wave reflection from the barrier a. What is the shape of the reflection? b. In what ways does it differ from the incident (incoming) wave? c. Compare this result to what you learned about reflected pulses from the wave on a spring demo? INTERFERENCE: 6) Allow the faucet to run. Feel free to adjust the frequency. a. Think back to the wave on a spring demo when multiple waves tried to occupy the spring at the same time (interference). What do you think the particularly bright and dark spots represent? 7) Show the graph and observe the last couple of waves in front of the wall. a. Once again, considering the......

Words: 665 - Pages: 3

Free Essay

Physics - Free Term Papers, Essays and Research Documents The Research Paper Factory JoinSearchBrowseSaved Papers Home Page » Science Project 1 In: Science Project 1 Project 1 Write an essay of 1500 words, giving credible references on the use of physics in your daily activities. You need to mention 5 or more activities where physics is used. Remember to follow the APA style and give references. Physics is used in so many ways that most people do not even realize that they are using it. Even a stay at home mom uses physics more than one would think. Daily activities that many people do include physics without thinking about it, such as driving a car, using a headrest in a car, walking and running, flushing the toilet, and washing and drying clothes. Driving a car has many different aspects of physics involved, but today only acceleration, speed, and velocity will be discussed. People talk in terms of physics everyday without even knowing that is what they are discussing. For example, “speed” limit, how quickly a car can “accelerate,” and when they add a direction, they are actually talking about the velocity of a vehicle because velocity has a magnitude and direction, not just magnitude. According to Barry Parker in Issac Newton School of Driving, “you are accelerating and decelerating most of the time when you take a trip through the busy streets of a city, either by stepping on the gas, braking, or turning the steering wheel.”......

Words: 490 - Pages: 2

Free Essay

Physics considered to be untrue) is his dropping balls of varing mass of the Leaning Tower of Pisa by which he showed that, contrary to Aristotle’s account, the speed of a falling object is independant of its mass. It is precisely this power — to overturn wrong ideas, even if though they have been believed true for centuries, and to suggest a more complete understanding — that makes experimentation so central to all of the sciences. This experimental focus was not the last development in the physics that we’ll be looking at, though it did help pave the way for it. This next and final (for our purposes) leap was due to Newton — using mathematics to describe physics. After that, classical mechanics was essentially complete, with “only” quite a few decades of improvements and polishing before the introduction of relativity and quantum mechanics. It is physics at this level, the state of the art of classical mechanics circa the mid 19th century, that we’ll be studying in this course. . Physics Timeline Dates | Characters | Theories and discoveries | 500 – 1 BC | Archimedes, Aristotle | Heliocentric theory, geometry | 1 – 1300 AD | Al-hazen, Ptolemy in Egypt | Optics, geocentric theory | 1301 – 1499 | Leonardo de Vinci, Nicolas Cusanus | Earth is in motion,Occam’s Razor | 1500 – 1599 | Nicolaus Copernicus,Tycho Brahe | Heliocentric theory revived, astronomy | 1600 – 1650 | Galileo Galilei, Johannes Kepler | Telescope,laws of planetary...

Words: 667 - Pages: 3

Premium Essay


...movement over the 6.0-meter distance? 1) 6.0 J 2) 90. J 3) 30. J 4) 15 J 2) F 81 3) 9F 4) 81F 13. Which quantity is a measure of the rate at which work is done? 1) momentum 2) energy 3) power 4) velocity Version 18 Midterm 2012 14. The diagram shows two bowling balls, A and B, each having a mass of 7.00 kilograms, placed 2.00 meters apart. 18. A force of 1 newton is equivalent to 1 1) kg 2 × m 2 s2 2) kg × m 2 s2 3) kg × m What is the magnitude of the gravitational force exerted by ball A on ball B? 1) 8.17 × 10 –10 N 3) 8.17 × 10 –9 N 2) 1.17 × 10 –10 N 4) 1.63 × 10 –9 10 –3 3) 1.5 × m 2) 6.6 × 10 2 m 4) 1.5 × 10 8 4) kg × m 2 N 15. At an outdoor physics demonstration, a delay of 0.50 seconds was observed between the time sound waves left a loudspeaker and the time these sound waves reached a student through the air. If the air is at STP, how far was the student from the speaker? 1) 1.7 × 10 2 m s2 m Base your answers to questions 19 and 20 on the information below. A stream is 30. meters wide and its current flows southward at 1.5 meters per second. A toy boat is launched with a velocity of 2.0 meters per second eastward from the west bank of the stream. 19. What is the magnitude of the boat’s resultant velocity as it crosses the stream? 1) 2.5 m/s 16. Which type of wave requires a material medium through which to travel? 1) sound 2) radio 3)......

Words: 1799 - Pages: 8

Free Essay


...COURSE PHYSICS 1 (CORE MODULES) Coordinators Dr. Oum Prakash Sharma Sh. R.S. Dass NATIONAL INSTITUTE OF OPEN SCHOOLING A-25, INSTITUTIONAL AREA, SECTOR-62, NOIDA-201301 (UP) COURSE DESIGN COMMITTEE CHAIRMAN Prof. S.C. Garg Former Pro-Vice Chancellor IGNOU, Maidan Garhi, Delhi MEMBERS Prof. A.R. Verma Former Director, National Physical Laboratory, Delhi, 160, Deepali Enclave Pitampura, Delhi-34 Dr. Naresh Kumar Reader (Rtd.) Deptt. of Physics Hindu College, D.U. Dr. Oum Prakash Sharma Asstt. Director (Academic) NIOS, Delhi Prof. L.S. Kothari Prof. of Physics (Retd.) Delhi University 71, Vaishali, Delhi-11008 Dr. Vajayshree Prof. of Physics IGNOU, Maidan Garhi Delhi Sh. R.S. Dass Vice Principal (Rtd.) BRMVB, Sr. Sec. School Lajpat Nagar, New Delhi-110024 Dr. G.S. Singh Prof. of Physics IIT Roorkee Sh. K.S. Upadhyaya Principal Jawahar Navodaya Vidyalaya Rohilla Mohammadabad (U.P.) Dr. V.B. Bhatia Prof. of Physics (Retd.) Delhi University 215, Sector-21, Faridabad COURSE DEVELOPMENT TEAM CHAIRMAN Prof. S.C. Garg Former Pro-Vice Chancellor IGNOU, Delhi MEMBERS Prof. V.B. Bhatia 215, Sector-21, Faridabad Prof. B.B. Tripathi Prof. of Physics (Retd.), IIT Delhi 9-A, Awadhpuri, Sarvodaya Nagar Lucknow-226016 Sh. K.S. Upadhyaya Principal Navodaya Vidyalaya Rohilla Mohammadabad, (U.P.) Dr. V.P. Shrivastava Reader (Physics) D.E.S.M., NCERT, Delhi EDITORS TEAM CHAIRMAN Prof. S.C. Garg Former Pro-Vice Chancellor IGNOU, Delhi MEMBERS Prof. B.B. Tripathi Prof. of Physics......

Words: 131353 - Pages: 526

Premium Essay


... PHYS 1313 S06 Prof. T.E. Coan Version: 16 Jan ’06 Introduction Physics makes both general and detailed statements about the physical universe and these statements are organized in such a way that they provide a model or a kind of coherent picture about how and why the universe works the way it does. These sets of statements are called “theories” and are much more than a simple list of “facts and figures” like you might find in an almanac or a telephone book (even though almanacs and telephone books are quite useful). A good physics theory is far more interested in principles than simple “facts.” Noting that the moon appears regularly in the night sky is far less interesting than understanding why it does so. We have confidence that a particular physics theory is telling us something interesting about the physical universe because we are able to test quantitatively its predictions or statements about the universe. Indeed, all physics (and scientific) theories have this “put up or shut up” quality to them. For something to be called a physics “theory” in the first place, it must be falsifiable and therefore must make quantitative statements about the universe that can be then quantitatively tested. These tests are called “experiments.” The statement, “My girlfriend is the most charming woman in the world,” however true it may be, has no business being in a physics theory because it simply cannot be quantitatively tested. If the experimental......

Words: 3271 - Pages: 14