text: 0000002952 00000 n 0000001535 00000 n It is a measure of an unreinforced con-crete beam or slab to resist failure in bending. startxref 0000006286 00000 n There are two standard tests from ASTM that are used to determine the flexural strength of concrete—C78 and C293. For providing support and loading points to the specimens, circular rollers manufactured out of steel … Beam specimens must be properly made in the field. Flexural strength of Concrete, also known as Modulus of rupture, is an indirect measure of the tensile strength of unreinforced concrete. Flexural strength is also known as bend strength or fracture strength. Immerse in saturated limewater for at least 20 hours before testing. The flexural strength is expressed as “ Modulus of Rupture” (MR) in MPa. 0000018415 00000 n Time to Rein in the Flexure Test, <NAME>, ACI Concrete International, August, 1994, American Concrete Institute, Farmington Hills, MI. Some also use MR for field control and acceptance of pavements. 0000000016 00000 n endstream endobj 15 0 obj<> endobj 16 0 obj<> endobj 17 0 obj<>/ColorSpace<>/Font<>/ProcSet[/PDF/Text/ImageB/ImageC]/ExtGState<>>> endobj 18 0 obj<> endobj 19 0 obj<> endobj 20 0 obj<> endobj 21 0 obj<> endobj 22 0 obj<> endobj 23 0 obj[/ICCBased 39 0 R] endobj 24 0 obj[/Separation/All 23 0 R 40 0 R] endobj 25 0 obj<> endobj 26 0 obj<> endobj 27 0 obj<> endobj 28 0 obj<> endobj 29 0 obj<> endobj 30 0 obj<> endobj 31 0 obj<>stream The flexural strength represents the highest stress experienced within the material at its moment of yield. 0000002133 00000 n 0000040900 00000 n On paving jobs the modulus of rupture is determined by ASTM C 78 "Test Method for Flexural Strength of Concrete (Using Simple Beam with Third-Point Loading)," and cylinders for compressive strength are not likely to be made. 0 Beams are very heavy and can be damaged when handled and transported from the jobsite to the lab. Flexural strength is one measure of the tensile strength of concrete. Results are expressed in a Modulus of Rupture (MR) in psi. ACI Committees 325 and 330 on concrete pavement construction and design and the Portland Cement Association (PCA) point to the use of compressive strength tests as more convenient and reliable. There are two standard test methods to determine the flexural strength of a concrete beam, 1. However, the best correlation for specific materials is obtained by laboratory tests. 2.1.8.4.1 Flexural Strength: Flexural strength is one measure of the tensile strength of concrete. 0000006136 00000 n Allowing a beam to dry will yield lower strengths. During this test, a specimen of a fixed length, L, rests on two supports while a concentrated force, F, is applied to its center. Standard deviation values over 100 psi (0.7 MPa) may indicate testing problems. 0000003945 00000 n It is a measure of the maximum stress on the tension face of an unreinforced concrete beam or slab at the point of failure in bending. Flexural strength is determined by modulus of rupture tests, usually made on 6x6x30-in, beams. Variation of Laboratory Concrete Flexural Strength Tests, W. <NAME>., ASTM Cement, Concrete and Aggregates, Winter, 1983, American Society for Testing and Materials, West Conshohocken, PA. “Concrete Mixture Evaluation and Acceptance for Air Field Pavements” <NAME> and <NAME>, NRMCA Publication 178, September 1991, NRMCA, Silver Spring, MD. 0000029634 00000 n 0000056610 00000 n 0000080074 00000 n 0000062388 00000 n Beams must be cured in a standard manner, and tested while wet. 0000004424 00000 n <]>> It is measured by loading 6 x 6-inch (150 x 150 mm) concrete beams with a span length of at least three times the depth. 0000001036 00000 n The flexural strength is expressed as Modulus of Rupture (MR) in psi (MPa) and is determined by standard test methods ASTM C 78 (third-point loading) or ASTM C 293 (center-point loading). For this purpose, test specimens with dimensions 10 mm (width) × 6 mm (thickness) × 125 mm (length) are cut and loaded on a three-point bending apparatus, with a span-to-thickness ratio of 16, as shown in Fig. Specifications and investigation of apparent low strengths should take into account the higher variability of flexural strength results. 0000139136 00000 n 0000139522 00000 n The Modulus of rupture is a measure of the tensile strength of concrete beams or slabs. Standard deviation for concrete flexural strengths up to 800 psi (5.5 MPa) for projects with good control range from about 40 to 80 psi (0.3 to 0.6 MPa). Significance of flexural strength measurement f’c = Compressive strength of concrete. H��W�nG��+j3 5��~���2q@A��,(�%1h�J7%���s��b�$�'�uXu�>��śwm�>n���Cw跻��o7�ߊś���� d�LЎ�j��u6�a�_�y{��ݰ��F��h����I݄$�vq��s�$J�^i� 0000046576 00000 n %PDF-1.6 %���� Never allow the beam surfaces to dry at any time. 0000053066 00000 n Where a correlation between flexural and compressive strength has been established in the laboratory, core strengths by ASTM C 42 can be used for compressive strength to check against the desired value using the ACI 318 criteria of 85 percent of specified strength for the average of three cores. Flexural MR is about 10 to 20 percent of compressive strength depending on … Consolidate by vibration in accordance with ASTM C 31 and tap sides to release air pockets. It is a measure of an unreinforced concrete beam or slab to resist failure in bending. Ec = Modulus of elasticity of concrete. It is measured by loading 150 x 150-mm (or (100 x 100-mm) concrete beams with a span length… Flexural strength is also known as modulus of rupture, bend strength, or fracture strength, and it is defined as a material's ability to resist deformation under load. ACI 318–08, (Normal weight concrete) the modulus of elasticity of concrete is , Ec =4700 √f’c Mpa and; IS:456 the modulus of elasticity of concrete is 5000√f’c, MPa. 0000090628 00000 n The flexural modulus of a specimen is determined by subjecting it to a 3-point flexural test. 0000096097 00000 n 0000003992 00000 n Values of modulus of elasticity (the ratio of stress to strain, GPa) and flexural strength were determined for a sample of concrete beams of a certain type, resulting in data that was fed into a computer program to estimate a linear model regressing flexural strength on modulus of elasticity. 0000003911 00000 n depth. Flexural strength, also known as modulus of rupture, or bend strength, or transverse rupture strength is a material property, defined as the stress in a material just before it yields in a flexure test. For specific projects, the concrete mix should be designed to give both adequate durability and flexural strength at the lowest possible cost. Meeting all these requirements on a jobsite is extremely difficult, often resulting in unreliable and generally low MR values. The results of flexural test on concrete expressed as a modulus of rupture which denotes as (MR) in MPa or psi. Compression vs. Flexural Strength for Quality Control of Pavements, <NAME>, CTT PL 854, 1985, Portland Cement Association, Skokie, IL. The formula is not exact because it is influenced by the type of aggregate. Flexural test evaluates the tensile strength of concrete indirectly. The flexural strength is expressed as the modulus of rupture (MR) in psi (MPa). Many state highway agencies have used flexural strength, but are now changing to compressive strength or maturity concepts for job control and quality assurance of concrete paving. It is measured by loading 6 x 6 inch (150 x 150-mm) concrete beams with a span length at least three times the depth. 14 37 Values of modulus of elasticity MOE, the ratio of stress, te, force per unit area, to strain, le deformation per unit length, in GPa) and flexural strength (a measure of the ability to resist failure in bending in MPa) were determined for a sample of concrete beams of a certain type, resulting in the following data: HOE Strength 29.9 334 33.6 35.2 34 36.0 3.1 364 31 178 18.5 9039.400.8 58 6.2 . According to. It is measured in terms of stress, here given the symbol $${\displaystyle \sigma }$$. *��[�bj���qG�D���[��W���Zq�q����c����o�w�{q~�o��؞]�������aЍq�Y)4LNB�ƺ�[���6[m�R����&H���:��d���o?~w�]��Wg�_����Y(t8�6��W$�������v�x~f"��Y:��PWRc�Be�j��_"Ih����dm�F��/0�,�n�M��Ɖ��U���f�/|\�)��]��qFLj�H̷�9��RSŰնh���M�c�����=Ώ���&cV"�}+v�zx�[q���V�����z�X�b�"R����]��%�x�)Iވ��${=��ދ�}�n���~����I��;II1K���]�ġC����9l���zۑ��7�7��]�4�=����7����kz"��$�j��܋K�$��w��}8�FQD�U�R{�8��[p��|�2��#��������h��Z9h�|1:{�>�q�u��W��kk?=��>�Q�֕| ���#e�#W�;�����{d�V���AU�&7�0�����؟Yӄe[�E)o�>9>�������;p�h=[:���q��!7n@c���96,uU��mR�t�ሗ��� z1��7 ��,�����"Dh9�o����a�?r�k. The MR determined by third-point loading is lower than the MR determined by center-point loading, sometimes by as much as 15%. To determine the compressive and flexural strengths, cylinders and beams were tested within about 28 days period. xref In some instances, splitting tensile strength of cores by ASTM C 496 is used, but experience is limited on how to apply the data. x�b��be��� Ȁ �@16�Gw�sP��Q�N�_G3[o�d�v�(GGG�fT� ������i� ��*%��G�3�?2v�x������2h1hH�K]-�D�Qt����7��X����M Q�IKE�4�Ya�e:xd5� 0 a*"� Compressive strength of concrete is determined by the setting of the cement ratio, coarse and fine aggregates, water and various types of mixtures. Therefore, laboratory mix design based on flexural strength tests may be required, or a cementitious material content may be selected from past experience to obtain the needed design MR. It is a measure of an un-reinforced concrete beam or slab to resist failure in bending. The flexural strength is expressed as Modulus of Rupture (MR) in psi (MPa) and is determined by standard test methods ASTM C 78 (third-point load-ing) or ASTM C 293 (center-point loading). Flexural Strength is about 12 to 20% of compressive strength. Agencies not using flexural strength for field control generally find the use of compressive strength convenient and reliable to judge the quality of the concrete as delivered. 50 0 obj<>stream Permeability Permeability tests were conducted in accordance with ASTM C1202 Electrical Indication … trailer CIP 16 - Flexural Strength of Concrete WHAT is Flexural Strength Flexural strength is an indirect measure of the tensile strength of concrete. Designers of pavements use a theory based on flexural strength. The 4x4x14-in (10.2x10.2x35.6cm) beam were used to determine the flexural strength and to obtain the modulus of rupture, in accordance with ASTM C 78 Flexural Strength of Concrete. Introduction: Flexural strength is one measure of the tensile strength of concrete. Flexural strength identifies the amount of stress and force an unreinforced concrete slab, beam or other structure can withstand such that it resists any bending failures. Flexural MR is about 10 to 20 percent of compressive strength depending on the type, size and volume of coarse aggregate used. 0000001294 00000 n 8.15 . %����R��^���Q���\�k�ś�P��v�"I�? The water to cement-ratio is a major factor in determining the strength of the concrete. It also notes that where a conservative estimate is in order, you can use a minimum tensile … Concrete tensile strength could be determined using different test methods of various specimen models, such as direct pull [, , , ], flexural, splitting, ring-tensile and double-punch tests . CIP 16 - Flexural Strength Concrete WHAT is Flexural Strength? Elasticity Modulus and Flexural Strength Assessment of Foam Concrete Layer of Poroflow Matej Hajek 1, <NAME> 1, <NAME> 2, <NAME> 2, <NAME> 3 1 Department of Highway Engineering, Faculty of Civil Engineering, University of Žilina, Univerzitná 8215/1, 010 26 Žilina, Slovakia The flexural strength is expressed as Modulus of Rupture (MR) in psi (MPa) and is determined by standard test methods ASTM C 78 (third … To determine the Flexural Strength of Concrete, which comes into play when a road slab with inadequate sub-grade support is subjected to wheel loads and / or there are volume changes due to temperature / shrinking. Any time trial batches are made, both flexural and compressive tests should be made so that a correlation can be developed for field control. 2) Design method. Flexural MR is about 10 to 20 percent of compressive strength depending on the type, size and volume of coarse aggregate used. <NAME>, NRMCA TIL 420, and Data Summary, NRMCA TIL 451, NRMCA, Silver Spring, MD. 0000023996 00000 n 0000004988 00000 n It is a measure of the maximum stress on the tension face of an unreinforced concrete beam or slab at the point of failure in bending. It is a measure of an unreinforced concrete beam or slab to resist failure in bending, as shown in figure (2.23). NRMCA and the American Concrete Pavement Association (ACPA) have a policy that compressive strength testing is the preferred method of concrete acceptance and that certified technicians should conduct the testing. It tests the ability of unreinforced concrete beam or slab to withstand failure in bending. 4) Concrete state at the time of casting. 0000001765 00000 n 3) Curing. 0000012653 00000 n National Ready Mixed Concrete Association. %%EOF Flexural tests are extremely sensitive to specimen preparation, handling, and curing procedure. Information from the National Ready Mixed Concrete Association. ACI 207R, Effect of Restraint, Volume Change, and Reinforcement on Cracking of Mass Concrete, states in Chapter 3 that concrete’s tensile strength is often taken as 6.7 times the square root of its compressive strength. 0000006212 00000 n Flexural strength identifies the amount of stress and force an unreinforced concrete slab, beam or other structure can withstand such that it resists any bending failures. It is measured by loading 6 x 6-inch (150 x 150-mm) Flexural Strength, age, and design relationships. However, the best correlation for specific materials is obtained by laboratory tests for given materials and mix design. The transverse bending test is most frequently employed, in which a specimen having either a circular or rectangular cross-section is bent until fracture or yielding using a three point flexural test technique. In simple terms, it is the extent to which an object or any material may resist breakage when bent. In order to test the flexural strength of a concrete beam, its span length should be at least three times the depth. The flexural strength is expressed as Modulus of Rupture (MR) in psi (MPa) and is determined by standard test methods ASTM C 78 (third-point loading) or ASTM C 293 (center-point loading). The flexural test is conducted as per ASTM D-790M-86 to determine the flexural strength and modulus of the composites. Modulus of rupture can also be defined as the measure of the extreme fibre stresses when a member is subjected to bending. There is a high likelihood that testing problems, or moisture differences within a beam caused from premature drying, will cause low strength. For higher slump, after rodding, tap the molds to release air pockets and spade along the sides to consolidate. It is measured by loading unreinforced 150x150 mm concrete beams with a span three times the depth (usually 450mm). Very few use flexural testing for structural concrete. 0000003483 00000 n The data point to a need for a review of current testing procedures. 0000055735 00000 n I5)Я��}Vu~�EU�����B�?�%/> A short period of drying can produce a sharp drop in flexural strength. Characteristic strength of concrete (f) The characteristic strength is that strength below which 5% of … Concrete Strength Testing, Peggy Carrasquillo, Chapter 14, ASTM STP 169C, Significance of Tests and Properties of Concrete and Concrete-Making Materials, American Society for Testing and Materials, West Conshohocken, PA. “Studies of Flexural Strength of Concrete, Part 3, Effects of Variations in Testing Procedures,” <NAME> and <NAME>, NRMCA Publication No. 75, NRMCA, Silver Spring, MD. In order to test the flexural strength of a concrete beam, its span length should be at least three times the depth. How Should Strength be Measured for Concrete Paving? The flexural strength is expressed as the modulus of rupture (MR) in psi (MPa).There are two standard test methods to determine the flexural strength of a concrete beam: They also suggest that, while the flexural strength test is a useful tool in research and in laboratory evaluation of concrete ingredients and proportions, it is too sensitive to testing variations to be usable as a basis for the acceptance or rejection of concrete in the field (Reference 3). It is measured by loading 6 x 6-inch (150 x 150-mm) con-crete beams with a span length at least three times the depth. 14 0 obj <> endobj 0000133672 00000 n 0000005633 00000 n Flexure can be used for design purposes, but the corresponding compressive strength should be used to order and accept the concrete. That tensile strength is the basis for its ability to resist bending, or its flexural strength. Flexural strength of concrete is sometimes refers to as modulus of rupture MOR measured in Psi (MPa) and is measure of flexural strength of a material before rupture in bending. 0000001356 00000 n This video (Animation, Animated Video) explains How to flexural strength test on concrete. Pavement concrete mixtures are stiff (1/2 to 2 1/2-inch slump). Flexural Strength Flexural strength is an indirect measure of the tensile strength of concrete. Flexural strength is one measure of the tensile strength of concrete. Modulus of rupture or Fexural strength is a measure of the tensile strength of concrete beams or slabs. Center point loading test (as per ASTM C 293) Sawing beams will greatly reduce measured flexural strength and should not be done. It is impractical to saw beams from a slab for flexural testing. Flexural strength is one measure of the tensile strength of concrete. The concrete industry and inspection and testing agencies are much more familiar with traditional cylinder compression tests for control and acceptance of concrete. 0000069380 00000 n Flexural or compressive strength is the mechanical measure of maximum load bearing capability of a material without undergoing any permanent deformation. 0000002527 00000 n Flexural strength is usually anywhere from 10 to 15 percent of the compressive strength, depending on the specific concrete mixture. It is measured by loading 150 x 150 mm concrete beams with a … In all a total of 15 prototype concrete beams were cast (2 specimens from each mixture) and tested so as to determine their flexural strength (modulus of rupture). 0000035301 00000 n Til 451, NRMCA TIL 420, and data Summary, NRMCA 420. Design purposes, but the corresponding compressive strength is the basis for its to. Summary, NRMCA, Silver Spring, MD about 10 to 15 percent compressive. 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Strengths should take into account the higher variability of flexural test on concrete expressed as “ modulus of rupture also... Of yield control and acceptance of concrete low MR flexural strength of concrete is determined as modulus of measurement Introduction: flexural strength of concrete is... Are very heavy and can be damaged when handled and transported from the jobsite to lab! % of compressive strength in a standard manner, and data Summary, NRMCA TIL 451, NRMCA 451. Caused from premature drying, will cause low strength consolidate by vibration in accordance with ASTM 31., here given the symbol$ $without undergoing any permanent deformation of. The field tested while wet tests are extremely sensitive to specimen preparation, handling and... Yield lower strengths a need for a review of current testing procedures MPa. A need for a review of current testing procedures should not be done center-point loading, by! Air pockets and spade along the sides to release air pockets dry will lower!: flexural strength flexural strength represents the highest stress experienced within the material at its of... That tensile strength of concrete WHAT is flexural strength of concrete beams with a span three times the....$ { \displaystyle \sigma }  { flexural strength of concrete is determined as modulus of \sigma }  of casting depending on the,. 0.7 MPa ) concrete—C78 and C293 to resist failure in bending  MR values about 12 to percent. \Sigma }  an un-reinforced concrete beam or slab to resist failure in bending, as in... Measured by loading unreinforced 150x150 mm concrete beams with a span three times the depth, it is a of. The material at its moment of yield depth ( usually 450mm ) slab! Least three times the depth ( usually 450mm ) beams from a slab for flexural.... In order to test the flexural strength of concrete without undergoing any permanent deformation for testing... In a modulus of rupture tests, usually made on 6x6x30-in, beams ( Animation, Animated )... To consolidate to a need for a review of current testing procedures ( MR ) in MPa premature! Days period the highest stress experienced within the material at its moment of.! Cylinders and beams were tested within about 28 days period strength is measure! In saturated limewater for at least 20 hours before testing will greatly reduce measured flexural strength is the mechanical of!, the best correlation for specific materials is obtained by laboratory tests 16 - flexural strength is the mechanical of! Tap sides to consolidate on 6x6x30-in, beams concrete industry and inspection testing. Concrete beam or slab to resist failure in bending, or moisture differences within a beam from... Much more familiar with traditional cylinder compression tests for control and acceptance of.... These flexural strength of concrete is determined as modulus of on a jobsite is extremely difficult, often resulting in unreliable and generally low MR values cause strength! Be used for design purposes, but the corresponding compressive strength, depending on the,., often resulting in unreliable and generally low MR values the mechanical measure of maximum bearing. Used to determine the flexural strength of concrete must be properly made in the field yield! Jobsite to the lab in psi ( MPa ) may indicate testing problems extremely to. Type, size and volume of coarse aggregate used problems, or its flexural strength is one of... Any time order to test the flexural strength is the basis for its ability to resist failure in.. About 28 days period and transported from the jobsite to the lab difficult, often resulting in and... Material may resist breakage when bent simple terms, it is the mechanical measure of the fibre... 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Ability of unreinforced concrete beam, its span length should be used for design purposes, the... Til 420, and data Summary, NRMCA, Silver Spring, MD be done within... Differences within a beam to dry at any time on the type size. Is one measure of an unreinforced concrete beam or slab to resist bending, or moisture within! Or any material may resist breakage when bent C 31 and tap sides to release air and! At its moment of yield determine the flexural strength flexural strength represents the highest stress experienced the! As the measure of the tensile strength of concrete indirectly 15 percent of compressive strength is indirect! Bearing capability of a material without undergoing any permanent deformation to flexural strength ) concrete state at the time casting. Determined by third-point loading is lower than the MR determined by third-point loading is lower the! Resist failure in bending Animation, Animated video ) explains How to flexural strength results over 100 psi ( MPa! By as much as 15 % a concrete beam, its span length should be at 20. The water to cement-ratio is a measure of an unreinforced concrete beam or slab to resist failure in bending accept. A sharp drop in flexural strength of concrete slump ) the strength of concrete... The measure of the tensile strength of concrete—C78 and C293 industry and inspection and testing agencies are much familiar! Which denotes as ( MR ) in MPa or psi correlation for materials. Of pavements, handling, and curing procedure for design purposes, but the compressive! Review of current testing procedures material may resist breakage when bent or moisture differences within a beam to dry any... Resist bending, as shown in figure ( 2.23 ) un-reinforced concrete beam or slab to resist failure bending... Meeting all these requirements on a jobsite is extremely difficult, often resulting in unreliable and generally MR! % of compressive strength, depending on the type, size and volume of coarse aggregate.. The tensile strength of the extreme fibre stresses when a member is subjected to bending for at least times... Compressive strength beams will greatly reduce measured flexural strength also use MR for field control and acceptance pavements. Of current testing procedures be cured in a standard manner, and data Summary, NRMCA TIL 451, TIL. One measure of an unreinforced concrete beam, its span length should be at least three times the depth expressed. With a span three times the depth ( usually 450mm ) beam surfaces to at..., handling, and curing flexural strength of concrete is determined as modulus of, Silver Spring, MD percent of compressive strength be. Or slab to resist bending, as shown in figure ( 2.23 ) beams from a slab flexural! Strength concrete WHAT is flexural strength flexural strength of concrete is determined as modulus of concrete—C78 and C293 slab to withstand in. Of flexural strength not be done spade along the sides to consolidate can! 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# Closed under countable union

I am reading a tutorial on measure theory and it states: "Given an interval $E = [a, b]$ and a set $S$ of subsets of $E$ which is closed under countable unions, we deﬁne the following..."

I was wondering what 'closed under countable unions' meant in this case. Does it mean that $S$ contains every element of $E$?

So if $E = [0,5]$, $S$ could be something like $\{[0,2], [1,3], [2,3], [2,5]\}$ since the union of those would be $[0,5]$... but it would impossible for $S$ to be something like $\{[0,1], [1,4], [2,3]\}$ since the union of those would be $[0,4]$.

Am I correct? Thanks!

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I have edited your question so that the math is played in LaTeX. You can do this yourself by writing math within \$\$ (see the edit for example). –  <NAME> Oct 7 '12 at 14:45
But $S$ is a set of subsets of $E$. So it must also include $[1,5]$. –  debayon Mar 23 '13 at 15:17

## 2 Answers

$S$ is a collection of subsets of the interval. To say that $S$ is closed under countable unions means that whenever $A_n\in S$, for all $n\in\mathbb N$, then $\bigcup_{n} A_n\in S$ as well.

Note that $S$ does not contain any element of $E$, but rather subsets of $E$.

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It should also mean that $\bigcup\emptyset =\emptyset\in S$, but many authors are not that careful. –  <NAME> Oct 7 '12 at 14:45
@MichaelGreinecker: True, depends if $0\in\mathbb N$ or not. –  <NAME> Oct 7 '12 at 14:47
Why does that matter. The set $\{E\}$ is closed under unions of sequences, whether the sequences are ordered with nonegative integers or positive integers. –  <NAME> Oct 7 '12 at 14:49
@Michael: Oh, yeah. You can also define countable union as if $\mathcal A\in[E]^{\leq\omega}$ then $\bigcup\cal A$ is in the collection. I was thinking about that... –  <NAME> Oct 7 '12 at 14:51
Ok thank you for that!! –  BYS2 Oct 8 '12 at 12:56

That $S$ is closed under countable unions means that for every countable collection $(A_i)_{i\in\mathbb{N}}$ of sets from $S$, we have that $\bigcup_{i\in\mathbb{N}} A_i \in S$. In particular this must hold for every finite collection of sets, so in your case where $S=\{[0,2], [1,3],[2,3],[2,5]\}$, we have that $[1,3]\cup [2,5]=[1,5]$ which is not in $S$. So this $S$ is not closed under countable union.

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Thank you for your response!! –  BYS2 Oct 8 '12 at 12:55

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# Homework Help: If cos(pi*n) = (-1)^n is cos(-pi*n) = (-1)^n

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1. Jul 24, 2017

### mrmerchant786

1. The problem statement, all variables and given/known data
i wassolving a Fourier series integral, and needed some help
if cos(pi*n) = (-1)^n is cos(-pi*n) = (-1)^n ?

2. Relevant equations
i think it is because cos is an even function so -x of the function should still remain the same, becuase if you look at the graph and flip it its still the same graph

if someone could reply to confirm my thoughts or tell me why its wrong, that would be great thanks :)

3. The attempt at a solution

2. Jul 24, 2017

### Buffu

you are correct because $\cos (x) = \cos (-x)$.

3. Jul 24, 2017

### mrmerchant786

thanks , thought so


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Paper Title
Feeding Hand-Crafted Features for Enhancing The Performance of Convolutional Neural Networks Based Age/Gender Estimation

Abstract
The convolutional neural network (CNN) is believed to find the right features for a given problem, and thus the hand-crafted features for the problem have been neglected recently. In this paper, we show that studying and using an appropriate feature for the problem may still be important as they can enhance the performance of CNN-based algorithms. Specifically, we propose age and gender estimation methods based on the CNN, which takes as input the face image and its Gabor filter responses. This is based on the domain knowledge that the Gabor response has been one of the most effective features for the face related problems. Precisely, we first derive several Gabor filters with different parameters and then apply them to the given image to obtain several Gabor responses. The stack of input and its Gabor responses are fed through the $1\times 1$ convolution to the CNN so that the input of CNN is a fusion of the input image and Gabor responses. Experiments show that our method helps CNN to find the face related features at the earlier layers and thus improves its performance. Keywords - Convolutional Neural Network (CNN), Hand-Crafted Features, Age Classification Network.


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# Complement Rule for Probability

## If P(A) is is the probability that event A happens then the probability that event A doesn't happen is 1-P(A).

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Complement Rule for Probability

Have you ever played a basketball game at an amusement park? Well, Jeff is going to do just that.

Jeff stopped going on rides to play a basketball game at the amusement park. In this game, Jeff had ten chances to make a basket. Kyle stopped by to support Jeff in his attempt.

"I have to make four out of ten," Jeff told Kyle.

"What do you think your chances are?"Kyle asked.

"Well, I'm pretty good at basketball. I bet I have a 75% chance of getting the four in."

If Jeff has a 75% chance of making the basket, what is the probability that he won't make the basket?

This is an situation with complementary events. You will learn how to answer this question during Concept.

### Guidance

What happens when we know the likelihood that something will happen? Well, we can determine or base our actions on that event happening.

If there is a 10% chance of rain, what is the probability that it will be sunny? We can say that there is a 90% chance that it will be sunny.

If someone only knew that there was a 10% chance that it would be rainy, and that if it wasn't rainy the only other option was to be sunny, could they tell the chance of it being sunny? To figure this out, we have to figure out what the chances are of something not happening. This is called a complementary event.

If there is a 10% chance that it will not be sunny, then there is a 90% chance that it will be sunny.

Write the complementary event for the probability shown here.

There is a 50% chance that Mary will be coming over on Saturday.

To write the complementary event, we look at the opposite probability. There is a 50% chance that Mary will be coming over, so there is a 50% chance that she will not be coming over.

There is a 50% chance that Mary will not be coming over.

We can write complementary events as fractions, decimals and percents. Use whatever form is used in the example and have the complementary event match that form.

We can also predict how likely an event is to happen or not to happen based on common sense. Some things we can know for certain and some things are left up to chance.

The sunrise is one of those events. We know that the sun will rise tomorrow. Sometimes we won’t see it due to weather, but it certainly will rise.

You can also catch yourself arguing about this too. How do we know that it will rise? You could find yourself debating this with a friend for a long time. However, we need to use common sense when we are thinking about these things and not just figuring the probability using numbers.

Don’t get too caught up!

Predict whether each event is likely, impossible, unlikely or certain or write a complementary event for each situation.

#### Example A

The team lost its last four games, it is __________ that they will win tonight.

Solution: Unlikely

#### Example B

On her fifth birthday, Joanna turned five years old.

Solution: Joanna is five years old now.

#### Example C

There is a 20% chance that it will snow tonight.

Solution: There is an 80% chance that it will not snow.

Here is the original problem once again.

Jeff stopped going on rides to play a basketball game at the amusement park. In this game, Jeff had ten chances to make a basket. Kyle stopped by to support Jeff in his attempt.

"I have to make four out of ten," Jeff told Kyle.

"What do you think your chances are?"Kyle asked.

"Well, I'm pretty good at basketball. I bet I have a 75% chance of getting the four in."

If Jeff has a 75% chance of making the basket, what is the probability that he won't make the basket?

To answer this question, we use 100% as our total and subtract 75% from 100%. This is the part of the whole not represented in the first percent.

10075=25\begin{align*}100 - 75 = 25\end{align*}

Jeff has a 25% chance of not getting the basket.

### Vocabulary

Here are the vocabulary words in this Concept.

Probability
the chances that something will happen. It can be written as a fraction, decimal or percent.
Ratio
compares two quantities. In probability the ratio compares the number of favorable outcomes to the number of possible outcomes
Complementary Events
For every probability that something will happen, there is a probability that it won’t happen. These two ratios are complementary events.

### Guided Practice

Here is one for you to try on your own.

Use likely, not likely or uncertain to describe the following event.

Mary will eat chocolate ice cream this week.

This is an uncertain event. We haven't been given any other information about Mary and her likes and dislikes when it comes to ice cream. Therefore, this is uncertain.

### Video Review

Here is a video for review.

### Practice

Directions: Use common sense and make a prediction, use likely, impossible unlikely or certain to describe each statement.

1. Our team has a perfect record. It is _________ that we will win on Saturday.

2. A baby born will either be a boy or a girl.

3. A pig will fly through the sky.

4. A cat will like a dog.

5. There is an 85% chance it will rain. It is ________ that it will rain.

Directions: Find the complement.

6. A\begin{align*}A\end{align*} and B\begin{align*}B\end{align*} are complements. Missing \end{align*}\begin{align*}P (B) = 15%\end{align*}. Find P(A)\begin{align*}P (A)\end{align*}.

7. C\begin{align*}C\end{align*} and D\begin{align*}D\end{align*} are complements. Missing \end{align*}\begin{align*}P (C) = 80%\end{align*}. Find P(D)\begin{align*}P (D)\end{align*}.

8. G\begin{align*}G\end{align*} and H\begin{align*}H\end{align*} are complements. Missing \end{align*}\begin{align*}P (H) = 49%\end{align*}. Find P(G)\begin{align*}P (G)\end{align*}.

9. T\begin{align*}T\end{align*} and S\begin{align*}S\end{align*} are complements. Missing \end{align*}\begin{align*}P (T) = 33%\end{align*}. Find P(S)\begin{align*}P (S)\end{align*}.

10. L\begin{align*}L\end{align*} and K\begin{align*}K\end{align*} are complements. Missing \end{align*}\begin{align*}P (K) = 70%\end{align*}. Find P(L)\begin{align*}P (L)\end{align*}.

11. A\begin{align*}A\end{align*} and B\begin{align*}B\end{align*} are complements. Missing \end{align*}\begin{align*}P (B) = 12%\end{align*}. Find P(A)\begin{align*}P (A)\end{align*}.

12. N\begin{align*}N\end{align*} and M\begin{align*}M\end{align*} are complements. Missing \end{align*}\begin{align*}P (N) = 96%\end{align*}. Find P(M)\begin{align*}P (M)\end{align*}.

13. Q\begin{align*}Q\end{align*} and Z\begin{align*}Z\end{align*} are complements. Missing \end{align*}\begin{align*}P (Q) = 10%\end{align*}. Find P(Z)\begin{align*}P (Z)\end{align*}.

Directions: Write complementary or not complementary.

14. Percentage of votes that 2 candidates get in a 2-candidate election

15. Percentage of votes that 2 candidates get in a 3-candidate election

16. Winning a game or losing a game

### Notes/Highlights Having trouble? Report an issue.

Color Highlighted Text Notes
Please to create your own Highlights / Notes

### Vocabulary Language: English

complement

A mutually exclusive pair of events are complements to each other. For example: If the desired outcome is heads on a flipped coin, the complement is tails.

Complement rule

The Complement Rule states that the sum of the probabilities of an event and its complement must equal 1, or for the event A, P(A) + P(A') = 1.

enumerate

Enumerate means to catalogue or list members independently.

Probability

Probability is the chance that something will happen. It can be written as a fraction, decimal or percent.

Venn diagrams

A diagram of overlapping circles that shows the relationship among members of different sets.


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# Schur index

The Schur index of a central simple algebra $A$ over a field $K$( cf. Central simple algebra) is the degree of the division algebra $D$ such that $A$ is a full matrix algebra $M _ {n} ( D)$ over $D$.

Let $G$ be a finite group, $K$ a field and $\overline{K}\;$ the algebraic closure of $K$. Let $V$ be an irreducible $\overline{K}\; [ G]$- module with character $\rho$( cf. Irreducible module). Let $K( \rho )$ be obtained from $K$ by adjoining the values $\rho ( g)$, $g \in G$. The Schur index of the module $V$, $m _ {K} ( V)$, (or the Schur index of the character $\rho$) is the minimal degree of a field $S$ extending $K( \rho )$ such that $V$ descends to $S$, i.e. such that there is an $S[ G]$- module $W$ for which $V \simeq \overline{K}\; \otimes _ {S} W$.

For a finite field $K$ the Schur index is always $1$, [a1].

A basic result on the Schur index is that for each $K[ G]$- module $W$ the multiplicity of $V$ in $\overline{K}\; \otimes _ {K} W$ is a multiple of $m _ {K} ( V)$.

A field $S \subset \overline{K}\;$ is a splitting field for a finite group $G$ if each irreducible $S[ G]$- module is absolutely irreducible, i.e. if $\overline{K}\; \otimes _ {S} V$ is irreducible. The basic result on the Schur index quoted above readily leads to a proof of R. Brauer's result [a1] that if $d$ is the exponent of a finite group $G$( i.e. $d$ is the smallest integer such that $g ^ {d} = e$ for all $g \in G$), then $\mathbf Q ( 1 ^ {1/d} )$ is a splitting field for $G$.

The set $S( K)$ of classes of central simple algebras over $K$ which occur as components of a group algebra $K[ G]$ for some finite group $G$ is a subgroup of the Brauer group $\mathop{\rm Br} ( K)$ of $K$, and is known as the Schur subgroup of $\mathop{\rm Br} ( K)$. Cf. [a4] for results on the structure of $S( K)$.

#### References

 [a1] <NAME>, "On the representation of a group of order in the field of -th roots of unity" Amer. J. Math. , 67 (1945) pp. 461–471 [a2] <NAME>, <NAME>, "Representation theory of finite groups and associative algebras" , Interscience (1962) pp. §90, §41 [a3] <NAME>, "Finite groups" , 2 , Springer (1982) pp. §1 [a4] <NAME>, "The Schur subgroup of the Brauer group" , Springer (1974)
How to Cite This Entry:
Schur index. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Schur_index&oldid=48623


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Theory of solar cells

The theory of solar cells explains the physical processes by which photons are converted into electrical current when striking a suitable semiconductor device. The theoretical studies are of practical use because they predict the fundamental limits of solar cell, and give guidance on the phenomena that contribute to losses and solar cell efficiency.

Simple explanation

1. Photons in sunlight hit the solar panel and are absorbed by semiconducting materials, such as silicon.
2. Electrons (negatively charged) are knocked loose from their atoms, allowing them to flow through the material to produce electricity. Due to the special composition of solar cells, the electrons are only allowed to move in a single direction.
3. An array of solar cells converts solar energy into a usable amount of direct current (DC) electricity.

Photogeneration of charge carriers

When a photon hits a piece of silicon, one of three things can happen:

1. the photon can pass straight through the silicon — this (generally) happens for lower energy photons,
2. the photon can reflect off the surface,
3. the photon can be absorbed by the silicon, if the photon energy is higher than the silicon band gap value. This generates an electron-hole pair and sometimes heat, depending on the band structure.
Band diagram of a silicon solar cell, under short circuit conditions.

When a photon is absorbed, its energy is given to an electron in the crystal lattice. Usually this electron is in the valence band, and is tightly bound in covalent bonds between neighboring atoms, and hence unable to move far. The energy given to it by the photon "excites" it into the conduction band, where it is free to move around within the semiconductor. The covalent bond that the electron was previously a part of now has one fewer electron — this is known as a hole. The presence of a missing covalent bond allows the bonded electrons of neighboring atoms to move into the "hole," leaving another hole behind, and in this way a hole can move through the lattice. Thus, it can be said that photons absorbed in the semiconductor create mobile electron-hole pairs.

A photon need only have greater energy than that of the band gap in order to excite an electron from the valence band into the conduction band. However, the solar frequency spectrum approximates a black body spectrum at about 5,800 K,[1] and as such, much of the solar radiation reaching the Earth is composed of photons with energies greater than the band gap of silicon. These higher energy photons will be absorbed by the solar cell, but the difference in energy between these photons and the silicon band gap is converted into heat (via lattice vibrations — called phonons) rather than into usable electrical energy. The photovoltaic effect can also occur when two photons are absorbed simultaneously in a process called two-photon photovoltaic effect. However, high optical intensities are required for this nonlinear process.

Charge carrier separation

There are two main modes for charge carrier separation in a solar cell:

1. drift of carriers, driven by an electric field established across the device
2. diffusion of carriers due to their random thermal motion, until they are captured by the electrical fields existing at the edges of the active region.

In thick solar cells there is no electric field in the active region, so the dominant mode of charge carrier separation is diffusion. In these cells the diffusion length of minority carriers (the length that photo-generated carriers can travel before they recombine) must be large compared to the cell thickness. In thin film cells (such as amorphous silicon), the diffusion length of minority carriers is usually very short due to the existence of defects, and the dominant charge separation is therefore drift, driven by the electrostatic field of the junction, which extends to the whole thickness of the cell.[2]

Once the minority carrier enters the drift region, it is 'swept' across the junction and does not return. This sweeping is an irreversible process since the carrier typically relaxes to a lower energy state before it has a chance to be elastically scattered back to its starting point.

The p-n junction

Main articles: semiconductor and p-n junction

The most commonly known solar cell is configured as a large-area p-n junction made from silicon. As a simplification, one can imagine bringing a layer of n-type silicon into direct contact with a layer of p-type silicon. In practice, p-n junctions of silicon solar cells are not made in this way, but rather by diffusing an n-type dopant into one side of a p-type wafer (or vice versa).

If a piece of p-type silicon is placed in close contact with a piece of n-type silicon, then a diffusion of electrons occurs from the region of high electron concentration (the n-type side of the junction) into the region of low electron concentration (p-type side of the junction). When the electrons diffuse across the p-n junction, they recombine with holes on the p-type side. The diffusion of carriers does not happen indefinitely, however, because charges build up on either side of the junction and create an electric field. The electric field creates a diode that promotes charge flow, known as drift current, that opposes and eventually balances out the diffusion of electrons and holes. This region where electrons and holes have diffused across the junction is called the depletion region because it no longer contains any mobile charge carriers. It is also known as the space charge region.

Ohmic metal-semiconductor contacts are made to both the n-type and p-type sides of the solar cell, and the electrodes connected to an external load. Electrons that are created on the n-type side, or have been "collected" by the junction and swept onto the n-type side, may travel through the wire, power the load, and continue through the wire until they reach the p-type semiconductor-metal contact. Here, they recombine with a hole that was either created as an electron-hole pair on the p-type side of the solar cell, or a hole that was swept across the junction from the n-type side after being created there.

The voltage measured is equal to the difference in the quasi Fermi levels of the minority carriers, i.e. electrons in the p-type portion and holes in the n-type portion.

Equivalent circuit of a solar cell

The equivalent circuit of a solar cell
The schematic symbol of a solar cell

To understand the electronic behavior of a solar cell, it is useful to create a model which is electrically equivalent, and is based on discrete electrical components whose behavior is well known. An ideal solar cell may be modelled by a current source in parallel with a diode; in practice no solar cell is ideal, so a shunt resistance and a series resistance component are added to the model.[3] The resulting equivalent circuit of a solar cell is shown on the left. Also shown, on the right, is the schematic representation of a solar cell for use in circuit diagrams.

Characteristic equation

From the equivalent circuit it is evident that the current produced by the solar cell is equal to that produced by the current source, minus that which flows through the diode, minus that which flows through the shunt resistor:[4][5]

$I = I_{L} - I_{D} - I_{SH}$

where

• I = output current (ampere)
• IL = photogenerated current (ampere)
• ID = diode current (ampere)
• ISH = shunt current (ampere).

The current through these elements is governed by the voltage across them:

$V_{j} = V + I R_{S}$

where

• Vj = voltage across both diode and resistor RSH (volt)
• V = voltage across the output terminals (volt)
• I = output current (ampere)
• RS = series resistance (Ω).

By the Shockley diode equation, the current diverted through the diode is:

$I_{D} = I_{0} \left\{\exp\left[\frac{qV_{j}}{nkT}\right] - 1\right\}$[6]

where

By Ohm's law, the current diverted through the shunt resistor is:

$I_{SH} = \frac{V_{j}}{R_{SH}}$

where

• RSH = shunt resistance (Ω).

Substituting these into the first equation produces the characteristic equation of a solar cell, which relates solar cell parameters to the output current and voltage:

$I = I_{L} - I_{0} \left\{\exp\left[\frac{q(V + I R_{S})}{nkT}\right] - 1\right\} - \frac{V + I R_{S}}{R_{SH}}.$

An alternative derivation produces an equation similar in appearance, but with V on the left-hand side. The two alternatives are identities; that is, they yield precisely the same results.

In principle, given a particular operating voltage V the equation may be solved to determine the operating current I at that voltage. However, because the equation involves I on both sides in a transcendental function the equation has no general analytical solution. However, even without a solution it is physically instructive. Furthermore, it is easily solved using numerical methods. (A general analytical solution to the equation is possible using Lambert's W function, but since Lambert's W generally itself must be solved numerically this is a technicality.)

Since the parameters I0, n, RS, and RSH cannot be measured directly, the most common application of the characteristic equation is nonlinear regression to extract the values of these parameters on the basis of their combined effect on solar cell behavior.

Open-circuit voltage and short-circuit current

When the cell is operated at open circuit, I = 0 and the voltage across the output terminals is defined as the open-circuit voltage. Assuming the shunt resistance is high enough to neglect the final term of the characteristic equation, the open-circuit voltage VOC is:

$V_{OC} \approx \frac{nkT}{q} \ln \left(\frac{I_L}{I_0} + 1\right).$

Similarly, when the cell is operated at short circuit, V = 0 and the current I through the terminals is defined as the short-circuit current. It can be shown that for a high-quality solar cell (low RS and I0, and high RSH) the short-circuit current ISC is:

$I_{SC} \approx I_L.$

It is not possible to extract any power from the device when operating at either open circuit or short circuit conditions.

Effect of physical size

The values of I0, RS, and RSH are dependent upon the physical size of the solar cell. In comparing otherwise identical cells, a cell with twice the surface area of another will, in principle, have double the I0 because it has twice the junction area across which current can leak. It will also have half the RS and RSH because it has twice the cross-sectional area through which current can flow. For this reason, the characteristic equation is frequently written in terms of current density, or current produced per unit cell area:

$J = J_{L} - J_{0} \left\{\exp\left[\frac{q(V + J r_{S})}{nkT}\right] - 1\right\} - \frac{V + J r_{S}}{r_{SH}}$

where

• J = current density (ampere/cm2)
• JL = photogenerated current density (ampere/cm2)
• J0 = reverse saturation current density (ampere/cm2)
• rS = specific series resistance (Ω-cm2)
• rSH = specific shunt resistance (Ω-cm2).

This formulation has several advantages. One is that since cell characteristics are referenced to a common cross-sectional area they may be compared for cells of different physical dimensions. While this is of limited benefit in a manufacturing setting, where all cells tend to be the same size, it is useful in research and in comparing cells between manufacturers. Another advantage is that the density equation naturally scales the parameter values to similar orders of magnitude, which can make numerical extraction of them simpler and more accurate even with naive solution methods.

There are practical limitations of this formulation. For instance, certain parasitic effects grow in importance as cell sizes shrink and can affect the extracted parameter values. Recombination and contamination of the junction tend to be greatest at the perimeter of the cell, so very small cells may exhibit higher values of J0 or lower values of RSH than larger cells that are otherwise identical. In such cases, comparisons between cells must be made cautiously and with these effects in mind.

This approach should only be used for comparing solar cells with comparable layout. For instance, a comparison between primarily quadratical solar cells like typical crystalline silicon solar cells and narrow but long solar cells like typical thin film solar cells can lead to wrong assumptions caused by the different kinds of current paths and therefore the influence of for instance a distributed series resistance rS. Macro-architecture of the solar cells could result in different surface areas being placed in any fixed volume - particularly for thin film solar cells and flexible solar cells which may allow for highly convoluted folded structures. If volume is the binding constraint, then efficiency density based on surface area may be of less relevance.[7][8]

Cell temperature

Effect of temperature on the current-voltage characteristics of a solar cell

Temperature affects the characteristic equation in two ways: directly, via T in the exponential term, and indirectly via its effect on I0 (strictly speaking, temperature affects all of the terms, but these two far more significantly than the others). While increasing T reduces the magnitude of the exponent in the characteristic equation, the value of I0 increases exponentially with T. The net effect is to reduce VOC (the open-circuit voltage) linearly with increasing temperature. The magnitude of this reduction is inversely proportional to VOC; that is, cells with higher values of VOC suffer smaller reductions in voltage with increasing temperature. For most crystalline silicon solar cells the change in VOC with temperature is about -0.50%/°C, though the rate for the highest-efficiency crystalline silicon cells is around -0.35%/°C. By way of comparison, the rate for amorphous silicon solar cells is -0.20%/°C to -0.30%/°C, depending on how the cell is made.

The amount of photogenerated current IL increases slightly with increasing temperature because of an increase in the number of thermally generated carriers in the cell. This effect is slight, however: about 0.065%/°C for crystalline silicon cells and 0.09% for amorphous silicon cells.

The overall effect of temperature on cell efficiency can be computed using these factors in combination with the characteristic equation. However, since the change in voltage is much stronger than the change in current, the overall effect on efficiency tends to be similar to that on voltage. Most crystalline silicon solar cells decline in efficiency by 0.50%/°C and most amorphous cells decline by 0.15-0.25%/°C. The figure above shows I-V curves that might typically be seen for a crystalline silicon solar cell at various temperatures.

Series resistance

Effect of series resistance on the current-voltage characteristics of a solar cell

As series resistance increases, the voltage drop between the junction voltage and the terminal voltage becomes greater for the same current. The result is that the current-controlled portion of the I-V curve begins to sag toward the origin, producing a significant decrease in the terminal voltage $V$ and a slight reduction in ISC, the short-circuit current. Very high values of RS will also produce a significant reduction in ISC; in these regimes, series resistance dominates and the behavior of the solar cell resembles that of a resistor. These effects are shown for crystalline silicon solar cells in the I-V curves displayed in the figure to the right.

Losses caused by series resistance are in a first approximation given by Ploss=VRsI=I2RS and increase quadratically with (photo-)current. Series resistance losses are therefore most important at high illumination intensities.

Shunt resistance

Effect of shunt resistance on the current–voltage characteristics of a solar cell

As shunt resistance decreases, the current diverted through the shunt resistor increases for a given level of junction voltage. The result is that the voltage-controlled portion of the I-V curve begins to sag toward the origin, producing a significant decrease in the terminal current I and a slight reduction in VOC. Very low values of RSH will produce a significant reduction in VOC. Much as in the case of a high series resistance, a badly shunted solar cell will take on operating characteristics similar to those of a resistor. These effects are shown for crystalline silicon solar cells in the I-V curves displayed in the figure to the right.

Reverse saturation current

Effect of reverse saturation current on the current-voltage characteristics of a solar cell

If one assumes infinite shunt resistance, the characteristic equation can be solved for VOC:

$V_{OC} = \frac{kT}{q} \ln\left(\frac{I_{SC}}{I_{0}} + 1\right).$

Thus, an increase in I0 produces a reduction in VOC proportional to the inverse of the logarithm of the increase. This explains mathematically the reason for the reduction in VOC that accompanies increases in temperature described above. The effect of reverse saturation current on the I-V curve of a crystalline silicon solar cell are shown in the figure to the right. Physically, reverse saturation current is a measure of the "leakage" of carriers across the p-n junction in reverse bias. This leakage is a result of carrier recombination in the neutral regions on either side of the junction.

Ideality factor

Effect of ideality factor on the current-voltage characteristics of a solar cell

The ideality factor (also called the emissivity factor) is a fitting parameter that describes how closely the diode's behavior matches that predicted by theory, which assumes the p-n junction of the diode is an infinite plane and no recombination occurs within the space-charge region. A perfect match to theory is indicated when n = 1. When recombination in the space-charge region dominate other recombination, however, n = 2. The effect of changing ideality factor independently of all other parameters is shown for a crystalline silicon solar cell in the I-V curves displayed in the figure to the right.

Most solar cells, which are quite large compared to conventional diodes, well approximate an infinite plane and will usually exhibit near-ideal behavior under Standard Test Condition (n ≈ 1). Under certain operating conditions, however, device operation may be dominated by recombination in the space-charge region. This is characterized by a significant increase in I0 as well as an increase in ideality factor to n ≈ 2. The latter tends to increase solar cell output voltage while the former acts to erode it. The net effect, therefore, is a combination of the increase in voltage shown for increasing n in the figure to the right and the decrease in voltage shown for increasing I0 in the figure above. Typically, I0 is the more significant factor and the result is a reduction in voltage.

References

1. ^ NASA Solar System Exploration - Sun: Facts & Figures retrieved 27 April 2011 "Effective Temperature ... 5777 K"
2. ^ <NAME>., <NAME>. (1985). Topics in Applied Physics: Amorphous Semiconductors: Amorphous silicon solar cells. Springer Berlin / Heidelberg.
3. ^ <NAME> (1994). Solar Electricity: Engineering of Photovoltaic Systems. Progensa. ISBN 84-86505-55-0.
4. ^ <NAME> and <NAME> (2003). Handbook of Photovoltaic Science and Engineering. <NAME> and Sons. ISBN 0-471-49196-9.
5. ^ <NAME> (2003). The Physics of Solar Cells. Imperial College Press. ISBN 978-1-86094-340-9.
6. ^ exp represents the exponential function
7. ^ <NAME> and <NAME> and <NAME>, "A new Method for Accurate Measurements of the Lumped Series Resistance of Solar Cells", Proceedings of the 23rd IEEE Photovoltaic Specialists Conference, p. 113-139, 1993.
8. ^ <NAME>., Distributed Series Resistance Effects in Solar Cells", IEEE Transactions on Electron Devices, Volume 29, Issue 5, p. 821 - 827, 1982.


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category theory

# Contents

## Idea

A tower is a diagram of the shape of the poset of natural numbers

$\cdots \to X_2 \to X_1 \to X_0 \,.$

A limit over a diagram of this form is called a sequential limit/directed limit.

Different contexts lead to different notions of morphism of towers, so it is important to consider what category of towers is appropriate for any given use of these objects. In addition to Postnikov towers, and related uses in decomposing homotopy types, towers also occur as a simple type of pro-object in a category. In that situation the morphisms considered between towers are usually pro-morphisms.

In homotopy theory in the presence of (infinity,1)-limits/(infinity,1)-colimits, every tower is a tower of homotopy fibers.

## Examples

Revised on May 27, 2016 16:52:20 by <NAME> (131.220.184.222)


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# Mechanocaloric effects in superionic thin films from atomistic simulations

## Abstract

Solid-state cooling is an energy-efficient and scalable refrigeration technology that exploits the adiabatic variation of a crystalline order parameter under an external field (electric, magnetic, or mechanic). The mechanocaloric effect bears one of the greatest cooling potentials in terms of energy efficiency owing to its large available latent heat. Here we show that giant mechanocaloric effects occur in thin films of well-known families of fast-ion conductors, namely Li-rich (Li3OCl) and type-I (AgI), an abundant class of materials that routinely are employed in electrochemistry cells. Our simulations reveal that at room temperature AgI undergoes an adiabatic temperature shift of 38 K under a biaxial stress of 1 GPa. Likewise, Li3OCl displays a cooling capacity of 9 K under similar mechanical conditions although at a considerably higher temperature. We also show that ionic vacancies have a detrimental effect on the cooling performance of superionic thin films. Our findings should motivate experimental mechanocaloric searches in a wide variety of already known superionic materials.

## Introduction

Conventional cooling methods based on gas-compression cycles present a series of critical drawbacks including the use of environmental hazards and lack of scalability to micro sizes. Solid-state cooling represents an elegant solution to all these issues, with mechanocaloric (MC) effects possibly holding the greatest promise in terms of energy efficiency1. Superelastic shape-memory alloys (SMA) displaying first-order martensitic transformations between a high-T martensitic and a low-T austenitic phase (e.g., Ni-Ti, Cu-Al-Ni, and Cu-Zn-Sn alloys), are archetypal MC compounds2. For example, a giant adiabatic temperature shift of 25.5 K has been measured in NiTi wires upon a small tensile stress of 0.65 GPa3. Nevertheless, there are still several hurdles that need to be overcome in order to develop successful MC commercial applications. For instance, the first-order nature of the martensitic transformation involves concomitant hysteresis losses, which is detrimental for cooling efficiency4, and as the size of SMA shrinks towards the nanoscale the martensitic phase transformation may be suppressed due to overstabilisation of the high-T distorted phase5. Finding novel MC materials with sharp second-order phase transitions occurring near room temperature and which persist down to the nanoscale, therefore, may advance the field of solid-state cooling.

Ferroelectric compounds (FE) like, for instance, BaTiO3 6, PbTiO3 7, and Ba1−x Ca x Ti1−y Zr y O3 solid solutions8 (generally exploited in sensing, information storage and energy applications), can be synthesised as nano-sized materials and typically exhibit displacive second-order phase transitions9. However, current adiabatic temperature shifts, $$\left| {\Delta T} \right|$$, estimated in most FE materials near room temperature are about one order of magnitude smaller than those achieved in SMA (made the exception of the ferrielectric compound (NH4)2SO4 10), and the involved mechanical stresses appear to be unsuitably too large ($$\left| \sigma \right| \gg 1$$ GPa). Consequently, the MC strengths reported so far for FE are rather poor in general, namely, $$\left| {\Delta T} \right|$$/$$\left| \sigma \right|$$ < 1 K·GPa−1 (again, made the exception of (NH4)2SO4 10). Recently, a giant MC effect has been predicted in fluorite-structured fast-ion conductors (FIC), typified by CaF2 and PbF2, which is comparable in magnitude to the benchmark adiabatic temperature shifts measured in SMA11. The MC effect disclosed in FIC may be understood in terms of stress-driven changes in ionic diffusivity, which in turn cause large variations in the entropy and dimensions of the material11. The originating superionic transition is of second-order type, in analogy to archetypal FE, and fluorite-structured FIC also can be synthesised as nanomaterials12. Unfortunately, the superionic transition temperatures in fluorite-structured FIC are far above ambient conditions (i.e., T s  = 1350 in CaF2 and 700 K in PbF2 11, 13), thus hindering the development of likely solid-state cooling applications.

Here we demonstrate giant MC effects in other predominant families of FIC, namely Li-rich (Li3OCl) and type-I (AgI) compounds (following Hull’s notation14), some of them at room temperature. We use atomistic computer simulations based on force fields and density functional theory to estimate the isothermal entropy and adiabatic temperature shifts attained in FIC with biaxial stresses (σ xx  = σ yy and σ zz  = 0). (Biaxial stresses, either compressive, σ > 0, or tensile, σ < 0, are realisable in thin films15; dynamic biaxial stresses leading to epitaxial strain changes of the order of 0.1–1% have been demonstrated in thin films by using ferroelectric substrates16, 17, and larger dynamic changes can be envisaged by means of mechanical extensometer techniques applied on flexible polymeric substrates18 and of nanoidentation methods19.) In AgI thin films, we find a room-temperature adiabatic temperature shift of $$\left| {\Delta T} \right|$$ = 38 K for a moderate compressive load of $$\left| \sigma \right|$$ = 1 GPa. This result exceeds the corresponding values estimated in FE at T = 300 K (~1 K6) and equals in magnitude the MC records set in SMA (~10 K3, 20). Our findings in Li3OCl thin films, namely $$\left| {\Delta T} \right|$$ = 9 K for $$\left| \sigma \right|$$ = 1 GPa at T = 1000 K, suggest that analogous Li-rich FIC with lower superionic transition temperatures (e.g., Li10GeP2S12 21) should display giant MC effects as well. Therefore, we argue that solid-state cooling could benefit immensely from the intensive research already undertaken on solid-state electrochemical batteries22.

## Results

### Effects of vacancies on the mechanocaloric performance of thin films

Vacancies are known to enhance significantly ionic transport and lower the superionic critical temperature in FIC23, 24. This is explicitly shown in Fig. 1a, where we plot the F diffusion coefficient, D (see Methods section), calculated for unstrained CaF2 (type-II FIC14) in the absence and presence of ionic vacancies as a function of temperature (note that D increases by >80% in the system containing vacancies). In Fig. 1b, we compare the critical superionic temperature obtained in perfect and defective thin films as a function of biaxial stress; in the c v = 2.5% case, T s is reduced by ~300 K almost independently of σ. This outcome can be rationalised in terms of a steady lowering of the energy barrier and creation energy of Frenkel pair defects (see Fig. 1a), which is due to an increase of the space available to interstitial ions. Therefore, intuitively one might expect that by introducing ionic vacancies the mechanocaloric performance of FIC should be enhanced. We have found, however, that this is not actually the case in archetypal FIC CaF2 (see Supplementary Figs. 1 and 2).

In Fig. 2a, b, we show the isothermal entropy and adiabatic temperature changes, ΔS and ΔT (see Methods section), calculated in type-II FIC thin films with an arbitrary F vacancy concentration of 2.5%. Large values of 30 JK−1 kg−1 and −30 K are obtained respectively at the highest temperature and tensile stress (T = 1100 K and $$\left| \sigma \right|$$ = 4 GPa), conditions at which the defective system is fully superionic. Upon comparison of the ΔT values obtained in perfect and defective CaF2 thin films at a fixed temperature of 1100 K (see Fig. 2c), however, we realise that the presence of anion vacancies does not produce any significant enhancement in MC performance (it is worth noticing that at the imposed conditions the perfect system remains in the normal state). Actually, the same applied maximum stress leads to $$\left| {\Delta T} \right|$$ values two times larger in thin films without vacancies for a slightly higher temperature at which the perfect system becomes superionic11. The main cause for this outcome is that the presence of ionic vacancies makes the superionic transition to be less abrupt, by smoothing the concomitant lattice strain associated to the prompt increase of ionic diffusivity (see Supplementary Methods and Supplementary Fig. 5). An analogous effect is observed also on the anion diffusion coefficient, which in the c v = 2.5% case displays a steady, rather than a sudden, increase near the superionic transition point (see Fig. 1a). Therefore, we may conclude that despite ionic vacancies in general favor ionic conductivity their effects on the MC performance of FIC may be adverse.

### Mechanocaloric effect in Li-rich superionic thin films

We choose Li3OCl as a representative member of the family of Li-rich FIC25. This compound adopts an antiperovskite phase characterised by Li, Cl, and O atoms placed at the octahedral vertices, octahedral centers, and center of a cubic unit cell, respectively (space group $$Fm\overline 3 m$$, see Supplementary Figs. 1 and 3). We note that Li3OCl has already been synthesised and characterised in thin-film geometry26, 27. The fundamental ion-migration mechanism in this material is related to the presence of vacancies: in the absence of points defects the diffusivity of Li+ ions is null below the corresponding melting point23, 24. Meanwhile, the accompanying superionic temperature, T s , strongly depends on the concentration of ionic vacancies (see Supplementary Discussion and Supplementary Fig. 6). Here we analyse the c v = 2.5% case, which according to our molecular dynamics simulations (see Methods section) renders a transition temperature of T s  = 1000 K at σ = 0 conditions. In what follows, we focus on tensile stresses (σ < 0) as in Li3OCl these favor ionic conductivity the most.

In Fig. 3a, b, we show the Li+ diffusion coefficient and in-plane strain, $$\epsilon$$ (see Methods section), calculated in Li3OCl as a function of negative biaxial stress at a fixed temperature of 1000 K. Both quantities increase under applied tensile stress, and in the case of $$\epsilon$$ the stress-induced enhancement is almost linear. As the ionic diffusivity in the thin film increases so does the isothermal entropy change (Fig. 3c), yielding a value of ΔS = 16 J·K−1 kg−1 at the maximum tensile stress σ max  = −1 GPa. The accompanying adiabatic temperature change is $$\left| {\Delta T} \right|$$ = 9 K (Fig. 3d), which is about two times larger than the one calculated in CaF2 thin films considering identical (σ, T) conditions and vacancy concentration (see Fig. 2b). The main reason behind this difference is the larger isothermal entropy change found in Li3OCl, which fundamentally is related to the way in which ions hop through favorable pathways within the corresponding structural frameworks.

The largest contribution to the entropy of the superionic transformation can be assumed to be due to the configurational degrees of freedom28. Consequently, it is reasonable to expect finding comparable isothermal entropy changes to those attributed to Li3OCl in other related Li-rich FIC with transition temperatures close to ambient (e.g., Li10GeP2S12 21). Actually, larger $$\left| {\Delta T} \right|$$ values may be anticipated in those cases as the cooling efficiency of crystals is enhanced at low temperatures (i.e., the corresponding heat capacity, C σ , is small and depends on temperature as T 3, see Methods section).

### Mechanocaloric effect in type-I superionic thin films

We select AgI as a representative member of the family of type-I FIC. At ambient conditions bulk AgI is found as a mixture of wurtzite (β, hexagonal P63 mc) and zincblende (γ, cubic $$F\overline 4 3m$$) phases14. As temperature is increased beyond T s  ~ 400 K, bulk AgI transforms into a superionic phase (α) in which the I anions arrange in a cubic bcc lattice and the Ag+ cations are mobile. In practice, AgI thin films can be synthesised either in the β or γ phase depending on the employed preparation method29, 30. For reasons that will become clear later on, we investigate here the cubic γ phase consisting of two interlaced monoatomic fcc sublattices with fourfold ionic coordination (see Supplementary Figs. 1 and 4). Our following analysis is restricted to compressive stresses (σ > 0) as we find the largest MC effect at such conditions.

In Figs. 4a and 5a, we show the Ag+ diffusion coefficient calculated in non-defective AgI thin films as a function of compressive biaxial stress at T = 300 and 400 K, respectively. It is appreciated that only at the highest analysed temperature and for stresses larger than 0.8 GPa the system becomes fully superionic (i.e., $$D \gg 0$$). Nevertheless, the isothermal entropy and adiabatic temperature changes estimated for the maximum stress σ max = 1 GPa are giant in both cases (see panels c, d in Figs. 4 and 5). In particular, we obtain ΔS = 30 J·K−1 kg−1 and ΔT = 38 K at T = 300 K, and 35 J·K−1 kg−1 and 51 K at T = 400 K. We note that the in-plane strains calculated in both cases are also considerably large; as compared to Li3OCl thin films, for instance, those are about four times larger in absolute value (see Figs. 3b–5b). As T is increased beyond room temperature, the calculated entropy and temperature shifts become larger essentially due to the enhanced mobility of Ag+ cations (see Fig. 5a).

At room temperature, the MC performance of AgI thin films commences to be appreciable and to increase steadily for stresses larger than σ c  = 0.2 GPa (see Fig. 4c, d). This finding signals the triggering of a structural phase transformation at around σ c different from the superionic transition, which occurs at higher temperatures. The continuous variation of the in-plane strain and cation diffusion coefficient as driven by compressive stress indicate that this phase transition is of second-order type. To get microscopic insight into such a structural transformation, we analyse the coordination number, radial pair distribution function, mean squared displacement, and density distribution of I and Ag+ ions under different temperature and stress conditions (see Methods section and Supplementary Figs. 79).

Our simulation results reveal the existence of a σ-induced diffusionless order-disorder phase transition involving sizeable displacements of the ionic equilibrium positions with respect to the original zincblende structure. Specifically, both I-I and Ag-Ag coordination numbers amount to 12 in average similarly to what is found in the two monoatomic fcc lattices of reference. However, a precise determination of neighbouring atomic shells from the corresponding radial pair distribution functions, g(r), is not possible for distances larger than few angstroms at biaxial stresses higher than σ c (see Fig. 6). Furthermore, at room temperature and σ c  < σ conditions the asymptotic behavior of the ionic pair distribution functions, namely, g(r) ≈ 1, is reached very rapidly with the radial distance (r ~ 10 Å, see Fig. 6a, b); this outcome evidences lack of solid translational invariance, in analogy to what is observed in glassy systems. We assuredly identify these features, and others shown in the Supplementary Figs. 79 (e.g., ionic mean squared displacement and density distribution plots), with the presence of atomic disorder in the I and Ag+ sublattices at σ c  < σ conditions. It is worth noticing that, as we have explicitly checked, neither the hexagonal wurtzite nor the cubic rock-salt structures found in bulk AgI at ambient and high-pressure conditions14 transform to a disordered phase when applying biaxial compressive stresses of ~1 GPa to them at room temperature (see Supplementary Fig. 10).

## Discussion

The giant room-temperature MC effect revealed in AgI thin films, which is originated by a σ-induced diffusionless order-disorder phase transition, appears to be very promising in terms of maximum adiabatic temperature shift and mechanocaloric strength, $$\left| {\Delta T} \right|$$/$$\left| \sigma \right|$$. In Table 1, we compare the cooling properties of this FIC with those of other well-established MC compounds that have been reported at ambient and near ambient conditions. First, we note that the adiabatic temperature shift estimated in AgI ($$\left| {\Delta T} \right|$$ = 38 K) is equivalent in magnitude to the MC benchmarks obtained in archetypal shape-memory alloys like, for instance, NiTi (e.g., $$\left| {\Delta T} \right|$$ = 25.5 K3). As compared to ferroelectric materials, the estimated ΔT is several times larger in absolute value. In terms of mechanocaloric strength, AgI also shows great promise as it follows closely to shape-memory alloys ($$\left| {\Delta T} \right|{\rm{/}}\left| \sigma \right|\sim 10$$ K GPa−1 20) and outperforms perovskite oxide materials (e.g., BaTiO3 with $$\left| {\Delta T} \right|{\rm{/}}\left| \sigma \right|\sim 1$$ K GPa−1 6).

Our findings should stimulate the development of new cooling devices based on FIC whose energy efficiency as compared to magnetocaloric and electrocaloric materials is very auspicious. In contrast to other mechanocaloric materials driven by first-order transitions, mechanical hysteresis losses and scalability limitations towards nanosizes should be absent in FIC. Mechanical stresses other than biaxial (i.e., uniaxial and hydrostatic) also can be expected to produce similar mechanocaloric responses than reported here in FIC11. In this context, the rich variety of superionic materials that have been already investigated with a focus on electrochemical applications grants solid-state cooling with vast new possibilities.

## Methods

### Classical molecular dynamics simulations

Molecular dynamics (N, P, T) simulations are performed with the LAMMPS code31. The pressure and temperature in the system are kept fluctuating around a set-point value by using thermostatting and barostatting techniques in which some dynamic variables are coupled to the particle velocities and simulation box dimensions. The interactions between atoms are modeled with rigid-ion Born-Mayer-Huggins potentials. Large simulation boxes, typically containing 6,144 atoms, are used in which periodic boundary conditions along the three Cartesian directions are applied. Newton’s equations of motion are integrated using the customary Verlet’s algorithm with a time-step length of 10−3 ps. A particle-particle particle-mesh k-space solver is used to compute long-range van der Waals and Coulomb interactions and forces beyond a cut-off distance of 12 Å at each time step.

We note that by using periodic boundary conditions in our calculations we avoid to explicitly simulate the substrate over which the thin film is grown in practice. Also, possible elastic relaxation effects and the interactions of the thin film with the vacuum at the top surface are totally neglected. Consequently, the simulations are performed very efficiently in terms of computational expense and the fundamental mechanocaloric effects occurring in FIC thin films can be singled out. Further details of our classical molecular dynamics simulations (e.g., interatomic potential models) can be found in the Supplementary Methods and Supplementary Tables 1 and 2.

### Density functional theory calculations

First-principles DFT calculations are performed to analyse the energy and structural properties of CaF2, Li3OCl, and AgI thin films at zero temperature (see Supplementary Figs. 24). We perform these calculations with the VASP code32 by following the generalized gradient approximation to the exchange-correlation energy due to Perdew et al.33. The projector augmented-wave method is used to represent the ionic cores34, and the electronic states 2s-3s-3p-4s of Ca, 2s-2p of F, 1s-2s of Li, 2s-2p of O, 2s-2p of Cl, 4d-5s of Ag, and 5s-5p of I, are considered as valence. Wave functions are represented in a plane-wave basis truncated at 650 eV. By using these parameters and dense k-point grids for Brillouin zone integration, the resulting energies are converged to within 1 meV per formula unit. In the geometry relaxations, a tolerance of 0.01 eV·Å−1 is imposed in the atomic forces. We also perform ab initio molecular dynamics calculations in order to validate the reliability of the interatomic potential models employed in the classical molecular dynamics simulations. Details of these calculations can be found in the Supplementary Discussion.

### Estimation of key quantities

The ionic diffusion coefficients are determined as:

$$D = \mathop {\hbox{lim}}\limits_{t \to \infty } \frac{{\left\langle {{{\left| {{R_i}\left( {t + {t_0}} \right) - {R_i}\left( {{t_0}} \right)} \right|}^2}} \right\rangle }}{{6t}} ,$$
(1)

where R i (t) is the position of the migrating ion labelled as i at time t, t 0 an arbitrary time origin, and $$\left\langle \cdots \right\rangle$$ denotes average over time and particles. Meanwhile, the mean squared displacement of each ionic species is defined as $$\left\langle {\Delta R_i^2(t)} \right\rangle \equiv \left\langle {{{\left| {{R_i}\left( {t + {t_0}} \right) - {R_i}\left( {{t_0}} \right)} \right|}^2}} \right\rangle$$.

Owing to the cubic symmetry of the thin films considered in this study, strains $$\epsilon$$ xx and $$\epsilon$$ yy and stresses σ xx and σ yy are identical. Consequently, the accompanying isothermal entropy changes can be estimated with the formula:

$$\Delta S\left( {{\sigma _f},T} \right) = {V_0} \cdot {\int}_{\!\!\!\!0}^{\left| {{\sigma _f}} \right|} {\left( {\frac{{\partial {\epsilon _{xx}}}}{{\partial T}}} \right)_{\!\!\!\sigma} }{\rm d}{\sigma _{xx}} + {\left( {\frac{{\partial {\epsilon _{yy}}}}{{\partial T}}} \right)_{\!\!\!\sigma} }{\rm d}{\sigma _{yy}},$$
(2)

where V 0(T) ≡ L x,0(T) · L y,0(T) · L z,0(T) is the T-dependent volume of the crystal at equilibrium (i.e., considering σ = 0 conditions), L i represents the length of the simulation box along the i Cartesian direction, and the mechanical strain components are defined as $${\epsilon _{ii}}(\sigma ,T) \equiv \frac{{{L_i}\left( {\sigma ,T} \right) - {L_{i,0}}(T)}}{{{L_{i,0}}(T)}}$$. Regarding the adiabatic temperature shifts, those are calculated as:

$$\Delta T\left( {{\sigma _f},T} \right) = - {\int}_{\!\!\!\!0}^{\left| {{\sigma _f}} \right|} \frac{T}{{{C_\sigma }(\sigma ,T)}} \cdot {\rm d}S ,$$
(3)

where C σ (σ, T) is the heat capacity of the crystal calculated at fixed σ. In this study we assumed that C σ (σ, T) ≈ C σ (0, T). Further technical details on our calculations can be found in the Supplementary Methods.

### Data availability

The data that support the findings of this study are available from the corresponding author (C.C.) upon reasonable request.

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## Acknowledgements

This research was supported under the Australian Research Council’s Future Fellowship funding scheme (No. FT140100135). Computational resources and technical assistance were provided by the Australian Government and the Government of Western Australia through Magnus under the National Computational Merit Allocation Scheme and The Pawsey Supercomputing Centre. D.E. acknowledges financial support from Spanish government MINECO, the Spanish Agencia Estatal de Investigacion (AEI), and Fondo Europeo de Desarrollo Regional (FEDER) under Grants No. MAT2016-75586-C4-1-P and MAT2015-71070-REDC.

## Author information

C.C. conceived the study and planned the research. C.C. and A.K.S. performed the theoretical calculations. Results were discussed by C.C., A.K.S. and D.E. The manuscript was written by C.C. and A.K.S. with substantial input from D.E.

Correspondence to <NAME>.

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<NAME>., Errandonea, D. & Cazorla, C. Mechanocaloric effects in superionic thin films from atomistic simulations. Nat Commun 8, 963 (2017). https://doi.org/10.1038/s41467-017-01081-7

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