formulasearchengine - User contributions [en]

Luttinger parameter

2014-01-29T15:15:02Z

132.187.31.76: /* Phenomenological Hamiltonian for the j=3/2 states */

The '''multislice''' algorithm is a method for the simulation of the interaction of an [[electron beam]] with matter, including all multiple elastic scattering effects. The method is reviewed in the book by Cowley.<ref name ="CowleyDP">{{cite book|title=Diffraction Physics, 3rd Ed|author = John M. Cowley|year=1995|publisher=North Holland Publishing Company}}</ref> The algorithm is used in the simulation of high resolution [[Transmission electron microscopy]] micrographs, and serves as a useful tool for analyzing experimental images.<ref name ="Kirkland">{{cite book|title=Advanced Computing in Electron Microscopy|author = Dr. Earl J. Kirkland}}</ref> Here we describe relevant background information, the theoretical basis of the technique, approximations used, and several software packages that implement this technique. Moreover, we delineate some of the advantages and limitations of the technique and important considerations that need to be taken into account for real-world use.

== Background ==

The multislice method has found wide application in electron crystallography. The mapping from a crystal structure to its image or diffraction pattern has been relatively well understood and documented. However, the reverse mapping from electron micrograph images to the crystal structure is generally more complicated. The fact that the images are two dimensional projections of three dimensional crystal structure it makes it tedious to compare these projections to all plausible crystal structures. Hence, the use of numerical techniques in simulating results for different crystal structure is integral to the field of electron microscopy and crystallography. Several software packages exist to simulate electron micrographs.

There are two widely used simulation techniques that exist in literature: the Bloch wave method, derived from Hans Bethe's original theoretical treatment of the Davisson-Germer experiment, and the multislice method. In this paper, we will primarily focus on the multislice method for simulation of diffraction patterns, including multiple elastic scattering effects. Most of the packages that exist implement the multislice algorithm along with Fourier analysis to incorporate electron lens aberration effects to determine electron microscope image and address aspects such as phase contract and diffraction contrast. For electron microscope samples in the form of a thin crystalline slab in the transmission geometry, the aim of these software packages is to provide a map of the crystal potential, however this inversion process is greatly complicated by the presence of multiple elastic scattering.

The first description of what is now known as the multislice theory was given in the classic paper by Cowley and Moodie .<ref name="Cowley">{{cite news|journal=Acta Crystallographica|author=J. M. Cowley and A. F. Moodie|year=1957|volume=10}}</ref> In this work, the authors describe scattering of electrons using a physical optics approach without invoking quantum mechanical arguments. Many other derivations of these iterative equations have since been given using alternative methods, such as Greens functions, differential equations, scattering matrices or path integral methods.

A summary of the development of a computer algorithm from the multislice theory of Cowley and Moodie for numerical computation was reported by Goodman and Moodie.<ref>P. Goodman and A. F. Moodie, Acta Cryst. 1974, A30, 280</ref> They also discussed in detail the relationship of the multislice to the other formulations. Specifically, using Zassenhaus's theorem, this paper gives the mathematical path from multislice to 1. Schroedingers equation (derived from the multislice), 2. Darwin's differential equations, widely used for diffraction contrast TEM image simulations - the Howie-Whelan equations derived from the multislice. 3. Sturkey's scattering matrix method. 4. the free-space propagation case, 5. The phase grating approximation, 6. A new "thick-phase grating" approximation, which has never been used, 7. Moodie's polynomial expression for multiple scattering, 8. The Feynman path-integral formulation, and 9. relationship of multislice to the Born series. The relationship between algorithms is summarized in Section 5.11 of Spence (2013),<ref name ="SpenceHREM">{{cite book|title=High-Resolution Electron Microscopy, 4th Ed.|author = John C. H. Spence|year=2013|publisher=Oxford University Press}}</ref> (see Figure 5.9).

== Theory ==

The form of multislice algorithm presented here has been adapted from Peng, Dudarev and Whelan 2003.<ref name ="Peng">{{cite book|title=High-Energy Electron Diffraction and Microscopy|author = L. M. Peng, S. L. Dudarev and M. J. Whelan|year=2003|publisher=Oxford Science Publications}}</ref> The multislice algorithm is an approach to solving the Schrödinger wave equation:

<math>\begin{align}
-\frac{\hbar^2}{2m} \frac{\partial^2\Psi(x,t)}{\partial x^2} + V(x,t)\Psi(x,t)
&=E\Psi(x,t)
\end{align}</math>

In 1957, Cowley and Moodie <ref name="Cowley"/> showed that the Schrödinger equation can be solved analytically to evaluate the amplitudes of diffracted beam. Subsequently, the effects of dynamical diffraction can be calculated and the resulting simulated image will exhibit good similarities with the actual image taken from a microscope under dynamical conditions. Furthermore, the multislice algorithm does not make any assumption about the periodicity of the structure, as a result this method can be used to simulate HREM images of aperiodic systems as well.

The following section will include a mathematical formulation of the Multislice algorithm. The Schrödinger equation can also be represented in the form of incident and scattered wave as:

<math>\begin{align}
\Psi({\mathbf{r}}) &= \Psi_{0}({\mathbf{r}}) + \int{G({\mathbf{r,r'}})V({\mathbf{r'}})\Psi({\mathbf{r'}})d{\mathbf{r'}}}
\end{align}</math>

where <math>G(\mathbf{r,r'})</math> is the Green’s function that represents the amplitude of the electron wave function at a point <math>\mathbf{r}</math> due to a source at point <math>\mathbf{r'}</math>.

Hence for an incident plane wave of the form <math>\Psi(r)=\exp(i\mathbf{k\cdot r})</math> the Schrödinger equation can be written as:

<math>\begin{align}
\Psi({\mathbf{r}}) = \exp(i{\mathbf{k\cdot r}}) - \frac{m}{2\pi\hbar^2}\int\frac{\exp(ik\cdot {\mathbf{|r-r'|}})}{{\mathbf{|r-r'|}}}
V({\mathbf{r'}})\Psi({\mathbf{r'}})dr'
\end{align}</math>

We then choose the coordinate axis in such a way that the incident beam hits the sample at (0,0,0) in the <math>\hat{z}</math>-direction. Now we consider wave-function with a modulation function <math>\phi({\mathbf{r}})</math> for the amplitude of the wave-function. Hence, the modulation function can be represented as:

<math>\begin{align}
\phi({\mathbf{r}}) &= 1 - \frac{m}{2\pi\hbar^2}\int{\frac{\exp[ik|{\mathbf{r-r'}}|-i{\mathbf{k}}\cdot({\mathbf{r-r'}})]}{|{\mathbf{r-r'}}|}V({\mathbf{r'})\phi({\mathbf{r'}})}dr'}
\end{align}</math>

Now we make substitutions with regards to the coordinate system we have adhered.

<math>\begin{align}
{\mathbf{k}} \cdot ({\mathbf{r-r'}}) &= k(z-z') \quad \& \quad |{\mathbf{r-r'}}|
\approx (z-z') + ({\mathbf{X-X'}})^2/{2(z-z')}
\end{align}</math>

<math>\begin{align}
\phi({\mathbf{r}}) = 1 -i\frac{\pi}{E\lambda} \int \int \limits_{z'=-\infty}^{z'=z}
V({\mathbf{X'}},z') \phi({\mathbf{X'}},z') \frac{1} {i\lambda (z-z')} \exp\left(ik\frac{|{\mathbf{X-X'}}|^2}{2(z-z')}\right)d{\mathbf{X'}}dz'
\end{align}</math>

where, <math>\lambda = 2\pi /k</math> is the wavelength of the electrons with energy <math>(E) = \hbar^2k^2/{2m}</math>

So far we have set up the mathematical formulation of wave mechanics without addressing the scattering in the material. The interaction constant is defined as

<math>\begin{align}
\sigma = \pi/E\lambda
\end{align}</math>

Further we also need to address the transverse spread which is done in terms of Fresnel propagation function

<math>\begin{align}
p({\mathbf{X}},z) = \frac{1}{iz\lambda} \exp\left(ik\frac{{\mathbf{X}}^2}{2z}\right)
\end{align}</math>

In multislice simulation the thickness of each slice over which the iteration is performed is usually small and as a result within a slice the potential field can be approximated to be constant <math>V({\mathbf{X'}},z)</math>. Subsequently, the modulation function can be represented as:

<math>\begin{align}
\phi({\mathbf{X}},z_{n+1}) = \int p({\mathbf{X}}-{\mathbf{X'}}, z_{n+1}-z_{n}) \phi({\mathbf{X}},z_{n})\exp\left(-i\sigma\int\limits_{z_{n}}^{z_{n+1}}V({\mathbf{X'}},z')dz'\right)dX'
\end{align}</math>

We can therefore represent the modulation function in the next slice

<math>\begin{align}
\phi_{n+1} = \phi({\mathbf{X}},z_{n+1}) = [q_{n}\phi_{n}]*p_{n}
\end{align}</math>

where, * represents convolution, <math>p_{n}=p({\mathbf{X}},z_{n+1}-z_{n})</math> and <math>q_{n}({\mathbf{X}})</math> defines the transmission function of the slice.

<math>\begin{align}
q_{n}({\mathbf{X}}) = \exp \{-i\sigma \int \limits_{z_{n}}^{z_{n+1}} V({\mathbf{X}},z')dz'\}
\end{align}</math>

Hence, the iterative application of the aforementioned procedure will provide a full interpretation of the sample in context. Further, it should be reiterated that no assumptions have been made on the periodicity of the sample apart from assuming that the potential <math>V(\mathbf{X},z)</math> is uniform within the slice. As a result, it is evident that this method in principle will work for any system. However, for aperiodic systems in which the potential will vary rapidly along the beam direction, the slice thickness has to be significantly small and hence will result in higher computational expense.

{| class="wikitable"
! align="right"| Data Points
!<math>\mathbf{log_2}</math>N
! Discrete FT
! Fast FT
! Ratio
|-
|64 ||6 ||4,096 ||384 ||10.7
|-
|128
|7
|16,384
|896
|18.3
|-
|256
|8
|65,536
|2,048
|32
|-
|512
|9
|262,144
|4,608
|56.9
|-
|1,024
|10
|1,048,576
|10,240
|102.4
|-
|2,048
|11
|4,194,304
|22,528
|186.2
|+ Table 1 - Computational efficiency of Discrete Fourier Transform compared to Fast Fourier Transform
|}

== Practical Considerations ==

The basic premise is to calculate diffraction from each layer of atoms using Fast Fourier Transforms (FFT) and multiplying each by a phase grating term. The wave is then multiplied by a propagator, inverse Fourier Transformed, multiplied by a phase grating term yet again, and the process is repeated. The use of FFTs allows a significant computational advantage over the Bloch Wave method in particular, since the FFT algorithm involves <math> N \log N</math> steps compared to the diagonalization problem of the Bloch wave solution which scales as <math>N^2</math> where <math>N</math> is the number of atoms in the system. (See Table 1 for comparison of computational time).

The most important step in performing a multislice calculation is setting up the unit cell and determining an appropriate slice thickness. In general, the unit cell usd for simulating images will be different from the unit cell that defines the crystal structure of a particular material. The primary reason for this due to aliasing effects which occur due wraparound errors in FFT calculations. The requirement to add additional “padding” to the unit cell has earned the nomenclature “super cell” and the requirement to add these additional pixels to the basic unit cell comes at a computational price.

To illustrate the effect of choosing a slice thickness that is too thin, let us consider a simple example. The Fresnel propagator describes the propagation of electron waves in the z direction (the direction of the incident beam) in a solid:

<math>\tilde{\phi}(\mathbf{u},z) = \tilde{\phi}(\mathbf{u},z=0)\exp(\pi i \lambda \mathbf{u}^2 z)</math>

Where <math>\mathbf{u}</math> is the reciprocal lattice coordinate, z is the depth in the sample, and lambda is the wavelength of the electron wave (related to the wave vector by the relation <math>k = 2\pi / \lambda</math>). Figure [fig:SliceThickness] shows vector diagram of the wave-fronts being diffracted by the atomic planes in the sample. In the case of the small-angle approximation (<math>\theta \sim</math> 100 mRad) we can approximate the phase shift as <math>d - S \approx \Delta z / \cos \theta - \Delta z\theta</math>. For 100 mRad the difference is fleeting <math>\cos(0.01) = 0.99995</math> so <math> \Delta z / \cos \theta \approx \Delta z\theta </math>. For small angles this approximation holds regardless of how many slices there are, although choosing a <math>\Delta z</math> greater than the lattice parameter (or half the lattice parameter in the case of perovskites) for a multislice simulation would be rather problematic.

[[File:MultisliceThickness.png|thumb|MultisliceThickness]]

Additional practical concerns are how to effectively include effects such as inelastic and diffuse scattering, quantized excitations (e.g. plasmons, phonons, excitons), etc. There was one code that took these things into consideration through a coherence function approach <ref name ="Physik">{{cite thesis|type=Ph.D.|title=A Coherence Function Approach to Image Simulation|publisher=Vom Fachbereich Physik Technischen Universitat Darmstadt|author=Heiko Muller|year=2000}}</ref> called Yet Another Multislice (YAMS), but the code is no longer available either for download or purchase.

==Available Software ==

There are several software packages available to perform multislice simulations of images. Among these is NCEMSS, NUMIS, MacTempas, and Kirkland . Other programs exist but unfortunately many have not been maintained (e.g. SHRLI81 by Mike O’Keefe of Lawrence Berkeley National Lab and Cerius2 of Accerlys). A brief chronology of multislice codes is given in Table 2, although this is by no means exhaustive.

{| class="wikitable"
! align="right" |Code Name
! Author
! Year Released
|-
|SHRLI
|O’Keefe
|1978
|-
|TEMPAS
|Kilaas
|1987
|-
|NUMIS
|Marks
|1987
|-
|NCEMSS
|O’Keefe & Kilaas
|1988
|-
|MacTEMPAS
|Kilaas
|1978
|-
|TEMSIM
|Kirland
|1988
|-
|HREMResearch
|Ishizuka
|2001
|-
|JEMS
|Stadelmann
|2004
|-
|JMULTIS
|Zuo
|1990
|+Table 2 - Timeline of various Multislice Codes
|}

== ACEM/JCSTEM ==

This software is developed by Professor Earl Kirkland of Cornell University. This code is freely available as an interactive Java applet and as standalone code written in C/C++. The Java applet is ideal for a quick introduction, with the caveat (and disclaimer) that the results are not accurate. The ACEM code accompanies an excellent text of the same name by Kirkland which describes several techniques for simulating electron micrographs (including multislice) in detail. Unfortunately, the main C/C++ routines use a command line interface (CLI) and thus may not be appropriate for beginners. http://people.ccmr.cornell.edu/~kirkland/

== NCEMSS ==

This package was released from the National Center for High Resolution Electron Microscopy. This program uses uses a mouse-drive graphical user interface and is written by Dr. Roar Kilaas and Dr. Mike O’Keefe of Lawrence Berkeley National Laboratory. While the code is no longer developed, the program is available through the Electron Direct Methods (EDM) package written by Professor Laurence Marks of Northwestern University. Debye-Waller factors can be included in as a parameter to account for diffuse scattering, although the accuracy is unclear (i.e. a good guess of the Debye-Waller factor is needed).

http://www.numis.northwestern.edu/edm/

== NUMIS ==

The Northwestern University Multislice and Imaging System (NUMIS) is a package is written by Professor Laurence Marks of Northwestern University. It uses a command line interface (CLI) and is based in UNIX. A structure file must be provided as input in order to run use this code, which makes it ideal for advanced users. The NUMIS multislice programs use the conventional multislice algorithm by calculating the wavefunction of electrons at the bottom of a crystal and simulating the image taking into account various instrument-specific parameters including <math>C_s</math> and convergence. This program is good to use if one already has structure files for a material that have been used in other calculations (for example, Density Functional Theory). These structure files can be used to general X-Ray structure factors which are then used as input for the PTBV routine in NUMIS. Microscope parameters can be changed through the MICROVB routine.

http://www.numis.northwestern.edu/ourwiki/index.php/Multislice

== MacTempas ==

This software is specifically developed to run in Mac OS X by Dr. Roar Kilaas of Lawrence Berkeley National Laboratory. It is designed to have a user-friendly user interface and has been well-maintained relative to many other codes (last update May 2013). It is available (for a fee) from http://www.totalresolution.com.

== JMULTIS ==

This is a software for multislice simulation was written in FORTRAN 77 by Dr. J. M. Zuo, while he was a postdoc research fellow at Arizona State University under the guidance of Prof. John Spence. The source code was published in the book of Electron Microdiffraction.<ref>Electron Microdiffraction, J.C. H. Spence and J. M. Zuo, Plenum, New York, 1992</ref> A comparison between multislice and Bloch wave simulations for ZnTe was also published in the book. A separate comparison between several multislice algorithms at the year of 2000 was reported in <ref>Koch, C. and J.M. Zuo, “Comparison of multislicecomputer programs for electron scattering simulations and the Bloch wavemethod”, Microscopy and Microanalysis,Vol. 6 Suppl. 2, 126-127, (2000).</ref>

== QSTEM: Quantitative TEM/STEM Simulations ==

Electron and Ion Microscopy, written by Professor Christopher Koch of Ulm University in Germany. Allows simulation of HAADF, ADF, ABF-STEM, as well as conventional TEM and CBED. The executable and source code is available as a free download at the Koch group website: http://elim.physik.uni-ulm.de/?page_id=834

== STEM-CELL ==

This is a code written by Dr Vincenzo Grillo of the Institute for Nanoscience (CNR) in Italy. This code is essentially a graphical frontend to the multislice code written by Kirkland, with more additional features. These include tools to generate complex crystalline structures, simulate HAADF images and model the STEM probe, as well as modeling of strain in materials. Tools for image analysis (e.g. GPA) and filtering are also available.
The code is updated quite often with new features and a user mailing list is maintained. Freely available at
http://tem-s3.nano.cnr.it/stemcell.htm

== References ==
<ref name="WilliamsAndCarter_v1">{{cite book|title=Transmission Electron Microscopy|author=Willams, D.B|author=Carter, C. B.|volume=1 - Basics|publisher=Plenum Press|year=1996|ISBN=0-306-45324-X}}</ref>
{{reflist}}

==External links==
* [http://people.ccmr.cornell.edu/~kirkland/ Multislice source code] (with tutorial) available under the [[GNU General Public Licence]]

[[Category:Microscopy]]
[[Category:Scientific modeling]]

Alfvén wave

2014-01-18T18:25:17Z

132.187.253.28:

{{multiple issues|
{{technical|date=January 2012}}
{{cleanup|reason=Acronyms and definitions being confused with each other|date=January 2013}}
}}

'''Digital Signal 1''' ('''DS1''', sometimes '''DS-1''') is a [[T-carrier]] signaling scheme devised by [[Bell Labs]].<ref>"How Bell Ran in Digital Communications" September 1996, webpage: [http://www.byte.com/art/9609/sec10/art6.htm BYTE-Bell]: Bell Labs scientists developed a time-division multiplexing scheme, T1.</ref> DS1 is a widely used standard in [[telecommunications]] in [[North America]] and [[Japan]] to transmit voice and data between devices. [[E-carrier|E1]] is used in place of T1 outside North America, Japan, and South Korea. DS1 is the logical bit pattern used over a physical T1 line; however, the terms "DS1" and "'''T1'''" are often used interchangeably.

==Bandwidth==
A DS1 [[electrical network|circuit]] is made up of twenty-four 8-bit [[channel (communications)|channels]] (also known as timeslots or [[DS0]]s), each channel being a 64 kbit/s [[DS0]] [[Time-division multiplexing|multiplexed]] carrier circuit.<ref>[http://replay.waybackmachine.org/20090129130729/http://justcircuits.com/ds1t1.html Just Circuits – T1 Made Simple]{{Dead link|date=June 2012}}</ref> A DS1 is also a [[full-duplex]] circuit, which means the circuit transmits and receives 1.544 [[Mbit/s]] concurrently. A total of 1.536 Mbit/s of [[Bandwidth (computing)|bandwidth]] is achieved by sampling each of the twenty-four 8-bit [[DS0]]s 8000 times per second. This sampling is referred to as 8-[[kHz]] sampling (See [[Pulse-code modulation]]). An additional 8 [[kbit/s]] of overhead is obtained from the placement of one framing bit, for a total of 1.544 Mbit/s, calculated as follows:

:<math>\begin{align}
& \left( 8\,\frac{\mathrm{bits}}{\mathrm{channel}} \times 24\,\frac{\mathrm{channels}}{\mathrm{frame}} + 1\,\frac{\mathrm{framing\ bit}}{\mathrm{frame}} \right)
\times 8\,000\,\frac{\mathrm{frames}}{\mathrm{second}} \\
= {} & 1\,544\,000\,\frac{\mathrm{bits}}{\mathrm{second}} \\
= {} & 1.544\,\frac{\mathrm{Mbit}}{\mathrm{second}}
\end{align}</math>

==DS1 frame synchronization==
{{see also|Synchronization in telecommunications}}
[[Frame synchronization]] is necessary to identify the [[timeslot]]s within each 24-channel frame. [[Synchronization]] takes place by allocating a framing, or 193rd, bit. This results in 8 kbit/s of framing data, for each DS1. Because this 8-kbit/s channel is used by the transmitting equipment as [[Overhead information|overhead]], only 1.536 Mbit/s is actually passed on to the user. Two types of framing schemes are [[Super Frame]] (SF) and [[Extended Super Frame]] (ESF). A Super Frame consists of twelve consecutive 193-bit frames, whereas an Extended Super Frame consists of twenty-four consecutive 193-bit frames of data. Due to the unique bit sequences exchanged, the framing schemes are not compatible with each other. These two types of framing (SF, and ESF) use their 8 kbit/s framing channel in different ways.

==Connectivity and alarms==
{{Unreferenced section||date=September 2008}}
'''Connectivity''' refers to the ability of the digital carrier to carry customer data from either end to the other. In some cases, the connectivity may be lost in one direction and maintained in the other. In all cases, the terminal equipment, i.e., the equipment that marks the endpoints of the DS1, defines the connection by the quality of the received framing pattern.

===Alarms===
Alarms are normally produced by the receiving terminal equipment when the framing is compromised. There are three defined [[alarm indication signal]] states, identified by a legacy color scheme: red, yellow and blue.

'''Red alarm''' indicates the alarming equipment is unable to recover the framing reliably. Corruption or loss of the signal will produce “red alarm”. Connectivity has been lost toward the alarming equipment. There is no knowledge of connectivity toward the far end.

'''Yellow alarm''', also known as remote alarm indication (RAI), indicates reception of a data or framing pattern that reports the far end is in “red alarm”. The alarm is carried differently in SF (D4) and ESF (D5) framing. For SF framed signals, the user bandwidth is manipulated and "bit two in every DS0 channel shall be a zero."<ref name="yel">American National Standards Institute, ''T1.403-1999'', ''Network and Customer Installation Interfaces – DS1 Electrical Interface'', p. 12</ref> The resulting loss of payload data while transmitting a yellow alarm is undesirable, and was resolved in ESF framed signals by using the data link layer. "A repeating 16-bit pattern consisting of eight 'ones' followed by eight 'zeros' shall be transmitted continuously on the ESF data link, but may be interrupted for a period not to exceed 100-ms per interruption."<ref name="yel"/> Both types of alarms are transmitted for the duration of the alarm condition, but for at least one second.

'''Blue alarm''', also known as alarm indication signal (AIS) indicates a disruption in the communication path between the terminal equipment and line repeaters or DCS. If no signal is received by the intermediary equipment, it produces an unframed, all-ones signal. The receiving equipment displays a “red alarm” and sends the signal for “yellow alarm” to the far end because it has no framing, but at intermediary interfaces the equipment will report “AIS” or [[Alarm indication signal|Alarm Indication Signal]]. AIS is also called “all ones” because of the data and framing pattern.

These alarm states are also lumped under the term Carrier Group Alarm (CGA). The meaning of CGA is that connectivity on the digital carrier has failed. The result of the CGA condition varies depending on the equipment function. Voice equipment typically coerces the robbed bits for signaling to a state that will result in the far end properly handling the condition, while applying an often different state to the customer equipment connected to the alarmed equipment. Simultaneously, the customer data is often coerced to a 0x7F pattern, signifying a zero-voltage condition on voice equipment. Data equipment usually passes whatever data may be present, if any, leaving it to the customer equipment to deal with the condition.

==Inband T1 versus T1 PRI==
Additionally, for voice T1s there are two main types: so-called "plain" or Inband T1s and PRI ([[Primary rate interface|Primary Rate Interface]]). While both carry voice telephone calls in similar fashion, PRIs are commonly used in call centers and provide not only the 23 actual usable telephone lines (Known as "B" channels for bearer) but also a 24th line (Known as the "D" channel for data<ref>Versadial, [http://www.versadial.com/call_recording_terms.html#t1pri Call recording encyclopedia], last accessed 19 April 2007</ref>) that carries signaling information. This special "D" channel carries: [[Caller ID]] (CID) and [[automatic number identification]] (ANI) data, required channel type (usually a B, or Bearer channel), call handle, [[Dialed Number Identification Service]] (DNIS) info, requested channel number and a request for response.<ref>Newton, H: "Newton's telecom dictionary", page 225. CMP books, 2004</ref>

Inband T1s are also capable of carrying CID and ANI information if they are configured by the carrier to do so but PRIs handle this more efficiently. While an inband T1 seemingly has a slight advantage due to 24 lines being available to make calls (as opposed to a PRI that has 23), each channel in an inband T1 must perform its own setup and tear-down of each call. A PRI uses the 24th channel as a data channel to perform all the overhead operations of the other 23 channels (including CID and ANI). Although an inband T1 has 24 channels, the 23 channel PRI can set up more calls faster due to the dedicated 24th signalling channel (D Channel).

==Origin of name==
{{Unreferenced section|date=November 2007}}
The name T1 came from the carrier letter assigned by AT&T to the technology. Essentially, the "T" is a part number that was assigned by AT&T. Just as there is the generally known [[L-carrier]] and N-carrier systems, T-carrier was the next letter available and T1 is the first level in the hierarchy. DS-1 meant "Digital Service – Level 1", and had to do with the service to be sent (originally 24 digitized voice channels over the T1). The terms T1 and DS1 have become synonymous and include a plethora of different services from voice to data to clear-channel pipes. The line speed is always consistent at 1.54 Mbit/s, but the payload can vary greatly.

==Alternative technologies==
'''Dark fiber''': [[Dark fiber]] refers to unused [[Optical fiber|fibers]], available for use. Dark fiber has been, and still is, available for sale on the wholesale market for both metro and wide area links, but it may not be available in all markets or city pairs.

Dark fiber capacity is typically used by network operators to build [[SONET]] and dense wavelength division multiplexing (DWDM) networks, usually involving meshes of [[self-healing rings]]. Now, it is also used by end-user enterprises to expand [[Ethernet]] local area networks, especially since the adoption of [[IEEE]] standards for [[Gigabit Ethernet]] and [[10 Gigabit Ethernet]] over single-mode fiber. Running Ethernet networks between geographically separated buildings is a practice known as "[[Wide Area Network|WAN]] elimination".

'''DSIC''' is a digital signal equivalent to two Digital Signal 1, with extra [[bit]]s to conform to a signaling standard of 3.152 Mbit/s. Few (if any) of these circuit capacities are still in use today. In the early days of digital and data transmission, the 3-Mbit/s data rate was used to link [[mainframe computers]] together. The physical side of this circuit is called TIC.{{cn|date=March 2013}}

==Semiconductor==
The T1/E1 protocol is implemented as a "line interface unit" in silicon. The semiconductor chip contains a decoder/encoder, loop backs, jitter attenuators, receivers, and drivers. Additionally, there are usually multiple interfaces and they are labeled as dual, quad, octal, etc., depending upon the number.

The transceiver chip's primary purpose is to retrieve information from the "line", i.e., the conductive line that transverses distance, by receiving the pulses and converting the signal which has been subjected to noise, [[jitter]], and other interference, to a clean digital pulse on the other interface of the chip.

==See also==
*[[T-carrier]]
*[[E-carrier]]
*[[Time-division multiple access]]
*[[Pulse-code modulation]]
*[[Federal Standard 1037C]]
*DS1 [[code|Encoding]] schemes: [[B8ZS]], [[HDB3]], [[Alternate Mark Inversion|AMI]]
*[[Line code]]
*[[Central Office Multiplexing]]
*[[Time-division multiplexing]]
*[[Multiplexing]]
*[[Physical layer]]
*[[Data frame]]
*[[Quantization (signal processing)]]
*[[Digital Signal 3]]
*[[Digital Signal 0]]

==Notes and references==
{{reflist}}
{{Use dmy dates|date=October 2011}}

[[Category:Telecommunications standards]]
[[Category:Multiplexing]]

[[he:T1]]

Debye length

2013-11-05T14:32:02Z

132.187.253.24: /* Typical values */

{{infobox person
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| known_for = Introducing the Drgganita system of astronomical computations
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| notable works = Drgganita, Goladipika, Grahanamandana
| occupation = Astronomer-mathematician
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| religion = [[Hindu]]
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'''Vatasseri Parameshvara Nambudiri ([[Malayalam]]: വടശ്ശേരി പരമേശ്വരന്‍)''' (ca.1380–1460)<ref name="Pingree">{{cite book|last=David Edwin Pingree|title=Census of the exact sciences in Sanskrit|publisher=American Philosophical Society|year=1981|series=A|volume=4|pages=187–192|isbn=978-0-87169-213-9}}</ref> was a major [[India]]n [[mathematician]] and [[astronomer]] of the [[Kerala school of astronomy and mathematics]] founded by [[Madhava of Sangamagrama]]. He was also an [[astrologer]]. Parameshvara was a proponent of [[observational astronomy]] in [[medieval India]] and he himself had made a series of [[eclipse]] observations to verify the accuracy of the computational methods then in use. Based on his eclipse observations, Parameshvara proposed several corrections to the astronomical parameters which had been in use since the times of [[Aryabhata]]. The computational scheme based on the revised set of parameters has come to be known as the ''Drgganita'' system. Parameshvara was also a prolific writer on matters relating to astronomy. At least 25 manuscripts have been identified as being authored by Parameshvara.<ref name="Pingree"/>

== Biographical details ==

Parameshvara was a [[Hindu]] of Bhrgu[[gotra]] following the [[Ashvalayana]]sutra of the Rgveda. Parameshvara's family name (''Illam'') was Vatasseri (also called Vatasreni) and his family resided in the village of Alathiyur (Sanskritised as ''Asvatthagrama'') in [[Tirur]], [[Kerala]]. Alathiyur is situated on the northern bank of the river [[Nila River|Nila]] (river [[Bharathappuzha]]) at its mouth in Kerala. He was a grandson of a disciple of [[Govinda Bhattathiri]] (1237–1295 CE), a legendary figure in the astrological traditions of [[Kerala]].

Parameshvara studied under teachers Rudra and Narayana, and also under [[Sangamagrama Madhava]] (c. 1350 – c. 1425) the founder of the [[Kerala school of astronomy and mathematics]]. [[Damodara]], another prominent member of the [[Kerala school of astronomy and mathematics|Kerala school]], was his son and also his pupil. Parameshvara was also a teacher of [[Nilakantha Somayaji]] (1444–1544) the author of the celebrated [[Tantrasamgraha]].

== Work ==
Parameshvara wrote commentaries on many mathematical and astronomical works such as those by [[Bhaskara I]] and [[Aryabhata]]. He made a series of eclipse observations over a 55 year period, and constantly attempted to compare these with the theoretically computed positions of the planets. He revised planetary parameters based on his observations.

Parameshvara's most significant contribution is his [[Mean value theorem|mean value type formula]] for the inverse interpolation of the sine. He was the first mathematician to give the [[radius]] of [[circle]] with an inscribed quadrilateral, an expression that is normally attributed to [[Lhuilier]] (1782), 350 years later. With the sides of the [[cyclic quadrilateral]] being ''a, b, c,'' and ''d'', the radius ''R'' of the circumscribed circle is:
:<math> R = \sqrt {\frac{(ab + cd) (ac + bd) (ad + bc)}{(a + b + c - d) (b + c + d - a) (c + d + a - b) (d + a + b - c)}}.</math>

==Works by Parameshvara==
The following works of Parameshvara are well-known.<ref>{{cite journal|last=A.K. Bag|date=May 1980|title=Indian literature on mathematics during 1400 - 1800 AD|journal=Indian Journal of History of Science|volume=15|issue=1|pages=79–93|url=http://www.new.dli.ernet.in/rawdataupload/upload/insa/INSA_1/20005af2_79.pdf}}</ref> A complete list of all manuscripts attributed to Parameshvara is available in Pingree.<ref name="Pingree"/>

*''Bhatadipika'' - Commentary on [[Āryabhaṭīya]] of [[Āryabhaṭa I]]
*''Karmadipika'' - Commentary on ''Mahabhaskariya'' of Bhaskara I
*''Paramesvari'' - Commentary on ''Laghubhaskariya'' of Bhaskara I
*''Sidhantadipika'' - Commentary on ''Mahabhaskariyabhashya'' of [[Govindasvāmi]]
*''Vivarana'' - Commentary on [[Surya Siddhanta]] and [[Lilāvati]]
*''Drgganita'' - Description of the ''Drk'' system (composed in 1431 CE)
*''Goladipika'' - Spherical geometry and astronomy (composed in 1443 CE)
*''Grahanamandana'' - Computation of eclipses (Its epoch is 15 July 1411 CE.)
*''Grahanavyakhyadipika'' - On the rationale of the theory of eclipses
*''Vakyakarana'' - Methods for the derivation of several astronomical tables

==References==
{{reflist}}

==Further reading==
*[[David Pingree]], Biography in Dictionary of Scientific Biography (New York 1970-1990).
*Bhaskara, Laghubhaskariyam : With Parameshvara's commentary (Poona, 1946).
*Bhaskara, Mahabhaskariyam: With Parameshvara's commentary called Karmadipika (Poona, 1945).
*Munjala, Laghumanasam : with commentary by Parameshvara (Poona, 1944).
*[[T.A. Sarasvati Amma]], Geometry in ancient and medieval India (Delhi, 1979).
*K Shankar Shukla, The Surya-siddhanta with the commentary of Parameshvara (Lucknow, 1957).
*[[Radha Charan Gupta]], Parameshvara's rule for the circumradius of a cyclic quadrilateral, Historia Math. 4 (1977), 67-74.
*[[Radha Charan Gupta]], A mean-value-type formula for inverse interpolation of the sine, Ganita 30 (1-2) (1979), 78—82.
*K Plofker, An example of the secant method of iterative approximation in a fifteenth-century Sanskrit text, Historia Math. 23 (3) (1996), 246-256.
*K K Raja, Astronomy and mathematics in Kerala, Brahmavidya 27 (1963), 136-143.
*{{cite journal|last=K. Chandra Hari|year=2003|title=Eclipse observations of Parameshvara, the 14 - 15 century astronomer of Kerala|journal=Indian Journal of History of Science|volume=38|issue=1|pages=43–57|url=http://www.new.dli.ernet.in/rawdataupload/upload/insa/INSA_1/20008275_43.pdf|accessdate=28 January 2010}}

==External links==
* {{cite encyclopedia | editor = Thomas Hockey et al | last = Achar | first = Narahari | title=Parameśvara of Vāṭaśśeri [Parmeśvara I] | encyclopedia = The Biographical Encyclopedia of Astronomers | publisher = Springer | year = 2007 | location = New York | page = 870 | url=http://islamsci.mcgill.ca/RASI/BEA/Paramesvara_of_Vatesseri_BEA.htm | isbn=978-0-387-31022-0}} ([http://islamsci.mcgill.ca/RASI/BEA/Paramesvara_of_Vatesseri_BEA.pdf PDF version])
*http://www-history.mcs.st-andrews.ac.uk/Biographies/Paramesvara.html
{{Indian mathematics}}




{{Persondata 
| NAME =Parameshvara
| ALTERNATIVE NAMES =
| SHORT DESCRIPTION =mathematician, astronomer
| DATE OF BIRTH =1380-00-00
| PLACE OF BIRTH =
| DATE OF DEATH =1425-00-00
| PLACE OF DEATH =
}}
[[Category:1380 births]]
[[Category:1425 deaths]]
[[Category:Indian Hindus]]
[[Category:Medieval Indian mathematicians]]
[[Category:14th-century mathematicians]]
[[Category:15th-century mathematicians]]
[[Category:15th-century astronomers]]
[[Category:Medieval Indian astronomers]]
[[Category:Medieval Indian astrologers]]
[[Category:Kerala school]]
[[Category:14th-century Indian people]]
[[Category:15th-century Indian people]]