Cross section (physics) - Knowledge (XXG)

43: 702: 5503: 3964: 737:, contact, etc.), the impact parameter and the scattering angle have a definite one-to-one functional dependence on each other. Generally the impact parameter can neither be controlled nor measured from event to event and is assumed to take all possible values when averaging over many scattering events. The differential size of the cross section is the area element in the plane of the impact parameter, i.e. 246: 1341: 7356: 5243: 186:, the intensity scattered at the forward and backward angles is greater than the intensity scattered sideways, so the forward differential scattering cross section is greater than the perpendicular differential cross section, and by adding all of the infinitesimal cross sections over the whole range of angles with integral calculus, we can find the total cross section. 5247: 7904: 2050: 2428: 7624: 3758: 6535:

experience a larger cross sectional area of water droplets than photons of higher energy. The halo around the Moon thus has a perimeter of red light due to lower energy photons being scattering further from the center of the Moon. Photons from the rest of the visible spectrum are left within the center of the halo and perceived as white light.

2257: 5061: 1187: 1114: 7190: 919: 5056: 2798:

Combining the scattering and absorption cross sections in this manner is often necessitated by the inability to distinguish them experimentally, and much research effort has been put into developing models that allow them to be distinguished, the Kubelka-Munk theory being one of the most important in

209:

of a given process depends strongly on experimental variables such as the density of the target material, the intensity of the beam, the detection efficiency of the apparatus, or the angle setting of the detection apparatus. However, these quantities can be factored away, allowing measurement of the

6228: 201:

for collisions of accelerated beams of one type of particle with targets (either stationary or moving) of a second type of particle. The probability for any given reaction to occur is in proportion to its cross section. Thus, specifying the cross section for a given reaction is a proxy for stating

232:

specifies the amount of optical power scattered from light of a given irradiance (power per area). It is important to note that although the cross section has the same units as area, the cross section may not necessarily correspond to the actual physical size of the target given by other forms of

1877:

with each other) is a measure of the interaction event between the two particles. The cross section is proportional to the probability that an interaction will occur; for example in a simple scattering experiment the number of particles scattered per unit of time (current of scattered particles

6534:

There is no simple relationship between the scattering cross section and the physical size of the particles, as the scattering cross section depends on the wavelength of radiation used. This can be seen when looking at a halo surrounding the Moon on a decently foggy evening: Red light photons

801:

It is a function of the scattering angle (and therefore also the impact parameter), plus other observables such as the momentum of the incoming particle. The differential cross section is always taken to be positive, even though larger impact parameters generally produce less deflection. In

7760: 4772: 1915: 2274: 5498:{\displaystyle \sigma _{\text{sc}}=\sigma _{\text{sc}}^{\text{(e)}}+\sigma _{\text{sc}}^{\text{(m)}}={\frac {1}{(4\pi \varepsilon \varepsilon _{0})^{2}}}\cdot {\frac {8\pi }{3}}k^{4}|\alpha ^{e}|^{2}+{\frac {1}{(4\pi \mu \mu _{0})^{2}}}\cdot {\frac {8\pi }{3}}k^{4}|\alpha ^{m}|^{2}} 3959:{\displaystyle W_{\text{i}}=-\oint _{A}\mathbf {\Pi } _{\text{i}}\cdot {\hat {\mathbf {r} }}dA\equiv 0,\qquad W_{\text{s}}=\oint _{A}\mathbf {\Pi } _{\text{s}}\cdot {\hat {\mathbf {r} }}dA,\qquad W_{\text{ext}}=\oint _{A}\mathbf {\Pi } _{\text{ext}}\cdot {\hat {\mathbf {r} }}dA.} 7479: 3021: 1754: 3584:

energy is absorbed within the sphere, otherwise energy is being created within the sphere. We will not consider this case here. If the host medium is non-absorbing, the energy must be absorbed by the particle. We decompose the total field into incident and scattered parts

925:

In situations where the scattering process is not azimuthally symmetric, such as when the beam or target particles possess magnetic moments oriented perpendicular to the beam axis, the differential cross section must also be expressed as a function of the azimuthal angle.

1632: 7749: 2110: 6953: 1861: 8221: 7450: 5735: 8057: 1336:{\displaystyle \sigma =\oint _{4\pi }{\frac {\mathrm {d} \sigma }{\mathrm {d} \Omega }}\,\mathrm {d} \Omega =\int _{0}^{2\pi }\int _{0}^{\pi }{\frac {\mathrm {d} \sigma }{\mathrm {d} \Omega }}\sin \theta \,\mathrm {d} \theta \,\mathrm {d} \varphi .} 4504: 4893: 1367:

The differential cross section is extremely useful quantity in many fields of physics, as measuring it can reveal a great amount of information about the internal structure of the target particles. For example, the differential cross section of

5566: 7351:{\displaystyle I\,\mathrm {d} \sigma =I\,\mathrm {d} x(x)=Ir\cos \alpha \,\mathrm {d} \alpha =I{\frac {r}{2}}\sin \left({\frac {\theta }{2}}\right)\,\mathrm {d} \theta =I{\frac {\mathrm {d} \sigma }{\mathrm {d} \theta }}\,\mathrm {d} \theta .} 995: 285:

of finite-sized particles there are collisions among particles that depend on their cross-sectional size. The average distance that a particle travels between collisions depends on the density of gas particles. These quantities are related by

5947: 4633: 5904: 5238:{\displaystyle \sigma _{\text{ext}}=\sigma _{\text{ext}}^{\text{(e)}}+\sigma _{\text{ext}}^{\text{(m)}}={\frac {1}{4\pi \varepsilon \varepsilon _{0}}}\cdot 4\pi k\Im (\alpha ^{e})+{\frac {1}{4\pi \mu \mu _{0}}}\cdot 4\pi k\Im (\alpha ^{m})} 1497: 7968: 817: 4681: 8127: 6387: 2633: 3475: 4272: 4147: 4057: 435:

If the particles in the gas interact by a force with a larger range than their physical size, then the cross section is a larger effective area that may depend on a variety of variables such as the energy of the particles.

7899:{\displaystyle \mathrm {d} \sigma =\mathrm {d} r(r)\times r\,\mathrm {d} \varphi ={\frac {a^{2}}{2}}\sin \left({\frac {\theta }{2}}\right)\cos \left({\frac {\theta }{2}}\right)\,\mathrm {d} \theta \,\mathrm {d} \varphi .} 6305:

in addition to the shape and size of the particle. The total amount of scattering in a sparse medium is determined by the product of the scattering cross section and the number of particles present. In terms of area, the

3635: 3334: 2938: 547: 2045:{\displaystyle {\begin{aligned}I_{\text{r}}&=I_{\text{i}}N\sigma ,\\\sigma &={\frac {I_{\text{r}}}{I_{\text{i}}}}{\frac {1}{N}}\\&={\text{probability of interaction}}\times {\frac {1}{N}}.\end{aligned}}} 3078: 2423:{\displaystyle S_{{\text{f}}{\text{i}}}=\delta _{{\text{f}}{\text{i}}}-2\pi i\delta \left(E_{\text{f}}-E_{\text{i}}\right)\delta \left(\mathbf {p} _{\text{i}}-\mathbf {p} _{\text{f}}\right)T_{{\text{f}}{\text{i}}}} 1647:. This general form is valid for any short-ranged, energy-conserving interaction. It is not true for long-ranged interactions, so there are additional complications when dealing with electromagnetic interactions. 3750: 4888: 6611: 3549: 7122: 7619:{\displaystyle \sigma =\int _{0}^{2\pi }{\frac {\mathrm {d} \sigma }{\mathrm {d} \theta }}\,\mathrm {d} \theta =\int _{0}^{2\pi }{\frac {r}{2}}\sin \left({\frac {\theta }{2}}\right)\,\mathrm {d} \theta =2r.} 6720: 6788: 8899: 3270: 1656: 5571: 2252:{\displaystyle {\frac {\mathrm {d} \sigma }{\mathrm {d} \Omega }}=\left(2\pi \right)^{4}m_{\text{i}}m_{\text{f}}{\frac {p_{\text{f}}}{p_{\text{i}}}}{\bigl |}T_{{\text{f}}{\text{i}}}{\bigr |}^{2},} 7666: 1920: 1531: 7661: 6524: 2793: 4819: 2497:. To avoid the need for conversion factors, the scattering cross section is expressed in cm, and the number concentration in cm. The measurement of the scattering of visible light is known as 6839: 4348: 1765: 233:

measurement. It is not uncommon for the actual cross-sectional area of a scattering object to be much larger or smaller than the cross section relative to some physical process. For example,

8138: 7374: 430: 328: 632: 6278: 2933: 2555:, shape, and size of the particle. The total amount of scattering in a sparse medium is proportional to the product of the scattering cross section and the number of particles present. 4508: 5740: 4890:(here we use the notation of magnetic polarizability in the manner of Bekshaev et al. rather than the notation of Nieto-Vesperinas et al.) expressed through the Mie coefficients as 2685: 7173: 5939: 1143:

is the thickness of the stationary target (SI unit: m). This formula assumes that the target is thin enough that each beam particle will interact with at most one target particle.

4676: 7979: 2535:= 10 b. The sum of the scattering, photoelectric, and pair-production cross-sections (in barns) is charted as the "atomic attenuation coefficient" (narrow-beam), in barns. 3582: 6983: 155:

that interact only upon contact, their scattering cross section is related to their geometric size. If the particles interact through some action-at-a-distance force, such as

1109:{\displaystyle {\frac {\mathrm {d} \sigma }{\mathrm {d} \Omega }}(\theta ,\varphi )={\frac {1}{nt\Delta \Omega }}{\frac {F_{\text{out}}(\theta ,\varphi )}{F_{\text{inc}}}}.} 4323: 3657: 5508: 3197: 3132: 2890: 2836: 1869:

A cross section is therefore a measure of the effective surface area seen by the impinging particles, and as such is expressed in units of area. The cross section of two

8606: 7469:

corresponds to scattering from the edge of the circle directly forward. This expression confirms the intuitive expectations that the mirror circle acts like a diverging

2863: 7007: 1421: 3684: 129:. In a way, it can be thought of as the size of the object that the excitation must hit in order for the process to occur, but more exactly, it is a parameter of a 7634:

The result from the previous example can be used to solve the analogous problem in three dimensions, i.e., scattering from a perfectly reflecting sphere of radius

4343: 914:{\displaystyle {\frac {\mathrm {d} \sigma }{\mathrm {d} (\cos \theta )}}=\int _{0}^{2\pi }{\frac {\mathrm {d} \sigma }{\mathrm {d} \Omega }}\,\mathrm {d} \varphi } 8745: 7915: 7035: 7051:

and a perfectly reflecting boundary. The beam consists of a uniform density of parallel rays, and the beam-circle interaction is modeled within the framework of

5051:{\displaystyle \alpha ^{e}=4\pi \varepsilon _{0}\cdot i{\frac {3\varepsilon }{2k^{3}}}a_{1},\qquad \alpha ^{m}=4\pi \mu _{0}\cdot i{\frac {3\mu }{2k^{3}}}b_{1}.} 4303: 3401: 3381: 3361: 1364:. The latter term may be confusing in contexts where multiple events are involved, since "total" can also refer to the sum of cross sections over all events. 8314: 8068: 6330: 6223:{\displaystyle {\frac {1}{4\pi \varepsilon \varepsilon _{0}}}\Im (\alpha ^{e})+{\frac {1}{4\pi \mu \mu _{0}}}\Im (\alpha ^{m})\geq {\frac {2k^{3}}{3}}\left} 2576: 8414:

Bekshaev, A Ya (2013-04-01). "Subwavelength particles in an inhomogeneous light field: optical forces associated with the spin and orbital energy flows".

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joining the reflection point of the ray with the center point of the mirror. Then the increase of the length element perpendicular to the beam is

3588: 4767:{\displaystyle \sigma _{\text{ext}}=\sigma _{\text{sc}}+\sigma _{\text{abs}}\qquad {\text{or}}\qquad Q_{\text{ext}}=Q_{\text{sc}}+Q_{\text{abs}}} 3275: 493: 8874: 567: 6297:, describes the likelihood of light being scattered by a macroscopic particle. In general, the scattering cross section is different from the 174:(see detailed discussion below). When a cross section is integrated over all scattering angles (and possibly other variables), it is called a 8792: 8337: 3029: 8782: 3689: 64: 6561: 4824: 3483: 7075: 6665: 8721: 8694: 8381: 8331: 6731: 3016:{\displaystyle Q_{\alpha }={\frac {\sigma _{\alpha }}{\sigma _{\text{geom}}}},\qquad \alpha ={\text{ext}},{\text{sc}},{\text{abs}}.} 86: 2570:. The sum of the absorption and scattering cross sections is sometimes referred to as the attenuation or extinction cross section. 1749:{\displaystyle \phi (\mathbf {r} )\;{\stackrel {r\to \infty }{\longrightarrow }}\;\phi _{-}(\mathbf {r} )+\phi _{+}(\mathbf {r} ).} 237:

can have light scattering cross sections for particular frequencies that are much larger than their actual cross-sectional areas.

3206: 8248: 8243: 380:

that interact by direct contact, as illustrated in Figure 1, then the effective cross section for the collision of a pair is

8404:

Bohren, Craig F., and Donald R. Huffman. Absorption and scattering of light by small particles. John Wiley & Sons, 2008.

1627:{\displaystyle \phi _{+}(\mathbf {r} )\;{\stackrel {r\to \infty }{\longrightarrow }}\;f(\theta ,\phi ){\frac {e^{ikr}}{r}},} 7744:{\displaystyle {\begin{aligned}r&=a\sin \alpha ,\\\mathrm {d} r&=a\cos \alpha \,\mathrm {d} \alpha ,\end{aligned}}} 103:

is a measure of the probability that a specific process will take place in a collision of two particles. For example, the

8869: 6948:{\displaystyle {\frac {d\sigma }{d\Omega }}=\left({\frac {q\,Q}{16\pi \varepsilon _{0}E\sin ^{2}(\theta /2)}}\right)^{2}} 1856:{\displaystyle {\frac {\mathrm {d} \sigma }{\mathrm {d} \Omega }}(\theta ,\phi )={\bigl |}f(\theta ,\phi ){\bigr |}^{2}.} 8864: 8253: 8216:{\displaystyle \sigma =\oint _{4\pi }{\frac {\mathrm {d} \sigma }{\mathrm {d} \Omega }}\,\mathrm {d} \Omega =\pi a^{2}.} 7445:{\displaystyle {\frac {\mathrm {d} \sigma }{\mathrm {d} \theta }}={\frac {r}{2}}\sin \left({\frac {\theta }{2}}\right).} 6480: 2749: 1383: 141: 4784: 719:

is the perpendicular offset of the trajectory of the incoming particle, and the outgoing particle emerges at an angle

667: 1866:

This has the simple interpretation as the probability density for finding the scattered projectile at a given angle.

386: 292: 57: 51: 582: 8894: 8844: 4278: 167: 8889: 6233: 5730:{\displaystyle \sigma _{\text{abs}}^{\text{(e)}}={\frac {1}{4\pi \varepsilon \varepsilon _{0}}}\cdot 4\pi k\left} 2895: 6659:

denote the radii of the scattering center and scattered sphere, respectively. The differential cross section is

2558:

In the interaction of light with particles, many processes occur, each with their own cross sections, including

68: 8884: 8233: 6317: 6298: 2559: 2544: 439:

Cross sections can be computed for atomic collisions but also are used in the subatomic realm. For example, in

2490:

are also widely used. Correspondingly, the differential cross section can be measured in units such as mb/sr.

1522:

of the wave function when the projectile and target are too far apart for the interaction to have any effect.

2648: 570:, or transformation to another species. The volumetric number density of scattering centers is designated by 8859: 8268: 7140: 7055:. Because the problem is genuinely two-dimensional, the cross section has unit of length (e.g., metre). Let 3337: 2543:

For light, as in other settings, the scattering cross section for particles is generally different from the

666:

measurement where a single particle is scattered off a single stationary target particle. Conventionally, a

234: 8052:{\displaystyle \sin \theta =2\sin \left({\frac {\theta }{2}}\right)\cos \left({\frac {\theta }{2}}\right),} 5911: 1525:

After scattering takes place it is expected that the wave function takes on the following asymptotic form:

4499:{\displaystyle Q_{\text{sc}}={\frac {2}{k^{2}a^{2}}}\sum _{n=1}^{\infty }(2n+1)(|a_{n}|^{2}+|b_{n}|^{2})} 1386:

that indicate the creation of metastable states and contain information about their energy and lifetime.

8909: 8288: 8273: 8263: 8258: 6812: 4638: 2510: 1369: 448: 104: 6397: 3554: 2639: 651: 213:

Differential and total scattering cross sections are among the most important measurable quantities in

6961: 8879: 8563: 8498: 8433: 8293: 5561:{\textstyle \sigma _{\text{abs}}=\sigma _{\text{abs}}^{\text{(e)}}+\sigma _{\text{abs}}^{\text{(m)}}} 4781:

Let us assume that a particle supports only electric and magnetic dipole modes with polarizabilities

2444: 1644: 149: 8814: 8809: 2638:

The total extinction cross section is related to the attenuation of the light intensity through the

1518:

coordinates between the projectile and the target. The arrow indicates that this only describes the

8904: 8283: 7643:

The plane perpendicular to the incoming light beam can be parameterized by cylindrical coordinates

6986: 6793:

In other words, the total scattering cross section is equal to the area of the circle (with radius

1887:) depends only on the number of incident particles per unit of time (current of incident particles 1870: 663: 183: 170:

limit of a function of some final-state variable, such as particle angle or energy, it is called a

31: 8318: 6647:

The following equations apply to two hard spheres that undergo a perfectly elastic collision. Let

4628:{\displaystyle Q_{\text{ext}}={\frac {2}{k^{2}a^{2}}}\sum _{n=1}^{\infty }(2n+1)\Re (a_{n}+b_{n})} 4308: 3640: 3383:) around the particle (the scatterer). The net rate of electromagnetic energy crosses the surface 8739: 8532: 8488: 8457: 8423: 5908:

For the case of a no-inside-gain particle, i.e. no energy is emitted by the particle internally (

5899:{\displaystyle \sigma _{\text{abs}}^{\text{(m)}}={\frac {1}{4\pi \mu \mu _{0}}}\cdot 4\pi k\left} 3343:

The main approach is based on the following. Firstly, we construct an imaginary sphere of radius

3137: 3086: 130: 2868: 2814: 2493:

When the scattered radiation is visible light, it is conventional to measure the path length in

7655:. In any plane of the incoming and the reflected ray we can write (from the previous example): 6803:) within which the center of mass of the incoming sphere has to arrive for it to be deflected. 2841: 1492:{\displaystyle \phi _{-}(\mathbf {r} )\;{\stackrel {r\to \infty }{\longrightarrow }}\;e^{ikz},} 811:

is not changed by the scattering process, and the differential cross section can be written as

8788: 8727: 8717: 8700: 8690: 8589: 8581: 8524: 8516: 8449: 8387: 8377: 8327: 8278: 6464: 6460: 2733: 2567: 2448: 1399: 1376: 6992: 8628: 8571: 8506: 8441: 7963:{\displaystyle \mathrm {d} \Omega =\sin \theta \,\mathrm {d} \theta \,\mathrm {d} \varphi .} 7052: 2729: 1395: 726: 711: 701: 222: 198: 156: 3662: 5942: 4328: 1896:), the characteristics of target (for example the number of particles per unit of surface 803: 693: 440: 214: 190: 7012: 8567: 8502: 8445: 8437: 2104:

before and after the collision respectively, the differential cross section is given by

17: 6438: 4288: 3386: 3366: 3346: 2892:

cross sections. These are normalized by the geometrical cross sections of the particle

2808: 2521: 1134: 452: 367: 355: 218: 194: 112: 108: 8359:

Nondestructive Testing Handbook Volume 4 Radiographic Testing, ASNT, 2002, chapter 22.

8122:{\displaystyle {\frac {\mathrm {d} \sigma }{\mathrm {d} \Omega }}={\frac {a^{2}}{4}}.} 6382:{\displaystyle \sigma =\sigma _{\text{abs}}+\sigma _{\text{sc}}+\sigma _{\text{lum}}.} 4345:, the problem has a precise solution. The scattering and extinction coefficients are 2628:{\displaystyle \sigma =\sigma _{\text{abs}}+\sigma _{\text{sc}}+\sigma _{\text{lum}}.} 2520:

can also be described in terms of scattering cross sections, in which case the square

8853: 8461: 8238: 7470: 7045:

The following example deals with a beam of light scattering off a circle with radius

6468: 6288:

In the context of scattering light on extended bodies, the scattering cross section,

2737: 2479: 1403: 938:

off a stationary target consisting of many particles, the differential cross section

206: 8616: 8536: 3470:{\displaystyle W_{\text{a}}=-\oint _{A}\mathbf {\Pi } \cdot {\hat {\mathbf {r} }}dA} 576:. Solving this equation exhibits the exponential attenuation of the beam intensity: 7062: 6321: 6302: 4267:{\displaystyle \mathbf {\Pi } _{\text{ext}}={\frac {1}{2}}\operatorname {Re} \left} 2552: 2498: 2464: 2061: 6989:. The total cross section is infinite unless a cutoff for small scattering angles 8550:

Nieto-Vesperinas, M.; Sáenz, J. J.; Gómez-Medina, R.; Chantada, L. (2010-05-14).

4142:{\displaystyle \mathbf {\Pi } _{\text{s}}={\frac {1}{2}}\operatorname {Re} \left} 4052:{\displaystyle \mathbf {\Pi } _{\text{i}}={\frac {1}{2}}\operatorname {Re} \left} 8839: 8834: 8611: 2506: 2471: 1174: 734: 464: 163:, their scattering cross section is generally larger than their geometric size. 126: 245: 6544: 6452: 6423: 6393: 3200: 2700: 2563: 2548: 2502: 2494: 2475: 563: 145: 8615:, 2nd ed. (the "Gold Book") (1997). Online corrected version: (2006–) " 8585: 8520: 8453: 8391: 8731: 8704: 8632: 6456: 3630:{\displaystyle \mathbf {E} =\mathbf {E} _{\text{i}}+\mathbf {E} _{\text{s}}} 2725: 2531: 1874: 1347: 1124: 730: 8593: 8528: 3329:{\displaystyle \eta ={\sqrt {\mu \mu _{0}/(\varepsilon \varepsilon _{0})}}} 1350:

when the type of cross section can be inferred from context. In this case,

8475:

Bliokh, Konstantin Y.; Bekshaev, Aleksandr Y.; Nori, Franco (2014-03-06).

8376:(Revised 2nd ed.). Ram Nagar, New Delhi: S. Chand & Company Ltd. 542:{\displaystyle {\frac {\mathrm {d} \Phi }{\mathrm {d} z}}=-n\sigma \Phi ,} 8576: 8551: 6448: 2721: 2642:, which says that attenuation is proportional to particle concentration: 2434: 2065: 1407: 676:

axis of this coordinate system aligned with the incident beam. The angle

1379:

may be used as the independent variable of differential cross sections.

136:

When two discrete particles interact in classical physics, their mutual

8511: 8476: 7178:

The differential relationship between incident and reflected intensity

3073:{\displaystyle \sigma _{\alpha }={\frac {W_{\alpha }}{I_{\text{inc}}}}} 2460: 1759:

The differential cross section is related to the scattering amplitude:

1650:

The full wave function of the system behaves asymptotically as the sum

444: 160: 777:. The differential cross section is the quotient of these quantities, 650:

is the total thickness of the material. For light, this is called the

7127:

The reflection angle of this ray with respect to the incoming ray is

7066: 3745:{\displaystyle W_{\text{a}}=W_{\text{i}}-W_{\text{s}}+W_{\text{ext}}} 2487: 2483: 686:, measured between the incident beam and the scattered beam, and the 374:

If the particles in the gas can be treated as hard spheres of radius

152: 4883:{\textstyle \mathbf {m} =(\mu \mu _{0})^{-1}\alpha ^{m}\mathbf {H} } 756:. The differential angular range of the scattered particle at angle 469:

If a beam of particles enters a thin layer of material of thickness

6606:{\displaystyle L_{\text{V}}={\frac {3.9}{C\sigma _{\text{scat}}}}.} 3544:{\textstyle \mathbf {\Pi } ={\frac {1}{2}}\operatorname {Re} \left} 2470:

In nuclear and particle physics, the conventional unit is the barn

30:"Scattering cross section" redirects here. Not to be confused with 8493: 8428: 7473:. The total cross section is equal to the diameter of the circle: 6301:

of a particle, as it depends upon the wavelength of light and the

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and, finally, the electric and magnetic absorption cross sections

2517: 1372:

provided strong evidence for the existence of the atomic nucleus.

447:

collides with nuclei in a reactor or other nuclear device, with a

244: 122: 7117:{\displaystyle \mathrm {d} x=r\cos \alpha \,\mathrm {d} \alpha .} 144:

to their relative motion within which they must meet in order to

111:

will be deflected by a given angle during an interaction with an

6715:{\displaystyle {\frac {d\sigma }{d\Omega }}={\frac {R^{2}}{4}},} 477: 27:

Probability of a given process occurring in a particle collision

6400:, which says that absorbance is proportional to concentration: 4281:. After that, all the integrals can be taken. In the case of a 1152:

may be recovered by integrating the differential cross section

6783:{\displaystyle \sigma _{\text{tot}}=\pi \left(r+R\right)^{2}.} 1406:(before scattering) is taken to be a plane wave with definite 802:

cylindrically symmetric situations (about the beam axis), the

282: 36: 8824: 8819: 6508: 2777: 1382:

Differential cross sections in inelastic scattering contain

700: 202:

the probability that a given scattering process will occur.

343: 265:, for collisions is related to the particle number density 3265:{\displaystyle I_{\text{inc}}=|\mathbf {E} |^{2}/(2\eta )} 2447:. The computation of the S-matrix is the main goal of the 125:) and is expressed in units of area, more specifically in 1643:

is some function of the angular coordinates known as the

3134:

is the energy flow through the surrounding surface, and

7462:

corresponds to backward scattering, and its minimum at

2714:

is the particle concentration as a number density, and

971:

is related to the flux of scattered particle detection

8900:

Scattering, absorption and radiative transfer (optics)

8829: 6236: 6230:

Equality occurs for non-absorbing particles, i.e. for

5914: 5511: 4827: 4787: 4641: 3486: 2898: 2871: 2844: 2817: 670:

is used, with the target placed at the origin and the

8477:"Extraordinary momentum and spin in evanescent waves" 8141: 8071: 7982: 7918: 7763: 7664: 7482: 7377: 7193: 7143: 7078: 7015: 6995: 6964: 6842: 6734: 6668: 6564: 6483: 6333: 5950: 5743: 5574: 5250: 5064: 4896: 4777:

Dipole approximation for the scattering cross section

4684: 4511: 4351: 4331: 4311: 4291: 4155: 4065: 3975: 3761: 3692: 3665: 3643: 3591: 3557: 3412: 3389: 3369: 3349: 3278: 3209: 3140: 3089: 3032: 2941: 2752: 2651: 2579: 2277: 2113: 1918: 1768: 1659: 1534: 1424: 1190: 1123:

is the finite angular size of the detector (SI unit:

998: 820: 585: 496: 389: 295: 8840:

IUPAC Goldbook – Definition: Collision Cross Section

8830:

Particle Data Group – The Review of Particle Physics

8552:"Optical forces on small magnetodielectric particle" 8487:(1). Springer Science and Business Media LLC: 3300. 8845:

ShimPlotWell cross section plotter for nuclear data

8835:

IUPAC Goldbook – Definition: Reaction Cross Section

4277:All the field can be decomposed into the series of 210:underlying two-particle collisional cross section. 8215: 8121: 8051: 7962: 7898: 7743: 7618: 7444: 7350: 7167: 7116: 7029: 7001: 6977: 6947: 6782: 6714: 6605: 6519:{\displaystyle A_{\lambda }=-\log {\mathcal {T}}.} 6518: 6381: 6272: 6222: 5933: 5898: 5729: 5560: 5497: 5237: 5050: 4882: 4813: 4766: 4670: 4627: 4498: 4337: 4317: 4297: 4266: 4141: 4051: 3958: 3744: 3678: 3651: 3629: 3576: 3543: 3469: 3395: 3375: 3355: 3328: 3264: 3191: 3126: 3072: 3015: 2927: 2884: 2857: 2830: 2788:{\displaystyle A_{\lambda }=-\log {\mathcal {T}}.} 2787: 2679: 2627: 2422: 2251: 2044: 1855: 1748: 1626: 1491: 1335: 1108: 913: 626: 541: 424: 342:is the cross section of a two-particle collision ( 322: 7009:is applied. This is due to the long range of the 4814:{\textstyle \mathbf {p} =\alpha ^{e}\mathbf {E} } 6543:The scattering cross section is related to the 2811:include efficiency coefficients for extinction 425:{\displaystyle \sigma =\pi \left(2r\right)^{2}} 323:{\displaystyle \sigma ={\frac {1}{n\lambda }},} 6634:, the scattering coefficient per unit length. 2501:, and is effective for particles of 2–50 2467:, a smaller unit is usually used in practice. 627:{\displaystyle \Phi =\Phi _{0}e^{-n\sigma z},} 8784:Introduction to Experimental Particle Physics 7361:The differential cross section is therefore ( 3199:is the intensity of the incident wave. For a 2524:is a convenient unit: 1 Å = 10 m = 2235: 2210: 1839: 1813: 929:For scattering of particles of incident flux 252:In a gas of particles of individual diameter 228:With light scattering off of a particle, the 8: 8315:International Bureau of Weights and Measures 6273:{\textstyle \Im (\varepsilon )=\Im (\mu )=0} 2928:{\textstyle \sigma _{\text{geom}}=\pi a^{2}} 189:Scattering cross sections may be defined in 2505:in diameter: as such, it is widely used in 148:from each other. If the particles are hard 8744:: CS1 maint: location missing publisher ( 8374:Biological instrumentation and methodology 6827:scatters off a fixed particle with charge 6392:The total cross section is related to the 6316:) is the sum of the cross sections due to 1873:(i.e. observed when the two particles are 1697: 1674: 1579: 1556: 1469: 1446: 1137:of the target particles (SI unit: m), and 8667: 8575: 8510: 8492: 8427: 8204: 8186: 8185: 8174: 8164: 8161: 8152: 8140: 8105: 8099: 8085: 8075: 8072: 8070: 8032: 8008: 7981: 7973:Together with the trigonometric identity 7949: 7948: 7940: 7939: 7919: 7917: 7885: 7884: 7876: 7875: 7861: 7837: 7816: 7810: 7799: 7798: 7775: 7764: 7762: 7726: 7725: 7698: 7665: 7663: 7596: 7595: 7581: 7561: 7552: 7547: 7532: 7531: 7520: 7510: 7507: 7498: 7493: 7481: 7425: 7405: 7391: 7381: 7378: 7376: 7337: 7336: 7325: 7315: 7312: 7298: 7297: 7283: 7263: 7249: 7248: 7213: 7212: 7198: 7197: 7192: 7142: 7103: 7102: 7079: 7077: 7019: 7014: 6994: 6969: 6963: 6939: 6921: 6906: 6893: 6877: 6871: 6843: 6841: 6771: 6739: 6733: 6698: 6692: 6669: 6667: 6591: 6578: 6569: 6563: 6507: 6506: 6488: 6482: 6370: 6357: 6344: 6332: 6235: 6206: 6196: 6172: 6167: 6160: 6151: 6148: 6136: 6126: 6102: 6097: 6090: 6081: 6078: 6061: 6051: 6039: 6020: 6001: 5989: 5970: 5951: 5949: 5919: 5913: 5885: 5880: 5873: 5864: 5855: 5835: 5829: 5817: 5781: 5762: 5753: 5748: 5742: 5716: 5711: 5704: 5695: 5686: 5666: 5660: 5648: 5612: 5593: 5584: 5579: 5573: 5552: 5547: 5534: 5529: 5516: 5510: 5489: 5484: 5477: 5468: 5462: 5443: 5431: 5421: 5399: 5390: 5385: 5378: 5369: 5363: 5344: 5332: 5322: 5300: 5291: 5286: 5273: 5268: 5255: 5249: 5226: 5195: 5176: 5164: 5133: 5114: 5105: 5100: 5087: 5082: 5069: 5063: 5039: 5026: 5008: 4996: 4977: 4963: 4950: 4932: 4920: 4901: 4895: 4875: 4869: 4856: 4846: 4828: 4826: 4806: 4800: 4788: 4786: 4758: 4745: 4732: 4722: 4715: 4702: 4689: 4683: 4662: 4652: 4640: 4616: 4603: 4569: 4558: 4545: 4535: 4525: 4516: 4510: 4487: 4482: 4475: 4466: 4457: 4452: 4445: 4436: 4409: 4398: 4385: 4375: 4365: 4356: 4350: 4330: 4310: 4290: 4253: 4248: 4238: 4233: 4228: 4218: 4213: 4203: 4198: 4193: 4171: 4162: 4157: 4154: 4128: 4123: 4113: 4108: 4103: 4081: 4072: 4067: 4064: 4038: 4033: 4023: 4018: 4013: 3991: 3982: 3977: 3974: 3936: 3934: 3933: 3924: 3919: 3912: 3899: 3874: 3872: 3871: 3862: 3857: 3850: 3837: 3806: 3804: 3803: 3794: 3789: 3782: 3766: 3760: 3736: 3723: 3710: 3697: 3691: 3670: 3664: 3644: 3642: 3621: 3616: 3606: 3601: 3592: 3590: 3562: 3556: 3551:is the time averaged Poynting vector. If 3531: 3522: 3517: 3495: 3487: 3485: 3450: 3448: 3447: 3439: 3433: 3417: 3411: 3388: 3368: 3348: 3315: 3300: 3294: 3285: 3277: 3245: 3239: 3234: 3228: 3223: 3214: 3208: 3177: 3172: 3166: 3149: 3139: 3115: 3098: 3088: 3062: 3052: 3046: 3037: 3031: 3005: 2997: 2989: 2971: 2961: 2955: 2946: 2940: 2919: 2903: 2897: 2876: 2870: 2849: 2843: 2822: 2816: 2807:Cross sections commonly calculated using 2776: 2775: 2757: 2751: 2724:. The absorbance of the radiation is the 2656: 2650: 2616: 2603: 2590: 2578: 2547:of the particle, and it depends upon the 2413: 2408: 2407: 2392: 2387: 2377: 2372: 2352: 2339: 2308: 2303: 2302: 2288: 2283: 2282: 2276: 2240: 2234: 2233: 2225: 2220: 2219: 2209: 2208: 2200: 2190: 2184: 2178: 2168: 2158: 2127: 2117: 2114: 2112: 2025: 2017: 1997: 1989: 1979: 1973: 1944: 1927: 1919: 1917: 1844: 1838: 1837: 1812: 1811: 1782: 1772: 1769: 1767: 1735: 1726: 1711: 1702: 1683: 1678: 1676: 1675: 1666: 1658: 1604: 1598: 1565: 1560: 1558: 1557: 1548: 1539: 1533: 1474: 1455: 1450: 1448: 1447: 1438: 1429: 1423: 1322: 1321: 1313: 1312: 1292: 1282: 1279: 1273: 1268: 1255: 1250: 1235: 1234: 1223: 1213: 1210: 1201: 1189: 1095: 1069: 1062: 1041: 1012: 1002: 999: 997: 903: 902: 891: 881: 878: 869: 864: 834: 824: 821: 819: 606: 596: 584: 510: 500: 497: 495: 416: 388: 302: 294: 166:When a cross section is specified as the 87:Learn how and when to remove this message 8775:Scattering Theory of Waves and Particles 8562:(11). The Optical Society: 11428–11443. 8367: 8365: 2474:, where 1 b = 10 m = 100 1346:It is common to omit the "differential" 271:, and mean free path between collisions 50:This article includes a list of general 8306: 2680:{\displaystyle A_{\lambda }=Cl\sigma ,} 8737: 8655: 8643: 8320:The International System of Units (SI) 7168:{\displaystyle \theta =\pi -2\alpha .} 6284:Scattering of light on extended bodies 5934:{\textstyle \sigma _{\text{abs}}>0} 3637:, and the same for the magnetic field 449:cross section that is energy-dependent 7630:Scattering from a 3D spherical mirror 6643:Elastic collision of two hard spheres 5058:Then the cross sections are given by 370:of the target particles (SI unit: m). 115:. Cross section is typically denoted 7: 7041:Scattering from a 2D circular mirror 6833:. The differential cross section is 5941:), we have a particular case of the 1902:), and the type of interaction. For 107:is a measure of probability that an 6815:, an incident particle with charge 6396:of the light intensity through the 4671:{\textstyle k=n_{\text{host}}k_{0}} 8825:BNL – National Nuclear Data Center 8612:Compendium of Chemical Terminology 8191: 8187: 8179: 8175: 8165: 8090: 8086: 8076: 7950: 7941: 7924: 7920: 7886: 7877: 7800: 7776: 7765: 7727: 7699: 7597: 7533: 7521: 7511: 7392: 7382: 7338: 7326: 7316: 7299: 7250: 7214: 7199: 7104: 7080: 6857: 6683: 6252: 6237: 6029: 5979: 5807: 5638: 5216: 5154: 4593: 4570: 4410: 2132: 2128: 2118: 1787: 1783: 1773: 1690: 1572: 1462: 1323: 1314: 1297: 1293: 1283: 1240: 1236: 1228: 1224: 1214: 1056: 1053: 1017: 1013: 1003: 904: 896: 892: 882: 835: 825: 593: 586: 533: 511: 505: 501: 459:Attenuation of a beam of particles 56:it lacks sufficient corresponding 25: 8687:Introduction to quantum mechanics 8326:(8th ed.), pp. 127–28, 7754:while the impact area element is 3577:{\displaystyle W_{\text{a}}>0} 3023:The cross section is defined by 451:and hence also with well-defined 6978:{\displaystyle \varepsilon _{0}} 4876: 4829: 4807: 4789: 4279:vector spherical harmonics (VSH) 4249: 4229: 4214: 4194: 4158: 4124: 4104: 4068: 4034: 4014: 3978: 3937: 3920: 3875: 3858: 3807: 3790: 3645: 3617: 3602: 3593: 3532: 3518: 3488: 3451: 3440: 3229: 2388: 2373: 1736: 1712: 1667: 1549: 1439: 1375:Instead of the solid angle, the 358:between collisions (SI unit: m), 41: 8343:from the original on 2021-06-04 6725:and the total cross section is 4972: 4727: 4721: 3894: 3832: 2982: 8787:. Cambridge University Press. 8761:Introduction to Quantum Theory 8754:Relativistic Quantum Mechanics 8249:Linear attenuation coefficient 8244:Luminosity (scattering theory) 7789: 7783: 7227: 7221: 7134:, and the scattering angle is 6929: 6915: 6261: 6255: 6246: 6240: 6203: 6180: 6168: 6152: 6133: 6110: 6098: 6082: 6045: 6032: 5995: 5982: 5881: 5865: 5823: 5810: 5712: 5696: 5654: 5641: 5485: 5469: 5428: 5405: 5386: 5370: 5329: 5306: 5232: 5219: 5170: 5157: 4853: 4836: 4622: 4596: 4590: 4575: 4493: 4483: 4467: 4453: 4437: 4433: 4430: 4415: 3941: 3879: 3811: 3455: 3321: 3305: 3259: 3250: 3235: 3224: 2699:is the attenuation at a given 1833: 1821: 1805: 1793: 1740: 1732: 1716: 1708: 1687: 1679: 1671: 1663: 1595: 1583: 1569: 1561: 1553: 1545: 1459: 1451: 1443: 1435: 1087: 1075: 1035: 1023: 989:in particles per unit time by 851: 839: 558:is the total cross section of 180:integrated total cross section 32:Radar scattering cross section 1: 8875:Experimental particle physics 8446:10.1088/2040-8978/15/4/044004 6422:is the absorbance at a given 3203:the intensity is going to be 483:of the beam will decrease by 241:Collision among gas particles 8820:IAEA – Nuclear Data Services 8752:J. D. Bjorken, S. D. Drell, 8685:Griffiths, David J. (2005). 8254:Mass attenuation coefficient 4318:{\displaystyle \varepsilon } 3652:{\displaystyle \mathbf {H} } 3338:impedance of the host medium 2803:Cross section and Mie theory 2068:of the colliding system are 8716:. Mill Valley, California. 8132:The total cross section is 6467:) of the reciprocal of the 3192:{\displaystyle \left=\left} 3127:{\displaystyle \left=\left} 2885:{\textstyle Q_{\text{abs}}} 2831:{\textstyle Q_{\text{ext}}} 2736:) of the reciprocal of the 2463:of total cross sections is 762:is the solid angle element 725:. For a given interaction ( 668:spherical coordinate system 8926: 8766:W. Greiner, J. Reinhardt, 8617:Scattering cross section, 7909:In spherical coordinates, 6437:is the concentration as a 2858:{\textstyle Q_{\text{sc}}} 2509:and in the measurement of 2019:probability of interaction 1356:may be referred to as the 658:Differential cross section 462: 172:differential cross section 29: 8689:. Pearson Prentice Hall. 7061:be the angle between the 6530:Relation to physical size 6299:geometrical cross section 4678:. These are connected as 3659:. Thus, we can decompose 2545:geometrical cross section 644:is the initial flux, and 8815:Scattering Cross Section 8712:Taylor, John R. (2005). 8234:Cross section (geometry) 6455:of the radiation is the 2056:Relation to the S-matrix 1402:scattering, the initial 1146:The total cross section 105:Rutherford cross-section 18:Scattering cross section 8768:Quantum Electrodynamics 8633:10.1351/goldbook.S05490 8269:Gamma ray cross section 7002:{\displaystyle \theta } 235:plasmonic nanoparticles 71:more precise citations. 8372:Bajpai, P. K. (2008). 8217: 8123: 8053: 7964: 7900: 7745: 7620: 7446: 7352: 7169: 7118: 7031: 7003: 6979: 6949: 6784: 6716: 6607: 6520: 6383: 6274: 6224: 5935: 5900: 5731: 5562: 5499: 5239: 5052: 4884: 4815: 4768: 4672: 4629: 4574: 4500: 4414: 4339: 4319: 4299: 4268: 4143: 4053: 3960: 3746: 3680: 3653: 3631: 3578: 3545: 3471: 3397: 3377: 3357: 3330: 3266: 3193: 3128: 3074: 3017: 2929: 2886: 2859: 2832: 2789: 2681: 2629: 2424: 2253: 2046: 1857: 1750: 1628: 1493: 1358:integral cross section 1337: 1110: 915: 705: 628: 543: 443:a "gas" of low-energy 426: 324: 278: 8810:Nuclear Cross Section 8781:R. C. Fernow (1989). 8481:Nature Communications 8289:Rutherford scattering 8274:Partial wave analysis 8264:Nuclear cross section 8259:Neutron cross section 8218: 8124: 8054: 7965: 7901: 7746: 7621: 7447: 7353: 7170: 7119: 7032: 7004: 6980: 6950: 6813:Rutherford scattering 6807:Rutherford scattering 6785: 6717: 6625:is sometimes denoted 6608: 6521: 6384: 6275: 6225: 5936: 5901: 5732: 5563: 5500: 5240: 5053: 4885: 4816: 4769: 4673: 4630: 4554: 4501: 4394: 4340: 4320: 4300: 4269: 4144: 4054: 3961: 3747: 3686:into the three terms 3681: 3679:{\displaystyle W_{a}} 3654: 3632: 3579: 3546: 3472: 3398: 3378: 3358: 3331: 3267: 3194: 3129: 3075: 3018: 2930: 2887: 2860: 2833: 2790: 2682: 2630: 2511:atmospheric pollution 2425: 2268:matrix is defined by 2254: 2047: 1858: 1751: 1629: 1494: 1370:Rutherford scattering 1338: 1111: 916: 704: 629: 544: 427: 325: 248: 8870:Dimensional analysis 8777:. McGraw Hill, 1966. 8646:, pp. 564, 574. 8577:10.1364/oe.18.011428 8294:Scattering amplitude 8139: 8069: 7980: 7916: 7761: 7662: 7480: 7375: 7191: 7141: 7076: 7013: 6993: 6962: 6840: 6732: 6666: 6562: 6545:meteorological range 6539:Meteorological range 6481: 6451:. The extinction or 6331: 6234: 5948: 5912: 5741: 5572: 5509: 5248: 5062: 4894: 4825: 4785: 4682: 4639: 4509: 4349: 4338:{\displaystyle \mu } 4329: 4309: 4289: 4153: 4063: 3973: 3759: 3690: 3663: 3641: 3589: 3555: 3484: 3410: 3387: 3367: 3347: 3276: 3207: 3138: 3087: 3030: 2939: 2896: 2869: 2842: 2815: 2750: 2649: 2577: 2445:Dirac delta function 2275: 2111: 1916: 1766: 1657: 1645:scattering amplitude 1532: 1422: 1188: 996: 818: 583: 494: 455:between collisions. 387: 293: 259:, the cross section 8865:Physical quantities 8714:Classical mechanics 8568:2010OExpr..1811428N 8503:2014NatCo...5.3300B 8438:2013JOpt...15d4004B 8284:Radar cross-section 7560: 7506: 7037:Coulomb potential. 7030:{\displaystyle 1/r} 6987:vacuum permittivity 6308:total cross section 5758: 5589: 5557: 5539: 5296: 5278: 5110: 5092: 4325:, and permeability 4243: 4208: 4118: 4028: 2539:Scattering of light 2262:where the on-shell 1520:asymptotic behavior 1362:total cross section 1278: 1263: 877: 184:Rayleigh scattering 176:total cross section 8512:10.1038/ncomms4300 8213: 8119: 8049: 7960: 7896: 7741: 7739: 7616: 7543: 7489: 7442: 7348: 7165: 7114: 7027: 6999: 6975: 6945: 6780: 6712: 6603: 6516: 6463:or, more usually, 6379: 6320:, scattering, and 6270: 6220: 5931: 5896: 5744: 5727: 5575: 5558: 5543: 5525: 5495: 5282: 5264: 5235: 5096: 5078: 5048: 4880: 4811: 4764: 4668: 4625: 4496: 4335: 4315: 4295: 4264: 4227: 4192: 4139: 4102: 4049: 4012: 3956: 3742: 3676: 3649: 3627: 3574: 3541: 3467: 3393: 3373: 3353: 3326: 3262: 3189: 3124: 3070: 3013: 2925: 2882: 2855: 2828: 2785: 2732:or, more usually, 2677: 2625: 2516:The scattering of 2420: 2249: 2042: 2040: 1853: 1746: 1624: 1489: 1390:Quantum scattering 1333: 1264: 1246: 1106: 911: 860: 706: 624: 562:events, including 539: 422: 320: 279: 182:. For example, in 131:stochastic process 8895:Scattering theory 8794:978-0-521-379-403 8416:Journal of Optics 8279:Particle detector 8183: 8114: 8094: 8040: 8016: 7869: 7845: 7825: 7589: 7569: 7529: 7433: 7413: 7400: 7334: 7291: 7271: 6933: 6861: 6742: 6707: 6687: 6598: 6594: 6572: 6373: 6360: 6347: 6213: 6143: 6071: 6027: 5977: 5922: 5862: 5788: 5756: 5751: 5693: 5619: 5587: 5582: 5555: 5550: 5537: 5532: 5519: 5456: 5438: 5357: 5339: 5294: 5289: 5276: 5271: 5258: 5202: 5140: 5108: 5103: 5090: 5085: 5072: 5033: 4957: 4761: 4748: 4735: 4725: 4718: 4705: 4692: 4655: 4552: 4519: 4392: 4359: 4298:{\displaystyle a} 4179: 4165: 4131: 4111: 4089: 4075: 4041: 4021: 3999: 3985: 3944: 3927: 3902: 3882: 3865: 3840: 3814: 3797: 3769: 3739: 3726: 3713: 3700: 3624: 3609: 3565: 3503: 3458: 3420: 3396:{\displaystyle A} 3376:{\displaystyle A} 3356:{\displaystyle r} 3324: 3217: 3183: 3175: 3170: 3152: 3118: 3068: 3065: 3008: 3000: 2992: 2977: 2974: 2906: 2879: 2865:, and Absorption 2852: 2825: 2619: 2606: 2593: 2568:photoluminescence 2551:of light and the 2449:scattering theory 2416: 2411: 2395: 2380: 2355: 2342: 2311: 2306: 2291: 2286: 2228: 2223: 2206: 2203: 2193: 2181: 2171: 2136: 2033: 2020: 2005: 1995: 1992: 1982: 1947: 1930: 1791: 1694: 1619: 1576: 1466: 1377:momentum transfer 1301: 1232: 1101: 1098: 1072: 1060: 1021: 900: 855: 519: 315: 97: 96: 89: 16:(Redirected from 8917: 8890:Particle physics 8798: 8749: 8743: 8735: 8708: 8671: 8665: 8659: 8653: 8647: 8641: 8635: 8625: 8604: 8598: 8597: 8579: 8547: 8541: 8540: 8514: 8496: 8472: 8466: 8465: 8431: 8411: 8405: 8402: 8396: 8395: 8369: 8360: 8357: 8351: 8350: 8349: 8348: 8342: 8325: 8311: 8222: 8220: 8219: 8214: 8209: 8208: 8190: 8184: 8182: 8178: 8172: 8168: 8162: 8160: 8159: 8128: 8126: 8125: 8120: 8115: 8110: 8109: 8100: 8095: 8093: 8089: 8083: 8079: 8073: 8058: 8056: 8055: 8050: 8045: 8041: 8033: 8021: 8017: 8009: 7969: 7967: 7966: 7961: 7953: 7944: 7923: 7905: 7903: 7902: 7897: 7889: 7880: 7874: 7870: 7862: 7850: 7846: 7838: 7826: 7821: 7820: 7811: 7803: 7779: 7768: 7750: 7748: 7747: 7742: 7740: 7730: 7702: 7654: 7648: 7639: 7625: 7623: 7622: 7617: 7600: 7594: 7590: 7582: 7570: 7562: 7559: 7551: 7536: 7530: 7528: 7524: 7518: 7514: 7508: 7505: 7497: 7468: 7461: 7451: 7449: 7448: 7443: 7438: 7434: 7426: 7414: 7406: 7401: 7399: 7395: 7389: 7385: 7379: 7367: 7357: 7355: 7354: 7349: 7341: 7335: 7333: 7329: 7323: 7319: 7313: 7302: 7296: 7292: 7284: 7272: 7264: 7253: 7217: 7202: 7183: 7174: 7172: 7171: 7166: 7133: 7123: 7121: 7120: 7115: 7107: 7083: 7060: 7053:geometric optics 7050: 7036: 7034: 7033: 7028: 7023: 7008: 7006: 7005: 7000: 6984: 6982: 6981: 6976: 6974: 6973: 6954: 6952: 6951: 6946: 6944: 6943: 6938: 6934: 6932: 6925: 6911: 6910: 6898: 6897: 6881: 6872: 6862: 6860: 6852: 6844: 6832: 6826: 6820: 6802: 6789: 6787: 6786: 6781: 6776: 6775: 6770: 6766: 6744: 6743: 6740: 6721: 6719: 6718: 6713: 6708: 6703: 6702: 6693: 6688: 6686: 6678: 6670: 6658: 6652: 6633: 6624: 6612: 6610: 6609: 6604: 6599: 6597: 6596: 6595: 6592: 6579: 6574: 6573: 6570: 6554: 6525: 6523: 6522: 6517: 6512: 6511: 6493: 6492: 6473: 6446: 6436: 6430: 6421: 6412: 6398:Beer–Lambert law 6388: 6386: 6385: 6380: 6375: 6374: 6371: 6362: 6361: 6358: 6349: 6348: 6345: 6315: 6296: 6279: 6277: 6276: 6271: 6229: 6227: 6226: 6221: 6219: 6215: 6214: 6212: 6211: 6210: 6201: 6200: 6178: 6177: 6176: 6171: 6165: 6164: 6155: 6149: 6144: 6142: 6141: 6140: 6131: 6130: 6108: 6107: 6106: 6101: 6095: 6094: 6085: 6079: 6072: 6067: 6066: 6065: 6052: 6044: 6043: 6028: 6026: 6025: 6024: 6002: 5994: 5993: 5978: 5976: 5975: 5974: 5952: 5940: 5938: 5937: 5932: 5924: 5923: 5920: 5905: 5903: 5902: 5897: 5895: 5891: 5890: 5889: 5884: 5878: 5877: 5868: 5863: 5861: 5860: 5859: 5840: 5839: 5830: 5822: 5821: 5789: 5787: 5786: 5785: 5763: 5757: 5754: 5752: 5749: 5736: 5734: 5733: 5728: 5726: 5722: 5721: 5720: 5715: 5709: 5708: 5699: 5694: 5692: 5691: 5690: 5671: 5670: 5661: 5653: 5652: 5620: 5618: 5617: 5616: 5594: 5588: 5585: 5583: 5580: 5567: 5565: 5564: 5559: 5556: 5553: 5551: 5548: 5538: 5535: 5533: 5530: 5521: 5520: 5517: 5504: 5502: 5501: 5496: 5494: 5493: 5488: 5482: 5481: 5472: 5467: 5466: 5457: 5452: 5444: 5439: 5437: 5436: 5435: 5426: 5425: 5400: 5395: 5394: 5389: 5383: 5382: 5373: 5368: 5367: 5358: 5353: 5345: 5340: 5338: 5337: 5336: 5327: 5326: 5301: 5295: 5292: 5290: 5287: 5277: 5274: 5272: 5269: 5260: 5259: 5256: 5244: 5242: 5241: 5236: 5231: 5230: 5203: 5201: 5200: 5199: 5177: 5169: 5168: 5141: 5139: 5138: 5137: 5115: 5109: 5106: 5104: 5101: 5091: 5088: 5086: 5083: 5074: 5073: 5070: 5057: 5055: 5054: 5049: 5044: 5043: 5034: 5032: 5031: 5030: 5017: 5009: 5001: 5000: 4982: 4981: 4968: 4967: 4958: 4956: 4955: 4954: 4941: 4933: 4925: 4924: 4906: 4905: 4889: 4887: 4886: 4881: 4879: 4874: 4873: 4864: 4863: 4851: 4850: 4832: 4820: 4818: 4817: 4812: 4810: 4805: 4804: 4792: 4773: 4771: 4770: 4765: 4763: 4762: 4759: 4750: 4749: 4746: 4737: 4736: 4733: 4726: 4723: 4720: 4719: 4716: 4707: 4706: 4703: 4694: 4693: 4690: 4677: 4675: 4674: 4669: 4667: 4666: 4657: 4656: 4653: 4634: 4632: 4631: 4626: 4621: 4620: 4608: 4607: 4573: 4568: 4553: 4551: 4550: 4549: 4540: 4539: 4526: 4521: 4520: 4517: 4505: 4503: 4502: 4497: 4492: 4491: 4486: 4480: 4479: 4470: 4462: 4461: 4456: 4450: 4449: 4440: 4413: 4408: 4393: 4391: 4390: 4389: 4380: 4379: 4366: 4361: 4360: 4357: 4344: 4342: 4341: 4336: 4324: 4322: 4321: 4316: 4304: 4302: 4301: 4296: 4273: 4271: 4270: 4265: 4263: 4259: 4258: 4257: 4252: 4242: 4237: 4232: 4223: 4222: 4217: 4207: 4202: 4197: 4180: 4172: 4167: 4166: 4163: 4161: 4148: 4146: 4145: 4140: 4138: 4134: 4133: 4132: 4129: 4127: 4117: 4112: 4109: 4107: 4090: 4082: 4077: 4076: 4073: 4071: 4058: 4056: 4055: 4050: 4048: 4044: 4043: 4042: 4039: 4037: 4027: 4022: 4019: 4017: 4000: 3992: 3987: 3986: 3983: 3981: 3965: 3963: 3962: 3957: 3946: 3945: 3940: 3935: 3929: 3928: 3925: 3923: 3917: 3916: 3904: 3903: 3900: 3884: 3883: 3878: 3873: 3867: 3866: 3863: 3861: 3855: 3854: 3842: 3841: 3838: 3816: 3815: 3810: 3805: 3799: 3798: 3795: 3793: 3787: 3786: 3771: 3770: 3767: 3751: 3749: 3748: 3743: 3741: 3740: 3737: 3728: 3727: 3724: 3715: 3714: 3711: 3702: 3701: 3698: 3685: 3683: 3682: 3677: 3675: 3674: 3658: 3656: 3655: 3650: 3648: 3636: 3634: 3633: 3628: 3626: 3625: 3622: 3620: 3611: 3610: 3607: 3605: 3596: 3583: 3581: 3580: 3575: 3567: 3566: 3563: 3550: 3548: 3547: 3542: 3540: 3536: 3535: 3527: 3526: 3521: 3504: 3496: 3491: 3476: 3474: 3473: 3468: 3460: 3459: 3454: 3449: 3443: 3438: 3437: 3422: 3421: 3418: 3402: 3400: 3399: 3394: 3382: 3380: 3379: 3374: 3362: 3360: 3359: 3354: 3335: 3333: 3332: 3327: 3325: 3320: 3319: 3304: 3299: 3298: 3286: 3271: 3269: 3268: 3263: 3249: 3244: 3243: 3238: 3232: 3227: 3219: 3218: 3215: 3198: 3196: 3195: 3190: 3188: 3184: 3182: 3181: 3176: 3173: 3168: 3167: 3158: 3154: 3153: 3150: 3133: 3131: 3130: 3125: 3123: 3119: 3116: 3107: 3103: 3102: 3079: 3077: 3076: 3071: 3069: 3067: 3066: 3063: 3057: 3056: 3047: 3042: 3041: 3022: 3020: 3019: 3014: 3009: 3006: 3001: 2998: 2993: 2990: 2978: 2976: 2975: 2972: 2966: 2965: 2956: 2951: 2950: 2934: 2932: 2931: 2926: 2924: 2923: 2908: 2907: 2904: 2891: 2889: 2888: 2883: 2881: 2880: 2877: 2864: 2862: 2861: 2856: 2854: 2853: 2850: 2837: 2835: 2834: 2829: 2827: 2826: 2823: 2794: 2792: 2791: 2786: 2781: 2780: 2762: 2761: 2742: 2719: 2713: 2707: 2698: 2686: 2684: 2683: 2678: 2661: 2660: 2640:Beer–Lambert law 2634: 2632: 2631: 2626: 2621: 2620: 2617: 2608: 2607: 2604: 2595: 2594: 2591: 2534: 2529: 2442: 2433:in terms of the 2429: 2427: 2426: 2421: 2419: 2418: 2417: 2414: 2412: 2409: 2402: 2398: 2397: 2396: 2393: 2391: 2382: 2381: 2378: 2376: 2362: 2358: 2357: 2356: 2353: 2344: 2343: 2340: 2314: 2313: 2312: 2309: 2307: 2304: 2294: 2293: 2292: 2289: 2287: 2284: 2267: 2258: 2256: 2255: 2250: 2245: 2244: 2239: 2238: 2231: 2230: 2229: 2226: 2224: 2221: 2214: 2213: 2207: 2205: 2204: 2201: 2195: 2194: 2191: 2185: 2183: 2182: 2179: 2173: 2172: 2169: 2163: 2162: 2157: 2153: 2137: 2135: 2131: 2125: 2121: 2115: 2103: 2094: 2085: 2076: 2051: 2049: 2048: 2043: 2041: 2034: 2026: 2021: 2018: 2010: 2006: 1998: 1996: 1994: 1993: 1990: 1984: 1983: 1980: 1974: 1949: 1948: 1945: 1932: 1931: 1928: 1908: 1901: 1895: 1886: 1862: 1860: 1859: 1854: 1849: 1848: 1843: 1842: 1817: 1816: 1792: 1790: 1786: 1780: 1776: 1770: 1755: 1753: 1752: 1747: 1739: 1731: 1730: 1715: 1707: 1706: 1696: 1695: 1693: 1682: 1677: 1670: 1642: 1633: 1631: 1630: 1625: 1620: 1615: 1614: 1599: 1578: 1577: 1575: 1564: 1559: 1552: 1544: 1543: 1513: 1507: 1498: 1496: 1495: 1490: 1485: 1484: 1468: 1467: 1465: 1454: 1449: 1442: 1434: 1433: 1414: 1396:time-independent 1355: 1342: 1340: 1339: 1334: 1326: 1317: 1302: 1300: 1296: 1290: 1286: 1280: 1277: 1272: 1262: 1254: 1239: 1233: 1231: 1227: 1221: 1217: 1211: 1209: 1208: 1180: 1172: 1171: 1169: 1168: 1165: 1162: 1151: 1142: 1132: 1122: 1115: 1113: 1112: 1107: 1102: 1100: 1099: 1096: 1090: 1074: 1073: 1070: 1063: 1061: 1059: 1042: 1022: 1020: 1016: 1010: 1006: 1000: 988: 970: 958: 957: 955: 954: 951: 948: 937: 920: 918: 917: 912: 907: 901: 899: 895: 889: 885: 879: 876: 868: 856: 854: 838: 832: 828: 822: 810: 797: 796: 794: 793: 790: 787: 776: 761: 755: 724: 718: 712:impact parameter 691: 684:scattering angle 681: 675: 652:Beer–Lambert law 649: 643: 633: 631: 630: 625: 620: 619: 601: 600: 575: 557: 548: 546: 545: 540: 520: 518: 514: 508: 504: 498: 486: 482: 475: 431: 429: 428: 423: 421: 420: 415: 411: 379: 365: 353: 341: 329: 327: 326: 321: 316: 314: 303: 276: 270: 264: 258: 223:particle physics 199:particle physics 157:electromagnetism 120: 99:In physics, the 92: 85: 81: 78: 72: 67:this article by 58:inline citations 45: 44: 37: 21: 8925: 8924: 8920: 8919: 8918: 8916: 8915: 8914: 8885:Nuclear physics 8850: 8849: 8806: 8801: 8795: 8780: 8736: 8724: 8711: 8697: 8684: 8680: 8675: 8674: 8666: 8662: 8654: 8650: 8642: 8638: 8624: 8618: 8605: 8601: 8549: 8548: 8544: 8474: 8473: 8469: 8413: 8412: 8408: 8403: 8399: 8384: 8371: 8370: 8363: 8358: 8354: 8346: 8344: 8340: 8334: 8323: 8313: 8312: 8308: 8303: 8298: 8229: 8200: 8173: 8163: 8148: 8137: 8136: 8101: 8084: 8074: 8067: 8066: 8028: 8004: 7978: 7977: 7914: 7913: 7857: 7833: 7812: 7759: 7758: 7738: 7737: 7706: 7695: 7694: 7672: 7660: 7659: 7650: 7644: 7635: 7632: 7577: 7519: 7509: 7478: 7477: 7463: 7456: 7455:Its maximum at 7421: 7390: 7380: 7373: 7372: 7362: 7324: 7314: 7279: 7189: 7188: 7179: 7139: 7138: 7128: 7074: 7073: 7056: 7046: 7043: 7011: 7010: 6991: 6990: 6965: 6960: 6959: 6902: 6889: 6882: 6873: 6867: 6866: 6853: 6845: 6838: 6837: 6828: 6822: 6816: 6809: 6794: 6756: 6752: 6751: 6735: 6730: 6729: 6694: 6679: 6671: 6664: 6663: 6654: 6648: 6645: 6640: 6632: 6626: 6623: 6617: 6587: 6583: 6565: 6560: 6559: 6553: 6547: 6541: 6532: 6484: 6479: 6478: 6471: 6442: 6432: 6426: 6419: 6414: 6406: 6401: 6366: 6353: 6340: 6329: 6328: 6311: 6295: 6289: 6286: 6232: 6231: 6202: 6192: 6179: 6166: 6156: 6150: 6132: 6122: 6109: 6096: 6086: 6080: 6077: 6073: 6057: 6053: 6035: 6016: 6006: 5985: 5966: 5956: 5946: 5945: 5943:Optical theorem 5915: 5910: 5909: 5879: 5869: 5851: 5841: 5831: 5813: 5806: 5802: 5777: 5767: 5739: 5738: 5710: 5700: 5682: 5672: 5662: 5644: 5637: 5633: 5608: 5598: 5570: 5569: 5512: 5507: 5506: 5483: 5473: 5458: 5445: 5427: 5417: 5404: 5384: 5374: 5359: 5346: 5328: 5318: 5305: 5251: 5246: 5245: 5222: 5191: 5181: 5160: 5129: 5119: 5065: 5060: 5059: 5035: 5022: 5018: 5010: 4992: 4973: 4959: 4946: 4942: 4934: 4916: 4897: 4892: 4891: 4865: 4852: 4842: 4823: 4822: 4796: 4783: 4782: 4779: 4754: 4741: 4728: 4711: 4698: 4685: 4680: 4679: 4658: 4648: 4637: 4636: 4612: 4599: 4541: 4531: 4530: 4512: 4507: 4506: 4481: 4471: 4451: 4441: 4381: 4371: 4370: 4352: 4347: 4346: 4327: 4326: 4307: 4306: 4305:, permittivity 4287: 4286: 4247: 4212: 4191: 4187: 4156: 4151: 4150: 4122: 4101: 4097: 4066: 4061: 4060: 4032: 4011: 4007: 3976: 3971: 3970: 3918: 3908: 3895: 3856: 3846: 3833: 3788: 3778: 3762: 3757: 3756: 3732: 3719: 3706: 3693: 3688: 3687: 3666: 3661: 3660: 3639: 3638: 3615: 3600: 3587: 3586: 3558: 3553: 3552: 3516: 3515: 3511: 3482: 3481: 3429: 3413: 3408: 3407: 3385: 3384: 3365: 3364: 3345: 3344: 3311: 3290: 3274: 3273: 3233: 3210: 3205: 3204: 3171: 3162: 3145: 3141: 3136: 3135: 3111: 3094: 3090: 3085: 3084: 3058: 3048: 3033: 3028: 3027: 2967: 2957: 2942: 2937: 2936: 2915: 2899: 2894: 2893: 2872: 2867: 2866: 2845: 2840: 2839: 2818: 2813: 2812: 2805: 2753: 2748: 2747: 2740: 2715: 2709: 2703: 2696: 2691: 2652: 2647: 2646: 2612: 2599: 2586: 2575: 2574: 2541: 2527: 2525: 2457: 2438: 2403: 2386: 2371: 2370: 2366: 2348: 2335: 2334: 2330: 2298: 2278: 2273: 2272: 2263: 2232: 2215: 2196: 2186: 2174: 2164: 2146: 2142: 2141: 2126: 2116: 2109: 2108: 2102: 2096: 2093: 2087: 2084: 2078: 2075: 2069: 2058: 2039: 2038: 2008: 2007: 1985: 1975: 1966: 1960: 1959: 1940: 1933: 1923: 1914: 1913: 1903: 1897: 1894: 1888: 1885: 1879: 1836: 1781: 1771: 1764: 1763: 1722: 1698: 1655: 1654: 1638: 1600: 1535: 1530: 1529: 1509: 1503: 1470: 1425: 1420: 1419: 1410: 1392: 1384:resonance peaks 1351: 1291: 1281: 1222: 1212: 1197: 1186: 1185: 1178: 1166: 1163: 1157: 1156: 1154: 1153: 1147: 1138: 1128: 1120: 1091: 1065: 1064: 1046: 1011: 1001: 994: 993: 978: 972: 960: 952: 949: 943: 942: 940: 939: 936: 930: 890: 880: 833: 823: 816: 815: 806: 804:azimuthal angle 791: 788: 782: 781: 779: 778: 763: 757: 738: 720: 714: 694:azimuthal angle 687: 677: 671: 660: 645: 642: 638: 602: 592: 581: 580: 571: 553: 509: 499: 492: 491: 484: 480: 470: 467: 461: 441:nuclear physics 404: 400: 399: 385: 384: 375: 361: 349: 337: 307: 291: 290: 272: 266: 260: 253: 243: 116: 93: 82: 76: 73: 63:Please help to 62: 46: 42: 35: 28: 23: 22: 15: 12: 11: 5: 8923: 8921: 8913: 8912: 8907: 8902: 8897: 8892: 8887: 8882: 8877: 8872: 8867: 8862: 8860:Atomic physics 8852: 8851: 8848: 8847: 8842: 8837: 8832: 8827: 8822: 8817: 8812: 8805: 8804:External links 8802: 8800: 8799: 8793: 8778: 8773:R. G. Newton. 8771: 8764: 8757: 8750: 8722: 8709: 8695: 8681: 8679: 8676: 8673: 8672: 8670:, p. 409. 8668:Griffiths 2005 8660: 8658:, p. 576. 8648: 8636: 8622: 8599: 8556:Optics Express 8542: 8467: 8406: 8397: 8382: 8361: 8352: 8332: 8305: 8304: 8302: 8299: 8297: 8296: 8291: 8286: 8281: 8276: 8271: 8266: 8261: 8256: 8251: 8246: 8241: 8236: 8230: 8228: 8225: 8224: 8223: 8212: 8207: 8203: 8199: 8196: 8193: 8189: 8181: 8177: 8171: 8167: 8158: 8155: 8151: 8147: 8144: 8130: 8129: 8118: 8113: 8108: 8104: 8098: 8092: 8088: 8082: 8078: 8060: 8059: 8048: 8044: 8039: 8036: 8031: 8027: 8024: 8020: 8015: 8012: 8007: 8003: 8000: 7997: 7994: 7991: 7988: 7985: 7971: 7970: 7959: 7956: 7952: 7947: 7943: 7938: 7935: 7932: 7929: 7926: 7922: 7907: 7906: 7895: 7892: 7888: 7883: 7879: 7873: 7868: 7865: 7860: 7856: 7853: 7849: 7844: 7841: 7836: 7832: 7829: 7824: 7819: 7815: 7809: 7806: 7802: 7797: 7794: 7791: 7788: 7785: 7782: 7778: 7774: 7771: 7767: 7752: 7751: 7736: 7733: 7729: 7724: 7721: 7718: 7715: 7712: 7709: 7707: 7705: 7701: 7697: 7696: 7693: 7690: 7687: 7684: 7681: 7678: 7675: 7673: 7671: 7668: 7667: 7631: 7628: 7627: 7626: 7615: 7612: 7609: 7606: 7603: 7599: 7593: 7588: 7585: 7580: 7576: 7573: 7568: 7565: 7558: 7555: 7550: 7546: 7542: 7539: 7535: 7527: 7523: 7517: 7513: 7504: 7501: 7496: 7492: 7488: 7485: 7453: 7452: 7441: 7437: 7432: 7429: 7424: 7420: 7417: 7412: 7409: 7404: 7398: 7394: 7388: 7384: 7359: 7358: 7347: 7344: 7340: 7332: 7328: 7322: 7318: 7311: 7308: 7305: 7301: 7295: 7290: 7287: 7282: 7278: 7275: 7270: 7267: 7262: 7259: 7256: 7252: 7247: 7244: 7241: 7238: 7235: 7232: 7229: 7226: 7223: 7220: 7216: 7211: 7208: 7205: 7201: 7196: 7176: 7175: 7164: 7161: 7158: 7155: 7152: 7149: 7146: 7125: 7124: 7113: 7110: 7106: 7101: 7098: 7095: 7092: 7089: 7086: 7082: 7042: 7039: 7026: 7022: 7018: 6998: 6972: 6968: 6956: 6955: 6942: 6937: 6931: 6928: 6924: 6920: 6917: 6914: 6909: 6905: 6901: 6896: 6892: 6888: 6885: 6880: 6876: 6870: 6865: 6859: 6856: 6851: 6848: 6808: 6805: 6791: 6790: 6779: 6774: 6769: 6765: 6762: 6759: 6755: 6750: 6747: 6738: 6723: 6722: 6711: 6706: 6701: 6697: 6691: 6685: 6682: 6677: 6674: 6644: 6641: 6639: 6636: 6630: 6621: 6614: 6613: 6602: 6590: 6586: 6582: 6577: 6568: 6551: 6540: 6537: 6531: 6528: 6527: 6526: 6515: 6510: 6505: 6502: 6499: 6496: 6491: 6487: 6439:number density 6417: 6404: 6390: 6389: 6378: 6369: 6365: 6356: 6352: 6343: 6339: 6336: 6293: 6285: 6282: 6269: 6266: 6263: 6260: 6257: 6254: 6251: 6248: 6245: 6242: 6239: 6218: 6209: 6205: 6199: 6195: 6191: 6188: 6185: 6182: 6175: 6170: 6163: 6159: 6154: 6147: 6139: 6135: 6129: 6125: 6121: 6118: 6115: 6112: 6105: 6100: 6093: 6089: 6084: 6076: 6070: 6064: 6060: 6056: 6050: 6047: 6042: 6038: 6034: 6031: 6023: 6019: 6015: 6012: 6009: 6005: 6000: 5997: 5992: 5988: 5984: 5981: 5973: 5969: 5965: 5962: 5959: 5955: 5930: 5927: 5918: 5894: 5888: 5883: 5876: 5872: 5867: 5858: 5854: 5850: 5847: 5844: 5838: 5834: 5828: 5825: 5820: 5816: 5812: 5809: 5805: 5801: 5798: 5795: 5792: 5784: 5780: 5776: 5773: 5770: 5766: 5761: 5747: 5725: 5719: 5714: 5707: 5703: 5698: 5689: 5685: 5681: 5678: 5675: 5669: 5665: 5659: 5656: 5651: 5647: 5643: 5640: 5636: 5632: 5629: 5626: 5623: 5615: 5611: 5607: 5604: 5601: 5597: 5592: 5578: 5546: 5542: 5528: 5524: 5515: 5492: 5487: 5480: 5476: 5471: 5465: 5461: 5455: 5451: 5448: 5442: 5434: 5430: 5424: 5420: 5416: 5413: 5410: 5407: 5403: 5398: 5393: 5388: 5381: 5377: 5372: 5366: 5362: 5356: 5352: 5349: 5343: 5335: 5331: 5325: 5321: 5317: 5314: 5311: 5308: 5304: 5299: 5285: 5281: 5267: 5263: 5254: 5234: 5229: 5225: 5221: 5218: 5215: 5212: 5209: 5206: 5198: 5194: 5190: 5187: 5184: 5180: 5175: 5172: 5167: 5163: 5159: 5156: 5153: 5150: 5147: 5144: 5136: 5132: 5128: 5125: 5122: 5118: 5113: 5099: 5095: 5081: 5077: 5068: 5047: 5042: 5038: 5029: 5025: 5021: 5016: 5013: 5007: 5004: 4999: 4995: 4991: 4988: 4985: 4980: 4976: 4971: 4966: 4962: 4953: 4949: 4945: 4940: 4937: 4931: 4928: 4923: 4919: 4915: 4912: 4909: 4904: 4900: 4878: 4872: 4868: 4862: 4859: 4855: 4849: 4845: 4841: 4838: 4835: 4831: 4809: 4803: 4799: 4795: 4791: 4778: 4775: 4757: 4753: 4744: 4740: 4731: 4714: 4710: 4701: 4697: 4688: 4665: 4661: 4651: 4647: 4644: 4624: 4619: 4615: 4611: 4606: 4602: 4598: 4595: 4592: 4589: 4586: 4583: 4580: 4577: 4572: 4567: 4564: 4561: 4557: 4548: 4544: 4538: 4534: 4529: 4524: 4515: 4495: 4490: 4485: 4478: 4474: 4469: 4465: 4460: 4455: 4448: 4444: 4439: 4435: 4432: 4429: 4426: 4423: 4420: 4417: 4412: 4407: 4404: 4401: 4397: 4388: 4384: 4378: 4374: 4369: 4364: 4355: 4334: 4314: 4294: 4283:uniform sphere 4262: 4256: 4251: 4246: 4241: 4236: 4231: 4226: 4221: 4216: 4211: 4206: 4201: 4196: 4190: 4186: 4183: 4178: 4175: 4170: 4160: 4137: 4126: 4121: 4116: 4106: 4100: 4096: 4093: 4088: 4085: 4080: 4070: 4047: 4036: 4031: 4026: 4016: 4010: 4006: 4003: 3998: 3995: 3990: 3980: 3967: 3966: 3955: 3952: 3949: 3943: 3939: 3932: 3922: 3915: 3911: 3907: 3898: 3893: 3890: 3887: 3881: 3877: 3870: 3860: 3853: 3849: 3845: 3836: 3831: 3828: 3825: 3822: 3819: 3813: 3809: 3802: 3792: 3785: 3781: 3777: 3774: 3765: 3735: 3731: 3722: 3718: 3709: 3705: 3696: 3673: 3669: 3647: 3619: 3614: 3604: 3599: 3595: 3573: 3570: 3561: 3539: 3534: 3530: 3525: 3520: 3514: 3510: 3507: 3502: 3499: 3494: 3490: 3478: 3477: 3466: 3463: 3457: 3453: 3446: 3442: 3436: 3432: 3428: 3425: 3416: 3392: 3372: 3352: 3323: 3318: 3314: 3310: 3307: 3303: 3297: 3293: 3289: 3284: 3281: 3261: 3258: 3255: 3252: 3248: 3242: 3237: 3231: 3226: 3222: 3213: 3187: 3180: 3165: 3161: 3157: 3148: 3144: 3122: 3114: 3110: 3106: 3101: 3097: 3093: 3081: 3080: 3061: 3055: 3051: 3045: 3040: 3036: 3012: 3004: 2996: 2988: 2985: 2981: 2970: 2964: 2960: 2954: 2949: 2945: 2922: 2918: 2914: 2911: 2902: 2875: 2848: 2821: 2804: 2801: 2796: 2795: 2784: 2779: 2774: 2771: 2768: 2765: 2760: 2756: 2694: 2688: 2687: 2676: 2673: 2670: 2667: 2664: 2659: 2655: 2636: 2635: 2624: 2615: 2611: 2602: 2598: 2589: 2585: 2582: 2540: 2537: 2482:units such as 2456: 2453: 2431: 2430: 2406: 2401: 2390: 2385: 2375: 2369: 2365: 2361: 2351: 2347: 2338: 2333: 2329: 2326: 2323: 2320: 2317: 2301: 2297: 2281: 2260: 2259: 2248: 2243: 2237: 2218: 2212: 2199: 2189: 2177: 2167: 2161: 2156: 2152: 2149: 2145: 2140: 2134: 2130: 2124: 2120: 2100: 2091: 2082: 2073: 2062:reduced masses 2057: 2054: 2053: 2052: 2037: 2032: 2029: 2024: 2016: 2013: 2011: 2009: 2004: 2001: 1988: 1978: 1972: 1969: 1967: 1965: 1962: 1961: 1958: 1955: 1952: 1943: 1939: 1936: 1934: 1926: 1922: 1921: 1892: 1883: 1864: 1863: 1852: 1847: 1841: 1835: 1832: 1829: 1826: 1823: 1820: 1815: 1810: 1807: 1804: 1801: 1798: 1795: 1789: 1785: 1779: 1775: 1757: 1756: 1745: 1742: 1738: 1734: 1729: 1725: 1721: 1718: 1714: 1710: 1705: 1701: 1692: 1689: 1686: 1681: 1673: 1669: 1665: 1662: 1635: 1634: 1623: 1618: 1613: 1610: 1607: 1603: 1597: 1594: 1591: 1588: 1585: 1582: 1574: 1571: 1568: 1563: 1555: 1551: 1547: 1542: 1538: 1500: 1499: 1488: 1483: 1480: 1477: 1473: 1464: 1461: 1458: 1453: 1445: 1441: 1437: 1432: 1428: 1391: 1388: 1344: 1343: 1332: 1329: 1325: 1320: 1316: 1311: 1308: 1305: 1299: 1295: 1289: 1285: 1276: 1271: 1267: 1261: 1258: 1253: 1249: 1245: 1242: 1238: 1230: 1226: 1220: 1216: 1207: 1204: 1200: 1196: 1193: 1173:over the full 1135:number density 1117: 1116: 1105: 1094: 1089: 1086: 1083: 1080: 1077: 1068: 1058: 1055: 1052: 1049: 1045: 1040: 1037: 1034: 1031: 1028: 1025: 1019: 1015: 1009: 1005: 976: 934: 923: 922: 910: 906: 898: 894: 888: 884: 875: 872: 867: 863: 859: 853: 850: 847: 844: 841: 837: 831: 827: 708: 707: 659: 656: 640: 635: 634: 623: 618: 615: 612: 609: 605: 599: 595: 591: 588: 550: 549: 538: 535: 532: 529: 526: 523: 517: 513: 507: 503: 460: 457: 453:mean free path 433: 432: 419: 414: 410: 407: 403: 398: 395: 392: 372: 371: 368:number density 359: 356:mean free path 347: 331: 330: 319: 313: 310: 306: 301: 298: 242: 239: 113:atomic nucleus 109:alpha particle 95: 94: 49: 47: 40: 26: 24: 14: 13: 10: 9: 6: 4: 3: 2: 8922: 8911: 8908: 8906: 8903: 8901: 8898: 8896: 8893: 8891: 8888: 8886: 8883: 8881: 8878: 8876: 8873: 8871: 8868: 8866: 8863: 8861: 8858: 8857: 8855: 8846: 8843: 8841: 8838: 8836: 8833: 8831: 8828: 8826: 8823: 8821: 8818: 8816: 8813: 8811: 8808: 8807: 8803: 8796: 8790: 8786: 8785: 8779: 8776: 8772: 8769: 8765: 8762: 8758: 8755: 8751: 8747: 8741: 8733: 8729: 8725: 8723:1-891389-22-X 8719: 8715: 8710: 8706: 8702: 8698: 8696:0-13-111892-7 8692: 8688: 8683: 8682: 8677: 8669: 8664: 8661: 8657: 8652: 8649: 8645: 8640: 8637: 8634: 8630: 8626: 8621: 8614: 8613: 8608: 8603: 8600: 8595: 8591: 8587: 8583: 8578: 8573: 8569: 8565: 8561: 8557: 8553: 8546: 8543: 8538: 8534: 8530: 8526: 8522: 8518: 8513: 8508: 8504: 8500: 8495: 8490: 8486: 8482: 8478: 8471: 8468: 8463: 8459: 8455: 8451: 8447: 8443: 8439: 8435: 8430: 8425: 8422:(4): 044004. 8421: 8417: 8410: 8407: 8401: 8398: 8393: 8389: 8385: 8383:9788121926331 8379: 8375: 8368: 8366: 8362: 8356: 8353: 8339: 8335: 8333:92-822-2213-6 8329: 8322: 8321: 8316: 8310: 8307: 8300: 8295: 8292: 8290: 8287: 8285: 8282: 8280: 8277: 8275: 8272: 8270: 8267: 8265: 8262: 8260: 8257: 8255: 8252: 8250: 8247: 8245: 8242: 8240: 8239:Flow velocity 8237: 8235: 8232: 8231: 8226: 8210: 8205: 8201: 8197: 8194: 8169: 8156: 8153: 8149: 8145: 8142: 8135: 8134: 8133: 8116: 8111: 8106: 8102: 8096: 8080: 8065: 8064: 8063: 8046: 8042: 8037: 8034: 8029: 8025: 8022: 8018: 8013: 8010: 8005: 8001: 7998: 7995: 7992: 7989: 7986: 7983: 7976: 7975: 7974: 7957: 7954: 7945: 7936: 7933: 7930: 7927: 7912: 7911: 7910: 7893: 7890: 7881: 7871: 7866: 7863: 7858: 7854: 7851: 7847: 7842: 7839: 7834: 7830: 7827: 7822: 7817: 7813: 7807: 7804: 7795: 7792: 7786: 7780: 7772: 7769: 7757: 7756: 7755: 7734: 7731: 7722: 7719: 7716: 7713: 7710: 7708: 7703: 7691: 7688: 7685: 7682: 7679: 7676: 7674: 7669: 7658: 7657: 7656: 7653: 7647: 7641: 7638: 7629: 7613: 7610: 7607: 7604: 7601: 7591: 7586: 7583: 7578: 7574: 7571: 7566: 7563: 7556: 7553: 7548: 7544: 7540: 7537: 7525: 7515: 7502: 7499: 7494: 7490: 7486: 7483: 7476: 7475: 7474: 7472: 7466: 7459: 7439: 7435: 7430: 7427: 7422: 7418: 7415: 7410: 7407: 7402: 7396: 7386: 7371: 7370: 7369: 7366: 7345: 7342: 7330: 7320: 7309: 7306: 7303: 7293: 7288: 7285: 7280: 7276: 7273: 7268: 7265: 7260: 7257: 7254: 7245: 7242: 7239: 7236: 7233: 7230: 7224: 7218: 7209: 7206: 7203: 7194: 7187: 7186: 7185: 7182: 7162: 7159: 7156: 7153: 7150: 7147: 7144: 7137: 7136: 7135: 7132: 7111: 7108: 7099: 7096: 7093: 7090: 7087: 7084: 7072: 7071: 7070: 7068: 7064: 7059: 7054: 7049: 7040: 7038: 7024: 7020: 7016: 6996: 6988: 6970: 6966: 6940: 6935: 6926: 6922: 6918: 6912: 6907: 6903: 6899: 6894: 6890: 6886: 6883: 6878: 6874: 6868: 6863: 6854: 6849: 6846: 6836: 6835: 6834: 6831: 6825: 6819: 6814: 6806: 6804: 6801: 6797: 6777: 6772: 6767: 6763: 6760: 6757: 6753: 6748: 6745: 6736: 6728: 6727: 6726: 6709: 6704: 6699: 6695: 6689: 6680: 6675: 6672: 6662: 6661: 6660: 6657: 6651: 6642: 6637: 6635: 6629: 6620: 6616:The quantity 6600: 6588: 6584: 6580: 6575: 6566: 6558: 6557: 6556: 6550: 6546: 6538: 6536: 6529: 6513: 6503: 6500: 6497: 6494: 6489: 6485: 6477: 6476: 6475: 6470: 6469:transmittance 6466: 6462: 6458: 6454: 6450: 6445: 6440: 6435: 6429: 6425: 6420: 6411: 6407: 6399: 6395: 6376: 6367: 6363: 6354: 6350: 6341: 6337: 6334: 6327: 6326: 6325: 6323: 6319: 6314: 6309: 6304: 6300: 6292: 6283: 6281: 6267: 6264: 6258: 6249: 6243: 6216: 6207: 6197: 6193: 6189: 6186: 6183: 6173: 6161: 6157: 6145: 6137: 6127: 6123: 6119: 6116: 6113: 6103: 6091: 6087: 6074: 6068: 6062: 6058: 6054: 6048: 6040: 6036: 6021: 6017: 6013: 6010: 6007: 6003: 5998: 5990: 5986: 5971: 5967: 5963: 5960: 5957: 5953: 5944: 5928: 5925: 5916: 5906: 5892: 5886: 5874: 5870: 5856: 5852: 5848: 5845: 5842: 5836: 5832: 5826: 5818: 5814: 5803: 5799: 5796: 5793: 5790: 5782: 5778: 5774: 5771: 5768: 5764: 5759: 5745: 5723: 5717: 5705: 5701: 5687: 5683: 5679: 5676: 5673: 5667: 5663: 5657: 5649: 5645: 5634: 5630: 5627: 5624: 5621: 5613: 5609: 5605: 5602: 5599: 5595: 5590: 5576: 5544: 5540: 5526: 5522: 5513: 5490: 5478: 5474: 5463: 5459: 5453: 5449: 5446: 5440: 5432: 5422: 5418: 5414: 5411: 5408: 5401: 5396: 5391: 5379: 5375: 5364: 5360: 5354: 5350: 5347: 5341: 5333: 5323: 5319: 5315: 5312: 5309: 5302: 5297: 5283: 5279: 5265: 5261: 5252: 5227: 5223: 5213: 5210: 5207: 5204: 5196: 5192: 5188: 5185: 5182: 5178: 5173: 5165: 5161: 5151: 5148: 5145: 5142: 5134: 5130: 5126: 5123: 5120: 5116: 5111: 5097: 5093: 5079: 5075: 5066: 5045: 5040: 5036: 5027: 5023: 5019: 5014: 5011: 5005: 5002: 4997: 4993: 4989: 4986: 4983: 4978: 4974: 4969: 4964: 4960: 4951: 4947: 4943: 4938: 4935: 4929: 4926: 4921: 4917: 4913: 4910: 4907: 4902: 4898: 4870: 4866: 4860: 4857: 4847: 4843: 4839: 4833: 4801: 4797: 4793: 4776: 4774: 4755: 4751: 4742: 4738: 4729: 4712: 4708: 4699: 4695: 4686: 4663: 4659: 4649: 4645: 4642: 4617: 4613: 4609: 4604: 4600: 4587: 4584: 4581: 4578: 4565: 4562: 4559: 4555: 4546: 4542: 4536: 4532: 4527: 4522: 4513: 4488: 4476: 4472: 4463: 4458: 4446: 4442: 4427: 4424: 4421: 4418: 4405: 4402: 4399: 4395: 4386: 4382: 4376: 4372: 4367: 4362: 4353: 4332: 4312: 4292: 4284: 4280: 4275: 4260: 4254: 4244: 4239: 4234: 4224: 4219: 4209: 4204: 4199: 4188: 4184: 4181: 4176: 4173: 4168: 4135: 4119: 4114: 4098: 4094: 4091: 4086: 4083: 4078: 4045: 4029: 4024: 4008: 4004: 4001: 3996: 3993: 3988: 3953: 3950: 3947: 3930: 3913: 3909: 3905: 3896: 3891: 3888: 3885: 3868: 3851: 3847: 3843: 3834: 3829: 3826: 3823: 3820: 3817: 3800: 3783: 3779: 3775: 3772: 3763: 3755: 3754: 3753: 3733: 3729: 3720: 3716: 3707: 3703: 3694: 3671: 3667: 3612: 3597: 3571: 3568: 3559: 3537: 3528: 3523: 3512: 3508: 3505: 3500: 3497: 3492: 3464: 3461: 3444: 3434: 3430: 3426: 3423: 3414: 3406: 3405: 3404: 3390: 3370: 3350: 3341: 3339: 3316: 3312: 3308: 3301: 3295: 3291: 3287: 3282: 3279: 3256: 3253: 3246: 3240: 3220: 3211: 3202: 3185: 3178: 3163: 3159: 3155: 3146: 3142: 3120: 3112: 3108: 3104: 3099: 3095: 3091: 3059: 3053: 3049: 3043: 3038: 3034: 3026: 3025: 3024: 3010: 3002: 2994: 2986: 2983: 2979: 2968: 2962: 2958: 2952: 2947: 2943: 2920: 2916: 2912: 2909: 2900: 2873: 2846: 2838:, scattering 2819: 2810: 2802: 2800: 2782: 2772: 2769: 2766: 2763: 2758: 2754: 2746: 2745: 2744: 2739: 2738:transmittance 2735: 2731: 2727: 2723: 2718: 2712: 2706: 2702: 2697: 2674: 2671: 2668: 2665: 2662: 2657: 2653: 2645: 2644: 2643: 2641: 2622: 2613: 2609: 2600: 2596: 2587: 2583: 2580: 2573: 2572: 2571: 2569: 2565: 2561: 2556: 2554: 2550: 2546: 2538: 2536: 2533: 2523: 2519: 2514: 2512: 2508: 2504: 2500: 2496: 2491: 2489: 2485: 2481: 2477: 2473: 2468: 2466: 2462: 2459:Although the 2454: 2452: 2450: 2446: 2441: 2436: 2404: 2399: 2383: 2367: 2363: 2359: 2349: 2345: 2336: 2331: 2327: 2324: 2321: 2318: 2315: 2299: 2295: 2279: 2271: 2270: 2269: 2266: 2246: 2241: 2216: 2197: 2187: 2175: 2165: 2159: 2154: 2150: 2147: 2143: 2138: 2122: 2107: 2106: 2105: 2099: 2090: 2081: 2072: 2067: 2063: 2055: 2035: 2030: 2027: 2022: 2014: 2012: 2002: 1999: 1986: 1976: 1970: 1968: 1963: 1956: 1953: 1950: 1941: 1937: 1935: 1924: 1912: 1911: 1910: 1906: 1900: 1891: 1882: 1876: 1872: 1867: 1850: 1845: 1830: 1827: 1824: 1818: 1808: 1802: 1799: 1796: 1777: 1762: 1761: 1760: 1743: 1727: 1723: 1719: 1703: 1699: 1684: 1660: 1653: 1652: 1651: 1648: 1646: 1641: 1621: 1616: 1611: 1608: 1605: 1601: 1592: 1589: 1586: 1580: 1566: 1540: 1536: 1528: 1527: 1526: 1523: 1521: 1517: 1512: 1506: 1486: 1481: 1478: 1475: 1471: 1456: 1430: 1426: 1418: 1417: 1416: 1413: 1409: 1405: 1404:wave function 1401: 1398:formalism of 1397: 1389: 1387: 1385: 1380: 1378: 1373: 1371: 1365: 1363: 1359: 1354: 1349: 1330: 1327: 1318: 1309: 1306: 1303: 1287: 1274: 1269: 1265: 1259: 1256: 1251: 1247: 1243: 1218: 1205: 1202: 1198: 1194: 1191: 1184: 1183: 1182: 1181:steradians): 1176: 1161: 1150: 1144: 1141: 1136: 1131: 1126: 1103: 1092: 1084: 1081: 1078: 1066: 1050: 1047: 1043: 1038: 1032: 1029: 1026: 1007: 992: 991: 990: 986: 982: 975: 968: 964: 947: 933: 927: 908: 886: 873: 870: 865: 861: 857: 848: 845: 842: 829: 814: 813: 812: 809: 805: 799: 786: 775: 771: 767: 760: 754: 750: 746: 742: 736: 735:gravitational 732: 728: 723: 717: 713: 703: 699: 698: 697: 695: 690: 685: 680: 674: 669: 665: 657: 655: 653: 648: 621: 616: 613: 610: 607: 603: 597: 589: 579: 578: 577: 574: 569: 565: 561: 556: 536: 530: 527: 524: 521: 515: 490: 489: 488: 487:according to 479: 474: 466: 458: 456: 454: 450: 446: 442: 437: 417: 412: 408: 405: 401: 396: 393: 390: 383: 382: 381: 378: 369: 364: 360: 357: 352: 348: 345: 340: 336: 335: 334: 317: 311: 308: 304: 299: 296: 289: 288: 287: 284: 275: 269: 263: 257: 251: 247: 240: 238: 236: 231: 230:cross section 226: 224: 220: 216: 211: 208: 207:reaction rate 205:The measured 203: 200: 196: 192: 187: 185: 181: 177: 173: 169: 164: 162: 158: 154: 151: 147: 143: 139: 138:cross section 134: 132: 128: 124: 119: 114: 110: 106: 102: 101:cross section 91: 88: 80: 70: 66: 60: 59: 53: 48: 39: 38: 33: 19: 8910:Spectroscopy 8783: 8774: 8767: 8760: 8753: 8713: 8686: 8678:Bibliography 8663: 8651: 8639: 8619: 8610: 8602: 8559: 8555: 8545: 8484: 8480: 8470: 8419: 8415: 8409: 8400: 8373: 8355: 8345:, retrieved 8319: 8309: 8131: 8061: 7972: 7908: 7753: 7651: 7645: 7642: 7636: 7633: 7464: 7457: 7454: 7364: 7360: 7180: 7177: 7130: 7126: 7057: 7047: 7044: 6957: 6829: 6823: 6817: 6810: 6799: 6795: 6792: 6724: 6655: 6649: 6646: 6627: 6618: 6615: 6548: 6542: 6533: 6443: 6433: 6427: 6415: 6409: 6402: 6391: 6322:luminescence 6312: 6307: 6303:permittivity 6290: 6287: 5907: 4780: 4282: 4276: 3968: 3479: 3342: 3082: 2806: 2797: 2716: 2710: 2704: 2692: 2689: 2637: 2557: 2553:permittivity 2542: 2515: 2499:nephelometry 2492: 2469: 2458: 2439: 2432: 2264: 2261: 2097: 2088: 2079: 2070: 2059: 1904: 1898: 1889: 1880: 1868: 1865: 1758: 1649: 1639: 1636: 1524: 1519: 1515: 1510: 1504: 1501: 1411: 1393: 1381: 1374: 1366: 1361: 1357: 1352: 1345: 1159: 1148: 1145: 1139: 1129: 1118: 984: 980: 973: 966: 962: 959:at an angle 945: 931: 928: 924: 807: 800: 784: 773: 769: 765: 758: 752: 748: 744: 740: 721: 715: 709: 688: 683: 678: 672: 661: 646: 636: 572: 559: 554: 551: 472: 468: 438: 434: 376: 373: 362: 350: 338: 332: 280: 273: 267: 261: 255: 249: 229: 227: 212: 204: 188: 179: 175: 171: 168:differential 165: 140:is the area 137: 135: 117: 100: 98: 83: 77:January 2022 74: 55: 8880:Measurement 8656:Taylor 2005 8644:Taylor 2005 6821:and energy 6449:path length 2799:this area. 2722:path length 2507:meteorology 2495:centimetres 1175:solid angle 662:Consider a 465:Attenuation 69:introducing 8905:Scattering 8854:Categories 8759:P. Roman, 8347:2021-12-16 8301:References 8062:we obtain 6453:absorbance 6424:wavelength 6394:absorbance 6318:absorption 4285:of radius 3201:plane wave 2809:Mie theory 2701:wavelength 2564:scattering 2560:absorption 2549:wavelength 2478:. Smaller 568:absorption 564:scattering 463:See also: 142:transverse 52:references 8740:cite book 8586:1094-4087 8521:2041-1723 8494:1308.0547 8462:119234614 8454:2040-8978 8429:1210.5730 8392:943495167 8198:π 8192:Ω 8180:Ω 8170:σ 8157:π 8150:∮ 8143:σ 8091:Ω 8081:σ 8035:θ 8026:⁡ 8011:θ 8002:⁡ 7990:θ 7987:⁡ 7955:φ 7946:θ 7937:θ 7934:⁡ 7925:Ω 7891:φ 7882:θ 7864:θ 7855:⁡ 7840:θ 7831:⁡ 7805:φ 7793:× 7770:σ 7732:α 7723:α 7720:⁡ 7689:α 7686:⁡ 7602:θ 7584:θ 7575:⁡ 7557:π 7545:∫ 7538:θ 7526:θ 7516:σ 7503:π 7491:∫ 7484:σ 7428:θ 7419:⁡ 7397:θ 7387:σ 7343:θ 7331:θ 7321:σ 7304:θ 7286:θ 7277:⁡ 7255:α 7246:α 7243:⁡ 7204:σ 7160:α 7154:− 7151:π 7145:θ 7109:α 7100:α 7097:⁡ 7063:light ray 6997:θ 6967:ε 6919:θ 6913:⁡ 6891:ε 6887:π 6858:Ω 6850:σ 6749:π 6737:σ 6684:Ω 6676:σ 6589:σ 6504:⁡ 6498:− 6490:λ 6457:logarithm 6368:σ 6355:σ 6342:σ 6335:σ 6259:μ 6253:ℑ 6244:ε 6238:ℑ 6194:μ 6190:μ 6187:π 6158:α 6124:ε 6120:ε 6117:π 6088:α 6049:≥ 6037:α 6030:ℑ 6018:μ 6014:μ 6011:π 5987:α 5980:ℑ 5968:ε 5964:ε 5961:π 5917:σ 5871:α 5853:μ 5849:μ 5846:π 5827:− 5815:α 5808:ℑ 5797:π 5791:⋅ 5779:μ 5775:μ 5772:π 5746:σ 5702:α 5684:ε 5680:ε 5677:π 5658:− 5646:α 5639:ℑ 5628:π 5622:⋅ 5610:ε 5606:ε 5603:π 5577:σ 5545:σ 5527:σ 5514:σ 5475:α 5450:π 5441:⋅ 5419:μ 5415:μ 5412:π 5376:α 5351:π 5342:⋅ 5320:ε 5316:ε 5313:π 5284:σ 5266:σ 5253:σ 5224:α 5217:ℑ 5211:π 5205:⋅ 5193:μ 5189:μ 5186:π 5162:α 5155:ℑ 5149:π 5143:⋅ 5131:ε 5127:ε 5124:π 5098:σ 5080:σ 5067:σ 5015:μ 5003:⋅ 4994:μ 4990:π 4975:α 4939:ε 4927:⋅ 4918:ε 4914:π 4899:α 4867:α 4858:− 4844:μ 4840:μ 4798:α 4713:σ 4700:σ 4687:σ 4594:ℜ 4571:∞ 4556:∑ 4411:∞ 4396:∑ 4333:μ 4313:ε 4245:× 4240:∗ 4210:× 4205:∗ 4185:⁡ 4159:Π 4120:× 4115:∗ 4095:⁡ 4069:Π 4030:× 4025:∗ 4005:⁡ 3979:Π 3942:^ 3931:⋅ 3921:Π 3910:∮ 3880:^ 3869:⋅ 3859:Π 3848:∮ 3824:≡ 3812:^ 3801:⋅ 3791:Π 3780:∮ 3776:− 3717:− 3529:× 3524:∗ 3509:⁡ 3489:Π 3456:^ 3445:⋅ 3441:Π 3431:∮ 3427:− 3363:(surface 3313:ε 3309:ε 3292:μ 3288:μ 3280:η 3257:η 3100:α 3054:α 3039:α 3035:σ 2984:α 2969:σ 2963:α 2959:σ 2948:α 2913:π 2901:σ 2773:⁡ 2767:− 2759:λ 2726:logarithm 2672:σ 2658:λ 2614:σ 2601:σ 2588:σ 2581:σ 2384:− 2364:δ 2346:− 2328:δ 2322:π 2316:− 2300:δ 2151:π 2133:Ω 2123:σ 2023:× 1964:σ 1954:σ 1875:colliding 1871:particles 1831:ϕ 1825:θ 1803:ϕ 1797:θ 1788:Ω 1778:σ 1724:ϕ 1704:− 1700:ϕ 1691:∞ 1688:→ 1680:⟶ 1661:ϕ 1593:ϕ 1587:θ 1573:∞ 1570:→ 1562:⟶ 1537:ϕ 1463:∞ 1460:→ 1452:⟶ 1431:− 1427:ϕ 1348:qualifier 1328:φ 1319:θ 1310:θ 1307:⁡ 1298:Ω 1288:σ 1275:π 1266:∫ 1260:π 1248:∫ 1241:Ω 1229:Ω 1219:σ 1206:π 1199:∮ 1192:σ 1085:φ 1079:θ 1057:Ω 1054:Δ 1033:φ 1027:θ 1018:Ω 1008:σ 909:φ 897:Ω 887:σ 874:π 862:∫ 849:θ 846:⁡ 830:σ 764:dΩ = sin 727:coulombic 664:classical 614:σ 608:− 594:Φ 587:Φ 534:Φ 531:σ 525:− 506:Φ 397:π 391:σ 346:unit: m), 312:λ 297:σ 250:Figure 1. 150:inelastic 8732:55729992 8705:53926857 8594:20589003 8537:15832637 8529:24598730 8338:archived 8317:(2006), 8227:See also 7065:and the 6638:Examples 6413:, where 3752:, where 3272:, where 2522:ångström 2480:prefixed 2435:S-matrix 1909:we have 1516:relative 1514:are the 1408:momentum 731:magnetic 445:neutrons 8564:Bibcode 8499:Bibcode 8434:Bibcode 6985:is the 6465:natural 6461:decadic 6447:is the 3336:is the 2734:natural 2730:decadic 2720:is the 2461:SI unit 2443:is the 2437:. Here 2066:momenta 2060:If the 1400:quantum 1394:In the 1170:⁠ 1155:⁠ 1133:is the 956:⁠ 941:⁠ 795:⁠ 780:⁠ 692:is the 682:is the 366:is the 354:is the 215:nuclear 191:nuclear 161:gravity 153:spheres 146:scatter 65:improve 8791: 8770:, 1994 8763:, 1969 8756:, 1964 8730: 8720: 8703: 8693: 8592: 8584: 8535: 8527: 8519: 8460: 8452: 8390: 8380: 8330: 7363:dΩ = d 7067:radius 6958:where 6441:, and 4635:Where 4149:, and 3969:where 3480:where 3083:where 2690:where 2566:, and 2530: 2518:X-rays 1637:where 1502:where 637:where 552:where 476:, the 333:where 221:, and 219:atomic 197:, and 195:atomic 54:, but 8607:IUPAC 8533:S2CID 8489:arXiv 8458:S2CID 8424:arXiv 8341:(PDF) 8324:(PDF) 2455:Units 1119:Here 281:In a 127:barns 123:sigma 8789:ISBN 8746:link 8728:OCLC 8718:ISBN 8701:OCLC 8691:ISBN 8623:scat 8590:PMID 8582:ISSN 8525:PMID 8517:ISSN 8450:ISSN 8388:OCLC 8378:ISBN 8328:ISBN 7649:and 7471:lens 6653:and 6631:scat 6622:scat 6593:scat 5926:> 5737:and 5568:are 4821:and 4654:host 3569:> 2973:geom 2905:geom 2486:and 2086:and 2064:and 1508:and 710:The 478:flux 8629:doi 8627:". 8572:doi 8507:doi 8442:doi 8023:cos 7999:sin 7984:sin 7931:sin 7852:cos 7828:sin 7717:cos 7683:sin 7640:. 7572:sin 7467:= 0 7460:= π 7416:sin 7274:sin 7240:cos 7184:is 7094:cos 6904:sin 6811:In 6741:tot 6581:3.9 6501:log 6410:Clσ 6372:lum 6346:abs 5921:abs 5755:(m) 5750:abs 5586:(e) 5581:abs 5554:(m) 5549:abs 5536:(e) 5531:abs 5518:abs 5293:(m) 5275:(e) 5107:(m) 5102:ext 5089:(e) 5084:ext 5071:ext 4760:abs 4734:ext 4717:abs 4691:ext 4518:ext 4274:. 4164:ext 3926:ext 3901:ext 3738:ext 3403:is 3216:inc 3151:inc 3064:inc 3007:abs 2991:ext 2935:as 2878:abs 2824:ext 2770:log 2618:lum 2592:abs 2528:000 1907:≪ 1 1360:or 1304:sin 1127:), 1097:inc 1071:out 977:out 935:inc 843:cos 560:all 283:gas 178:or 159:or 8856:: 8742:}} 8738:{{ 8726:. 8699:. 8609:, 8588:. 8580:. 8570:. 8560:18 8558:. 8554:. 8531:. 8523:. 8515:. 8505:. 8497:. 8483:. 8479:. 8456:. 8448:. 8440:. 8432:. 8420:15 8418:. 8386:. 8364:^ 8336:, 7368:) 6884:16 6798:+ 6619:Cσ 6555:: 6474:: 6431:, 6408:= 6359:sc 6324:: 6294:sc 6280:. 5288:sc 5270:sc 5257:sc 4747:sc 4724:or 4704:sc 4358:sc 4182:Re 4092:Re 4059:, 4002:Re 3506:Re 3340:. 2999:sc 2851:sc 2743:: 2708:, 2605:sc 2562:, 2532:pm 2526:10 2513:. 2503:μm 2488:μb 2484:mb 2476:fm 2451:. 2095:, 2077:, 1905:Nσ 1415:: 1179:4π 1167:dΩ 1125:sr 1121:ΔΩ 953:dΩ 798:. 792:dΩ 743:= 733:, 729:, 696:. 654:. 566:, 485:dΦ 344:SI 225:. 217:, 193:, 133:. 8797:. 8748:) 8734:. 8707:. 8631:: 8620:σ 8596:. 8574:: 8566:: 8539:. 8509:: 8501:: 8491:: 8485:5 8464:. 8444:: 8436:: 8426:: 8394:. 8211:. 8206:2 8202:a 8195:= 8188:d 8176:d 8166:d 8154:4 8146:= 8117:. 8112:4 8107:2 8103:a 8097:= 8087:d 8077:d 8047:, 8043:) 8038:2 8030:( 8019:) 8014:2 8006:( 7996:2 7993:= 7958:. 7951:d 7942:d 7928:= 7921:d 7894:. 7887:d 7878:d 7872:) 7867:2 7859:( 7848:) 7843:2 7835:( 7823:2 7818:2 7814:a 7808:= 7801:d 7796:r 7790:) 7787:r 7784:( 7781:r 7777:d 7773:= 7766:d 7735:, 7728:d 7714:a 7711:= 7704:r 7700:d 7692:, 7680:a 7677:= 7670:r 7652:φ 7646:r 7637:a 7614:. 7611:r 7608:2 7605:= 7598:d 7592:) 7587:2 7579:( 7567:2 7564:r 7554:2 7549:0 7541:= 7534:d 7522:d 7512:d 7500:2 7495:0 7487:= 7465:θ 7458:θ 7440:. 7436:) 7431:2 7423:( 7411:2 7408:r 7403:= 7393:d 7383:d 7365:θ 7346:. 7339:d 7327:d 7317:d 7310:I 7307:= 7300:d 7294:) 7289:2 7281:( 7269:2 7266:r 7261:I 7258:= 7251:d 7237:r 7234:I 7231:= 7228:) 7225:x 7222:( 7219:x 7215:d 7210:I 7207:= 7200:d 7195:I 7181:I 7163:. 7157:2 7148:= 7131:α 7129:2 7112:. 7105:d 7091:r 7088:= 7085:x 7081:d 7058:α 7048:r 7025:r 7021:/ 7017:1 6971:0 6941:2 6936:) 6930:) 6927:2 6923:/ 6916:( 6908:2 6900:E 6895:0 6879:Q 6875:q 6869:( 6864:= 6855:d 6847:d 6830:Q 6824:E 6818:q 6800:R 6796:r 6778:. 6773:2 6768:) 6764:R 6761:+ 6758:r 6754:( 6746:= 6710:, 6705:4 6700:2 6696:R 6690:= 6681:d 6673:d 6656:r 6650:R 6628:b 6601:. 6585:C 6576:= 6571:V 6567:L 6552:V 6549:L 6514:. 6509:T 6495:= 6486:A 6472:T 6459:( 6444:l 6434:C 6428:λ 6418:λ 6416:A 6405:λ 6403:A 6377:. 6364:+ 6351:+ 6338:= 6313:σ 6310:( 6291:σ 6268:0 6265:= 6262:) 6256:( 6250:= 6247:) 6241:( 6217:] 6208:2 6204:) 6198:0 6184:4 6181:( 6174:2 6169:| 6162:m 6153:| 6146:+ 6138:2 6134:) 6128:0 6114:4 6111:( 6104:2 6099:| 6092:e 6083:| 6075:[ 6069:3 6063:3 6059:k 6055:2 6046:) 6041:m 6033:( 6022:0 6008:4 6004:1 5999:+ 5996:) 5991:e 5983:( 5972:0 5958:4 5954:1 5929:0 5893:] 5887:2 5882:| 5875:m 5866:| 5857:0 5843:6 5837:3 5833:k 5824:) 5819:m 5811:( 5804:[ 5800:k 5794:4 5783:0 5769:4 5765:1 5760:= 5724:] 5718:2 5713:| 5706:e 5697:| 5688:0 5674:6 5668:3 5664:k 5655:) 5650:e 5642:( 5635:[ 5631:k 5625:4 5614:0 5600:4 5596:1 5591:= 5541:+ 5523:= 5491:2 5486:| 5479:m 5470:| 5464:4 5460:k 5454:3 5447:8 5433:2 5429:) 5423:0 5409:4 5406:( 5402:1 5397:+ 5392:2 5387:| 5380:e 5371:| 5365:4 5361:k 5355:3 5348:8 5334:2 5330:) 5324:0 5310:4 5307:( 5303:1 5298:= 5280:+ 5262:= 5233:) 5228:m 5220:( 5214:k 5208:4 5197:0 5183:4 5179:1 5174:+ 5171:) 5166:e 5158:( 5152:k 5146:4 5135:0 5121:4 5117:1 5112:= 5094:+ 5076:= 5046:. 5041:1 5037:b 5028:3 5024:k 5020:2 5012:3 5006:i 4998:0 4987:4 4984:= 4979:m 4970:, 4965:1 4961:a 4952:3 4948:k 4944:2 4936:3 4930:i 4922:0 4911:4 4908:= 4903:e 4877:H 4871:m 4861:1 4854:) 4848:0 4837:( 4834:= 4830:m 4808:E 4802:e 4794:= 4790:p 4756:Q 4752:+ 4743:Q 4739:= 4730:Q 4709:+ 4696:= 4664:0 4660:k 4650:n 4646:= 4643:k 4623:) 4618:n 4614:b 4610:+ 4605:n 4601:a 4597:( 4591:) 4588:1 4585:+ 4582:n 4579:2 4576:( 4566:1 4563:= 4560:n 4547:2 4543:a 4537:2 4533:k 4528:2 4523:= 4514:Q 4494:) 4489:2 4484:| 4477:n 4473:b 4468:| 4464:+ 4459:2 4454:| 4447:n 4443:a 4438:| 4434:( 4431:) 4428:1 4425:+ 4422:n 4419:2 4416:( 4406:1 4403:= 4400:n 4387:2 4383:a 4377:2 4373:k 4368:2 4363:= 4354:Q 4293:a 4261:] 4255:s 4250:H 4235:i 4230:E 4225:+ 4220:i 4215:H 4200:s 4195:E 4189:[ 4177:2 4174:1 4169:= 4136:] 4130:s 4125:H 4110:s 4105:E 4099:[ 4087:2 4084:1 4079:= 4074:s 4046:] 4040:i 4035:H 4020:i 4015:E 4009:[ 3997:2 3994:1 3989:= 3984:i 3954:. 3951:A 3948:d 3938:r 3914:A 3906:= 3897:W 3892:, 3889:A 3886:d 3876:r 3864:s 3852:A 3844:= 3839:s 3835:W 3830:, 3827:0 3821:A 3818:d 3808:r 3796:i 3784:A 3773:= 3768:i 3764:W 3734:W 3730:+ 3725:s 3721:W 3712:i 3708:W 3704:= 3699:a 3695:W 3672:a 3668:W 3646:H 3623:s 3618:E 3613:+ 3608:i 3603:E 3598:= 3594:E 3572:0 3564:a 3560:W 3538:] 3533:H 3519:E 3513:[ 3501:2 3498:1 3493:= 3465:A 3462:d 3452:r 3435:A 3424:= 3419:a 3415:W 3391:A 3371:A 3351:r 3322:) 3317:0 3306:( 3302:/ 3296:0 3283:= 3260:) 3254:2 3251:( 3247:/ 3241:2 3236:| 3230:E 3225:| 3221:= 3212:I 3186:] 3179:2 3174:m 3169:W 3164:[ 3160:= 3156:] 3147:I 3143:[ 3121:] 3117:W 3113:[ 3109:= 3105:] 3096:W 3092:[ 3060:I 3050:W 3044:= 3011:. 3003:, 2995:, 2987:= 2980:, 2953:= 2944:Q 2921:2 2917:a 2910:= 2874:Q 2847:Q 2820:Q 2783:. 2778:T 2764:= 2755:A 2741:T 2728:( 2717:l 2711:C 2705:λ 2695:λ 2693:A 2675:, 2669:l 2666:C 2663:= 2654:A 2623:. 2610:+ 2597:+ 2584:= 2472:b 2465:m 2440:δ 2415:i 2410:f 2405:T 2400:) 2394:f 2389:p 2379:i 2374:p 2368:( 2360:) 2354:i 2350:E 2341:f 2337:E 2332:( 2325:i 2319:2 2310:i 2305:f 2296:= 2290:i 2285:f 2280:S 2265:T 2247:, 2242:2 2236:| 2227:i 2222:f 2217:T 2211:| 2202:i 2198:p 2192:f 2188:p 2180:f 2176:m 2170:i 2166:m 2160:4 2155:) 2148:2 2144:( 2139:= 2129:d 2119:d 2101:f 2098:p 2092:f 2089:m 2083:i 2080:p 2074:i 2071:m 2036:. 2031:N 2028:1 2015:= 2003:N 2000:1 1991:i 1987:I 1981:r 1977:I 1971:= 1957:, 1951:N 1946:i 1942:I 1938:= 1929:r 1925:I 1899:N 1893:i 1890:I 1884:r 1881:I 1851:. 1846:2 1840:| 1834:) 1828:, 1822:( 1819:f 1814:| 1809:= 1806:) 1800:, 1794:( 1784:d 1774:d 1744:. 1741:) 1737:r 1733:( 1728:+ 1720:+ 1717:) 1713:r 1709:( 1685:r 1672:) 1668:r 1664:( 1640:f 1622:, 1617:r 1612:r 1609:k 1606:i 1602:e 1596:) 1590:, 1584:( 1581:f 1567:r 1554:) 1550:r 1546:( 1541:+ 1511:r 1505:z 1487:, 1482:z 1479:k 1476:i 1472:e 1457:r 1444:) 1440:r 1436:( 1412:k 1353:σ 1331:. 1324:d 1315:d 1294:d 1284:d 1270:0 1257:2 1252:0 1244:= 1237:d 1225:d 1215:d 1203:4 1195:= 1177:( 1164:/ 1160:σ 1158:d 1149:σ 1140:t 1130:n 1104:. 1093:F 1088:) 1082:, 1076:( 1067:F 1051:t 1048:n 1044:1 1039:= 1036:) 1030:, 1024:( 1014:d 1004:d 987:) 985:φ 983:, 981:θ 979:( 974:F 969:) 967:φ 965:, 963:θ 961:( 950:/ 946:σ 944:d 932:F 921:. 905:d 893:d 883:d 871:2 866:0 858:= 852:) 840:( 836:d 826:d 808:φ 789:/ 785:σ 783:d 774:φ 772:d 770:θ 768:d 766:θ 759:θ 753:b 751:d 749:φ 747:d 745:b 741:σ 739:d 722:θ 716:b 689:φ 679:θ 673:z 647:z 641:0 639:Φ 622:, 617:z 611:n 604:e 598:0 590:= 573:n 555:σ 537:, 528:n 522:= 516:z 512:d 502:d 481:Φ 473:z 471:d 418:2 413:) 409:r 406:2 402:( 394:= 377:r 363:n 351:λ 339:σ 318:, 309:n 305:1 300:= 277:. 274:λ 268:n 262:σ 256:r 254:2 121:( 118:σ 90:) 84:( 79:) 75:( 61:. 34:. 20:)

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Index