Cross section (physics)

32: 691: 235: 5492: 3953: 726:, 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. 1330: 7345: 5232: 175:, 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. 5236: 7893: 2039: 2417: 7613: 3747: 6524:

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.

2246: 5050: 1176: 1103: 7179: 908: 5045: 2787:

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

198:

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

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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

221:

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

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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

6523:

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

790:

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

7749: 4761: 1904: 2263: 5487:{\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}} 3948:{\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.} 7468: 3010: 1743: 3573:

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

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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.

1621: 7738: 2099: 6942: 1850: 8210: 7439: 5724: 8046: 1325:{\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 .} 4493: 4882: 1356:

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

5555: 7340:{\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 .} 984: 274:

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

5936: 4622: 5893: 5227:{\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})} 1486: 7957: 806: 4670: 8116: 6376: 2622: 3464: 4261: 4136: 4046: 424:

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.

7888:{\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 .} 6294:

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

3624: 3323: 2927: 536: 2034:{\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}}} 3067: 2412:{\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}}}} 1636:. 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. 3739: 4877: 6600: 3538: 7111: 7608:{\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.} 6709: 6777: 3259: 1645: 5560: 2241:{\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},} 7655: 1909: 1520: 7650: 6513: 2782: 4808: 2486:. 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 6828: 4337: 1754: 222:

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,

8127: 7363: 419: 317: 621: 6267: 2922: 2544:, 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. 4497: 5729: 4879:(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 2674: 7162: 5928: 1132:

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.

4665: 8888: 7968: 2524:= 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. 3571: 6972: 144:

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

1098:{\displaystyle {\frac {\mathrm {d} \sigma }{\mathrm {d} \Omega }}(\theta ,\varphi )={\frac {1}{nt\Delta \Omega }}{\frac {F_{\text{out}}(\theta ,\varphi )}{F_{\text{inc}}}}.} 4312: 3646: 5497: 3186: 3121: 2879: 2825: 1858:

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

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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

2852: 6996: 1410: 3673: 118:. 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 7623:

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

4332: 903:{\displaystyle {\frac {\mathrm {d} \sigma }{\mathrm {d} (\cos \theta )}}=\int _{0}^{2\pi }{\frac {\mathrm {d} \sigma }{\mathrm {d} \Omega }}\,\mathrm {d} \varphi } 7904: 7024: 7040:

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

5040:{\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}.} 4292: 3390: 3370: 3350: 1353:. 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. 8057: 6319: 6212:{\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} 2565: 8403:

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

3577: 4756:{\displaystyle \sigma _{\text{ext}}=\sigma _{\text{sc}}+\sigma _{\text{abs}}\qquad {\text{or}}\qquad Q_{\text{ext}}=Q_{\text{sc}}+Q_{\text{abs}}} 3264: 482: 8734: 6286:, describes the likelihood of light being scattered by a macroscopic particle. In general, the scattering cross section is different from the 163:(see detailed discussion below). When a cross section is integrated over all scattering angles (and possibly other variables), it is called a 8326: 3018: 8771: 8303: 3678: 6550: 4813: 3472: 7064: 6654: 6720: 3005:{\displaystyle Q_{\alpha }={\frac {\sigma _{\alpha }}{\sigma _{\text{geom}}}},\qquad \alpha ={\text{ext}},{\text{sc}},{\text{abs}}.} 2559:. The sum of the absorption and scattering cross sections is sometimes referred to as the attenuation or extinction cross section. 1738:{\displaystyle \phi (\mathbf {r} )\;{\stackrel {r\to \infty }{\longrightarrow }}\;\phi _{-}(\mathbf {r} )+\phi _{+}(\mathbf {r} ).} 226:

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

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

8393:

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

1616:{\displaystyle \phi _{+}(\mathbf {r} )\;{\stackrel {r\to \infty }{\longrightarrow }}\;f(\theta ,\phi ){\frac {e^{ikr}}{r}},} 7733:{\displaystyle {\begin{aligned}r&=a\sin \alpha ,\\\mathrm {d} r&=a\cos \alpha \,\mathrm {d} \alpha ,\end{aligned}}} 92:

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

6937:{\displaystyle {\frac {d\sigma }{d\Omega }}=\left({\frac {q\,Q}{16\pi \varepsilon _{0}E\sin ^{2}(\theta /2)}}\right)^{2}} 1845:{\displaystyle {\frac {\mathrm {d} \sigma }{\mathrm {d} \Omega }}(\theta ,\phi )={\bigl |}f(\theta ,\phi ){\bigr |}^{2}.} 8205:{\displaystyle \sigma =\oint _{4\pi }{\frac {\mathrm {d} \sigma }{\mathrm {d} \Omega }}\,\mathrm {d} \Omega =\pi a^{2}.} 7434:{\displaystyle {\frac {\mathrm {d} \sigma }{\mathrm {d} \theta }}={\frac {r}{2}}\sin \left({\frac {\theta }{2}}\right).} 6469: 2738: 4773: 708:

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

8710: 8683: 8370: 8320: 1855:

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

375: 281: 75: 53: 571: 46: 8833: 6222: 5719:{\displaystyle \sigma _{\text{abs}}^{\text{(e)}}={\frac {1}{4\pi \varepsilon \varepsilon _{0}}}\cdot 4\pi k\left} 2884: 6648:

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

2547:

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

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

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

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

8858: 2637: 559:, or transformation to another species. The volumetric number density of scattering centers is designated by 7129: 7044:. Because the problem is genuinely two-dimensional, the cross section has unit of length (e.g., metre). Let 2532:

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

655:

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

8853: 8242: 8041:{\displaystyle \sin \theta =2\sin \left({\frac {\theta }{2}}\right)\cos \left({\frac {\theta }{2}}\right),} 5900: 1514:

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

1372: 130: 4488:{\displaystyle Q_{\text{sc}}={\frac {2}{k^{2}a^{2}}}\sum _{n=1}^{\infty }(2n+1)(|a_{n}|^{2}+|b_{n}|^{2})} 1375:

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

656: 4627: 8883: 4267: 3543: 202:

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

156: 6950: 8878: 5550:{\textstyle \sigma _{\text{abs}}=\sigma _{\text{abs}}^{\text{(e)}}+\sigma _{\text{abs}}^{\text{(m)}}} 4770:

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

8803: 8798: 2627:

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

1507:

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

8873: 8222: 7632:

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

6782:

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

6306: 6287: 2548: 2533: 1876:) depends only on the number of incident particles per unit of time (current of incident particles 159:

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

40: 8307: 6636:

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

4617:{\displaystyle Q_{\text{ext}}={\frac {2}{k^{2}a^{2}}}\sum _{n=1}^{\infty }(2n+1)\Re (a_{n}+b_{n})} 4297: 3629: 3372:) around the particle (the scatterer). The net rate of electromagnetic energy crosses the surface 8848: 8257: 5897:

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

5888:{\displaystyle \sigma _{\text{abs}}^{\text{(m)}}={\frac {1}{4\pi \mu \mu _{0}}}\cdot 4\pi k\left} 3332:

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

3326: 3126: 3075: 223: 2857: 2803: 2482:

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

7644:. In any plane of the incoming and the reflected ray we can write (from the previous example): 6792:) within which the center of mass of the incoming sphere has to arrive for it to be deflected. 2830: 1481:{\displaystyle \phi _{-}(\mathbf {r} )\;{\stackrel {r\to \infty }{\longrightarrow }}\;e^{ikz},} 800:

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

57: 8898: 8277: 8262: 8252: 8247: 6981: 6801: 2499: 1358: 437: 93: 7952:{\displaystyle \mathrm {d} \Omega =\sin \theta \,\mathrm {d} \theta \,\mathrm {d} \varphi .} 8868: 8552: 8487: 8422: 8282: 6386: 3651: 2628: 2433: 1633: 640: 138: 4317: 1885:), the characteristics of target (for example the number of particles per unit of surface 8: 8893: 8272: 7001: 6975: 1859: 652: 172: 20: 8556: 8491: 8434: 8426: 2093:

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

8728: 8521: 8477: 8446: 8412: 4277: 3375: 3355: 3335: 2881:

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

119: 8348:

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

8111:{\displaystyle {\frac {\mathrm {d} \sigma }{\mathrm {d} \Omega }}={\frac {a^{2}}{4}}.} 6371:{\displaystyle \sigma =\sigma _{\text{abs}}+\sigma _{\text{sc}}+\sigma _{\text{lum}}.} 4334:, the problem has a precise solution. The scattering and extinction coefficients are 2617:{\displaystyle \sigma =\sigma _{\text{abs}}+\sigma _{\text{sc}}+\sigma _{\text{lum}}.} 2509:

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

8777: 8716: 8706: 8689: 8679: 8578: 8570: 8513: 8505: 8450: 8438: 8376: 8366: 8316: 8267: 7034:

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

6453: 6449: 6277:

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

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

8605: 8525: 3459:{\displaystyle W_{\text{a}}=-\oint _{A}\mathbf {\Pi } \cdot {\hat {\mathbf {r} }}dA} 565:. Solving this equation exhibits the exponential attenuation of the beam intensity: 8617: 8600: 8560: 8495: 8430: 7041: 4256:{\displaystyle \mathbf {\Pi } _{\text{ext}}={\frac {1}{2}}\operatorname {Re} \left} 2718: 1384: 715: 700: 211: 187: 145: 6978:. The total cross section is infinite unless a cutoff for small scattering angles 8539:

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

5931: 4131:{\displaystyle \mathbf {\Pi } _{\text{s}}={\frac {1}{2}}\operatorname {Re} \left} 4041:{\displaystyle \mathbf {\Pi } _{\text{i}}={\frac {1}{2}}\operatorname {Re} \left} 792: 682: 429: 203: 179: 8828: 8823: 152:, their scattering cross section is generally larger than their geometric size. 6427: 2797: 1123: 441: 356: 344: 207: 183: 101: 97: 8842: 8604:, 2nd ed. (the "Gold Book") (1997). Online corrected version: (2006–) " 8574: 8509: 8442: 8380: 8227: 7459: 6457: 2726: 2510: 2468: 1392: 690: 195: 8720: 8693: 8621: 8302: 3619:{\displaystyle \mathbf {E} =\mathbf {E} _{\text{i}}+\mathbf {E} _{\text{s}}} 8582: 8517: 7051: 6310: 6291: 3318:{\displaystyle \eta ={\sqrt {\mu \mu _{0}/(\varepsilon \varepsilon _{0})}}} 2541: 2487: 2453: 2050: 1339:

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

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

8365:(Revised 2nd ed.). Ram Nagar, New Delhi: S. Chand & Company Ltd. 531:{\displaystyle {\frac {\mathrm {d} \Phi }{\mathrm {d} z}}=-n\sigma \Phi ,} 8565: 8540: 8538: 2631:, which says that attenuation is proportional to particle concentration: 2495: 2460: 1163: 723: 665:

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

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

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When two discrete particles interact in classical physics, their mutual

8500: 8465: 7167:

The differential relationship between incident and reflected intensity

6533: 6441: 6412: 6382: 3189: 3062:{\displaystyle \sigma _{\alpha }={\frac {W_{\alpha }}{I_{\text{inc}}}}} 2689: 2552: 2537: 2491: 2483: 2464: 1748:

The differential cross section is related to the scattering amplitude:

1639:

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

552: 134: 766:. The differential cross section is the quotient of these quantities, 639:

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

7116:

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

6445: 3734:{\displaystyle W_{\text{a}}=W_{\text{i}}-W_{\text{s}}+W_{\text{ext}}} 2714: 2520: 1863: 1336: 1113: 719: 675:, measured between the incident beam and the scattered beam, and the 363:

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

4872:{\textstyle \mathbf {m} =(\mu \mu _{0})^{-1}\alpha ^{m}\mathbf {H} } 745:. The differential angular range of the scattered particle at angle 458:

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

6595:{\displaystyle L_{\text{V}}={\frac {3.9}{C\sigma _{\text{scat}}}}.} 6437: 3533:{\textstyle \mathbf {\Pi } ={\frac {1}{2}}\operatorname {Re} \left} 2710: 2459:

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

2423: 2054: 1396: 234: 19:"Scattering cross section" redirects here. Not to be confused with 8482: 8417: 7462:. The total cross section is equal to the diameter of the circle: 6290:

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

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provided strong evidence for the existence of the atomic nucleus.

436:

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

7106:{\displaystyle \mathrm {d} x=r\cos \alpha \,\mathrm {d} \alpha .} 2449: 433: 149: 133:

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

100:

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

6704:{\displaystyle {\frac {d\sigma }{d\Omega }}={\frac {R^{2}}{4}},} 16:

Probability of a given process occurring in a particle collision

7055: 6389:, which says that absorbance is proportional to concentration: 4270:. After that, all the integrals can be taken. In the case of a 2476: 2472: 1141:

may be recovered by integrating the differential cross section

141: 6772:{\displaystyle \sigma _{\text{tot}}=\pi \left(r+R\right)^{2}.} 1395:(before scattering) is taken to be a plane wave with definite 791:

cylindrically symmetric situations (about the beam axis), the

2506: 111: 8813: 8808: 1371:

Differential cross sections in inelastic scattering contain

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the probability that a given scattering process will occur.

466: 254:, for collisions is related to the particle number density 3254:{\displaystyle I_{\text{inc}}=|\mathbf {E} |^{2}/(2\eta )} 2436:. The computation of the S-matrix is the main goal of the 114:) and is expressed in units of area, more specifically in 8626: 6272: 1632:

is some function of the angular coordinates known as the

271: 8650: 7618: 6631: 3123:

is the energy flow through the surrounding surface, and

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corresponds to backward scattering, and its minimum at

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is the particle concentration as a number density, and

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is related to the flux of scattered particle detection

332: 8889:

Scattering, absorption and radiative transfer (optics)

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

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

8638: 8466:"Extraordinary momentum and spin in evanescent waves" 8130: 8060: 7971: 7907: 7752: 7653: 7471: 7366: 7182: 7132: 7067: 7004: 6984: 6953: 6831: 6723: 6657: 6553: 6472: 6322: 5939: 5732: 5563: 5239: 5053: 4885: 4766:

Dipole approximation for the scattering cross section

4673: 4500: 4340: 4320: 4300: 4280: 4144: 4054: 3964: 3750: 3681: 3654: 3632: 3580: 3546: 3401: 3378: 3358: 3338: 3267: 3198: 3129: 3078: 3021: 2930: 2741: 2640: 2568: 2266: 2102: 1907: 1757: 1648: 1523: 1413: 1179: 1112:

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

987: 809: 574: 485: 447: 378: 284: 8829:

IUPAC Goldbook – Definition: Collision Cross Section

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Particle Data Group – The Review of Particle Physics

8541:"Optical forces on small magnetodielectric particle" 8476:(1). Springer Science and Business Media LLC: 3300. 8834:

ShimPlotWell cross section plotter for nuclear data

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IUPAC Goldbook – Definition: Reaction Cross Section

8532: 4266:All the field can be decomposed into the series of 199:underlying two-particle collisional cross section. 8463: 8204: 8110: 8040: 7951: 7887: 7732: 7607: 7433: 7339: 7156: 7105: 7018: 6990: 6966: 6936: 6771: 6703: 6594: 6508:{\displaystyle A_{\lambda }=-\log {\mathcal {T}}.} 6507: 6370: 6261: 6211: 5922: 5887: 5718: 5549: 5486: 5226: 5039: 4871: 4802: 4755: 4659: 4616: 4487: 4326: 4306: 4286: 4255: 4130: 4040: 3947: 3733: 3667: 3640: 3618: 3565: 3532: 3458: 3384: 3364: 3344: 3317: 3253: 3180: 3115: 3061: 3004: 2916: 2873: 2846: 2819: 2777:{\displaystyle A_{\lambda }=-\log {\mathcal {T}}.} 2776: 2668: 2616: 2411: 2240: 2033: 1844: 1737: 1615: 1480: 1324: 1097: 902: 615: 530: 413: 331:is the cross section of a two-particle collision ( 311: 6998:is applied. This is due to the long range of the 4803:{\textstyle \mathbf {p} =\alpha ^{e}\mathbf {E} } 8840: 229: 6532:The scattering cross section is related to the 2800:include efficiency coefficients for extinction 2791: 414:{\displaystyle \sigma =\pi \left(2r\right)^{2}} 312:{\displaystyle \sigma ={\frac {1}{n\lambda }},} 6623:, the scattering coefficient per unit length. 2490:, and is effective for particles of 2–50 2456:, a smaller unit is usually used in practice. 616:{\displaystyle \Phi =\Phi _{0}e^{-n\sigma z},} 8773:Introduction to Experimental Particle Physics 8360: 7350:The differential cross section is therefore ( 3188:is the intensity of the incident wave. For a 2513:is a convenient unit: 1 Å = 10 m = 2224: 2199: 1828: 1802: 918:For scattering of particles of incident flux 646: 241:In a gas of particles of individual diameter 217:With light scattering off of a particle, the 8769: 8304:International Bureau of Weights and Measures 6518: 6262:{\textstyle \Im (\varepsilon )=\Im (\mu )=0} 2917:{\textstyle \sigma _{\text{geom}}=\pi a^{2}} 178:Scattering cross sections may be defined in 8396: 2494:in diameter: as such, it is widely used in 2044: 137:from each other. If the particles are hard 8733:: CS1 maint: location missing publisher ( 8363:Biological instrumentation and methodology 6816:scatters off a fixed particle with charge 6381:The total cross section is related to the 6305:) is the sum of the cross sections due to 1862:(i.e. observed when the two particles are 1686: 1663: 1568: 1545: 1458: 1435: 1126:of the target particles (SI unit: m), and 8673: 8656: 8564: 8499: 8481: 8457: 8416: 8174: 7962:Together with the trigonometric identity 7937: 7928: 7873: 7864: 7787: 7714: 7584: 7520: 7325: 7286: 7237: 7201: 7186: 7091: 6866: 3540:is the time averaged Poynting vector. If 2796:Cross sections commonly calculated using 2713:. The absorbance of the radiation is the 2536:of the particle, and it depends upon the 1310: 1301: 1223: 891: 155:When a cross section is specified as the 76:Learn how and when to remove this message 8764:Scattering Theory of Waves and Particles 8551:(11). The Optical Society: 11428–11443. 8402: 8356: 8354: 6795: 2463:, where 1 b = 10 m = 100 1335:It is common to omit the "differential" 260:, and mean free path between collisions 233: 39:This article includes a list of general 6527: 2669:{\displaystyle A_{\lambda }=Cl\sigma ,} 8841: 8700: 8644: 8632: 8309:The International System of Units (SI) 7157:{\displaystyle \theta =\pi -2\alpha .} 6273:Scattering of light on extended bodies 5923:{\textstyle \sigma _{\text{abs}}>0} 3626:, and the same for the magnetic field 2527: 438:cross section that is energy-dependent 8351: 7619:Scattering from a 3D spherical mirror 6632:Elastic collision of two hard spheres 5047:Then the cross sections are given by 1378: 359:of the target particles (SI unit: m). 104:. Cross section is typically denoted 7030:Scattering from a 2D circular mirror 6822:. The differential cross section is 5930:), we have a particular case of the 1891:), and the type of interaction. For 96:is a measure of probability that an 25: 6804:, an incident particle with charge 6385:of the light intensity through the 4660:{\textstyle k=n_{\text{host}}k_{0}} 13: 8814:BNL – National Nuclear Data Center 8601:Compendium of Chemical Terminology 8180: 8176: 8168: 8164: 8154: 8079: 8075: 8065: 7939: 7930: 7913: 7909: 7875: 7866: 7789: 7765: 7754: 7716: 7688: 7586: 7522: 7510: 7500: 7381: 7371: 7327: 7315: 7305: 7288: 7239: 7203: 7188: 7093: 7069: 6846: 6672: 6497: 6241: 6226: 6018: 5968: 5796: 5627: 5205: 5143: 4582: 4559: 4399: 2766: 2121: 2117: 2107: 1776: 1772: 1762: 1679: 1561: 1451: 1312: 1303: 1286: 1282: 1272: 1229: 1225: 1217: 1213: 1203: 1045: 1042: 1006: 1002: 992: 893: 885: 881: 871: 824: 814: 689: 582: 575: 522: 500: 494: 490: 448:Attenuation of a beam of particles 45:it lacks sufficient corresponding 14: 8910: 8792: 8676:Introduction to quantum mechanics 8315:(8th ed.), pp. 127–28, 7743:while the impact area element is 3566:{\displaystyle W_{\text{a}}>0} 3012:The cross section is defined by 440:and hence also with well-defined 6967:{\displaystyle \varepsilon _{0}} 4865: 4818: 4796: 4778: 4268:vector spherical harmonics (VSH) 4238: 4218: 4203: 4183: 4147: 4113: 4093: 4057: 4023: 4003: 3967: 3926: 3909: 3864: 3847: 3796: 3779: 3634: 3606: 3591: 3582: 3521: 3507: 3477: 3440: 3429: 3218: 2377: 2362: 1725: 1701: 1656: 1538: 1428: 1364:Instead of the solid angle, the 347:between collisions (SI unit: m), 30: 8666: 8332:from the original on 2021-06-04 6714:and the total cross section is 4961: 4716: 4710: 3883: 3821: 2971: 8776:. Cambridge University Press. 8750:Introduction to Quantum Theory 8743:Relativistic Quantum Mechanics 8589: 8387: 8342: 8296: 8238:Linear attenuation coefficient 8233:Luminosity (scattering theory) 7778: 7772: 7216: 7210: 7123:, and the scattering angle is 6918: 6904: 6250: 6244: 6235: 6229: 6192: 6169: 6157: 6141: 6122: 6099: 6087: 6071: 6034: 6021: 5984: 5971: 5870: 5854: 5812: 5799: 5701: 5685: 5643: 5630: 5474: 5458: 5417: 5394: 5375: 5359: 5318: 5295: 5221: 5208: 5159: 5146: 4842: 4825: 4611: 4585: 4579: 4564: 4482: 4472: 4456: 4442: 4426: 4422: 4419: 4404: 3930: 3868: 3800: 3444: 3310: 3294: 3248: 3239: 3224: 3213: 2688:is the attenuation at a given 1822: 1810: 1794: 1782: 1729: 1721: 1705: 1697: 1676: 1668: 1660: 1652: 1584: 1572: 1558: 1550: 1542: 1534: 1448: 1440: 1432: 1424: 1076: 1064: 1024: 1012: 978:in particles per unit time by 840: 828: 547:is the total cross section of 169:integrated total cross section 21:Radar scattering cross section 1: 8864:Experimental particle physics 8435:10.1088/2040-8978/15/4/044004 8289: 6411:is the absorbance at a given 3192:the intensity is going to be 472:of the beam will decrease by 230:Collision among gas particles 8809:IAEA – Nuclear Data Services 8741:J. D. Bjorken, S. D. Drell, 8674:Griffiths, David J. (2005). 8243:Mass attenuation coefficient 4307:{\displaystyle \varepsilon } 3641:{\displaystyle \mathbf {H} } 3327:impedance of the host medium 2792:Cross section and Mie theory 2057:of the colliding system are 7: 8705:. Mill Valley, California. 8215: 8121:The total cross section is 6626: 6456:) of the reciprocal of the 3181:{\displaystyle \left=\left} 3116:{\displaystyle \left=\left} 2874:{\textstyle Q_{\text{abs}}} 2820:{\textstyle Q_{\text{ext}}} 2725:) of the reciprocal of the 2452:of total cross sections is 751:is the solid angle element 714:. For a given interaction ( 657:spherical coordinate system 10: 8915: 8755:W. Greiner, J. Reinhardt, 8606:Scattering cross section, 7898:In spherical coordinates, 6426:is the concentration as a 2847:{\textstyle Q_{\text{sc}}} 2498:and in the measurement of 2008:probability of interaction 1345:may be referred to as the 647:Differential cross section 451: 161:differential cross section 18: 8678:. Pearson Prentice Hall. 7050:be the angle between the 6519:Relation to physical size 6288:geometrical cross section 4667:. These are connected as 3648:. Thus, we can decompose 2534:geometrical cross section 633:is the initial flux, and 8804:Scattering Cross Section 8701:Taylor, John R. (2005). 8223:Cross section (geometry) 6444:of the radiation is the 2443: 2045:Relation to the S-matrix 1391:scattering, the initial 1135:The total cross section 94:Rutherford cross-section 8757:Quantum Electrodynamics 8622:10.1351/goldbook.S05490 8258:Gamma ray cross section 6991:{\displaystyle \theta } 224:plasmonic nanoparticles 60:more precise citations. 8361:Bajpai, P. K. (2008). 8206: 8112: 8042: 7953: 7889: 7734: 7609: 7435: 7341: 7158: 7107: 7020: 6992: 6968: 6938: 6773: 6705: 6596: 6509: 6372: 6263: 6213: 5924: 5889: 5720: 5551: 5488: 5228: 5041: 4873: 4804: 4757: 4661: 4618: 4563: 4489: 4403: 4328: 4308: 4288: 4257: 4132: 4042: 3949: 3735: 3669: 3642: 3620: 3567: 3534: 3460: 3386: 3366: 3346: 3319: 3255: 3182: 3117: 3063: 3006: 2918: 2875: 2848: 2821: 2778: 2670: 2618: 2413: 2242: 2035: 1846: 1739: 1617: 1482: 1347:integral cross section 1326: 1099: 904: 694: 617: 532: 432:a "gas" of low-energy 415: 313: 267: 8799:Nuclear Cross Section 8770:R. C. Fernow (1989). 8470:Nature Communications 8278:Rutherford scattering 8263:Partial wave analysis 8253:Nuclear cross section 8248:Neutron cross section 8207: 8113: 8043: 7954: 7890: 7735: 7610: 7436: 7342: 7159: 7108: 7021: 6993: 6969: 6939: 6802:Rutherford scattering 6796:Rutherford scattering 6774: 6706: 6614:is sometimes denoted 6597: 6510: 6373: 6264: 6214: 5925: 5890: 5721: 5552: 5489: 5229: 5042: 4874: 4805: 4758: 4662: 4619: 4543: 4490: 4383: 4329: 4309: 4289: 4258: 4133: 4043: 3950: 3736: 3675:into the three terms 3670: 3668:{\displaystyle W_{a}} 3643: 3621: 3568: 3535: 3461: 3387: 3367: 3347: 3320: 3256: 3183: 3118: 3064: 3007: 2919: 2876: 2849: 2822: 2779: 2671: 2619: 2500:atmospheric pollution 2414: 2257:matrix is defined by 2243: 2036: 1847: 1740: 1618: 1483: 1359:Rutherford scattering 1327: 1100: 905: 693: 618: 533: 416: 314: 237: 8859:Dimensional analysis 8766:. McGraw Hill, 1966. 8635:, pp. 564, 574. 8566:10.1364/oe.18.011428 8283:Scattering amplitude 8128: 8058: 7969: 7905: 7750: 7651: 7469: 7364: 7180: 7130: 7065: 7002: 6982: 6951: 6829: 6721: 6655: 6551: 6534:meteorological range 6528:Meteorological range 6470: 6440:. The extinction or 6320: 6223: 5937: 5901: 5730: 5561: 5498: 5237: 5051: 4883: 4814: 4774: 4671: 4628: 4498: 4338: 4327:{\displaystyle \mu } 4318: 4298: 4278: 4142: 4052: 3962: 3748: 3679: 3652: 3630: 3578: 3544: 3473: 3399: 3376: 3356: 3336: 3265: 3196: 3127: 3076: 3019: 2928: 2885: 2858: 2831: 2804: 2739: 2638: 2566: 2434:Dirac delta function 2264: 2100: 1905: 1755: 1646: 1634:scattering amplitude 1521: 1411: 1177: 985: 807: 572: 483: 444:between collisions. 376: 282: 248:, the cross section 8854:Physical quantities 8703:Classical mechanics 8557:2010OExpr..1811428N 8492:2014NatCo...5.3300B 8427:2013JOpt...15d4004B 8273:Radar cross-section 7549: 7495: 7026:Coulomb potential. 7019:{\displaystyle 1/r} 6976:vacuum permittivity 6297:total cross section 5747: 5578: 5546: 5528: 5285: 5267: 5099: 5081: 4314:, and permeability 4232: 4197: 4107: 4017: 2528:Scattering of light 2251:where the on-shell 1509:asymptotic behavior 1351:total cross section 1267: 1252: 866: 173:Rayleigh scattering 165:total cross section 8501:10.1038/ncomms4300 8202: 8108: 8038: 7949: 7885: 7730: 7728: 7605: 7532: 7478: 7431: 7337: 7154: 7103: 7016: 6988: 6964: 6934: 6769: 6701: 6592: 6505: 6452:or, more usually, 6368: 6309:, scattering, and 6259: 6209: 5920: 5885: 5733: 5716: 5564: 5547: 5532: 5514: 5484: 5271: 5253: 5224: 5085: 5067: 5037: 4869: 4800: 4753: 4657: 4614: 4485: 4324: 4304: 4284: 4253: 4216: 4181: 4128: 4091: 4038: 4001: 3945: 3731: 3665: 3638: 3616: 3563: 3530: 3456: 3382: 3362: 3342: 3315: 3251: 3178: 3113: 3059: 3002: 2914: 2871: 2844: 2817: 2774: 2721:or, more usually, 2666: 2614: 2505:The scattering of 2409: 2238: 2031: 2029: 1842: 1735: 1613: 1478: 1379:Quantum scattering 1322: 1253: 1235: 1095: 900: 849: 695: 613: 551:events, including 528: 411: 309: 268: 171:. For example, in 120:stochastic process 8884:Scattering theory 8783:978-0-521-379-403 8405:Journal of Optics 8268:Particle detector 8172: 8103: 8083: 8029: 8005: 7858: 7834: 7814: 7578: 7558: 7518: 7422: 7402: 7389: 7323: 7280: 7260: 6922: 6850: 6731: 6696: 6676: 6587: 6583: 6561: 6362: 6349: 6336: 6202: 6132: 6060: 6016: 5966: 5911: 5851: 5777: 5745: 5740: 5682: 5608: 5576: 5571: 5544: 5539: 5526: 5521: 5508: 5445: 5427: 5346: 5328: 5283: 5278: 5265: 5260: 5247: 5191: 5129: 5097: 5092: 5079: 5074: 5061: 5022: 4946: 4750: 4737: 4724: 4714: 4707: 4694: 4681: 4644: 4541: 4508: 4381: 4348: 4287:{\displaystyle a} 4168: 4154: 4120: 4100: 4078: 4064: 4030: 4010: 3988: 3974: 3933: 3916: 3891: 3871: 3854: 3829: 3803: 3786: 3758: 3728: 3715: 3702: 3689: 3613: 3598: 3554: 3492: 3447: 3409: 3385:{\displaystyle A} 3365:{\displaystyle A} 3345:{\displaystyle r} 3313: 3206: 3172: 3164: 3159: 3141: 3107: 3057: 3054: 2997: 2989: 2981: 2966: 2963: 2895: 2868: 2854:, and Absorption 2841: 2814: 2608: 2595: 2582: 2557:photoluminescence 2540:of light and the 2438:scattering theory 2405: 2400: 2384: 2369: 2344: 2331: 2300: 2295: 2280: 2275: 2217: 2212: 2195: 2192: 2182: 2170: 2160: 2125: 2022: 2009: 1994: 1984: 1981: 1971: 1936: 1919: 1780: 1683: 1608: 1565: 1455: 1366:momentum transfer 1290: 1221: 1090: 1087: 1061: 1049: 1010: 889: 844: 508: 304: 86: 85: 78: 8906: 8879:Particle physics 8787: 8738: 8732: 8724: 8697: 8660: 8654: 8648: 8642: 8636: 8630: 8624: 8614: 8593: 8587: 8586: 8568: 8536: 8530: 8529: 8503: 8485: 8461: 8455: 8454: 8420: 8400: 8394: 8391: 8385: 8384: 8358: 8349: 8346: 8340: 8339: 8338: 8337: 8331: 8314: 8300: 8211: 8209: 8208: 8203: 8198: 8197: 8179: 8173: 8171: 8167: 8161: 8157: 8151: 8149: 8148: 8117: 8115: 8114: 8109: 8104: 8099: 8098: 8089: 8084: 8082: 8078: 8072: 8068: 8062: 8047: 8045: 8044: 8039: 8034: 8030: 8022: 8010: 8006: 7998: 7958: 7956: 7955: 7950: 7942: 7933: 7912: 7894: 7892: 7891: 7886: 7878: 7869: 7863: 7859: 7851: 7839: 7835: 7827: 7815: 7810: 7809: 7800: 7792: 7768: 7757: 7739: 7737: 7736: 7731: 7729: 7719: 7691: 7643: 7637: 7628: 7614: 7612: 7611: 7606: 7589: 7583: 7579: 7571: 7559: 7551: 7548: 7540: 7525: 7519: 7517: 7513: 7507: 7503: 7497: 7494: 7486: 7457: 7450: 7440: 7438: 7437: 7432: 7427: 7423: 7415: 7403: 7395: 7390: 7388: 7384: 7378: 7374: 7368: 7356: 7346: 7344: 7343: 7338: 7330: 7324: 7322: 7318: 7312: 7308: 7302: 7291: 7285: 7281: 7273: 7261: 7253: 7242: 7206: 7191: 7172: 7163: 7161: 7160: 7155: 7122: 7112: 7110: 7109: 7104: 7096: 7072: 7049: 7042:geometric optics 7039: 7025: 7023: 7022: 7017: 7012: 6997: 6995: 6994: 6989: 6973: 6971: 6970: 6965: 6963: 6962: 6943: 6941: 6940: 6935: 6933: 6932: 6927: 6923: 6921: 6914: 6900: 6899: 6887: 6886: 6870: 6861: 6851: 6849: 6841: 6833: 6821: 6815: 6809: 6791: 6778: 6776: 6775: 6770: 6765: 6764: 6759: 6755: 6733: 6732: 6729: 6710: 6708: 6707: 6702: 6697: 6692: 6691: 6682: 6677: 6675: 6667: 6659: 6647: 6641: 6622: 6613: 6601: 6599: 6598: 6593: 6588: 6586: 6585: 6584: 6581: 6568: 6563: 6562: 6559: 6543: 6514: 6512: 6511: 6506: 6501: 6500: 6482: 6481: 6462: 6435: 6425: 6419: 6410: 6401: 6387:Beer–Lambert law 6377: 6375: 6374: 6369: 6364: 6363: 6360: 6351: 6350: 6347: 6338: 6337: 6334: 6304: 6285: 6268: 6266: 6265: 6260: 6218: 6216: 6215: 6210: 6208: 6204: 6203: 6201: 6200: 6199: 6190: 6189: 6167: 6166: 6165: 6160: 6154: 6153: 6144: 6138: 6133: 6131: 6130: 6129: 6120: 6119: 6097: 6096: 6095: 6090: 6084: 6083: 6074: 6068: 6061: 6056: 6055: 6054: 6041: 6033: 6032: 6017: 6015: 6014: 6013: 5991: 5983: 5982: 5967: 5965: 5964: 5963: 5941: 5929: 5927: 5926: 5921: 5913: 5912: 5909: 5894: 5892: 5891: 5886: 5884: 5880: 5879: 5878: 5873: 5867: 5866: 5857: 5852: 5850: 5849: 5848: 5829: 5828: 5819: 5811: 5810: 5778: 5776: 5775: 5774: 5752: 5746: 5743: 5741: 5738: 5725: 5723: 5722: 5717: 5715: 5711: 5710: 5709: 5704: 5698: 5697: 5688: 5683: 5681: 5680: 5679: 5660: 5659: 5650: 5642: 5641: 5609: 5607: 5606: 5605: 5583: 5577: 5574: 5572: 5569: 5556: 5554: 5553: 5548: 5545: 5542: 5540: 5537: 5527: 5524: 5522: 5519: 5510: 5509: 5506: 5493: 5491: 5490: 5485: 5483: 5482: 5477: 5471: 5470: 5461: 5456: 5455: 5446: 5441: 5433: 5428: 5426: 5425: 5424: 5415: 5414: 5389: 5384: 5383: 5378: 5372: 5371: 5362: 5357: 5356: 5347: 5342: 5334: 5329: 5327: 5326: 5325: 5316: 5315: 5290: 5284: 5281: 5279: 5276: 5266: 5263: 5261: 5258: 5249: 5248: 5245: 5233: 5231: 5230: 5225: 5220: 5219: 5192: 5190: 5189: 5188: 5166: 5158: 5157: 5130: 5128: 5127: 5126: 5104: 5098: 5095: 5093: 5090: 5080: 5077: 5075: 5072: 5063: 5062: 5059: 5046: 5044: 5043: 5038: 5033: 5032: 5023: 5021: 5020: 5019: 5006: 4998: 4990: 4989: 4971: 4970: 4957: 4956: 4947: 4945: 4944: 4943: 4930: 4922: 4914: 4913: 4895: 4894: 4878: 4876: 4875: 4870: 4868: 4863: 4862: 4853: 4852: 4840: 4839: 4821: 4809: 4807: 4806: 4801: 4799: 4794: 4793: 4781: 4762: 4760: 4759: 4754: 4752: 4751: 4748: 4739: 4738: 4735: 4726: 4725: 4722: 4715: 4712: 4709: 4708: 4705: 4696: 4695: 4692: 4683: 4682: 4679: 4666: 4664: 4663: 4658: 4656: 4655: 4646: 4645: 4642: 4623: 4621: 4620: 4615: 4610: 4609: 4597: 4596: 4562: 4557: 4542: 4540: 4539: 4538: 4529: 4528: 4515: 4510: 4509: 4506: 4494: 4492: 4491: 4486: 4481: 4480: 4475: 4469: 4468: 4459: 4451: 4450: 4445: 4439: 4438: 4429: 4402: 4397: 4382: 4380: 4379: 4378: 4369: 4368: 4355: 4350: 4349: 4346: 4333: 4331: 4330: 4325: 4313: 4311: 4310: 4305: 4293: 4291: 4290: 4285: 4262: 4260: 4259: 4254: 4252: 4248: 4247: 4246: 4241: 4231: 4226: 4221: 4212: 4211: 4206: 4196: 4191: 4186: 4169: 4161: 4156: 4155: 4152: 4150: 4137: 4135: 4134: 4129: 4127: 4123: 4122: 4121: 4118: 4116: 4106: 4101: 4098: 4096: 4079: 4071: 4066: 4065: 4062: 4060: 4047: 4045: 4044: 4039: 4037: 4033: 4032: 4031: 4028: 4026: 4016: 4011: 4008: 4006: 3989: 3981: 3976: 3975: 3972: 3970: 3954: 3952: 3951: 3946: 3935: 3934: 3929: 3924: 3918: 3917: 3914: 3912: 3906: 3905: 3893: 3892: 3889: 3873: 3872: 3867: 3862: 3856: 3855: 3852: 3850: 3844: 3843: 3831: 3830: 3827: 3805: 3804: 3799: 3794: 3788: 3787: 3784: 3782: 3776: 3775: 3760: 3759: 3756: 3740: 3738: 3737: 3732: 3730: 3729: 3726: 3717: 3716: 3713: 3704: 3703: 3700: 3691: 3690: 3687: 3674: 3672: 3671: 3666: 3664: 3663: 3647: 3645: 3644: 3639: 3637: 3625: 3623: 3622: 3617: 3615: 3614: 3611: 3609: 3600: 3599: 3596: 3594: 3585: 3572: 3570: 3569: 3564: 3556: 3555: 3552: 3539: 3537: 3536: 3531: 3529: 3525: 3524: 3516: 3515: 3510: 3493: 3485: 3480: 3465: 3463: 3462: 3457: 3449: 3448: 3443: 3438: 3432: 3427: 3426: 3411: 3410: 3407: 3391: 3389: 3388: 3383: 3371: 3369: 3368: 3363: 3351: 3349: 3348: 3343: 3324: 3322: 3321: 3316: 3314: 3309: 3308: 3293: 3288: 3287: 3275: 3260: 3258: 3257: 3252: 3238: 3233: 3232: 3227: 3221: 3216: 3208: 3207: 3204: 3187: 3185: 3184: 3179: 3177: 3173: 3171: 3170: 3165: 3162: 3157: 3156: 3147: 3143: 3142: 3139: 3122: 3120: 3119: 3114: 3112: 3108: 3105: 3096: 3092: 3091: 3068: 3066: 3065: 3060: 3058: 3056: 3055: 3052: 3046: 3045: 3036: 3031: 3030: 3011: 3009: 3008: 3003: 2998: 2995: 2990: 2987: 2982: 2979: 2967: 2965: 2964: 2961: 2955: 2954: 2945: 2940: 2939: 2923: 2921: 2920: 2915: 2913: 2912: 2897: 2896: 2893: 2880: 2878: 2877: 2872: 2870: 2869: 2866: 2853: 2851: 2850: 2845: 2843: 2842: 2839: 2826: 2824: 2823: 2818: 2816: 2815: 2812: 2783: 2781: 2780: 2775: 2770: 2769: 2751: 2750: 2731: 2708: 2702: 2696: 2687: 2675: 2673: 2672: 2667: 2650: 2649: 2629:Beer–Lambert law 2623: 2621: 2620: 2615: 2610: 2609: 2606: 2597: 2596: 2593: 2584: 2583: 2580: 2523: 2518: 2431: 2422:in terms of the 2418: 2416: 2415: 2410: 2408: 2407: 2406: 2403: 2401: 2398: 2391: 2387: 2386: 2385: 2382: 2380: 2371: 2370: 2367: 2365: 2351: 2347: 2346: 2345: 2342: 2333: 2332: 2329: 2303: 2302: 2301: 2298: 2296: 2293: 2283: 2282: 2281: 2278: 2276: 2273: 2256: 2247: 2245: 2244: 2239: 2234: 2233: 2228: 2227: 2220: 2219: 2218: 2215: 2213: 2210: 2203: 2202: 2196: 2194: 2193: 2190: 2184: 2183: 2180: 2174: 2172: 2171: 2168: 2162: 2161: 2158: 2152: 2151: 2146: 2142: 2126: 2124: 2120: 2114: 2110: 2104: 2092: 2083: 2074: 2065: 2040: 2038: 2037: 2032: 2030: 2023: 2015: 2010: 2007: 1999: 1995: 1987: 1985: 1983: 1982: 1979: 1973: 1972: 1969: 1963: 1938: 1937: 1934: 1921: 1920: 1917: 1897: 1890: 1884: 1875: 1851: 1849: 1848: 1843: 1838: 1837: 1832: 1831: 1806: 1805: 1781: 1779: 1775: 1769: 1765: 1759: 1744: 1742: 1741: 1736: 1728: 1720: 1719: 1704: 1696: 1695: 1685: 1684: 1682: 1671: 1666: 1659: 1631: 1622: 1620: 1619: 1614: 1609: 1604: 1603: 1588: 1567: 1566: 1564: 1553: 1548: 1541: 1533: 1532: 1502: 1496: 1487: 1485: 1484: 1479: 1474: 1473: 1457: 1456: 1454: 1443: 1438: 1431: 1423: 1422: 1403: 1385:time-independent 1344: 1331: 1329: 1328: 1323: 1315: 1306: 1291: 1289: 1285: 1279: 1275: 1269: 1266: 1261: 1251: 1243: 1228: 1222: 1220: 1216: 1210: 1206: 1200: 1198: 1197: 1169: 1161: 1160: 1158: 1157: 1154: 1151: 1140: 1131: 1121: 1111: 1104: 1102: 1101: 1096: 1091: 1089: 1088: 1085: 1079: 1063: 1062: 1059: 1052: 1050: 1048: 1031: 1011: 1009: 1005: 999: 995: 989: 977: 959: 947: 946: 944: 943: 940: 937: 926: 909: 907: 906: 901: 896: 890: 888: 884: 878: 874: 868: 865: 857: 845: 843: 827: 821: 817: 811: 799: 786: 785: 783: 782: 779: 776: 765: 750: 744: 713: 707: 701:impact parameter 680: 673:scattering angle 670: 664: 641:Beer–Lambert law 638: 632: 622: 620: 619: 614: 609: 608: 590: 589: 564: 546: 537: 535: 534: 529: 509: 507: 503: 497: 493: 487: 475: 471: 464: 420: 418: 417: 412: 410: 409: 404: 400: 368: 354: 342: 330: 318: 316: 315: 310: 305: 303: 292: 265: 259: 253: 247: 212:particle physics 188:particle physics 146:electromagnetism 109: 88:In physics, the 81: 74: 70: 67: 61: 56:this article by 47:inline citations 34: 33: 26: 8914: 8913: 8909: 8908: 8907: 8905: 8904: 8903: 8874:Nuclear physics 8839: 8838: 8795: 8790: 8784: 8726: 8725: 8713: 8686: 8669: 8664: 8663: 8655: 8651: 8643: 8639: 8631: 8627: 8613: 8607: 8594: 8590: 8537: 8533: 8462: 8458: 8401: 8397: 8392: 8388: 8373: 8359: 8352: 8347: 8343: 8335: 8333: 8329: 8323: 8312: 8301: 8297: 8292: 8287: 8218: 8193: 8189: 8175: 8163: 8162: 8153: 8152: 8150: 8141: 8137: 8129: 8126: 8125: 8094: 8090: 8088: 8074: 8073: 8064: 8063: 8061: 8059: 8056: 8055: 8021: 8017: 7997: 7993: 7970: 7967: 7966: 7938: 7929: 7908: 7906: 7903: 7902: 7874: 7865: 7850: 7846: 7826: 7822: 7805: 7801: 7799: 7788: 7764: 7753: 7751: 7748: 7747: 7727: 7726: 7715: 7695: 7687: 7684: 7683: 7661: 7654: 7652: 7649: 7648: 7639: 7633: 7624: 7621: 7585: 7570: 7566: 7550: 7541: 7536: 7521: 7509: 7508: 7499: 7498: 7496: 7487: 7482: 7470: 7467: 7466: 7452: 7445: 7444:Its maximum at 7414: 7410: 7394: 7380: 7379: 7370: 7369: 7367: 7365: 7362: 7361: 7351: 7326: 7314: 7313: 7304: 7303: 7301: 7287: 7272: 7268: 7252: 7238: 7202: 7187: 7181: 7178: 7177: 7168: 7131: 7128: 7127: 7117: 7092: 7068: 7066: 7063: 7062: 7045: 7035: 7032: 7008: 7003: 7000: 6999: 6983: 6980: 6979: 6958: 6954: 6952: 6949: 6948: 6928: 6910: 6895: 6891: 6882: 6878: 6871: 6862: 6860: 6856: 6855: 6842: 6834: 6832: 6830: 6827: 6826: 6817: 6811: 6805: 6798: 6783: 6760: 6745: 6741: 6740: 6728: 6724: 6722: 6719: 6718: 6687: 6683: 6681: 6668: 6660: 6658: 6656: 6653: 6652: 6643: 6637: 6634: 6629: 6621: 6615: 6612: 6606: 6580: 6576: 6572: 6567: 6558: 6554: 6552: 6549: 6548: 6542: 6536: 6530: 6521: 6496: 6495: 6477: 6473: 6471: 6468: 6467: 6460: 6431: 6421: 6415: 6408: 6403: 6395: 6390: 6359: 6355: 6346: 6342: 6333: 6329: 6321: 6318: 6317: 6300: 6284: 6278: 6275: 6224: 6221: 6220: 6195: 6191: 6185: 6181: 6168: 6161: 6156: 6155: 6149: 6145: 6140: 6139: 6137: 6125: 6121: 6115: 6111: 6098: 6091: 6086: 6085: 6079: 6075: 6070: 6069: 6067: 6066: 6062: 6050: 6046: 6042: 6040: 6028: 6024: 6009: 6005: 5995: 5990: 5978: 5974: 5959: 5955: 5945: 5940: 5938: 5935: 5934: 5932:Optical theorem 5908: 5904: 5902: 5899: 5898: 5874: 5869: 5868: 5862: 5858: 5853: 5844: 5840: 5830: 5824: 5820: 5818: 5806: 5802: 5795: 5791: 5770: 5766: 5756: 5751: 5742: 5737: 5731: 5728: 5727: 5705: 5700: 5699: 5693: 5689: 5684: 5675: 5671: 5661: 5655: 5651: 5649: 5637: 5633: 5626: 5622: 5601: 5597: 5587: 5582: 5573: 5568: 5562: 5559: 5558: 5541: 5536: 5523: 5518: 5505: 5501: 5499: 5496: 5495: 5478: 5473: 5472: 5466: 5462: 5457: 5451: 5447: 5434: 5432: 5420: 5416: 5410: 5406: 5393: 5388: 5379: 5374: 5373: 5367: 5363: 5358: 5352: 5348: 5335: 5333: 5321: 5317: 5311: 5307: 5294: 5289: 5280: 5275: 5262: 5257: 5244: 5240: 5238: 5235: 5234: 5215: 5211: 5184: 5180: 5170: 5165: 5153: 5149: 5122: 5118: 5108: 5103: 5094: 5089: 5076: 5071: 5058: 5054: 5052: 5049: 5048: 5028: 5024: 5015: 5011: 5007: 4999: 4997: 4985: 4981: 4966: 4962: 4952: 4948: 4939: 4935: 4931: 4923: 4921: 4909: 4905: 4890: 4886: 4884: 4881: 4880: 4864: 4858: 4854: 4845: 4841: 4835: 4831: 4817: 4815: 4812: 4811: 4795: 4789: 4785: 4777: 4775: 4772: 4771: 4768: 4747: 4743: 4734: 4730: 4721: 4717: 4711: 4704: 4700: 4691: 4687: 4678: 4674: 4672: 4669: 4668: 4651: 4647: 4641: 4637: 4629: 4626: 4625: 4605: 4601: 4592: 4588: 4558: 4547: 4534: 4530: 4524: 4520: 4519: 4514: 4505: 4501: 4499: 4496: 4495: 4476: 4471: 4470: 4464: 4460: 4455: 4446: 4441: 4440: 4434: 4430: 4425: 4398: 4387: 4374: 4370: 4364: 4360: 4359: 4354: 4345: 4341: 4339: 4336: 4335: 4319: 4316: 4315: 4299: 4296: 4295: 4294:, permittivity 4279: 4276: 4275: 4242: 4237: 4236: 4227: 4222: 4217: 4207: 4202: 4201: 4192: 4187: 4182: 4180: 4176: 4160: 4151: 4146: 4145: 4143: 4140: 4139: 4117: 4112: 4111: 4102: 4097: 4092: 4090: 4086: 4070: 4061: 4056: 4055: 4053: 4050: 4049: 4027: 4022: 4021: 4012: 4007: 4002: 4000: 3996: 3980: 3971: 3966: 3965: 3963: 3960: 3959: 3925: 3923: 3922: 3913: 3908: 3907: 3901: 3897: 3888: 3884: 3863: 3861: 3860: 3851: 3846: 3845: 3839: 3835: 3826: 3822: 3795: 3793: 3792: 3783: 3778: 3777: 3771: 3767: 3755: 3751: 3749: 3746: 3745: 3725: 3721: 3712: 3708: 3699: 3695: 3686: 3682: 3680: 3677: 3676: 3659: 3655: 3653: 3650: 3649: 3633: 3631: 3628: 3627: 3610: 3605: 3604: 3595: 3590: 3589: 3581: 3579: 3576: 3575: 3551: 3547: 3545: 3542: 3541: 3520: 3511: 3506: 3505: 3504: 3500: 3484: 3476: 3474: 3471: 3470: 3439: 3437: 3436: 3428: 3422: 3418: 3406: 3402: 3400: 3397: 3396: 3377: 3374: 3373: 3357: 3354: 3353: 3337: 3334: 3333: 3304: 3300: 3289: 3283: 3279: 3274: 3266: 3263: 3262: 3234: 3228: 3223: 3222: 3217: 3212: 3203: 3199: 3197: 3194: 3193: 3166: 3161: 3160: 3155: 3151: 3138: 3134: 3130: 3128: 3125: 3124: 3104: 3100: 3087: 3083: 3079: 3077: 3074: 3073: 3051: 3047: 3041: 3037: 3035: 3026: 3022: 3020: 3017: 3016: 2994: 2986: 2978: 2960: 2956: 2950: 2946: 2944: 2935: 2931: 2929: 2926: 2925: 2908: 2904: 2892: 2888: 2886: 2883: 2882: 2865: 2861: 2859: 2856: 2855: 2838: 2834: 2832: 2829: 2828: 2811: 2807: 2805: 2802: 2801: 2794: 2765: 2764: 2746: 2742: 2740: 2737: 2736: 2729: 2704: 2698: 2692: 2685: 2680: 2645: 2641: 2639: 2636: 2635: 2605: 2601: 2592: 2588: 2579: 2575: 2567: 2564: 2563: 2530: 2516: 2514: 2446: 2427: 2402: 2397: 2396: 2392: 2381: 2376: 2375: 2366: 2361: 2360: 2359: 2355: 2341: 2337: 2328: 2324: 2323: 2319: 2297: 2292: 2291: 2287: 2277: 2272: 2271: 2267: 2265: 2262: 2261: 2252: 2229: 2223: 2222: 2221: 2214: 2209: 2208: 2204: 2198: 2197: 2189: 2185: 2179: 2175: 2173: 2167: 2163: 2157: 2153: 2147: 2135: 2131: 2130: 2116: 2115: 2106: 2105: 2103: 2101: 2098: 2097: 2091: 2085: 2082: 2076: 2073: 2067: 2064: 2058: 2047: 2028: 2027: 2014: 2006: 1997: 1996: 1986: 1978: 1974: 1968: 1964: 1962: 1955: 1949: 1948: 1933: 1929: 1922: 1916: 1912: 1908: 1906: 1903: 1902: 1892: 1886: 1883: 1877: 1874: 1868: 1833: 1827: 1826: 1825: 1801: 1800: 1771: 1770: 1761: 1760: 1758: 1756: 1753: 1752: 1724: 1715: 1711: 1700: 1691: 1687: 1672: 1667: 1665: 1664: 1655: 1647: 1644: 1643: 1627: 1593: 1589: 1587: 1554: 1549: 1547: 1546: 1537: 1528: 1524: 1522: 1519: 1518: 1498: 1492: 1463: 1459: 1444: 1439: 1437: 1436: 1427: 1418: 1414: 1412: 1409: 1408: 1399: 1381: 1373:resonance peaks 1340: 1311: 1302: 1281: 1280: 1271: 1270: 1268: 1262: 1257: 1244: 1239: 1224: 1212: 1211: 1202: 1201: 1199: 1190: 1186: 1178: 1175: 1174: 1167: 1155: 1152: 1146: 1145: 1143: 1142: 1136: 1127: 1117: 1109: 1084: 1080: 1058: 1054: 1053: 1051: 1035: 1030: 1001: 1000: 991: 990: 988: 986: 983: 982: 967: 961: 949: 941: 938: 932: 931: 929: 928: 925: 919: 892: 880: 879: 870: 869: 867: 858: 853: 823: 822: 813: 812: 810: 808: 805: 804: 795: 793:azimuthal angle 780: 777: 771: 770: 768: 767: 752: 746: 727: 709: 703: 683:azimuthal angle 676: 666: 660: 649: 634: 631: 627: 595: 591: 585: 581: 573: 570: 569: 560: 542: 499: 498: 489: 488: 486: 484: 481: 480: 473: 469: 459: 456: 450: 430:nuclear physics 405: 393: 389: 388: 377: 374: 373: 364: 350: 338: 326: 296: 291: 283: 280: 279: 261: 255: 249: 242: 232: 105: 82: 71: 65: 62: 52:Please help to 51: 35: 31: 24: 17: 12: 11: 5: 8912: 8902: 8901: 8896: 8891: 8886: 8881: 8876: 8871: 8866: 8861: 8856: 8851: 8849:Atomic physics 8837: 8836: 8831: 8826: 8821: 8816: 8811: 8806: 8801: 8794: 8793:External links 8791: 8789: 8788: 8782: 8767: 8762:R. G. Newton. 8760: 8753: 8746: 8739: 8711: 8698: 8684: 8670: 8668: 8665: 8662: 8661: 8659:, p. 409. 8657:Griffiths 2005 8649: 8647:, p. 576. 8637: 8625: 8611: 8588: 8545:Optics Express 8531: 8456: 8395: 8386: 8371: 8350: 8341: 8321: 8294: 8293: 8291: 8288: 8286: 8285: 8280: 8275: 8270: 8265: 8260: 8255: 8250: 8245: 8240: 8235: 8230: 8225: 8219: 8217: 8214: 8213: 8212: 8201: 8196: 8192: 8188: 8185: 8182: 8178: 8170: 8166: 8160: 8156: 8147: 8144: 8140: 8136: 8133: 8119: 8118: 8107: 8102: 8097: 8093: 8087: 8081: 8077: 8071: 8067: 8049: 8048: 8037: 8033: 8028: 8025: 8020: 8016: 8013: 8009: 8004: 8001: 7996: 7992: 7989: 7986: 7983: 7980: 7977: 7974: 7960: 7959: 7948: 7945: 7941: 7936: 7932: 7927: 7924: 7921: 7918: 7915: 7911: 7896: 7895: 7884: 7881: 7877: 7872: 7868: 7862: 7857: 7854: 7849: 7845: 7842: 7838: 7833: 7830: 7825: 7821: 7818: 7813: 7808: 7804: 7798: 7795: 7791: 7786: 7783: 7780: 7777: 7774: 7771: 7767: 7763: 7760: 7756: 7741: 7740: 7725: 7722: 7718: 7713: 7710: 7707: 7704: 7701: 7698: 7696: 7694: 7690: 7686: 7685: 7682: 7679: 7676: 7673: 7670: 7667: 7664: 7662: 7660: 7657: 7656: 7620: 7617: 7616: 7615: 7604: 7601: 7598: 7595: 7592: 7588: 7582: 7577: 7574: 7569: 7565: 7562: 7557: 7554: 7547: 7544: 7539: 7535: 7531: 7528: 7524: 7516: 7512: 7506: 7502: 7493: 7490: 7485: 7481: 7477: 7474: 7442: 7441: 7430: 7426: 7421: 7418: 7413: 7409: 7406: 7401: 7398: 7393: 7387: 7383: 7377: 7373: 7348: 7347: 7336: 7333: 7329: 7321: 7317: 7311: 7307: 7300: 7297: 7294: 7290: 7284: 7279: 7276: 7271: 7267: 7264: 7259: 7256: 7251: 7248: 7245: 7241: 7236: 7233: 7230: 7227: 7224: 7221: 7218: 7215: 7212: 7209: 7205: 7200: 7197: 7194: 7190: 7185: 7165: 7164: 7153: 7150: 7147: 7144: 7141: 7138: 7135: 7114: 7113: 7102: 7099: 7095: 7090: 7087: 7084: 7081: 7078: 7075: 7071: 7031: 7028: 7015: 7011: 7007: 6987: 6961: 6957: 6945: 6944: 6931: 6926: 6920: 6917: 6913: 6909: 6906: 6903: 6898: 6894: 6890: 6885: 6881: 6877: 6874: 6869: 6865: 6859: 6854: 6848: 6845: 6840: 6837: 6797: 6794: 6780: 6779: 6768: 6763: 6758: 6754: 6751: 6748: 6744: 6739: 6736: 6727: 6712: 6711: 6700: 6695: 6690: 6686: 6680: 6674: 6671: 6666: 6663: 6633: 6630: 6628: 6625: 6619: 6610: 6603: 6602: 6591: 6579: 6575: 6571: 6566: 6557: 6540: 6529: 6526: 6520: 6517: 6516: 6515: 6504: 6499: 6494: 6491: 6488: 6485: 6480: 6476: 6428:number density 6406: 6393: 6379: 6378: 6367: 6358: 6354: 6345: 6341: 6332: 6328: 6325: 6282: 6274: 6271: 6258: 6255: 6252: 6249: 6246: 6243: 6240: 6237: 6234: 6231: 6228: 6207: 6198: 6194: 6188: 6184: 6180: 6177: 6174: 6171: 6164: 6159: 6152: 6148: 6143: 6136: 6128: 6124: 6118: 6114: 6110: 6107: 6104: 6101: 6094: 6089: 6082: 6078: 6073: 6065: 6059: 6053: 6049: 6045: 6039: 6036: 6031: 6027: 6023: 6020: 6012: 6008: 6004: 6001: 5998: 5994: 5989: 5986: 5981: 5977: 5973: 5970: 5962: 5958: 5954: 5951: 5948: 5944: 5919: 5916: 5907: 5883: 5877: 5872: 5865: 5861: 5856: 5847: 5843: 5839: 5836: 5833: 5827: 5823: 5817: 5814: 5809: 5805: 5801: 5798: 5794: 5790: 5787: 5784: 5781: 5773: 5769: 5765: 5762: 5759: 5755: 5750: 5736: 5714: 5708: 5703: 5696: 5692: 5687: 5678: 5674: 5670: 5667: 5664: 5658: 5654: 5648: 5645: 5640: 5636: 5632: 5629: 5625: 5621: 5618: 5615: 5612: 5604: 5600: 5596: 5593: 5590: 5586: 5581: 5567: 5535: 5531: 5517: 5513: 5504: 5481: 5476: 5469: 5465: 5460: 5454: 5450: 5444: 5440: 5437: 5431: 5423: 5419: 5413: 5409: 5405: 5402: 5399: 5396: 5392: 5387: 5382: 5377: 5370: 5366: 5361: 5355: 5351: 5345: 5341: 5338: 5332: 5324: 5320: 5314: 5310: 5306: 5303: 5300: 5297: 5293: 5288: 5274: 5270: 5256: 5252: 5243: 5223: 5218: 5214: 5210: 5207: 5204: 5201: 5198: 5195: 5187: 5183: 5179: 5176: 5173: 5169: 5164: 5161: 5156: 5152: 5148: 5145: 5142: 5139: 5136: 5133: 5125: 5121: 5117: 5114: 5111: 5107: 5102: 5088: 5084: 5070: 5066: 5057: 5036: 5031: 5027: 5018: 5014: 5010: 5005: 5002: 4996: 4993: 4988: 4984: 4980: 4977: 4974: 4969: 4965: 4960: 4955: 4951: 4942: 4938: 4934: 4929: 4926: 4920: 4917: 4912: 4908: 4904: 4901: 4898: 4893: 4889: 4867: 4861: 4857: 4851: 4848: 4844: 4838: 4834: 4830: 4827: 4824: 4820: 4798: 4792: 4788: 4784: 4780: 4767: 4764: 4746: 4742: 4733: 4729: 4720: 4703: 4699: 4690: 4686: 4677: 4654: 4650: 4640: 4636: 4633: 4613: 4608: 4604: 4600: 4595: 4591: 4587: 4584: 4581: 4578: 4575: 4572: 4569: 4566: 4561: 4556: 4553: 4550: 4546: 4537: 4533: 4527: 4523: 4518: 4513: 4504: 4484: 4479: 4474: 4467: 4463: 4458: 4454: 4449: 4444: 4437: 4433: 4428: 4424: 4421: 4418: 4415: 4412: 4409: 4406: 4401: 4396: 4393: 4390: 4386: 4377: 4373: 4367: 4363: 4358: 4353: 4344: 4323: 4303: 4283: 4272:uniform sphere 4251: 4245: 4240: 4235: 4230: 4225: 4220: 4215: 4210: 4205: 4200: 4195: 4190: 4185: 4179: 4175: 4172: 4167: 4164: 4159: 4149: 4126: 4115: 4110: 4105: 4095: 4089: 4085: 4082: 4077: 4074: 4069: 4059: 4036: 4025: 4020: 4015: 4005: 3999: 3995: 3992: 3987: 3984: 3979: 3969: 3956: 3955: 3944: 3941: 3938: 3932: 3928: 3921: 3911: 3904: 3900: 3896: 3887: 3882: 3879: 3876: 3870: 3866: 3859: 3849: 3842: 3838: 3834: 3825: 3820: 3817: 3814: 3811: 3808: 3802: 3798: 3791: 3781: 3774: 3770: 3766: 3763: 3754: 3724: 3720: 3711: 3707: 3698: 3694: 3685: 3662: 3658: 3636: 3608: 3603: 3593: 3588: 3584: 3562: 3559: 3550: 3528: 3523: 3519: 3514: 3509: 3503: 3499: 3496: 3491: 3488: 3483: 3479: 3467: 3466: 3455: 3452: 3446: 3442: 3435: 3431: 3425: 3421: 3417: 3414: 3405: 3381: 3361: 3341: 3312: 3307: 3303: 3299: 3296: 3292: 3286: 3282: 3278: 3273: 3270: 3250: 3247: 3244: 3241: 3237: 3231: 3226: 3220: 3215: 3211: 3202: 3176: 3169: 3154: 3150: 3146: 3137: 3133: 3111: 3103: 3099: 3095: 3090: 3086: 3082: 3070: 3069: 3050: 3044: 3040: 3034: 3029: 3025: 3001: 2993: 2985: 2977: 2974: 2970: 2959: 2953: 2949: 2943: 2938: 2934: 2911: 2907: 2903: 2900: 2891: 2864: 2837: 2810: 2793: 2790: 2785: 2784: 2773: 2768: 2763: 2760: 2757: 2754: 2749: 2745: 2683: 2677: 2676: 2665: 2662: 2659: 2656: 2653: 2648: 2644: 2625: 2624: 2613: 2604: 2600: 2591: 2587: 2578: 2574: 2571: 2529: 2526: 2471:units such as 2445: 2442: 2420: 2419: 2395: 2390: 2379: 2374: 2364: 2358: 2354: 2350: 2340: 2336: 2327: 2322: 2318: 2315: 2312: 2309: 2306: 2290: 2286: 2270: 2249: 2248: 2237: 2232: 2226: 2207: 2201: 2188: 2178: 2166: 2156: 2150: 2145: 2141: 2138: 2134: 2129: 2123: 2119: 2113: 2109: 2089: 2080: 2071: 2062: 2051:reduced masses 2046: 2043: 2042: 2041: 2026: 2021: 2018: 2013: 2005: 2002: 2000: 1998: 1993: 1990: 1977: 1967: 1961: 1958: 1956: 1954: 1951: 1950: 1947: 1944: 1941: 1932: 1928: 1925: 1923: 1915: 1911: 1910: 1881: 1872: 1853: 1852: 1841: 1836: 1830: 1824: 1821: 1818: 1815: 1812: 1809: 1804: 1799: 1796: 1793: 1790: 1787: 1784: 1778: 1774: 1768: 1764: 1746: 1745: 1734: 1731: 1727: 1723: 1718: 1714: 1710: 1707: 1703: 1699: 1694: 1690: 1681: 1678: 1675: 1670: 1662: 1658: 1654: 1651: 1624: 1623: 1612: 1607: 1602: 1599: 1596: 1592: 1586: 1583: 1580: 1577: 1574: 1571: 1563: 1560: 1557: 1552: 1544: 1540: 1536: 1531: 1527: 1489: 1488: 1477: 1472: 1469: 1466: 1462: 1453: 1450: 1447: 1442: 1434: 1430: 1426: 1421: 1417: 1380: 1377: 1333: 1332: 1321: 1318: 1314: 1309: 1305: 1300: 1297: 1294: 1288: 1284: 1278: 1274: 1265: 1260: 1256: 1250: 1247: 1242: 1238: 1234: 1231: 1227: 1219: 1215: 1209: 1205: 1196: 1193: 1189: 1185: 1182: 1162:over the full 1124:number density 1106: 1105: 1094: 1083: 1078: 1075: 1072: 1069: 1066: 1057: 1047: 1044: 1041: 1038: 1034: 1029: 1026: 1023: 1020: 1017: 1014: 1008: 1004: 998: 994: 965: 923: 912: 911: 899: 895: 887: 883: 877: 873: 864: 861: 856: 852: 848: 842: 839: 836: 833: 830: 826: 820: 816: 697: 696: 648: 645: 629: 624: 623: 612: 607: 604: 601: 598: 594: 588: 584: 580: 577: 539: 538: 527: 524: 521: 518: 515: 512: 506: 502: 496: 492: 449: 446: 442:mean free path 422: 421: 408: 403: 399: 396: 392: 387: 384: 381: 361: 360: 357:number density 348: 345:mean free path 336: 320: 319: 308: 302: 299: 295: 290: 287: 231: 228: 102:atomic nucleus 98:alpha particle 84: 83: 38: 36: 29: 15: 9: 6: 4: 3: 2: 8911: 8900: 8897: 8895: 8892: 8890: 8887: 8885: 8882: 8880: 8877: 8875: 8872: 8870: 8867: 8865: 8862: 8860: 8857: 8855: 8852: 8850: 8847: 8846: 8844: 8835: 8832: 8830: 8827: 8825: 8822: 8820: 8817: 8815: 8812: 8810: 8807: 8805: 8802: 8800: 8797: 8796: 8785: 8779: 8775: 8774: 8768: 8765: 8761: 8758: 8754: 8751: 8747: 8744: 8740: 8736: 8730: 8722: 8718: 8714: 8712:1-891389-22-X 8708: 8704: 8699: 8695: 8691: 8687: 8685:0-13-111892-7 8681: 8677: 8672: 8671: 8658: 8653: 8646: 8641: 8634: 8629: 8623: 8619: 8615: 8610: 8603: 8602: 8597: 8592: 8584: 8580: 8576: 8572: 8567: 8562: 8558: 8554: 8550: 8546: 8542: 8535: 8527: 8523: 8519: 8515: 8511: 8507: 8502: 8497: 8493: 8489: 8484: 8479: 8475: 8471: 8467: 8460: 8452: 8448: 8444: 8440: 8436: 8432: 8428: 8424: 8419: 8414: 8411:(4): 044004. 8410: 8406: 8399: 8390: 8382: 8378: 8374: 8372:9788121926331 8368: 8364: 8357: 8355: 8345: 8328: 8324: 8322:92-822-2213-6 8318: 8311: 8310: 8305: 8299: 8295: 8284: 8281: 8279: 8276: 8274: 8271: 8269: 8266: 8264: 8261: 8259: 8256: 8254: 8251: 8249: 8246: 8244: 8241: 8239: 8236: 8234: 8231: 8229: 8228:Flow velocity 8226: 8224: 8221: 8220: 8199: 8194: 8190: 8186: 8183: 8158: 8145: 8142: 8138: 8134: 8131: 8124: 8123: 8122: 8105: 8100: 8095: 8091: 8085: 8069: 8054: 8053: 8052: 8035: 8031: 8026: 8023: 8018: 8014: 8011: 8007: 8002: 7999: 7994: 7990: 7987: 7984: 7981: 7978: 7975: 7972: 7965: 7964: 7963: 7946: 7943: 7934: 7925: 7922: 7919: 7916: 7901: 7900: 7899: 7882: 7879: 7870: 7860: 7855: 7852: 7847: 7843: 7840: 7836: 7831: 7828: 7823: 7819: 7816: 7811: 7806: 7802: 7796: 7793: 7784: 7781: 7775: 7769: 7761: 7758: 7746: 7745: 7744: 7723: 7720: 7711: 7708: 7705: 7702: 7699: 7697: 7692: 7680: 7677: 7674: 7671: 7668: 7665: 7663: 7658: 7647: 7646: 7645: 7642: 7636: 7630: 7627: 7602: 7599: 7596: 7593: 7590: 7580: 7575: 7572: 7567: 7563: 7560: 7555: 7552: 7545: 7542: 7537: 7533: 7529: 7526: 7514: 7504: 7491: 7488: 7483: 7479: 7475: 7472: 7465: 7464: 7463: 7461: 7455: 7448: 7428: 7424: 7419: 7416: 7411: 7407: 7404: 7399: 7396: 7391: 7385: 7375: 7360: 7359: 7358: 7355: 7334: 7331: 7319: 7309: 7298: 7295: 7292: 7282: 7277: 7274: 7269: 7265: 7262: 7257: 7254: 7249: 7246: 7243: 7234: 7231: 7228: 7225: 7222: 7219: 7213: 7207: 7198: 7195: 7192: 7183: 7176: 7175: 7174: 7171: 7151: 7148: 7145: 7142: 7139: 7136: 7133: 7126: 7125: 7124: 7121: 7100: 7097: 7088: 7085: 7082: 7079: 7076: 7073: 7061: 7060: 7059: 7057: 7053: 7048: 7043: 7038: 7027: 7013: 7009: 7005: 6985: 6977: 6959: 6955: 6929: 6924: 6915: 6911: 6907: 6901: 6896: 6892: 6888: 6883: 6879: 6875: 6872: 6867: 6863: 6857: 6852: 6843: 6838: 6835: 6825: 6824: 6823: 6820: 6814: 6808: 6803: 6793: 6790: 6786: 6766: 6761: 6756: 6752: 6749: 6746: 6742: 6737: 6734: 6725: 6717: 6716: 6715: 6698: 6693: 6688: 6684: 6678: 6669: 6664: 6661: 6651: 6650: 6649: 6646: 6640: 6624: 6618: 6609: 6605:The quantity 6589: 6577: 6573: 6569: 6564: 6555: 6547: 6546: 6545: 6539: 6535: 6525: 6502: 6492: 6489: 6486: 6483: 6478: 6474: 6466: 6465: 6464: 6459: 6458:transmittance 6455: 6451: 6447: 6443: 6439: 6434: 6429: 6424: 6418: 6414: 6409: 6400: 6396: 6388: 6384: 6365: 6356: 6352: 6343: 6339: 6330: 6326: 6323: 6316: 6315: 6314: 6312: 6308: 6303: 6298: 6293: 6289: 6281: 6270: 6256: 6253: 6247: 6238: 6232: 6205: 6196: 6186: 6182: 6178: 6175: 6172: 6162: 6150: 6146: 6134: 6126: 6116: 6112: 6108: 6105: 6102: 6092: 6080: 6076: 6063: 6057: 6051: 6047: 6043: 6037: 6029: 6025: 6010: 6006: 6002: 5999: 5996: 5992: 5987: 5979: 5975: 5960: 5956: 5952: 5949: 5946: 5942: 5933: 5917: 5914: 5905: 5895: 5881: 5875: 5863: 5859: 5845: 5841: 5837: 5834: 5831: 5825: 5821: 5815: 5807: 5803: 5792: 5788: 5785: 5782: 5779: 5771: 5767: 5763: 5760: 5757: 5753: 5748: 5734: 5712: 5706: 5694: 5690: 5676: 5672: 5668: 5665: 5662: 5656: 5652: 5646: 5638: 5634: 5623: 5619: 5616: 5613: 5610: 5602: 5598: 5594: 5591: 5588: 5584: 5579: 5565: 5533: 5529: 5515: 5511: 5502: 5479: 5467: 5463: 5452: 5448: 5442: 5438: 5435: 5429: 5421: 5411: 5407: 5403: 5400: 5397: 5390: 5385: 5380: 5368: 5364: 5353: 5349: 5343: 5339: 5336: 5330: 5322: 5312: 5308: 5304: 5301: 5298: 5291: 5286: 5272: 5268: 5254: 5250: 5241: 5216: 5212: 5202: 5199: 5196: 5193: 5185: 5181: 5177: 5174: 5171: 5167: 5162: 5154: 5150: 5140: 5137: 5134: 5131: 5123: 5119: 5115: 5112: 5109: 5105: 5100: 5086: 5082: 5068: 5064: 5055: 5034: 5029: 5025: 5016: 5012: 5008: 5003: 5000: 4994: 4991: 4986: 4982: 4978: 4975: 4972: 4967: 4963: 4958: 4953: 4949: 4940: 4936: 4932: 4927: 4924: 4918: 4915: 4910: 4906: 4902: 4899: 4896: 4891: 4887: 4859: 4855: 4849: 4846: 4836: 4832: 4828: 4822: 4790: 4786: 4782: 4763: 4744: 4740: 4731: 4727: 4718: 4701: 4697: 4688: 4684: 4675: 4652: 4648: 4638: 4634: 4631: 4606: 4602: 4598: 4593: 4589: 4576: 4573: 4570: 4567: 4554: 4551: 4548: 4544: 4535: 4531: 4525: 4521: 4516: 4511: 4502: 4477: 4465: 4461: 4452: 4447: 4435: 4431: 4416: 4413: 4410: 4407: 4394: 4391: 4388: 4384: 4375: 4371: 4365: 4361: 4356: 4351: 4342: 4321: 4301: 4281: 4273: 4269: 4264: 4249: 4243: 4233: 4228: 4223: 4213: 4208: 4198: 4193: 4188: 4177: 4173: 4170: 4165: 4162: 4157: 4124: 4108: 4103: 4087: 4083: 4080: 4075: 4072: 4067: 4034: 4018: 4013: 3997: 3993: 3990: 3985: 3982: 3977: 3942: 3939: 3936: 3919: 3902: 3898: 3894: 3885: 3880: 3877: 3874: 3857: 3840: 3836: 3832: 3823: 3818: 3815: 3812: 3809: 3806: 3789: 3772: 3768: 3764: 3761: 3752: 3744: 3743: 3742: 3722: 3718: 3709: 3705: 3696: 3692: 3683: 3660: 3656: 3601: 3586: 3560: 3557: 3548: 3526: 3517: 3512: 3501: 3497: 3494: 3489: 3486: 3481: 3453: 3450: 3433: 3423: 3419: 3415: 3412: 3403: 3395: 3394: 3393: 3379: 3359: 3339: 3330: 3328: 3305: 3301: 3297: 3290: 3284: 3280: 3276: 3271: 3268: 3245: 3242: 3235: 3229: 3209: 3200: 3191: 3174: 3167: 3152: 3148: 3144: 3135: 3131: 3109: 3101: 3097: 3093: 3088: 3084: 3080: 3048: 3042: 3038: 3032: 3027: 3023: 3015: 3014: 3013: 2999: 2991: 2983: 2975: 2972: 2968: 2957: 2951: 2947: 2941: 2936: 2932: 2909: 2905: 2901: 2898: 2889: 2862: 2835: 2827:, scattering 2808: 2799: 2789: 2771: 2761: 2758: 2755: 2752: 2747: 2743: 2735: 2734: 2733: 2728: 2727:transmittance 2724: 2720: 2716: 2712: 2707: 2701: 2695: 2691: 2686: 2663: 2660: 2657: 2654: 2651: 2646: 2642: 2634: 2633: 2632: 2630: 2611: 2602: 2598: 2589: 2585: 2576: 2572: 2569: 2562: 2561: 2560: 2558: 2554: 2550: 2545: 2543: 2539: 2535: 2525: 2522: 2512: 2508: 2503: 2501: 2497: 2493: 2489: 2485: 2480: 2478: 2474: 2470: 2466: 2462: 2457: 2455: 2451: 2448:Although the 2441: 2439: 2435: 2430: 2425: 2393: 2388: 2372: 2356: 2352: 2348: 2338: 2334: 2325: 2320: 2316: 2313: 2310: 2307: 2304: 2288: 2284: 2268: 2260: 2259: 2258: 2255: 2235: 2230: 2205: 2186: 2176: 2164: 2154: 2148: 2143: 2139: 2136: 2132: 2127: 2111: 2096: 2095: 2094: 2088: 2079: 2070: 2061: 2056: 2052: 2024: 2019: 2016: 2011: 2003: 2001: 1991: 1988: 1975: 1965: 1959: 1957: 1952: 1945: 1942: 1939: 1930: 1926: 1924: 1913: 1901: 1900: 1899: 1895: 1889: 1880: 1871: 1865: 1861: 1856: 1839: 1834: 1819: 1816: 1813: 1807: 1797: 1791: 1788: 1785: 1766: 1751: 1750: 1749: 1732: 1716: 1712: 1708: 1692: 1688: 1673: 1649: 1642: 1641: 1640: 1637: 1635: 1630: 1610: 1605: 1600: 1597: 1594: 1590: 1581: 1578: 1575: 1569: 1555: 1529: 1525: 1517: 1516: 1515: 1512: 1510: 1506: 1501: 1495: 1475: 1470: 1467: 1464: 1460: 1445: 1419: 1415: 1407: 1406: 1405: 1402: 1398: 1394: 1393:wave function 1390: 1387:formalism of 1386: 1376: 1374: 1369: 1367: 1362: 1360: 1354: 1352: 1348: 1343: 1338: 1319: 1316: 1307: 1298: 1295: 1292: 1276: 1263: 1258: 1254: 1248: 1245: 1240: 1236: 1232: 1207: 1194: 1191: 1187: 1183: 1180: 1173: 1172: 1171: 1170:steradians): 1165: 1150: 1139: 1133: 1130: 1125: 1120: 1115: 1092: 1081: 1073: 1070: 1067: 1055: 1039: 1036: 1032: 1027: 1021: 1018: 1015: 996: 981: 980: 979: 975: 971: 964: 957: 953: 936: 922: 916: 897: 875: 862: 859: 854: 850: 846: 837: 834: 831: 818: 803: 802: 801: 798: 794: 788: 775: 764: 760: 756: 749: 743: 739: 735: 731: 725: 724:gravitational 721: 717: 712: 706: 702: 692: 688: 687: 686: 684: 679: 674: 669: 663: 658: 654: 644: 642: 637: 610: 605: 602: 599: 596: 592: 586: 578: 568: 567: 566: 563: 558: 554: 550: 545: 525: 519: 516: 513: 510: 504: 479: 478: 477: 476:according to 468: 463: 455: 445: 443: 439: 435: 431: 426: 406: 401: 397: 394: 390: 385: 382: 379: 372: 371: 370: 367: 358: 353: 349: 346: 341: 337: 334: 329: 325: 324: 323: 306: 300: 297: 293: 288: 285: 278: 277: 276: 273: 264: 258: 252: 246: 240: 236: 227: 225: 220: 219:cross section 215: 213: 209: 205: 200: 197: 196:reaction rate 194:The measured 192: 189: 185: 181: 176: 174: 170: 166: 162: 158: 153: 151: 147: 143: 140: 136: 132: 128: 127:cross section 123: 121: 117: 113: 108: 103: 99: 95: 91: 90:cross section 80: 77: 69: 59: 55: 49: 48: 42: 37: 28: 27: 22: 8899:Spectroscopy 8772: 8763: 8756: 8749: 8742: 8702: 8675: 8667:Bibliography 8652: 8640: 8628: 8608: 8599: 8591: 8548: 8544: 8534: 8473: 8469: 8459: 8408: 8404: 8398: 8389: 8362: 8344: 8334:, retrieved 8308: 8298: 8120: 8050: 7961: 7897: 7742: 7640: 7634: 7631: 7625: 7622: 7453: 7446: 7443: 7353: 7349: 7169: 7166: 7119: 7115: 7046: 7036: 7033: 6946: 6818: 6812: 6806: 6799: 6788: 6784: 6781: 6713: 6644: 6638: 6635: 6616: 6607: 6604: 6537: 6531: 6522: 6432: 6422: 6416: 6404: 6398: 6391: 6380: 6311:luminescence 6301: 6296: 6292:permittivity 6279: 6276: 5896: 4769: 4271: 4265: 3957: 3468: 3331: 3071: 2795: 2786: 2705: 2699: 2693: 2681: 2678: 2626: 2546: 2542:permittivity 2531: 2504: 2488:nephelometry 2481: 2458: 2447: 2428: 2421: 2253: 2250: 2086: 2077: 2068: 2059: 2048: 1893: 1887: 1878: 1869: 1857: 1854: 1747: 1638: 1628: 1625: 1513: 1508: 1504: 1499: 1493: 1490: 1400: 1382: 1370: 1363: 1355: 1350: 1346: 1341: 1334: 1148: 1137: 1134: 1128: 1118: 1107: 973: 969: 962: 955: 951: 948:at an angle 934: 920: 917: 913: 796: 789: 773: 762: 758: 754: 747: 741: 737: 733: 729: 710: 704: 698: 677: 672: 667: 661: 650: 635: 625: 561: 548: 543: 540: 461: 457: 427: 423: 365: 362: 351: 339: 327: 321: 269: 262: 256: 250: 244: 238: 218: 216: 201: 193: 177: 168: 164: 160: 157:differential 154: 129:is the area 126: 124: 106: 89: 87: 72: 66:January 2022 63: 44: 8869:Measurement 8645:Taylor 2005 8633:Taylor 2005 6810:and energy 6438:path length 2788:this area. 2711:path length 2496:meteorology 2484:centimetres 1164:solid angle 651:Consider a 454:Attenuation 58:introducing 8894:Scattering 8843:Categories 8748:P. Roman, 8336:2021-12-16 8290:References 8051:we obtain 6442:absorbance 6413:wavelength 6383:absorbance 6307:absorption 4274:of radius 3190:plane wave 2798:Mie theory 2690:wavelength 2553:scattering 2549:absorption 2538:wavelength 2467:. Smaller 557:absorption 553:scattering 452:See also: 131:transverse 41:references 8729:cite book 8575:1094-4087 8510:2041-1723 8483:1308.0547 8451:119234614 8443:2040-8978 8418:1210.5730 8381:943495167 8187:π 8181:Ω 8169:Ω 8159:σ 8146:π 8139:∮ 8132:σ 8080:Ω 8070:σ 8024:θ 8015:⁡ 8000:θ 7991:⁡ 7979:θ 7976:⁡ 7944:φ 7935:θ 7926:θ 7923:⁡ 7914:Ω 7880:φ 7871:θ 7853:θ 7844:⁡ 7829:θ 7820:⁡ 7794:φ 7782:× 7759:σ 7721:α 7712:α 7709:⁡ 7678:α 7675:⁡ 7591:θ 7573:θ 7564:⁡ 7546:π 7534:∫ 7527:θ 7515:θ 7505:σ 7492:π 7480:∫ 7473:σ 7417:θ 7408:⁡ 7386:θ 7376:σ 7332:θ 7320:θ 7310:σ 7293:θ 7275:θ 7266:⁡ 7244:α 7235:α 7232:⁡ 7193:σ 7149:α 7143:− 7140:π 7134:θ 7098:α 7089:α 7086:⁡ 7052:light ray 6986:θ 6956:ε 6908:θ 6902:⁡ 6880:ε 6876:π 6847:Ω 6839:σ 6738:π 6726:σ 6673:Ω 6665:σ 6578:σ 6493:⁡ 6487:− 6479:λ 6446:logarithm 6357:σ 6344:σ 6331:σ 6324:σ 6248:μ 6242:ℑ 6233:ε 6227:ℑ 6183:μ 6179:μ 6176:π 6147:α 6113:ε 6109:ε 6106:π 6077:α 6038:≥ 6026:α 6019:ℑ 6007:μ 6003:μ 6000:π 5976:α 5969:ℑ 5957:ε 5953:ε 5950:π 5906:σ 5860:α 5842:μ 5838:μ 5835:π 5816:− 5804:α 5797:ℑ 5786:π 5780:⋅ 5768:μ 5764:μ 5761:π 5735:σ 5691:α 5673:ε 5669:ε 5666:π 5647:− 5635:α 5628:ℑ 5617:π 5611:⋅ 5599:ε 5595:ε 5592:π 5566:σ 5534:σ 5516:σ 5503:σ 5464:α 5439:π 5430:⋅ 5408:μ 5404:μ 5401:π 5365:α 5340:π 5331:⋅ 5309:ε 5305:ε 5302:π 5273:σ 5255:σ 5242:σ 5213:α 5206:ℑ 5200:π 5194:⋅ 5182:μ 5178:μ 5175:π 5151:α 5144:ℑ 5138:π 5132:⋅ 5120:ε 5116:ε 5113:π 5087:σ 5069:σ 5056:σ 5004:μ 4992:⋅ 4983:μ 4979:π 4964:α 4928:ε 4916:⋅ 4907:ε 4903:π 4888:α 4856:α 4847:− 4833:μ 4829:μ 4787:α 4702:σ 4689:σ 4676:σ 4583:ℜ 4560:∞ 4545:∑ 4400:∞ 4385:∑ 4322:μ 4302:ε 4234:× 4229:∗ 4199:× 4194:∗ 4174:⁡ 4148:Π 4109:× 4104:∗ 4084:⁡ 4058:Π 4019:× 4014:∗ 3994:⁡ 3968:Π 3931:^ 3920:⋅ 3910:Π 3899:∮ 3869:^ 3858:⋅ 3848:Π 3837:∮ 3813:≡ 3801:^ 3790:⋅ 3780:Π 3769:∮ 3765:− 3706:− 3518:× 3513:∗ 3498:⁡ 3478:Π 3445:^ 3434:⋅ 3430:Π 3420:∮ 3416:− 3352:(surface 3302:ε 3298:ε 3281:μ 3277:μ 3269:η 3246:η 3089:α 3043:α 3028:α 3024:σ 2973:α 2958:σ 2952:α 2948:σ 2937:α 2902:π 2890:σ 2762:⁡ 2756:− 2748:λ 2715:logarithm 2661:σ 2647:λ 2603:σ 2590:σ 2577:σ 2570:σ 2373:− 2353:δ 2335:− 2317:δ 2311:π 2305:− 2289:δ 2140:π 2122:Ω 2112:σ 2012:× 1953:σ 1943:σ 1864:colliding 1860:particles 1820:ϕ 1814:θ 1792:ϕ 1786:θ 1777:Ω 1767:σ 1713:ϕ 1693:− 1689:ϕ 1680:∞ 1677:→ 1669:⟶ 1650:ϕ 1582:ϕ 1576:θ 1562:∞ 1559:→ 1551:⟶ 1526:ϕ 1452:∞ 1449:→ 1441:⟶ 1420:− 1416:ϕ 1337:qualifier 1317:φ 1308:θ 1299:θ 1296:⁡ 1287:Ω 1277:σ 1264:π 1255:∫ 1249:π 1237:∫ 1230:Ω 1218:Ω 1208:σ 1195:π 1188:∮ 1181:σ 1074:φ 1068:θ 1046:Ω 1043:Δ 1022:φ 1016:θ 1007:Ω 997:σ 898:φ 886:Ω 876:σ 863:π 851:∫ 838:θ 835:⁡ 819:σ 753:dΩ = sin 716:coulombic 653:classical 603:σ 597:− 583:Φ 576:Φ 523:Φ 520:σ 514:− 495:Φ 386:π 380:σ 335:unit: m), 301:λ 286:σ 239:Figure 1. 139:inelastic 8721:55729992 8694:53926857 8583:20589003 8526:15832637 8518:24598730 8327:archived 8306:(2006), 8216:See also 7054:and the 6627:Examples 6402:, where 3741:, where 3261:, where 2511:ångström 2469:prefixed 2424:S-matrix 1898:we have 1505:relative 1503:are the 1397:momentum 720:magnetic 434:neutrons 8553:Bibcode 8488:Bibcode 8423:Bibcode 6974:is the 6454:natural 6450:decadic 6436:is the 3325:is the 2723:natural 2719:decadic 2709:is the 2450:SI unit 2432:is the 2426:. Here 2055:momenta 2049:If the 1389:quantum 1383:In the 1159:⁠ 1144:⁠ 1122:is the 945:⁠ 930:⁠ 784:⁠ 769:⁠ 681:is the 671:is the 355:is the 343:is the 204:nuclear 180:nuclear 150:gravity 142:spheres 135:scatter 54:improve 8780: 8759:, 1994 8752:, 1969 8745:, 1964 8719: 8709: 8692: 8682: 8581: 8573: 8524: 8516: 8508: 8449: 8441: 8379: 8369: 8319: 7352:dΩ = d 7056:radius 6947:where 6430:, and 4624:Where 4138:, and 3958:where 3469:where 3072:where 2679:where 2555:, and 2519: 2507:X-rays 1626:where 1491:where 626:where 541:where 465:, the 322:where 210:, and 208:atomic 186:, and 184:atomic 43:, but 8596:IUPAC 8522:S2CID 8478:arXiv 8447:S2CID 8413:arXiv 8330:(PDF) 8313:(PDF) 2444:Units 1108:Here 270:In a 116:barns 112:sigma 8778:ISBN 8735:link 8717:OCLC 8707:ISBN 8690:OCLC 8680:ISBN 8612:scat 8579:PMID 8571:ISSN 8514:PMID 8506:ISSN 8439:ISSN 8377:OCLC 8367:ISBN 8317:ISBN 7638:and 7460:lens 6642:and 6620:scat 6611:scat 6582:scat 5915:> 5726:and 5557:are 4810:and 4643:host 3558:> 2962:geom 2894:geom 2475:and 2075:and 2053:and 1497:and 699:The 467:flux 8618:doi 8616:". 8561:doi 8496:doi 8431:doi 8012:cos 7988:sin 7973:sin 7920:sin 7841:cos 7817:sin 7706:cos 7672:sin 7629:. 7561:sin 7456:= 0 7449:= π 7405:sin 7263:sin 7229:cos 7173:is 7083:cos 6893:sin 6800:In 6730:tot 6570:3.9 6490:log 6399:Clσ 6361:lum 6335:abs 5910:abs 5744:(m) 5739:abs 5575:(e) 5570:abs 5543:(m) 5538:abs 5525:(e) 5520:abs 5507:abs 5282:(m) 5264:(e) 5096:(m) 5091:ext 5078:(e) 5073:ext 5060:ext 4749:abs 4723:ext 4706:abs 4680:ext 4507:ext 4263:. 4153:ext 3915:ext 3890:ext 3727:ext 3392:is 3205:inc 3140:inc 3053:inc 2996:abs 2980:ext 2924:as 2867:abs 2813:ext 2759:log 2607:lum 2581:abs 2517:000 1896:≪ 1 1349:or 1293:sin 1116:), 1086:inc 1060:out 966:out 924:inc 832:cos 549:all 272:gas 167:or 148:or 8845:: 8731:}} 8727:{{ 8715:. 8688:. 8598:, 8577:. 8569:. 8559:. 8549:18 8547:. 8543:. 8520:. 8512:. 8504:. 8494:. 8486:. 8472:. 8468:. 8445:. 8437:. 8429:. 8421:. 8409:15 8407:. 8375:. 8353:^ 8325:, 7357:) 6873:16 6787:+ 6608:Cσ 6544:: 6463:: 6420:, 6397:= 6348:sc 6313:: 6283:sc 6269:. 5277:sc 5259:sc 5246:sc 4736:sc 4713:or 4693:sc 4347:sc 4171:Re 4081:Re 4048:, 3991:Re 3495:Re 3329:. 2988:sc 2840:sc 2732:: 2697:, 2594:sc 2551:, 2521:pm 2515:10 2502:. 2492:μm 2477:μb 2473:mb 2465:fm 2440:. 2084:, 2066:, 1894:Nσ 1404:: 1168:4π 1156:dΩ 1114:sr 1110:ΔΩ 942:dΩ 787:. 781:dΩ 732:= 722:, 718:, 685:. 643:. 555:, 474:dΦ 333:SI 214:. 206:, 182:, 122:. 8786:. 8737:) 8723:. 8696:. 8620:: 8609:σ 8585:. 8563:: 8555:: 8528:. 8498:: 8490:: 8480:: 8474:5 8453:. 8433:: 8425:: 8415:: 8383:. 8200:. 8195:2 8191:a 8184:= 8177:d 8165:d 8155:d 8143:4 8135:= 8106:. 8101:4 8096:2 8092:a 8086:= 8076:d 8066:d 8036:, 8032:) 8027:2 8019:( 8008:) 8003:2 7995:( 7985:2 7982:= 7947:. 7940:d 7931:d 7917:= 7910:d 7883:. 7876:d 7867:d 7861:) 7856:2 7848:( 7837:) 7832:2 7824:( 7812:2 7807:2 7803:a 7797:= 7790:d 7785:r 7779:) 7776:r 7773:( 7770:r 7766:d 7762:= 7755:d 7724:, 7717:d 7703:a 7700:= 7693:r 7689:d 7681:, 7669:a 7666:= 7659:r 7641:φ 7635:r 7626:a 7603:. 7600:r 7597:2 7594:= 7587:d 7581:) 7576:2 7568:( 7556:2 7553:r 7543:2 7538:0 7530:= 7523:d 7511:d 7501:d 7489:2 7484:0 7476:= 7454:θ 7447:θ 7429:. 7425:) 7420:2 7412:( 7400:2 7397:r 7392:= 7382:d 7372:d 7354:θ 7335:. 7328:d 7316:d 7306:d 7299:I 7296:= 7289:d 7283:) 7278:2 7270:( 7258:2 7255:r 7250:I 7247:= 7240:d 7226:r 7223:I 7220:= 7217:) 7214:x 7211:( 7208:x 7204:d 7199:I 7196:= 7189:d 7184:I 7170:I 7152:. 7146:2 7137:= 7120:α 7118:2 7101:. 7094:d 7080:r 7077:= 7074:x 7070:d 7047:α 7037:r 7014:r 7010:/ 7006:1 6960:0 6930:2 6925:) 6919:) 6916:2 6912:/ 6905:( 6897:2 6889:E 6884:0 6868:Q 6864:q 6858:( 6853:= 6844:d 6836:d 6819:Q 6813:E 6807:q 6789:R 6785:r 6767:. 6762:2 6757:) 6753:R 6750:+ 6747:r 6743:( 6735:= 6699:, 6694:4 6689:2 6685:R 6679:= 6670:d 6662:d 6645:r 6639:R 6617:b 6590:. 6574:C 6565:= 6560:V 6556:L 6541:V 6538:L 6503:. 6498:T 6484:= 6475:A 6461:T 6448:( 6433:l 6423:C 6417:λ 6407:λ 6405:A 6394:λ 6392:A 6366:. 6353:+ 6340:+ 6327:= 6302:σ 6299:( 6280:σ 6257:0 6254:= 6251:) 6245:( 6239:= 6236:) 6230:( 6206:] 6197:2 6193:) 6187:0 6173:4 6170:( 6163:2 6158:| 6151:m 6142:| 6135:+ 6127:2 6123:) 6117:0 6103:4 6100:( 6093:2 6088:| 6081:e 6072:| 6064:[ 6058:3 6052:3 6048:k 6044:2 6035:) 6030:m 6022:( 6011:0 5997:4 5993:1 5988:+ 5985:) 5980:e 5972:( 5961:0 5947:4 5943:1 5918:0 5882:] 5876:2 5871:| 5864:m 5855:| 5846:0 5832:6 5826:3 5822:k 5813:) 5808:m 5800:( 5793:[ 5789:k 5783:4 5772:0 5758:4 5754:1 5749:= 5713:] 5707:2 5702:| 5695:e 5686:| 5677:0 5663:6 5657:3 5653:k 5644:) 5639:e 5631:( 5624:[ 5620:k 5614:4 5603:0 5589:4 5585:1 5580:= 5530:+ 5512:= 5480:2 5475:| 5468:m 5459:| 5453:4 5449:k 5443:3 5436:8 5422:2 5418:) 5412:0 5398:4 5395:( 5391:1 5386:+ 5381:2 5376:| 5369:e 5360:| 5354:4 5350:k 5344:3 5337:8 5323:2 5319:) 5313:0 5299:4 5296:( 5292:1 5287:= 5269:+ 5251:= 5222:) 5217:m 5209:( 5203:k 5197:4 5186:0 5172:4 5168:1 5163:+ 5160:) 5155:e 5147:( 5141:k 5135:4 5124:0 5110:4 5106:1 5101:= 5083:+ 5065:= 5035:. 5030:1 5026:b 5017:3 5013:k 5009:2 5001:3 4995:i 4987:0 4976:4 4973:= 4968:m 4959:, 4954:1 4950:a 4941:3 4937:k 4933:2 4925:3 4919:i 4911:0 4900:4 4897:= 4892:e 4866:H 4860:m 4850:1 4843:) 4837:0 4826:( 4823:= 4819:m 4797:E 4791:e 4783:= 4779:p 4745:Q 4741:+ 4732:Q 4728:= 4719:Q 4698:+ 4685:= 4653:0 4649:k 4639:n 4635:= 4632:k 4612:) 4607:n 4603:b 4599:+ 4594:n 4590:a 4586:( 4580:) 4577:1 4574:+ 4571:n 4568:2 4565:( 4555:1 4552:= 4549:n 4536:2 4532:a 4526:2 4522:k 4517:2 4512:= 4503:Q 4483:) 4478:2 4473:| 4466:n 4462:b 4457:| 4453:+ 4448:2 4443:| 4436:n 4432:a 4427:| 4423:( 4420:) 4417:1 4414:+ 4411:n 4408:2 4405:( 4395:1 4392:= 4389:n 4376:2 4372:a 4366:2 4362:k 4357:2 4352:= 4343:Q 4282:a 4250:] 4244:s 4239:H 4224:i 4219:E 4214:+ 4209:i 4204:H 4189:s 4184:E 4178:[ 4166:2 4163:1 4158:= 4125:] 4119:s 4114:H 4099:s 4094:E 4088:[ 4076:2 4073:1 4068:= 4063:s 4035:] 4029:i 4024:H 4009:i 4004:E 3998:[ 3986:2 3983:1 3978:= 3973:i 3943:. 3940:A 3937:d 3927:r 3903:A 3895:= 3886:W 3881:, 3878:A 3875:d 3865:r 3853:s 3841:A 3833:= 3828:s 3824:W 3819:, 3816:0 3810:A 3807:d 3797:r 3785:i 3773:A 3762:= 3757:i 3753:W 3723:W 3719:+ 3714:s 3710:W 3701:i 3697:W 3693:= 3688:a 3684:W 3661:a 3657:W 3635:H 3612:s 3607:E 3602:+ 3597:i 3592:E 3587:= 3583:E 3561:0 3553:a 3549:W 3527:] 3522:H 3508:E 3502:[ 3490:2 3487:1 3482:= 3454:A 3451:d 3441:r 3424:A 3413:= 3408:a 3404:W 3380:A 3360:A 3340:r 3311:) 3306:0 3295:( 3291:/ 3285:0 3272:= 3249:) 3243:2 3240:( 3236:/ 3230:2 3225:| 3219:E 3214:| 3210:= 3201:I 3175:] 3168:2 3163:m 3158:W 3153:[ 3149:= 3145:] 3136:I 3132:[ 3110:] 3106:W 3102:[ 3098:= 3094:] 3085:W 3081:[ 3049:I 3039:W 3033:= 3000:. 2992:, 2984:, 2976:= 2969:, 2942:= 2933:Q 2910:2 2906:a 2899:= 2863:Q 2836:Q 2809:Q 2772:. 2767:T 2753:= 2744:A 2730:T 2717:( 2706:l 2700:C 2694:λ 2684:λ 2682:A 2664:, 2658:l 2655:C 2652:= 2643:A 2612:. 2599:+ 2586:+ 2573:= 2461:b 2454:m 2429:δ 2404:i 2399:f 2394:T 2389:) 2383:f 2378:p 2368:i 2363:p 2357:( 2349:) 2343:i 2339:E 2330:f 2326:E 2321:( 2314:i 2308:2 2299:i 2294:f 2285:= 2279:i 2274:f 2269:S 2254:T 2236:, 2231:2 2225:| 2216:i 2211:f 2206:T 2200:| 2191:i 2187:p 2181:f 2177:p 2169:f 2165:m 2159:i 2155:m 2149:4 2144:) 2137:2 2133:( 2128:= 2118:d 2108:d 2090:f 2087:p 2081:f 2078:m 2072:i 2069:p 2063:i 2060:m 2025:. 2020:N 2017:1 2004:= 1992:N 1989:1 1980:i 1976:I 1970:r 1966:I 1960:= 1946:, 1940:N 1935:i 1931:I 1927:= 1918:r 1914:I 1888:N 1882:i 1879:I 1873:r 1870:I 1840:. 1835:2 1829:| 1823:) 1817:, 1811:( 1808:f 1803:| 1798:= 1795:) 1789:, 1783:( 1773:d 1763:d 1733:. 1730:) 1726:r 1722:( 1717:+ 1709:+ 1706:) 1702:r 1698:( 1674:r 1661:) 1657:r 1653:( 1629:f 1611:, 1606:r 1601:r 1598:k 1595:i 1591:e 1585:) 1579:, 1573:( 1570:f 1556:r 1543:) 1539:r 1535:( 1530:+ 1500:r 1494:z 1476:, 1471:z 1468:k 1465:i 1461:e 1446:r 1433:) 1429:r 1425:( 1401:k 1342:σ 1320:. 1313:d 1304:d 1283:d 1273:d 1259:0 1246:2 1241:0 1233:= 1226:d 1214:d 1204:d 1192:4 1184:= 1166:( 1153:/ 1149:σ 1147:d 1138:σ 1129:t 1119:n 1093:. 1082:F 1077:) 1071:, 1065:( 1056:F 1040:t 1037:n 1033:1 1028:= 1025:) 1019:, 1013:( 1003:d 993:d 976:) 974:φ 972:, 970:θ 968:( 963:F 958:) 956:φ 954:, 952:θ 950:( 939:/ 935:σ 933:d 921:F 910:. 894:d 882:d 872:d 860:2 855:0 847:= 841:) 829:( 825:d 815:d 797:φ 778:/ 774:σ 772:d 763:φ 761:d 759:θ 757:d 755:θ 748:θ 742:b 740:d 738:φ 736:d 734:b 730:σ 728:d 711:θ 705:b 678:φ 668:θ 662:z 636:z 630:0 628:Φ 611:, 606:z 600:n 593:e 587:0 579:= 562:n 544:σ 526:, 517:n 511:= 505:z 501:d 491:d 470:Φ 462:z 460:d 407:2 402:) 398:r 395:2 391:( 383:= 366:r 352:n 340:λ 328:σ 307:, 298:n 294:1 289:= 266:. 263:λ 257:n 251:σ 245:r 243:2 110:( 107:σ 79:) 73:( 68:) 64:( 50:. 23:.

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