2447:
1248:
299:
1523:
837:
2249:
The following table lists some typical values for air at different pressures at room temperature. Note that different definitions of the molecular diameter, as well as different assumptions about the value of atmospheric pressure (100 vs 101.3 kPa) and room temperature (293.17 K vs 296.15 K or even
63:
Imagine a beam of particles being shot through a target, and consider an infinitesimally thin slab of the target (see the figure). The atoms (or particles) that might stop a beam particle are shown in red. The magnitude of the mean free path depends on the characteristics of the system. Assuming
3095:
The effective mean free path of a nucleon in nuclear matter must be somewhat larger than the nuclear dimensions in order to allow the use of the independent particle model. This requirement seems to be in contradiction to the assumptions made in the theory ... We are facing here one of the
1052:
1997:
171:
685:
1871:
of a molecule is defined in terms of the mean free path. Typically, gas molecules do not behave like hard spheres, but rather attract each other at larger distances and repel each other at shorter distances, as can be described with a
1374:
452:
704:
2166:
2808:
1243:{\displaystyle {\overline {\mathbf {v} _{\rm {relative}}^{2}}}={\overline {(\mathbf {v} _{1}-\mathbf {v} _{2})^{2}}}={\overline {\mathbf {v} _{1}^{2}+\mathbf {v} _{2}^{2}-2\mathbf {v} _{1}\cdot \mathbf {v} _{2}}}.}
1367:
1808:
1889:
3564:
2566:
2658:(HVL): a material with a thickness of one HVL will attenuate 50% of photons. A standard x-ray image is a transmission image, an image with negative logarithm of its intensities is sometimes called a
2959:
938:, is the average distance the particle travels between collisions with other moving particles. The derivation above assumed the target particles to be at rest; therefore, in reality, the formula
1608:
2240:
1003:
relative to the velocities of an ensemble of identical particles with random locations. In that case, the motions of target particles are comparatively negligible, hence the relative velocity
3374:
Mengual, O.; Meunier, G.; Cayré, I.; Puech, K.; Snabre, P. (1999). "TURBISCAN MA 2000: multiple light scattering measurement for concentrated emulsion and suspension instability analysis".
2073:
1675:
3032:
1042:
912:
113:
981:
513:
364:
1715:
294:{\displaystyle {\mathcal {P}}({\text{stopping within }}dx)={\frac {{\text{Area}}_{\text{atoms}}}{{\text{Area}}_{\text{slab}}}}={\frac {\sigma nL^{2}\,dx}{L^{2}}}=n\sigma \,dx,}
1308:
1279:
1549:
556:
3243:
2839:
2698:
2721:
1518:{\displaystyle v_{\rm {rel}}={\sqrt {\overline {\mathbf {v} _{\rm {relative}}^{2}}}}={\sqrt {\overline {\mathbf {v} _{1}^{2}+\mathbf {v} _{2}^{2}}}}={\sqrt {2}}v.}
3352:
3322:
2600:
2020:
1862:
1842:
1735:
1001:
379:
43:) travels before substantially changing its direction or energy (or, in a specific context, other properties), typically as a result of one or more successive
2629:
with probabilities depending on their energy, as a result their distribution changes in process called spectrum hardening. Because of spectrum hardening, the
832:{\displaystyle \langle x\rangle {\overset {\text{def}}{=}}\int _{0}^{\infty }xd{\mathcal {P}}(x)=\int _{0}^{\infty }{\frac {x}{\ell }}e^{-x/\ell }\,dx=\ell .}
2872:, however, the film thickness can be smaller than the predicted mean free path, making surface scattering much more noticeable, effectively increasing the
2085:
2487:
is the average distance a photon travels between collisions with atoms of the target material. It depends on the material and the energy of the photons:
2733:
1047:
If, on the other hand, the beam particle is part of an established equilibrium with identical particles, then the square of relative velocity is:
165:. The probability that a beam particle will be stopped in that slab is the net area of the stopping atoms divided by the total area of the slab:
3517:
3482:
1992:{\displaystyle \ell ={\frac {\mu }{\rho }}{\sqrt {\frac {\pi m}{2k_{\text{B}}T}}}={\frac {\mu }{p}}{\sqrt {\frac {\pi k_{\text{B}}T}{2m}}},}
1313:
2850:
1743:
315:
The drop in beam intensity equals the incoming beam intensity multiplied by the probability of the particle being stopped within the slab:
56:
536:
distance traveled by a beam particle before being stopped. To see this, note that the probability that a particle is absorbed between
3452:
3194:
3168:
3474:
64:
that all the target particles are at rest but only the beam particle is moving, that gives an expression for the mean free path:
2493:
3559:
2908:
2576:
370:
1557:
2596:
2584:
2174:
2456:= 1 to 100. The discontinuities are due to low density of gas elements. Six bands correspond to neighborhoods of six
3068:
In particle physics the concept of the mean free path is not commonly used, being replaced by the similar concept of
2649:
2883:
or ballistic transport. In such scenarios electrons alter their motion only in collisions with conductor walls.
2879:
Electron mobility through a medium with dimensions smaller than the mean free path of electrons occurs through
309:
2025:
1616:
2996:
2724:
1873:
1006:
927:
852:
131:
70:
941:
2618:
464:
2243:
321:
1876:. One way to deal with such "soft" molecules is to use the Lennard-Jones σ parameter as the diameter.
1684:
458:
1284:
1255:
3418:
3266:
3125:
2880:
2671:
2701:
680:{\displaystyle d{\mathcal {P}}(x)={\frac {I(x)-I(x+dx)}{I_{0}}}={\frac {1}{\ell }}e^{-x/\ell }dx.}
3237:
3207:
3069:
2990:
In an otherwise empty cavity, the mean free path of a single particle bouncing off the walls is:
1821:
1530:
3513:
3478:
3448:
3444:
3391:
3318:
3190:
3164:
3120:
2818:
1880:
691:
3184:
3505:
3499:
3426:
3409:
Young, Robert W. (July 1959). "Sabine
Reverberation Equation and Sound Power Calculations".
3383:
3274:
3102:
3077:
2655:
1868:
2824:
2683:
3348:
3344:
3314:
3073:
2896:
2706:
2461:
3466:
2079:
is the dynamic viscosity. This expression can be put into the following convenient form
1551:
times the number with stationary targets. Therefore, the following relationship applies:
447:{\displaystyle {\frac {dI}{dx}}=-In\sigma {\overset {\text{def}}{=}}-{\frac {I}{\ell }},}
3535:
3422:
3270:
3319:"Tables of X-Ray Mass Attenuation Coefficients and Mass Energy-Absorption Coefficients"
3135:
3088:
2861:
2842:
2677:
2287:
2005:
1847:
1827:
1720:
986:
3387:
3072:. In particular, for high-energy photons, which mostly interact by electron–positron
2446:
3553:
3540:
3293:
3278:
3057:
2451:
2263:
1678:
847:
2161:{\displaystyle \ell ={\frac {\mu }{p}}{\sqrt {\frac {\pi R_{\rm {specific}}T}{2}}},}
16:
Average distance travelled by a moving particle between impacts with other particles
3083:
Independent-particle models in nuclear physics require the undisturbed orbiting of
2865:
2450:
Mean free path for photons in energy range from 1 keV to 20 MeV for elements with
3049:
is a constant related to the shape of the cavity. For most simple cavity shapes,
2599:
can be looked up or calculated for any material and energy combination using the
2873:
2626:
2610:
2480:
2472:
843:
3509:
2979:
2348:
2267:
44:
3096:
fundamental problems of nuclear structure physics which has yet to be solved.
1867:
In practice, the diameter of gas molecules is not well defined. In fact, the
2869:
2803:{\displaystyle \mu ={\frac {q\tau }{m}}={\frac {q\ell }{m^{*}v_{\rm {F}}}},}
2469:
2457:
2325:
2298:
2291:
55:
3395:
2617:
is more complicated, because photons are not mono-energetic, but have some
2864:
of the charge carrier. The Fermi velocity can easily be derived from the
2622:
2273:
2259:
935:
152:. The typical number of stopping atoms in the slab is the concentration
36:
28:
3347:; Seltzer, S. M.; Chang, J.; Coursey, J. S.; Sukumar, R.; Zucker, D. S.
3084:
2891:
If one takes a suspension of non-light-absorbing particles of diameter
2592:
20:
3430:
3060:
in acoustics, using a geometrical approximation of sound propagation.
1717:(effective cross-sectional area for spherical particles with diameter
1362:{\displaystyle {\overline {\mathbf {v} _{1}\cdot \mathbf {v} _{2}}}=0}
3140:
3130:
2484:
1883:
as the actual gas being considered. This leads to a mean free path
1803:{\displaystyle \ell ={\frac {k_{\text{B}}T}{{\sqrt {2}}\pi d^{2}p}},}
40:
2250:
300 K) can lead to slightly different values of the mean free path.
2634:
2607:
2445:
3543:: Calculate mean free path for mixtures of gases using VHS model
2281:
2277:
533:
32:
1879:
Another way is to assume a hard-sphere gas that has the same
519:
is the distance traveled by the beam through the target, and
750:
565:
177:
2978:
can be evaluated numerically for spherical particles using
3294:"NIST: Note - X-Ray Form Factor and Attenuation Databases"
2640:
Sometimes one measures the thickness of a material in the
2676:
In macroscopic charge transport, the mean free path of a
2561:{\displaystyle \ell =\mu ^{-1}=((\mu /\rho )\rho )^{-1},}
2625:. As photons move through the target material, they are
2954:{\displaystyle \ell ={\frac {2d}{3\Phi Q_{\text{s}}}},}
126:
is the number of target particles per unit volume, and
3565:
Scattering, absorption and radiative transfer (optics)
3080:
is used much like the mean free path in radiography.
2999:
2911:
2868:
via the non-relativistic kinetic energy equation. In
2827:
2736:
2709:
2686:
2496:
2177:
2088:
2028:
2008:
1892:
1850:
1830:
1746:
1723:
1687:
1619:
1560:
1533:
1377:
1316:
1287:
1258:
1055:
1009:
989:
944:
855:
707:
559:
467:
382:
324:
174:
73:
3161:
Introduction to Plasma
Physics and Controlled Fusion
3045:
is the total inside surface area of the cavity, and
528:
is the beam intensity before it entered the target;
1603:{\displaystyle \ell =({\sqrt {2}}\,n\sigma )^{-1},}
532:is called the mean free path because it equals the
3189:, 3rd. edition, Cambridge University Press, 1990,
3026:
2953:
2833:
2802:
2715:
2692:
2560:
2234:
2160:
2067:
2014:
1991:
1856:
1836:
1802:
1729:
1709:
1669:
1602:
1543:
1517:
1361:
1302:
1273:
1242:
1036:
995:
975:
906:
831:
679:
507:
446:
358:
293:
107:
3257:Jennings, S (1988). "The mean free path in air".
2654:) of photons. This concept is closely related to
2235:{\displaystyle R_{\rm {specific}}=k_{\text{B}}/m}
3411:The Journal of the Acoustical Society of America
842:The fraction of particles that are not stopped (
3093:
3056:This relation is used in the derivation of the
3353:National Institute of Standards and Technology
3323:National Institute of Standards and Technology
2601:National Institute of Standards and Technology
1737:), it may be shown that the mean free path is
3498:Blatt, John M.; Weisskopf, Victor F. (1979).
8:
3242:: CS1 maint: multiple names: authors list (
3186:The mathematical theory of non-uniform gases
1527:This means that the number of collisions is
983:holds for a beam particle with a high speed
714:
708:
27:is the average distance over which a moving
3163:(1st ed.). Plenum Press. p. 156.
3467:"The Mean Free Path of Nucleons in Nuclei"
3232:. Krieger Publishing Company. p. 414.
3228:Vincenti, W. G. and Kruger, C. H. (1965).
3091:before they interact with other nucleons.
3006:
2998:
2939:
2918:
2910:
2826:
2787:
2786:
2776:
2761:
2743:
2735:
2708:
2685:
2546:
2528:
2507:
2495:
2224:
2218:
2183:
2182:
2176:
2117:
2116:
2105:
2095:
2087:
2053:
2041:
2027:
2007:
1965:
1954:
1944:
1928:
1909:
1899:
1891:
1849:
1829:
1785:
1771:
1760:
1753:
1745:
1722:
1701:
1686:
1655:
1643:
1629:
1618:
1588:
1577:
1570:
1559:
1534:
1532:
1502:
1486:
1481:
1476:
1466:
1461:
1456:
1451:
1436:
1409:
1408:
1403:
1399:
1383:
1382:
1376:
1341:
1336:
1326:
1321:
1317:
1315:
1294:
1289:
1286:
1265:
1260:
1257:
1225:
1220:
1210:
1205:
1192:
1187:
1182:
1172:
1167:
1162:
1158:
1143:
1133:
1128:
1118:
1113:
1106:
1092:
1065:
1064:
1059:
1056:
1054:
1015:
1014:
1008:
988:
964:
943:
894:
887:
874:
865:
854:
813:
803:
796:
782:
776:
771:
749:
748:
736:
731:
717:
706:
658:
651:
637:
626:
582:
564:
563:
558:
495:
488:
478:
466:
431:
418:
383:
381:
346:
323:
281:
264:
252:
246:
233:
222:
217:
210:
205:
202:
185:
176:
175:
173:
93:
72:
2902:, the mean free path of the photons is:
2252:
2068:{\displaystyle \rho =mp/(k_{\text{B}}T)}
54:
3151:
1670:{\displaystyle n=N/V=p/(k_{\text{B}}T)}
1310:are random and uncorrelated, therefore
918:is equal to the thickness of the slab.
3349:"XCOM: Photon Cross Sections Database"
3235:
3208:"Mean Free Path, Molecular Collisions"
3027:{\displaystyle \ell ={\frac {FV}{S}},}
1037:{\displaystyle v_{\rm {rel}}\approx v}
907:{\displaystyle T=I/I_{0}=e^{-x/\ell }}
108:{\displaystyle \ell =(\sigma n)^{-1},}
3230:Introduction to physical gas dynamics
2971:is the scattering efficiency factor.
2644:. Material with the thickness of one
976:{\displaystyle \ell =(n\sigma )^{-1}}
308:is the area (or, more formally, the "
7:
2246:, equal to 287 J/(kg*K) for air.
508:{\displaystyle I=I_{0}e^{-x/\ell }}
2932:
2788:
2205:
2202:
2199:
2196:
2193:
2190:
2187:
2184:
2139:
2136:
2133:
2130:
2127:
2124:
2121:
2118:
1431:
1428:
1425:
1422:
1419:
1416:
1413:
1410:
1390:
1387:
1384:
1087:
1084:
1081:
1078:
1075:
1072:
1069:
1066:
1022:
1019:
1016:
777:
737:
359:{\displaystyle dI=-In\sigma \,dx.}
14:
2075:is the density of ideal gas, and
1710:{\displaystyle \sigma =\pi d^{2}}
2637:spectrum changes with distance.
1477:
1457:
1404:
1337:
1322:
1303:{\displaystyle \mathbf {v} _{2}}
1290:
1274:{\displaystyle \mathbf {v} _{1}}
1261:
1221:
1206:
1183:
1163:
1129:
1114:
1060:
694:(or average, or simply mean) of
3210:. Hyperphysics.phy-astr.gsu.edu
1844:is the pressure of the gas and
3296:. Physics.nist.gov. 1998-03-10
3183:S. Chapman and T. G. Cowling,
2723:, a value directly related to
2577:linear attenuation coefficient
2543:
2536:
2522:
2519:
2460:. Also shown are locations of
2062:
2046:
1664:
1648:
1585:
1567:
1140:
1109:
961:
951:
761:
755:
618:
603:
594:
588:
576:
570:
371:ordinary differential equation
196:
182:
90:
80:
1:
3388:10.1016/S0039-9140(99)00129-0
3041:is the volume of the cavity,
1864:is the absolute temperature.
3471:Models of the Atomic Nucleus
3279:10.1016/0021-8502(88)90219-4
3064:Nuclear and particle physics
2597:mass attenuation coefficient
2585:mass attenuation coefficient
1493:
1442:
1369:, and the relative speed is
1348:
1232:
1150:
1098:
3501:Theoretical Nuclear Physics
3443:Davis, D. and Patronis, E.
3107:Theoretical nuclear physics
1544:{\displaystyle {\sqrt {2}}}
457:whose solution is known as
3583:
3473:(2 ed.). Heidelberg:
3445:"Sound System Engineering"
3259:Journal of Aerosol Science
2669:
3536:Mean free path calculator
3510:10.1007/978-1-4612-9959-2
2660:number of mean free paths
2648:will attenuate to 37% (1/
2642:number of mean free paths
934:of a particle, such as a
3465:Cook, Norman D. (2010).
846:) by the slab is called
310:scattering cross-section
156:times the volume, i.e.,
137:The area of the slab is
3159:Chen, Frank F. (1984).
2725:electrical conductivity
2700:is proportional to the
2613:the calculation of the
2390:10 cm – 1 km
2022:is the molecular mass,
1874:Lennard-Jones potential
928:kinetic theory of gases
922:Kinetic theory of gases
122:is the mean free path,
3112:
3101:John Markus Blatt and
3028:
2955:
2835:
2804:
2717:
2694:
2562:
2465:
2415:Extremely high vacuum
2236:
2162:
2069:
2016:
1993:
1858:
1838:
1804:
1731:
1711:
1671:
1604:
1545:
1519:
1363:
1304:
1275:
1244:
1038:
997:
977:
908:
833:
681:
509:
448:
360:
295:
109:
60:
47:with other particles.
3560:Statistical mechanics
3029:
2956:
2836:
2834:{\displaystyle \tau }
2805:
2718:
2695:
2693:{\displaystyle \ell }
2621:of energies called a
2595:of the material. The
2563:
2449:
2244:specific gas constant
2237:
2163:
2070:
2017:
1994:
1859:
1839:
1805:
1732:
1712:
1672:
1605:
1546:
1520:
1364:
1305:
1276:
1245:
1039:
998:
978:
909:
834:
682:
510:
449:
361:
296:
187:stopping within
110:
58:
3541:Gas Dynamics Toolbox
3447:(1997) Focal Press,
3126:Ballistic conduction
3053:is approximately 4.
2997:
2909:
2881:ballistic conduction
2825:
2734:
2716:{\displaystyle \mu }
2707:
2684:
2672:Ballistic conduction
2494:
2175:
2086:
2026:
2006:
1890:
1848:
1828:
1744:
1721:
1685:
1617:
1558:
1531:
1375:
1314:
1285:
1256:
1053:
1007:
987:
942:
853:
705:
557:
465:
380:
322:
172:
143:, and its volume is
134:area for collision.
71:
3423:1959ASAJ...31..912Y
3291:Based on data from
3271:1988JAerS..19..159J
2702:electrical mobility
1491:
1471:
1441:
1197:
1177:
1097:
781:
741:
3070:attenuation length
3024:
2951:
2831:
2800:
2713:
2690:
2603:(NIST) databases.
2558:
2483:of mono-energetic
2466:
2410:1 km – 10 km
2395:Ultra-high vacuum
2370:0.1 – 100 mm
2232:
2158:
2065:
2012:
1989:
1854:
1834:
1822:Boltzmann constant
1800:
1727:
1707:
1667:
1600:
1541:
1515:
1475:
1455:
1402:
1359:
1300:
1271:
1240:
1181:
1161:
1058:
1034:
993:
973:
904:
829:
767:
727:
677:
505:
444:
356:
291:
105:
61:
3519:978-1-4612-9961-5
3484:978-3-642-14736-4
3431:10.1121/1.1907816
3317:; Seltzer, S. M.
3121:Scattering theory
3019:
2946:
2942:
2795:
2756:
2434:
2433:
2309:Ambient pressure
2221:
2153:
2152:
2103:
2056:
2015:{\displaystyle m}
1984:
1983:
1968:
1952:
1939:
1938:
1931:
1907:
1857:{\displaystyle T}
1837:{\displaystyle p}
1795:
1776:
1763:
1730:{\displaystyle d}
1658:
1575:
1539:
1507:
1497:
1496:
1446:
1445:
1351:
1235:
1153:
1101:
996:{\displaystyle v}
790:
725:
724:
692:expectation value
645:
632:
461:and has the form
439:
426:
425:
401:
270:
228:
225:
220:
213:
208:
188:
130:is the effective
51:Scattering theory
3572:
3524:
3523:
3495:
3489:
3488:
3462:
3456:
3441:
3435:
3434:
3406:
3400:
3399:
3371:
3365:
3364:
3362:
3360:
3340:
3334:
3333:
3331:
3329:
3311:
3305:
3304:
3302:
3301:
3289:
3283:
3282:
3254:
3248:
3247:
3241:
3233:
3225:
3219:
3218:
3216:
3215:
3204:
3198:
3181:
3175:
3174:
3156:
3110:
3103:Victor Weisskopf
3078:radiation length
3033:
3031:
3030:
3025:
3020:
3015:
3007:
2960:
2958:
2957:
2952:
2947:
2945:
2944:
2943:
2940:
2927:
2919:
2840:
2838:
2837:
2832:
2809:
2807:
2806:
2801:
2796:
2794:
2793:
2792:
2791:
2781:
2780:
2770:
2762:
2757:
2752:
2744:
2722:
2720:
2719:
2714:
2699:
2697:
2696:
2691:
2656:half-value layer
2567:
2565:
2564:
2559:
2554:
2553:
2532:
2515:
2514:
2462:absorption edges
2297:number density (
2253:
2241:
2239:
2238:
2233:
2228:
2223:
2222:
2219:
2210:
2209:
2208:
2167:
2165:
2164:
2159:
2154:
2148:
2144:
2143:
2142:
2107:
2106:
2104:
2096:
2074:
2072:
2071:
2066:
2058:
2057:
2054:
2045:
2021:
2019:
2018:
2013:
1998:
1996:
1995:
1990:
1985:
1982:
1974:
1970:
1969:
1966:
1956:
1955:
1953:
1945:
1940:
1937:
1933:
1932:
1929:
1919:
1911:
1910:
1908:
1900:
1869:kinetic diameter
1863:
1861:
1860:
1855:
1843:
1841:
1840:
1835:
1809:
1807:
1806:
1801:
1796:
1794:
1790:
1789:
1777:
1772:
1769:
1765:
1764:
1761:
1754:
1736:
1734:
1733:
1728:
1716:
1714:
1713:
1708:
1706:
1705:
1676:
1674:
1673:
1668:
1660:
1659:
1656:
1647:
1633:
1609:
1607:
1606:
1601:
1596:
1595:
1576:
1571:
1550:
1548:
1547:
1542:
1540:
1535:
1524:
1522:
1521:
1516:
1508:
1503:
1498:
1492:
1490:
1485:
1480:
1470:
1465:
1460:
1453:
1452:
1447:
1440:
1435:
1434:
1407:
1401:
1400:
1395:
1394:
1393:
1368:
1366:
1365:
1360:
1352:
1347:
1346:
1345:
1340:
1331:
1330:
1325:
1318:
1309:
1307:
1306:
1301:
1299:
1298:
1293:
1280:
1278:
1277:
1272:
1270:
1269:
1264:
1252:In equilibrium,
1249:
1247:
1246:
1241:
1236:
1231:
1230:
1229:
1224:
1215:
1214:
1209:
1196:
1191:
1186:
1176:
1171:
1166:
1159:
1154:
1149:
1148:
1147:
1138:
1137:
1132:
1123:
1122:
1117:
1107:
1102:
1096:
1091:
1090:
1063:
1057:
1043:
1041:
1040:
1035:
1027:
1026:
1025:
1002:
1000:
999:
994:
982:
980:
979:
974:
972:
971:
917:
913:
911:
910:
905:
903:
902:
898:
879:
878:
869:
838:
836:
835:
830:
812:
811:
807:
791:
783:
780:
775:
754:
753:
740:
735:
726:
722:
718:
697:
686:
684:
683:
678:
667:
666:
662:
646:
638:
633:
631:
630:
621:
583:
569:
568:
549:
539:
531:
527:
518:
514:
512:
511:
506:
504:
503:
499:
483:
482:
459:Beer–Lambert law
453:
451:
450:
445:
440:
432:
427:
423:
419:
402:
400:
392:
384:
365:
363:
362:
357:
312:") of one atom.
307:
300:
298:
297:
292:
271:
269:
268:
259:
251:
250:
234:
229:
227:
226:
223:
221:
218:
215:
214:
211:
209:
206:
203:
189:
186:
181:
180:
164:
155:
151:
142:
129:
125:
121:
114:
112:
111:
106:
101:
100:
3582:
3581:
3575:
3574:
3573:
3571:
3570:
3569:
3550:
3549:
3547:
3532:
3527:
3520:
3497:
3496:
3492:
3485:
3477:. p. 324.
3464:
3463:
3459:
3442:
3438:
3408:
3407:
3403:
3373:
3372:
3368:
3358:
3356:
3343:Berger, M. J.;
3342:
3341:
3337:
3327:
3325:
3313:
3312:
3308:
3299:
3297:
3292:
3290:
3286:
3256:
3255:
3251:
3234:
3227:
3226:
3222:
3213:
3211:
3206:
3205:
3201:
3182:
3178:
3171:
3158:
3157:
3153:
3149:
3117:
3111:
3100:
3074:pair production
3066:
3058:Sabine equation
3008:
2995:
2994:
2988:
2977:
2970:
2935:
2928:
2920:
2907:
2906:
2897:volume fraction
2889:
2859:
2823:
2822:
2782:
2772:
2771:
2763:
2745:
2732:
2731:
2705:
2704:
2682:
2681:
2674:
2668:
2542:
2503:
2492:
2491:
2444:
2439:
2437:In other fields
2304:Mean free path
2214:
2178:
2173:
2172:
2112:
2108:
2084:
2083:
2049:
2024:
2023:
2004:
2003:
1975:
1961:
1957:
1924:
1920:
1912:
1888:
1887:
1846:
1845:
1826:
1825:
1819:
1781:
1770:
1756:
1755:
1742:
1741:
1719:
1718:
1697:
1683:
1682:
1651:
1615:
1614:
1584:
1556:
1555:
1529:
1528:
1454:
1378:
1373:
1372:
1335:
1320:
1319:
1312:
1311:
1288:
1283:
1282:
1259:
1254:
1253:
1219:
1204:
1160:
1139:
1127:
1112:
1108:
1051:
1050:
1010:
1005:
1004:
985:
984:
960:
940:
939:
924:
915:
883:
870:
851:
850:
792:
703:
702:
695:
647:
622:
584:
555:
554:
541:
537:
529:
526:
520:
516:
484:
474:
463:
462:
393:
385:
378:
377:
320:
319:
305:
260:
242:
235:
216:
204:
170:
169:
157:
153:
144:
138:
132:cross-sectional
127:
123:
119:
89:
69:
68:
53:
17:
12:
11:
5:
3580:
3579:
3576:
3568:
3567:
3562:
3552:
3551:
3545:
3544:
3538:
3531:
3530:External links
3528:
3526:
3525:
3518:
3490:
3483:
3457:
3436:
3401:
3366:
3345:Hubbell, J. H.
3335:
3315:Hubbell, J. H.
3306:
3284:
3249:
3220:
3199:
3176:
3169:
3150:
3148:
3145:
3144:
3143:
3138:
3136:Knudsen number
3133:
3128:
3123:
3116:
3113:
3098:
3065:
3062:
3035:
3034:
3023:
3018:
3014:
3011:
3005:
3002:
2987:
2984:
2975:
2968:
2962:
2961:
2950:
2938:
2934:
2931:
2926:
2923:
2917:
2914:
2888:
2885:
2862:Fermi velocity
2857:
2851:effective mass
2843:mean free time
2830:
2811:
2810:
2799:
2790:
2785:
2779:
2775:
2769:
2766:
2760:
2755:
2751:
2748:
2742:
2739:
2712:
2689:
2678:charge carrier
2667:
2664:
2646:mean free path
2631:mean free path
2615:mean free path
2569:
2568:
2557:
2552:
2549:
2545:
2541:
2538:
2535:
2531:
2527:
2524:
2521:
2518:
2513:
2510:
2506:
2502:
2499:
2477:mean free path
2443:
2440:
2438:
2435:
2432:
2431:
2428:
2425:
2422:
2419:
2416:
2412:
2411:
2408:
2405:
2402:
2399:
2396:
2392:
2391:
2388:
2385:
2382:
2379:
2376:
2372:
2371:
2368:
2365:
2362:
2359:
2356:
2355:Medium vacuum
2352:
2351:
2345:
2342:
2339:
2336:
2333:
2329:
2328:
2322:
2319:
2316:
2313:
2310:
2306:
2305:
2302:
2295:
2288:number density
2285:
2271:
2257:
2231:
2227:
2217:
2213:
2207:
2204:
2201:
2198:
2195:
2192:
2189:
2186:
2181:
2169:
2168:
2157:
2151:
2147:
2141:
2138:
2135:
2132:
2129:
2126:
2123:
2120:
2115:
2111:
2102:
2099:
2094:
2091:
2064:
2061:
2052:
2048:
2044:
2040:
2037:
2034:
2031:
2011:
2000:
1999:
1988:
1981:
1978:
1973:
1964:
1960:
1951:
1948:
1943:
1936:
1927:
1923:
1918:
1915:
1906:
1903:
1898:
1895:
1853:
1833:
1817:
1811:
1810:
1799:
1793:
1788:
1784:
1780:
1775:
1768:
1759:
1752:
1749:
1726:
1704:
1700:
1696:
1693:
1690:
1666:
1663:
1654:
1650:
1646:
1642:
1639:
1636:
1632:
1628:
1625:
1622:
1611:
1610:
1599:
1594:
1591:
1587:
1583:
1580:
1574:
1569:
1566:
1563:
1538:
1514:
1511:
1506:
1501:
1495:
1489:
1484:
1479:
1474:
1469:
1464:
1459:
1450:
1444:
1439:
1433:
1430:
1427:
1424:
1421:
1418:
1415:
1412:
1406:
1398:
1392:
1389:
1386:
1381:
1358:
1355:
1350:
1344:
1339:
1334:
1329:
1324:
1297:
1292:
1268:
1263:
1239:
1234:
1228:
1223:
1218:
1213:
1208:
1203:
1200:
1195:
1190:
1185:
1180:
1175:
1170:
1165:
1157:
1152:
1146:
1142:
1136:
1131:
1126:
1121:
1116:
1111:
1105:
1100:
1095:
1089:
1086:
1083:
1080:
1077:
1074:
1071:
1068:
1062:
1033:
1030:
1024:
1021:
1018:
1013:
992:
970:
967:
963:
959:
956:
953:
950:
947:
932:mean free path
923:
920:
901:
897:
893:
890:
886:
882:
877:
873:
868:
864:
861:
858:
840:
839:
828:
825:
822:
819:
816:
810:
806:
802:
799:
795:
789:
786:
779:
774:
770:
766:
763:
760:
757:
752:
747:
744:
739:
734:
730:
721:
716:
713:
710:
688:
687:
676:
673:
670:
665:
661:
657:
654:
650:
644:
641:
636:
629:
625:
620:
617:
614:
611:
608:
605:
602:
599:
596:
593:
590:
587:
581:
578:
575:
572:
567:
562:
524:
502:
498:
494:
491:
487:
481:
477:
473:
470:
455:
454:
443:
438:
435:
430:
422:
417:
414:
411:
408:
405:
399:
396:
391:
388:
367:
366:
355:
352:
349:
345:
342:
339:
336:
333:
330:
327:
302:
301:
290:
287:
284:
280:
277:
274:
267:
263:
258:
255:
249:
245:
241:
238:
232:
201:
198:
195:
192:
184:
179:
116:
115:
104:
99:
96:
92:
88:
85:
82:
79:
76:
59:Slab of target
52:
49:
25:mean free path
15:
13:
10:
9:
6:
4:
3:
2:
3578:
3577:
3566:
3563:
3561:
3558:
3557:
3555:
3548:
3542:
3539:
3537:
3534:
3533:
3529:
3521:
3515:
3511:
3507:
3503:
3502:
3494:
3491:
3486:
3480:
3476:
3472:
3468:
3461:
3458:
3454:
3453:0-240-80305-1
3450:
3446:
3440:
3437:
3432:
3428:
3424:
3420:
3416:
3412:
3405:
3402:
3397:
3393:
3389:
3385:
3382:(2): 445–56.
3381:
3377:
3370:
3367:
3354:
3350:
3346:
3339:
3336:
3324:
3320:
3316:
3310:
3307:
3295:
3288:
3285:
3280:
3276:
3272:
3268:
3264:
3260:
3253:
3250:
3245:
3239:
3231:
3224:
3221:
3209:
3203:
3200:
3196:
3195:0-521-40844-X
3192:
3188:
3187:
3180:
3177:
3172:
3170:0-306-41332-9
3166:
3162:
3155:
3152:
3146:
3142:
3139:
3137:
3134:
3132:
3129:
3127:
3124:
3122:
3119:
3118:
3114:
3108:
3104:
3097:
3092:
3090:
3086:
3081:
3079:
3075:
3071:
3063:
3061:
3059:
3054:
3052:
3048:
3044:
3040:
3021:
3016:
3012:
3009:
3003:
3000:
2993:
2992:
2991:
2985:
2983:
2981:
2974:
2967:
2948:
2936:
2929:
2924:
2921:
2915:
2912:
2905:
2904:
2903:
2901:
2898:
2894:
2886:
2884:
2882:
2877:
2875:
2871:
2867:
2863:
2856:
2852:
2848:
2844:
2828:
2820:
2816:
2797:
2783:
2777:
2773:
2767:
2764:
2758:
2753:
2749:
2746:
2740:
2737:
2730:
2729:
2728:
2726:
2710:
2703:
2687:
2679:
2673:
2665:
2663:
2661:
2657:
2653:
2652:
2647:
2643:
2638:
2636:
2632:
2628:
2624:
2620:
2616:
2612:
2609:
2604:
2602:
2598:
2594:
2590:
2586:
2582:
2578:
2574:
2555:
2550:
2547:
2539:
2533:
2529:
2525:
2516:
2511:
2508:
2504:
2500:
2497:
2490:
2489:
2488:
2486:
2482:
2478:
2474:
2471:
2463:
2459:
2455:
2454:
2448:
2441:
2436:
2429:
2426:
2423:
2420:
2417:
2414:
2413:
2409:
2406:
2403:
2400:
2397:
2394:
2393:
2389:
2386:
2383:
2380:
2377:
2374:
2373:
2369:
2366:
2363:
2360:
2357:
2354:
2353:
2350:
2346:
2343:
2340:
2337:
2334:
2331:
2330:
2327:
2323:
2320:
2317:
2314:
2311:
2308:
2307:
2303:
2300:
2296:
2293:
2289:
2286:
2283:
2279:
2275:
2272:
2269:
2265:
2261:
2258:
2256:Vacuum range
2255:
2254:
2251:
2247:
2245:
2229:
2225:
2215:
2211:
2179:
2155:
2149:
2145:
2113:
2109:
2100:
2097:
2092:
2089:
2082:
2081:
2080:
2078:
2059:
2050:
2042:
2038:
2035:
2032:
2029:
2009:
1986:
1979:
1976:
1971:
1962:
1958:
1949:
1946:
1941:
1934:
1925:
1921:
1916:
1913:
1904:
1901:
1896:
1893:
1886:
1885:
1884:
1882:
1877:
1875:
1870:
1865:
1851:
1831:
1823:
1816:
1797:
1791:
1786:
1782:
1778:
1773:
1766:
1757:
1750:
1747:
1740:
1739:
1738:
1724:
1702:
1698:
1694:
1691:
1688:
1680:
1679:ideal gas law
1661:
1652:
1644:
1640:
1637:
1634:
1630:
1626:
1623:
1620:
1597:
1592:
1589:
1581:
1578:
1572:
1564:
1561:
1554:
1553:
1552:
1536:
1525:
1512:
1509:
1504:
1499:
1487:
1482:
1472:
1467:
1462:
1448:
1437:
1396:
1379:
1370:
1356:
1353:
1342:
1332:
1327:
1295:
1266:
1250:
1237:
1226:
1216:
1211:
1201:
1198:
1193:
1188:
1178:
1173:
1168:
1155:
1144:
1134:
1124:
1119:
1103:
1093:
1048:
1045:
1031:
1028:
1011:
990:
968:
965:
957:
954:
948:
945:
937:
933:
929:
921:
919:
899:
895:
891:
888:
884:
880:
875:
871:
866:
862:
859:
856:
849:
845:
826:
823:
820:
817:
814:
808:
804:
800:
797:
793:
787:
784:
772:
768:
764:
758:
745:
742:
732:
728:
719:
711:
701:
700:
699:
693:
674:
671:
668:
663:
659:
655:
652:
648:
642:
639:
634:
627:
623:
615:
612:
609:
606:
600:
597:
591:
585:
579:
573:
560:
553:
552:
551:
548:
544:
535:
523:
500:
496:
492:
489:
485:
479:
475:
471:
468:
460:
441:
436:
433:
428:
420:
415:
412:
409:
406:
403:
397:
394:
389:
386:
376:
375:
374:
372:
353:
350:
347:
343:
340:
337:
334:
331:
328:
325:
318:
317:
316:
313:
311:
288:
285:
282:
278:
275:
272:
265:
261:
256:
253:
247:
243:
239:
236:
230:
199:
193:
190:
168:
167:
166:
163:
160:
150:
147:
141:
135:
133:
102:
97:
94:
86:
83:
77:
74:
67:
66:
65:
57:
50:
48:
46:
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3379:
3375:
3369:
3359:19 September
3357:. Retrieved
3338:
3328:19 September
3326:. Retrieved
3309:
3298:. Retrieved
3287:
3262:
3258:
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3229:
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3202:
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2467:
2452:
2401:8×10 – 8×10
2381:8×10 – 8×10
2375:High vacuum
2361:8×10 – 8×10
2248:
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546:
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2727:, that is:
2680:in a metal
2666:Electronics
2611:radiography
2481:pencil beam
2473:radiography
2458:noble gases
2442:Radiography
2338:220 – 8×10
2332:Low vacuum
369:This is an
3554:Categories
3417:(7): 918.
3300:2011-11-08
3265:(2): 159.
3214:2011-11-08
3147:References
2980:Mie theory
2870:thin films
2670:See also:
2627:attenuated
2430:>10 km
2347:0.1 – 100
2242:being the
1613:and using
844:attenuated
45:collisions
3238:cite book
3001:ℓ
2986:Acoustics
2933:Φ
2913:ℓ
2829:τ
2778:∗
2768:ℓ
2750:τ
2738:μ
2711:μ
2688:ℓ
2548:−
2540:ρ
2534:ρ
2526:μ
2509:−
2505:μ
2498:ℓ
2470:gamma-ray
2421:<8×10
2321:2.7 × 10
2318:2.7 × 10
2299:Molecules
2292:Molecules
2110:π
2098:μ
2090:ℓ
2030:ρ
1959:π
1947:μ
1914:π
1905:ρ
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1894:ℓ
1881:viscosity
1779:π
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958:σ
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778:∞
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729:∫
715:⟩
709:⟨
690:Thus the
664:ℓ
653:−
643:ℓ
598:−
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490:−
437:ℓ
429:−
416:σ
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3475:Springer
3396:18967735
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3115:See also
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3085:nucleons
2623:spectrum
2407:10 – 10
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2378:10 – 10
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2335:300 – 1
2324:64 – 68
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515:, where
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3419:Bibcode
3376:Talanta
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2895:with a
2860:is the
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2633:of the
2593:density
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2583:is the
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2485:photons
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2358:1 – 10
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540:and
534:mean
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219:Area
207:Area
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