2846:
2478:
2841:{\displaystyle {\begin{aligned}{}_{11}^{22}{\text{Na}}_{11}\left(3^{+}\right)&\rightarrow {}_{10}^{22}{\text{Ne}}_{12}\left(2^{+}\right)+\beta ^{+}+\nu _{\text{e}}&t_{1/2}&=2.6\,{\text{years}}\\{}_{49}^{115}{\text{In}}_{66}\left({\frac {9}{2}}^{+}\right)&\rightarrow {}_{50}^{115}{\text{Sn}}_{65}\left({\frac {1}{2}}^{+}\right)+\beta ^{-}+{\bar {\nu }}_{\text{e}}&t_{1/2}&=10^{14}\,{\text{years}}\end{aligned}}}
25:
245:
68:
1917:
assumption that the weak vector current responsible for the decay is conserved. Another observation is that the Fermi transitions illustrate how the nucleons inside the nucleus interact as free particles despite being surrounded by mesons mediating the nuclear force. This is useful in considering the barrier tunnelling mechanism involved with alpha decay and in deriving the
3356:
3212:
279:
seen in weak interactions. The Gamow–Teller theory was necessary for the inclusion of parity violation by modifying the matrix element to include vector and axial-vector couplings of fermions. This formed the matrix element that completed the Fermi theory of β decay and described parity violation,
1916:
The
Conservation of Vector Current hypothesis was created out of the Gamow–Teller theory. The Fermi decay is the result of a vector current and is dominant in the decay of the neutron to a proton while the Gamow–Teller decay is an axial-current transition. Conservation of Vector Current is the
2856:
A calculation of the β emission decay rate is quite different from a calculation of α decay. In α decay the nucleons of the original nucleus are used to form the final state α particle (He). In β decay the β and neutrino particles are the result of a nucleon transformation into its isospin
782:
transition, that is, the selection rules for beta decay caused by such a transition involve no parity change of the nuclear state. The spin of the parent nucleus can either remain unchanged or change by ±1. However, unlike the Fermi transition, transitions from spin 0 to spin 0 are excluded.
1387:
Due to the existence of the 2 possible final states, each β decay is a mixture of the two decay types. This essentially means that some of the time the remaining nucleus is in an excited state and other times the decay is directly to the ground state. Unlike Fermi transitions, Gamow–Teller
1388:
transitions occur via an operator that operates only if the initial nuclear wavefunction and final nuclear wavefunction are defined. The
Isospin and Angular Momentum selection rules can be deduced from the operator and the identification of allowed and forbidden decays can be found.
3221:
3072:
974:
1701:
1095:
1494:
492:
3089:
1799:
1592:
2428:
2305:
3081:. This assumption appears to be true based on the very short time scale (10 s) it takes for the formation of quasi-stationary nuclear states compared with the time it takes for a β decay (half lives ranging from seconds to days).
1890:
1183:
576:
38:
This article still needs revision to make it generally about transitions, to simplify and elaborate the discussion to be perhaps less technical to non experts, and to reduce the redundancies with the beta decay
1280:
1811:
One can measure the angular distributions of β particles with respect to the axis of nuclear spin polarization to determine what the mixture is between the two decay types (Fermi and Gamow–Teller).
2483:
1348:
673:
3351:{\displaystyle {\hat {H}}_{\text{int}}={\begin{cases}G_{V}{\hat {1}}{\hat {\tau }}&{\text{Fermi decay}}\\G_{A}{\hat {\sigma }}{\hat {\tau }}&{\text{Gamow–Teller Decay}}\end{cases}}}
2964:
832:
724:
1603:
2467:
1215:
993:
611:
1401:
824:
378:
394:
257:
showing the 4-point fermion vector current, coupled under Fermi's coupling constant, "Gf". Fermi's theory was the first theoretical effort in describing nuclear decay rates for
3207:{\displaystyle \left|M_{i,f}\right|^{2}=\left\langle \psi _{\text{Daughter}}\phi _{\beta }\psi _{\nu }\right|{\hat {H}}_{\text{int}}\left|\psi _{\text{Parent}}\right\rangle }
2021:
1985:
1956:
1709:
1502:
2936:
2317:
2956:
2200:
2189:
2163:
1374:
1306:
214:
184:
2903:
2875:
The light decay products can have continuous energy distributions. (before assuming the α carried away most of the energy was usually a good approximation).
1824:
275:
which was
Lorentz-invariant and involved a 4-point fermion vector current. However, this did not incorporate parity violation within the matrix element in
342:
In the Fermi transition, the electron and neutrino emitted from the β-decay parent nucleus have spin vectors which are anti-parallel to one another.
86:
3529:
1107:
500:
3077:
From this analysis we can conclude that the Gamow–Teller nuclear transition from 0 → ±1 is a weak perturbation of the system's interaction
1990:
Forbidden decays are those which are substantially more improbable, due to parity violation, and as a result have long decay times.
3456:
Saw, E. L.; Yap, C. T. (1988-11-03). "The Fermi to Gamow–Teller mixing ratio of the β decay of Mn and time-reversal invariance".
1220:
2872:
The β electron and neutrino are relativistic (nuclear decay energy is usually not enough to make heavy α nucleus relativistic).
3067:{\displaystyle W={\frac {2\pi }{\hbar }}\left|M_{i,f}\right|^{2}\times {\text{(Phase Space)}}={\frac {\ln 2}{t_{1/2}}}}
969:{\displaystyle \Delta I=I_{f}-I_{i}={\begin{cases}0&I_{i}=I_{f}=0\\1&I_{i}=0{\text{ and }}I_{f}=1\end{cases}}}
3382:
1311:
616:
104:
3078:
1696:{\displaystyle {}_{11}^{21}{\text{Na}}_{10}\rightarrow {}_{10}^{21}{\text{Ne}}_{11}^{*}+\beta ^{+}+\nu _{\text{e}}}
681:
1090:{\displaystyle {}_{2}^{6}{\text{He}}_{4}\rightarrow {}_{3}^{6}{\text{Li}}_{3}+\beta ^{-}+{\bar {\nu }}_{\text{e}}}
2440:
1489:{\displaystyle {}_{11}^{21}{\text{Na}}_{10}\rightarrow {}_{10}^{21}{\text{Ne}}_{11}+\beta ^{+}+\nu _{\text{e}}}
1188:
584:
487:{\displaystyle {}_{8}^{14}{\text{O}}_{6}\rightarrow {}_{7}^{14}{\text{N}}_{7}^{*}+\beta ^{+}+\nu _{\text{e}}}
42:
789:
351:
228:
186:. As a result, the total angular momentum of the nucleus is unchanged by the transition. By contrast, in a
2879:
The β decay rate calculation was developed by Fermi in 1934 and was based on Pauli's neutrino hypothesis.
747:), the physical laws would be the same if the interaction was reflected in a mirror. Hence the sum of a
3593:
3252:
876:
3588:
3573:
3552:
3500:
3442:
280:
neutrino helicity, muon decay properties along with the concept of lepton universality. Before the
1794:{\displaystyle I_{i}={\frac {3}{2}}^{+}\Rightarrow I_{f}={\frac {5}{2}}^{+}\Rightarrow \Delta I=1}
1587:{\displaystyle I_{i}={\frac {3}{2}}^{+}\Rightarrow I_{f}={\frac {3}{2}}^{+}\Rightarrow \Delta I=0}
3568:
2000:
2423:{\displaystyle {\vec {I}}={\vec {L}}+{\vec {S}}={\vec {1}}+{\vec {1}}\Rightarrow \Delta I=0,1,2}
1961:
1932:
3598:
3377:(University of Chicago Press ed.). Chicago: University of Chicago Press. p. 366-367.
2882:
1918:
1815:
276:
254:
216:, leading to a change in angular momentum between the initial and final states of the nucleus.
82:
2300:{\displaystyle {\vec {I}}={\vec {L}}+{\vec {S}}={\vec {1}}+{\vec {0}}\Rightarrow \Delta I=0,1}
3603:
2908:
3544:
3492:
3434:
2941:
3407:
323:
8:
3545:
3493:
3435:
2168:
2142:
1903:
for neutron decay while non-mirror nuclear decays tend to be an order of magnitude less.
1353:
1285:
768:
193:
163:
3411:
3523:
3473:
2888:
736:
305:
219:
The theoretical work in describing these transitions was done between 1934 and 1936 by
1885:{\displaystyle y\equiv {\frac {g_{\text{F}}M_{\text{F}}}{g_{\text{GT}}M_{\text{GT}}}}}
3477:
3378:
748:
3465:
3415:
744:
740:
327:
297:
285:
3216:
with the interaction
Hamiltonian forming 2 separate states from the perturbation.
293:
190:
transition, the spins of the two emitted particles are parallel, with total spin
118:
160:, the spins of the two emitted particles are anti-parallel, for a combined spin
3084:
The matrix element between parent and daughter nuclei in such a transition is:
331:
289:
281:
126:
3516:
Radioactivity & particle physics and, Radioactive fallout & technology
3419:
3582:
3375:
Principles of stellar evolution and nucleosynthesis : with a new preface
1805:
224:
150:
134:
779:
752:
309:
268:
220:
1958:) are often referred to as the "superallowed" decays while Gamow–Teller (
1808:", nuclei in which the numbers of protons and neutrons are interchanged.
786:
In terms of total nuclear angular momentum, the Gamow–Teller transition (
146:
1178:{\displaystyle I_{i}=0^{+}\rightarrow I_{f}=1^{+}\Rightarrow \Delta I=1}
571:{\displaystyle I_{i}=0^{+}\rightarrow I_{f}=0^{+}\Rightarrow \Delta I=0}
3469:
760:
756:
334:
which was required for describing high energy particle cross-sections.
261:. The Gamow–Teller theory was a necessary extension of Fermi's theory.
258:
130:
2023:
systems can be non-zero (in the center-of-mass frame of the system).
138:
244:
142:
3398:
Franz
Osterfeld (1992). "Nuclear spin and isospin excitations".
771:
of the interaction, and in this case pseudovectors and vectors
301:
272:
1814:
The mixture can be expressed as a ratio of matrix elements (
3344:
2869:
The β electron and neutrino did not exist before the decay.
1275:{\displaystyle \pi (Y_{\ell \,m})=(-1)^{\ell }\Rightarrow }
962:
2905:
is given by a transition matrix element (or "amplitude")
2026:
Below are the observed selection rules for beta decay:
156:
There are several types of beta decay transition. In a
2472:
The half life of the decay increases with each order:
1911:
380:
no change in the total angular momentum of the nucleus
322:) of the four-fermion interaction. From there modern
234:
3224:
3092:
2967:
2944:
2911:
2891:
2481:
2443:
2320:
2203:
2171:
2145:
2003:
1964:
1935:
1827:
1712:
1606:
1505:
1404:
1382:
1356:
1314:
1288:
1223:
1191:
1110:
996:
835:
792:
684:
619:
587:
503:
397:
354:
267:
Beta decay had been first described theoretically by
196:
166:
77:
may be too technical for most readers to understand
3350:
3206:
3066:
2950:
2938:weighted by the phase space and Planck's constant
2930:
2897:
2840:
2461:
2422:
2299:
2194:So for the "first-forbidden" transitions you have
2183:
2157:
2015:
1979:
1950:
1899:for mirror nuclei is on the order of the value of
1884:
1793:
1695:
1586:
1488:
1368:
1342:
1300:
1274:
1209:
1177:
1089:
968:
818:
718:
667:
605:
570:
486:
372:
208:
178:
3580:
3574:Transition Probabilities and Fermi's Golden Rule
3542:
3513:
3490:
3432:
133:. When undergoing beta decay, a nucleus emits a
3397:
1343:{\displaystyle \beta +{\bar {\nu }}_{\text{e}}}
668:{\displaystyle \pi (Y_{\ell \,m})=(-1)^{\ell }}
719:{\displaystyle {}_{7}^{14}{\text{N}}_{7}^{*}}
1376:states that couple to an even parity state.
763:and the corresponding nuclear transitions,
3528:: CS1 maint: location missing publisher (
730:
2828:
2637:
2462:{\displaystyle \Delta \pi =1\Rightarrow }
1238:
1210:{\displaystyle \Delta \pi =0\Rightarrow }
634:
606:{\displaystyle \Delta \pi =0\Rightarrow }
105:Learn how and when to remove this message
89:, without removing the technical details.
34:needs attention from an expert in Physics
3518:. University of California. p. 303.
2865:). Below is a list of the differences:
1906:
1818:relates transitions to matrix elements)
743:interactions (which are invariant under
3372:
3581:
3455:
1987:) decays are simple "allowed" decays.
819:{\displaystyle I_{i}\rightarrow I_{f}}
373:{\displaystyle \Delta I=0\Rightarrow }
45:may be able to help recruit an expert.
87:make it understandable to non-experts
1895:The interesting observation is that
61:
18:
3551:(2nd ed.). Wiley-VCH. p.
3499:(2nd ed.). Wiley-VCH. p.
3441:(2nd ed.). Wiley-VCH. p.
2124:fourth-forbidden (no parity change)
2096:second-forbidden (no parity change)
1924:
1912:Conservation of weak vector current
735:In nuclear transitions governed by
337:
326:was developed, which described the
235:The weak interaction and beta decay
13:
2444:
2396:
2279:
1965:
1936:
1779:
1572:
1383:Mixed Fermi and Gamow–Teller decay
1192:
1163:
836:
588:
556:
355:
282:Standard Model of Particle Physics
14:
3615:
3562:
2984:
778:The Gamow–Teller transition is a
755:is not meaningful. However, the
243:
66:
23:
2469:(parity violating transition).
2110:third-forbidden (parity change)
2082:first-forbidden (parity change)
3536:
3507:
3484:
3449:
3426:
3391:
3366:
3328:
3316:
3283:
3271:
3232:
3174:
2885:says that the transition rate
2779:
2702:
2534:
2456:
2393:
2387:
2372:
2357:
2342:
2327:
2276:
2270:
2255:
2240:
2225:
2210:
2139:Each of the above have Fermi (
1776:
1743:
1633:
1569:
1536:
1431:
1328:
1269:
1260:
1250:
1244:
1227:
1204:
1160:
1134:
1075:
1023:
803:
656:
646:
640:
623:
600:
553:
527:
424:
367:
1:
3360:
2851:
1804:The above reaction involves "
125:is the change in state of an
3547:Introductory Nuclear Physics
3495:Introductory Nuclear Physics
3437:Introductory Nuclear Physics
229:George Washington University
141:, transforming the original
7:
3569:Fermi Theory of Beta Decay
3373:Clayton, Donald D. (1983).
2016:{\displaystyle \beta +\nu }
36:. The specific problem is:
10:
3620:
1993:Now the angular momentum (
1980:{\displaystyle \Delta I=1}
1951:{\displaystyle \Delta I=0}
3543:Samuel S.M. Wong (2004).
3514:Willard F. Libby (1981).
3491:Samuel S.M. Wong (2004).
3433:Samuel S.M. Wong (2004).
3420:10.1103/RevModPhys.64.491
3400:Reviews of Modern Physics
3338:Gamow–Teller Decay
1282:the final Li 1 state has
300:, determined the correct
292:, and also independently
3458:Zeitschrift für Physik A
2931:{\displaystyle M_{i,f}}
731:Gamow–Teller transition
145:into one with the same
3352:
3208:
3068:
2952:
2951:{\displaystyle \hbar }
2932:
2899:
2842:
2463:
2424:
2301:
2185:
2159:
2017:
1981:
1952:
1886:
1795:
1697:
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1344:
1302:
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1211:
1179:
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3209:
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2160:
2018:
1982:
1953:
1907:Physical consequences
1887:
1796:
1698:
1589:
1491:
1371:
1345:
1303:
1277:
1217:parity is conserved:
1212:
1180:
1092:
971:
821:
721:
670:
613:parity is conserved:
608:
573:
489:
375:
211:
181:
123:beta decay transition
3222:
3090:
2965:
2942:
2909:
2889:
2479:
2441:
2318:
2201:
2169:
2165:) and Gamow–Teller (
2143:
2001:
1962:
1933:
1825:
1710:
1604:
1503:
1402:
1354:
1312:
1286:
1221:
1189:
1108:
994:
833:
790:
726:= excited state of N
682:
617:
585:
501:
395:
352:
330:in terms of massive
194:
164:
137:and a corresponding
3412:1992RvMP...64..491O
2883:Fermi's Golden Rule
2718:
2660:
2550:
2499:
2184:{\displaystyle S=1}
2158:{\displaystyle S=0}
1816:Fermi's golden rule
1666:
1649:
1620:
1447:
1418:
1369:{\displaystyle S=1}
1301:{\displaystyle L=1}
1039:
1010:
715:
698:
457:
440:
411:
277:Fermi's Golden Rule
255:Fermi's interaction
209:{\displaystyle S=1}
179:{\displaystyle S=0}
43:WikiProject Physics
16:Physical phenomenon
3470:10.1007/BF01295458
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2013:
1977:
1948:
1929:The Fermi decays (
1919:Geiger–Nuttall law
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1791:
1693:
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1607:
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1434:
1405:
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1026:
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603:
568:
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370:
324:electroweak theory
206:
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153:(nuclear charge).
3594:Quantum mechanics
3339:
3331:
3319:
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3286:
3274:
3241:
3235:
3197:
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3139:
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3026:
2987:
2898:{\displaystyle W}
2832:
2788:
2782:
2744:
2723:
2686:
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2641:
2604:
2555:
2504:
2390:
2375:
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2136:
1880:
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1625:
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1528:
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1331:
1084:
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703:
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60:
59:
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3505:
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3395:
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3370:
3357:
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3333:
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3321:
3320:
3312:
3309:
3308:
3295:
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3275:
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3211:
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3140:
3137:
3123:
3122:
3117:
3113:
3112:
3073:
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3056:
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3024:
3019:
3018:
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2810:
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2805:
2790:
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2767:
2755:
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2737:
2730:
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2190:
2188:
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2028:
2022:
2020:
2019:
2014:
1986:
1984:
1983:
1978:
1957:
1955:
1954:
1949:
1925:Forbidden decays
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1883:
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1407:
1375:
1373:
1372:
1367:
1349:
1347:
1346:
1341:
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866:
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759:, which governs
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439:
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423:
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417:
414:
410:
405:
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379:
377:
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371:
338:Fermi transition
328:weak interaction
321:
298:Murray Gell-Mann
286:George Sudarshan
249:
247:
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238:
215:
213:
212:
207:
185:
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158:Fermi transition
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27:
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19:
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3589:Nuclear physics
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2264:
2250:
2249:
2235:
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2199:
2198:
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2144:
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2002:
1999:
1998:
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1011:
1005:
1000:
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960:
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940: and
938:
926:
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888:
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871:
862:
858:
849:
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830:
810:
806:
797:
793:
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788:
787:
741:electromagnetic
733:
710:
705:
700:
693:
688:
686:
683:
680:
679:
659:
655:
630:
626:
618:
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614:
586:
583:
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430:
428:
418:
413:
412:
406:
401:
399:
396:
393:
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353:
350:
349:
340:
313:
294:Richard Feynman
284:was developed,
242:
237:
195:
192:
191:
165:
162:
161:
119:nuclear physics
111:
100:
94:
91:
83:help improve it
80:
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56:
50:
47:
41:
28:
24:
17:
12:
11:
5:
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3576:
3571:
3564:
3563:External links
3561:
3559:
3558:
3535:
3506:
3483:
3464:(3): 285–287.
3448:
3425:
3406:(2): 491–557.
3390:
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2038:
2033:
2012:
2009:
2006:
1976:
1973:
1970:
1967:
1947:
1944:
1941:
1938:
1926:
1923:
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1037:
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1008:
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865:
861:
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852:
848:
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838:
813:
809:
805:
800:
796:
767:depend on the
732:
729:
728:
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708:
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691:
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651:
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645:
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637:
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409:
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389:
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290:Robert Marshak
263:
262:
251:
250:
236:
233:
205:
202:
199:
175:
172:
169:
149:but differing
127:atomic nucleus
113:
112:
74:
72:
65:
58:
57:
31:
29:
22:
15:
9:
6:
4:
3:
2:
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3600:
3599:Radioactivity
3597:
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3405:
3401:
3394:
3386:
3384:0-226-10953-4
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3085:
3082:
3080:
3057:
3053:
3049:
3045:
3040:
3037:
3034:
3028:
3025:(Phase Space)
3020:
3015:
3010:
3005:
3002:
2999:
2995:
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1848:
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1806:mirror nuclei
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807:
798:
794:
784:
781:
776:
774:
770:
766:
762:
758:
754:
750:
746:
742:
738:
711:
706:
694:
689:
678:
677:
676:
660:
652:
649:
643:
635:
631:
627:
620:
597:
594:
591:
565:
562:
559:
548:
544:
540:
535:
531:
522:
518:
514:
509:
505:
497:
496:
475:
471:
466:
462:
458:
453:
448:
436:
431:
419:
407:
402:
391:
390:
387:
384:
383:
364:
361:
358:
348:
347:
346:
343:
335:
333:
329:
325:
320:
316:
311:
307:
303:
299:
295:
291:
287:
283:
278:
274:
270:
260:
256:
253:
252:
246:
241:
240:
232:
230:
226:
225:Edward Teller
222:
217:
203:
200:
197:
189:
173:
170:
167:
159:
154:
152:
151:atomic number
148:
144:
140:
136:
135:beta particle
132:
128:
124:
120:
109:
106:
98:
88:
84:
78:
75:This article
73:
64:
63:
54:
44:
40:
35:
32:This article
30:
21:
20:
3604:George Gamow
3546:
3538:
3515:
3509:
3494:
3486:
3461:
3457:
3451:
3436:
3428:
3403:
3399:
3393:
3374:
3368:
3218:
3215:
3086:
3083:
3076:
2881:
2878:
2857:complement (
2855:
2471:
2437:Notice that
2436:
2433:
2430:Gamow–Teller
2310:
2193:
2138:
2068:Gamow–Teller
2047:
2041:
2035:
2025:
1994:
1992:
1989:
1928:
1915:
1900:
1896:
1894:
1813:
1810:
1803:
1596:
1386:
785:
780:pseudovector
777:
772:
764:
753:pseudovector
734:
580:
344:
341:
332:gauge bosons
318:
314:
310:axial vector
266:
221:George Gamow
218:
188:Gamow-Teller
187:
157:
155:
122:
116:
101:
92:
76:
48:
37:
33:
3293:Fermi decay
3079:Hamiltonian
345:This means
304:structure (
147:mass number
129:undergoing
95:August 2016
51:August 2016
3583:Categories
3361:References
2958:such that
2852:Decay rate
2191:) decays.
2032:Transition
1350:state has
761:beta decay
757:weak force
259:beta decay
131:beta decay
3524:cite book
3478:120281084
3329:^
3326:τ
3317:^
3314:σ
3284:^
3281:τ
3272:^
3233:^
3192:ψ
3175:^
3158:ν
3154:ψ
3148:β
3144:ϕ
3134:ψ
3038:
3021:×
2985:ℏ
2981:π
2946:ℏ
2780:¯
2777:ν
2765:−
2761:β
2703:→
2599:ν
2586:β
2535:→
2457:⇒
2448:π
2445:Δ
2434:systems.
2397:Δ
2394:⇒
2388:→
2373:→
2358:→
2343:→
2328:→
2280:Δ
2277:⇒
2271:→
2256:→
2241:→
2226:→
2211:→
2011:ν
2005:β
1997:) of the
1966:Δ
1937:Δ
1832:≡
1780:Δ
1777:⇒
1744:⇒
1685:ν
1672:β
1663:∗
1634:→
1573:Δ
1570:⇒
1537:⇒
1478:ν
1465:β
1432:→
1329:¯
1326:ν
1316:β
1270:⇒
1265:ℓ
1254:−
1236:ℓ
1225:π
1205:⇒
1196:π
1193:Δ
1164:Δ
1161:⇒
1135:→
1076:¯
1073:ν
1061:−
1057:β
1024:→
856:−
837:Δ
804:→
769:chirality
712:∗
661:ℓ
650:−
632:ℓ
621:π
601:⇒
592:π
589:Δ
557:Δ
554:⇒
528:→
476:ν
463:β
454:∗
425:→
368:⇒
356:Δ
271:original
3201:⟩
3138:Daughter
3129:⟨
1392:Examples
1308:and the
980:Examples
385:Examples
139:neutrino
3408:Bibcode
2130:3, 4, 5
2116:2, 3, 4
2102:1, 2, 3
2088:0, 1, 2
775:added.
269:Fermi's
143:nuclide
81:Please
3476:
3381:
3196:Parent
751:and a
749:vector
745:parity
737:strong
308:minus
306:vector
302:tensor
273:ansatz
248:
3474:S2CID
2863:p → n
2859:n → p
2831:years
2640:years
2307:Fermi
2054:Fermi
1185:also
826:) is
581:also
39:page.
3530:link
3379:ISBN
2311:and
2074:0, 1
765:does
739:and
296:and
288:and
223:and
121:, a
3553:192
3501:200
3466:doi
3462:332
3443:198
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3240:int
3182:int
2861:or
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1597:or
773:are
227:at
117:In
85:to
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2547:22
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2133:0
2119:1
2105:0
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2077:0
2063:0
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1865:GT
1658:11
1653:Ne
1646:21
1641:10
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1624:Na
1617:21
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1444:21
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675:.
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2223:L
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2179:1
2176:=
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2099:2
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1757:=
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1240:m
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1228:(
1202:0
1199:=
1173:1
1170:=
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1130:+
1126:0
1122:=
1117:i
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1083:e
1066:+
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1036:6
1031:3
1019:4
1007:6
1002:2
957:1
954:=
949:f
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911:0
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869:=
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644:=
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168:S
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102:(
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93:(
79:.
53:)
49:(
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