Tarqalish nazariyasi

2-BOB. NAZARIYA
2.28 tenglamada berilgan Shrodinger tenglamasining burchak komponentlari burchakka bog'liqligi bo'lmagan barcha potensiallar uchun bir xil bo'ladi. Markaziy potentsial bog'liqlikni ko'rsatadigan qolgan radial tenglamani yangi funktsiyani kiritish bilan biroz soddalashtirish mumkin,
(2.29)
Bu bilan 2.25a tenglama soddalashtiriladi
(2.30)
Bu tenglama ko'pincha radial Shrodinger tenglamasi deb ataladi. Radial Shrodinger tenglamasini yanada yechish uchun potentsial uchun aniq funktsiya talab qilinadi.

2.2.2 Differensial kesma va tarqalish amplitudalari

Yadro astrofizikasida eksperimental natijalar va nazariy hisob-kitoblar o'rtasidagi uchrashuv nuqtasi ko'pincha differensial kesma hisoblanadi. Differensial kesma, dσ/dΩ, nur yo'nalishidan o'lchangan θ qutb burchaklari va azimut burchagi φ uchun ba'zi bir potentsial V(r) bilan tarqalgan zarrachalarning burchak taqsimotini tavsiflaydi [24]. Tarqalish potentsiali V(r) 2.22 tenglamada ko'rsatilgan bilan bir xil. Bir zarraning boshqasiga sochilishi uchun r ikki zarracha orasidagi nisbiy koordinatani ifodalaydi.

Differensial kesmani o‘lchangan tarqoq burchak oqimining ning steradian birligidagi zarrachalar oqimining hodisa oqimiga nisbati sifatida ifodalash, , vaqt birligidagi zarrachalar birligida ifodani beradi.

(2.31)
Oqim maydon birligidagi vaqt birligidagi zarrachalar sonining o'lchovi sifatida belgilanishi mumkin va zarrachalarning ehtimollik zichligi tezligiga ko'paytirilishiga tengdir.
(2.32)
Yaxshi kollimatsiyalangan, bir xil energiya zarralarining tushayotgan nuri tekis to'lqin sifatida ifodalanishi mumkin. Nur faqat A amplitudasi bilan +zˆ yo'nalishida bo'ladigan koordinatalar tizimini tanlab, tushayotgan to'lqin funksiyasi quyidagicha ifodalanadi.
2-BOB. NAZARIYA
(2.33)
Resulting in an incident flux of
(2.34)
Where . The wavefunction for the scattered particles need only be expressed at values of large r, at the detector, outside the range of the scattering potential. An outgoing spherical wave will asymptotically be proportional to at large r, and vary with the angles θ and φ. Confining the angular dependence of the scattered wave in the function f(θ, φ), the scattered wave can be written as
(2.35)
From this the scattered flux is found to be
(2.36)
The scattered angular flux per steradian is related to scattered flux by a factor of resulting in
(2.37)
Putting these expressions for the incident and scattered flux into Equation 2.31 results in a differential cross section of
(2.38)
In the case of elastic scattering the wave number ratio is one and we are left with
(2.39)
The amplitude A of each of the functions is irrelevant as we see the differential cross section is independent of the overall normalization it provides. The function f(θ, φ) is called the scattering amplitude which has units of length, and in general is complex valued [24].
CHAPTER 2. THEORY
2.2.3 Phase shifts
Consider the Schr¨odinger equation (2.22) with boundary conditions for a typical elastic scattering experiment. An incident flux ji originating from a source hits a target, and a scattered flux js radiates outward in all directions. A solution of the form
(2.40)
will exist where the incident wave, represents the incident beam, and is an outgoing scattered wave [25]. The label ”asymptotic” is used to indicate that this is a solution located in free space outside the range of the interaction potential. Putting in the expression for the incident and scattered waves from equations 2.33 and 2.35 and setting the normalization to one we have
(2.41)
The solution to the Schr¨odinger equation worked out in section 2.2.1 can be expressed as a series of partial waves,
(2.42)
with the spherical harmonics given in Equation 2.28 and the function u(r) satisfying the radial Schr¨odinger equation in Equation 2.30. As previously shown, the incident wave can be expressed as a plane wave, which is the same solution as if the scattering potential were identically zero. The normalization of the incident wave was shown to cancel in the calculation of the differential cross section, and thus will be set to unity here as it will be of no consequence in further calculations. The plane wave solution can then be expanded as
(2.43)
Where is a spherical Bessel function, and is a Legendre polynomial. The set of Legendre polynomials forms an orthogonal and complete set over angles 0 ≤ θ ≤ π, satisfying the orthogonality and normalization conditions [24]
(2.44)
CHAPTER 2. THEORY
Note the radial dependence of Equation 2.43 is located only in the Bessel function thus a similar solution to Equation 2.22 with a scattering potential will be of the form
(2.45)
where the radial function satisfies the partial wave equation,
(2.46)
with substitutions and . Outside the range of the potential U(r), this differential equation is satisfied by the spherical Bessel functions, (kr) and (kr). Since we need only consider the solutions outside the range of the potential when solving for the differential cross section we can take the solution as a linear combination of the two functions [25],
(2.47)
In the limit of kr → the asymptotic forms of the Bessel functions are
(2.48a)
(2.48b)
giving rise to the corresponding asymptotic expression for ,
(2.49)
Comparing the asymptotic limits of the zero scattering solution given in Equation 2.43 and the scattering solution of Equation 2.45, the apparent effect a short range scattering potential has is expressed as a phase shift of the radial function by the factor δ` at large r. Substituting into Equation 2.41 the results of equations 2.43 and 2.45 with corresponding asymptotic limits given by equations 2.48a and 2.49 respectively, and using Euler’s formula to express the sine functions in terms of complex exponentials we obtain
CHAPTER 2. THEORY