Geometric Shapes of Nuclei
5.2 The Rutherford Cross-Section
58 5 Geometric Shapes of Nuclei
Fig. 5.2 Angular
dependence of the scattering energy of electrons normalised to beam energy, E0=E, in elastic
electron-nucleus scattering.
The curves show this dependence for two different beam energies (0:5and 10 GeV) and for two nuclei with different masses (AD1 andAD50)
E= 0.5 GeV A= 5 0 E = 10 GeV A= 5 0
E= 0.5 GeV A =1
E= 10 GeV A =1 0
0.2 0.4 0.6 0.8 1 1.2
E'/E
0˚ 50˚ 100˚ 150˚
θ
off protons. The scattering energyE0 then varies between 10 GeV ( 0ı) and 445 MeV (D180ı) (cf. Fig.5.2).
5.2 The Rutherford Cross-Section 59 Scattering off an extended charge distribution Consider the case of a target so heavy that the recoil is negligible. We can then use three-momenta. IfZeis small, i.e., if
Z˛ 1 ; (5.17)
theBorn approximationcan be applied, and the wave functions i and f of the incoming and of the outgoing electron can be described by plane waves
iD p1
V eipx=„ f D p1
V eip0x=„: (5.18) We can sidestep any difficulties related to the normalisation of the wave functions by considering only a finite volumeV. We need this volume to be large compared to the scattering centre, and also large enough that the discrete energy states in this volume can be approximated by a continuum. The physical results have, of course, to be independent ofV.
We consider an electron beam with a density ofnaparticles per unit volume. With the volume of integration chosen to be sufficiently large, the normalisation condition is given by
Z
V
j ij2dV DnaV where VD Na
na ; (5.19)
i.e.,Vis the normalisation volume that must be chosen for a single beam particle.
According to (4.20), the reaction rateW is given by the product of the cross- section and the beam particle velocity va divided by the above volume. When applying the golden rule (4.19), we get
va
V DWD 2
„ ˇˇh fjHintj iiˇˇ2 dn dEf
: (5.20)
Here,Ef is the total energy (kinetic energy and rest mass) of the final state. Since we neglect the recoil and since the rest mass is a constant, dEf DdE0DdE.
The densitynof possible final states in phase space (cf. (4.16)) is dn.jp0j/D 4jp0j2djp0j V
.2„/3 : (5.21)
Therefore the cross-section for the scattering of an electron into a solid angle element d˝is
dva1 V D 2
„ ˇˇh fjHintj iiˇˇ2Vjp0j2djp0j .2„/3dEf
d˝ : (5.22)
60 5 Geometric Shapes of Nuclei The velocityvacan be replaced, to a good approximation, by the velocity of light c. For large electron energies,jp0j E0=capplies, and we obtain
d
d˝ D V2E02
.2/2.„c/4ˇˇh fjHintj iiˇˇ2 : (5.23) The interaction operator for a chargeein an electric potential isHint D e.
Hence, the matrix element is h fjHintj ii D e
V Z
eip0x=„.x/eipx=„d3x: (5.24) Defining themomentum transferqby
qDpp0; (5.25)
we may re-write the matrix element as h fjHintj ii D e
V Z
.x/eiqx=„d3x: (5.26)
Green’s theorem permits us to use a clever trick here: for two arbitrarily chosen scalar fieldsu andv, which fall off fast enough at large distances, the following equation holds for a sufficiently large integration volume:
Z
.u4vv4u/d3xD0 ; with 4 D r2: (5.27)
Inserting
eiqx=„D„2
jqj2 4eiqx=„ (5.28)
into (5.26), we may rewrite the matrix element as h fjHintj ii D e„2 Vjqj2 Z
4.x/eiqx=„d3x: (5.29)
The potential.x/and the charge density%.x/are related by Poisson’s equation 4.x/D%.x/
"0 : (5.30)
In the following, we will assume the charge density %.x/ to be static, i.e. independent of time.
5.2 The Rutherford Cross-Section 61
We now define a charge distribution functionf by%.x/DZef.x/which satisfies the normalisation conditionR
f.x/d3xD1, and re-write the matrix element as h fjHintj ii D e„2
"0Vjqj2 Z
%.x/eiqx=„d3x D Z4˛„3c
jqj2V Z
f.x/eiqx=„d3x: (5.31) The integral
F.q/D Z
eiqx=„f.x/d3x (5.32)
is the Fourier transform of the charge functionf.x/, normalised to the total charge.
It is called theform factorof the charge distribution. The form factor contains all the information about the spatial distribution of the charge of the object being studied.
We will discuss form factors and their meaning in the following chapters in some detail.
To calculate the Rutherford cross-section we, by definition, neglect the spatial extension – i.e., we replace the charge distribution by aı-function. Hence, the form factor is fixed to unity. By inserting the matrix element into (5.23) we obtain
d d˝
Rutherford
D 4Z2˛2.„c/2E02
jqcj4 : (5.33)
The1=q4-dependence of the electromagnetic cross-section implies very low event rates for electron scattering with large momentum transfers. The event rates drop off so sharply that small measurement errors inqcan significantly falsify the results.
Since recoil is neglected in Rutherford scattering, the electron energy and the magnitude of its momentum do not change in the interaction:
EDE0; jpj D jp0j: (5.34) The magnitude of the momentum transferqis therefore
p´ θ/2 q p
jqj D2 jpjsin
2 : (5.35)
62 5 Geometric Shapes of Nuclei
Fig. 5.3 Sketch of elastic electron scattering off a nucleus with charge Ze
q p´
p e
Ze
If we recall thatED jpjcis a good approximation we obtain the relativistic Rutherford scattering formula
d d˝
Rutherford
DZ2˛2.„c/2
4E2sin4 2 : (5.36)
The classical Rutherford formula (5.16) may be obtained from (5.33) by applying non- relativistic kinematics:pDmv,EkinDmv2=2andE0mc2.
Field-theoretical considerations Figure 5.3 is a pictorial representation of a scattering process. In the language of field theory, the electromagnetic interaction of an electron with the charge distribution is mediated by the exchange of a photon, the field quantum of this interaction. The photon which does not itself carry any charge, couples to the charges of the two interacting particles. In the transition matrix element, this yields a factorZeeand in the cross-section we have a term.Ze2/2. The three-momentum transferqdefined in (5.25) is the momentum transferred by the exchanged photon. Hence the reduced de Broglie wavelength of the photon is
– D „ jqj D „
jpj 1
2sin2 : (5.37)
If– is considerably larger than the spatial extent of the target particle, internal structures cannot be resolved, and the target particle may be considered to be point- like. The Rutherford cross-section from (5.33) was obtained for this case.
In the form (5.33), the dependence of the cross-section on the momentum transfer is clearly expressed. To lowest order the interaction is mediated by the exchange of a single photon. Since the photon is massless, the propagator (4.23) in the matrix element is1=Q2, or1=jqj2in a non-relativistic approximation. The propagator enters the cross-section squared which leads to the characteristic fast1=jqj4fall-off of the cross-section.
If the Born approximation condition (5.17) no longer holds, then our simple picture must be modified. Higher order corrections (exchange of several photons) must be included and more complicated calculations (phase shift analyses) are necessary.