Form factor (physics)

In nuclear and particle physics , the form factor is a factor in the cross-section for elastic collisions . It is the Fourier transform of the electrical charge distribution of the target particle (e.g. atomic nucleus ) and depends on the momentum transmitted during the scattering . The form factor indicates how the spread depends on the momentum transfer. By measuring the form factor with different pulse transmissions, conclusions can be drawn about the charge distribution of the target. ${\displaystyle F}$

In the case of inelastic collisions , the structure functions take the place of the form factor .

Form factor in Rutherford scattering

The Rutherford scattering formula , only for the scattering of a particle on a point charge ( Coulomb applies), can be extended for extended charge distributions. The differential cross section then looks like this

${\displaystyle \left({\frac {\mathrm {d} \sigma }{\mathrm {d} \Omega }}\right)_{\theta }=\left({\frac {\mathrm {d} \sigma }{\mathrm {d} \Omega }}\right)_{\text{Coul}}\cdot |F({\vec {q}})|^{2},}$

where is the shape factor of the charge distribution. ${\displaystyle F}$

It depends on the momentum transfer of the incident particle

${\displaystyle {\vec {q}}={\vec {p}}-{\vec {p}}\,{}^{\prime }}$

and contains all information about the spatial distribution of the charge in the scattering center. So you can use the measurement of the cross-section of certain scattering processes depending on the momentum transfer in order to make statements about the form of the scattering potential by subsequent comparison with theoretical models.

In the Born approximation (i.e. the potential of the interaction is so weak that the initial and final state can be treated approximately as plane waves ) the form factor results as a Fourier transform of the charge distribution function normalized to the total charge : ${\displaystyle f}$

${\displaystyle F({\vec {q}})=\int f({\vec {x}})\cdot e^{\mathrm {i} {\vec {q}}\cdot {\vec {x}}/\hbar }\,\mathrm {d} ^{3}x}$

With

• the imaginary unit ${\displaystyle \mathrm {i} }$
• the reduced Planck quantum of action ${\displaystyle \hbar .}$

The charge distribution function is defined as:

${\displaystyle f({\vec {x}})={\frac {\rho ({\vec {x}})}{Z\cdot e}},}$

in which

• ${\displaystyle \rho ({\vec {x}})}$ the static charge density
• ${\displaystyle Z}$the atomic number and
• ${\displaystyle e}$is the elementary charge ;

it satisfies the normalization condition

${\displaystyle \int f({\vec {x}})\,\mathrm {d} ^{3}x=1}$.

Often you only have a radial dependence, so that you can not but indicates, because has no directionality. ${\displaystyle F({\vec {q}})}$ ${\displaystyle F(q^{2})}$ ${\displaystyle q^{2}=|{\vec {q}}|^{2}}$

The form factor contains the information about the charge distribution and thus about the charge density of interest . It is determined experimentally by measuring cross-sections, and from this the charge distribution or charge density is calculated. The result for heavier cores is a charge distribution that is almost constant in the inner area and drops over an area of ​​2.4 fm on the outside . With light nuclei such as ⁴ He , ⁶ Li or ⁹ Be , a constant charge density cannot yet develop in the interior of the nucleus; a Gaussian charge distribution is observed here . ${\displaystyle F}$ ${\displaystyle f}$ ${\displaystyle \rho }$

Form factors of nucleons

When determining the form factors of the nucleons , much smaller structures have to be resolved. This requires a smaller De Broglie wavelength and thus correspondingly higher energies, so that more precise calculations are required because approximations are no longer valid. In addition, in contrast to the Rutherford scattering section, the treatment is now relativistic with four-vectors instead of vectors. In addition, electrical and magnetic form factors designated by and appear here . For the differential cross section one obtains the Rosenbluth formula , which goes back to MN Rosenbluth : ${\displaystyle G_{E}}$ ${\displaystyle G_{M}}$

${\displaystyle {\frac {\mathrm {d} \sigma }{\mathrm {d} \Omega }}=\left({\frac {\mathrm {d} \sigma }{\mathrm {d} \Omega }}\right)_{\text{Mott}}\cdot \left[{\frac {G_{E}^{2}(Q^{2})+\tau \cdot G_{M}^{2}(Q^{2})}{1+\tau }}+2\tau \cdot G_{M}^{2}(Q^{2})\cdot \tan ^{2}(\theta /2)\right]}$

With:

• ${\displaystyle \left(\mathrm {d} \sigma /\mathrm {d} \Omega \right)_{\text{Mott}}}$the Mott cross section
• ${\displaystyle Q^{2}=-q^{2}}$the negative square of the transmitted quad pulse
• ${\displaystyle \tau =Q^{2}/4M^{2}c^{2}}$the probability of a spin-flip in the scatter
• ${\displaystyle \theta }$the scattering angle .

If the cross-section has been measured at a fixed angle for several scattering angles, a Rosenbluth plot is made in which the -axis and the -axis are plotted. The Rosenbluth formula is then of the linear form ${\displaystyle Q^{2}}$ ${\displaystyle \tan ^{2}(\theta /2)}$ ${\displaystyle x}$ ${\displaystyle (d\sigma /d\Omega ):\left(\mathrm {d} \sigma /\mathrm {d} \Omega \right)_{\text{Mott}}}$ ${\displaystyle y}$

${\displaystyle y(x)=A\cdot x+B,}$

The magnetic and electrical form factors can be calculated from the gradient and the axis intercept : ${\displaystyle A=2\tau \cdot G_{M}^{2}(Q^{2})}$ ${\displaystyle B={\frac {G_{E}^{2}(Q^{2})+\tau \cdot G_{M}^{2}(Q^{2})}{1+\tau }}}$

${\displaystyle \Rightarrow G_{M}(Q^{2})={\sqrt {\frac {A}{2\tau }}}}$

and

${\displaystyle \Rightarrow G_{E}(Q^{2})={\sqrt {B(1+\tau )-{\frac {A}{2}}}}.}$

The experimental results show an exponential decrease for both form factors, which fits neither a point-like particle nor a homogeneous sphere. This gives an indication of a more complex internal structure of the nucleons.

Individual evidence

1. ↑ Bogdan Povh, Klaus Rith, Christoph Scholz, Frank Zetsche: Particles and Cores , 8th Edition, Springer Verlag 2009, Chapter 5.4: Form factors of the cores
2. ↑ MN Rosenbluth: High Energy Elastic Scattering of Electrons on Protons , Phys. Rev. (1950), Volume 79, Page 615
3. ↑ Bogdan Povh, Klaus Rith, Christoph Scholz, Frank Zetsche: Particles and Cores , 8th Edition, Springer Verlag 2009, Chapter 6.1: Form factors of the nucleon , in particular page 81