Structure function

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In nuclear and particle physics , the structural functions and or dimensionless , (and ) occur in deep inelastic scattering processes on nuclei and nucleons ( proton and neutron ). They indicate how strong the scatter is, depending on the energy transmitted between the scatter partners and the momentum . By measuring them, conclusions can be drawn about the internal structure of the collision partners, in particular about the momentum distributions of the quarks contained in the nucleons . ${\displaystyle W_{1}}$ ${\displaystyle W_{2}}$ ${\displaystyle F_{1}}$ ${\displaystyle F_{2}}$ ${\displaystyle F_{3}}$

With the help of the structure functions in deep-inelastic electron-nucleon scattering, the Parton model was developed and verified, i. H. the model for protons and neutrons composed of quarks. In addition, the spin and the electrical charge of the quarks can be determined experimentally using the structure functions.

In elastic scattering processes, the electrical and magnetic form factors are the analogues of the structural functions.

Experimental determination

Analogous to the Rosenbluth formula for elastic scattering processes, the following applies to the double differential cross section :

${\displaystyle {\frac {d^{2}\sigma }{d\Omega \,dE^{\prime }}}=\left({\frac {d\sigma }{d\Omega }}\right)_{\mathrm {Mott} }\left[2W_{1}(Q^{2},\nu )\,\tan ^{2}(\theta /2)+W_{2}(Q^{2},\nu )\right]}$

are there

• ${\displaystyle \left({\frac {d\sigma }{d\Omega }}\right)_{\mathrm {Mott} }}$the Mott - cross section
• ${\displaystyle Q^{2}}$the transmitted quadruple momentum , in the example of electron scattering with the quadruple momentum of the electron before and after the scattering ${\displaystyle Q^{2}=-q^{2}=(p_{e}-p_{e}^{\prime })^{2}}$ ${\displaystyle p_{e}}$ ${\displaystyle p_{e}^{\prime }}$
• ${\displaystyle \nu =E-E^{\prime }}$the transmitted energy in the laboratory system
• ${\displaystyle \theta }$the scattering angle
• ${\displaystyle W_{1}(Q^{2},\nu )}$and the structure functions. ${\displaystyle W_{2}(Q^{2},\nu )}$

If the cross-section is measured at fixed and for different scattering angles and plots on the x-axis and on the y-axis in analogy to the Rosenbluth plot , the double differential cross-section takes on the following linear form: ${\displaystyle Q^{2}}$ ${\displaystyle \nu }$ ${\displaystyle \tan ^{2}(\theta /2)}$ ${\displaystyle {\frac {d^{2}\sigma }{d\Omega \,dE^{\prime }}}:\left({\frac {d\sigma }{d\Omega }}\right)_{\mathrm {Mott} }=2W_{1}(Q^{2},\nu )\,\tan ^{2}(\theta /2)+W_{2}(Q^{2},\nu )}$

${\displaystyle y(x)=2W_{1}\cdot x+W_{2}}$

With

• the slope ${\displaystyle 2W_{1}}$
• the y-intercept . ${\displaystyle W_{2}}$

This has to be repeated for many values ​​of and in order to determine the structure functions. ${\displaystyle Q^{2}}$ ${\displaystyle \nu }$

Dimensionless structural functions

Often the dimensionless structure functions are given instead of and : ${\displaystyle W_{1}}$ ${\displaystyle W_{2}}$

${\displaystyle {\begin{alignedat}{2}F_{1}(x,Q^{2})&=M\cdot c^{2}&&\cdot W_{1}(Q^{2},\nu )\\F_{2}(x,Q^{2})&=\nu &&\cdot W_{2}(Q^{2},\nu )\end{alignedat}}}$

which depend on the Bjorken scale (also Bjorken 's scale variable) ( is the mass of the target - for example a proton - and the four-momentum of the target). This is a measure of the inelasticity. ${\displaystyle x={\frac {Q^{2}}{2Pq}}={\frac {Q^{2}}{2M\nu }}}$ ${\displaystyle M}$ ${\displaystyle P}$

In the inelastic scattering of neutrinos by nucleons, a third structural function occurs, which explicitly takes into account the parity violation of the neutrinos. ${\displaystyle F_{3}^{\nu N}}$

Structure functions and parton model

The dimensionless structure functions and depend on the Bjorken scale , but only very weakly on the four-momentum transfer ( scale invariance ). From this it follows that the nucleons consist of smaller point-like particles (partons). ${\displaystyle F_{1}}$ ${\displaystyle F_{2}}$ ${\displaystyle x}$ ${\displaystyle Q^{2}}$

Determination of the quark spin

The dimensionless structure functions satisfy the Callan-Gross relationship . This means that the partons are particles with spin 1/2. ${\displaystyle F_{2}(x)=2x\,F_{1}(x)}$

If the partons had spin 0, then this would be because this structure function corresponds to the magnetic form factor. ${\displaystyle F_{1}(x)=0}$

Determination of the electric charge of the quarks

In order to determine the third-number electric charge of the quarks, one compares the measured structure functions from electron-nucleon scattering and from neutrino -nucleon scattering. ${\displaystyle F_{2}^{eN}(x)}$ ${\displaystyle F_{2}^{\nu N}(x)}$

• Electron-nucleon scattering: Since electrons do not participate in the strong interaction , the scattering of electrons on nucleons can only take place on the electric charge z of the quarks. The structure function must therefore depend on z:

${\displaystyle F_{2}^{eN}(x)=x\cdot \sum _{f}z_{f}^{2}\left(q_{f}(x)+{\bar {q}}_{f}(x)\right)}$

The sum runs over all relevant quark types, i.e. u, d and s quarks. All other types of quark are too heavy to contribute. indicates the electrical charge of the respective type of quark in units of the elementary charge . and denote the momentum distributions of the quarks and antiquarks. ${\displaystyle z_{f}}$ ${\displaystyle q_{f}(x)}$ ${\displaystyle {\bar {q}}_{f}(x)}$

• Neutrino-nucleon scattering: Since neutrinos neither take part in the strong interaction nor in the electromagnetic force , the electrical charge of the quarks does not affect the structure function at this point :

${\displaystyle F_{2}^{\nu N}(x)=x\cdot \sum _{f}\left(q_{f}(x)+{\bar {q}}_{f}(x)\right)}$

The quark charge can be determined by comparing the measurement results of these two structural functions. It agrees with the predicted third-digit values.