\(\newcommand{\footnotename}{footnote}\)
\(\def \LWRfootnote {1}\)
\(\newcommand {\footnote }[2][\LWRfootnote ]{{}^{\mathrm {#1}}}\)
\(\newcommand {\footnotemark }[1][\LWRfootnote ]{{}^{\mathrm {#1}}}\)
\(\newcommand \ensuremath [1]{#1}\)
\(\newcommand {\LWRframebox }[2][]{\fbox {#2}} \newcommand {\framebox }[1][]{\LWRframebox } \)
\(\newcommand {\setlength }[2]{}\)
\(\newcommand {\addtolength }[2]{}\)
\(\newcommand {\setcounter }[2]{}\)
\(\newcommand {\addtocounter }[2]{}\)
\(\newcommand {\cline }[1]{}\)
\(\newcommand {\directlua }[1]{\text {(directlua)}}\)
\(\newcommand {\luatexdirectlua }[1]{\text {(directlua)}}\)
\(\newcommand {\protect }{}\)
\(\def \LWRabsorbnumber #1 {}\)
\(\def \LWRabsorbquotenumber "#1 {}\)
\(\def \mathchar {\ifnextchar "\LWRabsorbquotenumber \LWRabsorbnumber }\)
\(\def \mathcode #1={\mathchar }\)
\(\let \delcode \mathcode \)
\(\let \delimiter \mathchar \)
\(\let \LWRref \ref \)
\(\renewcommand {\ref }{\ifstar \LWRref \LWRref }\)
\(\newcommand {\intertext }[1]{\text {#1}\notag \\}\)
\(\newcommand {\mathllap }[2][]{{#1#2}}\)
\(\newcommand {\mathrlap }[2][]{{#1#2}}\)
\(\newcommand {\mathclap }[2][]{{#1#2}}\)
\(\newcommand {\mathmbox }[1]{#1}\)
\(\newcommand {\clap }[1]{#1}\)
\(\newcommand {\LWRmathmakebox }[2][]{#2}\)
\(\newcommand {\mathmakebox }[1][]{\LWRmathmakebox }\)
\(\newcommand {\cramped }[2][]{{#1#2}}\)
\(\newcommand {\crampedllap }[2][]{{#1#2}}\)
\(\newcommand {\crampedrlap }[2][]{{#1#2}}\)
\(\newcommand {\crampedclap }[2][]{{#1#2}}\)
\(\newenvironment {crampedsubarray}[1]{}{}\)
\(\newcommand {\crampedsubstack }{}\)
\(\newcommand {\smashoperator }[2][]{#2\limits }\)
\(\newcommand {\adjustlimits }{}\)
\(\newcommand {\SwapAboveDisplaySkip }{}\)
\(\require {extpfeil}\)
\(\Newextarrow \xleftrightarrow {10,10}{0x2194}\)
\(\Newextarrow \xLeftarrow {10,10}{0x21d0}\)
\(\Newextarrow \xhookleftarrow {10,10}{0x21a9}\)
\(\Newextarrow \xmapsto {10,10}{0x21a6}\)
\(\Newextarrow \xRightarrow {10,10}{0x21d2}\)
\(\Newextarrow \xLeftrightarrow {10,10}{0x21d4}\)
\(\Newextarrow \xhookrightarrow {10,10}{0x21aa}\)
\(\Newextarrow \xrightharpoondown {10,10}{0x21c1}\)
\(\Newextarrow \xleftharpoondown {10,10}{0x21bd}\)
\(\Newextarrow \xrightleftharpoons {10,10}{0x21cc}\)
\(\Newextarrow \xrightharpoonup {10,10}{0x21c0}\)
\(\Newextarrow \xleftharpoonup {10,10}{0x21bc}\)
\(\Newextarrow \xleftrightharpoons {10,10}{0x21cb}\)
\(\newcommand {\LWRdounderbracket }[3]{\underset {#3}{\underline {#1}}}\)
\(\newcommand {\LWRunderbracket }[2][]{\LWRdounderbracket {#2}}\)
\(\newcommand {\underbracket }[1][]{\LWRunderbracket }\)
\(\newcommand {\LWRdooverbracket }[3]{\overset {#3}{\overline {#1}}}\)
\(\newcommand {\LWRoverbracket }[2][]{\LWRdooverbracket {#2}}\)
\(\newcommand {\overbracket }[1][]{\LWRoverbracket }\)
\(\newcommand {\LATEXunderbrace }[1]{\underbrace {#1}}\)
\(\newcommand {\LATEXoverbrace }[1]{\overbrace {#1}}\)
\(\newenvironment {matrix*}[1][]{\begin {matrix}}{\end {matrix}}\)
\(\newenvironment {pmatrix*}[1][]{\begin {pmatrix}}{\end {pmatrix}}\)
\(\newenvironment {bmatrix*}[1][]{\begin {bmatrix}}{\end {bmatrix}}\)
\(\newenvironment {Bmatrix*}[1][]{\begin {Bmatrix}}{\end {Bmatrix}}\)
\(\newenvironment {vmatrix*}[1][]{\begin {vmatrix}}{\end {vmatrix}}\)
\(\newenvironment {Vmatrix*}[1][]{\begin {Vmatrix}}{\end {Vmatrix}}\)
\(\newenvironment {smallmatrix*}[1][]{\begin {matrix}}{\end {matrix}}\)
\(\newenvironment {psmallmatrix*}[1][]{\begin {pmatrix}}{\end {pmatrix}}\)
\(\newenvironment {bsmallmatrix*}[1][]{\begin {bmatrix}}{\end {bmatrix}}\)
\(\newenvironment {Bsmallmatrix*}[1][]{\begin {Bmatrix}}{\end {Bmatrix}}\)
\(\newenvironment {vsmallmatrix*}[1][]{\begin {vmatrix}}{\end {vmatrix}}\)
\(\newenvironment {Vsmallmatrix*}[1][]{\begin {Vmatrix}}{\end {Vmatrix}}\)
\(\newenvironment {psmallmatrix}[1][]{\begin {pmatrix}}{\end {pmatrix}}\)
\(\newenvironment {bsmallmatrix}[1][]{\begin {bmatrix}}{\end {bmatrix}}\)
\(\newenvironment {Bsmallmatrix}[1][]{\begin {Bmatrix}}{\end {Bmatrix}}\)
\(\newenvironment {vsmallmatrix}[1][]{\begin {vmatrix}}{\end {vmatrix}}\)
\(\newenvironment {Vsmallmatrix}[1][]{\begin {Vmatrix}}{\end {Vmatrix}}\)
\(\newcommand {\LWRmultlined }[1][]{\begin {multline*}}\)
\(\newenvironment {multlined}[1][]{\LWRmultlined }{\end {multline*}}\)
\(\let \LWRorigshoveleft \shoveleft \)
\(\renewcommand {\shoveleft }[1][]{\LWRorigshoveleft }\)
\(\let \LWRorigshoveright \shoveright \)
\(\renewcommand {\shoveright }[1][]{\LWRorigshoveright }\)
\(\newenvironment {dcases}{\begin {cases}}{\end {cases}}\)
\(\newenvironment {dcases*}{\begin {cases}}{\end {cases}}\)
\(\newenvironment {rcases}{\begin {cases}}{\end {cases}}\)
\(\newenvironment {rcases*}{\begin {cases}}{\end {cases}}\)
\(\newenvironment {drcases}{\begin {cases}}{\end {cases}}\)
\(\newenvironment {drcases*}{\begin {cases}}{\end {cases}}\)
\(\newenvironment {cases*}{\begin {cases}}{\end {cases}}\)
\(\newcommand {\MoveEqLeft }[1][]{}\)
\(\def \LWRAboxed #1!|!{\fbox {\(#1\)}&\fbox {\(#2\)}} \newcommand {\Aboxed }[1]{\LWRAboxed #1&&!|!} \)
\( \newcommand {\LWRABLines }[1][\Updownarrow ]{#1 \notag \\}\newcommand {\ArrowBetweenLines }{\ifstar \LWRABLines \LWRABLines } \)
\(\newcommand {\shortintertext }[1]{\text {#1}\notag \\}\)
\(\newcommand {\vdotswithin }[1]{\hspace {.5em}\vdots }\)
\(\newcommand {\LWRshortvdotswithinstar }[1]{\vdots \hspace {.5em} & \\}\)
\(\newcommand {\LWRshortvdotswithinnostar }[1]{& \hspace {.5em}\vdots \\}\)
\(\newcommand {\shortvdotswithin }{\ifstar \LWRshortvdotswithinstar \LWRshortvdotswithinnostar }\)
\(\newcommand {\MTFlushSpaceAbove }{}\)
\(\newcommand {\MTFlushSpaceBelow }{\\}\)
\(\newcommand \lparen {(}\)
\(\newcommand \rparen {)}\)
\(\newcommand {\ordinarycolon }{:}\)
\(\newcommand {\vcentcolon }{\mathrel {\mathop \ordinarycolon }}\)
\(\newcommand \dblcolon {\vcentcolon \vcentcolon }\)
\(\newcommand \coloneqq {\vcentcolon =}\)
\(\newcommand \Coloneqq {\dblcolon =}\)
\(\newcommand \coloneq {\vcentcolon {-}}\)
\(\newcommand \Coloneq {\dblcolon {-}}\)
\(\newcommand \eqqcolon {=\vcentcolon }\)
\(\newcommand \Eqqcolon {=\dblcolon }\)
\(\newcommand \eqcolon {\mathrel {-}\vcentcolon }\)
\(\newcommand \Eqcolon {\mathrel {-}\dblcolon }\)
\(\newcommand \colonapprox {\vcentcolon \approx }\)
\(\newcommand \Colonapprox {\dblcolon \approx }\)
\(\newcommand \colonsim {\vcentcolon \sim }\)
\(\newcommand \Colonsim {\dblcolon \sim }\)
\(\newcommand {\nuparrow }{\mathrel {\cancel {\uparrow }}}\)
\(\newcommand {\ndownarrow }{\mathrel {\cancel {\downarrow }}}\)
\(\newcommand {\bigtimes }{\mathop {\Large \times }\limits }\)
\(\newcommand {\prescript }[3]{{}^{#1}_{#2}#3}\)
\(\newenvironment {lgathered}{\begin {gathered}}{\end {gathered}}\)
\(\newenvironment {rgathered}{\begin {gathered}}{\end {gathered}}\)
\(\newcommand {\splitfrac }[2]{{}^{#1}_{#2}}\)
\(\let \splitdfrac \splitfrac \)
\(\newcommand {\LWRoverlaysymbols }[2]{\mathord {\smash {\mathop {#2\strut }\limits ^{\smash {\lower 3ex{#1}}}}\strut }}\)
\(\newcommand{\alphaup}{\unicode{x03B1}}\)
\(\newcommand{\betaup}{\unicode{x03B2}}\)
\(\newcommand{\gammaup}{\unicode{x03B3}}\)
\(\newcommand{\digammaup}{\unicode{x03DD}}\)
\(\newcommand{\deltaup}{\unicode{x03B4}}\)
\(\newcommand{\epsilonup}{\unicode{x03F5}}\)
\(\newcommand{\varepsilonup}{\unicode{x03B5}}\)
\(\newcommand{\zetaup}{\unicode{x03B6}}\)
\(\newcommand{\etaup}{\unicode{x03B7}}\)
\(\newcommand{\thetaup}{\unicode{x03B8}}\)
\(\newcommand{\varthetaup}{\unicode{x03D1}}\)
\(\newcommand{\iotaup}{\unicode{x03B9}}\)
\(\newcommand{\kappaup}{\unicode{x03BA}}\)
\(\newcommand{\varkappaup}{\unicode{x03F0}}\)
\(\newcommand{\lambdaup}{\unicode{x03BB}}\)
\(\newcommand{\muup}{\unicode{x03BC}}\)
\(\newcommand{\nuup}{\unicode{x03BD}}\)
\(\newcommand{\xiup}{\unicode{x03BE}}\)
\(\newcommand{\omicronup}{\unicode{x03BF}}\)
\(\newcommand{\piup}{\unicode{x03C0}}\)
\(\newcommand{\varpiup}{\unicode{x03D6}}\)
\(\newcommand{\rhoup}{\unicode{x03C1}}\)
\(\newcommand{\varrhoup}{\unicode{x03F1}}\)
\(\newcommand{\sigmaup}{\unicode{x03C3}}\)
\(\newcommand{\varsigmaup}{\unicode{x03C2}}\)
\(\newcommand{\tauup}{\unicode{x03C4}}\)
\(\newcommand{\upsilonup}{\unicode{x03C5}}\)
\(\newcommand{\phiup}{\unicode{x03D5}}\)
\(\newcommand{\varphiup}{\unicode{x03C6}}\)
\(\newcommand{\chiup}{\unicode{x03C7}}\)
\(\newcommand{\psiup}{\unicode{x03C8}}\)
\(\newcommand{\omegaup}{\unicode{x03C9}}\)
\(\newcommand{\Alphaup}{\unicode{x0391}}\)
\(\newcommand{\Betaup}{\unicode{x0392}}\)
\(\newcommand{\Gammaup}{\unicode{x0393}}\)
\(\newcommand{\Digammaup}{\unicode{x03DC}}\)
\(\newcommand{\Deltaup}{\unicode{x0394}}\)
\(\newcommand{\Epsilonup}{\unicode{x0395}}\)
\(\newcommand{\Zetaup}{\unicode{x0396}}\)
\(\newcommand{\Etaup}{\unicode{x0397}}\)
\(\newcommand{\Thetaup}{\unicode{x0398}}\)
\(\newcommand{\Varthetaup}{\unicode{x03F4}}\)
\(\newcommand{\Iotaup}{\unicode{x0399}}\)
\(\newcommand{\Kappaup}{\unicode{x039A}}\)
\(\newcommand{\Lambdaup}{\unicode{x039B}}\)
\(\newcommand{\Muup}{\unicode{x039C}}\)
\(\newcommand{\Nuup}{\unicode{x039D}}\)
\(\newcommand{\Xiup}{\unicode{x039E}}\)
\(\newcommand{\Omicronup}{\unicode{x039F}}\)
\(\newcommand{\Piup}{\unicode{x03A0}}\)
\(\newcommand{\Varpiup}{\unicode{x03D6}}\)
\(\newcommand{\Rhoup}{\unicode{x03A1}}\)
\(\newcommand{\Sigmaup}{\unicode{x03A3}}\)
\(\newcommand{\Tauup}{\unicode{x03A4}}\)
\(\newcommand{\Upsilonup}{\unicode{x03A5}}\)
\(\newcommand{\Phiup}{\unicode{x03A6}}\)
\(\newcommand{\Chiup}{\unicode{x03A7}}\)
\(\newcommand{\Psiup}{\unicode{x03A8}}\)
\(\newcommand{\Omegaup}{\unicode{x03A9}}\)
\(\newcommand{\alphait}{\unicode{x1D6FC}}\)
\(\newcommand{\betait}{\unicode{x1D6FD}}\)
\(\newcommand{\gammait}{\unicode{x1D6FE}}\)
\(\newcommand{\digammait}{\mathit{\unicode{x03DD}}}\)
\(\newcommand{\deltait}{\unicode{x1D6FF}}\)
\(\newcommand{\epsilonit}{\unicode{x1D716}}\)
\(\newcommand{\varepsilonit}{\unicode{x1D700}}\)
\(\newcommand{\zetait}{\unicode{x1D701}}\)
\(\newcommand{\etait}{\unicode{x1D702}}\)
\(\newcommand{\thetait}{\unicode{x1D703}}\)
\(\newcommand{\varthetait}{\unicode{x1D717}}\)
\(\newcommand{\iotait}{\unicode{x1D704}}\)
\(\newcommand{\kappait}{\unicode{x1D705}}\)
\(\newcommand{\varkappait}{\unicode{x1D718}}\)
\(\newcommand{\lambdait}{\unicode{x1D706}}\)
\(\newcommand{\muit}{\unicode{x1D707}}\)
\(\newcommand{\nuit}{\unicode{x1D708}}\)
\(\newcommand{\xiit}{\unicode{x1D709}}\)
\(\newcommand{\omicronit}{\unicode{x1D70A}}\)
\(\newcommand{\piit}{\unicode{x1D70B}}\)
\(\newcommand{\varpiit}{\unicode{x1D71B}}\)
\(\newcommand{\rhoit}{\unicode{x1D70C}}\)
\(\newcommand{\varrhoit}{\unicode{x1D71A}}\)
\(\newcommand{\sigmait}{\unicode{x1D70E}}\)
\(\newcommand{\varsigmait}{\unicode{x1D70D}}\)
\(\newcommand{\tauit}{\unicode{x1D70F}}\)
\(\newcommand{\upsilonit}{\unicode{x1D710}}\)
\(\newcommand{\phiit}{\unicode{x1D719}}\)
\(\newcommand{\varphiit}{\unicode{x1D711}}\)
\(\newcommand{\chiit}{\unicode{x1D712}}\)
\(\newcommand{\psiit}{\unicode{x1D713}}\)
\(\newcommand{\omegait}{\unicode{x1D714}}\)
\(\newcommand{\Alphait}{\unicode{x1D6E2}}\)
\(\newcommand{\Betait}{\unicode{x1D6E3}}\)
\(\newcommand{\Gammait}{\unicode{x1D6E4}}\)
\(\newcommand{\Digammait}{\mathit{\unicode{x03DC}}}\)
\(\newcommand{\Deltait}{\unicode{x1D6E5}}\)
\(\newcommand{\Epsilonit}{\unicode{x1D6E6}}\)
\(\newcommand{\Zetait}{\unicode{x1D6E7}}\)
\(\newcommand{\Etait}{\unicode{x1D6E8}}\)
\(\newcommand{\Thetait}{\unicode{x1D6E9}}\)
\(\newcommand{\Varthetait}{\unicode{x1D6F3}}\)
\(\newcommand{\Iotait}{\unicode{x1D6EA}}\)
\(\newcommand{\Kappait}{\unicode{x1D6EB}}\)
\(\newcommand{\Lambdait}{\unicode{x1D6EC}}\)
\(\newcommand{\Muit}{\unicode{x1D6ED}}\)
\(\newcommand{\Nuit}{\unicode{x1D6EE}}\)
\(\newcommand{\Xiit}{\unicode{x1D6EF}}\)
\(\newcommand{\Omicronit}{\unicode{x1D6F0}}\)
\(\newcommand{\Piit}{\unicode{x1D6F1}}\)
\(\newcommand{\Rhoit}{\unicode{x1D6F2}}\)
\(\newcommand{\Sigmait}{\unicode{x1D6F4}}\)
\(\newcommand{\Tauit}{\unicode{x1D6F5}}\)
\(\newcommand{\Upsilonit}{\unicode{x1D6F6}}\)
\(\newcommand{\Phiit}{\unicode{x1D6F7}}\)
\(\newcommand{\Chiit}{\unicode{x1D6F8}}\)
\(\newcommand{\Psiit}{\unicode{x1D6F9}}\)
\(\newcommand{\Omegait}{\unicode{x1D6FA}}\)
\(\let \digammaup \Digammaup \)
\(\renewcommand {\digammait }{\mathit {\digammaup }}\)
\(\newcommand {\smallin }{\unicode {x220A}}\)
\(\newcommand {\smallowns }{\unicode {x220D}}\)
\(\newcommand {\notsmallin }{\LWRoverlaysymbols {/}{\unicode {x220A}}}\)
\(\newcommand {\notsmallowns }{\LWRoverlaysymbols {/}{\unicode {x220D}}}\)
\(\newcommand {\rightangle }{\unicode {x221F}}\)
\(\newcommand {\intclockwise }{\unicode {x2231}}\)
\(\newcommand {\ointclockwise }{\unicode {x2232}}\)
\(\newcommand {\ointctrclockwise }{\unicode {x2233}}\)
\(\newcommand {\oiint }{\unicode {x222F}}\)
\(\newcommand {\oiiint }{\unicode {x2230}}\)
\(\newcommand {\ddag }{\unicode {x2021}}\)
\(\newcommand {\P }{\unicode {x00B6}}\)
\(\newcommand {\copyright }{\unicode {x00A9}}\)
\(\newcommand {\dag }{\unicode {x2020}}\)
\(\newcommand {\pounds }{\unicode {x00A3}}\)
\(\newcommand {\iddots }{\unicode {x22F0}}\)
\(\newcommand {\utimes }{\overline {\times }}\)
\(\newcommand {\dtimes }{\underline {\times }}\)
\(\newcommand {\udtimes }{\overline {\underline {\times }}}\)
\(\newcommand {\leftwave }{\left \{}\)
\(\newcommand {\rightwave }{\right \}}\)
\(\newcommand {\toprule }[1][]{\hline }\)
\(\let \midrule \toprule \)
\(\let \bottomrule \toprule \)
\(\newcommand {\cmidrule }[2][]{}\)
\(\newcommand {\morecmidrules }{}\)
\(\newcommand {\specialrule }[3]{\hline }\)
\(\newcommand {\addlinespace }[1][]{}\)
\(\newcommand {\LWRsubmultirow }[2][]{#2}\)
\(\newcommand {\LWRmultirow }[2][]{\LWRsubmultirow }\)
\(\newcommand {\multirow }[2][]{\LWRmultirow }\)
\(\newcommand {\mrowcell }{}\)
\(\newcommand {\mcolrowcell }{}\)
\(\newcommand {\STneed }[1]{}\)
\( \newcommand {\multicolumn }[3]{#3}\)
\(\newcommand {\tothe }[1]{^{#1}}\)
\(\newcommand {\raiseto }[2]{{#2}^{#1}}\)
\(\newcommand {\ang }[2][]{(\mathrm {#2})\degree }\)
\(\newcommand {\num }[2][]{\mathrm {#2}}\)
\(\newcommand {\si }[2][]{\mathrm {#2}}\)
\(\newcommand {\LWRSI }[2][]{\mathrm {#1\LWRSInumber \,#2}}\)
\(\newcommand {\SI }[2][]{\def \LWRSInumber {#2}\LWRSI }\)
\(\newcommand {\numlist }[2][]{\mathrm {#2}}\)
\(\newcommand {\numrange }[3][]{\mathrm {#2\,\unicode {x2013}\,#3}}\)
\(\newcommand {\SIlist }[3][]{\mathrm {#2\,#3}}\)
\(\newcommand {\SIrange }[4][]{\mathrm {#2\,#4\,\unicode {x2013}\,#3\,#4}}\)
\(\newcommand {\tablenum }[2][]{\mathrm {#2}}\)
\(\newcommand {\ampere }{\mathrm {A}}\)
\(\newcommand {\candela }{\mathrm {cd}}\)
\(\newcommand {\kelvin }{\mathrm {K}}\)
\(\newcommand {\kilogram }{\mathrm {kg}}\)
\(\newcommand {\metre }{\mathrm {m}}\)
\(\newcommand {\mole }{\mathrm {mol}}\)
\(\newcommand {\second }{\mathrm {s}}\)
\(\newcommand {\becquerel }{\mathrm {Bq}}\)
\(\newcommand {\degreeCelsius }{\unicode {x2103}}\)
\(\newcommand {\coulomb }{\mathrm {C}}\)
\(\newcommand {\farad }{\mathrm {F}}\)
\(\newcommand {\gray }{\mathrm {Gy}}\)
\(\newcommand {\hertz }{\mathrm {Hz}}\)
\(\newcommand {\henry }{\mathrm {H}}\)
\(\newcommand {\joule }{\mathrm {J}}\)
\(\newcommand {\katal }{\mathrm {kat}}\)
\(\newcommand {\lumen }{\mathrm {lm}}\)
\(\newcommand {\lux }{\mathrm {lx}}\)
\(\newcommand {\newton }{\mathrm {N}}\)
\(\newcommand {\ohm }{\mathrm {\Omega }}\)
\(\newcommand {\pascal }{\mathrm {Pa}}\)
\(\newcommand {\radian }{\mathrm {rad}}\)
\(\newcommand {\siemens }{\mathrm {S}}\)
\(\newcommand {\sievert }{\mathrm {Sv}}\)
\(\newcommand {\steradian }{\mathrm {sr}}\)
\(\newcommand {\tesla }{\mathrm {T}}\)
\(\newcommand {\volt }{\mathrm {V}}\)
\(\newcommand {\watt }{\mathrm {W}}\)
\(\newcommand {\weber }{\mathrm {Wb}}\)
\(\newcommand {\day }{\mathrm {d}}\)
\(\newcommand {\degree }{\mathrm {^\circ }}\)
\(\newcommand {\hectare }{\mathrm {ha}}\)
\(\newcommand {\hour }{\mathrm {h}}\)
\(\newcommand {\litre }{\mathrm {l}}\)
\(\newcommand {\liter }{\mathrm {L}}\)
\(\newcommand {\arcminute }{^\prime }\)
\(\newcommand {\minute }{\mathrm {min}}\)
\(\newcommand {\arcsecond }{^{\prime \prime }}\)
\(\newcommand {\tonne }{\mathrm {t}}\)
\(\newcommand {\astronomicalunit }{au}\)
\(\newcommand {\atomicmassunit }{u}\)
\(\newcommand {\bohr }{\mathit {a}_0}\)
\(\newcommand {\clight }{\mathit {c}_0}\)
\(\newcommand {\dalton }{\mathrm {D}_\mathrm {a}}\)
\(\newcommand {\electronmass }{\mathit {m}_{\mathrm {e}}}\)
\(\newcommand {\electronvolt }{\mathrm {eV}}\)
\(\newcommand {\elementarycharge }{\mathit {e}}\)
\(\newcommand {\hartree }{\mathit {E}_{\mathrm {h}}}\)
\(\newcommand {\planckbar }{\mathit {\unicode {x210F}}}\)
\(\newcommand {\angstrom }{\mathrm {\unicode {x212B}}}\)
\(\let \LWRorigbar \bar \)
\(\newcommand {\bar }{\mathrm {bar}}\)
\(\newcommand {\barn }{\mathrm {b}}\)
\(\newcommand {\bel }{\mathrm {B}}\)
\(\newcommand {\decibel }{\mathrm {dB}}\)
\(\newcommand {\knot }{\mathrm {kn}}\)
\(\newcommand {\mmHg }{\mathrm {mmHg}}\)
\(\newcommand {\nauticalmile }{\mathrm {M}}\)
\(\newcommand {\neper }{\mathrm {Np}}\)
\(\newcommand {\yocto }{\mathrm {y}}\)
\(\newcommand {\zepto }{\mathrm {z}}\)
\(\newcommand {\atto }{\mathrm {a}}\)
\(\newcommand {\femto }{\mathrm {f}}\)
\(\newcommand {\pico }{\mathrm {p}}\)
\(\newcommand {\nano }{\mathrm {n}}\)
\(\newcommand {\micro }{\mathrm {\unicode {x00B5}}}\)
\(\newcommand {\milli }{\mathrm {m}}\)
\(\newcommand {\centi }{\mathrm {c}}\)
\(\newcommand {\deci }{\mathrm {d}}\)
\(\newcommand {\deca }{\mathrm {da}}\)
\(\newcommand {\hecto }{\mathrm {h}}\)
\(\newcommand {\kilo }{\mathrm {k}}\)
\(\newcommand {\mega }{\mathrm {M}}\)
\(\newcommand {\giga }{\mathrm {G}}\)
\(\newcommand {\tera }{\mathrm {T}}\)
\(\newcommand {\peta }{\mathrm {P}}\)
\(\newcommand {\exa }{\mathrm {E}}\)
\(\newcommand {\zetta }{\mathrm {Z}}\)
\(\newcommand {\yotta }{\mathrm {Y}}\)
\(\newcommand {\percent }{\mathrm {\%}}\)
\(\newcommand {\meter }{\mathrm {m}}\)
\(\newcommand {\metre }{\mathrm {m}}\)
\(\newcommand {\gram }{\mathrm {g}}\)
\(\newcommand {\kg }{\kilo \gram }\)
\(\newcommand {\of }[1]{_{\mathrm {#1}}}\)
\(\newcommand {\squared }{^2}\)
\(\newcommand {\square }[1]{\mathrm {#1}^2}\)
\(\newcommand {\cubed }{^3}\)
\(\newcommand {\cubic }[1]{\mathrm {#1}^3}\)
\(\newcommand {\per }{/}\)
\(\newcommand {\celsius }{\unicode {x2103}}\)
\(\newcommand {\fg }{\femto \gram }\)
\(\newcommand {\pg }{\pico \gram }\)
\(\newcommand {\ng }{\nano \gram }\)
\(\newcommand {\ug }{\micro \gram }\)
\(\newcommand {\mg }{\milli \gram }\)
\(\newcommand {\g }{\gram }\)
\(\newcommand {\kg }{\kilo \gram }\)
\(\newcommand {\amu }{\mathrm {u}}\)
\(\newcommand {\nm }{\nano \metre }\)
\(\newcommand {\um }{\micro \metre }\)
\(\newcommand {\mm }{\milli \metre }\)
\(\newcommand {\cm }{\centi \metre }\)
\(\newcommand {\dm }{\deci \metre }\)
\(\newcommand {\m }{\metre }\)
\(\newcommand {\km }{\kilo \metre }\)
\(\newcommand {\as }{\atto \second }\)
\(\newcommand {\fs }{\femto \second }\)
\(\newcommand {\ps }{\pico \second }\)
\(\newcommand {\ns }{\nano \second }\)
\(\newcommand {\us }{\micro \second }\)
\(\newcommand {\ms }{\milli \second }\)
\(\newcommand {\s }{\second }\)
\(\newcommand {\fmol }{\femto \mol }\)
\(\newcommand {\pmol }{\pico \mol }\)
\(\newcommand {\nmol }{\nano \mol }\)
\(\newcommand {\umol }{\micro \mol }\)
\(\newcommand {\mmol }{\milli \mol }\)
\(\newcommand {\mol }{\mol }\)
\(\newcommand {\kmol }{\kilo \mol }\)
\(\newcommand {\pA }{\pico \ampere }\)
\(\newcommand {\nA }{\nano \ampere }\)
\(\newcommand {\uA }{\micro \ampere }\)
\(\newcommand {\mA }{\milli \ampere }\)
\(\newcommand {\A }{\ampere }\)
\(\newcommand {\kA }{\kilo \ampere }\)
\(\newcommand {\ul }{\micro \litre }\)
\(\newcommand {\ml }{\milli \litre }\)
\(\newcommand {\l }{\litre }\)
\(\newcommand {\hl }{\hecto \litre }\)
\(\newcommand {\uL }{\micro \liter }\)
\(\newcommand {\mL }{\milli \liter }\)
\(\newcommand {\L }{\liter }\)
\(\newcommand {\hL }{\hecto \liter }\)
\(\newcommand {\mHz }{\milli \hertz }\)
\(\newcommand {\Hz }{\hertz }\)
\(\newcommand {\kHz }{\kilo \hertz }\)
\(\newcommand {\MHz }{\mega \hertz }\)
\(\newcommand {\GHz }{\giga \hertz }\)
\(\newcommand {\THz }{\tera \hertz }\)
\(\newcommand {\mN }{\milli \newton }\)
\(\newcommand {\N }{\newton }\)
\(\newcommand {\kN }{\kilo \newton }\)
\(\newcommand {\MN }{\mega \newton }\)
\(\newcommand {\Pa }{\pascal }\)
\(\newcommand {\kPa }{\kilo \pascal }\)
\(\newcommand {\MPa }{\mega \pascal }\)
\(\newcommand {\GPa }{\giga \pascal }\)
\(\newcommand {\mohm }{\milli \ohm }\)
\(\newcommand {\kohm }{\kilo \ohm }\)
\(\newcommand {\Mohm }{\mega \ohm }\)
\(\newcommand {\pV }{\pico \volt }\)
\(\newcommand {\nV }{\nano \volt }\)
\(\newcommand {\uV }{\micro \volt }\)
\(\newcommand {\mV }{\milli \volt }\)
\(\newcommand {\V }{\volt }\)
\(\newcommand {\kV }{\kilo \volt }\)
\(\newcommand {\W }{\watt }\)
\(\newcommand {\uW }{\micro \watt }\)
\(\newcommand {\mW }{\milli \watt }\)
\(\newcommand {\kW }{\kilo \watt }\)
\(\newcommand {\MW }{\mega \watt }\)
\(\newcommand {\GW }{\giga \watt }\)
\(\newcommand {\J }{\joule }\)
\(\newcommand {\uJ }{\micro \joule }\)
\(\newcommand {\mJ }{\milli \joule }\)
\(\newcommand {\kJ }{\kilo \joule }\)
\(\newcommand {\eV }{\electronvolt }\)
\(\newcommand {\meV }{\milli \electronvolt }\)
\(\newcommand {\keV }{\kilo \electronvolt }\)
\(\newcommand {\MeV }{\mega \electronvolt }\)
\(\newcommand {\GeV }{\giga \electronvolt }\)
\(\newcommand {\TeV }{\tera \electronvolt }\)
\(\newcommand {\kWh }{\kilo \watt \hour }\)
\(\newcommand {\F }{\farad }\)
\(\newcommand {\fF }{\femto \farad }\)
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Polarisation: Licht ist transversale Welle, d. h. Schwingungsebenen stehen senkrecht zur Ausbreitungsrichtung des Lichts, Polarisation = Schwingungsebene des elektrischen Felds, Polarisationen lassen sich zerlegen in eine gewichtete Summe (Linearität der elektromagnetischen Wellen)
lineare Polarisation: Schwingung in einer konstanten Ebene, kann vernachlässigt werden
zirkulare Polarisation: andere Lösungen der Maxwell-Gleichungen, Rotation der momentanen Schwingungsebene des elektrischen Felds mit Lichtfrequenz
unpolarisiertes Licht: die Photonen des Lichts haben eine „zufällige“ Polarisation, bei den meisten natürlichen und künstlichen Lichtquellen der Fall (Sonne, Glühbirne)
Polarisation durch Reflexion: Fresnel-Gleichungen besagen, dass Reflexionsfaktor
(z. B. an Glasscheibe) von Einfallswinkel, den Brechzahlen und der Polarisation abhängen, also ist im reflektierten Licht eine der beiden Polarisationsrichtungen stärker vertretenPolarisationsfilter: können Reflexionen vermindern, Anwendung Sonnenbrillen (Wassersportler), Bildaufnahmen vom Himmel, um Himmel abzudunkel, Kontrast zu Wolken größer (Licht des Himmels ist teilpolarisiert)
Polarisationsfolien: Absorption von Lichtwellen durch Stromfluss, analog auch Drahtgitterpolarisator für Mikrowellen, elektrisches Feld bewirkt Wechselstrom im Gitter, dieses erzeugt ein Gegenfeld, das genau stark wie das einfallende Feld ist und dieses auslöscht, leider ist Licht zu kurzwellig für Gitterpolarisatoren, Edwin Land entwickelte 1938 aber eine preiswerte Herstellungsmethode für Gitter mit sehr kleinem Abstand,
Anwendung in praktisch allen LCDs (Flüssigkristalldisplays), z. B. Armbanduhr, Taschenrechner, Laptopgekreuzte Polarisatoren: führen zu einer kompletten Auslöschung des Lichts, ein zusätzlicher (schiefer) Zwischenpolarisator führt dazu, dass ein Teil des Lichts die Anordnung passiert
Polarisation des Himmels: maximal zu ca. \(\SI {75}{\percent }\) polarisiert (wenn Sonne tief steht und keine Bewölkung vorhanden ist), Grund: Rayleigh-Streuung an Luftmolekülen erhält die Schwingungsebene des elektromagnetischen Felds, Polarisation variiert mit Sonnenposition, beobachtetem Gebiet und Partikel in Atmosphäre, Anwendung in Fotografie (Verstärkung des Kontrasts Himmel – Wolken), da Wolken aufgrund der Mehrfachstreuung stark depolarisieren
Doppelbrechung: ein senkrecht auf die Grenzfläche eines doppelbrechenden Materials einfallender Lichtstrahl wird teilweise gebrochen (widerspricht dem Brechungsgesetz), d. h. für den außerordentlichen Strahl ergibt sich ein Brechungswinkel ungleich Null, die beiden Strahlen sind unterschiedlich polarisiert, Grundlage ist, dass sich unterschiedlich polarisiertes Licht in doppelbrechenden Medien (z. B. Kalkspat, Quarz) unterschiedlich schnell fortpflanzt und daher unterschiedliche Brechungsindizes gelten
Farbeffekte aufgrund Doppelbrechung: laufen die Lichtwellen danach durch einen Polarisator, passieren zwar nur Teile der beiden Wellen den Polarisator, diese sind dann aber gleich polarisiert, sodass sie interferieren können, lokale Dickenschwankungen oder Änderungen der Geometrie führen zu unterschiedlichen Farben (z. B. Zuckerschicht auf Glas, Cockpitfenster, Plexiglasbox)
Anwendungen: Spannungsdoppelbrechung (mechanische Spannungen sichtbar
machen), Feststellen von Konstruktionsfehlern
Haidinger-Schmetterling: Auge kann (direkt) eigentlich keine Polarisation feststellen, mit ein wenig Übung ist das aber tatsächlich möglich, polarisiertes Licht zeigt sich dabei als eine leichte (kaum wahrnehmbare) schmetterlingsartige Figur, vier Kreise, zwei gelbe und zwei blau, wobei die gelben Teile in die Richtung senkrecht zur Polarisation zeigen, Ausdehnung von ca. \(4^\circ \), Erklärung nicht vollständig geklärt, aber Lutein (ein Pigment im gelben Fleck) ist doppelbrechend und absorbiert Licht mit einer Polarisationsrichtung parallel zur Molekülachse besonders stark, aufgrund der kreisförmigen Anordnung der Moleküle auf der Netzhaut ergibt sich auf unterschiedlichen Bereichen der Netzhaut eine unterschiedlich starke Abschwächung des Lichts
Polarisation bei Tieren: manche Tiere (Bienen, einige Käfer und Spinnen) können Polarisation sehen und benutzen sie zur Orientierung
3D-Kino: Polarisationsfilter in den Brillengläsern sorgen dafür, dass jedes Auge das richtige Bild empfängt
Dichroismus: unterschiedliche Absorption für unterschiedliche Polarisation (Turmalin)
Zuckerlösung: rotiert die Schwingungsebene von einfallendem, linear polarisiertem Licht
Moiré: Überlagerung (multiplikativ oder additiv) zweier Gitter mit leicht anderer Frequenz (z. B. auch unterschiedliche Entfernung zum Beobachter), sowohl orts- als auch zeitbezogen möglich, bspw. wenn Räder eines Autos/Rotor eines Helikopters still zu stehen oder sogar rückwärts zu drehen scheinen, Farbmaske eines Digitalfotos kann zu farbigen Moirés führen, analog bei Scans