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Schwarzer Körper (Teil II)

Wechselwirkung von Licht und Materie gibt Einblick in die Welt der Quanten

Licht und Materie stehen in permanenter Wechselwirkung und ändern ihre Energie. Bekannte Phänomene wie Absorption, Reflexion, Streuung oder Brechung von Licht führen uns das eindrucksvoll vor Augen. Hierbei sind Körper die sämtliches Licht absorbieren und nicht reflektieren absolut schwarz. Die Sonne ist ein idealer schwarzer Körper, ihre Helligkeit resultiert allein auf Grund ihrer thermischen Wärmeentwicklung. Aber auch alle anderen Körper strahlen. Ein Spiegel reflektiert, Glassplitter brechen das Licht.

Herumirrende Photonen

Herumirrende Photonen

Zufällige Wegstrecke von Photonen innerhalb und beim Verlassen der Sonne.

© lactamme

Nach dem Kirchhoffschen Strahlungsgesetz ist für jeden Körper – im thermischen Gleichgewicht – bei jeder Wellenlänge und in jeder Richtung das Emissionsvermögen für thermische Strahlung proportional zu seinem Absorptionsvermögen. Eine schwarze Fläche heizt sich im Sonnenlicht leichter auf als eine weiße, dafür gibt sie die (Wärme-)Strahlung leichter ab – schwarz eloxierte Kühlbleche. Da nun das Absorptionsvermögen des schwarzen Körpers bei jeder Wellenlänge den größtmöglichen Wert annimmt, gilt dies auch für sein Emissionsvermögen. Ein beliebiger realer Körper kann bei keiner Wellenlänge mehr thermische Strahlung aussenden als ein schwarzer Körper, der daher eine ideale thermische Strahlungsquelle darstellt. Da sein Spektrum außerdem von keinen anderen Parametern als der Temperatur abhängt, insbesondere von keinen Materialeigenschaften, stellt er eine für zahlreiche theoretische und praktische Zwecke nützliche Referenzquelle dar. 3 Der deutsche Physiker Gustav Robert Kirchhoff formulierte das Strahlungsgesetz 1859 während er das Verfahren der Spektroskopie entwickelte. Es bildete den Grundstein der Untersuchung der Wärmestrahlung und so auch von Max Plancks Quantenhypothese. 4

Hohlraumstrahler

Zur Aufnahme des Emissionsspektrums eines schwarzen Körpers sollte man dementsprechend einen idealen schwarzen Körper untersuchen. Da ein solcher in der Natur aber fast nicht vorkommt, realisiert man ihn annähernd über einen Hohlraumstrahler mit einer kleinen Eintrittsöffnung. Dieser ist von innen mit einem schwarzen Material (Ruß) beschichtet, das beinahe alle einfallende Strahlung absorbiert. Der Rest wird so lange zwischen den Wänden des Strahlers hin- und her reflektiert, bis auch dieser absorbiert wird.

Modell eines schwarzen Strahlers

Modell eines schwarzen Körpers

Man realisiert einen idealen schwarzen Körper durch einen Hohlraumstrahler mit kleiner Öffnung in der Wandung.

© Uni-Dortmund

Dadurch erscheint die Öffnung von außen bei mäßigen Temperaturen als nahezu ideal schwarz. Dabei sollte das Loch möglichst klein sein, damit es das thermische Gleichgewicht nicht stört. Ein solcher Hohlraumstrahler strahlt über alle Frequenzen Energie ab. Sein Spektrum hat ein Intensitätsmaximum, das sich bei steigenden Temperaturen zu kürzeren Wellenlängen hin verschiebt. 1

Stefan-Boltzmann Gesetz

Ende des 19. Jh. war das Interesse an der Wärmestrahlung sehr groß und man versuchte mit Hilfe der damals recht jungen Physik der Wärme – der Thermodynamik oder statistischen Physik – diese Strahlung zu erklären. Josef Stefan und Ludwig Boltzmann gelang es, die Abhängigkeit der gesamten Strahlungsenergie von der Temperatur vorherzusagen. 5 Dieses Gesetz wird häufig auch als Boltzmannsches «T hoch 4 Gesetz» bezeichnet, da es nur wenige physikalische Gesetze mit so hohen Potenzen gibt. Diese hohe Potenz bewirkt, dass bei hohen Temperaturen der größte Teil der Wärmeabgabe eines Körpers über die Strahlung erfolgt, bei niedrigen Temperaturen dagegen die Konvektion und die Wärmeleitung überwiegt. 2 Für die genaue Verteilung der Wärmestrahlung aber gab es nur Ansätze, die nicht zufriedenstellend waren. Das Rayleigh-Jeanssche Strahlungsgesetz besagte sogar, dass die Strahlung für kleine Wellenlängen unendlich wird. Dieser Schwachpunkt der sonst recht brauchbaren Theorie ist als Ultraviolett-Katastrophe bekannt. 5 Das Stefan-Boltzmann-Gesetz liefert die Gesamtenergie, die ein schwarzer Körper über alle Wellenlängen integriert ausstrahlt. 6

Max Planck

Nur mit Widerwillen stellte er im Oktober 1900 seine berühmte Formel zur Strahlung schwarzer Körper auf. Um die damals bekannten Experimente exakt beschreiben zu können, musste er eine Hilfsgröße h einführen, die sich nur dadurch erklären ließ, dass Strahlung nicht kontinuierlich abgegeben und aufgenommen wird, sondern häppchenweise, eben in Vielfachen von h. Diese Annahme stand in eklatantem Widerspruch zu dem Diktum, die Natur mache keine Sprünge. Daher hoffte Planck sein «Wirkungsquantum» als bald durch mathematische Tricks wieder aus der Welt schaffen zu können. Doch der Quantengeist, den Planck gerufen hatte, war nicht mehr zu bändigen. Fünf Jahre später schrieb der junge Albert Einstein dem ominösen h praktische physikalische Realität zu und erklärte damit erfolgreich den fotoelektrischen Effekt. Trifft Licht auf eine Fotoplatte, erfolgt die Energieübertragung ebenfalls nicht kontinuierlich, sondern quantenweise. Diese Deutung, die Einstein später den Nobelpreis einbrachte, wurde ausgerechnet von Planck abgelehnt. Noch 1913 urteilte er, mit der Hypothese der Lichtquanten sei Einstein "in seinen Spekulationen über das Ziel hinausgeschossen". 7

Strahlungsbereich

Erst bei sehr hohen Temperaturen liegt das Maximum der Kurve in dem Wellenlängenbereich, der dem menschlichen Auge zugänglich ist. Naturgemäß entspricht die zugehörige Temperatur der Oberflächentemperatur der Sonne von 5800K. Bei niedrigeren Temperaturen liegt das Maximum der Strahlung im so genannten Infrarot-Bereich. Dies bezeichnen wir als Wärmestrahlung, da alle heißen Körper auf der Erde in diesem Bereich ihre maximale Abstrahlung besitzen. Da jeder reale Körper eine geringere Emission besitzt als ein schwarzer Körper, muss dessen Strahlungsverteilung über der Wellenlänge stets unterhalb der Planckschen Strahlungskurve bei gleicher Temperatur verlaufen. Dabei ist der Verlauf meistens sehr unregelmäßig über der Wellenlänge verteilt. Da der Absorptionskoeffizient gleich dem Emissionskoeffizienten ist, besitzt dieser auch die gleiche Wellenlängen Abhängigkeit. 2

Nicht wirklich schwarz

Der schwarze Körper erscheint dem Auge trotz der Strahlungsemission lediglich deshalb dunkel, weil das Auge in diesem Wellenlängenbereich nicht empfindlich ist. Bei einer Temperatur von 5800 K emittiert ein schwarzer Körper eine Strahlungsleistung von 64 MW/m2. Der Körper erscheint dem Auge jetzt hell leuchtend, da ein Teil der Strahlung in den optischen Spektralbereich fällt. Der Körper wird trotzdem als schwarzer Strahler bezeichnet, weil er nach wie vor alle auf ihn fallende Strahlung absorbiert. Die ausgesandte Strahlung besteht vollständig aus der vom Körper selbst thermisch emittierten Strahlung, sie enthält keinerlei reflektierte Umgebungsstrahlung. Da das Emissionsvermögen unabhängig von der Abstrahlrichtung den maximalen Wert annehmen muss, strahlt ein schwarzer Körper gleichmäßig in alle Richtungen – er ist ein Lambert-Strahler und sendet vollständig diffuse Strahlung aus. 3

Das Plancksche Strahlungsgesetz

Plancksches Strahlungsspektrum

Plancksches Strahlungsspektrum

Mit steigender Temperatur verschiebt sich das Maximum der Frequenzverteilung hin zu höheren Frequenzen bzw. kürzeren Wellenlängen.

© Stiftgymnasium Melk

Die Intensität und Frequenzverteilung der von einem schwarzen Körper ausgesandten elektromagnetischen Strahlung wird durch das Plancksche Strahlungsgesetz beschrieben. Mit steigender Temperatur verschiebt sich das Maximum der Frequenzverteilung zu höheren Frequenzen bzw. kürzeren Wellenlängen – Wiensches Verschiebungsgesetz. Die gesamte ausgestrahlte Energie ist proportional zur vierten Potenz der absoluten Temperatur des schwarzen Körpers – Stefan-Boltzmann-Gesetz. So emittiert beispielsweise ein schwarzer Körper bei einer Temperatur von 300 K (Raumtemperatur) pro m2 Oberfläche eine Strahlungsleistung von etwa 460 Watt. Kein realer Körper kann bei derselben Temperatur mehr thermische Strahlung aussenden. 3

Quantelung von Energie

Die Gesamtstrahlung eines schwarzen Körpers kann natürlich nicht auf alle Wellenlängen gleichmäßig verteilt sein, da dann immer eine unendliche Energiemenge – Fläche unter der Kurve – vorliegen müßte. Man beobachtet vielmehr eine Wellenlänge, bei der die Energieabstrahlung maximal ist, darüber und darunter nimmt die Energieabstrahlung stark ab. Der erste, der die Verteilung der Strahlungsenergie auf die Wellenlängen exakt beschreiben konnte war Max Planck. Dafür musste er annehmen, dass die Energie nicht kontinuierlich abgestrahlt wurde, sondern immer in Energiepaketen zusammengefasst ist – den sogenannten Quanten. 2


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