Next we go to the small scale of mechanicaloscillation. This time we take a sodium chloride crystal, which has sodium ionsand chlorine ions next to each other, as we described in an early chapter.These ions are electrically charged, alternately plus and minus. Now there isan interesting oscillation possible. Suppose that we could drive all the pluscharges to the right and all the negative charges to the left, and let go; theywould then oscillate back and forth, the sodium lattice against the chlorinelattice. How can we ever drive such a thing? That is easy, for if we apply anelectric field on the crystal, it will push the plus charge one way and theminus charge the other way! So, by having an external electric field we canperhaps get the crystal to oscillate. The frequency of the electric fieldneeded is so high, however, that it corresponds to infrared radiation!So we try to find a resonance curve by measuring the absorption of infraredlight by sodium chloride. Such a curve is shown in Fig. 23–7. Theabscissa is not frequency, but is given in terms of wavelength, but that is justa technical matter, of course, since for a wave there is a definite relationbetween frequency and wavelength; so it is really a frequency scale, and acertain frequency corresponds to the resonant frequency.
下面，我们来看，小尺度的力学震荡。这次，我们取一个钠氯晶体，它有钠离子和氯离子，挨在一起，正如在前面某章，我们所描述。这些离子，是带电的，交替地带正电和负电。现在，一个有趣的震荡，就是可能的。假设我们可以把所有的正电荷，都拉到右边，所有的负电荷，都拉到左边，然后，让它们开始运动；它们就会前后震荡，钠栅格，相对于氯栅格。我们如何驱动这样一个事情呢？很容易，因为，如果我们把电场，应用到晶体上，它将会把正电荷推向一边，负电荷推向另一边！于是，通过有一个外部的电场，我们或许就可以让晶体去振荡。然而，所需的电场频率，非常高，以至于，相当于是红外辐射！所以我们尝试，通过测量红外线被钠氯化合物所吸收，来找到一条共振曲线。这种曲线如图23-7。横坐标不是频率，而是波长，但这只是一个技术问题，当然，由于对一个波来说，在频率和波长之间，有确定的关系；所以，它确实是一个频率尺度，某确定频率，相应于共振的频率。

Fig. 23–7.Transmission of infrared radiationthrough a thin (0.17 μ m) sodium chloride film. [After R. B. Barnes, Z. Physik 75,723 (1932). Kittel, Introduction to Solid State Physics, Wiley, 1956.] 图23-7 红外辐射穿过一个薄膜(0.17 μm)，薄膜由钠和氯化物构成。[在R. B. Barnes之后, Z. 物理学 75, 723 (1932). Kittel, 《固体物理学介绍》, Wiley出版社, 1956.]

But what about the width? What determinesthe width? There are many cases in which the width that is seen on the curve isnot really the natural width γthat one would have theoretically. There are two reasons why there canbe a wider curve than the theoretical curve. 宽度又如何呢？什么决定着宽度呢？在很多情况下，曲线上所看到的宽度，并不是一条曲线理论上应该有的自然宽度。实际曲线，比理论上的曲线，要宽，为何如此？有两个原因。If the objects do not all have the same frequency, as might happenif the crystal were strained in certain regions, so that in those regions theoscillation frequency were slightly different than in other regions, then whatwe have is many resonance curves on top of each other; so we apparently get awider curve. 如果各对象，所拥有的频率，并不相同，例如，在某些区域，晶体会受到挤压，这时，这种情况就会发生；所以，在这些区域，震荡的频率，与其他地区相比，就会略有不同，因此，我们得到的结果就是，很多共振曲线，相互重叠，所以很明显，我们会得到一个更宽的曲线。The other kind of width is simply this: perhaps we cannot measurethe frequency precisely enough—if we open the slit of the spectrometer fairlywide, so although we thought we had only one frequency, we actually had acertain range Δω , then we may not have the resolving power needed to see a narrowcurve. Offhand, we cannot say whether the width in Fig. 23–7 isnatural, or whether it is due to inhomogeneities in the crystal or the finitewidth of the slit of the spectrometer.
其他类型的宽度，简单地说，就是这样：或许我们无法足够准确地测量频率--如果我们分光计的裂缝，打开的不是足够宽，于是，虽然我们想，我们只有一个频率，但实际上，我们是有一定范围的Δω，因此，要想看到一个窄的曲线，我们可能没有这种解决能力。不假思索地说吧，我们不能说，图23-7中的宽度，是否是自然的，或者，是否它可归于晶体不同质性，或归于分光计裂缝的有限宽度。

Fig. 23–8.Magnetic energy loss inparamagnetic organic compound as function of applied magnetic field intensity.[Holden et al., Phys. Rev. 75, 1614 (1949)] 图23-8在顺磁体有机化合物中的磁性能量损失，此化合物，是作为应用磁场密度的函数的。[Holden et al.,物理评论. 75, 1614(1949)]
Now we turn to a more esoteric example, andthat is the swinging of a magnet. If we have a magnet, with north and southpoles, in a constant magnetic field, the N end of the magnet will be pulled oneway and the S end the other way, and there will in general be a torque on it,so it will vibrate about its equilibrium position, like a compass needle. 现在，让我们转向一个更专业的例子，这是一个磁铁的摆动。如果我们有一个磁铁，它有南北极，放在一个恒定的磁场中，磁铁的N端，会被吸向一侧，S端则会向另一侧，通常，会有一个扭力，作用于其上，于是，它将会在其平衡位置附近振动，就像指南针一样。However, the magnets we are talking about are atoms. Theseatoms have an angular momentum, the torque does not produce a simple motion inthe direction of the field, but instead, of course, a precession. Now,looked at from the side, any one component is “swinging,” and we can disturb ordrive that swinging and measure an absorption. 然而，我们现在我们所谈论的磁铁，却是原子。这些原子，有一个角动量，扭力在场的方向上，所产生的，并不是一个简单的运动，当然是一个进动。现在，从一侧看，任何一个分量，都在“摇摆”，我们可以扰乱或驱动这个摇摆，和测量吸收。The curve in Fig. 23–8represents a typical such resonance curve. What has been done here is slightlydifferent technically. The frequency of the lateral field that is used to drivethis swinging is always kept the same, while we would have expected that theinvestigators would vary that and plot the curve. They could have done it thatway, but technically it was easier for them to leave the frequency ωfixed, and change the strength of the constant magnetic field, whichcorresponds to changing ω0 in our formula. They have plotted the resonance curve against ω0. Anyway, this is a typical resonance with a certain ω0and γ.
图23-8中的曲线，代表了这种共振曲线的一个典型。这里所做的，技术上，略有不同。用来驱动这个摆动的，是一个侧面的场，其频率，总是被保持为一样，而我们则在期待着，研究者们，可以改变频率，画出曲线。他们确实可以改变频率，但是，技术上说，这样做更容易些：让频率ω固定，而去改变恒定磁场的强度，这相当于在我们的公式中，改变了ω0 。她们画了，对于ω0的共振曲线。不管怎么说，这是一个典型的共振，具有一定的ω0和γ。

Fig. 23–9.The intensity of gamma-radiationfrom lithium as a function of the energy of the bombarding protons. The dashedcurve is a theoretical one calculated for protons with an angularmomentum ℓ=0 . [Bonner and Evans, Phys. Rev. 73, 666 (1948)] 图23-9 锂的伽马辐射密度，作为质子轰击的能量函数。虚线是对于质子的理论上计算出来的辐射密度，其角动量为ℓ=0。[Bonner and Evans, 物理评论 73, 666 (1948)]
Now we go still further. Our next examplehas to do with atomic nuclei. The motions of protons and neutrons in nuclei areoscillatory in certain ways, and we can demonstrate this by the followingexperiment. We bombard a lithium atom with protons, and we discover that acertain reaction, producing γ-rays, actually has a very sharp maximum typical of resonance. 现在，我们要走的更远一点。我们的下一个例子，将要处理原子核。对于原子核中的质子和中子，其运动，是以某种方式震荡，我们可以通过下面的实验，来演证这点。我们用质子，轰击锂原子，我们发现，某些反应，会产生伽马射线，实际上，是一种共振，其曲线，非常尖锐，最大典型的。 We note in Fig. 23–9,however, one difference from other cases: the horizontal scale is not afrequency, it is an energy! The reason is that in quantum mechanics whatwe think of classically as the energy will turn out to be really related to afrequency of a wave amplitude. When we analyze something which in simplelarge-scale physics has to do with a frequency, we find that when we doquantum-mechanical experiments with atomic matter, we get the correspondingcurve as a function of energy. In fact, this curve is a demonstration of thisrelationship, in a sense. It shows that frequency and energy have some deep interrelationship,which of course they do.
然而，在图23-9中，我们注意到，有一点，与其他情况不同：水平尺度，并非频率，它是能量！原因是这样，在经典力学中，有些东西，被我们思考为能量，而在量子力学中，则发现，这些东西，将与波幅的频率有关。在简单的大尺度物理中，有些东西，要与频率打交道，当我们分析这些东西时，我们发现，当我们用原子级别的东西，做量子力学实验时，我们得到的相应的曲线，是能量的函数{？}。事实上，这个曲线，在某种意义上，就是这种关系的演证。它显示出，频率和能量，有某种非常深的相互关系，它们确实有这种关系。

Fig. 23–11.Momentum dependence of the crosssection for the reactions (a) K−+p→Λ+π++π− and (b) K−+p→K¯¯¯¯0+n . The lower curves in (a) and (b) represent the presumednonresonant backgrounds, while the upper curves contain in addition thesuperposed resonance. [Ferro-Luzzi et al., Phys. Rev. Lett. 8, 28 (1962)]
图23-11 对于下面反应的横截面的动量依赖：(a) K−+p→Λ+π++π− 和(b) K−+p→K¯¯¯¯0+n 。 在(a)和 (b)中，较低的曲线，代表了假设的非共振的背景，而较高的曲线，则包含了叠加在{较低曲线}上面的共振。[Ferro-Luzzi et al., 物理评论. 信件. 8, 28 (1962)]
Finally, if we look in an issue of the PhysicalReview, say that of January 1, 1962, will we find a resonance curve?Every issue has a resonance curve, and Fig. 23–11 isthe resonance curve for this one. 最后，如果我们查看一下，《物理评论》的某一期，比如说1962年1月1日那一期，我们会发现一个共振曲线吗？每期都有一个共振曲线，图23-11就是那一期的。This resonancecurve turns out to be very interesting. It is the resonance found in a certain reactionamong strange particles, a reaction in which a K− and a proton interact. 你会发现，这个共振曲线，非常有趣。一些奇怪的粒子之间，会发生反应，在一个反应中，K−和质子，会相互作用，此曲线，就是在这个反应中发现的。Theresonance is detected by seeing how many of some kinds of particles come out,and depending on what and how many come out, one gets different curves, but ofthe same shape and with the peak at the same energy. We thus determine thatthere is a resonance at a certain energy for the K− meson.此共振是这样被探测到的：通过查看，在这个反应中，某类粒子，能产生出来多少，及依赖于什么，及{对于此依赖}会有多少个产生出来？每一种依赖，都会有不同的曲线，但曲线的形状，是同样的，并在同样的能量下，达到峰值。这样，我们就确定了，对于K−介子，在某确定能量处，有共振。Thatpresumably means that there is some kind of a state, or condition,corresponding to this resonance, which can be attained by putting together a K−and a proton. This is a new particle, or resonance. 这大概就意味着，有某种状态、或条件，相应于这个共振，通过把K−和质子，放在一起，可以达到此状态。这是一个新的粒子，或共振。Today we do not know whether to call a bump like this a “particle”or simply a resonance. When there is a very sharp resonance, itcorresponds to a very definite energy, just as though there were aparticle of that energy present in nature. 今天，我们并不知道，把这样一个隆起，应该称为“粒子”、还是共振。当有一个很尖锐的共振时，它就相当于一个非常确定的能量，就好像有一个这种能量的粒子，在自然中表现出来一样。 When the resonance gets wider, then we do not know whether to saythere is a particle which does not last very long, or simply a resonance in thereaction probability. In the second chapter, this point is made about theparticles, but when the second chapter was written this resonance was notknown, so our chart should now have still another particle in it!当共振变得更宽时，我们并不知道，是否该说：有了一个生命期不长的粒子；还是简单地说：大概是有了一个共振。在第二章，关于粒子，我们说过这个观点，但是，当写第二章时，我们还不知道，有这个共振，所以，我们的图表中，现在还应该有另外一个粒子。

1 Chapter24.Transients第24章 瞬态
24–1The energy of an oscillator 24-1 振荡器的能量
Although this chapter is entitled “transients,”certain parts of it are, in a way, part of the last chapter on forcedoscillation. One of the features of a forced oscillation which we have not yetdiscussed is the energy in the oscillation. Let us now consider thatenergy.
虽然这一章，被冠以“瞬态”，但是，它的一部分，在某种意义上，还是关于上一章强制震荡的。强制震荡，有一个特性，我们尚未讨论，它就是震荡中的能量。现在，让我们来考虑这个能量。

In a mechanical oscillator, how muchkinetic energy is there? It is proportional to the square of the velocity. Nowwe come to an important point. Consider an arbitrary quantity A , which may be the velocity or something else that we want to discuss.When we write A=A^eiωt , a complex number, the true and honest A , in the physical world, is only the real part; therefore if,for some reason, we want to use the square of A , it is not right to square the complex number and then take the realpart, because the real part of the square of a complex number is not just thesquare of the real part, but also involves the imaginary part. So whenwe wish to find the energy we have to get away from the complex notation for awhile to see what the inner workings are.
在一个力学震荡中，有多少动能呢？它正比于矢速的平方。现在，我们就来到了一个重要的点。考虑一个任意的量A，它可以是我们想讨论的矢速，也可以是其他的量。当我们写下A=A^eiωt时，这是一个复数，而真正的和诚实的A，亦即物理世界中的量，只是其实部；因此，如果由于某种原因，我们想使用A的平方，那么，对此复数取平方，然后取其实部，并不正确，因为，一个复数的平方的实部，并不是其实部的平方，而是包括了虚部。所以，当我们希望找到能量时，我们必须暂时离开复数表示法一会儿，去看看内部的工作机制是什么。

Now the true physical A is the real part of A0ei(ωt+Δ), that is, A=A0cos(ωt+Δ) , where A^ , the complex number, is written as A0eiΔ. Now the square of this real physical quantity is A2=A20cos2(ωt+Δ) . The square of the quantity, then, goes up and down from a maximum tozero, like the square of the cosine. The square of the cosine has a maximumof 1 and a minimum of 0 , and its average value is 1/2 .
现在，真实的物理量A，就是A0ei(ωt+Δ)的实部，也就是说，A=A0cos(ωt+Δ)，这里，A^是复数，被写作 A0eiΔ。现在，这个真实的物理量的平方，就是 A2=A20cos2(ωt+Δ)。因此，这个量的平方，就在最大值和零之间，上下游走，就像余弦的平方一样。余弦的平方，最大为1，最小为零，平均为1/2。

In many circumstances we are not interestedin the energy at any specific moment during the oscillation; for a large numberof applications we merely want the average of A2 , the mean of the square of A over a period of time large compared with the period of oscillation.In those circumstances, the average of the cosine squared may be used, so wehave the following theorem: if A is represented by a complex number, then the mean of A2is equal to A20/2. Now A20 is the square of the magnitude of the complex A^ . (This can be written in many ways—some people like to write |A^|2; others write, A^A^∗ , A^ times its complex conjugate.) We shall use this theorem several times.
在很多情形中，对于震荡的任何特殊时期的能量，我们都不感兴趣；对于大量的应用来说，我们只想得到A2的平均值，即A的平方，在一个时间周期内的平均值，此时间周期，与震荡的周期相比，足够大。在这些情形中，余弦平方的平均，可以使用，于是，我们就有下面的定理：如果A被一个复数所代表，那么，A2的平均，就等于A20/2。现在，A20就是复数A^的大小的平方。（这可用很多方式来写--有些人喜欢写成|A^|2 ; 其他人则喜欢A^A^∗，即A^乘以其共轭复数。这个定理，我们将会使用好几次。

Now let us consider the energy in a forcedoscillator. The equation for the forced oscillator is
现在，让我们考虑一个强制震荡中的能量。一个强制震荡的方程为：
(24.1)
In our problem, of course, F(t) is a cosine function of t . Now let us analyze the situation: how much work is done by theoutside force F ? The work done by the force per second, i.e., the power, is the forcetimes the velocity. (We know that the differential work in a time dtis Fdx , and the power is Fdx/dt .) Thus
当然，在我们的问题中，F(t)是 t的一个余弦函数。现在，让我们分析这个情况：外力F究竟做了多少功？此力每秒所做的功、亦即、功率，就是力乘以矢速。（我们知道，在时间dt所做的功{?微小的功}，就是 Fdx，功率就是 Fdx/dt。）这样
(24.2)

But the first two terms on the right can also be written as d/dt[(1/2)m(dx/dt)2+(1/2)mω20x2], as is immediately verified by differentiating. That is to say, theterm in brackets is a pure derivative of two terms that are easy to understand—oneis the kinetic energy of motion, and the other is the potential energy of thespring. Let us call this quantity the stored energy, that is, the energystored in the oscillation. Suppose that we want the average power over manycycles when the oscillator is being forced and has been running for a long time.In the long run, the stored energy does not change—its derivative gives zeroaverage effect. In other words, if we average the power in the long run, allthe energy ultimately ends up in the resistive term γm(dx/dt)2. There is some energy stored in the oscillation, but thatdoes not change with time, if we average over many cycles. Therefore the meanpower ⟨P⟩ is
但是，右边的前两项，可以写作d/dt[(1/2)m(dx/dt)2+(1/2)mω20x2]，可以通过微分，被立即核实。也就是说，方括号中的项，是两项的纯导数，这两项，很容易理解--一个是运动的动能，另一个是弹簧的势能。让我们把这个量，称为存储的能量，也就是说，在震荡器中所存储的能量。假设当振荡器被强制运动，并且已经运行了很长时间，而我们想得到关于很多循环的平均功率。从长远看，存储的能量，并不变化—其导数，给出的是，零平均效果。换句话说，如果我们对功率，求长期的平均值，那么，所有的能量，最终都会终结于抵抗项γm(dx/dt)2中。有些能量，存储在震荡器中，如果我们对很多循环求平均的话，那么，这些能量，并不随时间变化。因此，平均功率 ⟨P⟩就是：
(24.3)

Using our method of writing complex numbers, and our theorem that ⟨A2⟩=(1/2)A20 , we may find this mean power. Thus if x=x^eiωt, then dx/dt=iωx^eiωt . Therefore, in these circumstances, the average power could bewritten as
利用我们写复数的方法，和定理⟨A2⟩=(1/2)A20，我们可以找到这个平均功率。这样，如果x=x^eiωt，那么，dx/dt=iωx^eiωt 。从而，在这些情形下，平均功率就可写作：
(24.4)
In the notation for electrical circuits, dx/dt is replaced by the current I (I is dq/dt , where q corresponds to x ), and mγ corresponds to the resistance R . Thus the rate of the energy loss—the power used up by the forcingfunction—is the resistance in the circuit times the average square of thecurrent: 在电路的表示法中，dx/dt被电流I代替(I 是dq/dt , 这里 q相当于 x ), 而 mγ，相当于电阻R。这样，能量的损失率--被强制功能所消耗的功率就--，就是电路中的电阻，乘以电流平方的平均值：
(24.5)
This energy, of course, goes into heating the resistor; it issometimes called the heating loss or the Joule heating. 这个能量，当然就是用来加热电阻的；有时也被称为加热损失、或焦耳加热。

Another interesting feature to discuss ishow much energy is stored. That is not the same as the power, becausealthough power was at first used to store up some energy, after that the systemkeeps on absorbing power, insofar as there are any heating (resistive) losses.At any moment there is a certain amount of stored energy, so we would like tocalculate the mean stored energy ⟨E⟩ also. We have already calculated what the average of (dx/dt)2is, so we find
另外一个要讨论的特性，就是有多少能量，被存储了起来。这与功率，不是同一个问题，因为，虽然开始时，功率被用来存储一些能量，但在其后，只要有任何加热（阻力）损失的话，系统就会不断地吸收能量。在任一瞬间，都有一定量的能量，被存储，所以，我们也想计算平均被存储的能量 ⟨E⟩。我们已经计算了(dx/dt)2的平均值是多少，所以，我们发现：
(24.6)
Now, when an oscillator is very efficient, and if ω is near ω0 , so that |x^| is large, the stored energy is very high—we can get a large storedenergy from a relatively small force. The force does a great deal of work ingetting the oscillation going, but then to keep it steady, all it has to do isto fight the friction. The oscillator can have a great deal of energy if thefriction is very low, and even though it is oscillating strongly, not muchenergy is being lost. The efficiency of an oscillator can be measured by howmuch energy is stored, compared with how much work the force does peroscillation. 现在，当一个振荡器，非常有效，且如果ω接近于ω0，那么，|x^|就很大，被存储的能量就很高，--我们可以从一个相对小的力出发，得到一个非常大的被存储的能量。让振荡器开始震荡，力要做大量的功，但是，其后要让震荡器平稳{运行}，最主要的，就是要战胜阻力。如果摩擦非常小，那么，振荡器就会有巨大的能量，就算震荡很强，也不会有多少能量被损失掉。一个振荡器的效率，可以通过下面两者的比较，来测量，一、被存储的能量有多少；二、每次震荡，力做了多少功。

How does the stored energy compare with theamount of work that is done in one cycle? This is called the Q of the system, and Q is defined as 2π times the mean stored energy, divided by the work done per cycle. (Ifwe were to say the work done per radian instead of per cycle, then the 2πdisappears.) 被存储的能量，与一个循环中所做的功的量，如何比较呢？这被称为系统的Q，Q被定义为：2π乘以被存储的能量的平均值，再除以每个循环所做的功。（如果我们说每个弧度所做的功，而不是每个循环所做的功，那么，2π就消失了。）
(24.7)
Q is not a very useful number unless it is very large. When it isrelatively large, it gives a measure of how good the oscillator is. People havetried to define Q in the simplest and most useful way; various definitions differ a bitfrom one another, but if Q is very large, all definitions are in agreement. The most generallyaccepted definition is Eq. (24.7),which depends on ω . For a good oscillator, close to resonance, we can simplify (24.7)a little by setting ω=ω0 , and we then have Q=ω0/γ , which is the definition of Q that we used before. 除非Q很大，否则，它就不是一个非常有用的数字。当它相对较大时，对于测量一个振荡器有多好，它给出了一个标准。人们曾尝试，以各种最简单的、和最有用的方式，来定义Q；各种定义之间，差别不大，但是，如果Q非常大，所有的定义，就是一致的。最被普遍接受的定义，就是方程（24.7），它依赖于ω。对于一个好的震荡器，接近于共振的，通过设置ω=ω0，我们可以把方程（24.7）简化一点儿，我们就有Q=ω0/γ，这个定义，我们以前用过。

What is Q for an electrical circuit? To find out, we merely have to translate Lfor m , R for mγ , and 1/C for mω20 (see Table 23–1). The Q at resonance is Lω/R , where ω is the resonance frequency. If we consider a circuit with a high Q, that means that the amount of energy stored in the oscillation isvery large compared with the amount of work done per cycle by the machinerythat drives the oscillations.
对于电路，Q是什么呢？要找出这个，我们只需把m换成L，把mγ换成R，把mω20换成1/C （见表23-1）。共振时的Q，就是Lω/R ，这里ω就是共振频率。如果我们考虑一个电路，拥有高的Q值，这就意味着，在此震荡中，被存储的能量，与每个循环中所做的功相比，非常大；此所做的功，是由驱动此震荡的机械做的。