Compare a matter wave hx0x yf;03phf7s bm/6 fj*asq $\Psi$ to (a) a wave on a string, (b) an EM wave. Discuss similarities and differences.

Explain why Bohr's theory of the atom is not compatible with cb kl.4gaab48 ,q ac5qquantum mechanics, particularlybac54aq k84cal, qg.b the uncertainty principle.

Explain why it is thatqkn(vas5 r l;zt0y5c. the more massive an object is, the easier it becomes to predict;5 cv rnl( sqatk0yz5. its future position.

In view of the uncertainty principle, why does a baseball seem to hav b69v :dr3agmke a well-defin 6v3g darbkm:9ed position and speed whereas an electron does not?

  Would it ever be possible te j,m yvm,xpn v 1zn,).jd0;ybo balance a very sharp needle precisely on its point? Explain.nm pyj, enm0j;vvby,dx1) z.,

  A cold thermometer is placed in a hot bowl of soup. Will thr; awce+ ap5 r2szin7b8:/bhwe temperature reading of the thermometer be the same as the temperature of the hot soup before the measuremenabbnwhwr7a/se+ c8 r 52i;:pz t was made? Explain.

  Does the uncertainty principle set a limit to how well y.9yp 7wq46ebqr 1l pfoou can make any single measurement of posi bq.yer4op6f 1 lwp97qtion?

If you knew the position of an object precisely, with no uncertum3g- mamf2sv px/czo8) dsao 5,8l:painty, how well would you know it o,/8vu5ozp:c d8gsxmmf pla -s3 )ma2s momentum?

  When you check the pressure in a tire, doesn't some air inevitably kq3bw,/q m xga)f)xk -t1u+f tescape? Is it possible to avoid this escape of air altogether? Wfkmkqb u/ -t)w ta,+x31)qfxghat is the relation to the uncertainty principle?

  It has been said that the ground-state e1nbki0v +;z6urb3 jf0ff w.ginergy in the hydrogen atom can be precisely known but the excited states have some uncertainty in their values (an "energy width"). Is this cons 3u.i f1 6zjb+ib;r gwkfnv00fistent with the uncertainty principle in its energy form? Explain.

Which model of the hydrogen atomzr53 ow5fkxz9b 8rhw5, the Bohr model or the quantum-mechanical model, predicw9rb f zoxkwh355r5z 8ts that the electron spends more time near the nucleus?

The size of atoms varies by only a factor of threehy:t+uau+ hw . or so, from largest to smallest, yet the number of electruw hyh:a.+tu +ons varies from one to over 100. Explain.

Excited hydrogen and ey6/w,c;qw0-h qt xh rwxcited helium atoms both radiate light as they jump down to the yr,ht 6h-/wwqc ;0w qx$n=1, l=0, m_l=0$ state. Yet the two elements have very different emission spectra. Why?

How would the periodic table look if there we4,6qwe a gpi sngx1)f(zgg9crmhw*9/re no electron spin but otherwise quantum mechanics were valid? Consider the first 20 g1rw qag g*enw hi cs96p,9fzx/mg)( 4elements or so.

  Which of the followinmnxx iuz- y4r5; a.ai6g electron configurations are not allowed:
(a) $1s^2\,2s^2\,2p^4\,3s^2\,4p^2$;  ----待选择----allowednot allowed 
(b) $1s^2\,2s^2\,2p^3\,3s^1$;  ----待选择----allowednot allowed 
(c) $1s^2\,2s^2\,2p^6\,3s^2\,3p^3\,4s^2\,4d^3\,4f^1$? ----待选择----allowednot allowed 
If not allowed, explain why.

Give the complete electron configurationm):u6+uurt- 3el*ks/ q a kgbs for a uranium atom (careful scrutiny across the periodic table on the insuuk3smq)/ kest *:u-g6lra+bide back cover will provide useful hints).

In what column of the periodic table would sui5ny9ji 3 byh9;.*2dpa u doyou expect to find the atom with each of td9 5ui.h uoanjy3 *s9iyb;p2dhe following configurations:
(a) $1s^22s^22p^63s^2$;
(b) $1s^22s^22p^63s^23p^6$;
(c) $1s^22s^22p^63s^23p^64s^1$;
(d) $1s^22s^22p^5$?

Why do chlorine and iodine exhi*jwim vol 3e. 59u4woebit similar properties?

Explain why potassium and sodiuapjzqw bj/6 7a2+5cv03m*ghr hsu7lchzq 3 k6m exhibit similar properties.

The ionization energy for neon j b olz2f9y;r(p;u oo;dv-yv; $(Z=10)$ is 21.6 eV, and that for sodium $(Z=11)$ is 5.1 eV. Explain the large difference.

Why does the cutoff wavelength/5fxx( p0md* wwrx .)cng.ce h in (Fig. Below) imply a photon nature for light? ex/fc.(g h )dxw0mwnrc*.5px

Why do we not expect perfect agreemlpg1c39p)e.rfy4onrl g *ox4 ent between measured values of X-ray line wavelengths and those calculated using felp po gn)4lrgrx4.*o39y1c Bohr theory?

How would you figure out which lines in a)of.b5) qssi5x3ndkw8x,nd vn X-ray spectrum correspond to $K_\alpha, K_\beta, L_\alpha$ etc., transitions?

Why do we expect electron transitions deec63cxpc(7cebl )pxp6,qm 1y zp within an atom to produce shorter wavelengths than transitxpz17q) ( ppl,y3 6c 6ccmxcebions by outer electrons?

Compare spontaneous emission to stimulated emission.

How does laser light differ from0 /xb+lwwha .hsgb;/ q ordinary light? How is it the same?

Explain how a 0.0005-W laser beam, photogranfdag t;boa himxs.j4 d j5 -k3m*76+bphed at a distance, can seem much stronger than a 1000-W str x3f m;-b4o+asj.d5nhi tdg6ba *jk7meet lamp at the same distance.

  Does the intensity of light from a laser fall off as the inverse square of t4;bfjd+c .r tul c9ir0he distance? 4tcb ucfd9.rl jr+0 ;iExplain. ----待选择----yesno 

  The neutrons in a para+k 0ifr tioh4v+5 onh+llel beam, each having kinetic energy $0.40 \, \text{eV}$ are directed through two slits $0.50 \, \text{mm}$ apart. How far apart will the interference peaks be on a screen $1.0 \, \text{m}$ away? [Hint: First find the wavelength of the neutron.]
$\Delta y$ =    $ \times 10^{-7}\ \text{m}$

  Bullets of mass $3.0 \, \text{g}$ are fired in parallel paths with speeds of $220 \, \text{m/s}$ through a hole $3.0 \, \text{mm}$ in diameter. How far from the hole must you be to detect a $1.0 \, \text{cm}$-diameter spread in the beam of bullets? L =    $ \times 10^{28}\ \text{m}$

  A proton is traveling with a speed ofq wz. 8wcu nv04zki;l; $(6.560 \pm 0.012) \times 10^3 \, \text{m/s}$. With what maximum accuracy can its position be ascertained? [Hint: $\Delta p \approx m \Delta v$]
$\pm$    $ \times 10^{-11}\ \text{m}$

  If an electron’s position can be measured to an acculfcc5f-o s)9 )wtn-1wu(0w xpx0 oodz racy of $1.0 \times 10^{-8} \, \text{m}$, how accurately can its speed be known?
$\Delta v $=    $\times 10^{3}\ \text{m/s}$

  An electron remains in an excited state suiu-q wpb.l( :jc+q- choo lh;4c,t3of an atom for typically $10^{-8} \, \text{s}$. What is the minimum uncertainty in the energy of the state (in eV)?
$\frac{\hbar}{\Delta t}$ =    $ \times 10^{-8}\ \text{eV}$

  The $Z^0$ boson, discovered in 1985, is the mediator of the weak nuclear force, and it typically decays very quickly. Its average rest energy is $91.19 \, \text{GeV}$, but its short lifetime shows up as an intrinsic width of $2.5 \, \text{GeV}$ (rest energy uncertainty). What is the lifetime of this particle?
$\frac{\hbar}{\Delta t}$=    $ \times 10^{-25}\ \text{s}$

  What is the uncertaint-a)agb-dl.qu vt5qw /y in the mass of a muon $(m=105.7 \, \text{MeV/c}^2)$, specified in $\text{eV/c}^2$, given its lifetime of $2.20 \times 10^{-6} \, \text{s}$? [Hint: $\Delta E \approx \hbar / \Delta t$ and $\Delta m = \Delta E / c^2$]
$\frac{\hbar}{\Delta t}$=    $ \times 10^{-10}\ \text{eV}$
${\Delta m}$ =    = $\times 10^{-10}\ \text{eV}$

  A free neutron $(m=1.67 \times 10^{-27} \, \text{kg})$ has a mean life of $900 \, \text{s}$. What is the uncertainty in its mass (in kg)?
${\Delta m}$=    $\times 10^{-54}\ \text{kg}$

  An electron and a $140$-g baseball are each traveling $150 \, \text{m/s}$ measured to an accuracy of $0.055\%$. Calculate and compare the uncertainty in position of each.
For the electron $\frac{\hbar}{m\Delta v}$=    $\times 10^{-3}\ \text{m}$
For the baseball $\frac{\hbar}{m\Delta v}$ =    $ \times 10^{-33}\ \text{m}$

  Estimate the lowest possible energy of a neutron containqz) .susv ntjp: ,*ve+ed in a typical nucleus of tz:nvsu.q+ ,p*j)s evradius $1.0 \times 10^{-15} \, \text{m}$. [Hint: A particle can have an energy at least as large as its uncertainty.]
$\frac{\hbar}{\Delta x} $=    $\times 10^{-19}\ \text{kg} \cdot \text{m/s}$
E =    Mev

  Use the uncertainty principle to shoghc3k.+ht 4 zbw that if an electron were present in the nuctc.3hhbz+4 gk leus $(r \approx 10^{-15} \, \text{m})$, its kinetic energy (use relativity) would be hundreds of MeV. (Since such electron energies are not observed, we conclude that electrons are not present in the nucleus.) [Hint: A particle can have an energy at least as large as its uncertainty.]
E $\approx$    Mev

  How accurately can the position of a1v:inmif0x mz/rj3 q* $3.00$-keV electron be measured assuming its energy is known to $1.00\%$?
$\frac{\hbar}{\Delta p} =$    $ \times 10^{-10}\ \text{m}$

  For $n=6$, what values can $l$ have?
$\ell$ =    or    or    or    or    or   

  For $n=5, l=3$, what are the possible values of $m_l$ and $m_s$?
$n_\ell$ = $\pm $    or $\pm $    or $\pm $    or $\pm $   
$m_s$ = $\pm $ 1/  

  How many electrons c h ehalka4q qq,:1u(o0y. d:bean be in the $n=6, l=3$ subshell?    elections

How many different states are possible for an electrobu rmt(rt;h -v i,ohhtd )/7i9n whose principal quantum number io()itvr -d trbt /m97, h;huhis $n=4$? Write down the quantum numbers for each state.

List the quantum numbhp ,7)o-nqyq )q q vd*mf.x3/xi+xqpo ers for each electron in the ground state of
(a) carbon $(Z=6)$
(b) magnesium $(Z=12)$.

List the quantum numbers for each electron in the g x+zpq 04yanq-round state of nitrogen $(Z=7)$.

  Suppose a certain hydrogen atdwqaj, (z+(tx1 5cczxom has $l=4$. What are the possible values for $n, m_l$, and $m_s$?
n $\ge $   
$m_\ell$ =    $\pm$    $\pm$    $\pm$    $\pm$   
$m_s$ = $\pm$   

  Calculate the magnitude of the angular momentt:g(mt*5l; bc zjq .pxum of an electron in the $n=4, l=3$ state of hydrogen.
L =    $\times 10^{-34}\ \text{kg} \cdot \text{m}^2/\text{s}$

  If a hydrogen atom ha zl 4w,:z1ijp4ir4b0fp3 gpjes $m_l=-3$, what are the possible values of $n, l$, and $m_s$?
$\ell \ge $   
n$\ge \ell$ +    (minimum 4)
$m_s$ = $\pm $   

  Show that there can be 18 electrons in a "g" subshewfo *e4yopa) (ll.    electrons

What is the full electron configuration in the ground state for elements withu uai5 *b (7iidi4kur* j9owu. $Z$ equal to (a) 27, (b) 36, (c) 38? [Hint: see the periodic table inside the back cover.]

What is the full electron configuration for (a) selenium (Se), (b) x ,;b:qayu2s/tn +mef gold (Au), (c) radium (Ra)? [Hint: see the periodic table bu;a q, fm+t s:2/yexninside the back cover.]

  A hydrogen atom is in th 3gkt kj7*/ho/ffjjk59o up;ve $6s$ state. Determine
(a) the principal quantum number, n =   
(b) the energy of the state, $E_6$ =    Mev
(c) the orbital angular momentum and its quantum number $l$ L =   
(d) the possible values for the magnetic quantum number.$m_\ell$ =   

  Estimate the binding energy of the third electron in u8 7z(woct 8hd1 bcej+lithium using the Bohr theory. [Hint: Th 8e7t1h+ cwbzcj(o u8dis electron has $n=2$ and "sees" a net charge of approximately $+1e$.] The measured value is $5.36 \, \text{eV}$.    eV

Show that the total angular momentum is am6q+ga8r )t-wp l+kzb-vot lvd,p/, zero for a filled subshell.

  For each of the following atomic transitions, state whether the trajlaswl+l r2(e v. 38fensition is allowed or forbidden, and if forbidden, what rule is being violat f+lwjar vels3 2e(l8.ed:
(a) $3p \to 4p$  ----待选择----forbiddenallowed 
(b) $1s \to 2p$  ----待选择----forbiddenallowed 
(c) $3d \to 2d$;  ----待选择----forbiddenallowed 
(d) $4d \to 2p$;  ----待选择----forbiddenallowed 
(e) $4s \to 2p$. ----待选择----forbiddenallowed 

  An excited H atom is in a qp3gigz .b6n7$6d$ state.
(a) Name all the states $(n, l)$ to which the atom is "allowed" to jump with the emission of a photon.
(b) How many different wavelengths are there (ignoring fine structure)?   

  What are the shortest-wavelength X-rays emitted by electrons striking the face o3g;mt pc 8b9w bzde43bdb8;3 g4wt mzpc39beb f a $33.5$-kV TV picture tube?$\lambda$ =    nm
What are the longest wavelengths?   nm

  If the shortest-wavelength bremsstrahlung )md af22 qde* 9 .4hsfsw-qnncX-rays emitted from an X-ray tube haveqed sn)-2m.n9q4c sdf2*awfh $\lambda=0.030 \, \text{nm}$, what is the voltage across the tube?    kV

Show that the cutoff wavp-1a*ry9ugjy a0 msh6 x)fdv-);savzelength $\lambda_0$ is given by $\lambda_0 = \frac{1240 \, \text{nm} \cdot \text{V}}{V}$ where $V$ is the X-ray tube voltage in volts.

  Estimate the X-ray wave6) ,rhpru *idx4x zy0wvc/o 4slength emitted when a Co $(Z=27)$ atom jumps from $n=2$ to $n=1$.
$\lambda $ =    nm

  Estimate the wavelength fo5 dn;mptb*of kgvcb46g(2 y h;r an $n=2$ to $n=1$ transition in iron $(Z=26)$. $\lambda $ =    nm

  Use the Bohr theory toelg(v6d8sr( taeb.4x estimate the wavelength for an $n=3$ to $n=1$ transition in molybdenum $(Z=42)$. The measured value is $0.063 \, \text{nm}$. Why do we not expect perfect agreement?
$\lambda $ =    nm
n =   

  A mixture of iron and an unknown material is bombarded with elect8qo7cbb 9phx +rons. The wavelength 9boxqp 8 7b+chof the $K_\alpha$ lines are $194 \, \text{pm}$ for iron and $229 \, \text{pm}$ for the unknown. What is the unknown material?
 ----待选择----copperironmagnesiumvanadiumchromiummanganese 

  A laser used to weld detached retinas pu18xlszl+zh/ bts out $28$-ms-long pulses of $640$-nm light which average $0.68$-W output during a pulse. How much energy can be deposited per pulse and how many photons does each pulse contain?
E =    J
N =    $\times 10^{16}\ \text{photons}$

  A low-power laser used in i,u2xz 9d:*h kdh,iesj, alan:;qn7xa physics lab might have a power of $0.50 \, \text{mW}$ and a beam diameter of $3.0 \, \text{mm}$. Calculate
(a) the average light intensity of the laser beam, I =    $W/m^2$
(b) compare it to the intensity of a lightbulb producing $40$-W light viewed from $2.0 \, \text{m}$.I =    $W/m^2$ The laser beam is more intense by a factor of   

  Estimate the angular cf diqxyy(g1u22g *3bbjio/ yi52d/i spread of a laser beam due to diffraction if the beam emerges thx(jb5yi/*yo2q b idg2c/y1i 23 gfuidrough a $3.0$-mm-diameter mirror. Assume that $\lambda = 694 \, \text{nm}$. What would be the diameter of this beam if it struck
$\theta$ =    $ \times 10^{-4}\ \text{rad}$
(a) a satellite $300 \, \text{km}$ above the Earth, D =   $ \times 10^{2}\ \text{m}$
(b) the Moon?D =   $ \times 10^{5}\ \text{m}$

  What is the wavelength of the He-Ne laser s j8b.ryu2r)e?$\lambda $ =    nm

Use the uncertainty principle to estimate the position uncertainty fnbi +1wekjphvk9h 5 8;or the electron in the ground state of the hydrogen atom. [Hint: Determine the momentum using the Bohr model of Section 27-12 and assume the momentum can be an;khjw1 9pn+8bk ve hi5ywhere between this value and zero.] How does this result compare to the Bohr radius?

  An electron in the $n=2$ state of hydrogen remains there on average about $10^{-8} \, \text{s}$ before jumping to the $n=1$ state.
(a) Estimate the uncertainty in the energy of the $n=2$ state. $\frac{\hbar}{\Delta t} =$    $ \times 10^{-8}\ \text{eV}$
(b) What fraction of the transition energy is this? $\frac{\Delta E}{E} =$    $ \times 10^{-9}$
(c) What is the wavelength, and width (in nm), of this line in the spectrum of hydrogen?
$\lambda$ =    nm
$\Delta \lambda =$    $ \times 10^{-7}\ \text{nm}$

  What are the largest and smallest possible values for the 1uqkw 97+ardovw5+p q angular momentum $L$ of an electron in the $n=5$ shell?
$\text{The smallest value of } L \text{ is}$ =   
$\text{The largest value of } L \text{ is}$ =    $\times 10^{-34}\ \text{kg} \cdot \text{m}^2/\text{s}$

  Estimate
(a) the quantum number $l$ for the orbital angular momentum of the Earth about the Sun, $\ell$ =    $ \times 10^{74}$
(b) the number of possible orientations for the plane of Earth's orbit. N =    $ \times 10^{74}$

  A $12$-g bullet leaves a rifle at a speed of $180 \, \text{m/s}$.
(a) What is the wavelength of this bullet? $\lambda $ =    $ \times 10^{-34}\ \text{m}$
(b) If the position of the bullet is known to an accuracy of $0.60 \, \text{cm}$ (radius of the barrel), what is the minimum uncertainty in its momentum? $\frac{\hbar}{\Delta y}$ =    $\times 10^{-32}\ \text{kg} \cdot \text{m/s}$

  Using the Bohr formula for the radius of an electron orbit,k3md s+x w6*oj estimate the average distance from the nuclew6sjdk m3* o+xus for an electron in the innermost $(n=1)$ orbit of a uranium atom $(Z=92)$. Approximately how much energy would be required to remove this innermost electron?
$r_{\text{uranium}} =$    $ \times 10^{-13}\ \text{m}$
$\text{the binding energy is }$    $ \ \text{keV}$

  An X-ray tube operatedla n8up38nhrf(g)8 3eo q*gd)n.sh xs at $95 \, \text{kV}$ with a current of $25 \, \text{mA}$ and nearly all the electron energy goes into heat. If the specific heat of the $0.085$-kg plate is $0.11 \, \text{kcal/kg} \cdot ^\circ \text{C}$, what will be the temperature rise per minute if no cooling water is used?
$\frac{\Delta T}{t} =$    $ \times 10^{3}\ ^\circ\text{C}/\text{min}$

  The ionization (binding) energy of p+ v -mf./4fcvuylg:mnco4 o/the outermost electron in boron is $8.26 \, \text{eV}$.
(a) Use the Bohr model to estimate the "effective charge," $Z_{\text{eff}}$, seen by this electron. $Z_{\text{eff}} $=   
(b) Estimate the average orbital radius. z =    $ \times 10^{-10}\ \text{m}$

  Use the Bohr theory to show that the Mqc.i;-8yix tm oseley plot(Fig. Below) can be written $\sqrt{\frac{1}{\lambda}} = a(Z - b)$, where $b \approx 1$, and evaluate $a$. a =    $\ \text{m}^{-\frac{1}{2}}$

(a) Show that the number of different statesgidop 2y8l;o etvc7ed6x(6 ,aab)0ul for a given value of $l$ is equal to $2(2l + 1)$.
(b) What is this number for $l=0, 1, 2, 3, 4, 5,$ and $6$?

Show that the number of different electron states possible for a given valuetpm*b ya .7we0 of ma7*e.bt wp0y$n$ is $2n^2$. (See Problem 50.)

  A beam of electrons with energy 45 kV is shot through two narrow sliue15cwxdulpcg 8 4-0wts in a barrier. The du4wexul1c- gw8 p5 c0slits are a distance $2.0\times10^{-6}\ \text{m}$ apart. If a screen is placed 35.0 cm behind the barrier, calculate the spacing between the "bright" fringes of the interference pattern produced on the screen. $\Delta y =$    $\ \mu\text{m}$

  The angular momentum in the hydrogen atob:2t+ertlvu7 y;cs 2 km is given both by the Bohr model and by quantum mechanics. C2:rb l; 2vkcs7teuy +tompare the results for $L = mvr$.
$L_{\text{Bohr}}$ =    $\hbar$
$L_{\text{QM}}$ =    $\hbar$ or $L_{\text{QM}}$ =    $\hbar$

  An 1100-kg car is traveling with 7o0:odjy 62 ar2zp jzsaxbp;) a speed of $(22 \pm 0.22)\ \text{m/s}$. With what maximum accuracy can its position be determined? $\Delta x =$    $\times 10^{-37}\ \text{m}$

  An atomic spectrum contains a lig-;p1s xj nn;tne with a wavelength centered at 488 nm. Careful measurements show the line x;snpn1; -gjtis really spread out between 487 and 489 nm. Estimate the lifetime of the excited state that produced this line.
$\Delta t = $    $\times 10^{-14}\ \text{s}$

  Protons are accelerated from rest across 550 V. They are then direckag t,gzb: vx3 pe-2snx80m,wted at two slits 0.70 mm apart. How far apart will the interf v3a2k pmb-w zegx0gnt:,, xs8erence peaks be on a screen 28 m away? $\Delta y$ =    nm

An electron and a proton, each initxikq,wa:0b4 k-tz- l bially at rest, are accelerated across the same voltage. Assuming that the uncz 4w,t0q xkb:kl-ab- iertainty in their position is given by their de Broglie wavelength, find the ratio of the uncertainty in their momentum.

  If the principal quantum number n were limited to the range f6 -z8m.oupcmirom 1 to 6, how many elements would we find in nature?.mz8p- mi6ocu   

  If your de Broglie wavelength were 0.50 m, how fast would you be moving if your 8 ip3x b1p xsf71uixprc,; qf1us3x1if7q;1b1 f ippxxp,8crmass is 75.0 kg? $v = $    $ \times 10^{-35}\ \text{m/s}$
Would you notice diffraction effects as you walk through a doorway?  ----待选择----yesno 
Approximately how long would it take you to walk through the doorway?$\Delta t \approx $    $ \times 10^{34}\ \text{s}$

  Suppose that the spectrum of an unknown element shows a series of lines with one9(:e0 3 v vyi2rd6-6dmle;sjqamvgmo out of every four matching a line from the Lyman seriesdjdmv9y e26lvsmr( 3v a0o6;-egmq i: of hydrogen. Assuming that the unknown element is an ion with Z protons and one electron, determine Z and the element in question. ----待选择----lithium berylliumboron 

  Photons of wavelength 0.154 nm are emitted from the surf4 3oc:ahv ti8iace of a certain metal when it is bombarded with high energy radiation. If this pho4: cv8i thai3oton wavelength corresponds to the $$K_\alpha$$ line, what is the element? ----待选择----nickelcopperzinc 

Show that the diffractive sprehky j79a; e- yzh0/hmqad of a laser beam, $$\theta \approx \lambda/D$$ , is precisely what you might expect from the uncertainty principle. [Hint: Since the beam's width is constrained by the dimension of the aperture D, the component of the light's momentum perpendicular to the laser axis is uncertain.]